‘I It I
.I. (u
A .«o‘...
. .D$q 'l‘l
,..w~..n.=.... $5,218,.
a. (Hunk...
._ .

It #0:.
D! 2-7:. V ‘(O
. «a...

.v. nitrw U». r. .1.

:LI .1 . s . ri...) lids . a ‘
5:11;»... f5 .1 . A .. II
p v

1'
1e. . V
4:1. 49:. K .a
.53-- .I unaihfih
IQ‘I‘I‘sul 1‘ In! I EVA!
. ‘ 3.7... 5.4m» .1:

.1

t.
. § ‘
fiMWruwvau .43
.mhwvflnuri. 3 A 1n».
. :15";
.- KI...

.7; - . . a
.1 in? “a..." .
13.9.7». 3.2.

(1!! At...

«a n Ad .1.
.1 $33.5”

96.. 4.!
, .1\ :3. 71.... iv.
, . £12.}...fi... .I
4. Sai,..‘: .I!

‘ 5. negrdJEL.
mafia.) hr‘l“

an.

.3
9.21.6}...
. . . cfnl’o

In

"P’: -
‘mm;
A.

c
I

hm.

O

9
.3. .I)
:c It... Q... . ..v§;..
:55: f..!- v. e
2|“. ans...“
. 1.. - .5 r?

I“.
i

,.

as. "'1‘ ”Lunar-r
{.lnlsa...‘ .

Hwniiufium.
}‘ 1 I a.
.3» .. W .t.

' '1: £1:
“33;”

’T‘ .

‘H
.~.

1.

b‘dumww... an...“
: 1 . . . : . . .0!) 5.5. 93"). r13...
‘ . I if 31 5.9 : i.

?v

 

w
{4’

lllllllllllllllllllllllllllilllllllll

3 1293 01396 2356

This is to certify that the

dissertation entitled

ESSAYS ON UNCERTAINTY AND SUSTAINABILITY
IN THE SEMI-ARID TROPICS

presented by

Takeshi Sakurai

has been accepted towards fulfillment
of the requirements for

ph, D. degree in Agrimltural Etxnmics

 

 

~

Date Aug. 3, ms Mm (52.4.4

 

Major professor

 

 

Date 47/16, . 3 H‘LC
] /

Do I I . 5' .

MSU is an Affirmative Action/Equal Opportunity Institution 0-12771

 

w HERARY
M‘Chlgan State
University

 

 

 

 

“66!!an BOXtommflbdnckufiMywmd.
TOAVOIDFINESMunonotbdmmM

DATE DUE DATE DUE DATE DUE

 

 

 

fire 2 ‘J ‘" "l
~J lev-
J
(
Off! A
.____.
V
|_ ,
0 “ '
p
A.

 

 

 

 

 

 

'53? f; up

 

 

 

 

 

————l——— ______I

 

MSUIoMWMMOWIW

 

ESSAYS ON UNCERTAINTY AND SUSTAINABILITY
IN THE SEMI-ARID TROPICS

By
Takeshi Sakurai

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Agricultural Economics

1995

ABSTRACT

ESSAYS ON UNCERTAINTY AND SUSTAINABILITY
IN THE SEMI-ARID TROPICS

By

Takeshi Sakurai

This study explores the effects of uncertainty on sustainability in rural areas of the
semi-arid tropics characterized by mixed crop and livestock farming subject to frequent
drought and lack of formal financial and insurance institutions. The fundamental hypotheses
are: (i) Uncertainty due to drought induces precautionary saving; (ii) Saving takes a form of
livestock holdings because livestock is a relatively safe asset and there is no formal saving
institution. This combined with open access to common-me resource results in overgrazing;
(iii) Formal drought insurance will reduce overgrazing. The household’s ability to smooth
consumption and the village’s ability to manage grazing land are the two key concepts of
sustainability in this study. They are examined theoretically and empirically in four
independent essays.

The first essay is a literature review. The second essay is original extensions of the
models found in the literature. Effects of risks on saving are examined in the models, and it is
found that precautionary demand for saving exists if saving is riskless.

In the third essay, using a static household portfolio model, conditions for
overgrazing are examined by comparing an aggregation of household equilibrium livestock
holdings with the social optimal livestock holding. Since households in this model are
engaged in both crop production and livestock husbandry at the same time, the model

predicts that overgrazing does not necessarily occur even under open-access. If crop

production risk is high enough, overgrazing will occur even if the expected return to crop
production is greater than that to livestock holdings.

The fourth essay empirically explores potential demand for a hypothetical formal
drought insurance scheme and the determinants of such demand using household data from
Burkina Faso. Effective demand is found in all agroclimatic zones, but its size is correlated
with the drought risk of the zone. The determinant analysis shows that those who have large
crop stocks tend to demand less insurance, while the effects of livestock holdings and off-
farm income are ambiguous. But some households keep large livestock holdings but are not
sufficiently self-insured Those households may purchase insurance to substitute for livestock

holdings.

ACKNOWLEDGMENTS

First,'l would like to thank Dr. A. Allan Schmid, my major professor, who encouraged
and guided me throughout my graduate studies. Without him, I would not have considered
pursuing a Ph.D.

I owe my dissertation to Dr. Thomas Reardon very much. I would like to express my
deep gratitude to him for having made me interested in sustainability issues in the West Africa
semi-arid tropics, and for carefully and patiently reading earlier drafts of each chapter of my
dissertation and giving me detailed comments as my dissertation supervisor. It was very
fortunate that I worked with him as a graduate research assistant in the World Bank research
project “Management of Drought Risks in Rural Areas”. The experience in this project is
very important for me: it was a valuable basis of my dissertation; moreover it determines in
which field and in what way I will be working after graduation.

I also thank the other members of my dissertation committee, Dr. John P. Hoehn, Dr.
John Staatz, and Dr. Theodore Tomasi, for their comments on my proposal.

In addition to the members of my dissertation committee, many people gave
comments on earlier versions.

Regarding chapter 4, I acknowledge valuable comments from Dr. Keijiro Otsuka of
Tokyo Metropolitan University and participants at the following seminars where earlier
versions of chapter 4 were presented: the Annual Meetings of the Agricultural Economics
Society of Japan, Fukuoka, Japan, April 1995: Workshop for Theoretical Economics in
Agriculture, Fukuoka, Japan, April 1995; the seminar at the National Research Institute of
Agricultural Economics, Tokyo, Japan, April 1995.

Chapter 5 originated from the World Bank research project. In the project I worked
with Dr. Peter Hazell of International Food Policy Research Institute, and Dr. Madhur Gautam

iv

and Dr. Harold Alderman of the World Bank, besides Dr. Thomas Reardon. In fact, since this
project was initiated by them, I owe them for the basic idea and method. Their comments on
the paper that I wrote with them for the project were very helpful and are also reflected in this
chapter. I thank them very much. Also my thanks go to International Crops Research Institute
for the Semi-Arid Tropics (ICRISAT), and especially Dr. Peter Matlon of West African Rice
Development Association, for provision of ICRISAT Burkina Faso data. I also thank Dr. John
H. Sanders of Purdue University for provision of rainfall data in the Sahel. I acknowledge
valuable comments on earlier versions from Dr. Jock Anderson of the World Bank, Dr.
Robert Myers, and Dr. John Strauss of Michigan State University, and participants at the
Agricultural Economics Department seminar, February 1994, at the Annual Meetings of
American Agricultural Economics Association, San Diego, California, August 1994, and at
the seminar at the National Research Institute of Agricultural Economics, Tokyo, Japan, April
1995.

This research and my graduate training at Michigan State University are partially
funded by the World Bank through the project mentioned above, and in addition by USAID,
via Food Security 11, Michigan State University. I thank those funding agencies very much I
also thank Dr. Valerie A. Kelly and Dr. Thomas S. Jayne with whom I worked in the Food
Security project for experience in different kinds of problem in Africa.

Finally, my deepest thanks go to my wife, Kumi, and my daughter, Mahoko, for their

help and patience during hard times at Michigan State University.

TABLE OF CONTENTS

LIST OF TABLES ............................................................................................................... xi

LIST OF FIGURES ........................................................................................................... xiii

Chapter 1 Introduction

1 The Problem .............................................................................................................. l
1.1 Uncertainty and Lack of Financial Market ..................................................... l
l .2 Sustainability ................................................................................................. 2
1.3 Hypotheses .................................................................................................... S
2 Objectives of the Study .............................................................................................. 6
2.1 Objectives and Contributions of Chapters 2 and 3 .......................................... 7
2.2 Objectives and Contributions of Chapter 4 ..................................................... 7
2.3 Objectives and Contributions of Chapter 5 ..................................................... 8
References ......................................................................................................... ‘ ............ 11

Chapter 2 Household Saving in LDCs: A Critical Review

0

1

Summary ................................................................................................................. 14
Introduction ............................................................................................................ 16
Evidence of Consumption Smoothing ...................................................................... 18
Household Saving Models ....................................................................................... 24
3.1 Borrowing Constraint and Uncertainty ......................................................... 24
3.2 Saving Under Certainty / Two-Period Model ................................................ 26
3.3 Saving Under Uncertainty / Two-Period Model ............................................ 27

3.3.1 Labor Income Uncertainty / Two-Period Model ............................... 27

. /’Ii

3.3.2 Interest Rate Uncertainty / Two-Period Model .................................. 30

3.3.3 Labor Income and Interest Rate Uncertainty / Two-Period Model....3l

3.4 Saving Under Certainty I Multiperiod Model ................................................ 32
3.5 Saving Under Uncertainty I Multiperiod Model ............................................ 33
3.5.1 Labor Income Uncertainty / Multiperiod Model ............................... 33
3.5.2 Interest Rate Uncertainty / Multiperiod Model .................................. 39

3.5.3 Labor Income and Interest Rate Uncertainty / Multiperiod Model....40

3.6 Stochastic Production / Multiperiod Model .................................................. 41
3.7 Flexibility and Production/Saving ................................................................ 44
4 Conclusions ............................................................................................................ 52
References ..................................................................................................................... 55
Appendix ...................................................................................................................... 59
A Determinants of Drought Shock in Burkina Faso ......................................... 59
B Full Insurance in Burkina Faso .................................................................... 60

Chapter 3 Household Saving in LDCs:
Extensions and New Structural Models

0 Summary ................................................................................................................. 61
1 Introduction ............................................................................................................ 63
2 Extensions of Saving Models ................................................................................... 66
2.1 Correlation of Two Types of Risk ................................................................ 66
2.2 Stochastic Production Model ....................................................................... 68
2.3 Flexibility and Production/Saving ................................................................ 71

3 Structural Models and Their Implications ................................................................ 73
3.1 Time Frame of Decision Making ................................................................. 75
3.2 A Simultaneous Decision Mode ................................................................... 76
3.2.1 The First Order Conditions .............................................................. 79

3.2.2 Capital and Labor Inputs ................................................................. 80

vii

3.2.3 Saving and Borrowing ..................................................................... 82

3.2.4 Crop Diversification ......................................................................... 83

3.2.5 Social Insurance .............................................................................. 84

3.2.6 Lifetime Budget Constraints ............................................................ 85

3.3 A Sequential Decision Model ....................................................................... 86
3.3.1 The First Order Conditions .............................................................. 86

3.3.2 Effects of Increasing Risk ................................................................ 89

4 Conclusions ............................................................................................................. 90
References ..................................................................................................................... 94
Appendix .................................................................................................................... 103
A Labor Income and Interest Rate Uncertainty / Two-Period Model .............. 103

B A Dynamic Household Model with Uncertain Production .......................... 104

C A Sequential Decision Model with Flexibility ............................................. 106

D Effects of Increasing Risk on Decisions in a Sequential Decision Model.... 108

Chapter 4 Equity and Sustainable Management of
Common-Pool Resources: The Case of Grazing Land
in the West Africa Semi-Arid Tropics

0 Summary ............................................................................................................... 110
1 Introduction .......................................................................................................... 113
2 Definitions and Approach ...................................................................................... 115
2.1 Definitions ................................................................................................. 115
2.2 Approach .................................................................................................. l 18
2.2.1 Relationship to Previous Work ....................................................... 118

2.2.2 Modeling Strategies ....................................................................... 119

3 The Model ............................................................................................................. 121
3. 1 Framework ................................................................................................ 12 1

3.2 The Household optimum under open Access ............................................. 124
3.3 lnterhousehold Distribution of Optimal Investment ................................... 125

viii

3.4 Effect of Borrowing Constraints ................................................................ 128
3.5 Aggregate Livestock Holdings under Open Access .................................... 129
3.6 The Social Optimum .................................................................................. 130
3.7 Comparison between Household and Social Optima ................................... 133

4 Application of the Model to grazing Land Management ........................................ 135
4.1 Physical Solutions ...................................................................................... 136
4.2 Institutional Solutions ................................................................................ 137
4.2. 1 Privatization ................................................................................... 137

4.2.2 Quota ............................................................................................ 139

4.2.3 Transferable Permits ...................................................................... 140

4.2.4 User Fee ........................................................................................ 142

4.3 Other Economic Solutions ......................................................................... 142
4.3.1 Off-farm Income ........................................................................... 142

4.3.2 Formal Drought Insurance ............................................................. 143

5 Conclusions ........................................................................................................... 143
References ................................................................................................................... 146
Appendix .................................................................................................................... 154
A Proofs of Propositions 1-19 ....................................................................... 154

B Proofs for Privatization .............................................................................. 166

C Proofs for Quota ........................................................................................ 168

D Proofs for Transferable Permits ................................................................. 170

E Proofs for User fee .................................................................................... 171

Chapter 5 Crop Production under Drought Risk and
Estimation of Demand for Formal Drought Insurance
in the Sahel

0 Summary ............................................................................................................... 177
1 Introduction .......................................................................................................... 179
2 The Model ............................................................................................................. 183

ix

2.1 Time Frame ............................................................................................... 183

2.2 A Dynamic Household Model ................................................................... 184
2.3 The First Order Conditions ........................................................................ 187

3 Data ....................................................................................................................... 191
4 Empirical Estimation ............................................................................................. 192

4.1 Production Function .................................................................................. 192
4.2 Estimation of Demand for Drought Insurance (¢) ................................... 193
4.3 Determinant of Demand for Drought Insurance (4)) ................................. 194

5 Results and Discussion ........................................................................................... 197

5.1 Production Function .................................................................................. 198
5.2 Estimation of Demand for Drought Insurance (4») ................................... 201
5.3 Determinant of Demand for Drought Insurance (¢) ................................. 202

6 Conclusions ........................................................................................................... 206

References ................................................................................................................... 209
Appendix .................................................................................................................... 221
A Change in Average Rainfall and Drought Probabilities ............................... 221

B Expected Rainfall Levels (mm) .................................................................. 223

C. Calculation of (30), (31), and (32) ............................................................. 224

D Mean and Standard Deviation of Wealth Variables ..................................... 225

E Derivations for Table 3 .............................................................................. 226

F Regression Results at 35% Drought Probability .......................................... 229
Chapter 6 Summary of Approach and Findings ............................................. 233

LIST OF TABLES

Chapter 2 Household Saving in LDCs: A Critical Review

Table 1 .

Effect of Increasing Risk on Saving and Consumption .................... 58

Chapter 3 Household Saving in LDCs:
Extensions and New Structural Models

Table 1
Table 2
Table 3
Table 4

Table 5

Relationship Between 7 and 5:, ....................................................... 95
Effect of Increasing Risk on Decisions (uncorrelated) ...................... 97
Effect of Increasing Risk on Decisions (correlated) .......................... 98
Effect of Increasing Risk on Decisions (simultaneous) ................... 101
Effect of Increasing Risk on Decisions (sequential) ........................ 102

Chapter 5 Crop Production under Drought Risk and
Estimation of Demand for Formal Drought Insurance

in the Sahel
Table 1 Comparison of Assumptions on Drought Probability ..................... 211
Table 2 Annual Village-Level Rainfalls and Probabilities ............................ 213
Table 3 Production Function Estimates with Fixed Effects .......................... 214
Table 4 Summary of Signs of Marginal Productivity .................................. 215
Table 5 Zone-level ¢ and Drought Probabilities ........................................ 216
Table 6 Determinants of 4» (full sample) .................................................... 218
Table 7 Determinants of 4» (zone) .............................................................. 219
Table 8 Determinants of p (stratified by livestock holdings) ...................... 220
Table A1 Segou (Sudanian zone, Mali), Drought Frequency ......................... 221
Table A2 Koutiala (Guinean zone, Mali), Drought Frequency ....................... 222
Table B1 Expected Rainfall (mm) ................................................................. 223
Table D1 Mean and Standard Deviation of Wealth variables .......................... 225

xi

Table El
Table E2
Table E3
Table E4
Table F1
Table F2

Table F3

Signs of Marginal Productivity of Labor ........................................ 227

Signs of Marginal Productivity of Capital ...................................... 227
Signs of Marginal Productivity of Crop Diversification .................. 228
Signs of Marginal Productivity of Cultivated Area ......................... 228
Determinants of 4) (full sample) .................................................... 230
Determinants of ¢ (zone) .............................................................. 231
Determinants of op (stratified by livestock holdings) ...................... 232

xii

LIST OF FIGURES

Chapter 3 Household Saving in LDCs:
Extensions and New Structural Models

Figure 1
Figure 2

Figure 3

Relationship Between F and 5:, ....................................................... 96
Time Frame of Simultaneous Decision Making ............................... 99
Time Frame of Sequential Decision Making .................................. 100

Chapter 4 Equity and Sustainable Management of
Common-Pool Resources: The Case of Grazing Land
in the West Africa Semi-Arid Tropics

Figure 1
Figure 2
Figure 3
Figure 4
Figure 5
Figure A1

Figure A2

Figure A3

Figure A4

Figure A5

Household’s Risk Aversion Coefficient (¢) and Wealth ................. 149

Household’s Relative Risk Aversion Coefficient (¢W) and Wealth150

Optimal Investments (case 1, m-g(*)+b>0) .................................... 151
Optimal Investments (case 2, m-g(*)+b>0) .................................... 152
Optimal Investments (case 3, m-g(*)+b50) .................................... 153

Determination of Relative Size of Crop and Livestock Investments 172

Comparison of Household and Social Optima
(case 1, m-g(*)+b<0) ...................................... _ ............................... l 73

Comparison of Household and Social Optima
(case 2, m-g(*)+b20) ..................................................................... 174

Comparison of Household and Social Optima
(case 3, m-g(“)+b20) ..................................................................... 175

Comparison of Household and Social Optima
(case 4, m-g("‘)+b20) ..................................................................... 176

Chapter 5 Crop Production under Drought Risk and
Estimation of Demand for Formal Drought Insurance

in the Sahel
Figure 1 Time Frame of Decision Making ................................................... 212
Figure 2 Interpretation of ¢ ........................................................................ 217

xiii

Chapter 1

Introduction

1 The Problem

The main issues of this dissertation are household behavior under uncertainty in the
semi-arid tropics (SAT) and its effects on common-pool resources. These two issues are
linked by two major characteristics of the SAT: (i) existence of drought risk, and (ii) lack of
formal credit and insurance institutions, where the agricultural system is characterized by
sedentary village settlements engaged in mixed crop and livestock production such as are
common in the West Africa semi-arid tropics (W ASAT).

An important concept underlying the links between the two issues is sustainability. In
the following sub-sections, I explain how they are linked by introducing the concept of
sustainability. Then, I provide hypotheses that are theoretically explored and empirically

tested in this dissertation.

1. l Uncertainty and Lack of Formal Financial Institutions

All human endeavors are constrained by our limited and uncertain knowledge - about
external events past, present, and future; about the laws of Nature, God, and man;
about our own productive and exchange opportunities; about how other people and
even we ourselves are likely to behave (Hirshleifer and Riley, 1992).

On the one hand, there are many sources of uncertainty facing households: some are
common over households, such as drought and volatile market prices, and some are
idiosyncratic to a specific household, such as illness and accidents. But among them natural
events most heavily affect people’s life in rural areas of developing countries where
livelihood depends on natural resources. In the SAT, drought is the major factor that

determines household behavior.

2

On the other hand, formal insurance, and credit and capital markets are severely
underdeveloped or are not accessible by the rural poor in the SAT (e. g. Aleem, 1993; Bell,
1993; Binswanger, 1986; Binswanger and Sillers, 1983; Braveman and Guasch, 1993;
Christensen, 1989; Hoff and Stiglitz, 1993; Matlon, 1990; Udry, 1993a).

Faced with drought risk and the lack of those markets, rural households in the SAT
engage in a variety of risk management/coping strategies to smooth interyear income and
consumption. Those strategies have been empirically examined recently, e. g., earning non-
agricultural income (Reardon et al., 1992); investment in livestock (Christensen, 1989;
Rosenzweig and Wolpin, 1993); crop storage (Udry, 1993b); transfers among households
(Lucas and Stark, 1985; Rosenzweig and Stark, 1989); informal credit (Udry, 1994); crop
diversification (Matlon, 1990; Norman, 1973). In addition, instead of focusing on a
particular strategy, overall consumption smoothing is also examined in risk-sharing models
(e.g. Deaton, 1994; Sakurai, 1995; Townsend, 1994).

However, the effects of risk management/coping strategies on common-pool resources
have been rarely examined yet. To examine the links between them, sustainability is the key
concept Sustainability can be defined at different levels, such as the household level and the
village level, to be explained in the next section. Sustainabilities at different levels can be
complementary in some case, but can be conflicting in other case. And the links can be seen

as the relationship between sustainabilities at different levels.

1 .2 Sustainability

Sustainability has become an important issue in international development, but its
definition differs depending on at what level and in what agricultural system the sustainability
is considered (Lele, 1991; Pezzy, r992; Ruttan, forthcoming). On the one hand, because the
concept of sustainability is related to the use of natural resources, it is often defined at global,
regional, or national level (e.g. Pearce and Atkinson, 1995). However, on the other hand, as
sustainability is based on household behavior, it can be defined at the household level. For

3
example, in the Sahel, maintaining a household‘s agricultural productivity by controlling soil
degradation is the main issue in agricultural sustainability (Reardon, forthcoming). The
literature usually focuses on the “vicious cycle" between rural poverty and environmental
degradation, but Reardon and Vosti (1995) argue that categorizing poverty and natural
resources is necessary to analyze the links, and that the links should be understood in terms of
household income and investment/saving behavior.

Combining the discussion in section 1.1 with the household-level sustainability
mentioned above, sustainability must be analyzed based on a household’s risk
management/coping strategies in the absence of formal credit and insurance in the SAT.
Because each household’s objective is to sustain under risk its well-being, or more narrowly,
to smooth consumption above a minimum level over a certain period, I define household-
level sustainability as the ability of the household to smooth consumption over time. This
comprises for example agricultural productivity by controlling soil erosion, saving. as well as
other strategies in crop production, in off-farm activities, and in livestock husbandry.
Consequently, a household’s risk response is a function of household characteristics and
other exogenous factors, although the objective is to smooth consumption. For example,
some households will use off-farm income to increase agricultural productivity, but others
will not; whether off-farm income is reinvested in farming will depend on nature of capital
and labor market and characteristics of off-farm income flow (Reardon et aL, 1994). Effects
of livestock income and crop diversification on agricultural productivity are unknown, too
(Reardon, forthcoming)‘.

By defining sustainability at the household level as a household’s ability to smooth
consumption, in addition to crop production, off -farm activities and livestock husbandry

become important for the household to achieve sustainability. For example, Reardon et al.

 

‘ However livestock as traction animals is known to incl-ease household’s agricultural productivity
and supply response in the Sahel (Savadogoet al.,1994a; Savadogo er al., 1994b).

4

(1992) find that households that diversify income more (into off-farm and livestock sources)
have more stable income and consumption.

Now I define sustainability of a given income earning activity as the maintenance of
its productivity in the long-run. Thus, household-level sustainability depends on the
sustainability of a mix of activities. The mix then depends on household characteristics.

The productivity, here, sustainability of off-farm activities and livestock husbandry is
determined at least partially at the village level, as there is interdepedency among households.
For example, off-farm activities (local and migratory) of a given household can benefit from
off-farm income at the village level as the latter increases off-farm income opportunity’. By
contrast, livestock husbandry by other villagers will have a negative externality for a particular
household. That is, since livestock is kept on open-access grazing land, overgrazing will
decrease availability of grass per an animal, and as a result productivity of livestock will
become low. This implies that household-level sustainability is also determined by village-
level factors that influence the sustainability of a given household activity.

There are many kinds of commons, such as ponds, grazing land, forest, and so on, in
rural villages, and the poor tend to rely more than do the rich for their livelihood on
common-me resources in developing countries (Dasgupta and Maler, 1994). But this
difference in dependence is relative, as the poor have smaller herds than do the rich, and thus
put less absolute pressure on open-access grazing lands in the WASAT (Reardon and Vosti,
1995). In the Sahel, livestock is about 30% overstocked due to population growth. This has
resulted in a severe degradation of grasslands (Houerou, 1989). Because livestock is a
substitute for formal savings, it is used for household risk management. And as a result of
each household’s behavior under risk, overgrazing and degradation are occurring in the

Sahel. This means that all households will finally lose the ability to smooth consumption in

 

3 This may sound similar to the concept of agricultural growth multipliers (e.g. Haggblade and
Hazel], 1989; Haggblade e1 01., 1989; Hopkins et al., 1994). But multipliers usually imply the effect of
exogenousincreasein tradablefarmsectorincomeonnonfarm income,andigncreeachhouseholddedsion
undamtaintyandintadependarceamonghomeholthinavillage.

5

the long-run, if they do not have a rule to control the use of grazing land. This is an
important example of links between a household’s risk management/coping strategies and the
household’s sustainability, with the links influenced by village-level factors. However, the
links have not been explored based on household behavior in the literature.

Village-level factors determine household-level sustainability; sustainability at the
village level can be defined as the village’s ability to support household-level sustainability. It
directly depends on village-level productivities of crop, livestock and off-farm activity, which
are determined by village factors such as natural resource endowment, infrastructure, market
development, and so on. But village-level sustainability may also depend on informal and
formal village mics that govern household behavior. One example is risk-sharing within
village through transfers and remittances between households. Townsend (1994) provides
evidence that households in an Indian village pool risk almost fully, while Udry (1994), using
data from a village in northern Nigeria, finds evidence against full risk-sharing, but in support
of the existence of partial risk-sharing. Institutions that govern the local commons also affect
the sustainability of local commons (Ostrom, 1990; Ostrom et al., 1994), which determines
household-level sustainability.

A lot of empirical evidence on informal village institutions have been found, but those
studies examine performance of existing institutions based on observed behavior. But the
question I raise in this dissertation is what kind of formal or informal institutions could
potentially be established, how effectively they would promote sustainability at both the
household level and the village level, and how sustainable the institutions themselves would

be.

1 .3 Hypotheses
Given discussion above, the links between household behavior under drought risk and
its effects on common-pool resources are hypothesized in this dissertation as follows. (a)

Households in the SAT tend to save for future uncertain income, that is, there is precautionary

6

demand for savings. (b) Savings take the form of livestock holdings because there is no
formal saving institutions and because livestock is a relatively safe asset in times of drought in
the SAT. (c) As a result of the precautiormry livestock holding, overgrazing occurs because
grazing land is an open-access, common-pool resource. (d) Formal financial institutions, such
as drought insurance, will substitute for livestock holdings and reduce overgrazing.

These links are explored theoretically and empirically guided by the following issues.
(1) How do uncertainty and borrowing constraints affect household saving? The effects break
down into two categories: (i) the effects of uncertainty and borrowing constraints on
household saving; (ii) the links between saving and other self-insurance strategies. However,
since the latter links determine the effect of uncertainty and borrowing constraints, (i) and (ii)
can be integrated. (2) What is the aggregate effect of household risk management/coping
strategies (in particular, livestock holding) on the sustainable use of common-pool resources
(in particular, on grazing land)? (3) Are there potential solutions for overgrazing of
common-me grazing land? Several solutions are considered, among them formal drought

insurance.

2 Objectives of the Study

The objectives of this study are to address the problems presented above and to test
the hypotheses presented above using household data from rural Burkina Faso in a series of
essays: Chapter 2: Household Saving in LDCs: A Critical Review; Chapter 3: Household
Saving in LDCs: Extensions and New Structural Models; Chapter 4: Equity and Sustainable
Management of Common-Pool Resources: the Case of Grazing Land in the West Africa Semi-
Arid Tropics; Chapter 5: Crop production under Drought Risk and Estimation of Demand for
Formal Drought Insurance in the Sahel. Finally, in chapter 6, the four essays are synthesized
to draw the conclusions.

More specific objectives and contributions of each chapter are discussed in the

following sub-sections.

7
2. 1 Objectives and Contributions of Chapters 2 and 3

Chapters 2 and 3 are closely related. Chapter 2 consists of two literature reviews on
household saving in rural areas in less developed countries. In chapter 3 I extend models
found in the literature reviews, and based on the reviews and the extensions, 1 develop new
agricultural household models. The objectives are to identify gaps in the literature, to fill the
gaps, and to obtain theoretical insights and empirically testable hypotheses based on the
models. Thus, these chapters serve as a basis of future empirical studies on household saving
in LDCs.

In chapter 2, I identify two types of gap in the literature: gaps in empirical studies and
gaps in theoretical studies. The gaps in empirical studies are identified by literature review on
empirical evidence of household consumption smoothing. The gaps in theoretical studies, on
the other hand, are identified by review of literature on household saving models, from a
static two-period model to a dynamic multiperiod model under uncertainty and flexibility.

. In chapter 3, I extend models to fill the theoretical gaps identified in chapter 2. Then,
I deve10p structural dynamic models of the agricultural household in the context of the
WASAT. The models incorporate agricultural production, off-farm activities, saving,
sequential decisions, and other risk management/coping strategies. These models themselves
contribute to the literature because most household saving models ignore endogenous
decisions for agricultural production although it is an important element of the agricultural
household model (Singh, et al., 1986), and because off-farm activities have seldom been

treated as a risk response in a household model.

2. 2 Objectives and Contributions of Chapter 4

In chapter 4, I focus on the implications of the theoretical discussions in chapter 2
and 3 of household saving behavior on overgrazing in the WASAT, where all households in
the village engage in both crop production and livestock husbandry, and keep livestock on

open-access grazing land. The objective of this chapter is to construct a theoretical framework

8

that can be used to analyze the problem of household saving in the WASAT, that is,
overgrazing. The motivation behind this is to challenge the assertion since Hardin (1968) that
overgrazing is inevitable in open-access grazing land, and to identify more precise conditions
for overgrazing in the context of the WASAT.

First, I construct a household model that explains the household’s decision to allocate
its wealth between crop production and livestock husbandry. To have tractability, a static
portfolio selection model is used. The model is different from those found in previous studies
on resource management in two ways: (i) Each household faces two investment opportunities:
risky crop production and safe livestock holdings; (ii) Households in a village are
heterogeneous in wealth and as a result heterogeneous in risk preferences. These two are
realistic assumptions in the WASAT, and because of them it becomes possible to examine the
effects of relative returns to the two activities and the effects of wealth distribution of the
village using the model. In this regard, this chapter makes a disciplinary contribution.

Second, answers to the following questions are derived from the model as theoretical
insights: (i) How do WASAT households behave under uncertainty in agricultural
production? (ii) Why is there overgrazing? By answering (i) and (ii), conditions for
overgrazing are identified. (iii) What property regimes would solve overgrazing? Are there
any solutions for overgrazing? Current property regime is assumed to be open-access, and in
this chapter privatization, quota, transferable-permit, and user fee schemes are considered But
in addition to them, physical solutions and other economic solutions are considered. (iv) What
would be the equity consequence of these regimes? These theoretical insights obtained from
the model contribute not only to provision of testable hypotheses for empirical studies, which

in turn would be useful to policy formulation in the WASAT.

2. 3 Objectives and Contributions of Chapter 5
In chapter 5 hypothetical formal drought insurance is considered as a solution to

overgrazing. If formal drought insurance is more cost-effective than current informal self-

9

insurance, it will not only benefit grazing land, but also improve household welfare and
contribute to economic growth. Thus, the primary objective of this chapter is to estimate
demand for (hypothetical) formal drought insurance in the Sahel.

The first hypothesis is that current informal self-insurance provides inadequate
protection against drought shock for households in the Sahel. This hypothesis is empirically
tested by estimating demand for hypothetical formal drought insurance given self-insurance.
A hypothetical insurance scheme called “rainfall lottery" is introduced (Gautam et 01., 1994;
Hazell, 1992). In this scheme, an insurance company can assign any rainfall cut-off point to
the lottery, and if annual rainfall is lower than the predetermined cut-off point, the insurance
company will pay an indemnity. This particular insurance scheme is designed to avoid moral
hazard and adverse selection (that cause a supply-side problem for crop insurance, especially
in LDCs). Thus, the question is whether there is demand for the drought insurance. Therefore,
one objective of this chapter is to estimate potential demand for this hypothetical drought
insurance.

The second hypothesis is that the a household’s demand for the formal drought
insurance depends on the household’s current self-insurance. This hypothesis is empirically
tested by explaining the demand by household asset variables and household characteristics.

To accomplish these objectives: First, a dynamic household model is constructed
following chapter 3; Second, a new method to estimate demand for formal drought insurance
is established. This econometric method uses observed behavior of household that informally
self-insures itself through its income and asset portfolio strategies without formal insurance.
The method is newly developed in this chapter, and makes a disciplinary contribution; Third,
potential demand for formal drought insurance is estimated using household data from rural
Burkina Faso in the WASAT. Fourth, determinants of demand for formal drought insurance
are examined. These analyses reveal how households in the WASAT manage drought risk and
cope with drought shock, and indirectly test the hypotheses on interlinkages of strategies

derived from the models in chapters 3 and 4. In particular, the effect of livestock holding on

10

demand for formal drought insurance is examined to see if it will reduce overgrazing. The

third and fourth results make contributions to policy formulation.

11

References

Aleem, Ifran, “Imperfect Information, Screening, and the Costs of Informal Lending: A
Study of a Rural Credit Market in Pakistan,” In: Karla Hoff, Avishay Braveman, and
Joseph E. Stiglitz, eds., The Economics of Rural Organization. Theory, Practice, and
Policy, New York, NY: Oxford University Press, 1993.

Bell, Clive, “Interaction between Institutional and Informal Credit Agencies in Rural India,”
In: Karla Hoff, Avishay Braveman, and Joseph E. Stiglitz, eds., The Economics of
Rural Organization. Theory, Practice, and Policy, New York, NY: Oxford University
Press, 1993.

Binswanger, Hans, “Risk Aversion, Collateral Requirements, and the Markets for Credit and
Insurance in Rural Areas,” In: P. Hazell, C. Pomareda, and A. Valdes, eds., Crop
Insurance for Agricultural Development, Baltimore: Johns Hopkins University Press,

1986.

Binswanger, Hans P. and Donald A. Sillers, “Risk Aversion and Credit Constraints in
Farmers’ Decision Making,” Journal of Development Studies, 20: 5-21, 1983.

Braveman, Avishay and J. Luis Guasch, “Administrative Failure in Rural Credit Programs,”
In: Karla Hoff, Avishay Braveman, and Joseph E. Stiglitz, eds., The Economics of
Rural Organization. Theory, Practice, and Policy, New York, NY: Oxford University
Press, 1993.

Christensen, Garry Neil, “Determinants of Private Investment in Rural Burkina Faso,” Ph. D.
Dissertation, Cornell University, 1989.

Dasgupta, Partha and Karl-Goran Maler, “Poverty, Institutions, and the Environmental
Resource Base,” The World Bank, World Bank Environment Paper, No. 9, 1994.

Deaton, Angus, “The Analysis of Household Surveys: Microeconomic Analysis for
Development Policy,” mimeo, 1994.

Gautam, Madhur, Peter Hazell, and Harold Alderman, “Management of Drought Risks in
Rural Areas,” Policy Research Working Paper, The World Bank, 1994.

Haggblade, Steven and Peter Hazell, “Agricultural Technology and Farm-Nonfarm Growth
linkages,” Agricultural Economics, 3: 345-364, 1989.

Haggblade, Steven, Peter Hazell, and James Brown, “Farm-Nonfarm Linkages in Rural Sub-
Saharan Africa,” World Development, 17: 1173-1201, 8, 1989.

Hardin, Garrett, “The Tragedy of the Commons,” Science, 162 (3859): 1243-1248,
December 13, 1968. Reprinted in Environmental Economics: A Reader, ed.
hgagrlztandya, Anil and Julie Richardson, 60-70, New York, NY: St. Martin’s Press,
1 .

Hazell, Peter, ”The Appropriate Role of Agricultural Insurance in Developing Countries,”
Journal of International Development, 4: 567-582, 6, 1992.

Hirshleifer, Jack and John G. Riley, The Analytics of Uncertainty and Information,
Cambridge, UK; Cambridge University Press, 1992

Hoff, Karla and. Joseph E. Stiglitz, “Imperfect Information and Rural Credit Market: Puzzles
and Policy Perspectives,” 1m Karla Hoff, Avishay Braveman, and Joseph E Stiglitz,

12

eds., The Economics of Rural Organization. Theory, Practice, and Policy, New York,
NY: Oxford University Press, 1993.

Hopkins, Jane, Valerie Kelly, and Christopher Delgado, “Farm-Nonfarm Linkages in the West
African Semi-Arid Tropics: New Evidence from Niger and Senegal,” Selected Paper
at the AAEA Annual Meetings, in San Diego, California, 1994.

Houerou, Henry Noel Le, The grazing Land Ecosystems of the African Sahel, Berlin,
Germany: Springer-Verlag, 1989.

Lele, Sharachchandra M., “Sustainable Development: A Critical Review,” World
Development, 19: 607-621, 6, 1991.

Lucas, Robert and Oded Stark, “Motivations to Remit: Evidence from Botswana,” Journal of
Political Economy, 95: 901-918, 1985.

Matlon, Peter, “Farmer Risk Management Strategies: The case of West-African Semi-Arid
Tropics,” In The World Bank’s Tenth Agriculture Sector Symposium in Washington
DC, The World Bank, 1990.

Norman, D. W., “Economic Analysis of Agricultural Production and Labour Utilization
Among the Hausa in the North of Nigeria,” Dept. of Agricultural Economics,
Michigan State University, African Rural Employment Paper, No. 4, January, 1973.

Ostrom, Elinor, Governing the Commons, Cambridge, UK: Cambridge University Press, 1990.

Ostrom, Elinor, Roy Gardner, and James Walker, Rules, Games, & Common-Pool Resources,
Ann Arbor, MI: The University of Michigan Press, 1994.

Pearce, David and Giles Atkinson, “Measuring Sustainable Development,” In: Daniel W.
Bromley, eds., The Handbook of Environmental Economics, Cambridge, MA: Basil
Blackwell, 1995.

Pezzy, John, “Sustainable Development Concepts, An Economic Analysis,” The World Bank,
World Bank Environment Paper, No. 2, 1992.

Reardon, Thomas, “Sustainability Issues for Agricultural Research Strategies in the Semi-arid
Tropics: Focus on the Sahel,” Agricultural Systems, forthcoming.

Reardon, Thomas, Eric Crawford, and Valerie Kelly, “Links Between Nonfarm Income and
Farm Investment in African Households: Adding the Capital Market Perspective,”
American Journal of Agricultural Economics, 76: 5, 1994.

Reardon, Thomas, Christopher Delgado, and Peter Matlon, “Determinants and Effects of

Income Diversification amongst Farm Households in Burkina Faso,” Journal of
Development Studies, 28: 264-277, 1992.

Reardon, Thomas and Stephen A. Vosti, “Links Between Rural Poverty and the Environment
in Developing Countries: Asset Categories and Investment Poverty,” World
Development, 23: 9, 1995.

Rosenzweig, Mark and Oded Stark, “Consumption Smoothing, Migration and Marriage:
Evidence from Rural India,” Journal of Political Economy, 97: 905-927, 1989.

Rosenzweig, Mark R. and Kenneth I. Wolpin, “Credit Market Constraints, Consumption
Smoothing, and the Accumulation of Durable Production Assets in Low-Income

Countries: Investments in Bullocks in India,” Journal of Political Economy, 101:
223-244, 2, 1993.

13

Ruttan, Vernon W., “Sustainable Growth in Agricultural Production: Poetry, Policy, and
Science,” In: Stephen A. Vosti and Thomas Reardon, eds., Agriculture Sustainability,
Growth, and Poverty Alleviation: Issues and Policies, Baltimore, MD: Johns Hopkins
University Press, forthcoming.

Sakurai, Takeshi, “Consumption Smoothing in Burkina Faso,” Draft, Michigan State
University, 1995.

Savadogo, Kimseyinga, Thomas Reardon, and Kyosti Pietola, “Farm Productivity in Burkina
Faso: Effects of Animal Traction and Nonfarm Income,” American Journal of
Agricultural Economics, 1994a.

Savadogo, Kimseyinga, Thomas Reardon, and Kyosti Pietola, “Mechanization and
Agricultural Supply response in the Sahel: A Farm-Level profit Function Analysis,”
Department of Agricultural Economics, Michigan State University, Staff Paper, 94-66,
October, 1994b.

Townsend, Robert M., “Risk and Insurance in Village India,” Econometrica, 62: 539-591, 3,
1994.

Udry, Christopher, “Credit Market in Northern Nigeria: Credit as Insurance in a Rural
Economy,” In: Karla Hoff, Avishay Braveman, and Joseph E. Stiglitz, eds., The
Economics of Rural Organization. Theory, Practice, and Policy, New York, NY:
Oxford University Press, 1993a.

Udry, Christopher, “Risk and Saving in Northern Nigeria,” manuscript, Department of
Economics, Northwestern University, 1993b.

Udry, Christopher, “Risk and Insurance in a Rural Credit Market: An Empirical Investigation
in Northern Nigeria,” Review of Economic Studies, 61: 495-526, 1994.

Chapter 2

Household Saving in LDCs: A Critical Review

0 Summary

In this chapter I conduct two literature reviews: on empirical studies on household
consumption smoothing and on household saving models.

In the first review, I focus on agricultural households in rural areas of LDCs. There
are three findings: (i) There is empirical evidence of several strategies that smooth household
consumption such as crop diversification, income diversification (particularly off-farm
activities), use of risk-decreasing inputs, social insurance, and savings (livestock holdings); (ii)
They are not perfect, and village insurance studies imply that households mix those strategies
to smooth consumption; (iii) But those strategies are not empirically examined well in
structural household models and their interactions are little known.

In the second review, models for household saving are reviewed. It starts from two-
period static models, and finally deals with multiperiod dynamic models as well as flexible
decision models. The main concern of this review is the effect of uncertainty and borrowing
constraints. There are four findings, as follows.

First, two types of risk are often considered in literature: exogenous labor income risk
and exogenous interest rate risk. The effects of increase in those risks are summarized in
Table 1. The table shows that if saving is riskless, an increase in either type of risk increases
precautionary demand for saving, given a convex marginal utility function (v"' >0) and
decreases precautionary demand for saving, given a concave marginal utility function
(v"' <0). On the other hand, if saving is risky, convexity of the marginal utility function

(v"' > 0) does not necessarily induce precautionary demand for saving when interest rate risk

14

15

increases. In reality, however, those two risks coexist and must be correlated. Nevertheless
there are few saving models that deal with multiple, correlated risks.

Second, in terms of precautionary demand for saving, there is little difference between
two-period models and multiperiod models (as summarized in Table 1). However, the
advantage of multiperiod models is dynamic simulation. Deaton (1991) shows, based on
dynamic simulations, that a borrowing constraint under exogenous income risk induces
precautionary demand for saving even if the marginal utility function is concave. This is an
important implication that can be derived only from multiperiod models.

Third, a production function should be incorporated into a savings model. In other
words, saving behavior should be considered in a structural model of agricultural household.
The importance of the production function for analyzing household saving is that it includes
household risk management strategies through utilizing risk-decreasing inputs, and as a result
a household’s input decisions may affect the saving decision. Crop diversification also may
be-a risk-decreasing input decision. However, I find that there are few studies that consider
risk-decreasing inputs in household saving models.

Fourth, flexibility in decision making is important, especially in a model with
agricultural production, because the effect of flexibility depends on whether the flexible input
is risk-increasing or risk-decreasing. In addition, incorporating flexibility will allow us to deal
realistically with off-farm activities. However, I find that there are few studies that deal with
both off-farm activities and saving in flexible decision models.

As stated in the third and fourth conclusions, non-linearity of the marginal effect of
risk and the flexibility of decision making are important concepts, but their effects cannot be
predicted a priori. Therefore, empirical studies that treat them properly will answer how they
affect household’s risk responses. Such empirical studies should be based on a household
structural model that incorporates not only those two but also other risk management

strategies.

l6

1 Introduction

This chapter consists of two reviews. In the first one, I review literature on empirical
evidence of household consumption smoothing in rural areas of developing countries. In the
second, I review literature on household saving models. Through those two reviews, gaps in
the literature are identified so that this chapter can serve as a basis for future theoretical and
empirical studies on household saving in LDCs.

Saving is conventionally taken to mean consumption forgone today for the purpose
of greater consumption in the future. Saving is considered important for economic growth of
LDCs because it provides for accumulation of capital. The view that capital formation is the
key to growth, called capital fundamentalism, was reflected in the development strategies and
plans of many countries (Gillis et al., 1992). For example, in an influential paper, Lewis
(1954) writes “The central problem in the theory of economic development is to understand
the process by which a community which was previously saving and investing 4 or 5 per cent
of its national income or less, converts itself into an economy where voluntary saving is
running at about 12 to 15 per cent of national income or more. This is the central problem
because the central fact of economic development is rapid capital accumulation (including
knowledge and skills with capital)”.

However, the saving behavior of households in developing countries will not
necessarily lead to the accumulation of capital for several reasons. First, household saving in
LDCs is to smooth consumption in the face of volatile and unpredictable income (Deaton,
1989). This is not only because agricultural production in LDCs is very uncertain, but also
because households rely heavily on saving due to lack of formal consumption credit and crop
insurance institutions. Second, productivity of investment is low in LDCs (Dasgupta, 1993).
This is especially true in rural areas where there are few formal financial institutions.
Household saving to smooth consumption takes the forms of crop stocks and livestock
holdings, which are not productive. In other words, uncertainty and lack of formal financial

institutions force households to save in liquid, but unproductive assets in rural areas of LDCs.

17

Introduction of banks is shown to shift the composition of savings toward capital, to reduce
socially unnecessary capital liquidation, and as a result to promote growth (Bencivenga and
Smith, 1991). On the other hand, there is evidence that household savings induced by
liquidity constraints promote growth in OECD countries (Jappelli and Pagano, 1994), where
productivity of investment is generally high.

Sustainability is an important issue of international development. It can be defined at
different levels, but the household’s sustainability can be seen as its ability to sustain its well-
being or to smooth consumption above a minimum level over a certain period. In this sense,
saving/dissaving is the way to achieve sustainability in the short-run. On the other hand,
however, the increase in and stabilization of agricultural and non-agricultural productivity will
both increase and smooth income, and as a result will smooth consumption in the long-run.
To achieve long-run sustainability, use of more agricultural inputs and investment both in
agricultural equipment, in maintenance and improvement of land (e. g., terracing, reforesting,
digging irrigation ditches, etc.), and in human capital are critical. But the problem is, as
mentioned above, that households save to smooth consumption in the short-run, and such
behavior will not contribute to economic development in the long-run.

Thus, to formulate development policy, one needs to know household saving
behavior: why and how they save, and what will change their behavior from unproductive to
productive saving. Theoretical and empirical work at the microeconomic level has been
accumulating recently (for examples, see the review by Deaton (1994)). Deaton (1994)
discusses the conventional life-cycle (LC) model and the permanent-income (PI) model, and
presents examples of empirical studies in LDCs based on those models. But according to
Deaton there are important cases that are not covered by either one. One case is when the
marginal utility of consumption is not linear under uncertainty. In this case households show
a precautionary motive for saving, that is, they save more when the future becomes more
uncertain. The other case is when there are borrowing constraints. To incorporate those

missing features, Deaton simulates a household whose income sources are stochastic labor

18

income and saving, and shows that they save to smooth consumption if they know that they
cannot borrow even when their income is low.

Since uncertainty and borrowing constraints are features of LDC rural economies,
Deaton’s model may explain saving behavior of LDC households better than LC and PI
models. But the assumption of stochastic, exogenous labor income is still unrealistic because
endogenous agricultural production is an important element of the agricultural household
model (Singh, et al., 1986). However, there are few empirical studies that consider household
saving in a structural model; an exception is Rosenzweig and Wolpin (1993), who estimate a
dynamic household model and show that saving/dissaving of bullock holdings as a buffer
stock causes underinvestment in bullocks as productive capital. This kind of empirical study
should be conducted more to understand agricultural household’s behavior in LDCs. To
build a basis for such empirical studies, this chapter reviews literature on household saving.

Two review sections follow this introduction: In section 2, household saving is viewed
as one of the strategies to manage and to cope with risk to smooth consumption under
uncertainty and market imperfection, which are common in rural areas in LDCs. Then, I
review empirical evidence of consumption smoothing including not only saving but also
other strategies. In section 3, household saving models are reviewed The review starts from a
two-period static model, then deals with more complicated models, such as multiperiod
dynamic models and sequential decision models. The effects of uncertainty and borrowing

constraints on savings are emphasized.

