THREE ESSAYS IN DEVELOPMENT ECONOMICS
By
Godwin Debrah

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
Economics--Doctor of Philosophy
Agricultural, Food and Resource Economics--Dual Major
2017

ABSTRACT
THREE ESSAYS IN DEVELOPMENT ECONOMICS
By
Godwin Debrah
A large fraction of people in developing countries are engaged in some form of agricultural
activity for a living, making it very important for any effort or program designed to address the
issues with poverty or inclusiveness, to not abandon smallholder farmers. This dissertation,
titled, Three Essays in Development Economics seeks to bring to the attention of policymakers
on what to do and how to boost the incomes of rural dwellers in developing countries
sustainably. I use data from Ghana for the first essay, the second essay is a theory paper and data
from Tanzania is used for the third essay.
The first essay, which is titled, Does the Inverse Farm Size-Productivity Hypothesis hold
among Larger Farms. New Evidence from Ghana, examines the relationship between the area
planted in hectares and three measures of productivity, over a wide range of medium and large
scale farm sizes that represent the fastest growing segment of farms in Africa and now account
for a significant fraction of total area cultivated in the region. The results when Ordinary Least
Squares estimation is used, point to a negative and significant relationship between farm scale
and productivity. However, there is no statistically significant result in favor of the negative
relationship once I instrument for the bias that may be due to measurement error in farm size
variable. This can guide policymakers on how to redistribute land efficiently.
Inability of lenders to price for risk, limits the amount of loans financial institutions are
able to give out. This is what the second essay titled, Joint Liability Lending with Correlated
Risks explores. Because the rural-poor have no collateral to access loans, joint liability lending

has been a strategy microfinance institutions have used to price for risk and improve repayment
rates. However, when project returns are correlated like those of farmers, joint liability lending
may not help the lender to effectively price for risk to improve efficiency and repayment rates.
The second essay thus, theoretically, characterizes the optimal lending contracts in such a
situation. I find that correlation reduces the efficiency of group-based joint liability lending
relative to independent risks’ case. Thus, correlation is bad for group lending. I also extend the
model to show that, in some instances, it may be better for banks to separate whom they serve;
serves either only borrowers with correlated project returns or borrowers with independent risk.
The third essay in this dissertation titled, Predictors of the Choice of Rural Nonfarm
Activity in Tanzania investigates the predictors of participation in the rural nonfarm economy
and the predictors of the choice between wage and self-employment conditional on participation.
While some authors argue that households choose self-employment because they have no wage
employment options, others argue self-employed people are self-selected entrepreneurs and
should have some support. Understanding the predictors of the choice of employment would
serve as a source of very relevant information to policymakers.
The results suggest that, assets are key predictors of households’ participation in
agricultural as well as in the rural nonfarm economy. For households that participate in the rural
nonfarm economy, the choice between wage employment and self-employment is also related to
the availability of assets. In sum, this dissertation uses both theoretical and empirical evidence to
proffer solutions or suggestions that would help reduce poverty in developing countries.

Copyright by
GODWIN DEBRAH
2017

I would like to dedicate this dissertation to my father Mr. Emmanuel Debrah and to the memory
of my late mum Mrs. Selina Agbodza Debrah. I am eternally grateful for your love and
extraordinary sacrifices that you made because of my sisters and me.

v

ACKNOWLEDGEMENTS

My profound gratitude goes to my major professors; Professor Christian Ahlin and Professor
Thomas Jayne. Words, certainly, cannot describe how grateful I am for the financial support I
received from Professor Thomas Jayne, without which I could not have finished my doctoral
studies at Michigan State University. I am also particularly grateful to Professor Christian Ahlin
for his support, patience and encouragement. You are forever part of my history and success
story. I extend sincere thanks to Dr. Andrew Dillon and Dr. Leah Lakdawala. I really benefited
from your valuable comments, directions and other support before and during my dissertation. I
thank the PI and co-PIs of the GISAIA project.
I would also like to express gratitude to some of my friends who were of tremendous help
to me during my doctoral studies at Michigan State University-- Dr. Benjamin Adu-Addai, Dr.
Daniel Duah, Dr. Felix Yeboah, Dr. Serge Adjognon, Yi Li, Fei Jia, Chewe Nkonde and John
Olwande just to mention a few. To my entire church family in Lansing, I say it was a great
blessing meeting all of you.

vi

TABLE OF CONTENTS

LIST OF TABLES ......................................................................................................................... ix
LIST OF FIGURES ....................................................................................................................... xi
KEY TO ABBREVIATIONS ....................................................................................................... xii
1

INTRODUCTION ................................................................................................................... 1

2 ESSAY 1: DOES THE INVERSE FARM SIZE-PRODUCTIVITY HYPOTHESIS HOLD
AMONG LARGER FARMS? NEW EVIDENCE FROM GHANA ............................................ 6
2.1
Introduction ...................................................................................................................... 6
2.2
Literature review .............................................................................................................. 9
2.3
Data ................................................................................................................................ 13
2.4
Theoretical and Empirical Framework ........................................................................... 15
2.4.1
Estimation ............................................................................................................... 17
2.5
Robustness Checks ......................................................................................................... 20
2.6
Results ............................................................................................................................ 22
2.7
Conclusion...................................................................................................................... 27
APPENDICES .............................................................................................................................. 30
Appendix A: Tables for Essay 1 .................................................................................... 31
Appendix B: Figures for Essay 1 ................................................................................... 46
3

Essay 2: Joint Liability Lending with Correlated Risk.......................................................... 52
3.1
Introduction .................................................................................................................... 52
3.2
Literature Review ........................................................................................................... 56
3.3
Baseline Model ............................................................................................................... 58
3.3.1
Economic environment ........................................................................................... 58
3.3.2
Individual Lending in a Static Environment ........................................................... 60
3.4
Group Lending with independent risks .......................................................................... 61
3.5
Group Lending with Correlated Risk ............................................................................. 64
3.6
Constant Mass Case (Case 1) ......................................................................................... 65
3.7
Constant-correlation Case (Case 2) ................................................................................ 69
3.8
Correlation in a Market of Mixed Borrowers................................................................. 74
3.8.1
Optimal Contracts ................................................................................................... 75
3.8.2
Graphical presentation of propositions 6 ................................................................ 79
3.9
Conclusion...................................................................................................................... 80
APPENDICES .............................................................................................................................. 83
Appendix A: Tables and Figures for Essay 2 ................................................................. 84
Appendix B: Proofs of Lemmas and Propositions in Essay 2........................................ 89
vii

4 ESSAY 3: PREDICTORS OF THE CHOICE OF RURAL NONFARM ACTIVITY IN
TANZANIA .................................................................................................................................. 99
4.1
Introduction .................................................................................................................... 99
4.2
Data Description and Patterns of Rural Income Diversification in Tanzania .............. 102
4.3
Estimating Participation in RNFE ................................................................................ 104
4.4
Results and Discussions ............................................................................................... 105
4.5
Conclusion.................................................................................................................... 113
APPENDIX ................................................................................................................................. 115
BIBLIOGRAPHY ....................................................................................................................... 124

viii

LIST OF TABLES

Table 2-1: Changes in Farm Structure in Ghana (1992 to 2013) ................................................. 31
Table 2-2: Sample Size by Area Planted in Hectares .................................................................. 31
Table 2-3: Household Demographics and Input Use by Operated Farm Area ............................ 32
Table 2-4: Descriptive Statistics ................................................................................................... 33
Table 2-5: Test for Efficiency of Family versus Hired Labor (OLS) ........................................... 34
Table 2-6: Test for Efficiency of Family versus Hired Labor (IV) .............................................. 34
Table 2-7: Cobb-Douglas Production Function Estimation......................................................... 35
Table 2-8: Estimates of the IR Valuing Family Labor at District Median Wages (OLS) ............ 35
Table 2-9: Valuing Family Labor at Shadow Wages (OLS) ....................................................... 36
Table 2-10: First Stage of IV Estimation ...................................................................................... 36
Table 2-11: Correcting for Measurement Error Bias in Area Planted (IV) ................................. 37
Table 2-12: Correcting for Measurement Error Using Shadow Wages ...................................... 37
Table 2-13: Estimates of the IR Using Operated Farm Size (OLS) ............................................. 38
Table 2-14: Estimates of the IR Using Operated Farm Size (IV) ................................................. 38
Table 2-15: Alternative Estimate of the IR (OLS)........................................................................ 39
Table 2-16: Estimates of the IR Using Total Factor Productivity ................................................ 40
Table 2-17: Computed Shadow Wages Versus District Median Wages ..................................... 40
Table 2-18: OLS Estimation of IR in Levels ................................................................................ 40
Table 2-19: Comparing Self-reported Farm Sizes ....................................................................... 41
Table 2-20: Summary of Previous Studies .................................................................................. 42
Table 2-21: OLS Results Using Sample between 1st and 99th Percentile ..................................... 45
ix

Table 3-1: Joint Output Distribution in the Presence of Correlation ............................................ 84
Table 4-1: Rural Tanzania Income Generating Activities .......................................................... 116
Table 4-2: Means and Percentages (2008 Survey Data) ............................................................. 118
Table 4-3: Means and Percentages (2010 Survey Data) ............................................................. 119
Table 4-4: Means and Percentages (2012 Survey Data) ............................................................. 120
Table 4-5: Predictors of participation in Non-Ag Wage Employment ....................................... 121
Table 4-6: Predictors of Participation in Non-Ag Self Employment .......................................... 122
Table 4-7: Predictors of Choice of Non-Agricultural Self-Employment.................................... 123

x

LIST OF FIGURES

Figure 2-1: Plot of Measures of Productivity and Area Planted in levels .................................. 46
Figure 2-2: Plot of Measures of Productivity and Area Planted in levels using shadow wages. 47
Figure 2-3: Gross Output per Hectare Against Area Planted ...................................................... 48
Figure 2-4: Plot of Measures of Productivity and Area Planted in logs .................................... 49
Figure 2-5: Distribution of Area Planted Variable ....................................................................... 50
Figure 2-6: Sensitivity Analysis.................................................................................................... 51
Figure 3-1: Correlated Borrowers ................................................................................................. 84
Figure 3-2: Mixed Pool of Borrowers for Lower k....................................................................... 85
Figure 3-3: Mixed Pool of Borrowers for Higher k ...................................................................... 86
Figure 3-4: K=0.61 Where Mixed Line Crosses Correlated Line to Lie Above it) ...................... 87
Figure 3-5: Plot of Joint Liability against Correlation among borrowers (C against V) .............. 88

xi

KEY TO ABBREVIATIONS

RNFE

Rural Nonfarm Economy

IMF

International Monetary Fund

FAO

Food and Agricultural Organization

LSMS

Living Standards Measurement Study

ISA

Integrated Survey on Agriculture

RIGA Rural Income Generating Activity
LFE

Linear Fixed Effects

CRE

Correlated Random Effects

MLE

Maximum Likelihood Estimation

TZS

Tanzanian Shillings

MFIs

Microfinance Institutions

IR

Inverse Relationship

xii

1

INTRODUCTION

Although economic growth in many developing countries have led to some decline in poverty,
inequality, unemployment and abject poverty still persist in many parts of the developing world
(See recent World Bank 2016 report). Efforts by development agencies and the World Bank to
eradicate extreme poverty and promote shared prosperity require that growth is inclusive. The
poor and vulnerable would need good health, education and the needy given some safety nets
including health insurance and access to modern technology. Majority of these poor households
live in the rural areas and engage in some agriculture activity. Thus, efforts to lift these millions
of people out of poverty would require that agricultural productivity be increased. Aggressive
industrialization pursued, so that positive structural transformation can be achieved. The bulging
youth and its attendant unemployment challenges in Sub-Saharan Africa, makes it imperative
that policymakers aggressively work on expanding their economies, and provide jobs for the
youth. Agricultural productivity while depending on technology and input used, also depends on
farm management and whether the farmers are making the most of the land available to them
amidst the challenges associated with acquiring land.
Despite the land issues that have bedeviled the Sub-Saharan Africa region, policymakers
can maximize productivity by ensuring that land is available to those with high farm
productivity. This can also be achieved by facilitating the existence of land markets. This
dissertation that focuses on farm productivity, microcredit and the rural nonfarm economy, sheds
light on how policymakers may be able to boost the income of rural dwellers most of whom
participate in some form of agricultural activity, towards the eradication of extreme poverty and
promoting shared prosperity. The first essay of this dissertation studies the relationship between

1

productivity and farm size. We focus on the fastest growing segment of farms in Sub-Saharan
Africa. Many previous studies focused on farm sizes that are less than 5 hectares.
The relationship has important ramifications for policy. If policymakers were to have a
large parcel of land, there is the need for evidence-based decisions to be made regarding how
that parcel of land could be reallocated. Would they allocate them in larger pieces to medium and
large-scale farmers or subdivide them into smaller pieces for smallholder farmers. The latter
could be motivated by the empirical regularity that small farms produce more per land area than
larger scale farmers do. I use data from southern Ghana on medium scale farmers to investigate
the inverse farm size and productivity hypothesis. This data is part of the Guiding Investments in
Sustainable Agricultural Intensification in Africa (GISAIA) project. I also investigate whether
labor market imperfection can drive the IR or not.
The results show a negative and significant relationship between scale/farm size and
productivity when Ordinary Least Squares estimation is used. However, there is no statistically
significant result in favor of the IR once I instrument for the bias that may be due to
measurement error in farm size variable. This means that the empirical regularity may not hold
over the range of medium scale farms considered in this study. This can guide policymakers on
how to allocate land and how to formulate policies that would enable high productive farmers to
get access to more land.

A plethora of the literature on inverse farm size-productivity

hypothesis, using data from Sub-Saharan Africa has focused almost exclusively on farm sizes
that are less than 5 hectares. By considering larger scale farmers, i update the literature with
newer evidence. I also find no evidence of labor market imperfection as the source of the IR.

2

The causes of low farm productivity are not limited to labor and land market
imperfections or missing markets. Credit market imperfections also lead to inability of farmers to
access loans to expand productivity. Even if labor or land markets work, many smallholder
farmers are still constrained financially. Most farmers would need financial assistance to be able
to purchase inputs including improved seeds. Aside the unavailability of collateral, farmers tend
to have correlated risks, which makes them a very special category of borrowers. This motivates
the second essay of the dissertation.
The second essay studies joint liability lending in the case where project returns are
correlated, (Besley, 1994) identified three major things that make rural credit markets in
developing countries less efficient compared to those in developed countries. These things were
listed as collateral security unavailability, covariant risk and under developed states of related
institutions. I approach the issue of correlated risks or correlated project returns theoretically.
The problem is important both empirically and theoretically. Theoretically, it would be
interesting to know how correlated risk affects the efficiency of the credit market. Empirically,
correlated risk is a pervasive reality. It would be interesting to know whether lenders treat
borrowers who are known to have correlated risks separately, or pools borrowers.
In this second essay, I derive and characterize the optimal lending contracts for a joint
liability lending when risks are correlated. When correlation is introduced, I find that the
parameter space for which lending can be fully efficient is smaller compared to the case where
risks are not correlated. This is partly due to the monotonicity constraint that prevents the lender
from raising the joint liability component of the contract above the interest rate, and partly
because of affordability constraints. The monotonicity and affordability constraints work against
a restoration of the implicit discount (premium) that joint liability offers (charges) safe (risky)
3

borrowers- a discount/premium that varies negatively with correlation. Thus, correlation is bad
for group lending. High correlation can lead to the exclusion of some potential borrowers such as
the safe uncorrelated from the market there by reducing efficiency.
The results show that under certain conditions, such as situations where project returns
are low and the fraction of correlated borrowers is high, fully efficient lending requires that the
banks for correlated borrowers and those for non-correlated borrowers be separated.
Governments or policymakers can set aside agricultural development banks that focus
exclusively on farmers and develop strategies that can help improve efficiency in terms of
outreach and repayment rates.
In the absence of credit or assets needed to go into entrepreneurial or non-agricultural
self-employment, rural dwellers can engage in subsistence activities, or work in wage
employment. The debate in the development economics literature on households’ choice of
employment motivates essay three. The third essay investigates the predictors of participation in
the rural nonfarm economy and the predictors of the choice between wage and self-employment
conditional on participation. While some authors have found that households choose selfemployment because they have no option, others think self-employed people in rural areas are
entrepreneurs and must be supported. That could mean wage workers are frustrated
entrepreneurs. Further research is needed to ascertain the causes of the choice of employment by
households. This is also necessary to accelerate governments’ efforts geared towards eradicating
extreme poverty.
The essay uses the World Bank LSMS data on Tanzania. In particular, the first three
waves of this data set (2008/2009, 2010/2011 and 2012/2013). The FAO has data on rural