2 Evidence of Consumption Smoothing

Households in developing countries make use of a wide variety of strategies to
smooth consumption. Since saving is one of such strategies, and since those strategies are
interacted with each other, we need to know the interaction to get a complete image of

household behavior under uncertainty. In this section, empirical evidence of consumption

smoothing is reviewed.

19

The strategies can be classified as risk management and risk coping. Risk
management is to reduce variability of income ex ante (before a risky outcome is known):
crop insurance, crop/plot diversification, income diversification, and utilization of risk-
decreasing technology. They reduce income risk, and as a result smooth consumption. Since
formal crop insurance is rarely available in developing countries, household’s strategies are
mainly the latter three.

Sakurai (1993) shows that crop diversification measured by a Simpson index of
planted area reduces crop income shock. in the Sahelian and the Sudanian zones in Burkina
Faso, but increases crop income shock in the Guinean zone (the most agro-climatically
favorable zone). See Appendix A for the details. This implies that crop diversification is an ex
ante risk management strategy in riskier areas where ex post risk coping is less available, but
in less risky areas where ex post risk coping is also developed, crop diversification is rather an
income-increasing strategy by diversifying to risky cash crops.

Non-farm income is a very significant part of total income in the WASAT: based on
several household surveys during 1980s, Reardon et al. (1994) conclude that non-farm
income (all income but cropping and livestock income) varies 20 to 64 percent of total
incomes and non-cropping income (all income but cropping income) ranges 31 to 83
percent. In Burkina Faso it is found that income diversification indexed as the ratio of off-
farm income and all income, significantly increases household income per adult equivalent,
and that income diversification reduces income risk and as a result reduces food consumption
risk (Reardon et al., 1992). But since their non-farm income includes both income before
harvest and income after harvest (sales of labor and livestock ex post), it is not the same as the
income diversification defined above as an ex ante strategy. Therefore, it is not surprising that
income diversification in Reardon et al. (1992) increases total income as well. The results give
evidence, however, that income diversification, whether ex ante or ex post, smoothes

consumption.

20

An input can be classified as either risk-increasing or risk-decreasing. If an input is
risk-increasing, the marginal productivity of the input increases as use of the risky input
increases, while if an input is risk-decreasing, the marginal productivity of the input decreases
as use of the risky input increases. Fertilizers are often risk-increasing, while pesticides and
herbicides are risk-decreasing (Ramaswani, 1993). In Burkina Faso it is found that land is a
risk-decreasing input: crop production per area decreases with total cropped land, but the
sensitivity of yield to rainfall variation is low when total cropped land is large (Sakurai et al.,
1994). Therefore, because animal traction helps area expansion (Savadogo et al., 1994),
animal traction may also be a risk-decreasing technology in that region.

Most studies on ex ante risk management except for Reardon et al. (1992), which
includes risk coping ex post, do not examine whether the reduction of income risk actually
reduces consumption risk. Since a household‘s strategy is a combination of ex ante risk
management and ex post risk coping, successful risk management in one activity may not
necessarily mean successful household consumption smoothing.

On the other hand, risk coping strategies can be classified as saving, or intertemporal
consumption smoothing, and risk-sharing, or consumption smoothing across households.
Saving includes borrowing and lending in a formal or an informal credit market,
accumulating and selling assets, and storing goods for future consumption. The assets to sell
may include labor, that is, households seek off-farm income sources, such as temporal
migration, after a bad harvest. On the other hand, insurance is a formal institution of risk-
sharing. In addition, several informal risk-sharing arrangements are found in developing
countries: state-contingent transfers and remittances between relatives and neighbors, credit
contracts with state-contingent repayments, share—tenancy, and so on. Informal risk-sharing as
a whole is called “social insurance”. There are a lot of studies on saving and social insurance
in developing countries, reviewed in Gersovitz (1988) and Deaton (1994). Most studies test
how well households smooth consumption, but tell little about how they achieve it. Examples

of saving and social insurance follow.

21

Paxson (1992) tests the permanent income hypothesis (PIH) based on the following

saving equation.
S... - a + £39.: + viz-1+ 07... +e...

If transitory income, FL, is saved and permanent income, 9,5,, is consumed as PIH implies,
fl - 0 and y - 1. Using time-series data from rice farmers in Thailand, she shows that y is
closed to unity but that B is positive and significantly different from zero although it is much
smaller than y . Her conclusion is tlmt farm households save a significantly higher fraction of
transitory income than nontransitory income, that is, PIH is partially supported. However, it
does not necessarily mean that the saving allows them to smooth consumption. Deaton
(1992a) uses two-year panel data from Cote d’Ivoire to test a permanent income model
assuming that income is a stationary stochastic process. He finds that the orthogonality
requirement of PIH is not supported by the data, that is, changes in consumption are affected
by lagged saving and income.

An important methodological issue to test PIH is how to statistically distinguish
permanent income from transitory income when only total income is available. Paxson uses a
rainfall variable to estimate transitory income, because transitory income is considered to
depend on the variability of rainfall. Another issue is how to define household saving, that is,
to distinguish investment and saving from total income. Paxson tries three different
definitions of saving and obtains different results.

Townsend (1994) uses lO-year panels of grain consumption and incomes from
several sources of households in the semi-arid tropics of southern India, and tests a full
insurance model for each household (133 households), based on the following equation,

C, -a +55, +yX, +02,+e,
where C, is consumption per adult equivalent of a household in year t, C, village average
consumption per adult equivalent, X, is any other variable to test, such as household

contemporaneous income in year t, and Z is a vector of demographic variables. Full

insurance implies tint fl - 1 and y - 0, that is, household consumption is not affected by its

22
own income. He shows that there is a tendency to reject B - O and accept fl - 1 out of 133
households, but not all of them. However, when panel data are used, it is shown that income
does matter in the determination of consumption. He concludes that full insurance is rejected,
but there is partial insurance because overall the effect of income on consumption is not
large. Deaton (1994) uses panel data from rural Cote d’lvoire, and shows that the coefficients
on individual income changes are significantly positive (y > O) in contradiction to full
insurance which would require them to be zero (y - 0). Moreover, in two of the three
regions surveyed there is no effect of the village average income on individual consumption
(fl - 0), which is also inconsistent with full insurance which would require it to be unity

(fl- 1). Note that he uses village average income for C, instead of village average
consumption.

As shown above, full insurance is in general rejected, but partial insurance is still
supported. This means that households smooth consumption. However, the test for full
insurance does not explain how households smooth consumption. The propensity of
consumption out of own income can be a measure of consumption smoothing, controlling
for village consumption: Sakurai ( 1995) finds that within the same agroecological zone, a
village with a good access to a major road has smaller propensity to consume than a village
without a good access to a major road in Burkina Faso household panel data. The same
pattern is observed in three different zones, the Sahelian, the Sudanian, and the Guinean. And
in villages with a good access to the road, full insurance models tend to be supported, at least
partially, while they are never supported in villages without good access (see Appendix B).
Townsend (1994) separates households into farmers and landless, and tests full insurance for
each category in each village. He finds that the landless are much less insured against income
shocks than are farmers, in all three villages. Sakurai’s and Townsend’s findings suggest that
consumption smoothing strategies other than village-level insurance may be working, such as
insurance over zones including temporal migration and remittance and intertemporal

smoothing through saving/dissaving. However, a full insurance model cannot distinguish

23

hypothetical “village insurance" from those risk coping strategies. This weakness of the full
insurance model is well discussed in Alderman and Paxson (1992).

Determinant analysis and tests based on structural household models will show how
agricultural households smooth consumption and why they cannot do it. However, only a few
studies have been done on those issues to date. Reardon et al. (1992) is an example of a study
of determinants. They examine determinants of income diversification (earning more off -
farm income relative to farm income) and find that assets such as land and livestock holdings
have positive effects. Then, they show that income diversification is an important determinant
of consumption smoothing: it reduces variability of income, and as a result smoothes
consumption. They compare the consumption smoothing effects of income diversification
among three different agroclimatical zones, and find that the effect is the strongest in the
Sahelian zone where crop production is the riskiest and income is the most diversified and the
weakest in the Sudanian zone where cropping is still dominant. Livestock holdings have little
effect on income smoothing as livestock income is included in off-farm income.

An example of a stntctural household model is Rosenzweig and Wolpin (1993). They
establish a dynamic model of farm households who have precautionary demand for saving,
have a floor of minimum consumption, invest in productive assets (bullocks and pumps), use
productive assets (bullocks) as a buffer stock, and cannot borrow. In addition, they implicitly
assume some kind of social insurance that works when household income falls below a
consumption floor. Using data from the semi-arid tropics of India, they estimate structural
parameters, and show in simulations that households cannot hold the optimal number of
bullocks due to dissaving under borrowing constraints in the presence of weather shocks, that
an actuarially-fair nonsubsidized weather insurance will not help households to hold the
optimal number of bullocks because they are already self-insured, and that exogenous
additional income will increase bullock holdings not only because they can now afford it but

also because they become less risk averse. In their model, household profits is a function of

24

the number of bullocks, the number of pumps, and a weather shock; no variable inputs, such
as labor, seed, or fertilizer are in the function. This may be too simplified.

In addition, in the West African semi-arid tropics, a structural household model must
somehow incorporate off-farm activities: labor and capital allocation between on-farm and
off-farm must be considered in a dynamic setting.

As reviewed in this section, there are several strategies that smooth household
consumption in rural areas of LDCs. They are not perfect, and village insurance studies imply
that households mix those strategies to smooth consumption. However, those strategies are not
examined well in structural household models and their interactions are little known.
Therefore, those strategies, not only saving but also crop diversification, income
diversification (off-farm activities), social insurance, use of risk-decreasing inputs, should be
incorporated explicitly or at least interpreted implicitly in structural models of agricultural

household to analyze household saving behavior.

3 Household Saving Models
In this section, saving models are reviewed in terms of uncertainty and borrowing
constraints. The review starts from two-period models, and then deals with multiperiod

models. Finally, flexible decision models are introduced.

3. 1 Borrowing Constraint and Uncertainty

There are two types of borrowing constraint to be considered lifetime borrowing
constraints and single-period borrowing constraints.

The lifetime borrowing constraint implies thatif there is no bequest motive at the end
of its lifetime, a household must consume all it has in the last period, assuming a standard
utility function with positive marginal utility. In the last period, the household cannot borrow
because it cannot repay the loan. In addition, if the household has a debt at the beginning of

the last period, the debt must be repaid in the last period. Thus, this constraint can also be

25

called a lifetime budget constraint. The household’s decision in the second-to-last period is
affected by this constraint in the last period. If there is uncertainty in the last period, the
household’s decision in the second to last period is affected more by the uncertainty so that
the household can avoid violating the lifetime borrowing constraint. Then, the second to last
period affects the third to last period, and so on. Thus by backward induction, the lifetime
borrowing constraint determines the household’s lifetime optimal strategy. Even in the case
of infinite life, the condition is expressed as the limit of expected discounted assets being
zero, and the household’s decision in each period is affected. In this respect, two-period
models have an advantage over multiperiod models because it is easy to incorporate the
lifetime borrowing constraint in two-period models.

However, since it is obvious that a household cannot borrow in the last period, the
lifetime borrowing constraint is not usually referred to as borrowing constraint. A borrowing
constraint usually means that a household cannot borrow at all or there is a ceiling above
which a household cannot borrow. This type of borrowing constraint can be called a “single-
period borrowing constraint”. This constraint is not necessarily binding, even if it always
exists. This constraint binds only when a household’s optimal consumption exceeds its
current wealth. In a two-period model, the single-period constraint exists only in the first
period and affects the household’s decision only when it is binding. On the other hand, in a
multiperiod model, it can be occasionally binding depending on the household’s strategies,
and under the existence of a potential binding constraint the household optimizes . Therefore,
the existence of a borrowing constraint affects household behavior, especially under
uncertainty, even if it is not always binding. Unlike two-period models, multiperiod models
can treat this problem of a “potential” binding constraint under uncertainty. Note that when
one says that there is no borrowing constraint, it means that a household can borrow in each
period as much as it wants. But because the borrowed money must be repaid in the future

(due to lifetime borrowing constraint), the household will not borrow limitlessly.

26

To summarize the discussion above: (i) A lifetime borrowing constraint is always
binding regardless of assumption and it rules out limitless borrowing; (ii) The existence of a
single-period borrowing constraint depends on assumption; (iii) Even if a single-period

borrowing constraint exists, the constraint is not always binding.

3.2 Saving Under Certainty / Two-Period Model

This review starts with the simplest saving model. This model is also the starting point
of saving models in the literature, and is too basic to have a specific reference.

A household faces a two-period horizon, 1 (present) and 2 (future). Let y, 'and y2
denote the household’s exogenous labor income for period 1 and 2 respectively. It is labor
income because the household has only labor to sell, and it is exogenous because its labor
supply is assumed to be inelastic. Let c, and c2 , respectively, denote present and future
consumption. In two-period models, non time-separable utility, U(c,, cz) , is sometimes used.
But to make it easier to extend the model to a multiperiod model, additively time-separable
utility is used here. In addition the initial wealth at the beginning of the first period, w,, is
assumed. The real interest rate, r , and the discount factor, 6 , is known, fixed, and exogenous
to the household.

Define saving in period 1 by s, I w, + y, - c,. Then initial wealth at the beginning of
the second period is given by w2 -(1+r)s,. Thus, the household’s income for period 2 is
equal to y, + w2 - yz +(1+ r)sl , which must be equal to second period consumption, c,. The

problem now can be written as

mcax u(c,) 4» M0,, + (1+ r)(w, + y, - c,)) (1)
Thus, from the first order condition, the famous Fisher’s law is obtained:

"7(9- - 6(1 + r) (2)

v (0,)

In the first period, there is no single-period borrowing constraint, that is, s, can be negative. If
the household borrows, the debt must be repaid with fixed interest, r , in the second period. In

the second period, everything is already planned as assumed so that the household does not

27

have to violate the lifetime borrowing constraint. It is easily shown that the maximum amount

that the household can borrow against future labor income is l—yL. Thus, naturally when y,

+7

is zero, the household cannot borrow in the first period.

If there is a single-period borrowing constraint, then s, z 0. If it is binding, the
household will choose 3, - 0. Then, y, -c, and y2 -cz, which is “consumption tracks
income perfectly”. As a result of this binding constraint, cl is lower than the optimum, and

c2 is higher than the optimum without the borrowing constraint. Therefore, Fisher’s law does

u' c
not hold, that is, -%1)3> 6(1+r). In this certain world, the borrowing constraint may
Cz

increases consumption variability, and consumption growth between two periods may become
larger, if it is binding. Although this borrowing constraint will reduce consumption in the first
period relative to the case without the binding constraint, it will never induce positive saving

more than without constraint.

3.3 Saving Under Uncertainty / Two-Period Model

In this section, two-period saving models under uncertainty are reviewed. There are
two kinds of uncertainty distinguished in the literature: stochastic exogenous labor income
and stochastic exogenous interest rate. First, the case with labor income uncertainty, and then,

the case with interest rate uncertainty are reviewed. -

3.3. 1 Labor Income Uncertainty I Two-Period Model
Labor income uncertainty in a two-period saving model was first considered by

Leland (1968) and Sandmo (1970). They assume income risk in the second period, that is, y2
in the certainty model is assumed to be stochastic denoted by 5",. Both use a non-separable
utility function given by U(c,, c2). They show that saving rate (the ratio of 3, against total
income in the first period) under uncertainty depends on the third derivative of the utility

function: if it is quadratic, since the third derivative is zero, uncertainty has no effect on the

28
saving rate, but if U,22 - U222 is negative, uncertainty increases the saving rate higher than the
optimum under certainty, that is, there is positive precautionary demand for saving. They also
mention that in a special case of additively separable utility function, the condition for
positive precautionary demand for saving is that the third derivative of the utility function is

I"

positive (v > 0) because the cross derivative of the separable utility function is zero. In
other words, the marginal utility function is convex'. The difference between Leland and
Sandmo is the measure of the “degree of risk”. Leland uses a Taylor-series approximation
for the utility function, while Sandmo introduces two parameters, additive shift and
multiplicative shift for stochastic labor income and considers a mean-preserving spread.

A more general measure of increasing risk is given by Rothschild and Sti glitz (1970).

Their idea is as follows: if an individual’s utility is a function of a control variable a and an

exogenous random variable 0 , then the individual solves the utility maximization problem

given by
max EU( 0, a) (3)
The optimal a must satisfy the first order condition given by
EdU(0, 00-0 (4)
da

 

0U . . . . . . . . . .
If — rs concave in 0 ( < 0), an increase in the variability of 0 wrll result in a

6a 6chth2

6U
decrease in the optimal choices of a , while if — is convex in 0 ( 1 > 0), increase in

6a 60:60
the variability of 0 will result in increase in the optimal choices of a (Rothscth and Stiglitz,
1971). Since this method is often used in this paper, I will name it the RS method hereafter.

 

‘ Assume a standard utility function, u(c),with u'(c) > O and u”(c) < O, (i) if u(c) has a
convex marginal utility, u'"( c) is positive and the expected mginal utility incremes when the variability
of C increases; (ii) if u(c) has a concave marginal utility, u"'( C) is negative and the expected marginal
utility deacoses when the variability of C increases. The difference in the effect of the variability on the
expected marginal utility causes the different effect of the variability on saving.

An example of the cme (i) is u(c) - Inc where c > 0 mid an example of the case (ii) is
Me) - c - c3 where O < c <1. It is difficult to find in example of a utility function that globally
satisfies thestandard assumption, u'(c) > O and u"(c) < O, as well as u'"(c) < O.

29
Kimball (1990) applies the RS method to the two-period saving problem under
income uncertainty considered by Leland and Sandmo. Following the two-period model
under certainty given by (1), this problem can be written as
mcax EU(532, c,) - u(c,) + (SEvGi2 + w,) - u(c,) + (SEN)?2 +(1+ r)(w, + y, — c,)) (5)
From the first order condition, the optimal condition similar to Fisher’s law is obtained as

follows.
u’(c,)

- 61 6
Ev'(c,) ( +r) ( )

The effect of increasing the variability of labor income depends on the third derivative of U .
a3

6‘0. 692’

decrease optimal consumption and will increase optimal saving in the first period (T able 1).

I

If is convex, an increase in risk will

 

is negative, that is, the marginal utility v

Therefore, there is precautionary demand for saving when the third derivative of utility
function is positive, which is the same conclusion as drawn by Leland and Sandmo. Kimball
(1990) shows that a measure of the strength of the precautionary demand for saving is

analogous to the Arrow-Pratt measure of risk aversion, and defines absolute prudence as

v!"

-—
n '

v

All the models presented above assume no single-period borrowing constraint in the
first period. Consequently, if the household borrows, the debt must be repaid in the second
period with interest at the same rate as the interest rate on saving. This lifetime borrowing
constraint is the same as under certainty. However, the household does not know exactly how
much labor income it will earn in the second period under uncertainty. Therefore the
household’s decision is based on the expectation of stochastic labor income, which affects the
optimal consumption level depending on the nature of utility function as shown above.
Consider the case of precautionary demand for saving: the optimal c, is lower than cl under
certainty. Therefore, it is less likely that the single-period borrowing constraint is binding
even if it exists. Thus if uncertainty increases saving in the first period, tmcertainty will make

the borrowing constraint being binding less likely than under certainty, while if uncertainty

30

decreases saving, uncertainty will make the borrowing constraint being binding more likely

than under certainty. But in either case, like the case under certainty, a borrowing constraint

u’ c
will induce suboptimality, -E—f(-’)-)- > 6(1 + r), if it is binding.
v 62

3.3.2 Interest Rate Uncertainty / Two-Period Model

In this section, interest rate uncertainty incorporated in a household portfolio
selection problem is considered. In the model, a household allocates its wealth among
consumption and savings with stochastic returns. Since a farm household is concerned with
consumption and stochastic production, this kind of portfolio model can be considered to be
a starting point of a farm household model. However, most portfolio models, including ones
presented below, assume linear returns to investment and no labor input.

A portfolio model found in Sandmo (1968) can be written as follows.

max u(c,) + 6E’v(m + a(1 + f)) (7)

where m is money or a riskless asset without interest, a is a risky asset with random return

rate 5" , and consumption in the second period is c2 - m+a(1+ 7). Assuming that the
household has an initial asset, w,, and has no other income source, then the budget constraint
is C2 - m+a(l+ 7')- w, -c, + a‘r'. Because there is no exogenous income in the second
period, the household cannot borrow in the first period in this model. Using a quadratic
utility function, Sandmo (1968) shows that increasing the variance of stochastic returns
keeping the mean constant (i) decreases consumption, 6,, (ii) decreases investment in risky
assets, 0 , and (iii) increases investment in riskless assets, m. Precautionary demand for saving
exists in this case, even though a quadratic utility function (whose third derivative is zero) is
used.

More generally, Rothschild and Stiglitz (1971) use a similar model with one risky
asset and no riskless asset, and show: (i) If 2v"(c,) + v"'(c2 )62 < 0, increasing risk decreases
the saving rate, 3, and either v'" >0 or v'" <0 can satisfy this condition; (ii) if

2v"(c,) + v'"(cz)c, >0 , increasing risk increases the saving rate, 3 , and v'" > O is necessary

31

for this condition. Therefore, in this case a positive third derivative of the utility function is
not sufficient for precautionary demand for saving. But note that saving in this model is
risky. In this model, borrowing in the first period is impossible, too. If the quadratic utility
function is used (as in Sandmo ( 1968)), 2v"(cz) + v'"(c2)c2 <0 is always the case because
v'" - 0. Sandmo’s result (ii) is consistent with Rothschild and Stiglitz’s result (i).

A special case is the constant relative risk aversion (CRRA) utility function, which is
also found in Rothschild Iand Stiglitz (1971). The CRRA utility function is given by

-a

c
u(c) - l— (8)

where a is the coefficient of relative risk aversion. The CRRA utility function has a positive
third derivative if a>0, which is warranted if the utility function satisfies standard
assumptions: u' > 0 and u" <0. Therefore, because u'" > 0, the CRRA function has
precautionary demand for saving at least in the case of stochastic labor income, and because

linto u' -00, it endogenously bounds optimal consumption away from negative or zero
C

consumption. Moreover, risk aversion and precautionary demand for saving are controlled by
the same parameter, a, which is not necessarily true in general. With a CRRA utility function,
they show that the condition 2v"(cz) +v'"(c,)c, <0 becomes a<1 and the condition
2v”(c,) +v'"(c,)c, >0 becomes o> 1. Therefore, (i) if a< 1, an increase in interest risk
decreases saving; (ii) if a > 1, an increase in interest risk increases saving. The results in this

section are summarized in Table 1.

3.3.3 Labor Income and Interest Rate Uncertainty / Two-Period Model

It will be easy to extend the two-period saving model with labor income uncertainty in
section 3.3.1 to a model assuming that both labor income and interest rate are uncertain.
However, to my knowledge, there are few such extensions except for multiperiod models,
such as, Skinner (1988) and Dasgupta (1993). Therefore, in chapter 3, I will develop a two-
period model with two uncertainty sources and will examine its property, especially when they

are correlated.

32
3.4 Saving under Certainty / Multiperiod Model
The multiperiod model is an easy extension of two-period model in section 3.2. This
is also a text book problem, but here I follow Sargent (1987). The model is generally written

T
mabe'u(c,), 0<o <1 (9)

q r-r
subject to w“, - (1 + r)(w, + y, - c,)
y, is exogenous labor income in period t. Borrowing is allowed in each period except in the
last period, and therefore there is a lifetime budget constraint as discussed in section 3.1. The

constraint can be written as

r r
2(1+r)"c, -2(1+r)"y,+w, (10)

t-l r-t
This rules out a strategy of infinite consumption supported by unbounded borrowing. If T is

infinite time, the right-hand side must converge to zero. Bellman’s equation becomes

v(W,. y.) - mCaX[u(C.) + Mm... y... )] (11)
And using the first order conditions, the Euler equation is obtained as

an an

T—6(l+r)-— (12)

r 1+1
Thus, the Euler equation is almost like Fisher’s law obtained in the two-period saving model
in section 3.2. In fact a lot of empirical studies based on multiperiod dynamic optimization
are based on the Euler equation even in a stochastic setting. Since the Euler equation is one of
the necessary conditions, a solution to the dynamic optimization problem above must satisfy
it. However, in addition to that, the lifetime budget constraint given by (10) is also necessary,
and when we solve the problem to obtain the optimal consumption path, we have to use both
conditions. In the two period model, the lifetime budget constraint is already imposed on the
first order conditions. But in a multiperiod model, imposing the lifetime constraint on the first
order conditions is in general difficult. An example of a solvable problem is using
u(c,) - Inc,; the optimal consumption of an infinite-life household given by Sargent (1987)

33

c, -(1-6) y,+2(l+r)”’y,+j+w, (13)

1-1

3.5 Saving Under Uncertainty / Multiperiod Model
As with the two-period models, there are three types of risk: labor income risk, interest

rate risk, and both.

3.5.1 Labor Income Uncertainty / Multiperiod Model

Under uncertainty, the Euler equation given by (12) becomes

31- 6(1+r)E, du

1 Hi

(14)

 

This is the same as (12) except for the expectation operator. Moreover, the Euler equation
above is very similar to the optimality condition for the two-period model under income
uncertainty in section 3.3.1. But as discussed in section 3.4, the Euler equation in a
multiperiod model does not incorporate the lifetime budget constraint, while the optimality
condition in a two-period models does.

If we assume a quadratic utility function, then the Euler equation given by (14)
becomes

c, - 6(l+ r)E,(c,,,) (15)

Hall (1978) tests (15) if no other variables in the information set help to predict c,,, once

lagged consumption, c,, has been included, and shows that lagged income is orthogonal to

current consumption. Then where is the lifetime budget constraint? The level of income at
which consumption is initially set depends on lifetime resources, determined at least in part by
current and future levels of labor income as shown below. A simple case of ( 15) is when
6(l+r) is unity, that is, c, - E(c,,,). Using this Euler equation with a lifetime budget
constraint, the consumption level of an infinite life household is given by Deaton (1992b).

r on -1
—W,+-l—+-;g(l+r) E,y,,, (16)

C-
' l+r

34

This is equivalent to (13) under uncertainty when 6(1-1- r) is unity. Equation (16) implies
that consumption is the annuity value of expected human and non-human wealth. Deaton
(1992b) takes this to be the permanent income hypothesis. Assumptions for (16) are infinite
life, quadratic preferences, and a constant real interest rate equal to the rate of time-preference
(6(l+r) - l). The problems of using a quadratic utility function are that consumption is
allowed to be negative and that there is no precautionary motive for saving. As discussed for
two-period models, precautionary saving depends on the third derivative of the utility
function, but with a quadratic function, the third derivative is always zero. It can be seen from
(16): even if stochastic labor income becomes more risky, if its expected mean does not
change, consumption is not affected at all.

The effect of increasing risk can be easily seen using the Euler equation given by
(14), which is equivalent to (6) of the two-period model. Thus, with respect to risk in
stochastic labor income, if u'" > 0, an increase in risk will increase precautionary saving,
while if u'" < 0 an increase in risk will decrease precautionary saving. However, there is a
difference between the optimality condition of two-period models and the Euler equation of
multiperiod models. In two-period models, risk in 9, directly affects c2 , because
c2 - y, + w,. In other words, if we say there is income risk in the second period, it is also
consumption risk in the second period. On the other hand, in the Euler equation of
multiperiod models, we do not know the relationship between income risk and consumption
' without making specific assumptions: even if we assume an increase in income risk in period
t+1, it does not necessarily mean consumption in period t+1 becomes more risky because
consumption t+1 is endogenous. Intuitively, the argument that consumption is affected by
income risk is plausible if a lifetime budget constraint exists: in the last period the household
has to consume all it has, like in the second period in a two-period model. Then, using
backward induction, in each period consumption will be affected by the requirement in the
last period under income uncertainty. It can be formally proved by going back to the

Bellman’s equation like (11) to show that

35

(17)

 

3 > 3 >
a "(5') -0 = a Win) _0
dc, < 07y! <

is true for all t (Sibley, 1975). Based on Sibley’s proof, we can now examine precautionary
demand for saving by examining the sign of the third derivative of utility function.

As discussed, quadratic utility functions do not show precautionary demand for
saving. But if we use a utility function that has non-zero third derivative, such as the CRRA
utility function given by (8), saving will depend on risk in labor income. The CRRA utility
function is often used when precautionary demand for saving is concerned. But it is
impossible to obtain a closed-form solution for the CRRA utility function unlike for
quadratic utility function. Zeldes (1989b) uses a CRRA utility function, and calculates the
optimal consumption path numerically using stochastic dynamic programming. He shows
that the optimal consumption in a CRRA utility function under stochastic labor income is
smaller than that in a quadratic utility function, and the optimal consumption of CRRA utility
function decreases as risk in stochastic income increases: that is, there is precautionary
demand for saving as expected

Applying the CRRA utility function given by (8) to the Euler equation given by (14),

equation (14) becomes

c,'° - 6(1+r)E,c,',‘,’ (18)
Approximation of (18) assuming mm is distributed normally is as follows.
cl
c
E,ln4C-fl-- a"[ln(1+r)+ln6]+é-a£,var(ln£’-’-L) (19)
c

l l

 

If the discount rate is expressed as 6 - , and if r and B are small enough to be

l+fl
approximated as ln(1 + r) - r and ln(l +fl) - B, then equation (19) can be written as

logic-4- o"[r -p]+ -;-ol§,vm'(ln£m)+e
C

l l

(20)

t+1

where an error term, nstead of an expectation operator, on the left-hand side. Equation (20) is
found in Carroll (1992). As discussed above, although consumption is endogenous, a lifetime

budget constraint may be able to translate labor income risk into consumption risk. Assuming

36

so, the variance terms in (19) and (20) are considered to come from labor income risk, and an
increase in income risk increases consumption growth. Therefore, increasing risk in labor
income implies larger variance, and as a result larger consumption growth, that is, higher
precautionary saving. Moreover, 0 determines the size of induced precautionary saving by
an increase in the variance. This is consistent with the fact that a is the measure of prudence
in the case of a CRRA utility function. Carroll’s interpretation is that consumers with less
wealth have less ability to buffer consumption against income shocks; thus they have higher
variance and faster consumption growth (Carroll, 1992). Not only wealth, but any variable
that helps predict the future variability of consumption will have a role in predicting the rate
of grth of consumption (Deaton, 1992b).

The multiperiod models under income uncertainty presented above assume no
borrowing constraints except for the lifetime one, as discussed. Since households can borrow
to smooth consumption in any period except in the last, the Euler equation given by (14) or
(18) is predicted to hold. However, tests based on the Euler equation are often rejected (e. g.
Flavin (1981) and Hansen and Singleton (1983)). Some authors suggest that the rejections
are due to borrowing constraints (single-period constraints). As discussed in the case of two-
period models, if there is a borrowing constraint and if it is binding for a household, the
household’s current consumption will be lower than the optimum resulting in higher

marginal utility of current period. Thus, with a binding borrowing constraint, the Euler

 

. . . an an
equation of the household 1s vrolated as 3- > 60 + r)E, . This may cause mixed results

r 1+1
when we use aggregate data to test the Euler equation.

Based on this idea, Zeldes (1989a) empirically tests the existence of the single-period
borrowing constraints using household panel data. In addition to the standard dynamic
household model given earlier in this section, he makes anassumption that there is a single-
period borrowing constraint expressed as w, z 0 , which means a household cannot be a net
borrower in each period. Under this assumption, this constraint is binding for some

households, while for others this constraint is not binding in a given single period. This

37
constraint is expressed as the Lagrange multiplier, 1,, which will be positive if binding and

zero if non-binding. Then, equation (18) can be written as
c,” -6(l+r,)E,c,',‘,’+A, (21)

Using an approximation similar to (19) he finally obtains the following equation to estimate:

lnE‘iL- a +HH+ T, + a"[ln(1 +r,)-1-9,+1 -9,]+ ln(1+A,)+e,,, (22)
c

l
where a is a constant, HH is household fixed effect that incorporates household’s discount

factor, 6 , 9M — 6, is difference in household’s characteristics that affect risk aversion, a.

c,,, is assumed to be a mean-zero error term. Therefore, if ln(1+ 1,) + c,,, is positive, A, will

be positive, which means that there is a binding constraint. The variance term appearing in
(19) and (20) does not appear in (22) because Zeldes assumes that the variance is constant for
a household over time and includes the variance in the household fixed effect. In other words,
he assumes that income risk does not change for each household. He splits the whole sample
into two groups based on a wealth/income ratio, and finds that there is evidence that the
borrowing constraint is binding in low wealth group.

The Euler equation tells us how a household allocates resources intertemporally and
gives testable implications, but cannot show how a household reacts dynamically to the
existence of single-period borrowing constraint that may bind potentially in the future.
Deaton (1991) takes a different approach based on simulations. He assumes the same
borrowing constraint as Zeldes (1989a) explained above, but considers the following

modified Euler equation:

m -max ___,__,_au(w +y ), 6(l+r)E, (MC’) (23)
(9C, 66’, km

where 6(l+r)<1 is assumed, which is the condition when the household will run down
initial assets under certainty, or the household is “impatient” (Deaton, 1991). Equation (23)

means that, under the single-period borrowing constraint (w, 2 0 for all t), if the constraint is

binding, the household will consume all it has and the household's current marginal utility is

du w +
equal ‘0 —£-;C—y’-), but if the constraint is not binding, the household's current marginal
l

38

(la c
utility is equal to the expected marginal utility of the next period, 6(1+ r)E,—(—-L*Jl, as

t+l

usual. Deaton and Laroque (1992) prove that there is a unique time-invariant consumption
function as c - f (w + y). This function is found by backward induction on equation (23)
assuming stochastic i.i.d. income and a CRRA utility function as in Zeldes (1989b) but
without a borrowing constraint. Deaton shows that consumption decreases as income risk
(standard deviation of stochastic income) increases and the degree of absolute prudence (a
in CRRA utility functions) increases as expected, and using the obtained consumption
function, he shows in a simulation that consumption is much smoother than income where
assets work as a buffer stock.

The results above imply that a household saves more with than without a borrowing
constraint, because the household thinks the borrowing constraint may bind in the future, that
is, the precautionary motive for saving is strengthened by the existence of borrowing
constraint. This will happen even if the third derivative of the utility function is negative, and
in this case the borrowing constraint will cancel the precautionary motive of consumption
(uncertainty increases current consumption in this case as explained). Deaton points out that
because of the borrowing restrictions, the consumption pattern changes in a fundamental way
from the case when borrowing is allowed. As a results, households usually hold assets and
satisfy the unrestricted Euler equation in most periods. Consequently, tests for borrowing
constraints that look for zero assets and violation of the Euler equations may have few
observations to work with, even when households are never allowed to borrow (Deaton, 1994).

In Deaton’s model a household receives only stochastic labor income and has only
choice between consumption and save. This model, even if it can explain a household
behavior in LDCs, may be too simplified to understand its behavior and to obtain some policy

implications.

39
3.5.2 Interest Rate Uncertainty / Multiperiod Model

A multiperiod model with interest rate uncertainty is reviewed in this section.

Levhari and Srinivasan (1969) consider an optimal portfolio selection problem in a
multiperiod model under uncertainty, in which a household allocates its initial wealth among
consumption, investment in asset A, investment in asset B. Based on their model, Rothschild
and Stiglitz (1971) examine the effect of increasing risk in interest rate. The problem is given
by

max 12,: 6'u(c,) (24)

1.0
subject to w”, - (1+ 'r')(w, - c,); w, z c, z 0
where w, — c, is saving, which must be positive because in the next period there is no other
income than this saving, and F is stochastic interest rate to the saving. The saving is to be
allocated between assets A with a rate of return 7,, and asset B with a rate of return of F8. If
a is the fraction invested in the asset A, then f - ail, +(l- a)FB. Using the CRRA utility

function given by (8). they show that the consumption decision is independent of the

. . . . , w -c ,
portfolio decrsron (the chorce of a), and that the savrngs rate (g) rs constant,
1

independent of the initial wealth, w,. Then, applying the RS method, they show the following:
(i) When a < 1, an increase in variability of 'r', reduces demand for asset i; (ii) when a > 1, an

increase in variability of F, has an ambiguous effect on demand for asset i; (iii) When a < 1,

an increase in the variability of 7 decreases the savings rate; (iv) when a > 1, an increase in
variability of 7 increases the savings rate. These results are consistent with the two-period
model in section 33.2.

Since there is no exogenous labor income, this model excludes borrowing. This is
why there is no difference in the effect of an increase in risk between two-period models and

multiperiod models.

40
3.5.3 Labor Income and Interest Rate Uncertainty I Multiperiod Model
Skinner (1988) combines stochastic returns to saving and stochastic labor income,
and using a CRRA utility function and Taylor series approximation, obtains a closed-form

solution for consumption as follows:

Inf-Cm- - 0'1? —fi + v]+ ln-é— (25)

l
where v is risk premium composed of variance of income, variance of interest rate, and their
covariance. r' is the expected interest rate. This model allows the household to borrow (there

is no single-period borrowing constraint), but takes the lifetime borrowing constraint into

account by adding lnh, where L, is lifetime resources and T1 is its anticipated present

i.

value. He shows in simulations that the life-cycle consumption pattern is smoother in the case
of uncertainty than in the case of certainty and that the closer is the earning process to a
random walk, the greater will be precautionary savings. In his model given in (25), positive
covariance between income risk and interest rate risk (included in v) increases savings, but in
his simulations he assumes the correlation is zero.

Another example is from Dasgupta (1993), where a household’s asset level moves
according to the equation

w“, - (1+ r)(w, + y, —c, — a, — b, ) + (1 + (3)0, + (l «ebb, (26)

under the borrowing constraints expressed as a, 20, b, 2 0, and W, 2 0 for all t. w, z 0

implies that the household cannot borrow at all. Equation (26) means that the household

allocates its wealth w, + 9, among consumption, 6,, investment in risky asset A, a and

t 9
investment in risky asset B, b,. Saving, given by s, - w, + y, -c, - a, - b, , also produces a
positive, fixed return. This is very similar to the portfolio selection problem given in section
3.5.2, but the difference is the stochastic, exogenous labor income, 5",, which is a tradition in

savings models and allows the household to borrow against future labor income. Note in

contrast that the household cannot borrow in the problem in section 3.5.2.

PIC

agile

41
A standard way gives the following three Euler equations with the Lagrange

multipliers (p, z 0, 6, z 0, and Fr 20) associated with the constraints a, 20, b, 2 0, and

w, 2 0 respectively:

6E,(-r + 61)u'(c,+,) + 4), - 0 (27)
6E,(—r + fi)u'( cm) + 0, - 0 (28)
u'(c,) - 6(1 + r)E,u’(c,,, ) + ,u, (29)

To have a positive investment in risky assets, 4!, - 0 and 0, - 0 are necessary. From the Euler
equations above, 13,6, EB >r is a necessary condition given mv(&, u'(c,,,))< 0 and
cov (3, u'(c,+,)) < 0.

A possible simple conclusion is that an increase in riskiness in consumption in t+1
increases the right-hand side of (29) if u'"(c,,,) > 0, and therefore the left-hand side of the

equation 0, will decline. But Dasgupta rejects such argument, because c,,, is endogenous, and

stops further discussion. However, because convexity of the marginal utility function implies
convexity of the marginal value function (Sibley, 1975), the RS method can be applied in this

case, [00.

3 .6 Stochastic Production / Multiperiod Model

In Dasgupta’s model in section 3.5.3, two risky assets are considered as agricultural
inputs, but the model assumes linear return to assets. However, the input-output relationship is
often non-linear (unlike in Dasgupta’s model), and consequently the effect of increasing risk
is also non-linear. This non-linearity determines whether an input is risk-increasing or risk-
decrcasing, and therefore the input decision is a part of household risk management that may
affect household saving. Therefore, the production function is important for household
saving models. In this section I review models that incorporate stochastic agricultural
production.

Roe and Graham-Tomasi (1986) consider a quite general dynamic model of

agricultural household under uncertainty. The model explicitly incorporates stochastic

42

agricultural production and applies the full income concept of Singh et al. (1986), and
assumes a perfect labor market. Their model assrunes a finite-life household and a bequest
motive, and they solve the problem using a constant absolute risk aversion (CARA) utility
function for only two periods, the last and the second-to-last. This means that their model is
reduced to a two-period model. Here, for consistency with previous multiperiod models, I
assume no bequest motive in the following presentation.

The problem can be written as

r
max 12,2 6‘u(X,,, X", 1,); z -(X,,, X", 1,, L,, A.) (30)
"' r-o
subject to
Qt+1 - Q(l1’ At; 2MI) (31)
ct " pthqt + puxn + b}: (32)

s... - w. - am. -b.z. -c.
- [a,:-i, + bl, + pq,Q(L,_,, A,_,; E“) + (1+ r)s,]-a,A, - b,L, -c,
(33)

where X“: consumption of an agricultural staple, pq,: its price, X": a good purchased in a
market, p": its price, 1,: leisure, 1,: on-farm labor input, 2:, endowment of labor, b,: wage,
A,: land use for own production, a, endowment of land, a,: land rental price, Sm: saving
from period t to period t+1, r: interest rate of saving, 0,: consumption that consists of staple
foods, purchased goods and opportunity cost of leisure, and Qm - Q(l.,, A,; 5,): household
agricultural production function with stochastic term. The output appears in the next period,
t+l. The difference between I: and 1., represents hired labor if negative or off-farm labor
supply if positive, and the difference between A, and A, represents rented land if negative or
lent land if positive. Also this model allows the household to borrow at the rate of r , which
results in a negative saving in that period. In chapter 3, I will examine the effects of
uncertainty on decision variables applying the RS method to this model, because Roe and

Graham-Tomasi do not do it in the article.

43

Another example is Rosenzweig and Wolpin ( 1993). They consider an empirically-
estimable structural model in the context of the Indian semi-arid tropics. Their model
assumes that a household’s profit depends on its bullock holding level, number of pumps for
irrigation, and a household shock, and constructs a similar model to that of Dasgupta’s
explained in section 3.5.3 except for exogenous labor income. That is, although they
consider a very specific input-output relationship, they assume a linear function.

In their model, a household maximizes a CRRA utility function with a consumption

floor, given by

 

T
max E, 26‘ )"a

t-l 1 ' m

1
(c, -c
a

subject to c, - II(B,, M,, c,) - pbb“, - p'"m,,, - cn“, > cu, + g (34)
where II(B,, [M,, c,) is profit function, R is number of bullocks, IV, is number of pumps,
b“, is net purchase of bullocks, m”, is purchase of pumps, n,,,, is breeding of a calf, and
pb, p", c are their unit costs respectively. The source of the shock is unpredictable weather,
but the model does not tell how the weather shock affects profitability. Since there is no
interaction terms between weather and inputs, the shock just shifts total profit linearly. Since
pumps are not assumed to be sold, bullocks are the only savings in this model. On the other
hand, bullocks are important for production. Moreover, since no negative holdings of
bullocks and pumps are allowed, the household cannot borrow. Because the model is
constructed to have a consumption floor as seen in (34), it must be assumed that the
household has a form of disaster insurance so that its consumption never reaches the floor,
although such mechanism is not explicitly modeled.

They estimate the profit function first, then using the estimated parameters they
estimate remaining structural parameters. By doing this they assume that the production
decision is separable from the consumption decision. Using those parameters, they simulate
life-cycle consumption patterns for an average household Their interpretation of the model
is mainly based on the simulations. Since their findings are explained in section 2 of this

chapter, they are not repeated here.

44

They do not discuss the properties of their model, but doing so will be important to
understanding the results. Their estimation of a (the coefficient of relative risk aversion) is
close to unity (0.96), but significantly different from unity. If the weather shock works like
interest rate risk, 0 <1 could imply precautionary demand for consumption (according to
discussion in previous sections). However, the shock variable can be seen as exogenous
income risk, because there are no interaction terms between the weather shock and productive
capital in the profit function, that is, the input decision does not change the size of risk. In this
case, therefore, precautionary demand depends only on the sign of v"'. Since they use the
CRRA utility function, 0> 0 is enough to have positive v"'. Therefore, an increasing in
weather risk (that is, profit risk) is predicted to increase investment in bullocks and breeding.
However, their assumption that the profit function has no interaction between weather and
productive capital (bullocks and especially pumps) may not be true, that is, the effect of
weather on profit could depend on the level of the productive capital. If such interactions
were incorporated, the household’s precautionary behavior would be different.

In this section, I review two agricultural household saving models in which
agricultural production is explicitly incorporated. However, neither model assumes non-
linearity of production and non-linearity of marginal effect of risk. In fact, I find no
household saving model that deals with agricultural production and its non-linearity,

especially of risk effect.

3.7 Flexibility and Production/Saving

In this section, models with flexibility of decision process or sequential decisions
under uncertainty are reviewed. When we analyze production under uncertainty, flexibility is
an important concept. In addition, the saving decision under uncertainty is also affected by
information. The concept is initially introduced in firm investment analysis where capital

investment is relatively less flexible than the labor input decision. Recently, a similar idea has

45

been applied to farm household models, where labor is also a flexible input in crop
production to adjust rainfall shock.

Hartman (1972) considers the effect of price uncertainty on investment of a price-
taking, risk-neutral firm that maximizes the sum of discounted profit. He introduces
flexibility by assuming that labor is a flexible input which the firm can decide to maximize

short-run profit in each period after price is observed. The problem is written as

mgx E}; 6'[13.F(K.. 1:09. 13.. a.» 4.1:“. it» a.) — Cu. 6.)] (35)

where F(K,, 1;) is the production function which is assumed to be homogeneous of degree
one’. K, is the capital input level determined by depreciated previous capital level and
investment in the current period, as K, - (1— d)K,_, + I,, L: is labor input optimized in the
short-run under certainty and is a function of output price, wage and capital input level,
C(I,, Q) is investment cost function, where q, is a stochastic factor, ,3, is stochastic output
price, and W, is the stochastic wage rate. By applying the RS method to the first order
condition, he shows that an increase in uncertainty either in output price or wage increases

investment.

Hartman (1976) generalizes the results above using a single-period model given by

max E[pF(K, L‘(K, p, W))-w L‘(K , p, W)-CK] (so)
or using the profit function g( K, p, w) , which is maximized in the short-run, it is written as
mgx E[s'(K. 1'5. w)-cK] (37)
where uncertainty comes from only the output price and the firm is risk-neutral. He shows: (i)
. a’g . . . . . .. .
1f ——2—,_> O, optrmal K 1ncreases as output pnce uncertainty 1ncreases, and (11) 1f
6K6 p
6’s . . . .
m < O, optrmal K decreases as output prrce uncertarnty 1ncreases. The result of

Hartman (1972) is the same as the first case, because he assumes tint the production function

 

1 Aproductionfunctionishomogeneousofdegreeonewhenfflfl-y'U) forallt>0,wheret
is ascalarand x is avectorofinputs. Becauseof this assumption, the short-um profit function can be

written” h(K,, pt, 19,)!- th(pr’ wt)'

46

is homogeneous of degree one. If a risk-neutral firm decides all input before the price is
observed, price uncertainty has no effect on output or on inputs. But as shown above, if
flexible a decision is allowed, uncertainty in the output price has a positive or a negative effect

on output or inputs as in the case of a risk-averse firm.
2

Hartman (1976) considers a case of risk-averse firm and shows: (1) if 6K; > 0 and

 

 

 

a3
6 K :2- < 0. then uncertainty in output price reduces the optimal capital input; (ii) if
P
528 838 . . . . . .
—. < and 2- > 0, then uncertainty 1n output prrce 1ncreases the optrmal capital
3K0? 6K6 p

input; (iii) other combinations of those derivatives give ambiguous results. The results of
Hartman’s flexible decision model are different from Sandmo’s earlier results that a risk-
averse firm reduces output if output price is uncertain under the assumption that the input
decision is made before the price is observed and the marginal profit is linear in the stochastic
output price (Sandmo, 1971).

Hartman’s results for a risk-averse firm presented above imply that the effects depend
on whether capital is risk-increasing or risk-decreasing. But since Hartman uses a Taylor
approximation and ignores the third derivative of the utility function, his results do not
depend on this third derivative. His result may not hold generally considering that this third
derivative determines prudence.