4

income generating activities (RIGA) based on the LSMS data. I start by analyzing predictors of
participation in rural non-agricultural wage and self-employment, and proceed further to
investigate the predictors of the type of employment households choose once they participate in
the rural nonfarm economy. Households are put into four groups. Group 1 has households that
did not engage in any nonfarm activity. Group 2 comprises of households that engaged in some
nonfarm activity consisting of only wage employment. Group 3 comprises of households that
engaged in some nonfarm activity consisting of only self-employment and group 4 comprises of
households that engaged in both wage and self-employment activities in the rural nonfarm
economy.
The results suggest that, wealth, land or assets are key predictors of households’
engagement in agricultural as well as the rural nonfarm economy. Whether households choose
wage employment or self-employment upon participating is also related to the availability of
such assets. The results seem to suggest that households are pulled into the rural nonfarm
economy rather than being pushed into it as a survival strategy. Some of the results also suggest
that reverse causality could be of less concern.
In sum, this dissertation uses both theoretical and empirical evidence to investigate ways
policymakers can influence the poverty statuses of rural households or households that rely
heavily on agriculture for their livelihoods. The essays guide African governments or
governments of developing countries on interventions that can be made in terms of land
allocation. Who gets credit and how credit can be made available to farmers to help them
increase productivity and hence their incomes. The dissertation shows wealth or assets, or credit
constraints can hinder the ability of rural dwellers to take advantage of the rural nonfarm
economy as a way of boosting their incomes.
5

2

ESSAY 1: DOES THE INVERSE FARM SIZE-PRODUCTIVITY HYPOTHESIS HOLD
AMONG LARGER FARMS? NEW EVIDENCE FROM GHANA

Introduction
Interest in the relationship between farm size and productivity in Africa, has been driven in
recent years by rising doubts about the potential of smallholder-led agriculture growth in Africa
e.g., (Collier & Dercon, 2014), and also by documented changes in farm size distributions
observed in many African countries and in particular the rising share of cultivated land on
medium-scale farms (Jayne et al., 2016; Jayne, Chapoto, Sitko, Nkonde, & Chamberlin, 2014).
In light of such trends, major policy debates are arising over how the region’s remaining prime
agricultural land should be allocated, especially in light of rising land scarcity and land prices
(Deininger, Savastano, & Xia, 2017; Holden & Bezu, 2016) and challenges associated with
access to land for young people (Sezu & Holden, 2014).
The inverse farm size relationship (IR) refers to the observation that small plots or farms
produce more output per unit area than larger plots or farms. The IR has been one of the
enduring justifications for supporting smallholder farmers in developing countries. Recent
studies upholding the inverse relationship between scale and productivity in Africa have
generally been based on samples of farms almost entirely cultivating less than 10 hectares (Ali &
Deininger, 2015; Barrett & Bellemare, 2010; Carletto, Savastano, & Zezza, 2013; Larson,
Otsuka, Matsumoto, & Kilic, 2014). For example, less than one percent of the farms contained
in the Living Standards Measurement Study-Integrated Survey on Agriculture (LSMS-ISA)
analyzed in the studies cited above cultivate more than 10 hectares. Little is known about the
relationship between output per unit area and farm size across the range of farms between 5 and

6

100 hectares, which have grown rapidly in recent years (Jayne et al., 2016).1 Consequently,
available evidence is unable to guide African governments’ efforts to promote agricultural
productivity through land policies that might encourage access to land to either small-scale or
large farm units. Accurate information about the relationship between farm size and farm
productivity for a wide range of farm scales can therefore provide valuable guidance into African
governments’ agricultural and land tenure strategies.
This paper examines the relationship between farm area planted and three measures of
productivity, over a spectrum of farm sizes ranging from 5 to 100 2 hectares in four districts of
southern Ghana. These measures of productivity are: gross value of output per hectare planted,
net value of output per hectare planted, and total factor productivity. Comparing the results of
these alternative measures provide the means to evaluate the sensitivity of IR results to the use of
partial vs. total factor productivity measures3. In another specification, I examine the robustness
of my results to defining farm size in terms of the net value of output per hectare of potentially
utilizable land (area planted plus fallow) in light of the possibility that, larger farms may utilize a
smaller proportion of their total landholdings, which might be considered as foregone potential
that could have been realized by others under alternative land distribution patterns.
The study makes several contributions to this literature. First, I investigate the IR
hypothesis among farms sizes that are larger than typically studied in Africa, by using a sample
of households that is statistically representative of agricultural households cultivating between 5

1

In Ghana, farms cultivating between 5 and 100 hectares have grown rapidly since the early 1990s and now account
for about half of all the farmland under cultivation (Table 2-1).
2
The data has four observations greater than 100 hectares.
3
The land productivity measure has been criticized as not being a true measure of productivity. Net profit has been
argued as a better measure (Binswanger et al 1995). Total factor productivity looks at the productivity of all inputs
and is able to reward (penalize) prudently (imprudently) managed farms. See table 2-20 for the various measures
used in previous literature and their results.

7

and 100 hectares in the districts covered. Secondly, while a number of studies have
conventionally measured productivity in terms of yield, I use both partial and total factor
productivity measures as well as net values of production per hectare planted (the latter
accounting for the costs of inputs and labor and therefore approximating profits per unit area). I
then examine the consistency or sensitivity of results to the type of productivity measure used.
The measures of productivity taken as a whole tend to point to a negative relationship when I use
the Ordinary Least Squares method of estimation.
Third, I examine model robustness using the alternative measures of farm productivity
and a different land area measures (area planted plus fallow land), and test for potential labor
market imperfections as a possible cause of the IR.
Because family labor forms a significant component of farm labor, the valuation of
family labor may affect the estimated relationship between scale and productivity. I investigate
the role of family labor valuation in influencing the farm size/productivity relationship by
computing shadow wages using the estimated production function and examining whether the
relationship between productivity and land size differs depending on how family labor is valued.
If labor market imperfections do not exist, I would expect the sign of the relationship between
productivity and farm size to be unaffected whether family labor is valued using computed
shadow wages or observed local wage rates. My estimation results suggest no evidence of such
labor market imperfection.
As robustness checks, I examine the role of measurement error in farm size using an
instrumental variable. I instrument for respondent reported area in the year of the survey (2014)
using the respondent’s subsequent report of 2013/14 area one year later. In summary, I find that,

8

area planted is significantly inversely related to productivity among farms operating between 5
and 40 hectares in southern Ghana4.
The rest of the paper is organized as follows. Section 2.2 reviews the literature. Section 2.3
describes the data. Section 2.4 presents the theoretical and empirical framework. Section 2.5
presents some robustness checks. Section 2.6 has the results and discussions from the
regressions. The last section concludes.

Literature review
The inverse relationship between farm size and productivity was observed by Chayanov in
Russia and later by (Sen, 1962) in India.

Since then and with a few salient exceptions, their

findings have been generally reinforced by subsequent research, at least within the range of
relatively small farm sizes that tend to be examined in the literature. The works of (Lau &
Yotopoulos, 1971), (Benjamin & Brandt, 2002), (Berry & Cline, 1979), (M.R. Carter, 1984),
(Barrett & Bellemare, 2010), (Barrett, 1996), (Heltberg, 1998), (Barrett & Bellemare, 2010),
(Larson, 2012), (Carletto et al., 2013), and (Ali & Deininger, 2015) found evidence of the
existence of the IR. While (Kawasaki, 2010) used data from Japan and argued that a positive
relationship between farm size and productivity could be obtained if there is little land
fragmentation, (Dorward, 1999) using data from Malawi and (Kimhi, 2006) using maize plot
data from Zambia, found a positive relationship. (Kevane, 1996), (Zaibet & Dunn, 1998), and
(Binswanger, Deininger, & Feder, 1995), found no evidence to support the IR.

4

This is when Ordinary Least Squares (OLS) estimation is used. There is no statistically significant result in favor
of the IR once I instrument for the bias that may be due to measurement error in farm size variable using
instrumental variable (IV) approach.

9

Many authors argue that the typical IR observed in the literature reflect either an omitted
variable problem especially with respect to land quality and/or market imperfections (with labor,
land and credit markets cited most frequently) or measurement error with respect to cultivated
area. Measurement error can lead to the IR if smallholder farmers systematically understate their
farm sizes and/or when larger-scale farmers systematically over-report their farm sizes. Recently,
(Carletto et al., 2013) found that smallholders tend to overestimate plot sizes, while large scale
farmers underestimate plot sizes.
In addition, they found that measurement error does not explain the IR, and that the IR is
even more compelling after accounting for measurement error. (Gourlay, Dillon, McGee, &
Oseni, 2016) also found evidence to support the IR. Their measurements of plot area were from
compass and rope, GPS, and farmer self-reported. These results are different from (Lamb, 2003),
who concludes that the IR in profits disappears after a dummy variable for share cropping and
double cropping were used as instrument to overcome the bias due to measurement error. (Lamb,
2003) argues that, in the absence of any significant systematic measurement error in reported
farm sizes, the IR stems from market imperfections and or inverse correlation between land
quality and farm size. (Holden & Fisher, 2013) conclude that measurement error accounts for
about 60% of the IR based on their study of farms smaller than 1 hectare in Malawi.
An example of how omitted variable bias can lead to the IR is given in (Chen et al., 2011).
The argument is that, in rural China where local authorities divide land so that all local
households get their minimum nutritional needs, very fertile lands are seen to be subdivided into
smaller sizes making it seem that small farmlands are more productive if land quality is omitted
in the analysis. (Assunção & Ghatak, 2003), (Bhalla & Roy, 1988), (Benjamin, 1995), (Chen et
al., 2011), have all contributed to the IR literature from the omitted variables bias perspective.
10

While (Assunção & Ghatak, 2003) focused on individual heterogeneity among farmers, land
quality measures are the main focus of the other studies. Recently, (Barrett and Bellemare.,
2010) concluded that the IR cannot be entirely explained by unobserved land quality differences.
Our study does not include soil quality measures. I include village dummies in all estimations to
control for unobserved soil quality differences across villages and regions, recognizing that
unobserved plot-level soil quality variation within villages will still be a concern.
The other major category of explanations for the IR is market imperfections. These are
generally motivated by the fact that smallholders tend to use family labor more intensely than
large farms in the presence of market imperfections. This makes labor-to-land, and output-toland ratios higher on smaller farms, hence creating the observed IR ( Carter, 1984; Carter &
Wiebe, 1990). When land markets function imperfectly, larger landowners arguably have higher
land-to-labor ratios. Because it is expensive to supervise labor, larger farms tend to have lower
output to land ratios.
In place of an inverse relationship, (Kevane, 1996), used data from western Sudan and
asserted that a positive relationship between productivity and farm size should be observed. In
Kevane’s work, if investment matters in production and market imperfections prevent
smallholders from accessing credit to take advantage of a possible increase in productivity due to
some initial investments, larger farms would be more productive.
Despite the plethora of studies in the literature, there is no clear consensus among
agricultural economists as to what drives the IR or over which range of farm sizes it might hold.
I follow (Ali & Deininger, 2015) and compute shadow wages from the production function to
investigate the labor market imperfection argument. First, I test for differences in productivity

11

between family and hired labor and secondly, I compare the relationship between productivity
and farm size when family labor is valued using the district median wage versus using the
estimated shadow wages. I am unable to reject similar productivity levels between hired and
family labor, and neither do I find evidence of labor market imperfection. Section 3 has more
discussions on this.
Very recently, (Bevis & Barrett, 2017) used a plot level data set from Uganda to argue that
there is an edge effect (productivity of land higher around the edges of plots) which explains the
puzzle. They attribute the source of this edge effect to behavioral mechanism. That is, farmers
might take proper care of the more visible edge of plots than the interior. The data for this
research however does not have the information that would allow me test this behavioral
mechanism.
Choice of functional form may be especially important when testing the IR over a wide
range of farm sizes. (Kimhi, 2006) found that when endogeneity of plot size is corrected using
Heckman selectivity correction procedure, the relationship becomes U-shaped. This U shape is
not unique to Kimhi’s work. (Carter & Wiebe, 1990) found a positive relationship between net
profits and farm size but a U-shape between output per acre and farm size. I thus run a regression
that allows the slopes to differ by the range of farm size (0-5 ha, 5-20 ha and greater than 20 ha)
to investigate this but found no significant results.
Results may also depend on assumptions about the nature of the production function. In the
case of (Ali & Deininger, 2015), both Cobb-Douglas and Translog production functions were
estimated for Rwandan farmers. They found that the negative relationship between profits and
farm size disappeared once profits accounted for the cost of family labor valued at the local wage

12

rate rather than at the lower shadow wage rate. Their work suggests that market imperfections
may indeed be driving the IR at least in small farm settings where labor tends to be a major
production input. Table 2-20 summarizes the methods and results of some selected literature.

Data
The data used in this study is from a survey conducted in southern Ghana between late July and
first week of August in 2014 as part of the Guiding Investments in Sustainable Agricultural
Intensification in Africa (GISAIA) project. Unlike most of northern Ghana which has recorded
massive increases in the number of medium and large-scale farms in recent years (Chapoto,
Mabiso, & Bonsu, 2013), most of southern Ghana is densely populated and faces acute land
pressures. Four districts were purposively chosen from four regions of southern Ghana known to
contain a relatively large number of medium and large-scale farms. The selected districts are
Offinso North in the Ashanti region, Bibiani-Anhwiaso in the Western region, Afram Plains
South in the Eastern region and Nkwanta North in the Volta region.
A simple random sampling method was used. Villages within districts were randomly
selected and then farmers within villages randomly selected. Twenty villages were sampled at
random from each selected district. Agricultural extension agents in those villages created a list
of farmers operating more than 5 hectares in the 2013/2014 season in each of these 20 villages.
Some of these households were subsequently found to have cultivated less than five hectares.
The full population of listed farmers operating over 20 hectares was contained in the sample.
By sampling the entire population of such farmers, sampling bias issues are avoided. A farm
or area planted or landholdings as used in this paper are all at the level of the household. In
13

addition, for the purpose of this work, small farms are defined as those cultivating 5 hectares or
less. Medium-scale farmers cultivate between 5 and 20 hectares. Large-scale farmers are those
operating more than 20 hectares of land.5 The sample of 503 farmers contains 57 farmers with
operated land area less than 5 hectares, 385 farmers with operated area between 5 and 20
hectares, while 61 farmers have operated area above 20 hectares. Operated farm sizes at the 5th
and 95th percentiles of the distribution are 5.3 and 41.9 hectares respectively. The results of this
study can therefore be considered reasonably representative of farms operating up to roughly 40
hectares but not the entire population of medium and large-scale farms. Figure 2-5 has the
distribution of the area planted or cropped area in hectares.
Where the household head of the sampled farm was not available at the time of the
interview, he or she was replaced randomly with another farmer from the list. Most of the 38
replaced farmers had traveled out of the village at the time we visited those villages. Table 2-2
displays information on the number of farmers categorized by area planted. The survey collected
information on all utilized fields in aggregates. Area measures are as reported by the farmers to
enumerators; GPS or compass and rope methods were not used.
Table 2-3 presents descriptive statistics on the sampled farmers. The mean landholding is
18.32 hectares while the mean area cultivated was 12.85 hectares. For the price data, a section of
the survey asked farmer respondents about the wages they paid on average for various
agricultural activities per day. I use the median agricultural wage rate in the data to calculate the
net value of production and also for testing the relative efficiency of hired versus family labor.
Median village-level crop sales prices obtained from the survey data were used to value crop

5

While farm size categorization is deemed reasonable for southern Ghana, I realize that it may not conform to those
of other countries or regions with very different farm size distributions.

14

output. Prices of a few crops that were not available in the survey were replaced with prices
from the regional markets, as reported on the website of the Ministry of Food and Agriculture in
Ghana.

Theoretical and Empirical Framework
This section presents a simple theoretical framework that guides the empirical approach to
estimating the relationship between land size and productivity. I consider a constant returns-toscale Cobb-Douglas production function6. Farm households are assumed to face competitive
prices and choose capital and labor to maximize profits given a fixed quantity of land.
Let output be given by

𝑌 = 𝐴𝐾 𝛼 𝑁 𝛾 𝐿1−𝛼−𝛾

(1)

where 𝐴 is total factor productivity , 𝐾 is amount of non-labor inputs used, 𝑁 is labor input, 𝐿
is fixed land quantity, 𝑟 is rental rate of capital and 𝑤 is the wage rate.
The problem is stated formally as:
𝑀𝑎𝑥 𝜋 = 𝐴𝐾 𝛼 𝑁 𝛾 𝐿1−𝛼−𝛾 − 𝑟𝐾 − 𝑤𝑁

(2)

From the first order conditions with respect to labor and capital use, the following can be
derived

𝑁=[

𝑤

𝛾𝐴𝐾𝛼 𝐿 1−𝛼−𝛾

]

1
𝛾−1

and 𝐾 = [

𝑟
𝛼𝐴𝑁𝛾 𝐿 1−𝛼−𝛾

1
𝛼−1

]

6

Decreasing returns to scale by itself can lead to the IR. Results are not different from (Ali & Deininger, 2015) who
used a translog production function in one specification of the empirical analysis. Many studies have not rejected the
appropriateness of the Cobb Douglas production function