Although Hartman deals with a single-period model, it can be seen as a two-period
model because of the flexibility: in the first period, the firm decides capital input under
uncertainty, and in the second period, the firm decides labor input under certainty. This
means that the firm gets perfect information at the beginning of the second period. However,
the information is not necessarily perfect, although more information is available in the
second period than in the first period. Thus, we can introduce a degree of information, or
“informativeness”. Epstein (1980) explores this idea in a three period model: in the second
period there is a temporal resolution of uncertainty. Epstein’s problem can be written in

general as follows:

47
max E, max Ez/r U(x,, x2, 2) (38)
11 32

In period 1 the decision about x, is made subject to prior expectations about the state of

period 3. The uncertain future environment is represented by a random variable 2 . Before
the start of period 2 new information about 2 becomes available through the observation of
another random variable I; which is correlated with E. The agent is a Bayesian decision
maker and revises his prior probability distribution about 2 after observing I; to make a
decision about x,. The amount of additional information about 2 provided by I; is a
parameter in the model. F is said to be more informative than 17' if every user of
information about 2 is at least as well off observing I" before making a decision as he would
be if he based his decision on an observation of 17'. This condition can be written as

mxax E, mfx Em, U(x,, x2, 2) 2 max E7, nrax E2”, U(x,, x2, 2) (39)

This can be rewritten as ~ ~

mgx E, J(x,, II) znrax E, J(x,, II') (40)
where J is the maximum expected utility given x, and fl, and fl is random variable 2
conditional on )7. Based on the definitions above, Epstein’s Theorem 1 is: Let x: and x,” be
the solution of left-hand side and right-hand side respectively. If 73-J- is concave in f1, then

x1

0 C. a] " 0 .0
x, s x, , and if d— is convex in II, then x, z x, . The proof of this theorem is similar to
x1

Rothschild and Stiglitz’s proof of the effect of increasing risk (Rothschild and Stiglitz, 1971).
This theorem implies that it depends on the model whether the prospect of greater
information in period 2 discourages the adoption of an irreversible decision in period 1 to
maintain some flexibility in period 2 to take advantage of the information or not
In the three-period framework, Hartman’s problem (1976) can be written as
mg E-cK+Err)1‘aIx [pF(K,L)-wL] (41)
Thus, J(K, w, p) as defined in (40) is equivalent to profit function g(K, {3, w) defined in

(37). According to Theorem 1, as I; becomes more informative, K . increases (decreases) if

t? .. ..
3% is convex (concave) in p. Hartman’s problem is the case when p is known in the

48

second period, that is, I7 is perfectly informative. As shown by Hartman, in this case an

.. . d ..
increase in uncertainty in p increases (decreases) K , if j;- is convex (concave) in p.

Epstein concludes that, in this case, more information and more uncertainty have qualitatively
similar effects, which means that prospect of more information in the second period is the
same as relatively more uncertainty in the first period. On the other hand, if I; has no
information about p,the problem is reduced to a simultaneous decision problem, and since
the firm is risk-neutral, uncertainty in the output price it has no effect on its input decision. '
Epstein (1980) also considers a consumption/saving decision problem in the three-
period framework, which is an extension of the two-period model of Sandmo (1968) and
Rothschild and Stiglitz (1971) (discussed in section 3.3.2). The problem is written as

max u(w, — s, ) + (SE 02?}. [u( rs - s,) + 6&(23, )] (42)

Osstswr

where w, denotes initial wealth, 3, denotes saving in the first period, 7 denotes a known, fixed
interest rate on the first-period saving, s2 denotes saving in the second period, 2 denotes

stochastic interest rate on the second-period saving, and 6 denotes discount rate. Like
previous examples, J(s,, Z)- max [u(rs, -s,) +6Ev(Zs,)] is defined as the maximum

01:; s ra

expected utility in the second period, and by Theorem 1, the prospect of more information in

. . . . . dJ . . ..
the future 1ncreases (decreases) savrngs 1n the first perlod 1f — ls convex (concave) ln 2.
SI

. . . . . . J . . - .
Epstein shows that wrth the CRRA utlllty functlon glven by (8), -Z— is convex in 2 if a < l
51

d] ..
and .07 is concave in 2 if a> l. The two-period problem of Rothschild and Stiglitz
l

discussed in section 3.3.2 is considered to be the case when F has no information about 2
because the stochastic interest rate is assumed to be identically, independently distributed. As
shown in section 3.3.2, their results are that an increase in uncertainty in the interest rate
decreases savings in the first period if a < l and increases savings in the first period if a > 1.
Epstein concludes that in this case less information and more uncertainty have qualitatively

similar effects.

49

Epstein’s general conclusion is that the relationship between the qualitative effects of
earlier resolution (more information) and reduced prior uncertainty for a given information
structure are model specific.

Although it is theoretically ambiguous, information or sequential decision making is
empirically important. Rose (1992) develops a two-period model for her empirical study of
the effect of weather risk on the labor market in rural India. She assumes that in the first
period an agricultural household plants prior to the realization of weather and, in the second
period after rainfall is known, the household decides how much labor to use for harvesting
when the market wage has also adjusted to rainfall. In a single-period model, labor allocation
between on-farm and off-farm activities must be decided prior to knowing rainfall, and no
flexibility is allowed after rainfall is known. On the other hand, in the two-period model, the
household can adjust harvest labor supply depending on the harvest: in a drought year the
household will shift labor from on-farm to off-farm, but the wage will be lower in a drought
year because labor supply to off-farm activities increases.

The model is constructed as follows: in the second period there is no uncertainty, and
the household solves

mLax w,I:, + h(s)w,l, +g(e)F(N - L, N- 11, X) + y (43)
where L, denotes off-farm labor supply in the first period, which is predetermined in the
second period, l.z denotes off-farm labor supply in the second period, N - 1:, denotes on-
farm labor supply in the first period, N — L, denotes on-farm labor supply in the second
period, X denotes a vector of fixed household characteristics, and y denotes exogenous
income. The household profit from crop production is given by a profit function,
g( e)F(N - 1:, N - [1, X), which depends on on-farm labor supply in the first and the
second period, household characteristics, and profit rate as a function of rainfall shock a
which is known in the second period..The profit rate is given by g( c) - y + a(1 -a), that is,
mean rainfall 7 plus random deviation from the mean rainfall, or rainfall shock a. The size

of the shock depends on the household’s ability to reduce shock, such as by irrigation,

50
denoted by a (O < a <1). In the first period, market wage rate is known and fixed to w,. In
the second period, market wage w, is affected by rainfall shock 8 through h( e) , which is
given by h( s) - y + a(1 - b) where b represents some factor that determines the response of
wage rate against rainfall shock, such as availability of off-farm employment. From the first
order condition of (43), the optimal L, is determined as L’, - 1.;(1‘1, e).

In the first period, the household’s problem is to maximize expected utility of income
given by

“if" E U[w,l1 + h(§)w,l;(L,, E) + g(E)F(N— 11, N- 1:,(11. 5). X)+ Y] (44)
Rose uses a CRRA utility function, and assumes that on-farm labors in the first period and in
the second period are complements. However, since this model is still so complicated, she
cannot obtain unambiguous results unlike in a single-period model without flexibility. Even
the effect of rainfall risk on off-farm labor supply in the first period is ambiguous although it
is positive in her single-period model. She finally uses a linear statistical model for the
empirical analysis, and shows the empirical results are largely consistent with the prediction
based on her single-period model and two-period model. Empirically, she finds that rainfall
risk increases off-farm labor supply in the first period, which confirms the prediction of the
single-period model.

The above model is similar to that of Hartman’s risk-averse firm (1976) discussed
above, and the case of perfect information in firm’s input decision in Epstein (1980), if we
see labor supply in the first period as capital input and labor supply in the second period as
labor input that maximizes short-run profit after price is known. As shown previously, even in
simple models like those of Hartman and of Epstein, the effect of uncertainty on the long-run
input decision is ambiguous, but the effects are systematically related to the type of
technology. the household uses. Following them and my previous review of various models, I
will extend Rose’s model including saving, and will discuss its properties in chapter 3.

Rose does not estimate structural parameters of her flexible labor input model, but

uses it to obtain hypotheses for her empirical analysis. Fafchamps ( 1993) estimates structural

51

parameters of a sequential labor decision model under uncertainty using household data from
rural Burkina Faso, where agriculture is rain-fed and labor is the only variable factor of crap
production after the rains have started in cropping season.

Fafchamps’s model is concerned with sequential decision regarding labor input to
crop production, and for simplicity ignores labor allocation between on-farrn and off-farm
activities. His model ignores year-to-year investment and saving, too. Consequently the cost
of labor is the marginal utility of leisure (unlike in Rose’s model). There are two labor
decisions, L,, planting labor, and I1, weeding labor. Cultivated acreage is assumed to be
predetermined, denoted by A. Therefore, his model is essentially reduced to Epstein’s three-
period model defined as (38).

Using nested constant elasticity of substitution (CES) functioml forms, Fafchamps

writes his model as
(14)

1 a
max Ephgmgx ER, ,T;{6[Y(l'l1) +(1'Y)(1'LI)IE+(1’6)Y:} P (45)

L1

subject to
n-uh mo
Y, - [aY," + (1 - a)l;' Fe,” (47)
Y. -b[fiY;’ +0411: lie” (48)

Given land, A, and initial shock, 12,, the household obtains the first information of crop
growing process, I’,, and makes planting and replanting decisions, L,. Then, the second
shock R, takes place after planting, which gives the second information of crop growing

process, Y2. Having Y,, the household decides how much weeding to perform, L,. Finally,
the third shock, R, , determines final crop output, Y3. The household is assumed to know all

the information: R, , R,, R,, Y,, Y2, Y3, A, L,, and L, However, the econometrician knows
only Y3, A, L,,and L,.
Fafchamps estimates the model above using an iterative nested algorithm, full

information maximum likelihood, which I will not discuss here. He compares the results with

52

the results of deterministic model, and concludes that the stochastic control model is superior
in terms of data fitting based on Vuong’s non-nested model specification test (1989). His
conclusions are (i) estimation results agree with farmer’ perceptions about the dependency of
planting labor on early rainfall and about the occurrence of labor shortages, and (ii) they
confirm the importance of flexibility in production and provide a possible explanation of
why weeding is often “insufficient” on large farms or in good years.

Although Fafchamps ( 1993) does not explain how the flexibility affects household
risk management, his results suggest that flexibility should be incorporated in agricultural
household model explicitly. In the context of the semi-arid tropics, an important implication
of flexibility is in labor allocation between on-farm and off-farm activities depending on
rainfall, as Rose shows. However, neither Fafchamps nor Rose consider household saving
behavior under flexible decision making. The effect of flexibility on saving should be

examined in models as well as empirical studies.

4 Conclusions

There is empirical evidence that there exist several strategies that smooth household
consumption, such as crop diversification, income diversification (particularly off-farm
activities), use of risk-decreasing inputs, social insurance, and savings (livestock holdings).
However, those strategies are not examined well in structural household models and their
interactions are little known. Because saving is one of those strategies, household saving must
be examined in the interlinkages of those risk management strategies. To analyze household
saving in this way, there are four gaps in the literature.
(1) Multiple Risks and Their Correlation

Most saving models deal with two types of risk: exogenous labor income risk and
exogenous interest rate risk. The effects of increase in those risks are summarized in Table 1.
It shows that if saving is riskless, an increase in either type of risk increases precautionary

demand for saving given convexity of the marginal utility function (v"' > 0) and decreases

53
precautionary demand for saving given concavity of the marginal utility function (v"' < 0).
On the other hand, if saving is risky, convexity of the marginal utility function (v"' > 0) does
not necessarily induce precautionary demand for saving when interest rate risk increases. In
reality, however, those two risks coexist and must be correlated in some way. Moreover, there
will be more than two risks, although drought risk dominates in the semi-arid tropics.
Therefore, multiple risks and their correlation should somehow be incorporated into the

model.
(2) Dynamic Models and Lifetime Budget Constraint

In terms of precautionary demand for saving, there is little difference between the
two-period model and the multiperiod model. This is because in most multiperiod models, the
Euler equations derived from the first order conditions are used to examine the effect of
uncertainty. The Euler equations are necessary conditions for the optimal path, but as
discussed in section 3.5.1 they ignore the lifetime borrowing constraint. Because it is very
difficult to obtain a closed form solution reflecting the lifetime borrowing constraint, unless a
very simple model is used as shown in section 3.5.1, most studies are based only on the Euler
equations. To incorporate the lifetime borrowing constraint in a stochastic dynamic model, a
dynamic simulation starting from the end period must be conducted. Zeldes (1989b) and
Deaton (1991) are examples of this kind of simulation. Deaton shows that the borrowing
constraint under exogenous income risk induces precautionary demand for saving even if
marginal utility function is concave. This is an important implication that can be derived only
from multiperiod models, and with this respect multiperiod models have an advantage over
two-period model.

However, it will be difficult to simulate complicated household models. Therefore, it is
a challenging problem how to deal with a lifetime bonowing constraint in modeling as well as
in empirical studies of household saving.

(3) Production Function

54

The production function should be incorporated into a saving model. In other words,
saving behavior should be considered in a structural household model. In Dasgupta’s model
in section 3.53, two risky assets are considered as agricultural inputs, but the model assumes
linear return to the assets. However, the input-output relationship is often non-linear unlike in
Dasgupta, and consequently the effect of increasing risk is also non-linear. This non-linearity
determines whether an input is risk-increasing or risk-decreasing, and therefore the input
decision is a part of household risk management that may affect household saving decision.

In this chapter, I reviewed two agricultural household saving models in which
agricultural production is explicitly incorporated. However, neither model assumes non-
linearity of production and non-linearity of marginal effect of risk. In fact, I find no
household saving model that deals with agricultural production and its non-linearity of risk
effect.

Crop diversification and risk-decreasing inputs, a part of household risk management
strategies as reviewed, will be treated in production function in a structural household model.
(4) Flexible Decision

Flexibility in decision making is important. Fafchamps’ example (1993) shows that a
flexible model fits household’s behavior better. Although it does not explain how the
flexibility affects a household’s risk management, the results suggest that flexibility should be
incorporated in agricultural household model explicitly. As Epstein (1980) concludes, the
effect of flexibility is model dependent.

In the context the semi-arid tropics, an important implication of flexibility is in labor
allocation between on-farm and off-farm activities depending on rainfall, as Rose (1992)
shows. However, neither Fafchamps nor Rose consider household saving behavior under
flexible decision making. The effect of flexibility on saving should be examined in models as

well as in empirical studies.

55

References

Alderman, Harold and Christina H. Paxson, “Do the Poor Insure? A Synthesis of the
Literature on Risk and Consumption in Developing Countries,” The World Bank,
Policy Research Working Paper, 1992.

Bencivenga, Valerie R. and Bruce D. Smith, “Financial Intermediation and Endogenous
Growth,” Review of Economic Studies, 195-209, 58, 1991.

Carroll, Christopher D., “The Buffer-Stock Theory of Saving: Some Macroeconomic
Evidence,” Brookings Papers on Economic Activity, 2: 61-156, 1992.

Dasgupta, Partha, An Inquiry into Well-Being and Destitution, Oxford, UK: Oxford University
Press, 1993.

Deaton, Angus, “Savings in Developing Countries: Theory and Review,” Proceedings of the
World Bank Annual Conference on Development Economics, 61-96, 1989.

Deaton, Angus, “Saving and Liquidity Constraints,” Econometrica, 59: 1221-1248, 5, 1991.

Deaton, Angus, “Saving and Income Smoothing in Cote d’Ivoire,” Journal of African
Economies, 1-24, 1, 1992a.

Deaton, Angus, Understanding Consumption, Oxford, UK: Oxford University Press, 1992b.

Deaton, Angus, “The Analysis of Household Surveys: Microeconomic Analysis for
Development Policy,” mimeo, 1994.

Deaton, Angus and Guy Laroque, “On the Behavior of Commodity Prices,” Review of
Economic Studies, 59: 1-23, 1992.

Epstein, Larry G., “Decision Making and the Temporal Resolution of Uncertainty,”
International Economic Review, 21: 269-283, 2, 1980.

Fafchamps, Marcel, “Sequential Labor Decisions Under Uncertainty: An estimable
Household Model of West-African Farmers,” Econometrica, 61: 1173-1197, 5, 1993.

Flavin, Marjorie, “The Adjustment of Consumption to Changing Expectations about Future
Income,” Journal of Political Economy, 89 974-1009, 1981.

Gersovitz, Mark, “Saving and Development,” In: Hollis Chenery and TN. Srinivasan, eds.,
Handbook of Development Economics, Amsterdam, The Netherlands: Elsevier Science
Publishers, 381-424, 1988.

Gillis, Malcolm, Dwight H. Perkins, Michael Roemer, and Donald R. Snodgrass, Economics of
Development, New York, NY: W. W. Norton & Company, 1992.

Hall, Robert E, “Stochastic Implications of the Life Cycle-Permanent Income Hypothesis:
Theory and Evidence,” Journal of Political Economy, 86: 971-988, 1978.

Hansen, Lars Peter and Kenneth J. Singleton, “Stochastic Consumption, Risk Aversion, and
the5 Temporal Behavior of Asset Returns,” Journal of Political Economy, 91: 249-
26 , 1983.

Hartman, Richard, “The Effect of Price and Cost Uncertainty on Investment,” Journal of
Economic Theory, 5: 258-266, 1972.

Ri

56

Hartman, Richard, “Factor Demand with Output Price Uncertainty,” American Economic
Review, 66: 675-681, 1976.

Jappelli, Tullio and Marco Pagano, “Saving, Growth, and Liquidity Constraints,” Quarterly
Journal of Economics, 83-109, 1994.

Kimball, Miles S., “Precautionary Saving in the Small and in the large,” Econometrica, 58:
53-73, 1, 1990.

Leland, Hayne E, “Saving and Uncertainty: The Precautionary Demand for Saving,”
Quarterly Journal of Economics, 82: 465-473, 1968.

Levhari, D. and T. N. Srinivasan, “Optimal Savings under Uncertainty,” Review of Economic
Study, 36: 153-163, 1969.

Lewis, Arthur W., “Economic Development with Unlimited Supplied Labor,” The
Manchester School, 22: 139-191, 1954.

Paxson, Christina H., “Using Weather Variability to Estimate the Response of Saving to
Transitory Income in Thailand,” American Economic Review, 82: 15-33, 1, 1992.

Ramaswani, Bharat, “Supply Response to Agricultural Insurance: Risk Reduction and Moral
Hazard Effects,” American Journal of Agricultural Economics, 75: 914-925, 4, 1993.

Reardon, Thomas, Christopher Delgado, and Peter Matlon, “Determinants and Effects of
Income Diversification amongst Farm Households in Burkina Faso,” Journal of
Development Studies, 28: 264-277, 1992.

Reardon, Thomas et al., “Is Income Diversification Agriculture-Led in the West African
Semi-Arid Tropics? The Nature, Causes, Effects, Distribution, and Production-
Linkages of Off-Farm Activities,” In: S. Wangwe A. Atsain, and A. G. Drabek, eds.,
Economic Policy Experience in Africa: What Have We Learned?, Nairobi, Kenya:
African Economic Research Consortium, 1994.

Roe, Terry and Theodore Graham-Tomasi, “Yield Risk in a Dynamic Model of the
Agricultural Household,” In: Inderjit Singh, Lyn Squire, and John Strauss, eds.,
Agricultural Household Models, Baltimore, MD: The Johns Hopkins University Press,
1986.

Rose, Elaina, “Ex Ante and Ex Post Labor Supply Responses to Risk in a Low Income
Area,” Mimeo, Department of Economics, University of Pennsylvania, 1992.

Rosenzweig, Mark R. and Kenneth I. Wolpin, “Credit Market Constraints, Consumption
Smoothing, and the Accumulation of Durable Production Assets in Low-Income
Countries: Investments in Bullocks in India,” Journal of Political Economy, 101:
223-244, 2, 1993.

Rothschild, Michael and Joseph E. Stiglitz, “Increasing risk: I A definition,” Journal of
Economic Theory, 2: 225-243, 1970.

Rothschild, Michael and Joseph E. Stiglitz, “Increasing Risk II: Its Economic
Consequences,” Journal of Economic Theory, 3: 66-84, 1971.

Sakurai, Takeshi, “Effect of Crop Diversification on Drought Shock in Burkina Faso,” Draft,
Michigan State University, 1993.

57

Sakurai, Takeshi, “Consumption Smoothing in Burkina Faso,” Draft, Michigan State
University, 1995.

Sakurai, Takeshi, Madhur Gautam, Thomas Reardon, Peter Hazell, and Harold Alderman,
“Potential Demand for Drought Insurance in the Sahel,” Department of Agricultural
Economics, Michigan State University, Staff Paper, No. 94-67, 1994.

Sandmo, Agnar, “Portfolio Choice in a Theory of Saving,” Swedish Journal of Economics,
2: 106-122, 1968.

Sandmo, Agnar, “The Effect of Uncertainty on Saving Decisions,” Review of Economic
Studies, 37: 353-360, 1970.

Sandmo, Agnar, “Competitive Firm Under Price Uncertainty,” American economic Review,
61: 65-73, 1971.

Sargent, Thomas 1., Dynamic Macroeconomic Theory, Cambridge, MA: Harvard University
Press, 1987.

Savadogo, Kimseyinga, Thomas Reardon, and Kyosti Pietola, “Determinants of Farm
Productivity and Supply Response in Burkina Faso,” Staff Paper 9479, Department
of Agricultural Economics, Michigan State University, 1994.

Sibley, David S., “Permanent and Transitory Income Effects in a Model of Optimal
Consumption with Wage Income Uncertainty,” Journal of Economic Theory, 11: 68-
82, 1975.

Skinner, Jonathan, “Risky Income, Life Cycle Consumption, and Precautionary Savings,”
Journal of Monetary Economics, 22: 237-255, 1988.

Townsend, Robert M., “Risk and Insurance in Village India,” Econometrica, 62: 539-591, 3,
1994.

Vuong, H. Q., “Likelihood Ratio Tests for Model Selection and Non-Nested Hypotheses,”
Econometrica, 57: 307-333, 1989.

Zeldes, Stephen P., “Consumption and Liquidity Constraints: An Empirical Investigation,”
Journal of Political Economy, 97: 305-346, 2, 19893.

Zeldes, Stephen P., “Optimal Consumption with Stochastic Income: Deviation from Certainty
Equivalence,” The Quarterly Journal of Economics, 275-298, May, 1989b.

Table 1 Effect of Increasing Risk on Saving and Consumption

. Single Risk Model

 

 

 

 

 

 

v!" > 0 v", <0
Sourceoijsk 2V’+a(l+r)v ’0 2V'+a(l+r)v <0 2v'+a(l+r)v <0
OI' OI’ Of
0>l 0<l a(1
Income (is) s11 s1
w/o risky asset c,j CI I
1nterest(f) a 1 a1 a ,
w/o riskless asset c,,) c, 1 C1 11
Interest (5") m 1: m1 ml,
With “I a, at
riskless asset
C11 C, i? C]?

 

 

 

 

 

 

Note : S is riskless saving, m is riskless asset, a is risky asset, and C, is consumption in the first
period

59

Appendix A

Determinants of Drought Shock in Burkina Faso

 

 

 

 

 

Table A1 Determinants of Drqulrt Shock
Zara ALL Sahelian Guinean
Sudanian
Variables coeff. (SigT) coeff. (SigT) coeff. (519T)
Endogenous
lnDlV -0.34 (0.00) -0.37 (0.00) 5.81 (0.12) '°9°'¢'°P “Mmmmn-
NagY 2.2 (0.44) 0.89 (0.87) -37 (0.13) mm! non-ac income (1045)
Asset
DCRSTAE -3.8 (0.07) -4.4 (0.06) 2.1 (0.79) crow stocHItT‘per/AE)
DLVSTAE -0.37 (0.29) -0.38 (0.30) 33 (0.15) livestock holding (1045 per/AE)
AREAAE 0.20 (0.00) 0.21 (0.00) -0.11 (0.73) mm“ “m 9“ “1‘“th-
Household
AGED-NH 0.003 (0.04) 0.003 (0.02) -0.03 (0.06) ”WIMW W
AT 0.06 (0.29) 0.06 (0.49) 0.21 (0.44) W "fiction dam-y
HHAE 0.01 (0.36) 0.004 (0.58) 0.09 (0.04) “1“ "Mm “20 01W
LEADER 0.03 (0.55) 0.06 (0.43) 0.33 (0.19) ‘ " "HM-“'9 “COW
NUMWN -0.08 (0.00) -0.06 (0.05) -0.06 (0.51) ‘0' W” 0' "4 W
SIMPLE 0.04 (0.53) 0.03 (0.70) 0.78 (0.06) ‘ "W” WW 0""
Plot
AWN-L 0-003 (0.16) 0.002 (0.33) -0.07 (0.12) M- 3m" “"69 1"" “'0‘"
DISTIDX 0.035 (0.13) 0.018 (0.57) -0.42 (0.11) W “9- 0'9““ “Wk"?
LTOP 0.11 (0.24) 0.18 (0.19) 0.27 (0.48) ”9 “WW '0‘“
Radon
ACCESS 0.33 (0.00) 0.36 (0.00) N/A 1 r m «be near wane
CONST. -0.46 (0.02) -0.60 (0.01) 3.85 (0.09)
# of cases 247 207 40
R square 0.411 0.387 0.355

 

 

 

Source: Sakurai, Takeshi, “Effect of Crop Diversification on Drought Shock in Burkina Faso,” Draft,
Michigan State University. 1993.

Note: Endogulous variables are instrumalted (two-stage least squmed estimation).
HH=household, AE=adult equivdent size of household.

Droughtshock indrought yeart(u,)is defmcdas follows,

“1

 

av. production in good years

av. production in good years - production in drught year t

 

 

60

Appendix B
Full Insurance in Burkina Faso
1 The model
c, -HI-I+al’,,+fic"",+yx, +9,

This is estimated at village level. c,,: household consumption per adult equivalent size of
household in year t (CFA), 1”,: household income per AB in year t (CFA), 5,: village average
consumption per AB in year t (CFA), HH: household fixed effect, X,: a vector of
households characteristics, age of household head, single conjugal unit, dependency ratio and

number of wives. a -0 and fl - 1 imply full insurance.

 

 

2 Results

Table 81 Full Insurance in Burkina Faso

Village a (own income) [3 (av. consumption) specifications

Sahelian Zone

Acces 0.036 (0.518)* 0.947 (6.920)” village ",, wrthout X,
0.035 (0.458)* 0.946 (5.518)” village 7,, with X,

NoAcces 0.722 (2.933) 0.365 (1.230) village 7,, wrthout X,
0.692 (2.083) 0.288 (0.682) village ',, with X,

Sudanian Zone

Access 0.073 (0.494)* 0.941 (2.834)” village ',, without X,
0.032 (0.161)* 2.256 (0.531) village ',, with X,

NoAcoess 0.259 (2.715) 0.605 (1.879)* village ',, without X,
0.270 (2.357) 0.833 (1.691)* village 7,, with X,

Guinean Zone

Access 0.063 (3.656) 1.021 (7.171)” village 7,, wrthout X,
0.066 (3.856) 0.824 (4.583)” village ",, with X,

NoAcces 0.204 (4.075) 0.878 (2.474)” village ',, without X,
0.198 (2.816) 0.557 (0.719) village ',, with X,

 

Source: Sakurai, Takeshi, “Consumption Smoothing in Burkina Faso,” Draft. Michigan State University,
1995.

Note: Number in parenthesis is t-statistics of the regressions. " indicates that test result supports full
insurance: " for a means that the coefficient is not significmtly different from zero. " or " for fl means
that the coefficient is not significantly different from unity.

 

Chapter 3

Household Saving in LDCs:
Extensions and New Structural Models

0 Summary

There are gaps in the literature on household saving in LDCs. The gaps are both in
empirical studies and in theoretical studies. The objectives of this chapter are to fill certain
gaps in the theoretical literature, and to constnlct household models that will be used for
empirical studies to fill the gaps.

There are three gaps in the theoretical literature on saving. These are drawn from the
literature review in chapter 2.

First, we have little knowledge about the case when there are more than two risks and
they are correlated The findings in this chapter are: When two risks are correlated, the results
under uncorrelated risks do not hold. Most cases are ambiguous. Even the most robust
assertion (that when the marginal utility function is convex, there is precautionary demand
for riskless saving) holds no more. The results depend on how the two risks are correlated. An
interesting finding is that in the case of multiple, correlated risks, higher order correlations
rather than the first order correlations (just whether positive or negative) are important
determinants of the effects of increasing risk.

Second, there are few studies that examine the role of production technology in
household risk management/coping strategies. Inputs can be classified as either risk-
increasing or risk-decreasing based on the non-linear relationship between marginal
productivity and risk. However, little is known about how risk affects the use of those inputs
in an agricultural household model including saving. In other words, input has seldom been

examined as a part of a household risk management/coping strategies. The findings in this

61

62

chapter are: (i) The effects depend on the non-linear response of output to risk. If the risk
effect is diminishing, a convex marginal utility is sufficient for precautionary demand for
saving. But if the marginal utility is concave, the effect of increasing risk is ambiguous; (ii)
When the marginal utility function is convex and an input is risk-decreasing, the effect of
increasing risk on the input use is positive, but otherwise, the effect is ambiguous; (iii) A risk-
incrcasing input is overused, while a risk-decreasing input is underused compared with risk-
neutral inputs.

Third, little is known about how flexibility of decision making affects household’s
risk management, such as saving. The findings in this chapter are: (i) Flexibility changes the
standard results obtained in an inflexible model such as one presented above. The effects of
increasing risk on riskless saving and input use are ambiguous; (ii) The changes are model-
specific. In the model presented in this chapter, if a household increases on-farm harvest
labor in a good year, the standard results will hold. But, since the off-farm wage increases in a
good year, we cannot tell which labor, on-farm or off-farm, a household will increase in a
good year a priori; (iii) However, when flexibility is allowed, the household’s production risk
is lowered (in other words, the household is better off under flexibility), and consequently the
household’s precautionary demand for saving will be lower under flexibility than under
inflexibility.

On the other hand, the greatest gap found in empirical studies is that the interlinkages
of household risk management/coping strategies are little known. This gap must be filled by
empirical studies based on structural agricultural household models that incorporate the risk
management/coping strategies. For this purpose, I develop new structural household models
that satisfy the requirements. Then, I examine the properties of the models, especially the
effects of increasing risks on: (i) crop diversification; (ii) use of risk-decreasing and risk-
increasing inputs; (iii) on and off-farm input (labor and capital) supplies; (iv) saving.

Moreover, the effect of social insurance on those decision variables, and the effect of flexible

63
decision making on those variables are analyzed based on the models. The followings are the
main results.

First, social insurance is incorporated in the models. There is no qualitative difference
in responses to increasing risk between the case of partial social insurance and the case of no
social insurance, while the case of perfect social insurance is obviously different from the
others. We see only a substitution effect in the case of perfect social insurance, while in the
case of partial social insurance, there are both substitution and income effects.

Second, crop diversification is incorporated in the models explicitly. An increase in
rainfall risk has an ambiguous effect on crop diversification. This means that it is ambiguous
which effect dominates, income effect or substitute effect.

Third, I compare a simultaneous decision model with a sequential decision model.
There are a few, but important differences summarized in four points as follows.

First, one important finding is that in a sequential decision model, current period risk
in off-farm activities and future period risk in off-farm activities have different effects on off-
farm input decisions. The effects of the current risk depend on whether the input is risk-
increasing or risk-decreasing, such as the case of risk in crop production. On the other hand,
the effects of the futlue risk is like an exogenous income risk.

Second, in sequential decision model, the sign of the marginal effect of the initial

6w
wealth of season 2 on the end wealth of season 2, expressed as, 71w“ , as well as its marginal
2r

response to risks, are important These must be empirically estimated

Three, an advantage of the sequential decision model is that we can treat exogenous
variables known at the beginning of season 2 as risky variables at the beginning of season 1
when decisions for season 1 are made. The interest rate and the wage rate of hired labor in
season 2 are treated as risky variables to examine the effect of increasing risk of those
variables on the decisions in season 1.

Four, I conclude that the sequential model is better although its empirical estimation

requires more data.

1 Introduction

There are gaps in the literature on household saving in LDCs identified in chapter 2.
The gaps are both in empirical studies and in theoretical studies. The objectives of this
chapter are to fill the gaps in theoretical literature, and to construct household models that will
be used for empirical studies to fill the gaps.

The greatest gap in empirical studies is that the interlinkages of household risk
management/coping strategies are little known. There is a lot of empirical evidence that there
exist several strategies that smooth household consumption in rural areas of LDCs, such as
crop diversification, income diversification (particularly off-farm activities), use of risk-
decreasing inputs, social insurance, and savings (such as livestock holdings). However, most
studies examine one particular strategy, or test consumption smoothing at the household level
without specifying specific strategies used. This gap must be filled by empirical studies based
on structural agricultural household models that incorporate those risk management/coping
strategies listed above, and in those studies household saving in LDCs will be understood
better. For this purpose, new structural models are needed.

Three gaps among the gaps in theoretical studies are particularly important to
construct the structural models proposed above.

Frrst, we have little knowledge about the case when there are more than two risks and
they are correlated. Most saving models deal with either exogenous labor income risk or
exogenous interest rate risk, but rarely deal with them at the same time, especially if they are
correlated. However, the case of multiple risk will be important empirically, for example, crop
production risk and off-farm activity risk may be different but correlated.

Second, there are few studies that examine the role of production technology in
household risk management/coping strategies. Only a few household saving models include
cr0p production explicitly. The gap is not just with respect to crop production, but the non-
linear relationship between the marginal productivity of input and risk. Inputs are sometimes

classified as either risk-increasing or risk-decreasing based on the non-linear relationship.

risk

65

However, little is known about how risks affect those inputs decisions and how those inputs
affect household saving. Empirically, this model will answer which a household prefers,
saving or purchasing fertilizer, for example. Crop diversification will also be treated in this
framework.

The third gap is related to flexibility in decision making. Labor is a flexible input for
agricultural household: Fafchamps (1993) considers labor input adjustment based on rainfall
during the cropping season; Rose (1992) considers labor allocation between on-farm and off-
farm activities after harvest is known. Both empirically show that flexible models fit
household behavior. However, it is little known how flexibility affects household’s risk
management, such as saving. There is no agricultural household model that deals with saving
in a flexible decision framework, to my knowledge.

In this chapter I fill the three gaps in theoretical studies.

For the first gap, my analysis is based on a two-period model under both income risk
and interest rate risk, which is a direct extension from two-period models that deal with a
single risk, exogenous labor income risk or exogenous interest rate risk (e. g. Rothschild and
Stiglitz, 1971; Sandmo, 1968; Sandmo, 1970). Applying a standard method given by
Rothschild and Stiglitz (1971), I examine the effect of increasing those risks assuming either
independence or correlation. Moreover, I try different assumptions for the correlation to see
if that matters.

For the second gap, I start from a dynamic agricultural household model by Roe and
Graham-Tomasi (1986). Since they do not examine the effect of increasing risk, I do it in this
chapter. In this case, there is only yield risk in the model. My particular interest related with
the gap is to know what effect increasing risk will have on risk-increasing and risk-decreasing
inputs.

An extension from Rose (1992) is used for the third gap. I modify her model and
add saving to the model. The objective of this modeling is to examine the effect of increasing

risk on saving as well as other decision variables in a flexible decision model.

66

Then, based on the results of three extensions, I develop structural dynamic models of
agricultural household in the context of the West Africa semi-arid tropics (WASAT). A
specific context is nwded to develop a dynamic model that has to specify timing of decisions
and availability of information. Because the objective of this modeling is to construct models
that can be used to fill the gap in empirical studies, the models should incorporate household
risk management/coping strategies. I develop two kinds of structural model. The first model is
a simultaneous decision model, in which a household decides labor and capital allocation, and
crOp diversification at the beginning of each harvest year, immediately after harvesting. The
second model is a sequential decision model, in which a household makes a decision twice in
a harvest year”. at the beginning of the dry season and at the beginning of the rainy season.
By doing so, the household can use information on off-farm income after the dry season, and
based on the information, the household makes decisions for crop production in the rainy
season.

Finally, I examine the properties of the two models, especially effects of increasing
risks on: (i) crop diversification; (ii) use of risk-decreasing and risk-increasing inputs; (iii)
on- and off-farm input (labor and capital) supplies; (iv) saving. Moreover, the effect of social
insurance on those decision variables, and effect of flexible decision making on those

variables are analyzed based on the models.

2 Extensions of Saving Models

In this section, I extend saving models to fill the remaining gaps in theoretical

literature as explained in section 1.

2. 1 Correlation of Two Types of Risk
I develop a two-period model with two kinds of risk: labor income risk and interest
rate risk.

The problem can be written as

67
max EU(F, 9,, m, a) - u(c,)+ 6Ev(c,) - u(w, +y, - m - a) + t5Ev(j"2 + m + a(1 + F))
(1)
In the first period, the household has initial wealth, w,, and labor income, y, , the sum of
which is allocated among consumption, 0,, riskless asset, m , and risky asset, a , with stochastic
interest rate, 5" . In the second period, the household has initial wealth given by

w, - m+a(l+ ?) and stochastic labor income, )72, the sum of which is consumed in the

second period. The first order conditions with respect to m and a respectively are

Ea—U---u’+0Ev'-0 (2)
6m

Efl-—u'+6E(l+f)v'-0 (3)
6a

Differentiating twice (2) and (3) with respect to f and y, respectively, the effects of
increasing risk are examined (following Rothschild and Stiglitz (1971), or the RS method
hereafter). The third derivatives used for the RS method are shown in Appendix A. There are
two cases: one assumes that income risk and interest rate risk are independent, and the other
assumes that they are correlated.

When the two risks are uncorrelated, the results are as follows: (1) Increasing risk in
either the interest rate or labor income increases holdings of riskless assets when v'" > 0 and
decreases holdings of riskless assets when v'" < 0. This is a standard result of saving under
income uncertainty; (ii) The effect of increasing risk in the interest rate on holdings of risky
assets is ambiguous when v'" >0, and negative when v'" <0; (iii) Increasing labor income
risk increases investment in risky assets when v'" > 0 and decreases investment in risky asset
when v'" < 0; (iv) Those results are not different from those when only one risk is considered
(in sections 3.3.1 and 3.3.2 of chapter 2). These results are summarized in Table 2.

Then, assume that two risks are correlated. As shown in Table 1, four cases are
considered. The relation of the two risks are non-linear as graphically presented in Figure 1.
The four cases simplify the reality, but it is much more realistic than assuming a linear

relationship. In the negatively correlated cases, exogenous income is considered like ordinary

68

gifts within a village or a local area; when a drought occurs, such gifts are discouraged and
the interest rate becomes high. On the other hand, in the positively correlated cases,
exogenous income may be remittances from non-household members and/or public food aid;
they are high when a drought damages local economy and interest rate becomes high.

The results are shown in Table 3', when the two risks are correlated, the results under
uncorrelated risks do not hold any more. Even the most robust assertion that when v'" >0 ,
there is precautionary demand for riskless saving, depends on how the two risks are
correlated. As seen from Table 3, most results are ambiguous. This suggests that the effects
should be empirically determined. But one interesting point obtained from Table 3 is that the
effect of increasing risk depends more on the second-order relationship between the two risks
than on the first-order relationship (that is, just whether positively or negatively correlated).
For example, when v'" > O , cases 1 and 3 are similar, while cases 2 and 4 are similar. A lesson
from this analysis is that when we consider multiple, correlated risks, we need to pay attention

to higher order correlations.

2.2 Stochastic Production Model

In this section, I examine the effect of increasing risk on production input decision.
This analysis is an extension of Roe and Graham-Tomasi (1986) who consider a quite general
dynamic model of agricultural household under uncertainty. For simplicity I assume no
bequest motive, unlike Roe and Graham-Tomasi.

The problem can be written as

r
m?“ 5'2 ‘5'"(Xw x". 4): Z 4X... X... 1.. A. A.) (4)
(-
subject to
QM! -Q(L,, Ar; star) (5)
Cr " pqrxqr + puX-c + bill (6)

w: " 0:3 4’ bl: 4’ pq¢Q(l1-v Al-I; 51-1)... (1+ ”Si-r

(7)
-s+m4+qg+q

69
where X“: consumption of an agricultural staple, p“: its price, X m: a good purchased in a
market, p": its price, 1,: leisure, L,: on-farm labor input, 1:, endowment of labor, 1),: wage.
A: land use for own production, 2,, endowment of land, a,: land rental price, SM: saving
from period t to period t+1, r: interest rate of saving, 0,: consumption that consists of staple
foods, purchased goods and opportunity cost of leisure, and Q,+1 - Q(l.,, A,', 5,): household
agricultural production function with stochastic term. The output appears in the next period,
t+l. The difference between I: and 1, represents hired labor if negative or off-farm labor

supply if positive, and the difference between E and A, represents rented land if negative or

lent land if positive. Also this model allows the household to borrow at the rate of r , which
results in a negative saving in that period.
Bellman’s equation is:
v(w,) - m‘ax u(c,) + 6E,v(w,,,) (8)
This equation is different from what is found in Roe and Graham-Tomasi (1986). Here I

assume that utility is a function of total consumption given by (6) and that the value function

is a function of initial wealth given by w,. From Bellman’s equation, the following first order

conditions are obtained. The first order condition with respect to s, , is:

-u'(c,) + 6(1+ r)E,v'(w,,,) - O (9)
The first order conditions with respect to 11 and A, respectively are

-b,u'(c,) + 6p,,,,E,v’(w,,,)% - 0 (10)

-a,u’(c,) + dpwE,v'(w,,,)-:—g- - O (l 1)

Equations (9)-(ll) are necessary conditions for the optimal path. Thus, consumption
decisions and production decisions are not separable.
Now, unlike Roe and Graham-Tomasi (1986), I examine effects of increasing risk by

applying the RS method to the first order conditions (9)-(l l). The second derivatives of the
first order conditions (9)—(1 l) with respect to E, are in Appendix B.

70
Some assumptions are made before analyzing the effects. First, with respect to the

2

effect of the risk on output: %>0 and F<O are assumed. This means that a
e! 8!

production shock increases output, but the effect is diminishing. They are plausible

assumptions considering 5, like a rainfall shock. Second, with respect to the effect of the risk

 

 

 

 

 

 

 

62 62
on the marginal productivity of inputs: There is no a priori assumption for Q. . Q. .
61,68, 6A, 68,
63Q 63Q . . . . .
——.5- and -2 . But here I assume that labor lS risk-rncreasrng and the effect rs
61,68, 6A, 68,
. . . . 6’Q 63Q . . . .
drmrmshing ( _ >0 and ,2 < 0) and that land IS nsk-decreasrng and the effect is
61,68, 61,68,
62 63
diminishing ( Q. < O and Q., > 0).
6A, 68, 6A, 68,

The results of the effect of increasing risk are summarized as follows: (i) When
v”' 20, saving increase when production risk increases; (ii) When v"’ <0, the effect of
increasing risk on saving is ambiguous; (iii) The result (i) is consistent with a standard
precautionary demand for saving, while the result (ii) is not consistent. These results critically
depend on the non-linear response of output against risk (3%,— <0); (iv) When v'" 20, the
effect of increasing risk on the use of risk—decreasing inputs is positive, but that on the use of
risk-increasing inputs is ambiguous; (v) When v"' < 0, the effect of increasing risk on input
use is ambiguous; (vi) If inputs are risk-neutral, the effect of increasing risk on input use is

positive when v'" z 0 , but is ambiguous when v"' < 0. Proofs are in Appendix B.

From (10) and (11) with the assumptions above, it can be shown that risk-increasing
labor is underused at the optimum, that is, p,,,,E, fig - (1 + r)b, > 0 , and that risk-decreasing

61.

land is overused at the optimum, that is, p,,,,E, 3% - (l + r)a, < 0 , relative to the optimum

when those inputs are risk-neutral (proofs are in Appendix B). Note that since input is one-
year lagged, there is an interest for input prices.
As seenfrom the first order conditions (9)-(l 1), input decisions are not separable

from consumption decisions, and the results above show that even if saving increases due to

71

increasing production risk, whether consumption increases or decreases is unknown.
Therefore, the type of the input, that is, whether it is risk-increasing or risk-decreasing, is
important to analyze an agricultural household’s behavior. To do so, we need to make more

specific assumptions in the context of the agricultural system with which we are concerned.

2.3 Flexibility and Production/Saving
To examine the effect of flexibility, I develop a model based on Rose (1992). The
model is given by
rlnait u(y + w,L, - 3,) + 6Ev(c,) (12)
where c,-(1+r)s,+F(N-L,, N-L;,R, X)+w,(R)L;
y, exogenous income; L,, off-farm labor supply in the first period; 1,, off-farm labor
supply in the second period; N - 1,, on-farm labor supply in the first period; N - 1,, on-
farm labor supply in the second period; 12, stochastic rainfall, unknown in the first period
and known in the second period; w,, fixed wage in the first period; WAR) , wage depending
on rainfall in the second period; X, a vector of fixed household characteristics; 3,, saving in

the first period; r , known and fixed interest rate for saving; c, , consumption in the second

period Note that this model follows Rose’s assumption that on-farm labor and off-farm labor
are perfect substitute setting leisure constant, and consequently when one increases, the other
always decreases.

Unlike Rose, 1 use additively separate utility here and add saving s, in the first period
The saving produces a fixed, known interest rs,. Following Rose, off-farm labor supply in the
second period L1 is optimized in the short-run,denoted by g. This model looks very much
like a two-period saving model, but the differences are that the model includes endogenous
and flexible labor allocation decisions, that there is a production in the second period, and
that stochastic labor income in the second period is endogenous.

The first order conditions with respect to s, and L, respectively are

-u'+6(l+r)Ev'-O (l3)

72
w,u' - 6Ev’F, - 0 (14)

where F2 - w, is used because L, is optimized in the short-run. From those equations,
Ev’(EF, — (l +r) w,) +cov(v', F,)- 0 is obtained Based on this, the following results are
obtained: (i) If first-period on-farm labor is risk-increasing, labor is underused (lower than
when labor is risk-neutral). This is predicted in the single-period model, but not by the two-
period flexible model given by Rose (1992); (ii) If first-period on-farm labor is risk-
decreasing, labor is overused (higher than when labor is risk-neutral). The proofs are in
Appendix C. The results are the same result as in section 2.2. This implies that the results
depend on whether first-period labor, e.g. ploughing, planting, and weeding, is risk-increasing
or risk-decreasing. I-Iowever, Rose assumes only the case of risk-increasing without
explanation.

Then, the effect of increasing risk is examined using the RS method: (iii) An increase
in rainfall risk has an ambiguous effect on riskless saving in the first period, regardless of the
sign of v'"; (iv) An increase in rainfall risk has an ambiguous effect on off-farm labor supply
in the first period, regardless of the sign of v"'. Results (iii) and (iv) are different from those
obtained in section 2.2 using inflexible model. Proofs are in Appendix C.

The ambiguity comes from the ambiguous sign of 26%; the off-farm labor supply in

the second period is optimized after knowing rainfall, that is, it depends on the rainfall. If a

6
household allocates more labor to on-farm activity (harvest) in a good year, the sign of 61?

should be negative. However, since the off-farm wage in the second period also increases in a
good year due to high demand for harvest labor, a household may allocate more labor to off-
a O
farm activity to in a good year. In this case, 3% should be positive. We cannot tell whether

6
3% is positive or negative a priori; rather it should be empirically determined. But if we
assume that 61? is negative, the results become as follows: (v) An increase in rainfall risk has

a positive effect on riskless saving in the first period when v"' z 0, but has an ambiguous

73

effect when v'" <0; (vi) An increase in rainfall risk has a positive effect on off-farm labor
supply in the first period when v"' 20 and off-farm labor is risk-decreasing. Otherwise, the
effect of increasing rainfall risk is ambiguous. Proofs are in Appendix C. The results (v) and
(vi) are the same as those in section 2.2.

Although the effects are ambiguous a priori, the household’s production risk is
lowered (in other words, the household is better-off under the flexibility) when flexibility is
allowed, and consequently the household’s precautionary demand for saving will be lower

under flexibility than under inflexibility.

3 Structural Models and Their Implications

In the following sections, dynamic household models are developed in the context of
the West Africa Semi-Arid Tropics (W ASAT). The objective is to construct household models
that will be used for empirical studies to fill the gaps in the literature. The greatest gaps in
empirical studies is that the interlinkages of household risk management/coping strategies are
little known. This gap must be filled by empirical studies based on structural agricultural
household models that incorporate risk management/coping strategies.

Therefore, the models I develop here must have (i) use of risk-decreasing and risk-
increasing inputs; (ii) on- and off-farm input (labor and capital) supplies; (iv) saving; (iv)
crop diversification; (v) social insurance. Moreover, following the discussion in sections 2.1—
2.3, (a) multiple, correlated risk, (b) non-linear production function, and (c) flexible decision
making, are incorporated into the models as follows.

(a) Multiple, Correlated Risks

I assume two risks: stochastic rainfall and stochastic exogenous income. Rainfall
affects crop production. Exogenous income can be independent of rainfall, but may be
correlated with it. Unlike a simple saving model without production, exogenous income is not
labor income from inelastic labor supply. I assume that exogenous income is from

remittances, gifts, and/or public food aid. Therefore, we can consider that the stochastic

‘74
exogenous income is “social insurance”. Thus, the degree of the correlation between rainfall
and exogenous income can be considered as the degree of social insurance.
(b) Non-linear Production Function

I assume a crop/off—farm production function, which is a function of labor and capital
inputs for on/off-farm activities, crop diversification, rainfall, and household characteristics.
By defining signs of second and third order derivatives, I make assumptions about responses
of the marginal productivities of those inputs and crop diversification against increasing risk.
In the WASAT, off-farm activities are not limited to agricultural employment. Therefore, I
assume that off-farm income is a function of off-farm labor and capital inputs, not just wage
times labor supply (unlike Rose (1992)).