15

After some algebra, the optimal levels of capital and labor use in terms of fixed land can be
written as
𝛼−1
1−𝛼−𝛾

𝑁∗ = [

𝑤
1
𝑟
𝛾𝐴[( )𝛼 ]𝛼−1
𝛼𝐴

]

𝛾−1
1−𝛼−𝛾

L

and

𝐾∗ = [

𝑟
1
𝑤
𝛾𝐴[( )𝛾 ]𝛾−1
𝛾𝐴

]

L

So that, the optimal output level can be written as
𝑌 ∗ = 𝐷𝐿

(3)
𝛾−1
1−𝛼−𝛾

Where 𝐷 = 𝐴 [
{

𝑟
1
𝑤
𝛾𝐴[(𝛾𝐴)𝛾 ]𝛾−1

𝛼

𝛾

]

𝛼−1
1−𝛼−𝛾

[

𝑤
𝑟

} {

1

]

𝛾𝐴[(𝛼𝐴)𝛼 ]𝛼−1

}

From (3), it can be seen that, under constant returns to scale, the coefficient of the land variable
in the regression should be equal to one when we take logs of both sides. Empirically, most
papers have used the log-linear specification in estimating the relationship. This is seen by taking
the natural log of both sides of (3) to get
𝑙𝑛𝑌 = 𝑙𝑛𝐷 + 𝑙𝑛𝐿

(4)

Therefore, a regression equation could be specified as;
𝑙𝑛𝑌 = 𝛽0 + 𝛽1 𝑙𝑛𝐿 + 𝜖

(5)

In which case, the null hypothesis is that 𝛽1 = 1 while 𝛽1 < 1 denotes the IR. This means that,
output rises less quickly as land size rises. Alternatively, one can go back to (3) and divide
through by 𝐿 before taking the natural log of both sides. That gives us

16

𝑌

𝑙𝑛 = 𝑙𝑛𝐷 + 0 ∙ 𝑙𝑛𝐿
𝐿

Thus a regression specification from this, which is actually the most common
specification in the literature, will be of the form;
𝑙𝑛𝑦 = 𝛽0 + 𝛽1 𝑙𝑛𝐿 + 𝜀

, where 𝑦 =

𝑌

(6)

𝐿

In this case, the null is that 𝛽1 = 0 , while 𝛽1 < 0 indicates a negative relationship
between output per unit area and land area. I use specification in (6) in this research. This model
is similar to that of (Assunção & Braido, 2007), except that in their model, farmers maximized
expected profits. The theoretical framework above can also be used to motivate a regression
model in which the dependent variable is the net value of production or profit per hectare. This is
straightforward because, the profit function can be written as a linear function in the land size
variable. As mentioned earlier, I present estimation results for both gross and net value of output
per hectare planted. I also show results for total factor productivity measure for comparison
purposes.

2.1.1

Estimation

To test the IR hypothesis, I run the regression of the form
𝑙𝑛𝑦𝑖 = 𝛽0 + 𝛽1 𝑙𝑛𝐿𝑖 + 𝑿𝒊 𝚪 + 𝜀𝑖

(7)

where 𝑦𝑖 is either total factor productivity or the gross or net value of output per hectare planted
for household 𝑖, 𝐿𝑖 is area planted and X is a vector of covariates which includes dummies for
villages and whether major staple crops are grown (rice, maize and yam, groundnut and soy), and

17

labor demand variables. Tables 2-3 and 2-4 have descriptive statistics of the variables used in the
analysis.

Labor Market Imperfections
I test whether labor market imperfections drive the IR. To do this, I first test whether the
efficiency of family labor is the same as that of hired labor. This enables me to see whether it
costs more to use hired labor or not, and to determine how that affects larger scale farmers or
gives advantage to smallholder farmers. (Feder, 1985) shows, “a model with no supervision
effects on labor productivity would predict no relationship between farm size and productivity”.
Data on labor used by the farmers was self-reported. Information was collected on all
plots that were used by the farmer but the information is not at the plot level. Farmers stated
lump sums of how much they expended on all utilized plots, both cash and in-kind expenditures
for hired and exchange labor. Following (Benjamin, 1992), I test for the differences in efficiency
using the equation 𝑙𝑛𝑁 = 𝛼 + 𝛽𝐿 + 𝛿𝑙𝑛𝑤 + (1 − 𝜃)

𝑁𝐻

(8)

𝑁

Where 𝑁 is total labor demand, 𝑤 is the price of labor, 𝑁 𝐻 is hired labor thus

𝑁𝐻
𝑁

is the fraction

of hired labor used. i test under the null hypothesis that 𝜃 = 1 or (1 − 𝜃) = 0. A negative sign
would suggest that hired labor is more efficient. By contrast, if hired labor were less efficient
than family labor, this would make large-scale farms that use hired labor relatively intensively
face a higher effective labor cost per worker than small farms. This can result in large farms
having a lower labor to land ratio compared to small-scale farms and consequently, have a lower
output to land ratio. Alternatively, the higher effective cost of labor can cause large-scale farmers
18

to substitute capital for labor. If this were to be the case, the large-scale farms would hire less
labor, so family members of small farms, who cannot work as wage laborers apply their excess
labor intensively on their own farms, hence driving down the shadow wage of family labor. This
is the classical explanation for how labor market imperfections may drive the IR.

The variable

𝑁𝐻
𝑁

may have to be treated as endogenous because of a possible division bias

or simultaneity. Results using ordinary least squares and instrumental variable estimations, fails
to reject similar productivity levels, although estimates become quite noisy in the instrumental
variable estimation. In a recent work by (Lafave & Thomas, 2014), separation or recursion in the
agricultural household model is rejected but this is not due to differential cost of hired versus
family labor. Suggesting that, markets are not complete and hired and family labor may not be of
the same efficiency.
Various measures of net production value are calculated in this paper in order to observe
any sensitivity of results to alternative labor valuation rules. I first compute the net value of
production valuing family and exchanged labor at the hired labor median wage rate in the
district. Estimation of the production function gives us the output elasticity of family labor.7 The
value of marginal product of family labor is estimated from the production function, which is
used as the shadow wage. I then examine the robustness8 of the relationship between net output
per hectare and farm size when family labor is valued at shadow wages or the median
agricultural wage rate in the district. This allows us to test the labor market imperfection, which
says that, when households have surplus labor due to labor market imperfections, they apply

𝑌̂
MPL=𝛽̂( ) , following (Jacoby, 1993) and also (Ali & Deininger, 2015).
𝐿
8
Getting shadow wages from the production function suggests bootstrapping for robust standard errors. I do this but
do not report. The bootstrapped standard errors are slightly larger than those from the OLS estimation.
7

19

them on their small farms intensely thereby making the marginal product of labor consistently
lower on small farms than the outside wage rate. Results derived from using the average
agricultural wage rate and the shadow wage should not be significantly different if labor market
imperfection does not exist.
For both sets of net value per hectare categories (based on observed local wage rates vs.
shadow prices), I derive three different measures of family labor costs. First, I value child labor
just as an adult laborer. In the second case, child labor is dropped from the analysis entirely so
that, only family adult labor is accounted for. In the third case, hired labor is the only type of
labor accounted for. These alternative approaches of valuing family labor provide a good
assessment of the potential role of labor market imperfections in influencing the nature of the
relationship between farm size and farm productivity.

Robustness Checks
Measurement error in area planted
Because recent studies have found that respondent reported farm size can be prone to
measurement error, studies examining the relationship between farm productivity and farm size
need to consider the extent to which their results are affected by potential measurement error
bias. To do this, and as robustness check on my main results, I use a subsequent measure of area
planted obtained from farmers a year after the main survey, as an instrument for the area planted
variable. I show below how this strategy works. The basic idea is that in a true model of the
form,
𝑦 = 𝛼 + 𝛽𝐿 + 𝜀 , 𝐿 ⊥ 𝜀,
20

if the regressor is measured inaccurately but there are two different inaccurate measures of it,
say,
𝐿1 = 𝐿 + 𝜀1 and 𝐿2 = 𝐿 + 𝜀2 where 𝐿 ⊥ 𝜀1 , 𝜀2 and 𝜀1 , 𝜀2 ⊥ 𝜀, 𝜀1 ⊥ 𝜀2
Without loss of generality, 𝐿2 can be used as an instrument for 𝐿1 . Then in the limit,
𝛽̂𝐼𝑉 =

𝑐𝑜𝑣(𝑦,𝐿2 )
𝐶𝑜𝑣(𝐿1 ,𝐿2 )

=

𝑝𝑙𝑖𝑚( 𝛽̂𝐼𝑉 ) → 𝛽 +

𝑐𝑜𝑣(𝛼+𝛽𝐿+𝜀,𝐿2 )
𝐶𝑜𝑣(𝐿1 ,𝐿2 )

𝑐𝑜𝑣(𝑦,𝜀2 )
𝑣𝑎𝑟(𝐿)

=

𝑐𝑜𝑣(𝛼+𝛽𝐿+𝜀,𝐿+𝜀2 )
𝐶𝑜𝑣(𝐿+𝜀1 ,𝐿+𝜀2 )

=

𝑐𝑜𝑣(𝛼+𝛽𝐿+𝜀,𝐿+𝜀2 )
𝑣𝑎𝑟(𝐿)

=𝛽

Because the last condition 𝜀1 ⊥ 𝜀2 may not necessarily hold9, results may not be consistent.
However, Table 2-19 shows evidence of mean reversion from the two measures of area planted
reported by the farmers. This is expected of random measurement errors; high reported values
tend to be followed by low reported values. This may also suggest that the instrument could be
plausible. I report the results as an attempt to address the potential measurement error bias.

Total Factor Productivity
I examine the sensitivity of our farm size/productivity relationship to whether partial or total
factor productivity measures are used. While the net value of output measures may most closely
approximate farmer profits per unit land, policy makers may be greatly interested in which scale
of farming provides the greater return to all factors of production. Total factor productivity is
difficult to measure but I calculate a representation of it. See (Li, Feng, & Fan, 2013) for details.
Assume the production function for simplicity is given by
9

A typical case is when there is reporter fixed effect of mis-reporting.

21

𝛼

𝛼

𝛼𝑁

𝑌𝑖 = 𝐴0 𝐾𝑖 𝑘 𝐿𝑖 𝐿 𝑁𝑖

exp(𝜀𝑖 )

where 𝑌𝑖 is the gross output level of farmer i. 𝐾𝑖 , 𝐿𝑖 , 𝑁𝑖 denote physical capital inputs, land and
labor inputs respectively whereas 𝛼𝑗 for j=K, L, N denote the output elasticities of capital, land
and labor respectively. Taking logs and simplifying yields
𝑙𝑛𝑌𝑖 = 𝑙𝑛𝐴0 + 𝛼𝐾 𝑙𝑛𝐾𝑖 + 𝛼𝐿 𝑙𝑛𝐿𝑖 + 𝛼𝑁 𝑙𝑛𝑁𝑖 + 𝜀𝑖 = 𝐵 + 𝛼𝐾 𝑙𝑛𝐾𝑖 + 𝛼𝐿 𝑙𝑛𝐿𝑖 + 𝛼𝑁 𝑙𝑛𝑁𝑖 + 𝜀𝑖
The coefficient of return to scale, CRTS= 𝛼𝐾 + 𝛼𝐿 + 𝛼𝑁
Define 𝛼𝑗∗ = 𝛼𝑗 /𝐶𝑅𝑇𝑆 for i= K, L, N and obtain total factor productivity as

TFPi =

𝑌𝑖
𝛼∗𝐾 𝛼∗𝐿 𝛼∗𝑁
𝐾𝑖 𝐿 𝑁𝑖

Results
I first estimated equation (8) to examine the relative productivity of family versus hired labor.
From the results in Table 2-5, the coefficient of interest is the fraction of labor that is hired. This
coefficient is positive but insignificant, suggesting that the fraction of hired labor has no
significant impact on the total amount of labor demanded or used. This also means that, we
cannot reject similar productivity of hired and family labor based on this test.
The results indicate no clear evidence that labor market imperfection adversely affects
large farms due to additional costs of supervision,10 associated with using hired labor. This type

10

If large differences existed, it would mean that, cost of supervising hired labor would be higher and this
corroborates the arguments that, supervision costs make larger scale farmers have a lower labor to land and
consequently, lower output to land ratio. See (Feder, 1985) for a detailed discussion of how supervision cost can
lead to a systematic relationship between productivity and land size.

22

of labor market imperfection could drive the IR if transaction costs associated with hired labor
were large, as discussed in detail in section 2.2. Although there could be bias in the estimation of
coefficients of interest in Table 2-5 due to simultaneity, or division bias, the average agricultural
wage variable is significant at 5% and carries the expected sign. The log of area planted variable
is also highly significant and has a positive sign. Gender and educational attainment of the
household head are used as instruments for the fraction of hired labor used (reported in table 26). Similar levels of productivity cannot be rejected between the two types of labor. Both OLS
and IV estimation results are similar to those obtained for rural Java by (Benjamin, 1992).
However, the IV estimation is quite noisy so I cannot talk of strong identification. I treat findings
in this paper as more of correlations.
After testing for possible differences in efficiency between family and hired labor, I
estimate the value of marginal product of family labor (shadow wages) from a Cobb Douglas
production function drawing on the works of (Ali & Deininger, 2015), and (Jacoby, 1993).
Table 2-17 shows how the estimated shadow wages compare to the district median wages. Table
2-7 has the results from the production function estimation. The returns to scale estimate is
reported in Table 2-7. I cannot reject constant returns to scale (CRS) production function at 1%
significance level. It can also be seen that the coefficients of family labor and exchange labor are
not significant while that of hired labor is highly significant. The remaining variables have the
expected sign.
To understand the basic nature of the relationships between farm size and farm
productivity, I run bivariate LOWESS regressions in both levels and logs and for gross and net
measures of output per hectare. These bivariate relationships are presented in Figures 2-1
through 2-4. All four figures suggest an inverse relationship between planted area and the
23

various measures of farm productivity for the range of farm sizes we concentrate on (5 to 40
hectares).
Table 2-8 contains the OLS results in which the dependent variables are the various gross
and net output per hectare measures discussed in Section 2.4.1. Table 2-8 results are considered
the baseline results to be compared to results from other estimations. NVP1 is the net value of
production that values family, communal and child labor using median wage of agricultural
activities in the respective districts. NVP2 is the same as NVP1 but does not include child labor
in its calculation. NVP3 uses only hired labor and does not include the cost of using other types
of labor. NVP4 and NVP5 correspond to NVP1 and NVP2 respectively, except that the former
uses the shadow price of family labor to value family labor. Table 2-9 contains estimation results
from using the calculated shadow wages to compute net value of production instead of the
median district wages used earlier.
Both sets of OLS results from Tables 2-8 and 2-9 uphold the IR.

The estimated

coefficients on log of land variable from the baseline results (table 2-8) are -0.31 and -0.53 for
gross value per hectare and net value per hectare measures of productivity. These estimates are
not much different from those obtained in recent studies such as (Ali & Deininger, 2015),
(Carletto et al., 2013) and (Gourlay et al., 2016). Regardless of whether family labor is valued at
shadow prices derived from the production function or from observed district median agricultural
wage rates, and regardless of whether family or family plus hired or just hired labor is counted in
the valuation of labor, the consistent finding is a significant inverse relationship between farm
size and productivity between 5 to 40 hectares of cultivated area.

24

Table 2-10 presents the first-stage estimation when possible measurement error in area
planted is taking into consideration. The results are significant and show that the instrument
performs well in explaining the variations in the area planted variable. The relevance assumption
required of a valid instrument appears to be well satisfied. Table 2-11 and Table 2-12 vary only
in that labor is valued according to district median wage rates in Table 2-11 and according to
shadow wages in Table 2-12. The negative relationship between farm size and productivity
continues to hold in these models, and the point estimate on the farm size variable remains
highly negative but now only weakly significant or in some cases insignificant relative to
analogous results in Tables 2-8 and 2-9. The instrumental variable regression point estimates
become noisy but do not change very much.
From Table 2-12, it can be seen that the estimates are smaller in magnitude than those in
Table 2-9 and they also remain negative but are now insignificant. In addition, compared to
baseline results in Table 2-8, the estimates are of similar magnitude but imprecisely measured.
Taken altogether, the results in Tables 2-8 through 2-12 indicate that labor market imperfection
do not appear to drive the IR. When Ordinary Least Squares estimation is used, the relationship
is negative but there is no statistically significant result in favor of the IR once I instrument for
the bias that may be due to measurement error in farm size variable.
Table 2-19 suggests that the farmers in the sample on relatively small holdings tended to
overstate their area planted in the subsequent (2015) year compared to the original survey year
(2014). As a result, this may have artificially inflated farm productivity at the lower part of the
farm size distribution if the 2014 figure is understated. However, the results shown provide
similar point estimates of the relationship albeit with less precision in the estimates. This result