(c) Flexible Decision Making

Idevelop two kinds of structural models. The first model is a simultaneous decision
model, in which a household decides labor and capital allocation, and crop diversification at
the beginning of each harvest year, immediately after harvesting. The second model is a
sequential decision model, in which a household makes a decision twice in a harvest year. at
the beginning of the dry season and at the beginning of the rainy season. By doing so, the
household can use information on off-farm income after the dry season, and based on the
information the household make decisions for crop production in the rainy season.

Note that the simultaneous decision model and the sequential decision model are
similar to Rose’s (1992) one-period model and two-period model respectively, but my
models deal not only with labor allocation but also with capital allocation and saving in a
dynamic framework. And as a result, information about wealth is the key in decision making
in my sequential decision model unlike Rose’s.

Both models are based on the concept of “harvest year”, which is explained in
section 3.1. Then in section 3.2, I develop a simultaneous decision model, and finally it is

modified to incorporate sequential decisions in section 323.

75
3. 1 Time Frame of Decision Making

There are two seasons in the WASAT: the rainy season and the dry season. Only in
the rainy season, crop production takes place. Crop production is rain-fed and unstable as
rainfall is very variable in the WASAT. Households are engaged in a variety of off -farm
activities to compensate shortfalls of crop income due to drought and to stabilize total
household income over years as explained in chapter 2. Off-farm activities mainly occur in
dry season after the harvest is known, but some of them are conducted in rainy season as well.

That crop output is a key variable leads us to the concept of “harvest year”, which
begins with the harvest at the beginning of the dry season and ends at the end of the rainy
season, with the simplifying assumption that all harvesting is done instantly at the beginning
of each harvest year, as shown in Figures 2 and 3. The two figures differ in assumptions about
information the household uses in decision making. Figure 2 corresponds to a simultaneous
decision model, in which the household makes all the decisions at the beginning of the dry
season of year t. Figure 3 corresponds to a sequential decision model, in which there are two
sequential decisions in year t: at the beginning of the dry season and at the beginning of the
rainy season.

Both models start from the dry season of year t. At the beginning of the dry season,
the household knows its initial wealth level which consists of the harvest of the previous year
t-l, carry-over wealth from year H, and stochastic exogenous income. Carry-over wealth
consists of crop stocks and livestock holdings. Stochastic exogenous income may depend on
the harvest of year t-l and household characteristics. It includes public food aid, gifts, and
remittances from non-household members, and acts as if social insurance. I assume that such
income is uncertain in the previous year, but at the beginning of year t, it becomes certain.
Off-farm income includes income from local employment, self-employment, and migration.
In both the simultaneous decision model and the sequential decision model, idiosyncratic risk

in off-farm income is assumed.

76

In the sequential decision model, decisions are made as follows. At the beginning of
the dry season, given knowledge about initial wealth as well as its characteristics and being
under idiosyncratic risk in the dry season and uncertainty about the future rainy season, the
household decides: (i) how to allocate family labor between leisure and off-farm activities in
the dry season; (ii) how to allocate its initial wealth among consuming, precautionary saving,
and off-farm activities in the dry season. Then, at the beginning of the rainy season, knowing
its new wealth level given by the sum of net off-farm earnings and saving and being under
rainfall risk and idiosyncratic risk, the household decides: (i) how to allocate family labor
among leisure, on—farm, and off-farm activities in the rainy season; (ii) how to allocate its
wealth among consuming, precautionary saving, off-farm activities, and purchasing variable
inputs for crop production in the rainy season; (iii) how to diversify crops.

In the simultaneous decision model, on the other hand, the household can make all

decisions including on-farm and off-farm activities at the beginning of the dry season.

3.2 A Simultaneous Decision Model
Iapply a standard intertemporal utility maximization problem given by a time-

separable lifetime utility function for household as follows:
1
max 12,; 6'u(c,,,, 1“,) (15)
-0

where t refers to harvest year, u( ) is the one-period utility function, 0, is real consumption in
year t, I, is leisure in year t, 6 is the discount factor, E, is the expectation operator
(conditional on information available at year t), and T is the end of the household’s planning
horizon. Standard assumptions about the utility function apply: u( ) is monotonically
increasing and concave in c, and 1,.

As is shown in Figure 3, year t starts with the harvest from rainy season in year H,
and therefore the household’s initial wealth is given by

w, -(1+ r)s,_, + Q_, + y, (16)

,7
where s,_, is saving or carry-over wealth from year H to year t and Q,_, is total income (on-
farm and off-farm) in year H, y, is stochastic, exogenous income that is realized and known
at the beginning of year t with certainty.

Since there is no formal saving institutions, the saving takes the form of crop stocks
and livestock holdings that produce no interest. But considering livestock, reproduction may
yield a positive interest rate, while death is a negative interest rate. This interest rate is
stochastic by nature, and is known only at the end of year t-l. However, here for simplicity
interest rate, r , is assumed to be known, fixed, and positive.

Total income, Q,_,, includes both crop income and off-farm income. Crop income

cannot be consumed in year t-l due to the time frame of this model, but part of off-farm
income can be consumed in year t-1. However, here Q,_, is assumed to be the sum of total
crop income and total off-farm income. Thus, to allow off-farm income earned in year t-l to
be consumed in year H, the consumed off-farm income is considered to be borrowing from
year t, denoted by b,_,. Since this borrowing is internal, naturally it is limited by the amount
of off-farm income in year t—l. Now, equation (16) will become
w, -(1+ r)s,-, + Q_, + y, - b,_, (17)

Equation (17) means that part of total off-farm income, denoted by b,_,. is already consumed
in year t-l and is not available as the initial wealth of year t.

If the household is allowed to borrow from external sources, borrowing is lnndled as

a subset of saving: if the saving decision in year H is negative (s,_, < 0), the household will

be a net borrower in year H. The interest rate of this borrowing is the same as that of saving.
If borrowing is not allowed, there will be a constraint as s, z 0 for all t. In addition, a lifetime
budget constraint must be imposed regardless of the single-period borrowing constraint.
Since the household’s finite life is determined to end in year T and no bequest motive is
assumed, all assets must be consumed in year T in this model. This constraint will be
discussed later, but for now, only intertemporal optimal allocations are considered as if the

household’s life is infinite.

78
y, includes gifts, public food aid, and remittances from non-household members to

be received in year t. It depends on household characteristics and is conditioned by the
previous year’s harvest, and therefore is predetermined at the beginning of year t. However, at
the beginning of year t, y”, is unknown because it depends on rainfall in year t.

In a simultaneous decision model, the initial wealth given by (17) is allocated at the
beginning of harvest year t among consumption, variable capital inputs for crop production

including hired labor, off-farm activities, and savings as follows:

b a 0
w,+I-;Lr--c,+K, +a,l:+K, +3, (18)

where K,“ is variable inputs for crop production including seeds and fertilizer, K," is variable
input for off-farm activities, a,L',' is cost for hired labor, that is, wage rate 0, times amount of
hired labor 1: , and b, is off-farm income consumed in year t as explained above.

Labor constraints are as follows:

N,+l:-I,+l,‘+l,° (19)
where N, is the exogenous, total time available to the household in year t, I: is hired labor
and I, is leisure that appears in the utility function. Equation (19) means that leisure is
determined by on-farm labor supply, 1: , hired labor, 1:, and off-farm labor supply
including local on-farm employment, 1:.

Total income generated in year t, 2, is a function of endogenous capital and labor
inputs for on-farm and off-farm activities and crop diversification, D,, in year t, and
exogenous or predetermined variables including stochastic yearly rainfall, R , idiosyncratic
risk in off-farm earnings, 5, , and a vector of household characteristics, x,:

Q.-Q<K:'.K.°. 11.1th 12.1%.. é”..X.) (20>

79
Crop diversification, D,, is a Simpson index'. D, is greater than or equal to zero and

less than unity: D, is exactly equal to zero when a household produces only one crop, and is

unity when a household produces infinite kinds of cr0ps, which is impossible.

This dynamic optimization problem for the household can be written as
V,(W,) ' max “(Cir It) +6Eivr+r(wi+i)

b d 0 a 0
-mrax u(w,+l—'—-K, -a,l,'—K, -3,,N,+Lf-L,—l,) (21)
r + r
+ 6E.V...«1+ r)s. + Q + 9... - b.)
where H, - (K,°, K,°, 1:, 1:, 1:, D,, S,, b,) is the vector of decision variables of harvest year
t, and V,( ) is the maximum value of future utility discounted back to t. Given the standard

assumptions for the utility function, the value function, V( ), is also monotonically increasing

and concave in w.

3.2.1 The First Order Conditions

The first order conditions for each decision variable given by TI, is as follows:

 

 

K,°: -iu-+6E, 6V iQ;-0 (22)
6c, 6w", 6K,
V
Kf: —fl+6E,-Qa—--o (23)
*1 8K: 6W“,
0 V
L, - in- 015:, L-T, - 0 (24)
(91. 3W.“ 311
,. u 6u 6V 6Q
: - — + — + (SE —— - 25
l’ 0' 6c, 01, ' 6w“, at: ( )
I: _ .61 6E, Qfl— _ (26)
all a: a 1+!
1),: Gigi-(1Q— - (27)
6W“, 60:
3,: -—§C£-+6(l+r)£’, 6V -0 (28)

i Mi

 

, D -,_2 planted areaforcropkUra)
' 1 household 3 total planted area ( ha)

V
b,: -—l-2u--6E,—a--O (29)
1+ r 6c, 6w”,

The effect of increasing risk is examined following the RS method in the following sub-

sections.

3.2.2 Capital and Labor Inputs

Effects of increasing risk on on-farm inputs, K: , 1: , and L: are examined first using
the RS method. From equations (22), (24), and (25), differentiating twice with respect to
stochastic variables, R E, , and y”, , respectively yields:

R:
‘lm(a~Q+ Liz—MI): _a_g_+ W(:_:_Q+a zyt+l)__ 6Q +2V"(_ 6Q +iy+L)— 32Q + VI 63Q~
6R, 6R aK,“ aR,’ 6R, aK,“ 3R, 6R, 6K,°6R M36];2
(30)
~. "(292 0"_Q V..6__’_Q5_Q_
8" (El—1(“Vo’lé',2 6K," (31)
~ . wig.
yul‘ V 6K,“ (32)

Before signing them, I have to make some more assumptions, (a) and (b).
(a) On 53,, or social insurance:
y,“ is stochastic exogenous income conditioned by the harvest of year t, and R.

There are three cases about the relationship of the two: (i) If 9“, acts like social insurance,

they will be negatively correlated (2?”- ”I <0). And if it is a perfect insurance,
6

372 + 29-1-0 will hold, while if it is a partial insurance, 22 + Q,_,, > 0 will hold. Note

6R 6R 6R 6R

that this does not necessarily smooth consumption. (ii) In isolated villages with few economic

links with external world, this relation will be the opposite because the drought shock affects

all the village economy. That is, a? ’ —‘-L>0 will hold Inthis case, however, 29—, im- >0 is
6R 6R 6R

6
still true. (iii) They can be independent, that rs, 35}?- -.0 In any case except for the perfect

81

az-
insurance, marginal productivity of rainfall should be diminishing, that is, 613+ (92,4 <
. . 32y}, . .
wrll hold although the srgn of —-L6R’ is ambiguous.

f

6
However, in signing (30), because the sign of 6);}: does not directly affect the signs

of (30), there are only two cases: one is the case of imperfect social insurance, that is, when

g+§2§l>0 holds, and the other is the case of perfect insurance, that is, when
6Q +6), ’ -0. In the latter case, only the effect of rainfall uncertainty is examined.

aR aR

(b) On risk-increasing and risk-decreasing inputs
2
(i) If K: is risk-increasing, 6K°6R > 0 will hold and its effect is diminishing with
l

 

a 62 .
< 0 (ii) If K is risk-decreasing, “Q. < 0 will hold and its effect rs

a’Q
(sinaR2 ' ' 6K, 6R,

increasing with risk, that is, 63Q~ > 0.
6K,°6Rz

 

 

risk, that is,

 

Based on the assumptions above, (30), (31), and (32) are signed. The results are

summarized in Table 4.

Then, effects of increasing risk on off-farm inputs, K,° , and I: are examined. From

equations (23) and (26), differentiating twice with respect to stochastic variables, R, 5,, and

i... . respectively yields:
"' 62 1..
I“ V,(6Q+ 22m) aQ+V"( -—Q +-a—-%L)-- 3Q (33)
5R. 3R aR’ 6R, 6K,"
‘ "(j—.9 29., WK;_’Q«?_Q v~"Q a__“_Q , a’Q
: " +2V V——. 34
8' 519’ 6e, 2 6K” as, 6K," 68', _+ algae} ( )
.. m 3Q
: v _
y“! 8K: (35)

(33)—(35) are signed as done for (30)-(32).
The results are summarized in Table 4. The following is found under imperfect social
insurance: (i) When V'" > 0, an increase. in rainfall variability increases off-farm inputs and

risk-decreasing on-farm inputs , but has an ambiguous effect on risk-increasing on-farm

82
inputs; (ii) When V'” > 0, an increase in idiosyncratic risk in off-farm earning increases on-
farm inputs and risk-decreasing off-farm inputs, but has an ambiguous effect on risk-
increasing off—farm inputs; (iii) When V’" > 0, an increase in social insurance risk increases
both on-farm and off-farm inputs; (iv) When V'" < 0, an increase in rainfall variability has an
ambiguous effect on both on-farm and off-farm inputs; (v) When V'” < 0, an increase in
idiosyncratic risk in off-farm earning has an ambiguous effect on both on-farm and off-farm
inputs; (vi) When V'" < 0, an increase in social insurance risk decreases both on-farm and
off-farm inputs.
From (22) and (28), the following equation is obtained:

61-55 5" aQ+ooov(-i’-V—. Q,» 6(1+r)E.

66, W”, 'aK: 3W”: 6K: 3W“,

62
The covariance will be positive if 6K 3- < 0, that is, K: is risk-decreasing, and it will be

 

 

 

(35)

 

a
l

 

62
negative if Q- >0, that is, K," is risk-increasing. If the covariance is positive,
6K,‘6R
Ea 6Q
—< 1 + r must hold, and if the covariance is negative, E, — >1 + r must hold. These
E‘6K,‘ ax:

imply that at the equilibrium, a risk-decreasing input for crop production is overused in a
sense its marginal productivity is less than the interest rate while a risk-increasing input for
crop production is underused. Similar results for labor input for crop production can be
obtained from (24) and (28). Moreover, capital input and labor input for off-farm activities
have the same properties, which can be obtained from (23) and (28), and (26) and (28)

respectively. These results are consistent with findings in section 2.3.

3.2.3 Saving and Borrowing

Similarly to section 3.2.2, the effect of increasing risk on 3, and b, is examined from
(28) and (29). From (28), differentiating twice with respect to stochastic variables, R, 8,, and
y“, , respectively yields:
62 "

~ m 6Q a? V" 6+2Q
R (1+” (6R + 6R )+(1+ r) (TR,2+ 0"R,2 ) (36)

83

com.

62
(1+ r)V"'(-:-Q;i-)2 + (l + r) V"???- (37)

r
5,: (l + r)V”' (38)
From (29), the same equations as above but with the opposite sign are obtained. The signs of
those equations are determined using the assumptions given in section 3.2.2.

Under imperfect social insurance (i) if V"' > 0, an increase in any risk will increase
saving and decrease borrowing; (ii) if V"' < 0, an increase in rainfall risk and idiosyncratic
risk have an ambiguous effect on saving and borrowing, but an increase in social insurance
risk increases saving and decreasing borrowing. The results are summarized in Table 4.

Equations (28) and (29) are the same. Thus, at the equilibrium, the marginal gain

from saving is equated to the marginal cost of internal borrowing from off-farm income. This

is because the interest rate for internal borrowing from off-farm income is the same as the

interest rate of saving. But b, is bounded by the amount of off-farm income, and

 

6u
consequently equation (29) will possibly become 6— < 6(1+ r) E, due to the binding
c

1 1+1
constraint This may affect the optimal level of other decision variables, but the first order
conditions (22)-(28) will still hold.

If the optimal internal borrowing from off-farm income exceeds off-farm income, the
household will have to borrow from external sources at the same interest rate as saving, and
consequently negative saving will be the optimum. However, if such borrowing is not allowed,

saving cannot be negative. Thus, if the borrowing constraint is binding, the household’s
6V 6::

<—

6‘9,” *3 .

 

saving in year t will be zero and equation (28) will become 6(1 + r)E,

3 . 2 . 4 Crop Diversification

Equation (27),; can be written as

6E—E,— “’9 owl, .19.)-0 (39)
awn] 6D, 6W“, 60!

84

 

. . 6Q . .
where 6 > 0. Thus, based on equation (39), if 35 > 0, the covariance must be negative
w

1+1 3
2

and if 32< 0, the covariance must be positive. In addition, T——. < O is assumed to
8 WM] 1

2

hold. Consequently, negative covariance requires .. > 0 and positive covariance
l l

 

2
requires .. < 0. Therefore, there are two cases that satisfy equation (39): one is

l I
0Q

 

2
— > 0 and j—Qr > 0 (crop diversification is output-enhancing and risk-increasing); the
6D, 6D, 6R,

62Q

other is fly-<0 and .. <0 (crop diversification is output-reducing and risk-
6D, 6R,

 

l

. . . . . . . . 6Q .
decreasrng). If crop drversrfrcatron is used to reduce crop production risk, 5 < 0 wrll hold.
I

But it is not necessarily the case, as shown Sakurai (1993).

The effect of increasing risk is examined using (30), (31), and (32) by replacing K:
with D,. Given the assumptions above, under imperfect insurance: (i) When V"' > 0, an
increase in risk in rainfall has an ambiguous effect on crop diversification; (ii) When V"' > 0,
an increase in idiosyncratic risk in off-farm activities increases crop diversification if crop
diversification is risk-increasing, and decreases crop diversification if crop diversification is
riskodecreasing; (iii) When V"' > 0, an increase in social insurance risk increases crop
diversification if crop diversification is risk-increasing and decreases crop diversification if
crop diversification is risk-decreasing; (v) When V"' < 0, (i) is the same, (ii) becomes

ambiguous, and (iii) becomes the opposite. The results are summarized in Tables 4.

3.2.5 Social Insurance

Social insurance is defined here as informal institutions that make the household total
income uncorrelated with rainfall, or reduce the correlation. There are several institutions for
social insurance observed in LDCs, but in this model the institutions are not specified; it is
assumed to reduce income risk due to rainfall costlessly. If this insurance requires some ex

ante cost, this model can be seen as a model for formal insurance.

85

The following is found under perfect social insurance: (i) An increase in rainfall risk

has no effect on saving, borrowing, and off—farm inputs; (ii) An increase in rainfall risk

6
decreases risk-increasing on-farm inputs and risk-increasing crop diversification (6-Dg > 0);
3

(iii) An increase in rainfall risk increases risk-decreasing on-farm inputs and risk-decreasing

6
crop diversification (£< 0); (iv) Those effects do not depend on the sign of V"'.
1

Although there is perfect insurance, the household responds to an increase in rainfall risk.
This is because the marginal productivities of on-farrn inputs and crop diversification are not
linearly related to rainfall; the response is seen as a pure substitution effect between inputs

keeping expected income constant.

3.2.6 Lifetime Budget Constraints

The household’s wealth at the end of its lifetime must be zero if no bequest motive is
assumed, and this constraint affect household’s behavior backward from the last period.
Intertemporal optimal conditions like (22)-(29) are necessary for a dynamic optimum, but
cannot give a solution without the lifetime constraint. If a lifetime constraint is imposed, the
household will be less likely to borrow even if borrowing is allowed in each period. It does
not affect the intertemporal optimal conditions, but the consumption level will become
smaller than without the constraint. Therefore, if we are concerned with the level, we need to
incorporate the lifetime constraint.

But here, the effects of risk are the main concern, and so we can ignore the lifetime
constraint. In addition, since households in rural LDCs are extended family with several
generations, it will be difficult to apply the concept of “lifetime” used for nuclear family in
developed countries. Deaton (1989) calls it a “dynastic” household, and assumes that the

household lives forever.

3.3 A Sequential Decision Model

In this section sequential decisions shown in Figure 3 are incorporated in the dynamic
model. Now the dry season is called “season 1” and the rainy season is called “season 2”.
At the beginning of season 1, the household’s decision is based on its initial wealth consisting
of harvest of year t-l, saving and exogenous income, and at the beginning of season 2, its
decision is based on the outcome of off-farm activities in season 1. Since there is uncertainty
in off-farm income of season 1, the household will be better off if it makes decisions for
season 2 with the information on season 1. The sequential decision model is set up as follows:
First, the optimal decision rule for season 2 is found in short-run utility maximization. This is
the two-period production/saving problem. Second, given the optimal decision rule for season

2, a standard dynamic optimization is solved for off-farm activities in season 1.

3.3. l The First Order Conditions

A standard intertemporal utility maximization problem given by a time-separable
lifetime utility function (the same as (15)) is applied. But in addition, utility for each season is
assumed to be additively separable, that is, u(c,, 1,) - u(cu, 1,,) + 6u(c,,, 12,) , where c,, and
1,, are consumption and leisure in season I of year t, and c,, and 1,, are consumption and
leisure in season 2 year t.

In season 2, the household is assumed to solve two-period production/saving problem
given the outcome of off-farm activities in season 1.

max u(c,,, l,,) + 6E,,v(w,,,,) (40)

The initial wealth of season 2 is given by w,, -(1+ r,,)3,,, which is the sum of saving and its
interest in season 1. Initial wealth and the time endowment are allocated respectively as
follows:

Wu-C2,+K;,+K;,+02,L:,+Sz,- N2,+L:,I12,+L:,+L;, (41)

l + b '
where K; and K;, are purchased variable input for crop production and off -farm activities

respectively in season 2 of year t, 1;, and 1;, are labor input for crop production and off-

87

farm activities respectively in season 2 of year t, L2, is hired labor for crop production in
season 2 of year t, 3,, is saving and b,, is internal borrowing as discussed in section 3.2.

Hired labor is valued at exogenous market wage, a,,, known to the household when the

decision is made. Then, at the beginning of the season I of the next year, the household
wealth becomes

wlm ' Q(K;’ K20" L1,, L1" [in D,,, R20 X,, 52:) '1' (1+ rzt)32r " by 4’ 57ml (42)
where Q is crop/off-farm income function in season 2. Crop income is assumed to depend on
stochastic rainfall, R“, and off-farm income is assumed to be independent of rainfall but has
an idiosyncratic risk, 82,. D,, is crop diversification index as defined in section 3.2, which

affects only crop production. 9",, is the stochastic, exogenous income known to the

household at the beginning of season I in year t+l. Its properties are as discussed in section

3.2.
The first order conditions are obtained by differentiating (37) with respect to 92,, a

vector of the decision variables of this problem: 6,, - (KL. Kin L2. 12,. 12,. D,,, 32,, c,,).

 

 

 

6'“ 3Q
K" : 43
" ac: W 6K; ( ’
o 614 69
K ,: , -0 44
2 68—2, “s—h’ aw,,,, 6K° ( )
a 0"" 3V 5Q
1,: -——+6E —-—- (45)
' all! ”a WIMI 61::
",: -a,,fl+-a—u+ 6E,—— 6v ‘7?- (46)
662, ‘91:: :Qawrm 611:
6u 66v
1;: ——+6E,, -——— -0 (47)
‘ all: win] 61;:
6v _6__Q_
D :
" E"— aw... all. (48’
32,: - jg- + (I + r,, )6E,, - O (49)
*2! Int
1 6u
b,: ——-6£,,-iv— -0 (50)

' 1 + b 6cm 6w,,,,

88

Those first order conditions (43)-(50) are very similar to the first order conditions
(22)-(29), and the discussion on the first order conditions derived from the simultaneous
decision model in section 3.2 is valid (I will not repeat it here).

From (43) to (50), the optimal level of the decision variables can be written as a

function of wealth at the beginning of season 2, w,,, the interest rate, r,,, the market wage,
a,,, household characteristics, X, , and distribution of stochastic variables, R“, 82,, and y,,,,.

Those variables, except for w,,, are exogenously given. The household’s expectation about

the levels of the exogenous stochastic variables at the beginning of season 2 of year t is the

same as that of the beginning of season 1 because no new information about those stochastic

variables is given at the beginning of season 2. Thus, the optimal 6,, will depend on the

household’s decision in season 1 through w,,.

The multiperiod utility maximization problem can be written as follows:
V<w..> - megx u(c... I.) + 6a.[u<c;.. 1.2) + 6v((1 + r..)s.‘. + Q‘ - 12;. mm]

where c,, -w,,-K,°,—s,, (51)
In ' Nu ' L1:
9,, - (S,,, K,°,, 1.7,) is a vector of decision variables in season 1, and superscript * means that
those variables are determined by w,, according to the optimal decision rule implied by the
first order conditions (43)-(50). w,, is determined by decision variables in season I, but it is
uncertain due to idiosyncratic risk:

w,, - (1 + r,,)3,, + F(Kf,, L’,’,, X,, 8") (52)
where F is off-farm income function in season 1. Thus, the household’s problem is to
allocate its initial wealth, w,,, among consumption in season 1, c,,, purchased variable input
for off-farm activities in season l, K,",, and saving in season I, 3,,, and to allocate its labor
endowment, Nu, between leisure in season I, 1,,, and labor input for off-farm activities in
season i, L‘,’,, to maximize equation (51).

The first order conditions with respect to 6,, can be simplified using the short-run

first order conditions (43)-(50) as follows:

89

              

 

6a 2 6w
: —— 6 l r v'-1‘—”--O (53)
6a 6F 6w
K”: -—+6 l+r ——-¥-"--0 54
1. ac" ( ">129" 6K” aw“ ( )
L7,: 611 2 u r a€fl&L_0 (55)
6’1! all! a”,

The household does not know the interest rate in season 2, r,, , and the hired labor wage rate

in season 2, dz, , at the beginning of season 1: they are assumed to be known at the beginning

of season 2. Therefore, in addition to rainfall risk and idiosyncratic risk in season 2 and
idiosyncratic risk in season 1, there are interest risk and hired wage risk for the household

who makes decisions at the beginning of season 1 in the optimization problem above.

3.3.2 Effects of Increasing Risk

From the first order conditions (53)-(55), the effect of increasing risk is examined
following the RS method. The second derivatives from the first order conditions are shown in
Appendix D.

The results depend on L631”, which is given by

2!

6w,,,, - 6Q 6K“ +_6__Q 65,+ _6__Q 6K° +_6__Q 65,... 6_Q_ 6B, +(1+ r106"?! - 6b,,
6W2, 6K; 6W,,+ 61;,6 ”'2: +7“: 6W1, +01%”? “’2: +‘9D2r 5W2: 6W2: 6W2,

(56)
whose sign is likely positive, but can be negative: the sign depends on the household’s
marginal propensity to save, to consume, and to invest out of w,, , as well as on the marginal
productivities of the investment. It can be negative, for example, when a household has very
low w,, due to drought in previous year and/or a failure in off-farm activity in dry season of

W . . . . . .
current year. Here, however, —‘-‘=‘- is assumed to be positive for srmplrcrty. Moreover, we
2:

need to specify whether this positive marginal effect is risk-increasing or risk-decreasing as

done before. Here, its is assumed to be risk-decreasing against all risk.

90

Given the assumptions above, effect of an increase in the following risks are
examined: (1) rainfall risk in crop production in season 2 under imperfect social insurance,
Ru; (2) R, under perfect insurance; (3) idiosyncratic risk in off-farm activities in season I,
8'”; (4) exogenous income risk, or risk in social insurance, in“; (5) interest rate risk in
season 2, r,,; (6) wage rate risk in season 2, a,,; (7) idiosyncratic risk in off-farm activities
season 2, 82,.

The results, except for (2) rainfall risk under perfect social insurance, (3)
idiosyncratic risk in off-farm activities in season 1, and (6) wage rate risk are: (i) saving in
season 1 increases when risks increase; (ii) off-farm inputs (capital and labor) in season I
increase when risks increase and when inputs are less the optimum. The results (i)-(ii) mean
that those risks have the same effect as the risk of future exogenous income. As for (3)
idiosyncratic risk in off-farm activities in season 1: (iii) saving in season 1 increases; (iv) off-
farm inputs increase if they are risk-decreasing. Otherwise, the effect on off-farm inputs are
ambiguous. The results (ii) and (iii) are different. As for (2) rainfall risk under perfect
insurance: (vi) an increase in risk has no effect on the household’s behavior in season 1.
Finally as for (6) wage rate risk in season 2: (vii) an increase in risk has an ambiguous effect
on the household’s behavior season 1 under the assumptions given above. The results are

summarized in Table 5.

4 Conclusions

From sections 2.1-2.3, three points are concluded.

First, when there are more than two risks, how they are correlated is an important
determinant of the effects of increasing risk. As shown in Table 3, most results are
ambiguous, but the second order relations determine the effects rather than the first order
relations. This finding is reflected in modeling in section 3.

Second, the effects of increasing risk depend on how the marginal productivity of

inputs responds to change in risk, and whether the input is risk-increasing or risk-decreasing.

91

However, as shown in section 2.2, this makes results ambiguous. Risk-increasing inputs often
show an ambiguous effect because benefit from increasing output and cost from increasing
risk are canceling each other. The conclusion is that the non-linearity of risk response is
important.

Third, introduction of flexibility changes the results. Most results obtained in an
inflexible model in section 2.2 do not hold in a flexible model in section 2.3. The changes
are model-specific, but in general in flexibility models, production risk in the second period
becomes similar to exogenous income risk.

Based on the findings listed above, I develop two structural household models in
section 3 and examine their properties to compare a simultaneous decision model and a
sequential decision model.

First, social insurance is incorporated in the models. As shown in Table 4, there is no

qualitative difference in responses to increasing risk between the case of partial social

- .. 6"
insurance (i-Xgi. <0 and 29-4- fléflo 0) and the case of no social insurance (—XL¢L >0
6R 6R 6R 6R

and £+QH>OL while the case of perfect social insurance (2&1fl<0 and
6R 6R 6R

g + 262.1%- - 0) is obviously different from others. As discussed in section 3.2.5, we see
only a substitution effect in the case of perfect social insurance. Thus, in the case of partial
insurance, we may see an income effect in addition to a substitution effect. Since perfect
insurance is an extreme case, this result predicts that the difference among villages in terms of
risk response will not be qualitative, but quantitatively distributed between perfect and
nothing, that is, the income effect differs.

Second, crop diversification is incorporated in the models explicitly. Crop
diversification is not necessarily a strategy that reduces risk by having a smaller expected
output. Rather, it can be an output-enhancing, risky strategy. This point is treated by

assuming that crop diversification can be either risk-increasing or risk-decreasing. Either case

can be optimal in the dynamic models considered. As shown in Table 4, an increase in

92

rainfall risk has an ambiguous effect on crop diversification. This means that it is ambiguous
which effect dominates, the income effect or the substitution effect. If there is perfect social
insurance, only the substitution effect will exist. In this case, as shown in Table 4, an increase
in rainfall risk decreases risk-increasing crop diversification and increase risk-decreasing crop
diversification, as expected. This finding implies that crop diversification must be treated
carefully in empirical studies. Otherwise, empirical studies will show mixed, ambiguous
results.

Third, I compare a simultaneous decision model with a sequential decision model.
There are a few, but important differences summarized in three points as follows.

(1) One important finding is that in the sequential decision model current period risk
in off-farm activities, 8”, and future period risk in off-farm activities, 82,, have different
effects on off—farm input decisions. The effects of the current risk depend on whether the
input is risk-increasing or risk-decreasing like the case of risk in crop production. On the
other hand, the effects of future risk do not depend on the technology because risk affects
current decisions just like exogenous income risk does. Therefore, the effects are the same as
exogenous income risk effects. Since off-farm activities are conducted mainly in season I, the
ambiguity of the effects of idiosyncratic risk in season I is important. This risk may
discourage off-farm activities, even if there is income risk in the future. This point can be
applied to saving as well, if saving is assumed to be risky. In fact, that saving is riskless may be
a strong assumption, especially if we assume crop stocks and livestock holdings as a substitute
for saving in the context of the WASAT, even though it may be true that such savings are
relatively riskless compared with other risky investment. Then, if saving is risky, the non-
linear relationship between risk and return will become important In the WASAT, livestock
holdings may be risk-increasing or risk-decreasing, which determines the effect of increasing
risk as shown previous sections.

6w
(2) In the sequential decision model, the sign of 74w“ and its marginal response
2:

against risks are important to examine the effect of uncertainty, as discussed in section 3.3.2.

93

6w
In Table 5, —3'-"- is assumed to be positive for simplicity, but it can be negative as argued,
21

, w
and therefore in empirical studies it must be estimated. Moreover, the magnitude of —‘-'-’l

2:
will also crucial to understand the household’s behavior under uncertainty.

(3) An advantage of the sequential decision model is that we can treat exogenous
variables known at the beginning season 2 as risky variables at the beginning of season I
when decisions for season 1 are made. In section 3.2.5, the interest rate and the wage rate of
hired labor in season 2 are treated as risky variables to examine the effect of increasing risk
of those variables on the decisions in season 1. Although the effect of wage rate risk is
ambiguous, the effect of interest rate risk is found to be the same as exogenous income risk.

(4) I conclude that the sequential model is better although its empirical estimation

requires more data.

94

References

Fafchamps, Marcel, “Sequential Labor Decisions Under Uncertainty: An estimable
Household Model of West-African Farmers,” Econometrica, 61: 1173-1197, 5, 1993.

Roe, Terry and Theodore Graham-Tomasi, “Yield Risk in a Dynamic Model of the
Agricultural Household,” In: Inderjit Singh, Lyn Squire, and John Strauss, eds.,
Agricultural Household Models, Baltimore, MD: The Johns Hopkins University Press,
1986.

Rose, Elaina, “Ex Ante and Ex Post Labor Supply Responses to Risk in a Low Income
Area,” Mimeo, Department of Economics, University of Pennsylvania, 1992.

Rothschild, Michael and Joseph E. Stiglitz, “Increasing Risk II: Its Economic
Consequences,” Journal of Economic Theory, 3: 66-84, 1971.

Sakurai, Takeshi, “Effect of Crop Diversification on Drought Shock in Burkina Faso,” Draft,
Michigan State University, 1993.

Sandmo, Agnar, “Portfolio Choice in a Theory of Saving,” Swedish Journal of Economics,
2: 106-122, 1968.

Sandmo, Agnar, “The Effect of Uncertainty on Saving Decisions,” Review of Economic
Studies, 37: 353-360, 1970.

95

Table 1 Relationship Between f and y,

 

 

 

Correlation Case

. i < 0 if— < O 2:; < O 'a—f'i: < O
Negatlve 1 a; r 6512 r 6;; 9 65,22
6y 6? 62" 627

' —-l<0,—<0,-—-3->0,-—_->0
We 2 6f 3;, 6r: ay,’
6; ar a" 5‘?

' ' 4 O r — > O r T1 > O, T < O
Posrtlve 3 6? 65,2 6r2 6y:

p " 4 ay,>0 £90 2:3-<0 Pit->0
OSItIVG a- r 65’: r 3;: r 85’:

 

Gmlnforeachcasearepresentedinfigurc 1.

 

Figure 1 Relationship Between i and 52,

Case 1 Case 2

Ni
V

 

 

 

 

:gl
:.<

Case 3 Case 4

VI
‘8

 

 

 

 

:gl
:5

 

 

 

 

 

Table 2 Effect of lncreasLnQ Risk on Decisions (uncorrelated)
V"' > 0 v"' < 0
Source of Risk 2V" + a(1 + F)v"' > 0 2v" + a(1+ f)v"' < 0 2v" 4» a(l + f)v"' < 0
Income ( g) m1 m 1
0‘1 a1
61* C,
Interest (F) m1 m1 ml
01 01 al
C11 Cr 1? C, 1

 

 

 

 

Note : m is riskless asset, a is risky asset, and C, is consumption in the first period.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Table 3 Effect of lncreasigg Risk on Decisions (correlated)
v"' > 0
CASE 1 CASE 2 CASE 3 CASE 4
Source of Risk (negafiVC) (negative) (Positive) (Positive)
lncome(y,) m1 m1? m1 m1?
a1? a1? a1? a1?
8,1? 8,1? 8,1? 8,1?
Interest (F) m1 m1? m1? ml
at?“ a1? a1? at?‘
C111,] C,$? C,I? 6:3?!
v'" < 0
CASE 1 CASE 2 CASE 3 CASE 4
Source of Risk (negative) (negative) (Positive) (Positive)
lncome(y,) m1? ml, m1? m1
a1? a1? a1? a1?
8,1? 8,1? 8,1? 8,1?
Interest (5") m1? m1 m1 m1?
a 1 ? a i ?’ a l a 1 ?
8,1? 611?: 8,1‘ 8,1?
Note: m is riskless asset. a isriskyasset,and c, is consrnnptioninthefirst period.
Note I:When g-Z-«l-a < 0 holds, a 1’ andasaresult €11 .Otherwise,theyaremnbigmus.

6
Note2:When §Z+a>0holds,al mdasaresult C;1.Otherwise,theymeunbiguous.
T

 

 

Figure 2 Time Frame of Simultaneous Decision Making

 

 

 

harvest year t
t- 1 -
dry season rainy season “'1
crop production
0”“th work some off-farm work
harvest planting harvest
of year t-i | Of year t

State: initial wealth
-harvest of year t-1+ saving from year t-I +exogenous income

Decisions: on/off-farm activities
wealth-cropping-l-off-farrn activities+saving+consumption
time-cropping+off-farm+leisure

 

 

 

 

100

 

 

State: after harvest

a cropping income
+saving from year t-l
+exogenous income

Decisions: off-farm

wealth-off-farm+saving
+consumption

time-off-farm+leisure

 

 

Figure 3 Time Frame of Sequential Decision Making

 

 

harvest year t
t' 1 - t+1
season 1 (dry) season 2 (rainy)
' crop production
off-farm work some off-farm work
harvest planting harvest
of year t-1 0“ year t

State: after dry season
- saving from season 1

Decisions: on/off-farm
wealth-on-farm+off-farm
+saving+consumption
time-on-farm+off-farm
+leisure

 

 

 

 

101

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Risk-Declarglgml .I. I 1

l: 13:}.

Note: R , rainfall risk; 8, , idiosyncratic risk in off-farm activities; y“, , social insurance risk.

Table 4 Effect of Increasing Risk on Decisions (simultaneous)
Utility Type ~ V"',> 0 ~ ~ V'" < 0 ~
”5" 5”“ Ellgl‘zfill 5‘ 9‘" srgn. Eagle; 5' I 9'“ .3222;
On-Farm Inputs (1,“, K“, L?) ,

Risk-10C.Lmbigu_ousl l l l I 1 1a nous uousl .l t

Risk-Dec.| t 1 t l t l l I m ml .1 t
Off-Farm Inputs (Lj’, K,°)

Risk-Inc. 4 Wm 1‘ noeffect mm .|, noeffect

Risk-Dec. 1‘ i 1 noeffect mam ,l, noeffect
m5 l 1’ 1‘ no effect mmm ,I, no effect
BorrowirLcL ,1 A ,|, no on.“ Mm m 1 no effect
Crop Diversification 7

Risk-Inc. l .3ng t I r I I I

 

J r

 

102

Table 5 Effect of Increasing Risk on Decisions ( uential)

 

 

 

 

 

 

 

 

 

 

1 Z 3 4 5 6 7
Source of Risk R2: 52. 52. 5’... '2. 02. 51.
(perfect
insurance)
, Savings in season 1 1 no effect 1 1 1 arrbjguom i

 

Cap. for Off-Farm

 

mm]; It I; l-m ....1

 

Risk-Increasiry} 1

Risk-Decreasing f

 

noeffectl T I T I T I'm!” 1

 

Labor for Off-farm 7

 

-o

 

 

 

 

 

Risk-Increasing]| 1 Inc effect I 1 I 1 I
Risk-Decreasing 1 I no effect 1 I t I i I mm 1‘
(V"' > 0 only)
Note: a—wfim > 0 is assumed.
68"
62W 63w
Columnlassumes—l—L< 0and-.—w”-L>O.
6R2: 6W2, aRzzxawzt
2
Column3assumes m<0and 2.3—“L>0.
6a,,6w2, 682,6W2,
6w 63w
Column 5 assumes __l.LL <0 and —2”-L > 0.
aruawz, 6r;6w2,
2
Column '7 assumes J—W 0, —-—Q‘I—<0 0. 19— >.0 and $7<0 for risk-
6e,,6K° 661,61? 681,61; 681,611,
2
increasing inputs, and 6 Q' 0, __63_Q|__> 0 —a—Ql—< 0 (1 Q. > 0 for risk-

 

 

... < .. . .. . ..
asuaK; 64.61:; anal; “" am:

decreasing inputs.

103

Appendix A

Labor Income and Interest Rate Uncertainty / Two-Period Model

This is used in section 2.1 to derive the results in Tables 2 and 3.

 

 

From (2),
.. 2 2...
630;, -6(iy}+a) v"'+6-a-:y%v" (A1)
6m67 6r 6r
2
~ 2..
ills-opwl) v" '"+da§7:-v (A2)
61W 2 yz
From (3),
3 ~ ..
all -6(1+r)(-a—y}+a a)zv "'+26(-€2}+a a")v +6(l+r)%v" (A3)
6a67 6r 6r
3 .. 2~ 2
_:_‘.’_-6(1+r)(1+a-:—-y -—:-)v'" +26—(1+a.‘35.)v" +6(l+r)a§:;-v" +6-5-9—jv '
6 “692 6% 2 2
(A4)
When we assume that income risk and interest rate risk are independent, 2x}- - 0 and
r
a’y ar 3’7
_l~2 - O are assumed in (Al) and (A3), and — -O and — - O are assumed In (A2) and
6’ a» ‘9”

(A4). On the other hand, when we assume that they are correlated, there are four cases as

shown in Table l and Figure l.

104

Appendix B

A Dynamic Household Model with Uncertain Production
In this appendix, effects of increasing risk in production are examined in the dynamic
model in section 2.2. Analysis starts from the first order conditions (9), (10), and (11).

(1) Examination of Effects of Risk

From (9)
2
"I 6Q v” _2__aQ
—: .. Bl
v (68,]+ 67,2 ( )
From (10)
2 2 3
v"’p “(2.2) 272+ wig-192nm M 6 Q 6 —Q— ——+ v'——6 Q., (82)
q 68, 6L, 68, 611 q 68, 61,68, 61,68,
And from (11)
2 62 2 3
vmpul( j?) aQ‘.""'a"'--Im-Q2”6Q+2vnpt+1a“'Q_----aQ +v’—a-Q:'7 (B3)
q , 6A, 68, 6A, 4 68 ,6A, 68, 6A, 68,

To evaluate the sign of (BU-(83), we need to make specific assumptions about the

 

 

2
production function. As mentioned in the text, the assumptions are 5Q>0, 3:772 <0,
8 8,
32 3 2 3
TQ->O, 6 Q.., <0, 6 Q. <0,and —a—T;>0.
68, 61., 6L, 68, 6A, 68, 6A, 68,

The sign of (81) is positive when v'" 20 , and ambiguous when v'" <0. The sign of
(82) is ambiguous regardless of the sign of v"'. The sign of (B3) is positive when v"' 20,
and ambiguous when v'" <0.

If both inputs are risk-neutral, (82) and (83) will become the same as (Bl), and
consequently the results will be that the sign of (82) and (83) is positive when v'" z 0, and

ambiguous when v"' <0.

(2) Proof of “labor is underused”

From (9) and (10), the following is obtained:

105

. 6Q . 6Q ,
pq,,,E,v E, E + p,,,,cov (v , 31—10-“ + r)b,E,v - 0 (B4)
In (84), the covariance is negative when labor is risk-increasing input. Therefore, form (84)
6Q
quE: 32' " (1 + r)br > 0 (BS)

is obtained. This means that labor is underused at the optimum relative to the optimum when

labor is risk-neutral (that is, the covariance is zero).

(3) Proof of “land is overused”

From (9) and (l 1), the following is obtained:

pq,+1£;V'E, 3% + pqr+l oov (V's %) - (l + r)brErv' - 0 (B6)

In (86), the covariance is positive when land is risk-decreasing input. Therefore, from (86)
6Q
qulEtE-(l +790, <0 (B7)

is obtained, This means that land is overused at the optimum relative to the optimum when

land is risk-neutral (that is, the covariance is zero).

 

106

Appendix C

A Sequential Decision Model with Flexibility
Proofs for the flexibility model in section 2.3 are below.
(1) Proofs of (i) and (ii)
From equations (13) and (14),
Ev'(EF,-(1+r)w,)+cov(v’, F,)- 0 (Cl)
is obtained. In (Cl), (i) when the covariance is negative, EF, -(1 + r) wI > 0 holds. That is,
on-farm labor in the first period is underused; (ii) when the covariance is positive,

EF, -(1 + r) w, <0 holds. That is, on-farm labor in the first period is overused. Thus, we

F,
need to determine the sign of the covariance, which depends on the signs of 33:2; and 33'
From 3% with F2 - w,, the following is obtained:
6_v’ 6F 6L'+ 65 "(6_F_+
-'v +—3- F,-—3- - - c2
a—R (aR aRL‘ aR W261?) aR 3R4) ( )

6v’

Therefore, the sign of -=- negative. This means that total income is high when rainfall is
6R

. . 6F . . .
high. Thus, (1) when fiL > O, the covariance rs negative, and consequently on-farm labor in
. . .. 6F . . .
the first period rs underused; (u) when 3;;- < 0, the covariance rs positive, and consequently
F
on-farm labor in the first period is overused. Moreover, $7;- > 0 implies that on-farm labor
in the first perrod rs risk-increasing, while g7;- < 0 implies that on-farm labor in the first

period is risk-decreasing.

(2) Proofs of (iii)

Differentiating the first order condition (13) with respect to 1% twice yields

a 2
vm(_ch_)2 + V" 8;;22 (C3)

aR aR aR ’aR‘W2 aR aR aR

107

62c, 62F 62w 61:34”; awgg (C4)
(3R2 3R2 aR aR

62c
is obtained. In (C4), the sign of 6% is ambiguous a priori. Therefore, the sign of if is
. . . . . all, .
ambiguous, and consequently the srgn of (C3) IS also ambiguous. However, if —61i rs

62c
negative, the sign of 3523- becomes negative, and consequently: if v"' 2 O , the sign of (C3) is

positive; if v'" < 0 , the sign of (C3) is ambiguous.

(3) Proofs of (iv)

Differentiating the first order condition (14) with respect to F twice yields

68 6c 6F 62c 6’F
-v’" —.-§-2F —2v'—1—l—v"—51F - '—l c5
(6R)F‘ 3R aR aR‘ v (9R2 ( )

6c
The sign of (C5) is ambiguous because the sign of 'a—Rzl is ambiguous as discussed above.

6 6 c
However, if 61? is negative, the sign of a—R} becomes negative, and consequently: when

6F 62F
v” '> O, and 51- < O and 31-; > O (on-farm labor m the first period is risk-decreasing and

its effect is diminishing), the sign of (C3) is negative; but other combinations of the signs give

ambiguous results.

Appendix D

Effects of Increasing Risk on Decisions in Sequential Decision Model

From the first order conditions, (53)-(55) effect of increasing risk can be examined

by differentiating twice with respect to stochastic variables following the RS method. The

derivatives are as follows.

 

 

 

 

 

From (53):
3
E“: WI( %2 )2 awri+1+vnwu+1 n+1 +2V" 2+1w2+1+IV {:v2q
0R2 5W2: 3R2. aRziaWz. aRziaw
(DI)
£22: VM(_a_“:-__lt+1)2 0W“, +1 +vH WH+ 1 2+ 1 +2V" 2+1 Wfl+| I €:w2+|
68,, 6w2, 68,, 68,,6w, 682,6W2,
(DZ)
.. ,,, 6w ,
Yrur: V _li_l. (D3)
6w”
T2,: V"'(Llul )2— awlr+1+ _1_awr+l)a w+12 (D4)
6r2, 6w”
an; Vm(._int)ziw_ti_t V~__tht_ini. 2V~_u+_1_ini_+yr.fl‘.’tni.
6a,, 6w” 003.5%.
(DS)
3
g"; Vm(__int)=§‘_".u V'a—lm_.m 2V~_ini._~_ini_ V izflni.
68,, 6w” 68,, 68,,6w1, 68,,6w,,
(D6)
Similarly from (54):
11' mm “W (D7)
1' 3K;
.. 6F
8 : D2 * D8
- 6F
: D3 * D9
Yrur ( ) 6K; ( )
6F
r : D4 * D10
1! ( ) 6K; ( )
6F
“2:3 U35)" (D11)

3K3

109

 

 

 

 

2
5,: (D6)* ”Z, +2(v"iw:ma—WJ'—*l+v'—a-‘3P—:L)+v'iW-M-—aaf?(mz)
3K1: 68,, 6W2, awhaslt 6W2, 6K;68,,
Finally from (55):
11' (Dl)* 3F (D13)
1' 61:,
£,,: (my-i;- (D14)
«9L.
.. 6F
Yrur: (D3)* 61.: (D15)
r,,: (D4) * 2; (D16)
«9L,
6F
0 : D * D17
2. ( 5) 61:, ( )
2
5,; (D6)* a1: ”(WEE-Law“ +V’—"—”—’-‘m)+V'iW-y—+L 63F (1)13)

 

 

al.. 62.. awn awnaéi awz. aliaéi.