25

may be in contrast to findings in (Carletto et al., 2013; Gourlay et al., 2016) who found that
failure to correct for respondent-reported area measurement error rather works against the IR.
Tables 2-13 and 2-14 present IR test results using operated land area (area planted plus
fallowed land) instead of just area planted. When available land is to be redistributed to
smallholders or medium or large scale farmers, policymakers may be interested in how farmers
use (or not) the entire land under their control and not simply the amount cultivated. This is an
important indicator of social efficiency in land utilization. Results show that the IR still exists but
is less statistically significant for the net value per hectare measures in the IV estimation, just as
was found in Table 2-11. It can also be seen from the last column in Table 2-13 that, if only hired
labor was included in computing costs of labor, one could have concluded that there is no
significant relationship, which could be misleading.
The fact that the IR is stronger and more precisely estimated in models of gross farm
output per hectare rather than net farm output per hectare may mean that relatively large farms
may be more efficient users of key inputs such as fertilizer, seed and hired labor, or may
substitute more efficient mechanization for manual land preparation.
Table 2-15 presents results from IR tests based on a production function approach, in
which other inputs to the production process are added to equation (7). Because these variables
are endogenous, we favor the earlier reported models but report Table 2-15 results as a
robustness check. These production function models produce statistically significant IR results
consistent with those reported earlier from Ordinary Least Squares estimations.
In Table 2-16, we compare the robustness of the partial productivity models reported so
far with those based on total factor productivity. Table 2-16 shows that both the OLS and the IV
26

estimation show a highly significant inverse relationship between farm size and total factor
productivity. The area elasticities of productivity from the OLS TFP results (-0.34) are near the
mid-point of those from the partial productivity models. The area elasticities from the IV results
(-1.16) are higher than the other estimates obtained and highly significant.
The weight of the evidence suggests that, regardless of whether total or partial
productivity measures are used, regardless of plausible alternative ways of valuing family labor,
regardless of whether farm size measures are based on area planted, area planted plus fallowed
land, or total landholding size, I observe a strong inverse relationship between scale and
productivity on farms between 5 and 40 hectares in southern Ghana when Ordinary Least
Squares estimation is used. However, there is no statistically significant result in favor of the IR
once I instrument for the bias that may be due to measurement error in farm size variable. I also
did some sensitivity analysis by dropping data points that appear to be outliers based on Figure
2-6 and in another instance, restrict the sample to area sizes between the 1st and 99th percentile.
The results are in table 2-21 and are not different from our main results.

Conclusion
This study examines the relationship between farm size and farm productivity, on farms
cultivating between 5 and 100 hectares in southern Ghana but with emphasis on 5 to 40 hectares
range. The study is unique in that, it covers the rapidly growing segment of “medium-scale”
farms in Africa, which now accounts for a significant fraction of total area cultivated in the
region ( Jayne et al., 2016). Most available studies examining the relationship between farm size
and productivity in Africa are based on small-scale farm samples with very few observations

27

over 10 hectares. This study therefore can inform contemporary policy discussions about the pros
and cons of promoting larger-farm scales in Africa e.g., (Collier & Dercon, 2014). I examine the
relationship between farm size and farm productivity using (i) both partial and total factor
productivity measures; (ii) alternative measures of farm size (area planted and area planted plus
fallowed); (iii) valuing family labor at local farm wage rates and at shadow wages to account for
the possibility of labor market imperfections driving the results; and (iv) using respondent based
area measures as an instrumental variable in an attempt to correct for measurement error bias.
I find that the inverse relationship between farm size and farm productivity is consistently
upheld through all permutations of these models. However, the instrumental variable estimation
shows widely varying area elasticities of productivity, which, while all consistently negative, are
often imprecisely measured.

Thus, there is no evidence in favor of the IR when I use

instrumental variable estimation to control for the bias that may be due to measurement error and
account for labor use.
What do these results mean for policy? An important emerging policy debate centers on
how unutilized land in Africa should be allocated to competing users. One school of thought
argues that small is still beautiful while another argues for favoring larger-scale farmers who are
often asserted to make more productive use of available land. My results can at least partially
guide these discussions but have several limitations. First, while the findings from Ghana uphold
the productivity advantages of relatively small farms, the largest portion of the data covers farms
only up to 40 hectares. Therefore, I am not in a position to assess the relative productivity of
small farms of, say 5 hectares, with large farms of 5000, 1000, or even 100 hectares.

28

Second, different farm scales may produce different general equilibrium effects that are
not examined here. (Mellor, 1976),

(Johnston & Kilby, 1975) and others have observed that

“unimodal” farm distribution patterns, such as those found in much of green revolution Asia,
have resulted in very different patterns of expenditures in local rural economies and hence
produce multiplier effects of different magnitudes to those generated under bi-modal farm
distribution systems such as those in much of Latin America. Given the potential importance of
these general equilibrium effects of alternative agrarian structures, comparisons of the relative
productivity of “small” versus “large” farms can provide highly important but incomplete
information to guide governments toward comprehensive land and agricultural policy strategies.

29

APPENDICES

30

Appendix A: Tables for Essay 1

Table 2-1: Changes in Farm Structure in Ghana (1992 to 2013)
Farm Size
% Growth in Number % of Total Operated Land
Number of Farms
Category
of Farms 1992 to 2013
1992
2013
1992
2013
0-2 ha
2-5 ha
5-10 ha
10-20 ha
20 -100 ha
Over 100 ha

1,458,540
578,890
116,800
38,690
18,980
-

1,582,034
998,651
320,411
117,722
37,421
1,740

Total

2,211,900

3,057,978

8.5
72.5
174.3
204.3
97.2
-

25.1
35.6
17.2
11.0
11.1
-

14.2
31.3
22.8
16.1
12.2
3.5

100

100

Source: (Jayne et al., 2016).
Table 2-2: Sample Size by Area Planted in Hectares
District

Region

Bibiani-Anhwiaso

Western

14

47

24

13

98

Nkwanta North

Volta

19

88

43

10

160

Afram Plains South Eastern

12

49

35

19

115

Offinso North

12

53

46

19

130

57

237

148

61

503

Total (n)

Ashanti

<=5 ha 5-10 ha 10-20 ha >20 ha Total

Source: Authors’ compilation from survey data.

31

Table 2-3: Household Demographics and Input Use by Operated Farm Area
Variable
<=5
Full Sample
5 to 20 ha
ha
(n=503)
(n=383)
(n=22)
Means and Percentages
Age of household head
45.51
47.27
44.91
Household head years in current settlement
38.18
39.32
38.04
Experience in farming
20.84
23.23
20.61
Male headed household (%)
95.00
91.00
94.00
Education of household head (%)
No formal education
46.67
31.82
50.68
Basic education
22.08
45.45
20.94
Secondary education
18.13
13.64
17.90
Tertiary education
13.13
9.09
10.47
Household head previously employed (%)
10.93
9.09
10.70
Household head attracted to farming:
Because parents were farmers (%)
49.50
40.91
51.96
Because farming is a business (%)
14.91
9.09
12.01
Household head applied for loan (%)
14.31
0.14
0.12
Operated area size (ha)
18.32
4.30
10.47
Area planted (ha)
12.85
4.30
8.90
Used fertilizer (%)
52.63
30.00
50.00
Fertilizer (kg/ha)
48.72
17.98
47.22
Number of crops grown
2.97
2.71
2.96
Number of fields
3.35
3.23
3.31
Used weedicide (%)
86.03
90.00
86.24
Used pesticide (%)
9.11
15.00
8.73
Used manure (%)
3.24
5.00
2.91
Used hired labor (%)
94.83
90.91
95.04
Used family labor (%)
74.55
68.18
78.33
Used communal labor (%)
16.90
9.09
18.02
Used mechanization (%)
74.55
45.45
74.15
Hired labor days per ha
36.62
71.02
35.41
Family labor days per ha
13.84
16.34
15.39
Communal labor days per ha
4.38
2.14
4.56
Household planted maize (=1)
85.49
81.82
84.86
Household planted yam (=1)
58.45
36.36
59.53
Household planted rice (=1)
17.50
22.73
17.75
Household planted groundnut (=1)
26.24
31.82
28.20
Household planted soy (=1)
0.60
4.55
0.26
Household planted cocoa (=1)
19.28
31.82
17.49
Source: Authors’ computation from survey data. Operated farm area is area planted plus fallow area..