Chapter 4

Equity and Sustainable Management of
Common-Pool Resources:
The Case of Grazing Land in the West Africa Semi-arid Tropics

0 Summary

In the West Africa Semi-arid Tropics (WASAT), it is common to find sedentary
settlement mixed crop and livestock system of production. Livestock is used as a major form
of saving because there are no formal saving or credit institutions. Since crop production is
uncertain due to frequent drought, households tend to keep livestock to insure income.
Livestock is kept on open—access grazing land, and overgrazing occurs due to lack of
governing rules. It is predicted that degradation of grazing land due to overgrazing will
decrease productivity of livestock, and consequently households will lose both income and
their ability to smooth income, which is the basis of households’ sustainability. However, few
studies have so far tried to analyze grazing land management in terms of households’
behavior under uncertainty. In this chapter, the decline of livestock productivity is assumed to
occur by a simplified mechanism: given fixed grazing land, the more livestock, the less grass
available per animal, and as a result the slower animal growth (lower livestock productivity).
Degradation of grazing land will further decrease the amount of grass per an animal, while an
introduction of feed-efficient animals will increase livestock productivity.

This chapter addresses four main questions: (1) How do WASAT households behave
under uncertainty, especially under uncertainty in agricultural production? (2) Why is there
overgrazing? (3) What property regimes would solve overgrazing? (4) What would be the
equity consequence of these regimes?

110

111

To answer these questions, a portfolio selection model that explains household’s
behavior under uncertainty is developed. The model is different from previous studies on
resource management in two points: (1) Each household faces two investment opportunities:
risky crop production and safe livestock holdings. (2) Households in a village are
heterogeneous in wealth and as a result heterogeneous in risk preference. Prices are not
allowed to change in this analysis. Optimal investments are derived from the model under an
open-access regime as a Nash-Coumot equilibrium.

The optimal livestock holding under an open-access regime is compared with the
social optimal livestock holding, where economic overgrazing is defined as a case when the
open-access optimum is greater than the social optimum. Then, the relationship between
wealth distribution in a village and the degree of overgrazing is examined by comparing two
villages with the same aggregated wealth, the same number of households, but a different
distribution of household wealth.

The following are the key findings from these analyses: (1) A household’s optimal
livestock holding is high when crop production risk is high and/or when the household’s
initial wealth is high; (2) Overgrazing is inevitable when the expected return to livestock
holdings is greater than the expected return to crop production; (3) Overgrazing does not
necessarily occur when the expected return to crop production is greater than the expected
return to livestock holdings. Crop production risk must reach a threshold level for
overgrazing to occur; (4) The degree of overgrazing increases when household wealth in the
village approaches equality keeping the average wealth constant In other words, the social
gain by moving from the household optimum to the social optimum is large when household
wealth is distributed equally. In this analysis wealth distribution, not wealth level determines
the effect. But this is true when household wealth is above a certain level; (5) Moving from
the household optimum to the social optimum is not necessarily a Pareto improvement, that is,
very rich and very poor households are likely to be worse off absolutely at the socially

optimal equilibrium; (6) Although overgrazing can be severe in a village with relatively equal

112

wealth distribution, a sustainable common property management rule will be established more
easily than in a heterogeneous village, according to conclusion (4) and (5).

Finally, using the model, several institutions for grazing land are examined. In these
analyses, households are assumed to be in the household optimum under an open access
regime, and welfare effects of institutions are considered. The findings are : (l) Privatization
will reduce overgrazing, but wealthier households are more likely to be better off absolutely
by privatization; (2) A quota will reduce overgrazing, but its welfare effect is ambiguous
although a Pareto improvement may be possible; (3) Transferable permits and user fees will
also reduce overgrazing, but their welfare effect for individual households is ambiguous; (4)
The feasibility of these institutions depends on transaction costs, monitoring costs, and
enforcing costs, which is because the grazing land is a common-pool resource. Together with
conclusion (5) in the previous paragraph, if village wealth is more equally distributed,
enforcement costs will be lower and consequently institutions, such as quotas, will be more
feasible; (5) Increase in livestock productivity (by conservation of grazing land and/or
improvement of animals) will be difficult due to free-riding and will not reduce overgrazing,.
But increase in crop and off-farm productivity (the expected return to them) and/or decrease
in their risk could reduce overgrazing by decreasing demand for livestock.

The last result above predicts that off-farm income will play an important role in
determining demand for livestock holdings, considering the difficulty in increasing
productivity of crop production However, the result is true only if a household’s wealth does
not change by an increase in productivity of off-farm income and/or an decrease in its risk. If
a household’s wealth increases by them, their effect on livestock holding is ambiguous
because increasing off-farm income decreases demand for precautionary demand for
livestock holdings on the one hand, but increasing household wealth increases the optimal
livestock holding on the other hand Therefore, risk in on- and off-farm activities needs to be
so small that households will invest in them and will reduce livestock holding in spite of

increasing total wealth.

113

1 Introduction

Livestock has an important role in many agricultural systems in rural areas of
developing countries. In the West Africa Semi-arid Tropics (W ASAT) especially, it is used not
only for animal traction but also as a major form of saving because there are no formal
saving or credit institutions. Since crop production is uncertain due to frequent drought in the
WASAT, households tend to keep livestock to insure income (Reardon et al., 1988).
Although livestock is important for income smoothing, not all households can accumulate
enough livestock to cope with income shortfall due to drought. Consequently, livestock
income makes income distribution more unequal (Reardon and Taylor, 1994).

The WASAT livestock population has increased over the last three decades, mainly
because of increasing human population'. Due to the sharp increase in stock numbers, in
conjunction with prolonged drought, the Sahel grasslands have undergone extremely severe
degradation. In northern Burkina Faso, the long-term carrying capacity of livestock of
rangelands is estimated at 9 hectares/TLU (tropical livestock unit), or 28 kg live weight per
hectare, while the actual stocking rate averages 6 hectares/TLU, that is, 33% overstocking. In
the Sahel as a whole, although carrying capacity varies, about 30% overstocking isestimated
(Houerou, 1989).

That grazing lands are “open access” has been an economic explanation of
overgrazing since the publication of Hardin’s “Tragedy of the Commons”: livestock is
private property, but cattle graze on the commons (Hardin, 1968):. “Open access” or “lack

of a communal management scheme for grazing land” was not a problem and even could be

 

‘ Growth rates in stock numbers between 1950 mid 1983 in this region are: cattle (2.52), sheep
(2.67), goats (2.24) and camels (1.90). The rate of demographic growth is 1.5 to 2.5% per annum among
pastoralists and 3.0 to 3.5% per mum among settled framers. These human and livestock growth rates me
very close; and therefore, the human to livestock ratio has remained unchanged from 1950 to 1983, 0.9
'I'LU (tropical livestock unit) per person in the overall population and about one TLU per inhabitant in the
rural population The growth rate of cultivated land is also about 2.5% per annum, which is very close to
the growth rates of both human and livestock population (Houerou, 1989).

2 Iwilldefine“openaccess”insection 2.

114

considered optimal when human population density was low in the WASAT’. Now, although
overgrazing and degradation are common, village rules have not adjusted to the new situation
and allow overgrazing by permitting open access to grazing land by villagers. It is predicted
that degradation of grazing land due to overgrazing will decrease available grass per animal,
and as a result, will lower animal growth rate. Economically, this means that average returns to
livestock holdings (or productivity of livestock) will be lower, and consequently households
will lose both income and their ability to smooth income. Since households’ sustainability‘
depends on their ability to insure consumption by smoothing income, maintaining
productivity of livestock, that is, sustainable management of grazing land, is important for
households’ sustainability in the WASAT.

However,-because productivity of livestock is not the only concern of households in
the WASAT, performance of grazing land management cannot be judged only by livestock
productivity. Even if villagers succeed in establishing new rules to achieve long-run
sustainable use of grazing land, if it reduces their ability to smooth income in the short-run,
such rules may not be adopted. In the short-run, overgrazing may be even better for them
than other welfare loss, such as severe income shortfall in drought years. Therefore, we should
not analyze the livestock husbandry independently of farm management of rural households
that are risk averse, face several income opportunities, and are probably under borrowing
constraints. However, few studies have so far tried to analyze grazing land management in
terms of households’ behavior under uncertainty.

Thus, this chapter addresses four main questions: (1) How do WASAT households

behave under uncertainty, especially under uncertainty in agricultural production? (2) Why is

 

3 Population densities in this region are rather low in absolute terms, but they have sharply
increased over the last four decades as the following statistics show: for example, Burkina Faso, 12.7
(1950), 19.8 (1970), 273 (1983), 32.8 (1990); Mali, 2.1 (1950), 2.8 (1970), 4.6 (1983), 6.9 (1990);
Niger, 1.7 (1950), 3.2 (1970), 4.6 (1983), 6.1 (1990); Senegd, 2.7 (1950), 21.8 (1970), 31.3 (1983), 37.5
(1990). (in inhabitants per kmz, from several sources)

“ Sustainability at the household level is defined as the ability of the household to smooth
consumption over time (see chapter 1).

115
there overgrazing? (3) What property regimes would solve overgrazing? (4) What would be
the equity consequence of these regimes? To answer these questions, a model that explains
household’s behavior under uncertainty is developed in section 3. In the model, a household
faces two investment opportunities: risky crop production and safe livestock holdings. The
optimal investments are derived from the model under an Open-access regime. In section 4,
using the model, several types of institutions for grazing land supported by other property
right regimes than open access, such as privatization, quotas, user fees, and transferable

permits are compared in terms of efficiency and equity.

2 Definitions and Approach
2. 1 Definitions

When we analyze resource management, it is important to distinguish physical
attributes of a resource and human institutions that govern its use. Schmid (1987) uses the
terms “situation” and “structure” to distinguish property right regimes from the resource
itself. The “situation” is the set of attributes of the resource and the “structure” is a rights
regime assigned to it. Following this definition, physical attributes of grazing land in the
WASAT are described first, and then rights regimes are considered.

Grazing land in the WASAT is interspersed with other farm land distant from where
villagers live. Villagers keep their privately-owned livestock on the grazing land. Exclusion of
others from using a particular spot of the grazing land is inherently difficult because fencing
and monitoring are costly relative to the gains from exclusion. Moreover, quality of grass is
not homogeneous, especially when a drought occurs. Therefore, considering drought risk, the
gain from exclusion is very low or negative. Schmid (1987) classifies this kind of situation as
a high-exclusion-cost good. On the other hand, since livestock is privately owned,
consumption of grass by the livestock of a villager reduces the availability of grass for others.
That is, there is rivalry or subtractability in use of the grazing land among villagers. Schmid

( 1987) classifies this kind of situation as an incompatible-use good In sum, grazing land in

116

the WASAT has two attributes: difficult exclusion and high subtractability. Ostrom el al.
(1994) classify goods that have those two attributes as common-pool resources or CPRs, and
call a case where a CPR is involved a CPR situation. In this chapter I use the term “common-
pool resource” for grazing land in the WASAT according to this definition.

However, CPR situations are not necessarily tragic: when the quantity demanded of
the resource is not sufficiently large, individual strategies do not produce suboptimal
outcomes, and only when CPR situations produce suboptimal outcomes, is it called a CPR
dilemma (Ostrom et al. 1994). This distinction between CPR situations and CPR dilemmas
implies that although grazing land in the WASAT is a common-me resource, it is not
necessarily destructively used.

Now, given the physical attributes, we will consider human institutions that govern the
use of the grazing land. Berkes and Farvar (1988) distinguishes a common-property resource
(the same as a common-pool resource as defined above) from a common-property regime,
and state that common-property resources may be held within: (1) open access; (2)
communal property; (3) state property; (4) private property. In addition, Stevenson (1991)
divides open access into two categories: open access to limited users and open access to
unlimited users. Resources open to only members are not common property, but open access
(the former case) by his definition because members can extract unlimitedly. On the other
hand, common property by his definition limits both users and amount of use by rules. As
Bromley (1991) points out, “common property” has been often used confusingly for
common-pool resources that are open access. But here, following Bromley, I restrict the term
common property to express a regime of common-pool resources that are communally
owned and managed, distinguishing it from open-access property, state property, and private
property.

To characterize the current regime of grazing land in the WASAT in general, “open
access limited to villagers” will be the most appropriate. There is no state-owned or privately-

owned grazing land. Grazing lands that are communally owned and managed may possibly

117

exist, but they are not many and they do not reduce the importance of analyzing the open-
access grazing lands in the WASAT. Therefore, in this chapter, grazing land is assumed to be
open access only to villagers. This means that every villager can use the grazing land as much
as he/she likes, but grazing land is open only to villagers-1.

Thus, the questions can be restated: ( 1) Are there CPR dilemmas? (2) Are there
institutionally feasible alternatives to improve the dilemma outcomes? (3) Are there any other
potential solutions?

First, as discussed above, if the quantity demanded of grazing land is not large, there
will be no dilemma. This may be obvious, but there has been little analysis on when the
demand is low enough to avoid the dilemma. Therefore, this is one of the questions to be
answered in section 3. Second, as classified by Berkes and Farvar (1988), there are three other
property right regimes for common-pool resources: communal property, state property, and
private property. In section 4, the performance of those regimes is predicted using the model
presented in section 3. Privatization, quota, transferable permit, and user fee schemes are
considered as institutional solutions assuming communal property or state property. These
institutions are concerned with how to control the use of grazing land, that is, how to assign
and enforce a quota to households. A user fee scheme controls a household’s livestock
holdings by adjusting the fee rate without assigning a quota. But if communication is allowed
among households, they may be able to reach a cooperative solution to implement a quota.
On the other hand, a transferable permit scheme is a non-cooperative solution that depends
on the market to allocate a quota. Finally, as mentioned above, reduction of demand for
grazing land will solve the CPR dilemma, even without new institutions. The model will also

tell how such solutions can be obtained.

 

5 There may be conflicts between villages and conflicts between peasants and herders. However, in
thispaperlfocusontheissueofresourcemanagementwithinavillagewhereallhouseholdsareengagedin
bothcropproductionandanimal husbandry. Theyarecommonin theWASAT.

118
2.2 Approach
2.2.1 Relationship to Previous Work

This chapter follows previous work on open-access resource management in that I
compare the open-access equilibrium with the socially optimal equilibrium and examine
effects of institutions on the open-access equilibrium, as reviewed next. However, there are
fundamental differences: (1) Households in my model face several income opportunities; (2)
Households are heterogeneous in wealth and as a result differs in risk preferences. These two
are main contributions of this chapter to the literature on resource management.

Resource management under open access (sometimes called common property, as
discussed above) has been well examined since Gordon’s static model of fishery management
(1954). He compares a socially optimal monopoly solution with an uncontrolled competitive
equilibrium in which the monopolist’s rent is dissipated, and concludes that the uncontrolled
equilibrium means a higher expenditure of effort, higher fish landings, and a lower
continuing fish population than in the socially optimal equilibrium. More generally,
exploiters in the uncontrolled competitive equilibrium equate the average product of the
variable input to the input’s real rental rate, while a single exploiter equates the marginal
product of the variable input to its real rental rate (Haveman, 1973; Weitzman, 1974).

Since Gordon, dynamics and uncertainty have been incorporated into the model.
Examples of dynamic models are Brown (1974), Clark and Munro (1975), and Dasgupta and
Heal (1979). In dynamic models, the discount rate is one of the important factors, while in
static models the discount rate is assumed to be zero.

However, here a static model under uncertainty is considered. Andersen (1982)
introduces price uncertainty in a static model of competitive commercial fishery. His
conclusion is that, for risk averse firms, price uncertainty independent of fishing activity
reduces the number of fishing units and total fishing effort when compared with certainty,
and that the total fishing effort in an open-access fishery under price uncertainty may be

smaller than in a socially optimal fishery if the variance of price is high and/or fishery firms

119

are highly risk averse. That is, over-exploitation under open access does not always hold
under uncertainty. Sandler and Sterbenz (1990) assume harvest uncertainty in a static model
and demonstrate that a fixed number of risk-averse firms faced with harvest uncertainty
owing to resource stock uncertainty reduce their exploitation of commons.

Although cropping/livestock husbandry households and cropping/fishing households
are common in rural areas in developing countries, there are few papers that discuss the issue
of common-pool resource management for those households. Wilson and Thompson (1993)
is one exception. They analyze ejidos, communal land holdings groups in Mexican semi-arid
highlands, producing both crops and livestock. They argue that communal grazing lands
have an advantage over private ownership because communal ownership reduces the
individual’s risk due to uneven distribution of rainfall within an ejido. However, their analysis
does not discuss the relationship between the livestock husbandry and the crop production.
Unlike in Wilson and Thompson, the relationship between these two activities is central to the
present chapter; it is assumed that a household’s uncertainty comes from crop production
and that livestock holdings which are less risky play the role of a hedge against risk.

In addition, most literature on common-me resource management including Wilson
and Thompson assumes homogeneous agents that all have the same wealth. By contrast, my
model assumes that households are heterogeneous in wealth and as a result are heterogeneous
in risk preference. This assumption allows me to analyze the effect of wealth distribution on

overgrazing and the effect of resource management rules on wealth distribution.

2 . 2 . 2 Modeling Strategies

My model is to be fully presented in section 3, but in this sub-section modeling
strategies are briefly explained. My model focuses on both common-pool resource
management and micro-behavior of agricultural households, and its key elements are (1)
major characteristics of WASAT households: uncertainty, risk aversion, and credit constraints,

(2) a standard portfolio selection model under uncertainty, e.g., Newbery and Stiglitz (1981),

120

Robinson and Barry (1987), and Hirshleifer and Riley (1992), (3) Just and Zilberman’s
treatment of heterogeneity of households in wealth and risk preference (1983, 1988a, 1988b),
and (4) Nash-Cournot solution to obtain optimal household behavior under an open-access
regime.

The behavior of each household is represented as a portfolio management problem in
which initial wealth (human and non-human except land) is allocated between risky crop
production and riskless livestock husbandry to maximize household utility. I follow Just and
Zilberman’s portfolio selection model of micro-behavior of agricultural households and their
treatment of heterogeneity of households in wealth and risk preference, where household’s
risk preference is a function of its initial wealth: e. g., the relationship between farm size and
technology adoption (1983), the effects of agricultural policies on income distribution
(1988a), and the effects of environmental policies on income distribution (1988b).

In addition, the subtractability of grazing land, an attribute of common-pool
resources, is incorporated by a livestock productivity function, which depends on total village
livestock holdings. I use a function given by Gibbons. Following Gibbons (1992), I apply a
Nash-Cournot solution to my model to obtain optimal household behavior under an open-
access regime in which each household behaves optimally assuming that other households
make optimal decisions“.

Overgrazing is defined as follows. First, I assume that the total grazing land available
to villagers is constant, livestock is homogeneous, its amount is continuous, and its price is
constant. Consequently investment in livestock (expressed in monetary terms) is proportional
. to the total amount of livestock (expressed in live weight), and since grazing land is constant,
the population density of livestock is proportional to the total investment in livestock.

Therefore, summing optimal livestock holdings over all households in the village is

 

‘ Since a static one-period model is used, Nash-Cournot equilibrium is a one-shot, noncooperative
solutionthatassumes nocommunication. 'l‘heineffrdencyoftheopen-access regimewill bereducedifthe
game is infinitely repeated (Friedman, 1990) and/or communication is dlowed (Ostrom er al., 1994).

12!

considered to express the stocking level of livestock. Second, the socially optimal livestock
holding is obtained as a monopoly solution, or a solution of a so-called social planner.
Finally, by comparing the two stocking levels, overgrazing is defined as a case when livestock
holdings at the open-access equilibrium are higher than at the socially optimal equilibrium.
The relationship between wealth distribution in a village and the degree of
overgrazing is examined by comparing two villages that have the same number of households
and the same aggregated wealth (as a result, average wealth per household is the same)’.
However, wealth is more widely distributed in one village (unequal village) than the other
(equal village). Because this comparison is done by spreading wealth while keeping the mean

constant, I call it “a mean-preserving spread of wealth” in this chapter.

3 The Model
3. 1 Framework

Consider initially a single household i with fixed productive capacity, W“ which is
simply assumed to be the household’s initial wealth except land to be invested in crop
production and livestock holdings. The productive capacity can take any form, either human
or non-human, but it is expressed in monetary terms, that is, the initial wealth is assumed to be

perfectly liquefiable. The household can allocate its initial wealth in any proportion between

them, that is, W, - A, +1... where A, is input for crop production and 1., is livestock
holdings”. A, and 11 are also in monetary terms, and their prices are assumed to be fixed.

Since cr0p production is risky, its return is given by «A, , where it is the stochastic net return

 

" These two conditions are enough when there are so many households that the wealth distribution
can be approximated as being continuous, or when wealth is evenly distributed even if it is discrete. When
wealth is discrete and is distributed unevenly, one more condition as follows is needed to compare wealth
distribution; wealth distribution is spread proportionally to the distance to the mean. In either case, this
analysis is concerned with the wealth distribution, not the level of wealth.

3 A,isnotnecessarilylinritedtoaopproduction,butcmbeconsideredtobesumofallinvestment
opporttmities except livestock. In this cme,A, includes not only crop production, but off -farm activities and
migration,mdbecauseitincludes riskyaopproductionmdbecauseotherincomesom’cecannotemcel the
risk perfectly. Alcan still be considered to berisky investment relative tol+

122

per unit of agricultural investment. Jr has a normal distribution with mean, m , and variance,
v.

In the livestock husbandry, the subtractability of the open-access grazing land is
incorporated following Gibbons (I992). The total livestock holding of the village determines
the level of subtractability of grass, and consequently determines the village productivity of
livestock. On the other hand, household i’s total cost to keep livestock depends only on its

holdings of livestock. The net return to livestock is given by
“(Em-bl. (1)
id
Equation (1) means that if household i decides to keep the livestock level at L, , the value of

the livestock will be [13(2 [1) - bl1 at the end of this period. This production includes both

growth of each animal and net births. The return to livestock husbandry is assumed to be
deterministic, for simplicity’. b is the cost per unit of livestock investment (variable cost of
keeping animals), and g( ) is the village productivity function. This productivity function
implies that the household’s return to livestock husbandry depends on the total livestock
holdings of the village. Since village grazing land is fixed, increasing total livestock in the
village decreases livestock productivity due to congestion, that is, g' < 0 , and the decrease in
productivity is accelerated by increasing total livestock due to degradation, that is, g" < 0. As

mentioned above, a fixed livestock price is assumed, so 11 and [13(2 11) - bl1 can also be
is

interpreted as the amount of livestock. Livestock holdings are similar to formal saving with
interest However, this interest rate is not exogenous, but is determined by the aggregation of
each household’s livestock holdings, that is, this interest rate is endogenous.

Thus, household i’s net wealth at the end of the period is given by

 

Y.- - M. +Lg<;A)-bl. <2)
9 Even if the return to livestock husbandry is stochastic, and has some positive correlation with crop

production, the following model will notchrmge much ifcropproduction is more risky thm livestock.

173
Because .1: is normally distributed, Y, is normally distributed. The household maximizes

expected utility of Y,. Except for the endogenous interest rate of livestock holding, the

maximization problem is the same as a static portfolio problem with a riskless asset. The

solution to this problem depends on the form of the utility function (Newbery and Stiglitz,

1981), but since Yi is assumed to be normally distributed, the utility function can be treated
as a function only of the mean and standard deviation of Y, (Hirshleifer and Riley, 1992).

Thus, maximizing the expected utility of Y. is equivalent to maximizing the certainty-

equivalent Y.- given by
l
CE(Y,) - EY, - 543,. var(Y,)

,, 1 (3)
- m. + “(21.) - bl. - am:

where m and v are mean and variance of Jr respectively, and d, is the coefficient of absolute

risk aversion at the expected income. The use of certainty-equivalent income rather than

expected utility provides analytic advantages (Robinson and Barry, 1987).
U "(Y0
WW9

 

The coefficient of absolute risk aversion, A, is given by - . Therefore,

household i’s d, is a function of household i’s expected final wealth. However, to obtain an
explicit solution, 4:, is assumed to be constant for each household in this chapter, that is, each

household has constant absolute risk aversion. Such a utility function is given by
U(Y,) - -ke""‘, where k is a constant. It has the standard property of a utility function:
U ' > O and U" < O. For distributional analysis, the household-specific of, is assumed to be a
function of the household’s initial wealth following Just and Zilberman (1988a; 1988b).
This means that in a cross-sectional comparison of households in the village, 4t, is given by
¢r - “W9 (4)

where W, is household i’s initial wealth. Initial wealth can be used here because each
household has constant absolute risk aversion during the single period of the model.
(However, since the household’s initial wealth differs in each period, the household’s absolute

risk aversion is not constant over periods, depending on initial wealth, as expressed in (4).)

124

6
In the following analysis, 33/- < O is assumed, that is, the wealthier the household, the

6‘
less risk averse (Newbery and Stiglitz, 1981). In this case though 3% < O is assumed, 43 must

be positive by definition. Consequently, lim 4: must be zero or some positive constant.

2 (9
Therefore, W > O is required. In addition, increasing relative risk aversion, {-y-‘t— > -1, is

W ¢
assumed, following Arrow (1971). This condition is equivalent to saying that ¢W is an

increasing function of W although 4t is a decreasing function of W. However, the sign of
__a‘<¢W> - 22g
aw" aw

 

a"2 .
+ W av; is still ambiguous because my only assumptions concemrng the

6’ 62
relationship between 45 and W are :—;< O and W >0. But now I assume 6(ng < 0.

Based on the discussion above, 4; and ¢W are graphed in Figure l and Figure 2 respectively

in which the following additional assumptions are reflected: lirlrod- 00, lim ¢- 0,

 

limopw -0, lim ¢W - 00. An example that meets these assumptions is 43 - W" where

"'0’

-l<a<O.

3.2 The Household Optimum under Open Access

The household optimum under open access is solved by n-player Nash equilibrium.
Assume that each household in the village decides investment in livestock holding and in crop
production simultaneously. Then, household i’s optimality condition is obtained by the first

order condition of (3) with respect to L under the constraint W, - A + L assuming that

other households behave optimally. The first order condition is

ac Y " . . " . .
—§(L_tl--m+g(;l.)-b+Lg'<§m+v¢A -0 (5)

From (5). the optimal household investment in crop production and livestock holdings are

obtained as

-m+g(21:)-b+g'<gl:)w
- -V¢ ”12142)

A

 

(6)

125
m-g(zl§)+b-V¢W
- ‘-

L' ..
-V¢+g'(Zl{)

 

(7)

Equations (6) and (7) imply that household i’s optimal A, and I1 depend on initial wealth,
W,, and the coefficient of absolute risk aversion, 4),. They also depend on village level

parameters, m, b, and v, and on the village livestock productivity function. Superscript *
means a household optimum. A. or I: can be negative. A negative optimum means the
household specializes in one activity and needs to borrow money. If there are borrowing or
liquidity constraints, that household will have a corner solution, which will be discussed in
section 3.4.

When households’ optima are compared in the village, the relationship between

optimal investment and initial wealth over households is revealed to be:

 

r n it ‘
fl_g<§m+vawz« (8)
aw -V¢+g’(21£)

-v¢_v.fl.A.

£__:2_vg_
6W -V¢+g'(2£§)

To derive (8) and (9), it is assumed that 2 I: is already given. Consequently, g(21: ) and
[-

i-l

(9)

{(2113 can be considered to be constant. In the following section, they are sometimes

written as g( *) and g'(“) respectively for simplicity.

3.3 lnterhousehold Distribution of the Optimal Investment

Using (6)-(9), how the household’s optimum changes with its wealth can be known.

There are two distinguishable cases: when m - g(*) + b > O and when m - g(*) + b s 0. The

former condition means that the expected return to crop production (m) is greater than the

I26
expected return to livestock holding (g( *) - b). The latter condition means that the expected

return to livestock holding is greater than or equal to the expected return to crop production.
When m - g( *) + b > O (crop returns are higher than livestock returns), the following

five propositions are obtained from the model. Their proofs are in Appendix A.

Proposition 1 : All households invest in crop production.

Proposition 2: There is a unique wealth level under which households do not invest in
livestock and do specialize in crop production.

Proposition 3: The smaller the difference in expected returns to crop production and to
livestock holdings, the less likely is specialization in crop production.

Proposition 4: The riskier is crop production, the less likely is specialization in crop
production.

Proposition 5: All households invest more in crop production than in livestock holdings when
the difi’erence in expected returns to crop production and to livestock holdings is
large. But when the drfierence in expected returns is small, there are some households
that invest more in livestock holdings than in crop production even with the initial

assumption that crop returns are higher than livestock returns.

The unique wealth level defined in proposition 2 is denoted by We. The propositions

. O . 0 A. . 0 . .
above,togetherwith lth -O, hm A -°°, ___6 >0 forall W, lrmL -O, lrm L -°°,
w-oo iv»- 6W w-oo w-o.

6L
”<0 for 0<W<Wc, and %>0 for We <W where We satisfies :iW-O. givea

graphic presentation of the relationship between optimal investments and households’ wealth
when the expected rettu'n to crop production is greater than that to livestock holdings (shown
in Figure 3 and Figure 4). There are two cases: one when A. always exceeds 1: (Figure 3)
and the other when A‘ is lower than L’ in a range of initial wealth (Figure 4).

127
When m - g(*) + b s O (livestock returns are higher than crop returns), propositions
similar to the case where m -g(*)+ b > O can be made as follows. Their proofs are in
Appendix A. Because proofs of them are very similar to those of propositions 1-5, details are

omitted.

Proposition 6: All households invest in livestock holdings.

Proposition 7: There is a unique wealth level under which households do not invest in crop
production and do specialize in livestock husbandry.

Proposition 8: The smaller the difi'erence in the expected return to crop production and to
livestock holdings, the less likely is specialization in livestock husbandry.

Proposition 9: The greater the sensitivity of livestock productivity to changes in total village
livestock holdings, the less likely is specialization in livestock husbandry”.

Proposition 10: Poorer households invest more in livestock holdings than in crop production,

while richer households invest more in crop production than in livestock holdings.

The unique wealth level given by proposition ‘7 is denoted by W4. The propositions

above together with limA.-0, limA.-°°, 2.2-<0 for O<W<Wd, £>O for
W-°° W-W 3W 0W

A C O
Wd<W, where W, satisfies L-O, limL -0, lim L -°°, £>O forall W,givea
6W VH0 8W

W-oo

graphic presentation of the relationship between optimal investments and households’ wealth
when the expected return to livestock holdings is greater than to crop production, which is

shown in Figure 5.

 

‘° The sensitivity is expressed in g'( *) , that is, it is determined by village stock level and the nature
or grass. Since g"( *) is negative, the sensitivity is high when village stock level is high.

128
3.4 Effects of Borrowing Constraints

In this section, effects of borrowing constraints on optimal investments are considered
based on the relationships obtained in section 3.3 (shown in Figures 3-5). As explained
previously, a negative optimal investment theoretically means that the household needs to
borrow to optimize its investment and that it specializes in one activity. This borrowing must
be paid with interest at the end of the period in this one-period model, which is expressed as a
negative return to the negative investment. Because of this cost of borrowing, limitless
borrowing is excluded from the model.

On the other hand, if households cannot borrow, the model predicts that relatively
poor households are forced to choose suboptimal behavior to specialize in one activity: if the
expected return to crop production is greater than to livestock holdings, households whose
initial wealth is equal to or less than WA will choose A. - W and l: -0 (Figures 3 and 4),
and if the expected return to livestock holdings is greater than or equal to the expected return
to crop production, households whose initial wealth is equal to or less than WL will choose
A. -O and l: -W (Figure 5).

Based on the discussion above, whether bonowing constraints exist or not, the model
predicts that households in the low wealth state will specialize in one activity. The question is
whether there is an important number of such households. This question cannot be answered
by the model, but rather requires empirical evidence. According to observation, most Sahel
households are engaged in both activities, which means that their initial wealth is high enough
not to require borrowing according to the model. Therefore, I assume that zero or negative
investment due to low wealth has no effect when l aggregate household investment to the
village level, although the model itself does not eliminate the possibility of negative
investment.

The static model has to ignore several effects of borrowing constraints that a dynamic
model can capture. First, borrowing constraints that are binding occasionally induce

precautionary saving as Deaton (1991) shows in a simulation. This may induce investment in

129

livestock more than without binding constraints. Second, even if the borrowing constraints are
not binding, that is, a household’s initial wealth is large enough not to borrow as discussed
above, if the household believes that there is some probability that its initial wealth may
occasionally be small enough to require that it borrow in the future, the household may try to
avoid such a binding situation by choosing a safer portfolio (Carroll, 1992; Deaton, 1992).
This behavior may also induce more investment in livestock than would be the case without

constraints.

3.5 Aggregate Livestock Holdings under Open Access

Village livestock holdings are the sum of the household optimal livestock holdings
obtained as equation (7). But aggregation does not make sense if there are households that
have negative livestock holdings at their optimum (borrowing for crop production), which is
possible as shown in Figures 3 and 4: however, as discussed in section 3.4, when village
livestock holdings are considered, I assume that there is no negative number, or that the
negative number effect is minimal.

The relationship between wealth distribution in a village and the village livestock
holdings under open access is analyzed by applying “a mean-preserving spread of wealth”

2 O
explained in section 2.2.2. If I: is a convex function of W, that is, if W >0, a mean-

preserving spread of wealth increases the village livestock holdings, and if I: is a concave
2

function of W , that is, if %? < 0, amean-preserving spread of wealth decreases the village

livestock holdings. However, based on equation (7) and the assumptions made above. it is
2 O

impossible to determine the sign of W in general, as seen in Figures 3-5. But by restricting

conditions, the following proposition is obtained; its proof and an example are in Appendix

A.

I30

Proposition 11: The more unequally wealth is distributed in the village, the smaller are
village livestock holdings under open access, given that household wealth is sufficiently large
for all households. How large is large enough for this to hold depends on the all factors that
determine optimal livestock holdings and the relationship between coefficient for absolute risk

aversion and household initial wealth.

Proposition 11 intuitively implies that, the richer a household is, the lower the
marginal increase in livestock holdings per unit wealth. Therefore, if wealth is transferred
from a poor household to a rich household (mean-preserving redistribution of wealth), the
sum of livestock of the two households will be lower than before the redistribution. This
relationship is based on only wealth distribution of the village, but it does not hold for a very
poor household, because the marginal increase in livestock holdings becomes low among the
very poor, that is, a very poor household will invest in crop production rather than in livestock
holdings.

The relationship between wealth distribution and village livestock holdings is from the
non-linear relationship between wealth and the household optimal livestock holdings, shown
in Figures 3-5 (see proof of proposition 11 in Appendix A). Thus, the key to the proposition

above is the concavity of the optimal livestock holding curve to household wealth.

3.6 The Social Optimum

Overgrazing is defined as a case where the sum of the household optimal livestock
holdings is greater than the social optimum at which social welfare is maximized Thus, the
social optimum is introduced in this section to compare with the household optimum
discussed in previous sections. The socially optimal allocation is obtained assuming that
village total wealth is allocated between crop production and livestock holding to maximize
the sum of household certainty-equivalent incomes. This is a problem of a “social planner”

who decides the optimal livestock holdings for the society as a whole and also the optimal

I31

crop production for each household. In other words, the social planner decides how to
allocate the social optimal livestock holdings to each household.

Village total income is the sum of household incomes

fix-n24+ihgtfia>-bih (10)

id id i-l
But since crop production is stochastic, the social planner maximizes certainty-equivalent
income as is done for the household's optimum. The social certainty-equivalent income is the

sum of household certainty-equivalent incomes as follows:

CE(ZY,)-mjA,+Zl,g(ZL,)-b21,—%v§¢,A,2 (ll)

r-I
Maximizing the sum of certainty-equivalent income given by (11) implies that the planner
weights each household equally and as a result social planner’s risk preference is uniquely
determined by this summation.
There are two kinds of budget constraint: for the village and for the individual

household. At the village level, the budget constraint is 2 W, - 2 A, + 2 L, and the budget

1"! ID! |-‘

constraint for each household is W, - A, + 1,. Thus, the social optimum is obtained from the

first order conditions with respect to Z L, and A under the budget constraints mentioned
-1

above. The first order condition with respect to 2 I1 is
I]

 

6CE( . Y2) n n n n 6L
in --m+g(2L,’)+ 2Eg'(211’)-b+ v2¢,4’ ,, -O
6211 hi -r i-l i-r BEL,
(12)
and the first order condition with respect to A is

dCE( n Y,) n n ,,
——§—-m-g(§l:)-§I:g'(§u)+b-v¢w-o (13)

6A
Superscript 8 means the social optimum. From (13), ¢A’ must be the same for all households

because other variables in the equation are common to the village. Thus, because ¢A’ is

I32

 

constant and because 2 -1, equation (12) is the same as equation (13). The
in!

5'11
62 1,
condition for the social optimum given by (13) is different from the condition for the
household optimum given by (5) because the social optimum requires ¢A’ to be constant.
However, because 43, is household specific, socially optimal A,’ differs over households.

¢A’ is not necessarily positive, but can be zero or negative. If it is not positive, all

households in the village specialize in livestock at the social optimum. Although this is

possible in this model, such a case is ignored for this analysis to make a sense, that is A,’ > O

is assumed here. This is supported by the following proposition. Its proof is in Appendix A.

Proposition 12: All households invest in crop production to some extent at the social
optimum when the expected return to livestock holdings is small enough even if it is greater

than the expected return to crop production.

When A’ is positive, 214: is also positive. However, the social optimal livestock

t-I
n

holdings, 211' , can be either positive, zero, or negative. It depends on the size of Z A,.': if

SA,’ <2W,,then 211’ >O,iffiA,’ - SW,Jhen 2L“; -O,andif:A,’ >2W,,then

2L: <0. Even if 21.: is zero or negative, there can be some households that invest in
.1 -1

livestock. However, in the following analysis, only the case where 2A: < 2W“ and

2 l,’ > O is dealt with. This assumption implies that the village as a whole will not be a net
in!
borrower.

With respect to the social optimum, the following propositions are obtained. Their

proofs are in Appendix A.

133
Proposition 13: There is a unique social optimum.
Proposition 14: The more unequal is household wealth in the village, the larger the socially
optimal livestock holdings.
Proposition 15: The socially expected return at the social optimum is lower when household
wealth is more unequally distributed in the village. This relationship is strengthened

when crop production risk is large.

Propositions l4 and 15 compare two villages that have the same number of
households and the same aggregated wealth (see footnote 7). However, wealth is more widely
distributed in one village (unequal village) than the other (equal village). Proposition 14
implies that the socially optimal livestock holdings are greater in the unequal village than in
the equal village. However, proposition 15 implies that the expected return at the social

optimum is lower in the unequal village than in the equal village.

3.7 Comparison between Household and Social Optima

Overgrazing is defined as a case where 2L: >21: , that is, the sum of the

i-I i-I

household’s optimal livestock holdings exceeds the social optimum. 21,. is obtained from
III

the first order condition given by equation (5) by summing over households:

n(-m+ g(*)- b) + ZCg'G‘) ”2M; -0 (14)
Equation (14) is to be compared with equation (13), which gives the social optimum.
Equation (13) is rewritten using oA' I K ’ as

-m+g(s)-lb+zl:g'(s)+vl(’ -O (15)
When n -1, (l4) and (15) are equal, which means the social optimum is the same as the

household optimum. Thus, in the following discussion, n z 2 is assumed. The following

propositions are obtained by comparing (14) and (15). Their proofs are in Appendix A.

W

W

Prop

It 011

ought

I34

Proposition 16: Overgrazing always occurs when the expected return to livestock holding is
greater than the expected return to crop production.

Proposition 1 7: Overgrazing does not necessarily occur when the expected return to livestock
holding is smaller than or equal to the expected return to crop production.

Proposition 18: When the expected return to crop production is greater than the expected
return to livestock holding, overgrazing will occur if crop production risk is high,
households are relatively equal in wealth and/or the difference between the two

returns are small.

The final question is whether all households will be better off in the social optimum.
In other words, is the change from the household optimum under open access to the social
optimum a Pareto improvement? Household i’s certainty-equivalent income at the household

optimum is obtained from equations (4) and (5) as
CE<YI> - mA.‘ + MEL?)— bl: - -:-V¢.(A.-')’

i-l

4:131:13 +év¢.(4‘)’ + minim-b)

i-I i-I

(16)

And household i’s certainty-equivalent income at the social optimum is obtained from
equations (4) and (13) as

I 3 n I 8 1 3

CEO?) - mA. Mezzo - bl, 7mm )’
.. I 1 , (17)
4.2131219 +-v¢.(4’)’ +W.(g(214)- b)

- .1 2 r-1
By comparing CE(Y,.) and CE(Y,’), it can be seen that CE(Y,’) is not necessarily greater
than CE(Y,.). Therefore, the social optimum is not always a Pareto improvement, although
society as a whole maximizes total certainty-equivalent income at the social optimum.

However, it is impossible to obtain explicit conditions for CE(Y,’) 2 CE (Y,.) based on (16)

and (17). The following proposition is obtained The proof is in Appendix A.

troll

tth'
[DUO
Mum
he cl
3m m

31m

“We

I35

Proposition 19: When there is overgrazing, there will be a certain level of initial wealth below
which no household will be better 017' at the social optimum, and another certain level

of initial wealth above which no household will be better off at the social optimum.

Proposition 19 implies that in a village whose wealth distribution is equal, the social
optimum will be a Pareto improvement, while in a village whose wealth distribution is unequal,
the social optimum will not be a Pareto improvement (the proof is not based on the wealth
distribution, but on the wealth level in this case unlike previous ones). In the case of non-
Pareto improvement, the poor who are more risk averse are forced to have a more risky
portfolio at the social optimum, while the rich who are less risk averse are forced to have a
portfolio with less expected return at the social optimum.

Section 3.7 together with pervious sections concludes: (1) Village livestock holdings
Imder an open-access regime is greater than village livestock holdings at the socially optimal
equilibrium when the expected return to livestock holdings is greater than the expected return
to crop production; (2) If the expected return to crop production is greater than the expected
return to livestock holdings, crop production risk must be high enough for overgrazing; (3)
The degree of overgrazing is high when household wealth in the village is relatively equal.
But in other words, the social gain at the social optimum is large when household wealth is
relatively equal; (4) The social optimum is not necessarily a Pareto improvement, that is, very
rich and very poor households are likely worse off at the socially optimal equilibrium if such
households exist in the village; (5) Although overgrazing can be severe in a relatively equal
village, a common property management rule will be established more easily than in a

heterogeneous village, according to conclusion (3) and (4).

4 Application of the Model to Grazing Land Management
In this section, I discuss potential solutions for overgrazing based on the model

presented in previous sections. They are classified into three categories: physical solutions,

I36

institutional solutions, and other economic solutions. The other economic solutions may need
new institutions, but the distinction between the institutional solutions and the other economic
solutions here is that the institutional solutions are involved in a change in property rights.
Although solutions for common-pool resource management have been considered in various
contexts, to my knowledge there has been no model for grazing land management in the

WASAT. Therefore, the following analyses are new in the theoretical literature.

4.1 Physical Solutions

Physical solutions include increase in productivity of livestock, increase in
productivity of crop production, and utilization of risk-decreasing technology in crop
production.

The increase in livestock productivity may be achieved by conservation investment to
increase carrying capacity, and/or by introduction of highly productive grass and/or feed-

efficient animals from outside. From equation (15), the social optimum is given by
" , m—g(s)+b-vK’
2 a -
-1

, (18)
g (s)

A conservation investment that lowers the negative effect of livestock population on

productivity will give a smaller negative g'(s) , and thus will increase the social optimum,

z I: . The open-access problem is seen in equation (18): the private cost of conservation
-1

investment that is included in b will easily offset the gain from conservation, a smaller
negative g'(s). Conservation investment is often suboptimal due to lack of land tenure and
lack of immediate ‘benefit (Clay and Reardon, 1993). Conservation investment will be more
difficult in grazing land because there is far less immediate benefit from theinvestment and
the benefit cannot be privatized if it is open access. In addition, since the grazing land is a
common-pool resource, investment in it requires coordination, which is difficult.

On the other hand, if the expected return to livestock (livestock productivity) becomes

much greater than that to crop production (crop productivity), the socially optimum livestock

137
holding will also rise; this is the case when g(s) - b is greater than m and m - g(s) + b is a
large negative number. However, if grazing land is open access, any improvements in
livestock productivity will also increase investment in livestock. As seen in equation (7), the
household optimal livestock holding will increase and, as propositions l6 and 17 state, when
the expected return to livestock is higher than that to crop production, overgrazing is
inevitable. Therefore, improvement of livestock productivity will not solve the problem under

open access.

As shown in propositions 16, 17, and 18, increase in crop productivity and/or decrease
in crop production risk will reduce overgrazing. Thus, introduction of highly productive
varieties and risk-decreasing technology, such as fertilizer and small-scale irrigation, and
control of soil erosion will solve the overgrazing physically. Theoretically it is true, and it is

of course another important issue. But it is not easy to achieve (Reardon, forthcoming).

4.2 Institutional Solutions
As discussed above, in a situation of open access, physical solutions have limited
power without appropriate institutions. Therefore, a change in property rights, such as

privatization, quotas, transferable permits, and user fees should be considered.

4.2.1 Privatization
Privatization is often considered to be a solution for the tragedy of the commons
because it internalizes the cost of degrading grazing land It can be introduced into the model

as follows. Assume that each household is allowed to privatize some specific area of village

grazing land based on its current share of village livestock holdings, -,,-l3—. For simplicity, all
21.

households are assumed to have some livestock (l: > O) and quality of grass is assumed to be

uniformly distributedu so that productivity of a piece of the total grazing land depends only

 

“ msmsumpdonavddstheproblanofhaaogeneomgrazinglmddesaibedinsecum2.1.

mil
in

his:

is o

in]! 1

hold
{term
that 1

an

138

on livestock density. Also assume all animals have the same feed-conversion efficiency. Since

the total grazing land of the village is assumed to be fixed, it can be assumed to be unity in

this analysis. Consequently, each household will have "I1 of grazing land and if household
2 1:

i’s optimal livestock holding under privatization is l," , the livestock density for household i

 

P n 0 LP n a
will be %2 l, and the livestock productivity function for household i will be g(2:7 2 L,).
i -I ‘ in

Under these assumptions, it can be shown that Lf < L; must hold. Its proof is in

Appendix B. This means that privatization of grazing land decreases optimal livestock

holdings for each household, and as a result total village livestock holdings, 21,”, also
l-l

becomes smaller.

At issue is whether privatization is a Pareto improvement, that is, whether
CE(Y,’)2 CE(Y,.) for all i. As shown in Appendix B, CE(Y,")2 CE(Y,.) does not always
hold but depends on the relative magnitudes of I; and if, household’s initial wealth, and the
features of the livestock productivity function. The proof suggests that wealthier households
that have larger optimal livestock holdings under open access are likely to be better off
absolutely by privatization.

However, the above comparison of certainty-equivalent incomes ignores the private
costs associated with privatization: allocation of village grazing land to each household will
cause high transaction costs, especially if quality is not uniform (unlike my assumption);
Even if it is somehow allocated, there would be high exclusion costs to stop animals invading
privatized grazing land Such additional costs can be seen included in b. If b under private
property is sufficiently greater than b under open access, no household will be better off by
the privatization (in the previous discussion b is assumed to be the same in both regimes, that
is, no exclusion cost is assumed). Therefore privatization of open-access grazing land will be

difficult.

139

4.2.2 Quota

A quota scheme may be suitable for grazing land because it restricts the total number
of livestock in the village on which livestock productivity of the village depends, but it avoids
the problem of physical allocation of grazing land in the case of privatization. However,
allocation of the quota is still a problem. In a quota scheme, villagers (or policy makers) have
to choose a desired level of village livestock holdings”. and decide an allocation rule in
which the desired level is achieved: a quota would be used to allocate livestock holdings to
each household, but it is not necessarily the least cost way.

The total amount of livestock that the village must give up, D, to achieve a desired

level of livestock, L0, is given by 2L; - LD - D. D is allocated among households by

i-I

z d, - D. Different allocation rules translate into different social costs associated with the
-r

quota. If the quota is proportional to current livestock holdings under open access, the
allocation rule will be d, - -,-,Q-['.—. This quota scheme is not a Pareto improvement, that is,
211'
some households will be worse off under this quota than under open access (the proof in
Appendix C). It depends on household wealth level, and the relationships between household
wealth and the optimal livestock holding under open access and between household wealth
and household preference. Although the model predicts an ambiguous effect on welfare, it
also shows that it may be possible to design a quota that will achieve a Pareto improvement,
and that it will be easier if household wealth and as a result preference is less heterogeneous.
In the model above, no monitoring and enforcement costs are taken into account. If
the costs are high, they will offset the benefit and will make the scheme infeasible; it will
happen especially if some households are worse off as shown above. Even if every household

is better off under the quota, each household will have incentive to violate the quota, and

 

‘2 The socially desired level is not necessarily the social optimum because the social optimum is not
known to villagers or policy makers; it is something they decide based on available information.

be!

m

0335

.5.
n}

at arm
ewes

140

therefore monitoring and enforcement costs will still be required. The costs partly depend on
the characteristics of the village: if it tends to be cooperative, the costs will be low. As shown
in proposition 19, a village whose wealth distribution is relatively equal will have more

incentive to cooperate because the gain from cooperation is greater.

4.2.3 Transferable Permits

The transferable permit is theoretically a cost effective way to achieve a desired level
of resource use (Baumol and Oates, 1988). Transferable permits are considered to be the way
to allocate a quota in the market instead of merely assigning it administratively. In the
previous section, an assigned quota is not assumed to be transferable, but here the quota can
be sold and bought. Consequently, this system is flexible to change in an individual
household’s economic condition. Moreover, the market price of the permits could increase if
demand increases. Therefore, the permits system can meet human population growth, which is
considered to be one of main causes of increasing total livestock holdings.

However, an assignment problem still remains in this transferable permit system.
Some government assignment of rights is antecedent to market trade so the market is not an
alternative to government in this sense. There are two ways to allocate permits. One is to
assume households share equal property rights, and the other is to assume the village
possesses the property rights. In the former case (equal property rights), each household is
given an equal number of permits initially, and they can then trade the permits. Thus,
households that want to keep more livestock than initially permitted must pay households that
want to keep less livestock than initially permitted. As a result, there is income transfer from
the rich to the poor because the poor have fewer animals to graze as shown in Figures 3-5.
Because the poor benefit from this scheme, they will have an incentive to monitor the rich if
monitoring cost does not exceed the benefits from the permits sold. This may not be realistic,

for example, they may not have the capacity to monitor, but it is possible at least theoretically.

Eli

bet

ii

1‘?

is?"

r: r

141
Even so, the benefit will not necessarily exceed the enforcement and monitoring costs, and
therefore its feasibility is not promised.

The latter case (village ownership) requires a village agency that sells permits, and any
household that wants to keep livestock must pay according to the size of its livestock
holdings. The permits can be traded between households to meet changing demand. Thus the
income transfer effect still exists, but is smaller than in the former case (equal property
rights), and similarly flexible to achieve least-cost allocation. The money collected by the
agency will be spent for monitoring and some investment in the commons. Again, if
monitoring costs exceeds the permits sales, this scheme is not feasible. This works like a user
fee, but the difference is that the permit scheme keeps the total number of livestock constant
or less, while a user fee cannot control the total number without adjusting the fee rate.

The scheme of transferable permits where households start with an equal allotment is
as follows. First, every household is entitled to an equal number of permits, which is given by

D

. . . . . L
a socrally desired level (see footnote 9) drvrded by the number of households, that rs, —.
n

Then, they trade them to achieve maximization of expected utility at the unit price p. It is
assumed that the desired level of livestock holdings, LD , is smaller than the level under open
access, 2L: , and there is no excess supply of permits. Thus, the optimal livestock holdings

l-l

for each household under the transferable permit scheme, l,"I , should sum to the desired level,

that is, 2L" - Lo and price of the permits, p , which is also endogenously determined in the

model, should be positive.

Comparing I,” with the household’s optimum under open access, L: , given by (7), it
is not possible to tell which is greater in general. Thus, although the transferable permit
. scheme enforces a reduction in total village livestock holdings, we cannot predict who will
increase and who will decrease livestock holding in this model (see Appendix D). Nor can we

tell who will be better off and who will be worse off. But obviously there will be some worse

142

off. Although there is an income transfer to the poor, the welfare effect is not clear because

they have to increase risky crop production.

4.2.4 User Fee

A user fee scheme is as follows. Let t denote the user fee rate proportional to
household i’s livestock holdings. t is given exogenously and is assumed to be positive. Then
household i has to pay tl,. Under this scheme, all households will reduce optimal livestock
holdings (see Appendix E). However, we cannot tell who becomes better off and who
becomes worse off. Again, a user fee requires monitoring and enforcement, and the collected
fee must be spent on these costs. However, who owns the grazing land, state or village, will
affect the costs: Communal ownership will be less costly than state ownership in monitoring
and enforcement, although it depends on characteristics of the community. But even if this
scheme works, the amount of livestock reduction cannot be controlled directly unlike in the

case of transferable permits, but is controlled indirectly by adjusting the fee rate.

4.3 Other Economic Solutions

Because livestock is used as an asset to smooth income in the WASAT, if cheaper (or
at least as cheap as livestock) alternatives are available, they will be able to reduce the number
of livestock, and as a result, the CPR dilemma will be reduced. As implied by propositions 17
and 18, off-farm activities, included in crop production in this model as note in footnote 6, is

one possible alternative. Another potential alternative is formal drought insurance.

4.3.1 Off-farm Income

Off-farm income is known to be a significant part of total income of households the
WASAT (Reardon et al., 1994),. The effect of off-farm income may be to increase the
expected return to crop production (higher m) and/or to reduce crop production risk (lower

v). Therefore, off-farm income will reduce the degree of overgrazing, according to

r

143

proposition 18. However, this is true only if a household’s wealth does not change by an
increase in productivity of off-farm income and an decrease in its risk. If a household’s
wealth increases by them, their effect on livestock holding is ambiguous because household
wealth increases the optimal livestock holding on the other hand. Because of these two
opposite effects, off-farm income will have ambiguous effects on livestock holdings in the
long-run. Christensen (1989) empirically finds that the effect of off-farm income on
livestock holdings is ambiguous. For off-farm income to reduce livestock holdings, risk in
crop production and off-farm activities should be so small that households invest in those
activities instead of livestock husbandry. The issue is riskiness, cost, and availability of other
self-insurance strategies relative to livestock holdings. However, the problem in the WASAT is

that there are few other strategies, either formal or informal.

4.3.2 Formal Drought Insurance

If formal financial institutions, such as drought insurance, consumption credit and
savings were available in the WASAT, households will reduce number of livestock. In the case
of formal drought insurance, it will reduce livestock holding if (i) it is less costly than
households’ self-insurance, and it is demanded by those who keep large livestock holdings;
(ii) it substitutes for their livestock holdings, which is one of their self-insurance strategies.
Therefore, these must be empirically examined to see if formal drought insurance will reduce
the pressure on grazing land. Probably, only the rich may be able to utilize such financial
institutions. But even if only the rich benefit from the financial institutions, they will work:
because in general the rich keep more livestock, if they reduce the amount, the poor may be

able to enjoy higher productivity of livestock and be better off.

5 Conclusions
Because I assume livestock is a riskless asset, household optimal livestock investment

under an open-access regime increases as households become more risk averse or crop

mill
hold
over

llpt

mp
over

ma}

man
oltr
lea
So i
dist
b ht

€th

144

production becomes more risky, as expected. In addition, I assume households have
increasing relative risk aversion, thus wealthier households tend to keep larger livestock
holdings. But regardless of the household’s risk attitude, if —m + g(*) - b> 0 holds, there is
overgrazing. This is the condition when the expected return to livestock is greater than the
expected return to crop production. This condition is not affected by crop production risk,
but the degree of overgrazing depends on risk. An important implication is that increasing
crop and/or off-farm productivity (higher returns to them, or higher tn) may reduce
overgrazing, even if grazing land is open access, and that decreasing their risks (smaller v)
may also reduce overgrazing.

When the expected return to livestock holdings, however, is smaller than the expected
return to crop production, overgrazing does not necessarily occur even if grazing land is
open access. Under this condition, if crop production risk is high, overgrazing is likely.
Wealth distribution in the village is one of factors that determine the degree of overgrazing.
So, if wealth is more equally distributed, overgrazing is more likely under open access. Wealth
distribution also determines distribution of cost and benefit of management scheme for
grazing land. In a village where there are very rich and very poor households, they are likely
to be worse off at the social optimum than at the household optimum. This suggests that
communal rule will be difficult to enforce if village wealth is unequal, although overgrazing
may be less severe in an unequal village.

A quota scheme will be an effective approach if it works with small costs relative to
the benefit. The costs depend on a community’s potential to cooperate. If it is not feasible,
other institutions that create more incentive to monitor neighbors or to generate revenue to
cover monitoring/enforcement cost will be required. Transferable permit and user fee
schemes are examined in this chapter. Both reduce total livestock holdings, but have
ambiguous effects on household investment and on household welfare if households are
heterogeneous. If the only alternative to livestock holdings is risky crop production, both

schemes will impose a high cost to households that reduce livestock holdings. Therefore other

I45

economic solutions mentioned in section 4.3 are important to implement those institutions. I

think that a mix of solutions will be a better way.

146

References

Andersen, Peder, “Commercial Fisheries under Price Uncertainty,” Journal of environmental
Economics and Management, 9: 11-28, 1982.

Arrow, Kenneth, “The Theory of Risk Aversion,” In: Kenneth Arrow, eds., Essays in the
Theory of Risk Bearing, Chicago, IL: Markham, 1971.

Baumol, William J. and Wallace B. Oates, The Theory of Environmental Policy, second edition,
Cambridge, UK: Cambridge University Press, 1988.

Berkes, Fikret and M. Taghi Farvar, “Introduction and Overview,” In: Fikret Berkes, eds.,
Common Property Resources, London, UK: Belhaven Press, 1988.

Bromley, Daniel W., “Testing for Common versus Private Property: Comment,” Journal of
Environmental Economics and Management, 21: 92-96, 1991.

Brown, Gardner Jr., “An Optimal Program for Managing Common Property Resources with
Congestion Externalities,” Journal of Political Economy, 82: 163-173, 1974.

Carroll, Christopher D., “The Buffer-Stock Theory of Saving: Some Macroeconomic
Evidence,” Brookings Papers on Economic Activity, 2: 61-156, 1992.

Christensen, Garry Neil, “Determinants of Private Investment in Rural Burkina Faso,” Ph. D.
Dissertation, Cornell University, 1989.

Clark, Colin W. and Gordon R. Munro, “The Economics of Fishing and Modern Capital
Theory: A Simplified Approach,” Journal of Environmental Economics and
Management, 2: 92-106, 1975.

Clay, Daniel C. and Thomas Reardon, “Determinants of Farm-level Conservation Investment
in Rwanda,” Mimeo, Michigan State University. 1993.

Dasgupta, Partha and Geoffrey Heal, Economic Theory and Exhaustible Resources, New York,
NY: Cambridge University Press, 1979.

Deaton, Angus, “Saving and Liquidity Constraints,” Econometrica, 59: 1221-1248, 5, 1991.

Deaton, Angus, “New Directions: Beyond Permanent Income,” In: eds., Understanding
Consumption, New York: NY: Oxford University Press, 177-213, 1992.‘

Friedman, James W., Game Theory with Applications to Economics, Oxford, UK: Oxford
University Press, 1990.

GibbonslsrRobert, Game Theory for Applied Economists, Princeton, NJ: Princeton University
ess, 1992.

Gordon, H. Scott, “The Economic Theory of A Common Property Resource: The Fishery,”
Journal of Political Economics, 62: 124-142, 1954.

Hardin, Garrett, “The Tragedy of the Commons,” Science, 162 (3859): 1243-1248,
December 13, 1968. Reprinted in Environmental Economics: A Reader, ed.
lvéarkand92 ya, Anil and Julie Richardson, 60-70, New York, NY: St. Martin’s Press,
1 .

Haveman, Robert H., “Common Property, Congestion, and Environmental Pollution,”
Quarterly Journal of economics, 87: 278-287, 1973.

I47

Hirshleifer, Jack and John G. Riley, The Analytics of Uncertainty and Information,
Cambridge, UK: Cambridge University Press, 1992.

Houerou, Henry Noel Le, The grazing Land Ecosystems of the African Sahel, Berlin,
Germany: Springer-Verlag, 1989.

Just, Richard E. and David Zilberman, “Stochastic Structure, Farm Size and Technology
Adoption in Developing Agriculture,” Oxford Economic Papers, 35: 307-328, 1983.

Just, Richard E. and David Zilberman, “The Effects of Agricultural Development Policies on
Income Distribution and Technological Change in Agriculture,” Journal of
Development Economics, 28: 193-216, l988a.

Just, Richard E. and David Zilberman, “A Methodology for Evaluating Equity Implications
of Environmental Policy Decision in Agriculture,” Land Economics, 64: 37-61, 1,
1988b.

Newbery, David MG. and Joseph E. Stiglitz, The Theory of Commodity Price Stabilization,
Oxford, UK: Oxford University Press, 1981.

Ostrom, Elinor, Roy Gardner, and James Walker, Rules, Games, & Common-Pool Resources,
Ann Arbor, MI: The University of Michigan Press, 1994.

Reardon, Thomas, “Sustainability Issues for Agricultural Research Strategies in the Semi-arid
Tropics: Focus on the Sahel,” Agricultural Systems, forthcoming.

Reardon, Thomas et al., “Is Income Diversification Agriculture-Led in the West African
Semi-Arid Tropics? The Nature, Causes, Effects, Distribution, and Production-
Linkages of Off-Farm Activities,” In: S. Wangwe A. Atsain, and A. G. Drabek, eds.,
Economic Policy Experience in Africa: What Have We Learned?, Nairobi, Kenya:
African Economic Research Consortium, 1994.

Reardon, Thomas, Peter Matlon, and Christopher Delgado, “Coping with Household-level
Food Insecurity in Drought-affected Areas of Burkina Faso,” World Development,
16: 1065-1074, 9, 1988.

Reardon, Thomas and J. Edward Taylor, “Agroclimatic Shock, Income Inequality, and
Poverty: Evidence from Burkina Faso,” Michigan State University, Department of
Agricultural Economics Staff Paper, No. 94-27, 1994.

Robinson, Lindon J. and Peter J. Barry, The Competitive Firm ’3 Response to Risk, New York,
NY: Macmillan Publishing Company, 1987.

Sandler, Todd and Frederic P. Sterbenz, “Harvest Uncertainty and the Tragedy of the
Commons,” Journal of Environmental Economics and Management, 18: 155-167,
1990.

Schmid, A. Allan, Property, Power, & Public Choice, New York, NY: Praeger Publishers,
1987.

Stevenson, Glenn G., Common Property Economics. A General Theory and Land Use
Applications, Cambridge, UK: Cambridge University Press, 1991.

Weitzman, Martin L, “Free Access vs Private Ownership as Alternative Systems for Managing
Common Property,” Journal of Economic Theory, 8: 225-234, 1974.

l'llsot

148

Wilson, Pail N. and Gary D. Thompson, “Common Property and Uncertainty: Compensating
Coalitions by Mexico’s Pastoral Ejidatarios,” Economic Development and Cultural
Change, 300-318, 1993.

149

Figure l Household’s Risk Aversion Coefficient (4:) and Wealth

 

 

. 640 62¢ . .
Assumptions: —<0,—-;>0,llm¢-°O,and lrm d-O
6W 6W til-'0 we“

150
Figure 2
Household’s Relative Risk Aversion Coefficient MW) and Wealth

¢W

 

 

W

mm), 61va

Assumptions: , ,
6W 6W

<0.,1313,(¢W)-o.andu;n.<¢m-oo

 

Figure 3

A*, L*

 

 

151

Optimal Investments (Case 1, m — g(*) + b > O)

45 degree li .

.‘--'--.-..---.'-.----.--

b-....--o-----o--- 9---.-.

 

 

s -'
,3

Expected return to crap production is greater than that to livestock.

. 61. a a
At WA,L -O,at WC,3W-O,andat WWA -L is minimum.

Always A. > 1:.

 

Figure 4

A*, L*

 

152

Optimal Investments (Case 2, m - g(*) + b > O)

45 degree li '

A*-—*

\

    

.------.--------.—-----—

t----- - o-------~
p-QOOQQQQOOOQOQC-Ooooonco- o.--
.

 

 

W

A

3

W

3

Expected return to crop production is greater than that to livestock.

. 6L . .
At WA,L -0,at WC,3-“-;-O,andat W_,A -L is minimum.

A’ 213 whenWsW, or W,sw,and A. <1: when W,<W<W,.

Figur

4*
‘l

153

Figure 5 Optimal Investments (Case 3, m - g(*) + b s O)

 

 

 

A*, L*
45 degree lin
A*
1 a:
. a L
w. E 5
0 : Wt. W.
W

Expected return to crop production is smaller than that to livestock.

. am
At w,, A -O,at WC, 6—W--o,.

A’<L’ when W<Wn andA’zL’when WnsW.

154

Appendix A

Proof of Proposition 1

In equation (6), the numerator is always negative because m - g( *) + b > 0 and g'( *) < 0,

and the denominator is always negative because v4: > 0 and g'(*)< 0. Therefore, A. is

always positive. Q.E.D.

Proof of Proposition 2

In equation (7), the denominator is always negative because v43 > 0 and g'(*) < 0, but the

. . m — g(*) + b . . . . .
numerator rs negative only when < ¢W. The left-hand srde of this inequality ls
v

 

a positive constant, and the right-hand side is an increasing function of wealth, W, as

. . m - g(*) + b . .
assumed. Therefore, there must be a unique W that satisfies - ¢W, which rs

 

denoted as W, . Thus, at and below WA , I: is non-positive, which means that the poorer

households do not invest in livestock and specialize in crop production. QED.

Proof of Proposition 3

As shown previously, the unique W, at and below which households do not invest in

. . . m - g(“‘) + b . . .
livestock rs determrned by - 40W. Therefore, the smaller rs the difference rn
v

 

expected returns, that is, m -g(*)+b is smaller, the smaller is WA. It means that most

households invest in livestock in addition to crop production when those returns are close._

Q.E.D.

Proof of Proposition 4
Similarly to the proof of proposition 3, if crop production becomes riskier, that is, if v
becomes higher, WA will be smaller. It means that only very poor households specialize in

crop production when crop production is very risky (because of proposition 2). Q.E.D.

155
Proof of Proposition 5
Difference between the optimal investment in crop production and the optimal investment in

livestock holdings is given by

-V¢ + g’(*)
In equation (Al), because the denominator is always negative, A. - L. is positive when the
numerator is negative, that is when [g'(*) + v¢]W < 2[m - g(*) + b]. Since 2[m - g( *) + b] is
a positive constant, this condition implies that when [g'(*)+ v¢]W is smaller than that

constant. all households regardless of initial wealth invest more in crop production than in
livestock. Let f(W) -[g'(") +v¢]W, then lvunof(VV) - O and Jim f(W) - -00. Its first
07¢

derivative is f(W) " g'(*) + V¢ + vW-a—v; whose limits are: lvunof'(VV) - 0° and

Jim f '(W) - g'(*) < 0. In addition, there is a unique W satisfying f '(W) - O, which is easy
34’ 8'(*)

to see if the condition is rewritten as p + WW - -

positive constant and the left-hand side is the first derivative of ¢W which is already assumed

where the right-hand side is a

to decrease monotonously from infinity to zero. Therefore, the unique W, say W.,, satisfying
f'(W) - 0 gives the maximum value of f(W) -[g'(*) +v¢]W, and the value must be
positive, that is, f(W_) -[g'(*) +v¢]WM >0. Thus, A. -L. is always positive when
f(W_) < 2[m - g(*) + b]. But if f(W_) > 2[m - g(“) + (21 holds, A’ - L’ will be negative in
some range of wealth around W. and is positive in the lower tail and the upper tail of wealth
distribution. In other words, only some households in the wealth range around W., invest
more in livestock holdings than in crop production, but only when f(Wn) > 2[m - g(“) + b]
is satisfied.

Using the first order condition, f (W) I [g'(*) + v¢]W can be written as

f(W.)--vW:,%, which does not depend on m, g'(’) or b.Therefore, when

2[m - g( *) + b] increases, or when the difference in expected returns to crop production and

to livestock holdings increases, no household invests more in livestock holdings than in crop

production. But when the difference decreases, some households invest more in livestock

bolt
cm;

This

its:

 

156

holdings than in crop production, and the wealth range where livestock investment exceeds
crop production investment becomes wider as the difference in expected returns decreases.

This proof of proposition 5 is graphically presented in Figure Al. QED.

Proof of Proposition 6

To show L. >0 for all W is very similar to the proof of proposition 1. QED.

Proof of Proposition 7
m - g( *) + b

, -W.Thus, A'sO for WsWL.
80")

There is a unique W, say WL, that satisfies

QED.

Proof of Proposition 8
m — g( *) + b

g'(*)

zero. It means that there are less households specializing in livestock husbandry. When

- WL, when m - g( *) + b becomes closer to zero, W1. also becomes closer to

m -g(*)+ b - 0, W1. becomes zero, that is, there is no household specialized in livestock

husbandry. QED.

Proof of Proposition 9
The sensitivity is expressed in g'(“). It is negative, which means that when total village

livestock holdings decrease, livestock productivity increases. Therefore, greater sensitivity

a _ 3
implies a smaller (or larger in absolute value) g'(‘) for a given 2 l: . In W - Wu
-1

g'(*)
when the absolute value of g'(‘) increases, WL becomes smaller. It means that there are less

households specializing in livestock husbandry. QED.

PM

is

ll

she

s’j

Fifi

W

.1
‘i

hint.

r."

en»

 

 

157
Proof of Proposition 10
Similarly to the proof of proposition 5, A. - L. is positive when
[g'(*) + v¢]W < 2[m - g(*) + b]. However, in this case, the right hand side of this inequality
is a negative constant or zero. According to Figure A1, when 2[m - g( *) + b] is negative or
zero. there is a unique, positive W , say W, , above which [g'( *) + v¢]W < 2[m - g( *) + b] is
satisfied. That is, A. < L’ for W< w, and A. 2 L’ for W2 w,,. QED.

Proof of Proposition 11

As discussed in section 3.4, most households are considered to have positive livestock
C O 62L.
holdings (L > 0). But even if L > 0 is assumed, we cannot sign 6W2 a priori. However, as

 

shown in Figures 3-5, %> 0 holds (for W> W, if m —g(*)+b> O and for all W if

U 2 I

. L d L
m — g(*) + b s 0) and 11m 3W - 0 also holds. This suggests that W < 0 must hold for

2
sufficiently large W and above. As discussed in the text, if d—W-z- <0 holds, a mean-

preserving spread of wealth decreases the village livestock holdings.

For example, if 4» - W" where -1< a < 0 is assumed (this function satisfies all
2

assumptions made for of as explained in section 3.1), it is possible to show that W <0

holds for all W if tn - g( *) + b s 0, or the expected return to crop production is smaller than
2 C

d L
to livestock holdings, and that W < 0 holds for sufficiently large W if m - g( *) + b > 0,
or the expected return to crop production is larger than to livestock holdings. In the latter

 

I
._ . a: '3 ‘
case, the exact condition for W cannot be obtained, but W -( ( )) is large enough for
v
a’L' , ,
W < 0 (denvatlons are not shown).

Proof of Proposition 12
When ¢A’ is positive, A’ will be positive because d is assumed to be positive, that is, all

households invest in crop production to some extent at the social optimum. In equation (13),

pit

Pr

 

158

when ¢A’ is positive, m — g(s) - 2 Eg'(s) + b must be positive. Because g'(s) is assumed to
be negative, if m -g(s) +b is positive, the condition for ¢A’ to be positive is met (the

expected return to crop production is greater than the expected return to livestock holdings).
On the other hand, when m - g(s) + b s 0, the sign of ¢A’ is ambiguous, but if the difference

in the expected returns is small enough, ¢A’ can be positive (the expected return to crop

production is smaller than the expected return to livestock holdings).

Proof of Proposition 13
The proposition is equivalent to saying that equation (13) gives a unique, positive ¢A’ under

assumptions of 2A: >0 and 21; >0. Let oA’ I K’, so equation (13) can be written as a
ll

function of K:
h(K)- --m -g(2W,-— -K2-—)— (2W,- K2—)g(2W,- -K2—)+b vK
¢i r-1¢
(A2)
Differentiation with respect to K gives
h' 2 — 2W,- K — " A3
00- 80“ ,‘+(M 2,)" ..¢.g()- v ( )

h'(K) is negative for positive K, because g'( ) and g"( ) are assumed to be negative and
2 W, - K2; -2 l, is assumed to be positive.
.1 n l -
When K- 0, function (A2) becomes h(0) - m-g(2w,) - (ZW,)g'(2W,)+ b.
u- l- III

This is the same as the left-hand side of equation (12) with a negative sign when

II n
2 l, - z W, . Therefore, when the socially optimal livestock holdings is smaller than total

W.
wealth, thatis, when ZE< 2W,, h(0)>0 holds. On the other hand, when K-g—L’

run It

159

W, 2W
function (A2) becomes MEL—f) - m— g(O) —O* g'(O) + b - v ‘7, 1 . Because

 

 

 

ul ¢i r- ¢i
2 W‘ min W
J%— z 1‘ - min(¢,W,) > O, the following inequality holds:

.1. max _
. ‘i ¢i

W:

h(",', 1 )sm-g(O)-O*g'(0)+b-vmin(¢,W,.) (A3)
.1 4’:

In equation (12), when 211 -O, the left-hand side is positive, because the optimal level

r-l

exists where O < 21;. This means that the right hand side of inequality (A3) is negative.

r-l
n n

w, a W.-
Therefore h(gx-T) < 0 holds. Thus, h'(K) < 0, 11(0) >0, and h( '7. 1 ) < 0 indicate that
.4». 2.4..

W!

there is a unique, positive K between 0 and 4&1-1— that gives h(K) - 0. Such K is now

2 is.
denoted by K’. Q.E.D.

Proof of Proposition 14
As a measure of inequality, the mean-preserving spread of wealth distribution is used
here. So, if A’ is concave in W, 2x: will be lower and 2 I; will be higher when wealth is

i-l l-I
distributed less equally.
Differentiating A' with respect to W yields
K,
s 6 —
aA ( 4.) 1 av

—-——-K’ -__
aw aw ( o” aw) (A4)

is

in

5665

 

 

 

 

160

 

K3
3 (92 ""'
52A (is) 2 (M 1 a__’¢
2 ' 2 'K (— 3 2 2 . (A5)
aw W 42 29W 42 aw

Because ¢ is assumed to be a convex function of W with 43' <0 and 4)" > 0, (A4) is

positive, and (A5) is negative. Therefore, A’ is concave in W. QED.

Proof of Proposition 15
The socially expected return is given in equation (1 1). When livestock holdings are allocated

optimally in the society, equation (1 1) can be written using K ’ as

Edgy)- -mg —+(2W K’21-]g(2m-K’§i)

i-l i-l¢ i-l

1
.-.1.-¢

Differentiating equation (A6) with respect to 21— results in

£213 -an ,2 (A7,
“722. ..

1
As proved for proposition l4, — is lower when wealth is more unequally distributed.
.1 i

-b(;W, -K’ ---]1-2-v(K”) (A6)

2. 4’.-

Therefore, equation (A7) implies that the socially expected return is lower when the village
wealth is more unequal. When v, crop production risk, is large, this effect is strengthened as

seen in equation (A7). QED.

Proof of Proposition 16
The condition above is written as -m+g(*)- b> O as already discussed in section 3.2.

Assume contrarily that 21.; s :1: , then 0 < g(s) s g(*) and g'(s) s g'(*)< 0 hold and
-l

i-l
consequently the following inequalities hold:
-m+g(s)-bs[—m+g(*)-b]<n(-m+ g(*)-b) (A8)

161

21:31.0 s [Edam] $2131") (A9)
thfirA: " VZ%A. - vK’Zi'; sz’-t§i%z VIC? z vK’

where the terms in [ ] are used only to prove the inequalities. Inequalities (A8), (A9) and

 

 

(A10)

(A10) imply that (14) and (15) cannot hold simultaneously, which contradicts the proofs that

both household and social optima exist. This contradiction is due to the additional

assumption of 21: $21; Therefore, 21: >21: must hold when —m+g(*)- b>0.
-l

id id r-l

That is, overgrazing always occurs. QED.

Proof of Proposition 17
Now the condition is -m + g(*) — bs 0. If 2 l1. 5 XL: is contrarily assumed, inequalities

(A9) and (A10) still bold, but inequality (A8) does not always hold. There are two possible
inequalities under those assumptions above:
02-m+g(*)-b>n(—m+g(*)-b)>-m+g(s)-b (All)
and
Oz-m+g(‘)-bz-m+g(s)-bzn(-m+g(*)-b) (A12)
If inequality (All) holds, then inequality (A8) also holds and consequently 21.: > :1:
-t -l

must hold, which means there is overgrazing. On the other hand, if inequality (A12) holds,
inequality (A8) does not hold, which means that either 21; > 21; or 21: 5211’ can be
.1 u u -

correct. That is, overgrazing does not necessarily occur when —m + g( *) - bs 0. QED.

162
Graphic Presentation of Proofs of Propositions 16 and 17
The proofs of propositions l6 and 17 are presented in graphs as follows. When
-m+g(*)- b> 0 (overgrazing always occurs according to proposition 16), the socially
optimal investment in crop production for each household, A’ - E , and the household
-m+ (* -b+ ’(* W
ivz+g’(fl ) 'm

graphed in Figure A2 as functions of household initial wealth, W. There is a unique W that

 

optimal investment in crop production under open access, A. -

equates A’ and A: which is given by
_ t _ 8
m g( ),+b VK -A’+C, (A13)
8 0‘)
W, is a sum of A’ and a positive constant, C,. Below W,, A’ > A', and above W,, A’ < A.

 

W,-A’+

as shown in Figure A2. The size of W, is associated with the degree of overgrazing; the larger
C,, the larger W,, and consequently wider range of wealth supports A' > A'. C, is large
when (l) —m+g(*)- b is large and/or absolute value of g'(*) is small, which is the
condition that many households specialize in livestock husbandry according to propositions 7
and 8, (2) v is large, which implies that crop production is risky, and (3) K8 is large, which
implies relatively equal distribution of wealth (see the proof at the end of this section). This
means that large village livestock holdings under open access according to section 3.5 and

small socially optimal livestock holdings according to proposition 14. These conditions are
reasonable for overgrazing. However, if all households have wealth greater than W,, there is

no overgrazing because 21: > 21;. This contradicts proposition 16. Therefore, it is
-l a-

considered that W, is determined by wealth distribution so that some households have wealth
below W,.

When -m + g(*)— bs O (overgrazing does not always occur according to
proposition 17), C, in equation (A13) can be either positive, zero, or negative depending on
m -g(*)+b-vK’.

When m - g(*) + b -— vK’ is negative, —m + g(*) — b must be small negative or zero

and/or vK’ must be large positive. Small negative or zero -m+g(*)- b is the condition

when
mph
,1de

him

holdll
lnthl;

Ill! t

atgali
associ
ma]
Willi
A4 is
5ng
Under,

~m+

£3110“

held

163

when specialization in crop production is unlikely according to proposition 3. Large v
implies risky crop production, and is also the condition when specialization in crop
production is unlikely according to proposition 4. These conditions are associated with large
livestock holdings under open access when the expected return to crop production is higher
than the expected return to livestock holdings. Large Ks implies that large village livestock
holdings under open access and small socially optimal livestock holdings, as explained above.

In this case C, is positive and there is a unique W, (Figure A3). Therefore, the consequence

will be either overgrazing or undergrazing.

On the other hand, when m -g(*) + b - vK’ is positive, which is the case of large
negative -m+g(*)- b and/or small positive vK', C, is'negative. These conditions are
associated with small livestock holdings under open access, and large livestock holdings at the
social optimum. In this case there are two possible patterns of A’ and A. depending on the
position of C, and curvatures of A’ and A. as shown in Figure A4 and Figure A5. Figure
A4 is the case when overgrazing is still possible, but is less likely than the case in Figure A3.
Figure A5 is the case when there is no W,, and A’< A. always holds. This means
undergrazing always occurs, when C, is small enough (or large negative enough), that is,
-m+ g(*)- b is positive and very large and/or vK’ is very small though positive. Thus,
proposition 17 can be conditioned as proposition 18.

The relationship between K’ and the wealth distribution of a village is explained as
follows. Totally differentiating equation (13) with respect to K’ and 2% and rearranging

.r ,

yield

[23(9) + (j W, - K ’2 %)g"(s)JK’

i-l

dK’ _ .r i
32%, [28'(3)+(2W. - K’Z%)g"(s)]z;l--v

Equation (A13) is positive, because both numerator and denominator are negative; g'(s) < 0 ,

 

(A14)

8"(S)<O. and (2W, —K’2i)>0. As shown in Proof of Proposition 14, i- is a

n! i

164

" 1
concave function of W , and consequently 2 — decreases with an increase in the spread of
i-l r’

wealth. Therefore, K " is large in a village with relatively equal wealth distribution.

Proof of Proposition 18

See the discussion about Figures A3-A5 above.

Proof of Proposition 19
When there is overgrazing, 21.: < 21.: hold by definition. Thus, 8(2 11’) > g(ZL; ) > O

r-l

and O > g'(le)> g’(zfl) hold. Consequently with respect to the second terms of (16)

i-l

and (17). W.(g(211’% b) > way?) -— b) always holds.
When A’> A., or W <W, in Figure A2 and A3, %v¢,(4’)’ >év¢a,(A:)2 always

holds, too. Therefore, CE( Yf) < CE ( Yi') requires sufficiently small 4’2 [1212 1:) relative
5.1

in!

to 4.131211.) in order to offset W,(g(2l,’)-b)>W,(g(2L:)-b) and

l :21 ‘2 ’n’In’ “pn‘ ’n’ ".
5mm.) >5v¢.(4) .For A. gauzmqmzm whom. A. 2194A”

i-l

necessary, because 0 > g'(2£)> g’(ZLE) is assumed. Because 2 I: > L: is considered to
. n -l
hold in most cases, 4’21; > A:L: also holds in most cases assuming A’ > A.. Thus, the
i-
conditions for CE(Y,’) < CE(Y:) will be that 1; and/or 4. are very small, that is, household
i’s initial wealth is very low.

On the other hand, when A' s A., or W 2 W, in Figures A2 and A3,
1 , l .
-2-v¢,(A, )2 < 5v¢,(A‘ )2 always holds. If the difference between the two sides are sufficiently

large, CE(Y,’) <CE(Y:) will hold regardless of the first terms in (16) and (17). That is,

households whose initial wealth is very large will be worse off at the social optimum. Even if

165

the difference between %v¢,(A,’)z and %v¢,(A:)2 is not sufficiently large, if
4’2 (fix; 1:) is sufficiently small relative to A'KgKZd), CE(Yf) < CE(Yi') will hold.
h

Since 0> ((211)) 3121;) and A’s A. are assumed, 21; <L: is necessary for

run!
i-i i-l

[£213le3 to be sufficiently small relative to A'IJgKSK). it: <1; is the

condition when household i’s livestock holdings exceed the livestock holdings that the village

as whole will have at the social optimum. It implies wealth distribution is extremely unequal.

Q.E.D.

166

Appendix B

Effects of Privatization
Proof of l1” 2 L,

Under this privatization scheme, household i’s certainty-equivalent income when grazing land
is privately owned is
" . 1
CE(Y,")-mA, +l1g(—:.—2L,)--bl1 --2-V¢A,2 (Bl)

And the first order condition with respect to l, is

dCE(Y.") If". 11"".,Lf". ,
———'—--—m+ (-—.' )+-.— (-.- )-b+v -O(BZ)
311 811;; 2.2“ 1121. M
From (5), the first order condition under open access is given by
-m+g(21;)+l;g’(21_;)-b+vM'-O - (BB)
M M

Comparing (B2) and (B3), it can be shown that household i’s optimal livestock holdings

under private ownership is smaller than under open access. Assuming contrarily that I: z 1;,

p n n
then 24.21., 221., holds. Under this condition, the following three inequalities hold:
i - :-

agmsafim fingxgijm<z:g'<%§z;>s1.:gxgz;). a n a

v“' s vdzf. If the three inequalities simultaneously hold, then either (B2) or (B3) cannot
be true. This is a contradiction to the fact that both optima exist. Therefore, I," < I: must

hold.

Condition for cm?) 2 CE(Y,')

Using the first order conditions under open access and private ownership respectively, the
certainty-equivalent income can be expressed as
O n O l O
CE(YD ' MAI + 58(2Lr)‘ bE "' 5v¢i(Ad )2

n a (B4)
- 14:1:ng +§v¢.(4‘>’ + “cough-b)

167

P " . l
CE(Yf) - mA.” + Hafiz.)- b1: viva-(4”?
11,, ,, 121, l I: .. (135)
-A,"-.- "<-—. ') -v.( 4")” W.( (—. 7)-b)
[12118 [WA +2 4'21. + 311211

Comparing (B4) and (B5), it can be concluded that CE( Y,”) 2 CE( Y:) does not always bold
but depends on the relative magnitudes of l; and If , household’s wealth and the features of

n p n
the livestock productivity function. Because A. <4”, g(EL)< 301-1721,) and
hi

‘ i-l

n p n
g'(zu)<g'(%21:) hold, CE(Y,”)>CE(Y,') holds for households that satisfies

ml

1; >%Zl; . Even if L: sit-21; , if the difference is small enough, CE(Yf): CE(K.)
i- i-

will still hold. This suggests that wealthier households that have larger optimal livestock

holdings, L: , under open access are likely to be better off by privatization.

168

Appendix C

Effects of Quota
Under this quota scheme, the certainty-equivalent income for household i under the quota

scheme is

CE(YI')-m(4+— WNW-£35- —)g( L:- D)- -b(1;-—,,— -§-D)-lV¢,(4+D—ll-)
21; 21; ~ 21; 21:

full i-l i-l (-
(C1)
The difference in certainty-equivalent incomes for household i between open access and

quota is
CE(Y?)-CE(Y..’)D

-(I;--D—-—)g(2L-D)-leg(21§)+D£(m+b-V¢tA'-%V¢t?i-)
is "1 '4 2. 2i

 

 

£421;- D)(g(ZLI- -D>- g<2I;»+DI:<g «213-44., .. D.)
EL 2’s
(C2)

From the middle expression to the last, the first order condition under open access,
m -g(*)+ b—v¢,A: —l,'g'(*) -0, is used.

The first term in the large parenthesis is positive and common for the all households.
Thus, the sign of equation (C2) depends on how large the second term is relative to the fixed I
first term, which is determined by household 1.; and ¢,l1'. It is impossible to determine
specific conditions when the latter negative term is greater than the former fixed positive term
(in absolute values), but (C2) implies that some households may have smaller certainty-

equivalent income in this quota scheme than under open access.

169
Since D changes the sizes of fixed positive term and the household-specific negative

term, it may be possible to design the quota through adjusting D so that all (or most)

households can be better off by the quota.

170

Appendix D

Effects of Transferable Permits
Under this transferable permit scheme, the budget constraint for household i is

D
W.+(-Ln—-I.>p-4+I. (DI)

Using (D1), household i‘s certainty-equivalent income is given by

n 1 n
CE(le-MW.-L(p+1)+§gl.)+l.g(2m-bI.--2-v¢.(m-I.(p+mfzw

l-l
(D2)

The first order condition with respect to Z1 is given by
_ _g ”.._ ..,"... _... g"... _g_
m(1+p n)+g(21,) b+1,g(21,)+v¢,(w, l1(p+l)+n;l1)(l+p n) 0

i-l i-l
(D3)
From (42), optimal livestock holdings for household i under the transferable permits scheme
is

m<1+p- grain» b—v¢.<W. +ffijux1+ p- f)
t- -1

 

I: - (D4)

-V¢.(p +1)(1+ p -f) + g'(z 11")

i-l

Comparing I: given by (D4) with the household’s optimum under open access, 1;, given by

('7), it is not possible to tell which is greater in general.

171

Appendix E

Effects of User Fee
The budget constraint under this user fee scheme for household i will be

W. - :1, - A. + 1, (En
where household i’s certainty-equivalent income is given by

CE(Y,') - m(Wi - (1 + t)l.,) + L,g(2 l1) - bl1 - i—wMWl - (l + t)L,. )2 (E2)

And the first order condition with respect to L, is

aCE(Y,')_ "._ .,"._ .
—6L (l+t)m+g(§l.,) b+l1g (21,) v¢,A, o (133)

Equation (E3) is to be compared to the first order condition under open access, given by
equation (5). Assume contrarily that I; 21;, then g(L:)sg(l:), g'(fl) s g'(L:), and
4' s A: must hold. But those conditions cannot satisfy both (5) and (F3) at the same time.

Therefore, 11' < L; is required.

Figure Al

f(W)

172

Determination of Relative Size of Crop and Livestock Investments

high 2(m-g(*)+b)

 

A'>L

A.>L

low 2(m-g(*)+b)

 

b»
-- . ”A

 

l“.

 

 

W

negative 2(m-g(‘)+b)

 

 

 

Figure A2

173

Comparison of Household and Social Optima (Case 1, m —g(*)+b< O)

A*,A S

 

 

45 degree 1'

A*

r

W

Expected return to crop production is smaller than that to livestock.

C,>O

11’: A. when WsW,,and A’<A. when W>W,

174
Figure A3

Comparison of Household and Social Optima (Case 2, m —g(*)+bz O)

 

 

A*,A 5
45 degree '
AS+ C e
‘Ae
AS
Ce
0 W, ‘
W

Expected return to crop production is greater than that to livestock.
C, > 0

A’zA' when WsW,,and A’<A' when W>W,

Figure A4

Comparison of Household and Social Optima (Case 3, m -g(*)+bz O)

 
 

 

 

 

A*, A5
45degree l' ,
111*
/ TA 5
I s
A +Ce
%
0 We! We: W
Ce

Expected return to crop production is greater than that to livestock.
C, sO, but close to zero. A' < A. when W< W A' 2 A.

11’

when W,l s Ws W,,, and A’ < A. when W,, < W.

176
Figure A5
Comparison of Household and Social Optima (Case 4, m -g(*)+bz O)

 

 

A*, A5
45 degree I' '
l
A*
T
A S
AS+C
0
. W
Ce
\

 

Expected return to crop production is greater than that to livestock.
C, s 0 , but close to zero.

Always A’ < A..

Chapter 5

Crop Production under Drought Risk and
Estimation of Demand for Formal Drought Insurance
in the Sahel

0 Summary

The Sahel is a drought-prone region. But there is currently no formal drought
insurance; rather, households informally insure themselves through their income and asset
portfolio strategies. The concern is that their self-insurance is costly and not effective when a
drought affects all households over large areas. Formal drought insurance, on the other hand,
can pool risks across zones, and its cost can be below self-insurance. This chapter explores the
level and determinants of potential demand in the Sahel for a hypothetical formal drought
insurance called rainfall lottery as follows.

First, the conceptual part consists of six steps. The paper: (i) Sets out a utility
maximizing model under budget and labor constraints with self-insurance; (ii) Derives the
first order conditions that represent “self-insurance equilibrium”, which we are observing;
(iii) Sets out a similar model, but with the hypothetical formal drought insurance, or the
rainfall lottery; (iv) Derives the first order conditions that represent “formal-insurance
equilibrium”, which we cannot observe; (v) Defines “equilibrium premium rate”. This is the
premium rate at which the household is indifferent between the self-insurance equilibrium
and the formal-insurance equilibrium; (vi) In the case of a rainfall lottery, the equilibrium
premium rate is expressed by the ratio of expected marginal values of a good year (non-
drought year) and a drought year, i.e. the utility trade-off between two types of year.

Second, the empirical part consists of six steps. The paper. (i) Estimates parameters of
the behavioral equations at the self-insurance equilibrium; (ii) Calculates the expected

177

I78

marginal values of a good year and a drought year using the parameters estimated in the first
step, and obtains the equilibrium premium rate as defined above. To have the expected values,
I use rainfall distribution to be defined in (iii) and a rainfall cut-off point that separates a
drought year from a good year to be defined in (iv); (iii) Assumes that annual rainfall is
normally distributed. Its mean and variance are obtained from long-run rainfall statistics and
are assumed not to change over years; (iv) Defines a drought year by setting a rainfall cut-off
point on the normal rainfall distribution. This point can be assigned arbitrarily, and if rainfall
of a year is below the cut-off point, the insurant will win the rainfall lottery; (v) Calculates a
drought probability for each cut-off point, given mean and variance of normally distributed
rainfall; (vi) Defines “effective demand” for the formal drought insurance as the case when
the estimated equilibrium premium rate is greater than the drought probability that the cut-
off point implies. Note that the effective demand means that the household will be willing to
pay more premium than in the case of actuarially fair drought insurance, and that the
insurance firm will be able to make positive gross revenues.

Third, the analysis part is as follows. This paper (i) Examines the existence of
effective demand for the formal drought insurance for several drought probabilities; (ii)
Examines how the individual equilibrium premium rates vary with household assets and
characteristics via determinant analysis; (iii) Exogenously stratifies the sample in various ways
to see if that affects determinants. Panel data on households from three agroclimatic zones in
Burkina Faso are used: the Sahelian (unfavorable agroclimatically), the Guinean (favorable
agroclimatically), and the Sudanian (between the two). From the analysis part, the following
findings are salient

First, effective demand for the hypothetical formal drought insurance is found in all
zones, which implies inadequacies in households’ current self-insurance strategies. But the
demand is correlated with agroclimatic zone because drought risk differs over zones: it is
highest in the Sahelian zone and lowest in the Guinean zone. The effective demand is

distributed in the most narrow range of drought probability in the Sahelian zone (1596-2596)

I79

and in the most wide range of drought probability in the Guinean zone (lO%-35%). This
implies that households are more heterogeneous in terms of the level of self-insurance in the
agroclimatically favorable zone. In the determinant analysis, a 15% drought probability is
chosen as that at which most households are considered to demand the insurance.

Second, the determinant analysis shows that the effective demand depends on how
households manage drought risk ex ante and ex post; there is much heterogeneity in the
strategies to self-insure even within zones. The results show that this is true with off-farm
income only in the middle livestock stratum and with livestock holdings only in the lower
livestock stratum. On the other hand, in the upper livestock stratum, neither livestock holdings
nor off-farm income has a significant effect on demand for formal drought insurance.
However, since in this stratum households on average show the largest demand for the
drought insurance, their self-insurance is not sufficient. This is a sign that formal drought
insurance can substitute for livestock holdings at least partially among those who keep

relatively large livestock holdings without changing their welfare.

1 Introduction

This chapter explores the level and determinants of effective demand for formal
drought insurance in the Sahel, the drought-prone region of the West African semi-arid
tropics (WASAT). There is currently no such formal insurance; rather, households informally
insure themselves through their income and asset portfolio strategies. By studying how
households allocate inputs to production under informal self -insurance, their willingness to
pay for a hypothetical formal drought insurance is inferred.

In the WASAT, rural households engage in a variety of risk management/coping
strategies to smooth interyear consumption in the face of unstable output and drought risk
(Matlon, 1990); e.g. income diversification (Reardon et al., 1992), cr0p diversification

(Matlon, 1990; Norman, 1973), livestock holdings (Christensen, 1989), crop storage (Udry,

{Ell
incl

{no}

firm

“P‘

{351

3.
M.

'33?
‘E

180

1993), and informal credit (Udry, 1994). However, this paper is concerned with formal
drought insurance for the following reasons:

First, most informal strategies do not fully insure in the case of drought, which affects
all households within a village or even over large areas. During severe droughts, informal
transfers within a rural community, such as state-contingent transfers and remittances among
friends and neighbors, will not be effective if the community as a whole does not have
enough food. In addition, local interest rates increase because many households seek credit at
the same time, local wages decline due to increase in labor supply, and livestock prices drop
because of distress sales of livestock. Few informal strategies are immune against the
covariability problem; exceptions are seasonal migration and remittance from non-household
members living far away or abroad‘.

Second, except for the few informal risk-sharing across zones that do not experience
drought simultaneously, a formal institution at this moment is public food aid. However,
expectation of food aid may be causing a moral hazard problem to households.

Third, existing WASAT household strategies for risk management may be neither
cost-effective nor shared by all households. It is natural that self-insurance costly. But here, a
few important costs associated with self-insurance are to be mentioned. One cost is that
livestock holdings and crop stocks as precautionary savings are not productive. If households
can invest in more productive activities, their welfare will increase in the long-run. The other
cost is the degradation of grazing land. Since livestock is kept on open-access grazing land,
household self-insurance strategy through livestock holdings is likely cause overgrazing.

To overcome the problems above, therefore, formal insurance will be needed Formal
insurance can pool risks across agroclimatic zones and will possibly be less costly than
current informal mechanisms. But formal insurance institutions are rarely developed in

drought-prone regions of developing countries; in Burkina Faso there is none. Supply-side

 

_‘ 'I'hereismcvidencetlmtvillagesthathaveagoodaccesstothem'banarea(capitalcity)m'ebetter
immedagainstdroughuiankinaFasoJ'hisrestdtsimpliesthatinformai mechanisms aaosszonesare
worldng(Sdmrm'. 1995).

181

causes of market failure, such as moral hazard and high transaction costs, have been relatively
well studied (Binswanger, 1986). To avoid the supply-side problems, a “rainfall lottery” has
been considered, in which if annual rainfall reported by the regional whether station is lower
than a level predetermined by an insurance company, the company will pay an indemnity
(Gautam et al., 1994; Hazell, 1992). This formal drought insurance scheme can require less
transaction cost because the insurance firm does not need to inspect drought damage of each
insurant, and can avoid moral hazard because indemnity payment does not depend on
individual’s drought damagez. Following them, this paper applies this hypothetical drought
insurance scheme.

Since the rainfall lottery is not a perfect insurance, the question is whether households
are willing to buy the hypothetical drought insurance as a substitute for and/or as a
supplement to the existing informal insurance strategies or not. If the answer is yes, then the
next question is who will buy it. This paper addresses those demand issues, which has not
been empirically examined.

Demand for the hypothetical drought insurance is inferred from household
production input decisions without formal insurance. This approach was developed by
Gautam e1 al. (1994), following from literature which infers risk attitudes from risk avoidance
behavior in agricultural production (Antle, 1987; Antle, 1989; Moscardi and Janvry, 1977).
The model has been modified in the context of the WASAT and used to estimate demand for
the hypothetical drought insurance (Sakurai et al., 1994). The method this chapter uses is
summarized as follows: (i) Set out a dynamic household model, in which a household
maximizes its utility under budget and labor constraints with self-insurance; (ii) Derive the
first order conditions that become the behavioral equations for precautionary saving, crop
production, and off-farm activities. They represent “self-insurance equilibrium”, which we

are observing; (iii) Set out a similar dynamic household model, but with the hypothetical

 

2 Exactly speaking, there will be small moral hand because a household will choose riskier
strategies withtheformddroughtinsmmcethmwithoutit.

182

formal drought insurance, or the rainfall lottery; (iv) Derive the first order conditions that
represent “formal-insurance equilibrium”, which we cannot observe; (v) Define
“equilibrium premium rate”. This is the premium rate at which the household is indifferent
between the self-insurance equilibrium and the formal-insurance equilibrium; (vi) In the case
of rainfall lottery, the equilibrium premium rate is expressed by the ratio of the expected,
marginal values of a good year (non-drought year) and a drought year. This is the marginal
rate of substitution between a good year and a drought year, measuring the utility trade-off
between the two types of year; (v) If the equilibrium premium rate estimated above is greater
than the probability of drought3, there exists “effective demand”.

The approach of this chapter significantly differs from previous papers in the
assumption about the household’s view of the world (summarized in Table 1). In Gautam et
al. (1994), for example, all households have the same binary view of the world: a good year
and a drought year, and believe the same, fixed probability of drought based on which each
household optimizes the intertemporal allocation of resources. The household’s definition of
a drought is shared by the insurance company which runs the rainfall lottery. That is, the
definition of a drought is exogenous to both households and to the insurance company. In
Sakurai et al. (1994) there is no a priori definition of a drought. Each household has a
binary view, but households’ subjective beliefs of the probability of drought differ although
they all know the true distribution of annual rainfall. This means that households define a
drought based on their welfare, and consequently their subjective probabilities change
according to their wealth and experience of drought. That is, they are Bayesian decision
makers: Based on (subjective) belief, each household optimizes the intertemporal allocation
of resources. On the other hand, the insurance company can set any arbitrary rainfall level to
define a drought. Therefore, a drought year by a household’s subjective judgment is not
necessarily a drought year by the insurance company’s definition. In this chapter, however,
households do not have a binary view of drought, but they all know the true distribution of

 

3 'I'heprobabilityofdroughtisdcfinedlaterinthissection.

96W
@133!
it c
htile

IN]

05;

“PE

'9

183

rainfall based on which each household optimizes the intertemporal allocation of resources. If
an insurance company provides a rainfall lottery, then households come to have a binary
assessment of drought with the probability of drought arbitrarily set by the company. The
new approach of this chapter can, thus, avoid the strong assumption that a household has a
binary view without formal insurance, either objective or subjective. This is important because
we cannot know whether households have a binary view or not and what probability they
believe, merely based on observation of households’ behavior without formal drought
insurance.

Based on that assumption, probability of a drought year is defined as follows: (i)
Annual rainfall is assumed to be distributed normally with fixed mean and variance obtained
from long-term rainfall statistics; (ii) Households and the insurance firm know this
distribution, but there is no objective definition of a drought year; (iii) The insurance firm
sets a rainfall cut-off point arbitrarily to define a drought year for the rainfall lottery.
Households who buy this lottery now have a binary view of the world; (iv) Since a normal
distribution is assumed, the probability of a drought year is easily calculated for a given cut-
off point. Thus, the cut—off point can be expressed as either a rainfall level or a drought
probability. Since they are equivalent, the drought probability is used instead of a rainfall cut-
off point; (v) Thus, a low drought probability implies a low cut-off rainfall level and low

expected rainfall in a drought year, that is, a severe drought occurring with low frequency.

2 The Model
2. 1 Time Frame

Roughly speaking, a household makes two major decisions about its income strategies
during a year". off-farm work (mainly done in the dry season in rural Burkina) and
agricultural production (done only in the rainy season). That agricultural output is a key
variable leads us to the concept of “harvest year”, which begins with the harvest at the

beginning of the dry season and ends at the end of the rainy season, with the simplifying

an

5638

Elli

COW

an
Inc!

“HP

xiii
Sat]
at

1M

Trim,

184

assumption that all harvesting is done instantly at the beginning of each harvest year. The
harvest year in the survey is defined to begin September 1 and to end August 31. The dry
season of harvest year t is named “season 1 of year t” and the rainy season of harvest year t
is named “season 2 of year t”, as shown in Figure 1.

At the beginning of season 1 of year t, the household knows its initial wealth which
consists of the harvest of the previous year t-l, carry-over wealth from year t-l, and off-farm
income. Carry-over wealth consists of crop stocks and livestock holdings. Off—farm income is
assumed to be “expected with certainty” at the beginning of season 1. Off-farm income
includes pensions, remittances from non-household members, and income from local
employment and migration. Given knowledge about initial wealth as well as its characteristics,
the household decides: (i) how to allocate family labor between on-farm and off-farm
activities in season 2; (ii) how to allocate its initial wealth among consuming, precautionary
saving, buying inputs for agricultural production, and buying (hypothetical) insurance if
available; and (iii) how much to diversify crops. Although agricultural production takes place
in season 2, it is assumed that decisions about agricultural production, such as input use and
crop diversification, are made simultaneously at the beginning of the harvest year.

In season 2 the household undertakes both agricultural production and off-farm work
according to the input decisions made in season 1. However, agricultural output also depends
on rainfall in season 2 and on household-specific shocks, which are stochastic. I assume here
that uncertainty in agricultural production comes from rainfall uncertainty, but the magnitude
of the drought shock on agricultural production depends on the household’s characteristics

and its decisions about input use and crop diversification.

2.2 A Dynamic Household Model
First a model without formal drought insurance is considered. I apply a standard
intertemporal utility maximization problem given by a time-separable lifetime utility function

for household i as follows:

185

T
max Euzbiklli(c‘ii+h Link) (1)

where i refers to household, t refers to year, U,( ) is the one-period utility function, C,, is real
consumption in year t, 1,, is leisure in year t, 6, is the discount factor, [3,, is the expectation
operator (conditional on information available at year t), and T is the end of the household’s
planning horizon. Standard assumptions about the utility function apply: U,( ) is

monotonically increasing and concave in C,, and L,,. See Gautam el al. (1994) for

theoretical considerations.
As is shown in Figure 1, year t starts with the harvest from season 2 in year H, and
therefore the household’s initial wealth is given by
We - 5..-. + Y.“ +Mt. (2)
where S,,_l is carry-over wealth from year H to year t and Y,,_, is agricultural output in year

t-l. M,, is wealth that is realized in year t with certainty, but is conditioned by year t-l’s state
(good or drought), and therefore is predetermined at the beginning of year t. M,, includes
off-farm income to be earned (with certainty) in season 1 of year t, plus gifts and public food
aid received in year t.

Labor constraints are as follows:

M+4-%+4+4 (a
where N,, is the exogenous, total time available to the household in year t, [1", is hired labor
and 1,, is leisure that appears in the utility function. Equation (3) means that leisure is
determined by on-farm labor supply ([1,), hired labor (11:) and off-farm labor supply

including local on-farm employment in season 2 (11",). Off-farm labor supply in season 1 is

predetermined at the time of decision making for year t and does not appear in the labor

constraints. This implies that off-farm labor supply in season 1 is already subtracted from

total available labor, N,,. The labor variables are expressed in monetary terms.

The budget constraints are as follows:
Mt+lz-C0+Kn+l:r+sir (4)

186
Equation (4) means that the household’s initial wealth (W,,) and off-farm income in season 2
(1;) are allocated in period t among consumption (C,, ), purchased farm inputs for
agricultural production (K,, ), hired labor (1.1;), and savings (S,,).
As defined in equation (2), initial wealth (W,,) includes carry-over wealth from year t-

1 (S,,_,), but in equation (4) I assume that there is no interest accrual. In rural Burkina Faso

there is no formal savings mechanism (rural bank): livestock holdings and crop stocks are
used as substitutes. Livestock may provide positive interest (reproduction), but on the other
hand it is subject to capital risks due to price fluctuation. To make the model simple, I assume
there is no capital gain or loss on wealth. Borrowing is handled as a subset of saving: if the
saving decision in period t (S,,) is negative, it implies that the household will be a net
borrower in year t. No interest for borrowing is assumed“.

Agricultural output in year t (Y,,) is a function of endogenous purchased farm inputs
and labor (K,, and 11],) and crop diversification (D,,) in year t, and exogenous or
predetermined variables including yearly rainfall (R,) and a vector of household/plot
characteristics (X,,):

r'..-F(11’..K...IJ...R.X..) (5)

Crop diversification (D,,) is a Simpson index‘. D,, is greater than or equal to zero and
less than unity; D,, is exactly equal to zero when a household produces only one crop, and is
unity when a household produces infinite kinds of crops, which is impossible.

Without formal drought insurance, the household forms its expectations based on a
given rainfall distribution. By using the equations above, the dynamic optimization problem

for the household can be written as

 

‘ Although no interest for borrowing is assllmed, the household cannot borrow limitlessly because
by the end of the household’s planning horizon (period T), all debt must be repaid. In other words, the
household is not free from life-time budget constraint. Moreover, since there is uncertainty in future
income, borrowing is riskier for the household than without uncertainty.

5 D - 1 _ 2 planted area for crop k (ha) 2
“ t household '3 total planted area (ha)

 

187
V.<W >- maxU.C< I...>+6.f_: V...(W.)f(R)dR <6)

01'
V,,(VV,,) - “g“(/KW“ + L; -Kit- Lil; -Sif’ Nit + L: " Lift -
e. (7)
+ 6.1,V...<S. + Y. + M...)/<R>dR

where 11,, - (D,,, [1],, 13,, 11",, K,,, S,,) is the vector of decision variables of period t, and
Vu() is the maximum value of future utility discounted back to t, and f(R) is the

probability density function of rainfall. Given the standard assumptions for the utility

function, the value function, V( ), is also monotonically increasing and concave in W.

2.3 The First Order Conditions
Assuming that households behave optimally without formal drought insurance, what

we observe is a “self-insurance equilibrium”, given by the first order conditions obtained by

differentiating equation (7) with respect to the control variables, II,,. They are as follows:
6V6 (W

Dir: ‘5 YWR 0 (8)
I; n+1 )ZDit
_a_,U (w )aY,

f l . i

2 +6 V”‘? 1+ 0 9
1“ al.. «9W... aI.’. 41.1.)... ‘ ’
[1,, a_,_U _a_,__U. _,__ o (,0)

76.. at.

,. 8U, 6U

: -—’--O 11
1.. EC" —+ a," ( )

. L___Vit+1(m:+l) LY"
K,,. “of: aw“ “K —f(R)dR- -o (12)
S,, -111: 61.. W,,..- 0 (.3,

Wit-+1
Under the hypothetical formal drought insurance scheme, or the rainfall lottery, the
insurance company sets a rainfall cut-off point denoted by R'. If the accumulated rainfall for
the year is below the cut-off point, the year is defined as a “drought year’, a “good year”
has rainfall greater than or equal to R'. Consequently the definition of drought year is
weather-station-speciflc rather than household-specific.

188

If this insurance scheme were available, household i would have a binary view of the
world, and be at a new equilibrium called “formal-insurance equilibrium”. Budget constraint
(4) would become

W,,+l,°,-C,,+K,,+L’,',+.S',,+pl,, (14)
where p1,, is the insurance premium that the household would pay if such a scheme were
available, and p is the known exogenous premium rate against total indemnity I,,. The total
indemnity, I,,, is decided by the household with other input decisions, II,,. If year t is a good
year, using superscript g, the initial wealth of year t+1 is W,,,l - S,, + Y: +M,f. And similarly,
if year t is a drought year, using superscript d, initial wealth in year t+1 is
W:,, I S,, + Y: + M: + I,,. These two equations go into the dynamic optimization given by
(7)-

The formal-insurance equiliirium is given by values (W,,,v W:,,,C, ,,I_,,, flu, I,,).

If I,, > O, the first order conditions will be

4
D'it 6,fW% ___:,:fd (m+ +6ij:: Wit+:V(W n+1 )‘a—Y:f8(R)dR' 0 (15)
‘VIt-oi ‘VI'HJ
.. MC in _._<__._._..W‘.> lam. V. .(W...)_. . _
IVA—LL ”fl—M 74’ “M of“ all. (”MR 0
(16)
,1... 6U.(C.,. 5,) _ aU,(é,,. I,“ ,) _ 0 (,7)
“ 6C. at.
1,:- _aU,(é,,.g,) av,(é,,.g.)_o (.3)
" 60.61..
U(C dV (W4 )_6_,,_Y 6V (W3, )6Y
K,,- -.a_1_u:£tl+5 5f;_251_m_ W,,“ aKudf (R)dR+ M5f;._li.fl:,lt_La_KI:f-8(R)a- -0
(19)
s,;— “U22" —I—L'-'—)+,df,‘7vaw (W )f"(R)dR+ dfi—UW—w-‘W (W )‘(de- o (20)
mt n+1
In: -:C-U(—C"AL+6 aff;_i£fl_ii_Lfd (Ina- -gi(Cit’ in! W“) "' 0 (21)

m1

where f (R) and f‘ (R) are truncated distribution density functions: that is, the rainfall
distribution density function, f ( R), is truncated by the rainfall cut-off point, R’.

189

If the first order condition (21) is evaluated at the self-insurance equilibrium, it will

become

3” ’I___I.( o f "91"": (R)dR -g.IC- I. W5.) (22)

“+1

Then, g,(C ,,, 11,, W:,,) s 0 shows no potential demand for the formal drought insurance at

the self-insurance equilibrium, while g,(C,,, L,,,W )>O shows potential demand.

it+1
Similarly, all the remaining first order conditions (15)-(20) are evaluated at the self-insurance
equilibrium. From (20) and (22), conditions for potential demand for the formal drought

insurance at the self-insurance equilibrium is expressed as
if: ‘7Vm;(wg+l) d(R)dR

n+1 ¢it
(23)
“fa Vi£+1(W m; )(f4 MM'I’ +1:aavig+1(wjt+1) f8(R)dR -1+¢it

Wr+1W't+1

 

where ¢,, is the marginal rate of substitution at the self-insurance equilibrium between

drought and good years for a given threshold R.. on is the utility weight for a drought year

if the weight for a good year is set to be unity, and measures the utility trade-off between two

types of years. ¢,, is defined by

V W“
L’s-Mir ( ).,R),,R v"
n+1 . (24)
¢“' av (W‘ ) T
12% «gym V

n+1

 

In (23), if it"; is equal to p, the household will be indifferent between purchasing and

if
not-Purchasing the formal drought insurance. Therefore, 11:— is called household i’s
+ it

“equilibrium premium rate” at the self-insurance equilibrium, and is denoted by p: The
demand is “effective”, when a household is willing to pay more premium than the premium
of an actuarially fair drought insurance. For the demand to be effective, it is necessary that
p; is greater than the probability of a drought year given by q - Pr(R s R.) . When effective
demand exists, the insurance firm can set premium rate higher than the probability of a

drought year so that it make enough revenue to cover administration costs and to make a

190

 

profit from the insurance scheme‘. From (23), the condition becomes q < p; - 4," . which
if

is equivalent to

_ —q— 25
¢..>1_q ( )

Using ¢n from (24), the first order conditions (15)-(20) evaluated at self-insurance
equilibrium give the self -insurance equilibrium relationships that show how the household
weights future cost and benef it".

Equation (15)Y can be rewritten as

62’
I! R dR- I!
f,,. -:-Y-D'“f( ) —¢-._. 60“

—f (R)dR (26)
The left hand side of equation (26) is the marginal cost of crop diversification, and the right
hand side is the marginal benefit of crop diversification accrued only in drought years. These

derivatives should have opposite signs, but we cannot assign a priori a sign to those

derivatives. Moreover, both sides can be zero: in this case household i chooses crop

diversification to maximize both good year and drought year outputs. If such D,, exists,
equation (26) will have no role in estimating ¢,,.
From (l6)-(20), the labor condition is obtained as
3
1+4», 1:574 (7:, (R)¢fl2+¢,,f;a aZf(R)dR (27)

The left hand side of equation (27) is the marginal return to off-farm labor, i.e. the wage rate
(=1), weighted by l and ¢“. The right hand side is the amarginal return to labor used in

agricultural production, weighted by l and d“. If I; Yf‘(R)dR—“—- 1, any 40,, is not

determined by this condition. To have a unique, positive d“, both f ’(R)dR >1 and

1:31:

 

‘ This is a mdition that the insurance firm can Me a positive, gross revenues. It is a necessary
condition for the firm to cover administration cost and to make a net profit, but not sufficient.

7 A critical assumption to derive the relationships is that the covariances between marginal value of

valuefuncticnmdthemarginal productivitiesofinputsm'ew'o, although they areaffectedby rainfall. That
wva( Vii-61 LY“ ____)- O (IN ( avlhl mv(_ anHl L) -0

aw, ’uaz) ’ aw,’ 011, aw,’ 3K,

       

191

1:311)“ (R)dR<l are required, assuming f;Y,,f‘(R)dR>f; Y,fd (R)dR This

condition means labor input is less than the optimum in a good year, and greater than the
optimum in a drought year.

Equating (l9) and (20) yields a saving equation as
°° (W,, R'a'lYud
1+¢.-. '1}. 6K, (R)dR+ Inf“ —-_-f" (R)dR (28)

 

 

The left side is the marginal return to savings, which is assumed to be unity, weighted by l

and ¢,,. The right hand side of equation (28) is the marginal agricultural product of
purchased variable inputs, weighted by 1 and ¢,,. This saving equation has implications for

determination of ¢,, similar to those discussed with respect to the labor equation.

3 Data

The household data used to estimate the models were collected in a farm household
survey in Burkina Faso conducted by International Crops Research Institute for the Semi-
Arid Tropics (ICRISAT). The panel data used cover three full harvest-years, 1981/82-1983/84
(see Figure l), with a sample of 89 households in each year, spread over five villages, in three
agroclimatic zones: the Sahelian, in the northwest; the Sudanian, in the Central Plateau; and
the Guinean, in the southwest

These three zones differ in rainfall level and variability. Annual rainfall data and
long-term statistics are shown in Table 2. The village-level rainfall data were observed by
survey enumerators in the surveyed villages, and the long-term averages and standard
deviations are from weather stations that cover those villages“. The Sahelian zone has the
lowest and most variable rainfall, so cropping is the riskiest. The Guinean zone is moderately-

favored agroclimatically, with medium to high rainfall that is least variable, thus cropping is

 

3 The weather stations are located in Djibo (Sahelian), Yako (Sudanian), and Boromo (Guinean).
Data are through 1983 and include 29 years for Djibo, 38 years for Yako, md 58 years for Boromo. Source:
Sivakumar and Gnoumou (1987).

192
much less risky. The Sudanian zone is between the two zones in rainfall level and variability.
Cropping is only rainfed in all three zones, and undertaken only by smallholders. Note that

most rainfall levels during the survey period are lower than the long-term average.

4 Empirical Estimation

The empirical analysis goes as follows: (i) Estimate parameters of the behavioral
equations at the self-insurance equilibrium; (ii) Calculate expected marginal values of a good
year and a drought year using the parameters estimated in the first step, and obtain the
equilibrium premium rate. To have expected values, we need to know rainfall distribution and
a rainfall cut-off point that separates a drought year from a good year. As noted before, the
cut-off point is expressed as a probability of drought; (iii) Examine whether there is effective
demand for the formal drought insurance for several drought probabilities; (iv) Examine how
the household equilibrium premium rate varies with household assets and characteristics via

determinant analysis; (v) Split the sample in various ways to see if that affects determinants.

4. 1 Production Function
For equation (5) a modified Cobb-Douglas function with interaction terms is

specified as follows:
In Ya ' (511111,, +flrln Kit +nrlnDn+PrlnAn +wR¢

" (
+a.&.1nl.£+fi.R.InK.+mR.1nD..+u.R.InA.+ 211ml. +e.

i-l

29)

where subscript i refers to household, v refers to village and t refers to year. The variables are:

Y“, household i’s agricultural output in 1000*ch CFA (FCFA) per hectare; 11’“ household
i’s farm labor input (both own and hired) per hectare (in 1000*FCFA) assuming that labor is
paid the market wage; .K,,, household i’s purchased variable inputs per hectare (in
1000*FCFA ), which includes purchased fertilizer, and seed (both own and purchased); D,,,
household i’s crop diversification index as defined in footnote 1; A“, household i‘s total

cultivated area (hectares); R, , rainfall in village v in year t (in 1000 mm); III-4 , a dummy for

193
household i, or household i’s fixed effect; c,,, the disturbance term. This specification
includes interaction terms between inputs and rainfall, which capture change in marginal
productivity depending on rainfall. This production function is estimated separately in fixed-

effect estimation in each agroclimatic zone (Table 3).

4.2 Estimation of Demand for Drought Insurance (:1)

Applying the production function to the optimality conditions (26), (27), and (28),

and solving for 4)“, results in the following three equations:
f:.(n. + an.)Y.ff(R)dR
-f:(m + n.R )Y. 3(de
¢,_f; (0 +a.I§)Y ..ff(R)dR- 11’. (31)
11. -f (a. + a.R.>Y.f;‘(R)dR
“m. +fi.R.)Y.f:< (.de K.
K. -f (fl. +fi.R.)Y./;’ (IMR

Since 40,, is obtained in each equation above, it now has superscripts D, L, or K. a, , a2 , A,

D
it

 

(30)

 

 

(3 2)

fl, , n, , and n, are parameters obtained from the estimation of the production function
specified as (29). 11,, and K,, are production input decisions, the same as defined in the
specification of the production function. They are endogenous and are assumed to be
decided at the beginning of each year before knowing the level of rainfall. R; is rainfall cut-
off point for zone 2, and f:(R) and f: (R) are truncated rainfall distribution density
functions for zone 2. To have expected values based on limited long-term rainfall statistics
(mean and variance) for each zone, rainfall is assumed to be normally distributed and
independent over years. Here we use long-term mean and variance because we think that they

are more appropriate for estimating demand for drought insurance’. The relationship

 

9 Asnotedinsection3andshowninTablel,averagerainfalldmingthesm'veyperiodislowerthan
the long-term average. Using rainfall series in this region, we examine how the change in rainfall
distribution affects frequency of drought defined by this scheme (Appendix A). The results suggest that the
insurance company should use the long-term distribution because the short- term distributions (in this case
lOyeu's) vary alotdepcnrhngonwhrchdecadetodroose.

194
between the assigned probability of a drought year, q , and the rainfall cut-off point, R; , in
each zone are shown in Appendix B. Using those R; , 4’: , El; , and 4’: are calculated for each
case in equations (30), (31), and (32) respectively. The methods are in Appendix C.

A unique 4’ attributed to the group is estimated in a system as follows:

¢.’.’ - ¢ + e? (33)
41!: - ¢ + e3 (34)
4’5 - 4’ + 65 (35)

The system (33)-(35) is estimated by iterated feasible generalized least squares and ¢ is
obtained as a coefficient of unity for three equations under equality restriction across
equations. Correlation in disturbances is assumed. The three zones are estimated together, and
then separately to know the differences between zones for each probability of drought given

in Appendix B. The results are shown in Table 5.

4.3 Determinants of Demand for Drought Insurance (4’)

¢ is estimated as a measure of risk aversion for the community. However, it varies
with household wealth across a community or even across years for a given household
depending on shocks received. Thus I examine determinants of 40. Within a household, ¢
changes each year depending on initial wealth. But because 43 is used as a measure of
demand for drought insurance (reflecting a fixed preference), I use two different
specifications to examine effects of household wealth. The first is a cross-sectional
specification, and the second one is a fixed-effect specification. The cross-sectional system

specifications as follows”:

“I: " Zgjffi + zgrxu + V: (36)
“I; " 2&1 11+ Egrxu +V: (37)

 

‘0 This is not perfectly cross-sectional. bee-Ire time-variant variables are used as in the pooled
regression. However. here it is named cross-sectional regression, because the time-invariant variables used
in equations are time-means to capture cross-section effects.

I95
¢if ' 2g) 11'" Zgrxw + v: (38)
i

o L x
¢u , ¢n , and 4’1: are calculated based on (30), (31), and (32) respectively, as explained above.

D L

v,, , v,, , and v: are disturbances. The system is estimated by iterated feasible generalized least

squares. Equality restrictions across equations are imposed on C, and 51 respectively, and

correlation in disturbances is assumed.

X

I, are time-invariant variables so as to reflect average characteristics rather than
time-varying decisions. The variable set includes a constant, the household’s average off-farm.
income in season 1 (local self-employment, local agriculture employment and migration
income), its average livestock holdings, its average crop production, its average amount of
cr0p gifts received from other households, the average amount of public food aid the
household received, its cultivated area, and overall soil quality of the household’s plots (how
many years a household continues to use the same plot on average weighted by plot area). All
variables are expressed in 1000 FCFA per adult equivalent (AE) except for cultivated area
which is in hectares per AE and overall soil quality which is in years. All of these variables are
part of the wealth that the household expects to have. Means and standard deviations of those
wealth variables are shown in Appendix D.

X n, are time-variant variables. They include the household’s characteristics and the
characteristics of the household’s plots. The household characteristics variables are the AE
size of the household, the age of the household head, the number of wives of the household
head, the dependency ratio (percentage children younger than 15), a dummy for compound
leader (1 if the leader of the household is also compound leader), a dummy for single
conjugal unit (1 if the household has a single conjugal unit), a dummy for animal traction (1
if the household uses animal traction). Household plot characteristics are: soil-type
diversification index (using a Simpson index), a toposequence index, and the average distance
between the residence and the plots weighted by plot size (Km). The soil-type diversification

index measures heterogeneity of soil types among the household’s plots. In general, the more

196

diversified the soil types on a given farm, the less risky is crop production. The toposequence
index is the household‘s average of plots’ toposequence index assigned as l=uppermost,
2=next to l, 3=mid~slope, 4=next to 5, 5=swamp. In general, the higher the toposequence
index, the less risky is crop production.

Characteristics of a household that increase its capacity to diversify income sources,
either within crop production or within overall income, will have a negative effect on 4: , and
those making the household more risk averse will have a positive effect on ¢. But those

effects are ambiguous a priori.

The fixed effect specification is as follows:

¢.’.’ - 29X... + u. + v: (39)
WI; ' 2;;er + u; + V: (40)
¢.’.‘ - 29X... + H V.’.‘ (41)

D L
u, is a household specific (fixed) effect, and the X“, are time-variant variables. 4’1: , ¢n , and

L

x
‘1: are the calculated 4: , and v: , v,, , and v: are disturbances. The system is estimated by

iterated feasible generalized least squares. Equality restrictions across equations are imposed

on g, and u, respectively and correlation in disturbances is assumed. There is no time-
invariant variable (X,,) other than u,. Therefore, u, captures the time-invariant part of
household and plot characteristics, as well as the household’s average wealth as a fixed
household characteristic. The time-variant variables X a: are the initial assets of each year:
off-farm income to be earned, livestock holdings, crop production of the previous year,
public food aid to be received, crop gifts from other households to be received (in 1000
FCFA per AE). and cultivated area in hectares per AE As explained above in the section. on
the dynamic household model, I assume that off-farm income in season 1, food aid, and crop
gifts are expected with certainty when a household makes a decision.

In this fixed effect specification, we can know the effects of change in the household
wealth level on demand for drought insurance. Because I use the agricultural output of the

previous year as an initial asset variable, data from the first harvest year has to be dropped.

197

Using those two specifications, I estimate the effects of exogenous or predetermined
variables on demand for drought insurance, using the full sample as well as using sub-groups.
Sub—groups are formed by zone or by stratification based on household average livestock

holdings per AE: below 25 percentile, above 75 percentile, and between the two.

5 Results and Discussion

41 is a function of the expected marginal productivities of drought and good years,
that is, the household’s input decisions to avoid drought risk directly determine demand for
drought insurance. Since the expected marginal productivities depend on rainfall distribution
and the drought probability assigned, it depends on them too. ¢ is estimated over a range of
drought probabilities, from 10% to 40%, to see how ¢ changes as assigned drought
probability, q, changes.

The relationship between group average 40 and q is interpreted as follows: When
considering a household group, except in the case of an extremely low q , the lower q , the
more households in the group have effective demand, and when q is above some high point,
no household demands the insurance hence the group’s average 4: is close to zero. This is
because a low q implies a severe drought, while a high q implies a mild drought as explained
in section 1. Note that since this s is a zone-level average, zero .1 does not necessarily mean

that no household demands the drought insurance.
The change of individual 41,, is interpreted based on the definition given by (24),

I
d I

111,, - —r, where V“' and V’ are marginal values of the value function given by the
Vs

dynamic optimization problem, equation (6). As intuitively seen in Figure 2, drought
probability, q, affects the combination of the two slopes and as a result gives a different ¢,,.
This relationship implies that a higher probability of drought makes the difference in

expected wealth between good and drought years smaller, and as a result, gives a smaller

I98

V—dr. If V“', is large enough relative to V8,, the household will demand the drought
v8
insurance, while if they are close, the household will not demand it.

Therefore, households that are highly risk-averse and/or have a large drought shock
have high ¢' s. Thus, the relationship between asset variables and p is considered as follows:

(i) Assets can have a positive effect. In this case, there will be two different situations.
One is the case when the asset allows households to make risky decisions because the asset
provides ability to smooth consumption. The other is the case when the asset is safe, but its
insurance effect is not sufficient. For example, a household will tend to keep a safe asset if the
household is very risk averse, but if the asset does not effectively smooth income, the
household will have high 43. This means that a more risk-averse household will allocate more
labor and capital to relatively safe assets and/or risk reducing activities, such as off-farm
activities and savings (livestock holdings) than to risky crop production. But if they do not
reduce the size of drought shock enough, the household will have a large ¢. These two
effects can be distinguished by regression specifications: the former is observed in the fixed
effect regression and the latter is observed in the cross—section regression.

(ii) Assets can be a negative effect This is the case when accumulated assets insure
household income effectively. For example, a household that is able to earn enough off-farm
income and/or to keep large livestock holdings will have a small drought shock, and also have
a small 4».

Thus, assets have two opposite effects on ¢. I name the case (i) "risk aversion effect”

and the case (ii) "income effect”. Because of those two effects, impacts of wealth variables,

especially off-farm income and livestock holdings, are a priori ambiguous.

5 . 1 Production Function
Table 3 presents the parameter estimates of the production function. Using estimated

parameters, I evaluate signs of marginal productivity and effects of rainfall on marginal

199

productivity at drought probabilities of 15% and 35% to show the production function is
reasonably estimated. The results are summarized in Table 4, and the derivations are in
Appendix E. Since the production function has interaction terms between inputs and rainfall
as shown equation (29), the marginal productivities depend on rainfall level. Therefore, it is
possible that a marginal productivity has different signs at 15% and 35% drought

probabilities“. The reason why 1 choose 15% and 35% drought probabilities is based on the

results of ¢ estimation explained in section 5.2.
2

anR

 

In general, when > 0, input Q is risk-increasing (a decrease in rainfall causes

a decrease in the marginal productivity of Q, or equivalently, the marginal productivity of
2

 

rainfall increases with increasing use of Q), and when a? R <0, input Q is risk-decreasing

(a decrease in rainfall causes an increase in the marginal productivity of Q, or equivalently,
the marginal productivity of rainfall decreases with increasing use of Q). The optimal
application of a risk-increasing (decreasing) input under risk averse preferences is always less
(more) than the risk-neutral level of input use (Ramaswani, 1993).

The results shown in Table 4 are interpreted as follows:
(i) Labor

With respect to labor input, positive marginal productivity in both good and drought
years is expected, but whether its use increases or decreases risk is ambiguous a priori.

In the Sahelian zone, labor has a negative marginal productivity and is risk-decreasing
in a good year, while it has a positive marginal productivity and is risk-increasing in a
drought year. Both are significant. The marginal productivity turns negative at 447 mm

rainfall and above, and at the mean rainfall (480 mm) it is still close to zero, although it is

 

aY
“ ThesignsinTable4areobtainedas follows; For example, toknow thesignofthe Sahelian 3L7
in drought years at 15% drought probability, first, from Appendix B expected level of rainfall in a drought
year at 15 % drought probability is found, which is 227 mm. Second, looking at the column that covers

61'
227m in Table E1, positive sign for the Sahelian W is found.

200

negative. Those signs imply that labor input is close to the optimum in a good year and is less
than optimum in a drought year at a given labor input level decided before knowing rainfall.

In the Sudanian and the Guinean zones, labor has a positive marginal productivity
and is risk-increasing. These signs imply that if a household uses more labor per hectare,
yield per hectare will be higher in both good and drought years, but greater labor use causes
the drought shock to productivity to be larger.

(ii) Capital (purchased variable inputs)

Capital also should have a positive marginal productivity in both good and drought
years, but whether its use increases or decreases risk is also ambiguous a priori. The estimates
are not significant in the Sahelian and the Guinean zones, probably because their use of
fertilizer is small and ineffective and/or seed is overused.

In the Sahelian zone, the signs imply that at a given capital input level, capital has a
negative marginal productivity and reduces risk in a drought year, while it has a positive
marginal productivity and increases risk in a good year. Therefore, capital acts as insurance in
a drought year.

In the Sudanian zone, capital has the same property as labor (see above).

In the Guinean zone, capital decreases risk by reducing output regardless of rainfall
level. This property is the same as the capital input in the Sahelian zone in a drought year.
(iii) Crop Diversification

The effect of crop diversification is ambiguous a priori, as explained in section 2.3,
although the standard view based on portfolio theory, such as Newbery and Stiglitz (1981),
predicts that diversification reduces the variance of the return by reducing the expected
return.

In all the three zones, estimates are significant, and crop diversification has similar
effects: in a drought year, it gives a low output and a low risk, while in a good year it gives a
high output and a high risk. Crop diversification acts as if an insurance in a drought year, but

acts as a gamble in a good year. In the Sahelian zone, even in a good year crop diversification

201

is insurance if rainfall is not so high. This result implies that crop diversification measured by
a Simpson index includes both risk-decreasing crop diversification, such as choosing safer
portfolio, and risk-increasing crop diversification, such as growing cash crop.

(iv) Land

Note that the signs for land input in Table 4 is the effect of total cultivated land
(hectare) on output per hectare, not total output. Land is assumed to be a fixed input, and
therefore the results should be interpreted as a cross-sectional comparison.

In the Sahelian zone, the estimates are significant, and land has the same property as
capital and crop diversification: insurance in a drought year. This may because a large
cultivated area increases the probability of survival of crops in a drought year.

In the Sudanian and the Guinean zones, the estimate is not significant. The negative

signs imply that land act as insurance regardless of rainfall level.

5.2 Estimation of Demand for Drought Insurance (:11)

¢'s are estimated for each zone separately and using the full sample (combined
zones). The results are shown in Table 5.

In the Sahelian zone there is effective demand for the insurance at between 15% and
25% probability and at other probabilities there is no demand. At 25%, 43 begins to drop, and
at 30% and above, it is zero or negative. This implies that most households are already
insured at this level of drought, which is consistent with the finding that the majority of
households in the Sahelian zone are self-insured for an occasional drought (Reardon et al.,
1992).

On the other hand, both the Sudanian and the Guinean zones have effective demand
for the insurance over all the range of drought probabilities. In the Sudanian zone, there is
effective demand at the full range of drought probabilities. d, with patterns similar to the
Sahelian’s, peaks at 15-25% drought probabilities and then drops at 30% (yet, unlike the

Sahelian zone, 4: stays positive through 40%). In the Guinean zone, 41 increases with drought

202

q ) decreases

 

probability, but the effectiveness of demand (the difference between 4) and

with the drought probability, as expected. The Guinean 4: pattern implies that expected
wealth is less sensitive to rainfall and/or households are less risk averse on average in the
Guinean zone compared with other zones.

Comparing the three zones, demand is highest in the Sahelian zone, smallest in the
Guinean zone and between the two in the Sudanian zone. Moreover, the Sahelian zone is the
most homogeneous, while the Guinean zone is the most heterogeneous in terms of assets and
other characteristics that drive insurance demand, or in other word, ability to self-insure.
Again, the Sudanian zone is in the middle. This pattern is consistent with the riskiness of crop
production in those zones.

Finally, 49 is estimated using the full sample (combined zones). Between 15% and
30%, p' s are very close to each other, and at 35%, ¢ is the largest. That is, demand for the
drought insurance is distributed over a wide range of drought probabilities. Consequently, for
the subsequent analysis, two probabilities of drought, 15% and 35%, are used; 15% is the first
peak of d , at which most households are considered to show effective demand for the
drought insurance, and 35% is the peak just before a sudden fall of d at which probably
some households (that are distinguishable from others in terms of risk management) still

show effective demand”.

5.3 Determinants of Demand for Drought Insurance (d)

Demand being “effective” at the zone level (that is, where zone-level ¢ exceeds

-—q—) does not mean that all households in the zone have effective demand; conversely, lack

1-4
of demand at the zone level can mask demand by certain groups within it. Determinant

analysis will help to understand who has demand for the formal drought insurance.

 

‘1 Comparing between 15% and 35% drought probabilities, at 15% drought probability drought is
severe, but it occurs infrequently, while at 35% drought probability drought is mild. but it occurs
frequently.

203

As discussed in section 4.3, two different regression specifications are used: cross-
section and fixed effects. The former specification explains 4) by average household wealth
variables so that we can compare across households. The latter specification explains change
in ¢ over time within a household by change in household wealth so that we can know how
4» changes each year depending on initial household wealth.

Table 6 shows regression results for the full sample using 4) evaluated at 15%
drought probability”. As explained in section 5.2, most households are considered to have
demand for the drought insurance at 15% drought probability. Coefficients for household
fixed effects are estimated in the fixed effect regressions, but the results are not shown in the
table; not all of them are statistically significant.

In both the cross-section regression and the fixed effect regression, crop production
has a negative significant effect and cultivated land has a positive significant effect. The
former implies that initial crop stock decreases demand for the drought insurance, as
expected. The latter implies that households with more land are more risk averse or land is a
risky asset. In full sample regression, the positive effect of cultivated land seems to reflect that
land per AB is the highest in the Sahelian zone where estimated ¢ is also the highest (see
Appendix D). The positive sign means that a risk averse household tends to cultivate a large
area, but risk-decreasing effect of land is not effective to smooth output and/or a household
that relies on more crop production (using more land) has a larger drought shock. In both
regressions, on the other hand, off-farm income and livestock holdings do not show
significant effect, which may be explained by the two opposing effects (risk averse and

income effects) of those variables as discussed above.

 

‘3 Similar analyses were done for a 35% probability of drought, but results are not discussed here
(tables of the results are in Appendix F). As explained in section 5.2, at 35% some households that are well
self-insured for this level drought show no demand Therefore, at 35% only drought-sensitive households are
considered to have effective demand for the drought insurance. Some regression results differ from those
when 15% drought probability is assumed. The changes of signs mid significance can be explained by the
assumption that only drought-sensitive households still demand the drought insurance at 35% drought
probability.

204

Public food aid significantly decreases demand for drought insurance in the fixed
effect regression, as expected . But in the cross—section regression food aid increases demand,
which implies that more food aid is given to households that demand drought insurance more
when we compare between households cross-sectionally, because such households suffer
larger drought shock. Actually in the Sahelian zone, public food aid per AB is the largest
(Appendix D).

In Table 7, regression results for each agroclimatic zone are shown. In the Sahelian
zone, the signs of initial crop stocks, cultivated area, and public food aid are similar to those
for the full sample, but the significance level is lower. In addition to those, inter-household
gifts have a significant positive effect in the cross-section specification, which means that
those who receive more gifts tend to demand drought insurance if other things are equal,
probably because they have larger drought shocks. This means that gifts in the Sahelian zone
work like state-contingent transfer (they go to households that have large shocks), but the
gifts themselves are not effective to smooth income.

In the Sudanian zone, initial crop stocks decrease demand for drought insurance in
the fixed-effect regression, as expected. But cultivated area has a negative significant effect in
the fixed-effect regression, too. The cultivated area result is contrary to those obtained for the
full sample, and also to those of the Sahelian zone and the Guinean zone. The negative effect
of cultivated area in the fixed effect regression implies that a household has a smaller shock
when it cultivates more land. The extra land acts as a buffer, by supplementing their limited
fertile land and stabilizing output over years. The result implies that the risk-decreasing effect
of land is in effect in the Sudanian zone. Much less public food aid was distributed in the
Sudanian ‘zone (compared to the Sahelian zone), but it has a significant effect unlike the
Sahelian zone.

In the Guinean zone, initial crop stocks have a positive significant effect in the fixed
effect regression. Cultivated area also has a positive significant effect in the fixed effect

regression, but a negative significant effect in the cross-section regression. The negative effect

205

of cultivated area implies that land has income effects when we compare between households.
This will be because more land means more room for cotton, the main cash crop in the
Guinean zone. On the other hand, the positive effects in the fixed-effect regression imply that
a household tends to have a large shock when it cultivates more land and has more crop
stocks, because the household can make risky decisions. This is consistent with the fact that
crop production is a major source of cash in this zone. Inter-household gifts have a negative,
significant effect in the cross-section regression, which means that a household that receives
more gifts on average is insured better when we compare between households. This is the
income effect unlike the case of the Sahelian zone. In the Guinean zone where more gifts are
observed than other zones, gifts may not be state-contingent transfers, but rather reflect
household wealth.

Although off-farm income and livestock holdings are important income sources in
the three zones, the results show no significant effect for them except for livestock holdings in
the Guinean zone. This is probably because risk aversion and income effects cancel each
other.

But exogenous stratification by wealth reveals significant effects of wealth variables".
The full sample is divided into three: lower 25 percentile of livestock holdings, upper 25
percentile, and between them. Average 40 is not significantly different from zero in the lower
stratum, and is the highest in the upper stratum (Table 8). In the lower stratum there is no
demand for drought insurance, while in other strata there is effective demand The difference

in d partially reflects that livestock holdings per AE are the largest in the Sahelian zone

 

'4 Splitting a sample of households on aprion' grounds by wealth variables is found in the literature
e.g., on liquidity constraints (Bernanke, 1984; Hayashi, 1985; Zeldes, 1989), and on the rate of time
preference (Lawrance, 1991). They split a sample to separate poor households from the rich, and examine if
those two groups differ. There is a possible selectivity bias in splitting a sample by endogenous wealth
variables. Hayashi (1985) uses Tobit to account for the bias. Zeldes (1989) is also aware of the bias, and
tries to split the sample bmed on predicted wealth by a logistic regression. But due to poor prediction. the
resultsbasedonpredictedwealtharenotsatisfactoryandheusestheresultsbasedonactual wealthinhis
paper. Using time-varying wealth variables. I-layashi and Zeldes are concerned with a liquidity constraint of
temporarily poor households. On the other hand, Bernanke 0984) and Lawrence (1991) do not consider the
bias. Lawrence divides the full sample into seven strata by average real labor income. In my case, I an
considering permanently (in a relative sense) poor households by stratifying the full sample by average
wealth, aid therefore selectivity bias is less problematic like in Lawrence.

206

where ¢ is also the highest. However, only half of the upper stratum households are from the
Sahelian zone. The results imply that a household that is highly risk averse or tends to have a
large shock keeps large livestock holdings, but the income smoothing effect is not sufficient.

In Table 8, the results of regression analyses for each stratum are shown. In
households with low livestock holdings, livestock holdings have a significant negative effect in
both specifications. This means that livestock holdings effectively insure in this household
group. But when a household has high off-farm income and/or high initial crop stocks, the
household tends to have a large drought shock because those allow the household to make
risky decisions. In the middle stratum, livestock holdings show only a cross-sectional effect; a
household that tends to be more risk averse or to have a larger shock, keeps more livestock
holdings when comparing between households. On the other hand, off-farm income has a
negative significant effect in the fixed-effect specification, that is, off-farm income is
effectively used for self-insurance. In the upper stratum, neither off-farm income nor
livestock holdings has a significant effect. Since those households on average demand the
drought insurance more than do other strata, the insignificant results do not mean that they
are already self-insured enough. Rather, the results imply (i) risk averse households or
households with high risk keep more livestock holdings than do those with low risk. (ii) But
because the level of livestock holding per AE and off-farm income per AE do not affect the
degree of insurance demand (the size of d), those households may need a better insurance.
Probably, their livestock holdings is larger that the minimum livestock holdings that give the
same level of self-insurance. Therefore, it will be possible that formal drought insurance
substitutes for some of livestock holdings making the household’s welfare at least unchanged

or even improved.

6 Conclusions
There is effective demand for the hypothetical formal drought insurance in all zones,

that is, although there are a lot of informal risk management mechanisms in the Sahel, but

207

they are not inadequate. This may not be surprising because we observe catastrophic drought
shocks during the early 1980’s. But such demand is very correlated with agroclimatic zone
because drought risk differs over zones. This was expected, and raised a question of the
feasibility of drought insurance due to a covariability problem. However, the determinant
analysis shows that the demand also depends strongly on how households manage risk ex
ante and ex post, and as a result there is a lot of heterogeneity even within an agroclimatic
zone. In addition to these findings, as shown in Table 2, drought does not necessarily occur
simultaneously over zones. Therefore, it can be concluded that it will be possible to avoid
covariability and the drought insurance scheme will be feasible.

With respect to the effects of initial assets on 41 , in general, the results support the
hypothesis that those who are wealthier and/or are more self-insured demand formal drought
insurance less, because they tend to suffer smaller income shocks from drought. But in some
cases the risk aversion effect dominates the income effect. In these cases, households that
suffer larger shocks keep more assets, but still demand a better insurance because their self-
insurance is insufficient.

Most households decrease demand for the formal drought insurance when they have
more initial crop stocks. However, if households tend to invest in off-farm activities and/or
livestock holdings when they have more crop output, the relation between demand for the
formal drought insurance and initial crop stocks will be positive. Such exceptions are
observed in the Guinean households and by households with fewer livestock (some
households belong to both strata).

Public food aid has an effect similar to that of initial crop stocks, but is significant
only in the full sample, in the Sudanian zone, and in the lower livestock stratum. Most
households decrease demand for drought insurance when food aid is available. Those
households have a larger drought shock and have a high 40 compared to others in the cross-

section.

208

Effects of cultivated area differ between zones although land is a risk-decreasing
input in all zones. In the Sahelian zone, land does not effectively smooth income, while in the
Sudanian zone land is effectively used for self-insurance. In the Guinean zone, land appear to
be seen as room for cash cropping.

The negative significant effects of off-farm income and livestock holdings support
the hypothesis that those who do not have self-insurance mechanisms demand the drought
insurance more, because they tend to suffer a large income shock from drought. The results
show that this is true with off-farm income in the middle livestock stratum and with livestock
holdings in the lower livestock stratum. On the other hand, in the upper livestock stratum,
neither livestock holdings nor off-farm income has no significant effect on demand for the
formal drought insurance. However, since in this stratum households on average show the
largest demand for the drought insurance, their self-insurance level is not sufficient. This is a
sign that the formal drought insurance can substitute for livestock holdings at least partially

among those who keep relatively large livestock holdings while not changing their welfare.

209

References

Antle, John M., “Econometric Estimation of Producer’s Risk Attitudes,” American Journal
of Agricultural Economics, 69: 509-522, 3, I987.

Antle, John M., “Nonstructural Risk Attitude Estimation,” American Journal of Agricultural
Economics, 71: 774-784, 3, 1989.

Bernanke, Ben 8., “Permanent Income, Liquidity, and Expenditure on Automobiles:
Evidence from Panel Data,” The Quarterly Journal of Economics, 99. 587-614,
August, 1984.

Binswanger, Hans, “Risk Aversion, Collateral Requirements, and the Markets for Credit and
Insurance in Rural Areas,” In: P. Hazell, C. Pomareda, and A. Valdes, eds., Crop
Insurance for Agricultural Development, Baltimore: Johns Hopkins University Press,
1986.

Christensen, Garry Neil, “Determinants of Private Investment in Rural Burkina Faso,” Ph. D.
Dissertation, Cornell University, 1989.

Gautam, Madhur, Peter Hazell, and Harold Alderman, “Management of Drought Risks in
Rural Areas,” Policy Research Working Paper, The World Bank, 1994.

Hayashi, Fumio, “The Effect of Liquidity Constraints on Consumption: A Cross-sectional
Analysis,” The Quarterly Journal of Economics, 100: 183-206, February, 1985.

Hazell, Peter, “The Appropriate Role of Agricultural Insurance in Developing Countries,”
Journal of International Development, 4: 567-582, 6, 1992.

Lawrance, Ernily C., “Poverty and the Rate of Time Preference: Evidence from Panel Data,”
Journal of Political Economy, 99: 54-77, 1, I991.

Matlon, Peter, “Farmer Risk Management Strategies: The case of West-African Semi-Arid
Tropics,” In The World Bank’s Tenth Agriculture Sector Symposium in Washington
DC, The World Bank, 1990.

Moscardi, Edgardo and Alain de Janvry, “Attitudes Toward Risk among Peasants: An
Econometric Approach,” American Journal of Agricultural Economics, 59. 710-716,
4. 1977.

Newbery, David MG. and Joseph E. Stiglitz, The Theory of Commodity Price Stabilization,
Oxford, UK: Oxford University Press, 1981.

Norman, D. W., “Economic Analysis of Agricultural Production and Labour Utilization
Among the Hausa in the North of Nigeria,” Dept. of Agricultural Economics,
Michigan State University, African Rural Employment Paper, No. 4, January, 1973.

Ramaswani, Bharat, “Supply Response to Agricultural Insurance: Risk Reduction and Moral
Hazard Effects,” American Journal of Agricultural Economics, 75: 914-925, 4, 1993.

Reardon, Thomas, Christopher Del gado, and Peter Matlon, “Determinants and Effects of
Income Diversification amongst Farm Households in Burkina Faso,” Journal of
Development Studies, 28: 264—277, 1992.

Sakurai, Takeshi, “Consumption Smoothing in Burkina Faso,” Draft, Michigan State
University, 1995.

210

Sakurai, Takeshi, Madhur Gautam, Thomas Reardon, Peter Hazell, and Harold Alderman,
”Potential Demand for Drought Insurance in the Sahel,” Department of Agricultural
Economics, Michigan State University, Staff Paper, No. 94-67, 1994.

Sivakumar, M.V.K. and Faustin Gnoumou, “Agroclimatology of West Africa: Burkina
Faso,” International Crops Research Institute for the Semi-Arid Tropics, Information
Bulletin, No. 23, 1987.

Udry, Christopher, “Risk and Saving in Northern Nigeria,” manuscript, Department of
Economics, Northwestern University, 1993.

Udry, Christopher, “Risk and Insurance in a Rural Credit Market: An Empirical Investigation
in Northern Nigeria,” Review of Economic Studies, 61: 495-526, 1994.

Zeldes, Stephen P., “Consumption and Liquidity Constraints: An Empirical Investigation,”
Journal of Political Economy, 97: 305-346, 2, 1989.

211

Table 1 Com arison of Assum tions on Drou ht Probabilit

  

  
 
  

 

   
 

       

 

 

 

 

 

 

 

 

household’s view w/o household distribution of
formal insurance probability of rainfall
drought
I
Gautam 3’ 01- binary, fixed objectivel, fixed not specified3
(1994)
Sakurai et al. binary. changing subjectivez, changing fixed (normal)4
(1994)
This continuous, fixed not specified fixed (normal)4
Chapter

 

Table 1 continued)

 

 

 

insurance scheme insurance firm’s household’s view

probability of under formal
drought insurance

‘ Gautam et al. binary same as household’s binary (unchanged)

(1994)

I Sakurai et al. binary arbitrary binary (same as

. insurance firm)

, (1994)

. This Chapter . binary arbitrary binary (same as

insurance firm)

 

 

 

 

Note 1: Based on physical state, such as annual rainfall.

Note 2: Based on individual homehold’s welfare status

Note 3: Instead of specifying rainfall distribution, a fixed, binary probability is used ‘

Note 4: Rainfall distribution need not be normal, but must be known to every one. In empirical analysis,
normd distributions are used for simplicity.

2l2

 

 

Figurel Time Frame of Decision Making
harvest year t
dry 59350" rainy season
t4 off-farm work

‘1"— "r

 

 

crop production

 

 

harvest planting
of year t-1

 

State: initial wealth

some off-farm work .

-harvest of year t-‘I+ saving from year t-l

+exogenous income

Decisions: on/off-farm activities

wealth-cropping+off-farrn activities-1-saving+consumption

time-cropping+off-farm+leisure
crop diversification

 

harvest
of year t

 

t+1

 

 

213

 

  

 

 

  

 

   
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Table 2 Annual Village-Level Rainfalls and the Probabilities

Zone _I mz 0‘z 02/": village 1981 1982 1983

Sahelian 480 163 0.34 village 1 454 382 476 !
(probability) (43.6) (28.5) (49.2) ,
village 2 513 347 454 I
( robability) (57.9) (21.6) (43.6) .

Sudanian I 724 181 0.25 village 3 709 586 654)
(probability) (47.8) (22.5) (34.6) '
village 4 541 552 442 i
(probability) (15.7) (17.2) (6.0) ,

Guinean 952 200 0.21 village 5 691 849 779
(probability) (0.95) (30.2) (19.2)
village 6 908 605 701 .
(probability) (41.3) (4. 1) (10.4) I, ‘

Rainfall (mm), Probability (%)

Note: The means (mz ) and the stardard deviations (az ) are based on long term rainfall data Village-level
rainfall mid its probability (probability that rainfall is below it) are shown in each cell. The short mean is
the average of rainfall over three years given in the table. Village 5 is not used in the analysis due to
incomplete production data.

214

 

 

 

 

 

 

 

 

 

 

 

 

Table 3 Production Function Estimates with Fixed Effects

Variable Sahelian Zone Sudanian Zone Guinean Zone
Labor ((11) 4.09 (3.64)” 0.82 (1.89)' 1.27 (2.69)"
Labor*Rainfall ((12) -915 (3.41)" -0.89 (1.21) -0.80 (1.33)
Capital (31) -O.18 (0.30) -017 (1.02) 0.27 (0.63)
CapitaPRainfalltfiz) 0.55 (0.40) 0.39 (1.52) -070 (1.17)
Crop Diversification (111) -049 (1.13) 4.95 (2.15)" 6.57 (1.98)’
Diversification*Rainfall (172) 0.93 (0.89) 9.10 (2.34)" 8.81 (1.86)‘
Total Area (#1) 0.88 (1.36) 0.26 (0.74) 0.79 (2.62)"
Total Area*Rainfall ([12) -2.38 (1.71)‘ -0.86 (1.29) -1.19 (2.86)"
Rainfall (in) 35.1 (4.44)” 10.8 (5.46)" 7.17 (4.02)"
R squared 0.79 0.80 0.85
Number of Cases 32*3 37*3 20‘3

 

 

 

 

 

 

Note: The dependent variable is household crop output per hectare. Labor md capital input are also divided
by total cultivated area Absolute values of t-statistics are in parentheses; "‘ means 10 % level of
significance and " means 5 % level of significmce. Coefficients for household dummies are not shown.
but we estimated significantly at 10 % level or better.

215

 

 

 

 

 

 

 

 

 

 

 

 

 

Table 4 Summary of Signs of Marginal Productivity
Sahelian Zone Sudanian Zone Guinean Zone
Drou ht Good Drou ht Good Drou ht Good
Drought Prob (%) 15 35 15 35 15 35 15 35 15 35 15 35
Expected Rainfall 227 308 525 573 443 533 774 827 i 641 740 1007 1066
ay 3 I! .3
Labor dL’ + + - - + + + + + + + +
azy II DI 8. 33 88 83 It I! 88 I. I! 83
a L’ d R + + - - + + + + + + + +
.ml fl 8. 88 I! It
Cap1 6K - - + + + + + + - - - -
azy 88 I! I! 33
—d K 6R - - + + + + + + - - - -
Crop aY
Diverse. — -8 -8 -8 8 -3 -8 8 3 -3 3 -8 3 8 8 8 3
d D + + + + +
ii; a a s a s a a a as as as as
dDdR - - - + - - + + - - + +
148 d a—Y 38 I! I. .8
n d A - - + + - - - - - - - -
62), .8 II 88 3.
Mali ' ' ’ + ' ° ' ’ ‘ ' ' ’

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

NotezTheprobabilityofadroughtyearissetat15%md35%respectively.Thosesignsarebasedonthe

estimated

is significant at 5 % level, " is marked, and at least 10 % level, "' is marked

showninTable 3,and the derivations are in Appendix E If theestimationofthe sign

 

III III.-
I

216

 

 

 

Table 5 Zone-Level ¢ and Drought Probabilities

l'Wlught

Probability ((1) 10% 15% 20% 25% 30% 35% 40%

14:; 0.1 1 1 0.176 0.250 0.333 0.429 0.538 0.667

Sahelian 4o 2.47 112" 104” 49.1" -12.5 -050 -168‘
(6.28) (173) (213) (18.9) (15.8) (937) (938)

Sudanian 45 8.07” 14.5” 20.2" 19.9" 4.89" 4.11" 2.95"
(0.48) (0.75) (1.06) (1.16) (0.72) (0.66) (0.62)

Guinean 4. 0.38" 0.44" 0.51” 0.58" 0.66” 0.73" 0.80"
(0.017) (0.015) (0.014) (0.015) (0.015) (0.016) (0.018)

Full Sample 41 6.63‘ 38.3" 35.7” 31.7" 31.6” 73.4“ 48.2"
(3.69) (7.07) (8.17) (6.91) (6.17) (4.98) (9.03)

 

 

 

Note : According to condition (25), when ¢ is greater than Tq— , there is effective demand. Standard

errors are in parenthesis; " means 10% level of significance and ** means 1% level of significance.

‘4

 

217

Figure 2 Interpretation of (p

Value

 

 

 

 

 

d
W W 9 Wealth

r
d

V

Note: ¢ is defined as ¢,, - —r. Expected wealth in agood year (W’)and in a drought year (W4)
Vs

depend on input decisions and the definition of drought given by the threshold level of rainfall. Two

marginfl values depend on the expected wealth levels and the curvature of the value function, that is. the
degree of risk aversion.

218

 

 

 

 

 

 

Table 6 Determinants of (P (full sample)

Probabrlrty of Droufl 15% he]
Specification Cross-Section Fixed Eff“!
0 Wealth Variables

off-farm income (FCFA/AB) -O.14 (0.42) 0.06 (0.20)
livestock holdings (FCFA/AB) -0.05 (0.66) -003 (0.47)
initial crop stock (FCFA/AB) -l.87 (2.34)" -113 (4.72)"
cultivated area (Ha/AE) 36.6 (3.00)’ " 22.9 (2.40)‘ ‘
public aid (FCFA/AB) 57.8 (4.19)" -148 (1.78)‘
private gifts (FCFA/AB) 2.13 (0.68) 8.14 (1.10)
0 Household Character.

adult equivalent size (AB) -2.63 (1.05) -

age of household head -0.23 (0.44) -
number of wives 1.22 (0.14) -
dependency ratio 4.35 (0.09) -

J compound leader 34.1 (2.19)" -
single conjugal unit -40.1 (2.17)" -
animal traction -6.11 (0.37) -

0 Plot Characteristics

years since last fallow -l.56 (2.10)" -
soil type diversification -103 (3.48)” -
toposequence index -11.8 (1.19) -
distance to plot (Km) -1.00 (0.19) -

 

. Constant 119 (2.19)” -
Lumber of cases I 267 I 178 I
Note: Absolute value of t-statistics is in parenthesis. “ meals 10 % level of significance and " means 5 %
level of sigrrificmce.

218

 

 

 

Table 6 Determinants of (I) (full sample)
Probability of Drought? 15%

Panel
Specification Cross-Section Fixed Effect
0 Wealth Variables
off-farm income (FCFA/AF.) -O.l4 (0.42) 0.06 (0.20)
livestock holdings (FCFA/AB) -0.05 (0.66) -0.03 (0.47)
initial crop stock (FCFA/AB) -l.87 (2.34)" -1.13 (4.72)"
cultivated area (Ha/AB) 36.6 (3.00)’ ' 22.9 (2.40)‘ ‘
public aid (FCFA/AB) 57.8 (4.19)' ’ - 148 (1.78)‘
private gifts (FCFA/AB) 2.13 (0.68) 8.14 (1.10)
0 Household Character.
adult equivalent size (AB) -263 (1.05) -
age of household head -0.23 (0.44) -
number of wives 1.22 (0.14) -
dependency ratio 4.35 (0.09) -

A compound leader
single conjugal unit
animal traction

0 Plot Characteristics
years since last fallow
soil type diversification
toposequence index
distance to plot (Km)

0 Constant

 

—_

 

number of cases

34.1 (2.19)"
40.1 (2.17)"

-6. l 1 (0.37)

-l.56 (2.10)"
-103 (3.48)"
-11.8 (1.19)
-1.00 (0.19)

119 (2.19)"

 

267

 

178

 

Note: Absolute value of t-statistics is in parenthesis. ‘ means 10 % level of significance and " means 5 %

level of significance.

 

219

Table 7 Determinants of 4) (zone)

 

 

 

 

 

 

Probability of Drought 15%
Panel
Specification Cross-Section Fixed Effect
0 Sahelian Zone, (6:112 (6.47)" n=96 n=64
off-farm income (FCFA/AB) -O.54 (0.59) -0.23 (0.26)
livestock holdings (FCFA/AB) 1 -0.13 (0.78) 0.05 (0.54)
initial crop stock (FCFA/AB) -5.68 (1.21) 2.38 (4.87)"
cultivated area (Ha/AB) 61.2 (1.54) 39.2 (2.09)' ‘
public aid (FCFA/AB) 54.7 (1.28) -186 (1.40)
m’vate Efts (FCFA/AB) 56.5 (2.90)" 16.4 (1.12)
0 Sudanian Zone, 91:14.5 (19.3)" n=111 n=74
off-farm income (FCFA/AB) -0.04 (0.92) 0.01 (0.10)
livestock holdings (FCFA/AB) 0.00 (0.02) -0.08 (1.18)
initial crop stock (FCFA/AB) 0.21 (1.05) -0.11 (1.93)‘
cultivated area (Ha/AB) -8.26 (3.01)" -7.61 (3.18)"
public aid (FCFA/AB) 1 1.5 (1.75)‘ -104 (3.21)’ '
E'vatc e'fts (FCFA/AB) 0.63 (0.84) 2.06 (1.07)
- Guinean Zone. «1:044 (293)" n=60 n=40
off-farm income (FCFA/AB) 0001 (1.11) -0.001 (1.64)
livestock holdings (FCFA/AB) 0.004 (1.41) -0005 (1.92)'
initial crop stock (FCFA/AB) -0001 (0.54) 0.003 (3.53)"
cultivated area (Ha/AB) -0.183 (1.93)’ 0.239 (6.35)"
public aid (FCFA/AF.) 0.118 (0.32) 0.745 (0.84)
, Evan: gifts (FCFA/AB) -0.011 (1.86)‘ 0.016 (0.63)

 

 

Note: Only wealth vmiables are showu for average wealth specifications. T-statistics are in parenthesis.
“ means 10 %level ofsignificanceand ” merms 5 % level ofsignificance.

220

Table 8 Determinants of ¢ (stratification by livestock holdings)

 

 

 

 

 

Probability of Droght 15%
fianel

§peciflcation Cross-Section Fixed ETC“
0 Lower Stratum, (21:-3.39 (0.70) n=66 n=44
off-farm income (FCFA/AB) 0.27 (0.53) 0.44 (2.35)"
livestock holdings (FCFA/AB) -104 (2.07)" -3.28 (6.54)”
initial crop stock (FCFA/AB) -211 (3.00)" 0.47 (5.24)"
cultivated area (Ha/A15) 44.4 (3.06)" -0.41 (0.10)
public aid (FCFA/AB) -129 (0.95) -221 (2.47)“
m'vate Efts (FCFA/AB) 2.82 (1.32) 11.4 (5.57)"
0 Middle Stratum, 0:526 (2.69)" n=135 n=90
off-farm income (FCFA/AB) -0.06 (0.52) -0.32 (4.39)"
livestock holdings (FCFA/AB) 0.57 (1.99)‘ -0.18 (1.36)
initial crop stock (FCFA/AB) 0.54 (1.66)‘ -0.09 (0.87)
cultivated area (Ha/AB) -15.6 (2.12)" 9.22 (2.49)”
public aid (FCFA/AH) 3.12 (0.61) 21.4 (1.14)

'vate 'fts (FCFA/AB) -0.51 (0.35) 2.34 (0.83)
. Upper Stratum, n=24.8 (4.15)" n=66 n=44
off-farm income (FCFA/AB) -0.50 (1.49) -0.33 (0.40)
livestock holdings (FCFA/AB) 0.07 (1.00) -0.00 (0.11)
initial crop stock (FCFA/AF.) -230 (2.54)‘ ’ -120 (3.27)' ‘
cultivated area (Ha/AB) 28.0 (3.42)‘ ' 39.9 (2.86)‘ "‘
public aid (FCFA/AB) ~27.8 (1.23) Na
Evan: gills (FCFA/AB) 10.3 (1.12) -13.7 (1.11)

 

 

 

Note: Only wealth variables are shown for average wealth specifications. T—statisties are in parenthesis.
‘mems10%levelofsignificmcemd"5%levelol'signifieance.

221

Appendix A

Change in Average Rainfall and Drought Probabilities

Table A1

Segou (Sudanian zone, Mali), Drought Frequency

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

long-term short-term (1980’s)
‘ grfigglgrob. 10% 20% 30% 40% 10% 20% 30% 40%
Rainfall 505 564 607 644 384 439 479 512
Mean
Decade Rainfall
1940 630 0.1 0.2 ~ 0.3 0.6 0 0 0.1 0.1
1950 801 0 0 0 0 0 0 0 0
1960 730 0 0.1 0.1 0.1 0 0 0 0
1970 638 0.1 0.3 0.3 0.4 0 0 0 0.2
1980 544 0.3 0.8 0.9 0.9 0 0.2 0.3 0.4
Total Frequency 0.1 0.28 0.3 0.38 O 0.04 0.08 0.14

 

 

Note : Long-term statistics are: averagei678 and stmdard deviation=135 based on 64 years of data. Short-
term statistics are: average=544 and standard deviation=125 based on 10 years of data. Each cell shows
obsa'ved relative frequency of drought in the decade (number of drought years/10), when drought is defined

by the threshold rainfall level given in the table.

Mean rainfall was higher during the 1950’s mid 1960’s, md lowest during the 1980’s. Even if we use long-
term statistics. there is only one drought at 40% probability of drought during 1950’s mid 1960’s. Using
the same threshold level of rainfall. during the 1980's there are more drought years than predicted; at 40%
probability of drought 9 out of 10 years are drought by the definition. However, in the long run. the
observed frequencies of drought are very close to what is predicted If we use short-term statistics based on
data from the 1980’s. it obviously gives a good prediction of frequency of drought in 1980's, but in other
decadesitdoesnotgiveagoodprodiction.

222

 

 

 

 

 

 

 

 

 

 

 

 

Table A2 Koutlala (Guinean zone, Mall),, Drought Frequency
long-term short—term (1980’s)

Drou ht Prob. 10% 20% 30% 40% 10% 20% 30% 40%

'I'Eresi‘lold

Rmnfall 724 802 859 907 610 676 723 763
Mean

Decade Rainfall

1930 921 0.1 0.3 0.4 0.5 0 0 0.1 0.2

1940* 1053 0 0.11 0.11 0.11 0 0 0 0

1950 1099 0 0 0 0.1 0 0 0 0

1960 1018 0.1 0.1 0.1 0.2 0 0 0.1 0.1

1970 864 0.1 0.2 0.6 0.7 0 0.1 0.1 0.2

1980 801 0.3 0.4 0.7 0.9 0.1 0.2 0.3 0.4

Total Frequency 0.10 0.19 0.32 0.42 0.02 0.05 0.08 0.15

 

 

 

 

 

 

 

 

 

 

Note 1: "' means that there is a missing year in the 1940’s.

 

Long-term statistics are: average=952 and standard deviation=l78 based on 70 years of data. Short-term
statistics are: average=801 and standard deviation=149 based on 10 years data. Each cell shows observed
relative frequency of drought (number of drought years/10), when drought is defined by the threshold rinfall
level given in the table. '

Mean rainfall was higher during the 1940’s, 1950’s. and 1960’s. The table shows that long-term statistics
we better tlmn short-term (1980’s), as discussed on the Sudanian table.

223

Appendix B

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Table B1 Ex cted Rainfall (mm)
Prob. (q) 10% 15% 20% 25% 30% 35% 40%
Truncation 2 -1.645 -1.282 -1.036 -0.842 -0.675 -0.385 -0253
Igahelian
mean 480
so 163
Cut-off Point 271 311 343 370 395 417 439
Rf 512 525 537 549 561 573 585
194 227 252 273 291 308 323
Sudanian
mean 724
so 181
Cut-off Point 492 536 572 602 629 654 678
R: 759 774 787 801 814 827 841
407 443 471 494 514 533 549
Guinean
mean 952
SD 200
Cut-off Point 696 745 784 817 847 875 901
R5 991 1007 1022 1037 1051 1066 1081
Rf 601 641 672 698 720 740 759

 

Rf (expected good rainfall) and Rf (expected drought rainfall) are calculated when the probability of a
drought year is q. In a standard normal distribution. truncation Z gives q probability of lower tail.

Therefore, 2‘ (expected zsooreforagoodyearfis given by

Z’-J:xf(xlx>B)dx-

3 l-<D(B)

3 f(x) I

where f(x) is the standard normal p.d.f.. <1>(x) is the standard normal c.d.f. and B=truncation 2. Z“
(expectedzscoreforadroughtyear)is obtn'nedinasimilarway. Using thesetwoncoresandlong-term.
zone-level rainfall and standard deviation, md assuming normd distribution ofrainfall. R: and R:
medeulfled foreach zone.

 

224

Appendix C

Calculation of (30), (31), and (32)
Using the production function given by (29), the expectations in (30)—(32) are

specified as follows.

I; Yrrff(R)dR - exp(a,ln 11’, +131 In K“ + mm D“ + [.11 In A“ + hiya.) 0
(C1)

1: exp[(012 in 1", +6, in K“ + nzlnD“ + u, may + mid/“Rm

and

f” Y.Jf(R)dR - exp(a, mg, + p, In K“ + n. 1nD,., + p, In A“ + 1.1111.) .
"° (C2)

f exp[(ct2 ln Li’, + [3, In K“ + n, In Q, + u, in A“ + w)R]/f(R)dR
where mm and 13R) are the probability density functions of rainfall for each zone. They

have a truncated normal distribution, and are given by

2 2
exp[_1_R_-_'s2_ exp -1522.

 

f(R) 20: 20: C3
‘ (1-¢(R;))o../i? qa,J2_a ‘ ’

and

 

(C4)

exp[- 2 2
d
R-
fz()- MEG) ROI—J-

where m; and 0, are long-term mean and variance of rainfall for each zone. shown in Table

 

1, and ‘NR: ) is the normal cumulative distribution function.

7 Table D]

225

Appendix D

Mean and Standard Deviation of Wealth Variables

 

I Full Sample

 

Sahelian

Sudanian

 

 

Guinean I

 

 

 

 

 

number of cases 96 11 1
Off-farm income (FCFA/AB) 15568 15930 1 1783 21990
(18604) (20191) (15090) (20258)
livestock (FCFA/AB) 39262 71500 283 22 7927
(86141) (131810) (36900) (5162)
Crop Production (FCFA/AB) 17254 14904 13611 27756
(9786) (6959) (5452) ( 12339)
Cultivated area (Ha/AF.) 1.219 1.579 0.952 1.108
(0.612) (0.751) (0.416) (0.322)
Public Food Aid (FCFA/AB) 92 231 20 5
(35) (39) (7.7) (3.5)
Private Food Gifts (FCFA/AB) 263 183 171 561
(135) (64. 5) (74.5) (242)

 

 

 

 

 

 

 

Note: Mean and stmdard deviation (in parenthesis) are in each cell. Means are significantly differult between

zones at0.1% level.

226

Appendix E

Derivations for Table 3
Q is a general input variable, that is, Q can be Lf , K, or D Differentiating the

production function given in (29) with respect to Q yields

aY Y

where x, and x2 are parameters corresponding to al. a2 . fl“ fiz, 1),, and 77,, respectively.

Then, differentiating the first derivative with respect to rainfall, R . yields
021' Y
- x + x + x -

where C - w + (12 In L’ + fl: In K + 1), In D + p, In A. C depends on the household’s input

 

(I32)

decisions and parameters, but independent of which variable to use for differentiation.

aY
Basedon(E1),if x2>0, then iY->0 when R>-£‘- and if xz<0, then --—>0
6Q ‘2 3Q

2

 

1

when R< -£L. Based on (E2), if x2C>O, then >0 when R> -£L-— and if
x, anR x, C

2

 

ch <0, then 66 YR >0 when R< -£L--(-:1-. C for each household is calculated using

“2
1
actual input level and the estimated parameters shown in Table 2. C is positive in most cases,
x l
and very small relative to the rainfall level given by - -‘- , and therefore C can be ignored to

x,
2

aQaR '

 

evaluate the sign of

Based on the general derivation with respect to input Q above, signs and the

significance levels of first and cross derivatives with respect to L’ , K , and D are determined

based on the estimations of - EL. The results are shown in Tables El through BB.
K,

In the case of land input, A, signs and significance levels of the first and cross
derivatives are determined based on E1 and E2, but it, - -(.zl ~fl‘ +111 and
x, - -a2 - fl, + u, are substituted Note that in this case, Y is output per hectare. The results

are in Table FA.

227

 

 

 

 

 

 

 

   

 

 

 

 

Table El Sim of Marginal Productivity of Labor
Rainfall (mm) <447 447 447-921 921 921-1596 1596 1596<
. 61’ 0 _ - - _ -
Sahellan '37? +
azY +/_ - - - - - -
dL’aR
. _ + + + 0 - - -
Sudanlan 3L,
321’ + + +/- - - - -
dL’aR
. LY
Gulnean 6L,
62); + + + + +/- - -
aL’aR

 

 

 

 

 

 

 

 

 

 

 

 

Table E2 Signs of Marginal Productivity of Capital

Rainfall (mm) 430 330 330-392 392 392430 430 430<
aY

S ahelian __ - 0 + + + + +

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

228

 

 

 

Table E3 Signs of Magginal Productivity of Crop Diversification
Rainfall (mm) <530 530 530—543 543 543-746 746 746<
. a}, - 0 + + + + +
Sahellan a D
6‘ 11’ -/+ + + + + +

Sudanian

Guinean

 

 

6D

+

6D§R
fl - - - O + + +

 

621’

6D

-/+

.1.

+

+

+

8D6R
LY’ - - - - - o +

 

 

(92Y

6D6R

 

 

 

 

 

-/+

 

 

 

 

 

 

Table E4 SLLns of Maginal Productivity of Cultivated Area
Rainfall (M) <-1067 -1067 -1067-486 486 486-2419 2419 2419<
. fiY - - - 0 + + +
Sahelian a A
all, ./+

Sudanian

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

229

Appendix F

Regression Results at 35% Drought Probability

As explained in section 5.2, two drought probabilities, 15% and 35%, are picked up to
compare. However, in the text, only results at 15% drought probability are used. In this
appendix, the results for 35% drought probability is presented. There are three tables that are
corresponding to Table 7, Table 8, and Table 9 respectively.

The detailed comparison between two result sets with different drought probabilities

are not conducted in this paper.

230

Table F1 Determinants of 4) (full sample)

 

 

 

Probability of Drought 35%
Panel

Specification Cross-Section Fixed Eff“!
0 Wealth Variables

off-farm income (FCFA/AF.) -0.31 (1.29) -0.46 (1.30)
livestock holdings (FCFA/AB) -014 (2.35)" -0.06 (0.86)
initial crop stock (FCFA/AB) 1.07 (1.92)’ -0.20 (0.70)
cultivated area (Ha/AB) -49.3 (5.80)" 25.4 (2.20)”
public aid (FCFA/AE) 6.50 (0.68) 603 (6.01)’ '
private gifts (FCFA/A13) -2.08 (0.95) 7.07 (0.79)

' Household Character.

adult equivalent size (AB) -2. 15 (1.22) -
age of household head 1.28 (3.49)" -
number of wives 1.29 (0.21) -
dependency ratio 26.2 (0.77) -
compound leader 12.4 (1.15) -
single conjugal unit 39.4 (3.06)” -
animal traction 9.85 (0.87) A -

0 Plot Characteristics

years since last fallow -l.26 (2.44)" -
soil type diversification 55.5 (2.68)" -
toposequence index -7.80 (1 . 16) -
distance to plot (Km) -5. 13 (1.37) -

 

 

 

0 Constant 60.7 (1.61 -

number of cases 267 178
7 Note: Only wealth variables are shown for average wealth specifications. T-statisties are in parenthesis.
I“means 10%levei ofsignificancemd “means5%levelofsignificmce.

 

231

Table F2 Determinants of ¢ (zone)

 

 

 

 

 

 

 

Probability of Drotliht 35%
Panel
Specification Cross-Section Fixed Effect
- Sahelian Zone, 91:-0.50 (0.05)" n=96 n=64
off-farm income (FCFA/AB) 0.10 (0.21) 0.05 (0.09)
livestock holdings (FCFA/A13) -0.14 (1.56) 0.02 (0.35)
initial crop stock (FCFA/AB) 4.83 (1.89)‘ -1.26 (4.03)"
cultivated area (l-la/AE) -49.6 (2.28)" 11.2 (0.94)
public aid (FCFA/AB) -5.86 (0.25) 72.1 (0.85)
'vate 'fts (FCFA/AB) 11.2 (1.05) -3.66 (0.39)
- Sudanian Zone, ¢=4.11 (6.23)" n=111 n=74
off-farm income (FCFA/AB) -0.01 (0.23) —0.002 (0.04)
livestock holdings (FCFA/AF.) -0.02 (1.00) -0.11 (1.49)
initial crop stock (FCFA/AB) 0.12 (0.65) -0.27 (4.15)"
cultivated area (Ha/AB) 3.24 (1.25) -5.15 (1.95)'
public aid (FCFA/AB) 10.8 (173)" -77.6 (2.17)' "
E'vate E'fts (FCFA/AB) 2.35 (3.31)“ -043 (0.20)
. Guinean Zone, o=0.73 (45.6)" n=60 =40
off-farm income (FCFA/AB) 0.001 ( 1.00) -0.001 (1.20)
livestock holdings (FCFA/AB) 0.002 (0.76) 0001 (0.17)
initial crop stock (FCFA/AB) -0007 (2.99)' ' -0004 (2.70)' ‘
cultivated area (Ha/AB) ‘ '-0.004 (0.04) 0.195 (3.33)‘ ‘
public aid (FCFA/AE) -0.365 (0.89) 1.771 (1.29)
private gifts (FCFA/A12) -0.007 (0.98) 0.018 (0.46)

 

Note: Only wealth variables we shown for average wealth specifications. T—statistics we in parenthesis.
* meals 10 % level ofsignificanceand " 5 % level of significance.

232

 

 

 

 

Table F3 Determinants of ¢ (stratification by livestock holdings)
Probability of Dmtfl 35% I
PaneT

Specification Cross-Section Fixed Effect
- Lower Stratum, (21:63.2 (6.01)" n=66 n=44
off-farm income (FCFA/AB) -050 (0.49) 1.24 (2.13)"
livestock holdings (FCFA/AB) -247 (0.24) -9.63 (6.12)"
initial crop stock (FCFA/AB) 2.32 (1.64) 1.07 (3.81)"
cultivated area (Ha/AB) 49.3 (1.69)‘ -34.6 (2.64)‘
public aid (FCFA/AB) 13.8 (0.50) -96.1 (0.34)
private gifts (FCFA/AB) -2.26 (0.53) -3.79 (0.59)
0 Middle Stratum, 16:94.9 (16.4)" n=135 n=90
off-farm income (FCFA/AF.) -0.46 (1.62) -0.61 (1.90)'
livestock holdings (FCFA/AB) 0.39 (0.54) -3.95 (6.82)"
initial crop stock (FCFA/AB) 2.46 (3.03)" -0.67 (1.50)
cultivated area (Ha/AB) -107 (5.81)’ ‘ 4.47 (0.27)
public aid (FCFA/AB) 44.3 (3.46)" 1031 (12.4)“
E'vate gifts (FCFA/AB) -4.47 (1.22) 19.3 (1.54)
0 Upper Stratum, 10:60.6 (733)" n=66 n=44
off-farm income (FCFA/AB) -0.52 ( 1.33) 0.72 (0.70)
livestock holdings (FCFA/AB) 0.07 (0.84) -0.03 (0.43)
initial crop stock (FCFA/AB) 1.17 (1.10) 0.60 ( 1.32)
cultivated area (Ha/AB) ~26.9 (2.79)‘ " 5.07 (0.29)
public aid (FCFA/AB) -83.2 (3.13)’ ' n/a

. private gifts (FCFA/AB) 18.5 (1.71)' 5.21 (0.34)

 

 

 

Note: Only wealth variables are shown for average wealth specifications. T-statisties are in parenthesis.
*means 10 %levelofsignificanceand "means 5%levelofsignificance.

 

Chapter 6

Summary of Approach and Findings

In this dissertation, I maintain the following hypotheses: (a) Households in the Semi-
arid Tropics (SAT) tend to save for future uncertain income, that is, there is precautionary
demand for savings; (b) Such savings take the form of livestock holdings because there is no
formal saving institutions and because livestock is a relatively safe asset in times of drought in
the SAT; (c) As a result of the precautionary livestock holding, overgrazing occurs because
grazing land is an open-access, common-pool resource; (d) Formal financial institutions, such
as drought insurance, will substitute for livestock holdings and reduce overgrazing.

Properties and conditions of these hypotheses are examined theoretically in chapters
24. Then, they are empirically tested in chapter 5. However, the tests are indirect and partial
because the objectives of chapter 5 are not primarily to test these hypotheses. In this chapter, I
summarize how these hypotheses are tested indirectly and draw conclusions of this

dissertation.

Hypothesis a: Households in the SAT tend to save for future uncertain income, that is, there
is precautionary demand for savings.

a. 1 Condition for Precautionary Savings

Saving is considered to be a riskless asset with a fixed interest rate in most models
reviewed in chapter 2. In this case, stochastic exogenous labor income is usually the only
source of risk. But sometimes risky savings are also considered In this case, risk comes from
the stochastic interest rate, but the model can include risky exogenous income as well.

The findings in the literature review in chapter 2 are summarized as follows. When
saving is riskless, an increase in exogenous income risk will increase optimal saving if the

marginal utility function is convex, while it will decrease optimal saving if the marginal utility

233

234

function is concave‘. This is the most standard assertion about precautionary demand for
saving. Even if risk is from a stochastic interest rate, the effects of increasing risk on riskless
saving are the same as those of income risk. However, when saving is risky, the effect of
increasing interest rate risk is not the same as in the case of riskless saving: if the marginal
utility function is convex, the effects are ambiguous, while if the marginal utility function is
concave, an increase in interest rate risk will decrease optimal saving. In the case of concave
marginal utility, the effects on risky saving are the same as those on riskless saving. The
ambiguity in the case of convex marginal utility is due to the mixture of prudence and risk-
aversion: a positive third derivative of the utility function (i.e. convex marginal utility
function) represents the prudence that increases optimal saving when risk increases; on the
other hand. a negative second derivative (i.e. concave utility) represents the risk-aversion that
decreases optimal holding of risky asset when risk increases. Therefore, if the household is
prudent enough, an increase in interest rate risk will increase even risky saving.

In short, convex marginal utility is a necessary condition for precautionary demand
for savings. It is a necessary and sufficient condition when saving is riskless, while it is just a

necessary condition when saving is risky.

a. 2 Robustllus of Precautionary Savings

In chapter 3, I examine the robustness of the above conclusions in several extensions
of the basic models presented in chapter 2.

First, I consider the case when the two risks (exogenous income risk and exogenous
interest rate risk) are correlated In this case, when one of the risks increases it is possible that
optimal riskless saving decreases even when the marginal utility function is convex, and that

optimal riskless saving increases even when the marginal utility function is concave. These

 

‘ The convexity and concavity of the marginal utility determine how the expected marginal utility
changeswhenriskchanges. Seefootnote 1 incbqlterZ.

235

results are not consistent with the results which are obtained when the risks are independent.
In addition, when the risks are correlated, the effects on risky saving are ambiguous.

Second, in agricultural household models, risky agricultural production is equivalent
to the risky saving in models presented in chapter 2. But the differences are (i) risky
agricultural production is non-linearly related to inputs and risks, while risky saving usually
assumes a linear return to investment; (ii) risky agricultural production requires multiple
inputs, such as labor, capital, and land, while risky saving usually assumes a single input,
money; (iii) labor income is not exogenous in an agricultural production, and therefore
exogenous income may not exist or we must assume something other than labor income.

In a dynamic agricultural household model, 1 consider both riskless saving and capital
and labor inputs for risky agricultural production, and examine the effects of increasing risk
in agricultural production. The results are: (i) optimal riskless saving increases when marginal
utility function is convex, but it is ambiguous when the marginal utility function is concave;
(ii) optimal use of inputs increase when the inputs are risk-decreasing and when the marginal
utility function is convex, but otherwise it is ambiguous. The ambiguity of the effect on
optimal use of risk-increasing inputs in the case of convex marginal utility is due to mixture
of prudence and risk-aversion; a risk-averse household reduces risk-increasing inputs when
risk increases.

Third, in the dynamic agricultural household model, I consider the case when two
correlated risks exist. In this model, the two risks are stochastic rainfall which affects
agricultural production and social insurance (or transfer) risk, which are equivalent to interest
rate risk and exogenous income risk respectively in the model presented first. These two risks
are correlated because rainfall shock determines social insurance. However, unlike the case
presented first, the correlation is one-way; rainfall affects social insurance, but there is no
effect of social insurance on rainfall. By introducing this correlation of risks, the results are
qualitatively the same as those in the single-risk case presented second if social insurance is

imperfect. However, when social insurance is perfect, risk-neutral decisions such as riskless

236

saving become insensitive to change in risk, while risk-decreasing (increasing) inputs increase
(decreases) when risk increases regardless of the sign of the third derivative of utility
function. These are substitution effects due to change in risk, or pure risk-aversion effects.

Fourth, the effects of increasing risk are examined in a sequential decision model. If
saving is riskless, the results are not different from those obtained in the simultaneous
decision model presented second; that is, when marginal utility is convex, optimal riskless
saving increases as risks increase. However, with respect to input decisions, the results depend
on the model, that is, they depend on the assumption of availability of information and timing
of decisions.

These four extensions above lead to the conclusion that precautionary demand for
savings and precautionary demand for risk-decreasing inputs exist when marginal utility is
convex. Even when risks are correlated, if the correlation is specific as defined above, the

results will be robust.

a. 3 Endogenous Risk

So far I have considered only exogenous risks. In two period models, this is
acceptable because exogenous risks can be immediately translated into consumption risk in
the second period. In multiperiod models, however, precautionary demand for savings
depends not directly on exogenous risks but on endogenous income and/or consumption
risks (or risk in utility derived from consumption and leisure) induced by the exogenous
risks. From this viewpoint, we can say that precautionary demand for savings depends on a
household’s ability to smooth income and/or consumption under exogenous risks. The
theoretical results presented in a. 1 and a. 2 are still true, but are just qualitative, and the
degree of precautionary demand for savings depends on household characteristics. Therefore,
wealth or anything that helps predict the future variability of consumption will have a role in

predicting the precautionary demand for savings.

237

This point is reflected in the agropastoral model in chapter 4. The model assumes that
crop production risk is constant over the households, and predicts that the more risky is crop
production, the more livestock a household will keep. But this risk can be regarded as income
risk that depends on household characteristics. By assuming 80, this model is consistent with
the view that precautionary demand for savings depends on household’s ability to reduce

exogenous risk.

a. 4 Evidence of Precautionary Dcnnnd for Savings

Here, I present empirical evidence of precautionary demand for savings by showing
that households in the Sahel have a convex marginal utility function. With a standard utility
function, if we can show that the decline of marginal utility is diminishing as wealth increases,
we can conclude that the marginal utility function is convex.

The convexity of the marginal utility function can be proven empirically based on
empirically estimated 41 , or the marginal rate of substitution of expected utility between a
good year and a drought year. As seen from Figure 2 in chapter 5, the ratio between two
slopes can be easily calculated using the estimated 41. Because this ratio decreases as drought
probability increases in the Sahelian and the Sudanian zones, I conclude that the marginal
value function is convex there. This suggests the existence of precautionary demand for

saving. But in the Guinean zone, the result suggests no precautionary demand for saving.

a. 5 Demand for Drought Insurance and Precautionary Demand for Savings

41 (the marginal rate of substitution of utility between a good year and a drought
year) measures demand for formal drought insurance, but it is a mixed measure of a
household’s degree of risk-aversion and a household’s ability to reduce crop production
risk. Thus, a high d can be due to both high risk-aversion and high expected risk in crop
production. These two effects are not consistent because a risk-averse household (high 41)

tends to choose less risky crop production (low 4:). Therefore if a household has a high d , it

238
may imply that the household cannot reduce exogenous risks effectively, even if it is risk-

averse. On the other hand, a low ¢ implies that a risk-averse household successfully reduces

exogenous risks.

There may be three cases of the relation between a household’s initial wealth and 43
as follows. (i) Wealth will give a low 41 if it allows a household to reduce exogenous risks.
This may include the case when a poor household is forced to choose risky crop production
due to binding constraints. (ii) Wealth will produce a high 40 if it induces risky decisions
because wealth provides the ability to smooth consumption. (iii) Wealth will produce a high
41 if a household facing great exogenous risk accumulates wealth, but the wealth does not
effectively reduce risk. (ii) and (iii) are distinguished by regression specifications: in fixed
effect regressions we can observe (ii), while in cross-section regressions we can observe (iii).
In addition to these three effects, change in wealth is likely to affect a household’s risk-
aversion, which also changes 4:. Taking this effect into account, wealth will make the
household less risk-averse, and as a result will reduce d. Therefore, (1) will be strengthened
and (ii) and (iii) will be weakened by this effect.

On the other hand, wealth will also have effects on precautionary demand for savings
because wealth provides the ability to reduce exogenous risk and to smooth consumption. In
case (i) above, wealth may reduce precautionary demand for savings because it reduces
exogenous risk, and therefore, a high d implies high precautionary demand for savings. In
the case (ii) and (iii), wealth may induce or at least not reduce precautionary demand for
savings because it increases or at least does not reduce income risk, and therefore, a high 4»
implies high precautionary demand for savings. Thus, in either case, p measures both
demand for formal drought insurance and precautionary demand for savings.

Following the discussion above, 41 in Table 5 of chapter 5 now can be regarded as an
index of precautionary demand for savings. Consequently, Table 5 can be interpreted as
follows: in the Sahelian and the Sudanian zones, precautionary demand for savings decreases

as rainfall risk decreases. This suggests that households in the Sahelian and the Sudanian

239

zones have little ability to smooth consumption and as a result rainfall risk induces
precautionary demand for savings. On the other hand, in the Guinean zone, as discussed
above 41 may not imply precautionary demand for savings, but rather it can be interpreted
that households in the zone have precautionary demand for dissaving, which increases as
rainfall risk decreases. This suggests that households in the Guinean zone tend to gamble
when they are in a less risky situation because they have enough ability to smooth
consumption. Moreover, comparison of precautionary demand for savings between the
Sahelian and the Sudanian zones is consistent with differences in riskiness in rainfall of the

“NO 20008.

It can be concluded from the empirical study that precautionary demand for savings

exists in the two relatively risky zones, and that it can be measured by 41.

Hypothesis b: Savings take the form of livestock holdings because there is no formal saving
institutions and because livestock is a relatively safe asset in times of drought in the SAT.

The following facts support the hypothesis that livestock holdings are precautionary
in the Sahel.

First, as shown in Appendix D of chapter 5, livestock holdings per adult equivalent are
highest in the Sahelian zone, and lowest in the Guinean zone. This implies that the more
rainfall risk, the more are livestock holdings relative to crop production. This indirectly
suggests that livestock is at least one form of precautionary savings.

Second, in cross-section regressions the effect of livestock holdings on d is positive
in the middle livestock holding stratum in Table 8 of chapter 5, which implies that when we
compare households, a household that has more livestock takes more risk. In addition, when
average 418 are compared among three strata in Table 8, it is found that the larger is the
average d , the larger is the average livestock holding. This implies that livestock holding is
precautionary saving, but it does not reduce exogenous risks effectively. These findings are

consistent with the case (iii) in a. 5.

240

Third, the fixed effect regressions in Tables 7 and 8 of chapter 5 show that livestock
holdings have a negative or an insignificant effect on 41: negative in the Guinean zone and
the lower livestock holding stratum and insignificant in other zones and strata. The negative
effect means that livestock holdings reduce exogenous risks, which is consistent with effective
precautionary saving, or the case (i) in a. 5. The case of an insignificant effect may occur in
one of three situations as follows: ( 1) although livestock holdings are precautionary, the
reduction of exogenous risks is not effective; (2) household’s livestock holdings are
excessively so large that there is no marginal effect of livestock on risk reduction; or (3)
livestock holdings make households less risk-averse (negative effect on d) but allows them to
gamble (positive effect on 41), and these two effects are canceling each other. But case (3) is
not precautionary saving.

Combining the cross-section and fixed effect results, (1) and (2) will be the case in the
Sahelian and the Sudanian zones. Therefore, the empirical results support the hypothesis that
livestock holdings are precautionary in those zones.

However, the evidence does not necessarily imply that livestock holdings are the only
precautionary savings. For example, off-farm income has a negative significant effect on p
in the middle livestock holding stratum. This implies that off-farm activities are also

precautionary for the households in this stratum.

Hypothesis c: As a result of the precautionary livestock holding, overgrazing occurs because
grazing land is an open-access, common-pool resource.

Empirical evidence presented in (a) and (b) supports that livestock holdings are
precautionary in the Sahel. This finding justifies the agropastoral model in chapter 4, in
which livestock holdings are assumed to be riskless saving to hedge against crop production
risk. One important finding from the model is that overgrazing does not necessarily occur
even if grazing land is open-access, and that overgrazing is likely to occur when crop

production risk is high and expected return to crop production is low.

241
However, I have no data on quality of grazing land and governing rules of common-
pool resources. And therefore, no empirical evidence is presented for this hypothesis.
Empirical examination of this hypothesis is left to future research. Thus, this is the weakest

link of the four hypotheses.

Hypothesis d: Formal financial institutions, such as drought insurance, will substitute for
livestock holdings and reduce overgrazing.

This is predicted by the agropastoral model in chapter 4; if agricultural production
risk decreases, livestock holdings will be lower. If drought insurance reduces crop production
risk, a household that buys formal drought insurance will reduce agricultural production risk
and as a result will reduce livestock holdings. Therefore, for formal drought insurance to
reduce overgrazing, a necessary condition is that most households demand the insurance.
This question is empirically answered in chapter 5 by examining demand for hypothetical
formal drought insurance.

Since there is no demand for formal drought insurance in the lower livestock holding
stratum, insurance will not reduce livestock holdings in this household group. But because
total livestock holdings in this stratum are small relative to those in other strata, those
households do not harm grazing land very much. On the other hand, in other strata, there is
effective demand for the formal drought insurance, and consequently if households buy the
insurance, livestock holdings may be reduced

But in those strata, livestock holdings have insignificant effect on 41. As discussed in
(c), there will be three cases: (1) Livestock holdings do not effectively reduce exogenous risk;
(2) There is an excess of livestock holdings (relative to the optimal self-insurance), probably
in the upper stratum; and (3) Livestock holding induces gambling behavior. In the case of (I)
and (2), households still demand formal drought insurance. This implies that livestock
holdings are not as effective insurance as is formal drought insurance. Therefore, households
will buy the insurance to be more effectively insured, if it is available, and they are likely to

reduce livestock holdings to finance it, especially if households have excess livestock

242

holdings. In case (3) when households demand formal drought insurance, it implies that they
will substitute the insurance for current self-insurance including livestock holdings.
Therefore, in either case, effective demand for the formal drought insurance implies that

households will reduce livestock holdings.

However, if there are other precautionary assets, livestock is not necessarily reduced
by the insurance. For example, in the middle livestock holding stratum, off-farm income has
a negative effect on ¢. This means that households may reduce off-farm activities if drought
insurance reduces crop production risk effectively, and may not reduce livestock holdings
very much.

Because households that have large livestock holdings are responsible for
overgrazing, to avoid overgrazing they have to reduce livestock holdings significantly. As
shown in chapter 5, households in the upper livestock holding stratum show strong demand
for formal drought insurance, but livestock holdings have no effect on the demand It may
imply that the households in this stratum tend to keep excess livestock holdings though they
do not reduce risks enough. Therefore, they are most likely to substitute formal drought
insurance for livestock holdings. This means that formal drought insurance will decrease

overgrazing significantly.

Conclusion

Empirical evidence supports the hypothetical story presented at the beginning of this
chapter, except for hypothesis c. It is regrettable that I cannot test hypothesis c in this
dissertation due to lack of data. However, all the theoretical and empirical explorations in this
dissertation provide a big picture of household behavior under the drought risk in the Sahel,

and will contribute to future empirical research.

“lillillllllllllm