32

Above 20
ha (n=98)
47.51
38.49
21.18
99.00
34.74
21.05
20.00
24.21
12.24
41.84
27.55
0.22
52.17
30.23
67.71
61.71
3.05
3.55
84.38
9.38
4.17
94.90
61.22
14.29
82.65
33.64
7.22
4.18
88.78
59.18
15.31
17.35
1.02
23.47

Table 2-4: Descriptive Statistics
Variable Label

Units of
Measurement
Ghana Cedis

5146.76

Standard
Deviation
32399

Sample
Size
503

Net value of production (NVP2) per
hectare (values family adult and hired
labor)

Ghana Cedis

5175.80

32413

503

Net value of production (NVP3) per
hectare (values only hired labor)

Ghana Cedis

5363.01

32446

503

Gross Value of production per hectare
Area planted
Area Operated
Hired labor
Family labor
Communal labor
Fertilizer cost
Chemical cost
Fertilizer use
Fraction of households planting maize
Fraction of household planting yam
Fraction of household planting rice
Fraction of household planting groundnut
Fraction of households planting soy
Fraction of households planting cocoa

Ghana Cedis
hectares
hectares
person-days
person-days
person-days
Ghana Cedis
Ghana Cedis
kilogram
percentage
percentage
percentage
percentage
percentage
percentage

6309.55
12.85
18.32
409.58
121.96
37.72
591.24
392.61
450.99
85.49
58.45
17.50
26.24
0.60
19.28

32496
15.70
59.60
823.36
200.94
98.70
1071.6
386.7
881.7
N/A
N/A
N/A
N/A
N/A
N/A

503
503
503
503
503
503
503
503
498
503
503
503
503
503
503

Net value of production (NVP1) per
hectare (valuing all labor, including
family adult and child, and hired)

Source: Authors’ computation from survey data.

33

Mean

Table 2-5: Test for Efficiency of Family versus Hired Labor (OLS)
Log (Total Labor Demand)
Log (Area planted )
0.47 **
(0.072)
Log(Average agric wage) -0.14**
(0.069)
Fraction of hired labor
0.17
(0.140)
Constant term
Village dummies
N
Adjusted R2

5.42***
(0.340)
Yes
493
0.2351

Standard errors are in parenthesis. * p < 0.10, ** p < 0.05, *** p < 0.01

Table 2-6: Test for Efficiency of Family versus Hired Labor (IV)
Log (Total Labor Demand)
Log (Area planted )
0.485***
(0.106)
Log (Average agric wage)
-0.136*
(0.077)
-0.052
Fraction of hired labor
(1.249)
Village dummies
Yes
Constant term
5.535***
(0.709)
492
N
Standard errors are in parenthesis. * p < 0.10, ** p < 0.05, *** p < 0.01
Gender and education were used as instruments for the fraction of hired labor used.

34

Table 2-7: Cobb-Douglas Production Function Estimation
Log (Gross Value of Production)
Log (Area planted )
0.576***
(0.09)
Log (Hired labor days)
0.093***
(0.03)
Log (Family labor days)
0.031
(0.02)
Log (Communal labor days)
0.027
(0.03)
Log (Chemical cost)
0.035
(0.03)
Log (Fertilizer cost)
0.057***
(0.02)
Shock in past 5 years
-0.206*
(0.12)
Crop dummies
Yes
Village dummies
Yes
Returns to scale (RTS)
0.82
P-value (Null: RTS=1)
0.9421
Constant term
7.633***
(0.46)
502
N
Adjusted R2
0.3544
Standard errors are in parentheses. * p < 0.10, ** p < 0.05, *** p < 0.01
Crop dummies are for maize, rice, yam, soy and groundnut.

Table 2-8: Estimates of the IR Valuing Family Labor at District Median Wages (OLS)
Log (Gross Value Output Per Hectare) Log (Net Value Output Per Hectare)
(1)
Log (Area planted)
Crop dummies
Village dummies
Constant term
N
Adjusted R2

-0.31***
(0.087)
Yes
Yes
8.04***
(0.401)
502
0.2323

(NVP1)

(NVP2)

(NVP3)

-0.53***
(0.157)
Yes
Yes
8.17***
(0.696)
385
0.1704

-0.53***
(0.156)
Yes
Yes
8.30***
(0.691)
388
0.1726

-0.25*
(0.138)
Yes
Yes
7.99***
(0.637)
428
0.1910

Standard errors are in parentheses * p < 0.10, ** p < 0.05, *** p < 0.01. Crop dummies are for maize, rice, yam, soy
and groundnut. I also as robustness checks use area under the main crops (maize, rice, yam, soy, and groundnut) in
place of crop dummies in all models. The results were not different. NVP1 is the net value of production that values
family, communal and child labor using median wage of agricultural activities in the respective districts. NVP2 is
the same as NVP1 but does not include child labor in its calculation. NVP3 uses only hired labor and does not
include the cost of using other types of labor.

35

Table 2-9: Valuing Family Labor at Shadow Wages (OLS)
Log (Net Value Per Hectare)
(NVP4)
(NVP5)
***
Log (Area planted )
-0.437
-0.442***
(0.152)
(0.150)
Village dummies
Yes
Yes
Crop dummies
Yes
Yes
Constant term
8.318***
8.331***
(0.680)
(0.672)
412
412
N
Adjusted R2
0.1702
0.1642
Standard errors are in parentheses. * p < 0.10, ** p < 0.05, *** p < 0.01.
Crop dummies are for maize, rice, yam, soy and groundnut.
NVP4 is the net value of production that values family, communal
and child labor at family shadow wages. NVP5 is the same as NVP4
but does not include child labor in its calculation.

Table 2-10: First Stage of IV Estimation
Log (Area Planted Per Hectare)
1
Instrument
0.136***
(0.033)
Crop dummies
Yes
Village dummies
Yes
Constant term
1.67***
(0.214)
Constant term
1.67***
(0.214)
475
N
Adjusted R2
0.2219
Standard errors are in parentheses. * p < 0.10, ** p < 0.05, *** p < 0.01
1
The instrument is the second measure of area planted collected in
2015 (see Section 3.4).
Crop dummies are for maize, rice, yam, soy and groundnut.

36

Table 2-11: Correcting for Measurement Error Bias in Area Planted (IV)
Log (Gross Value Per Hectare Planted) Log (Net Value Per Hectare Planted)
(1)
NVP1
NVP2
NVP3
**
Log (Area planted)
-0.81
-0.48
-0.68
-0.43
(0.408)
(0.533)
(0.537)
(0.538)
Crop dummies
Yes
Yes
Yes
Yes
Village dummies
Yes
Yes
Yes
Yes
Constant term
9.08***
8.17***
8.57***
8.53***
(0.858)
(1.260)
(1.267)
(1.278)
475
366
369
406
N
R2
0.308
0.348
0.345
0.354
Standard errors are in parentheses. * p < 0.10, ** p < 0.05, *** p < 0.01
NVP1 is the net value of production that values family, communal and child labor using
Median wage of agricultural activities in the respective districts. NVP2 is the same as
NVP1 but does not include child labor in its calculation. NVP3 uses only hired labor and
does not include the cost of using other types of labor. The instrument is the second measure
of area planted collected in 2015 (see Section 3.4).
Crop dummies are for maize, rice, yam, soy and groundnut.

Table 2-12: Correcting for Measurement Error Using Shadow Wages
Log (Net value Per Hectare)
(NVP4)
Log Area planted )
Crop dummies
Village dummies
Constant term
N
R2

-0.13
(0.603)
Yes
Yes
7.79***
(1.439)
391
0.3282

(NVP5)
-0.15
(0.596)
Yes
Yes
7.83***
(1.420)
391
0.3334

Standard errors are in parentheses. * p < 0.10, ** p < 0.05, *** p < 0.01. The instrument
is the second measure of area planted collected in 2015. (see Section 3.4). NVP4 is
the net value of production that values family, communal and child labor at family
shadow wages. NVP5 is the same as NVP4 but does not include child labor
in its calculation. Crop dummies are for maize, rice, yam, soy and groundnut.

37

Table 2-13: Estimates of the IR Using Operated Farm Size (OLS)
Gross Value Per Hectare Net Value of Production
(1)
NVP1
NVP2
NVP3
***
***
***
Log (Area operated)
-0.406
-0.536
-0.551
-0.079
(0.083)
(0.139)
(0.138) (0.125)
Crop dummies
Yes
Yes
Yes
Yes
Village dummies
Yes
Yes
Yes
Yes
***
***
***
Constant term
7.927
7.987
8.158
7.638***
(0.421)
(0.703)
(0.698) (0.650)
502
385
388
428
N
R2
0.361
0.354
0.353
0.350
Standard errors are in parentheses. * p < 0.10, ** p < 0.05, *** p < 0.01
NVP1 is the net value of production that values family, communal and child labor using median
wage of agricultural activities in the respective districts. NVP2 is the same as NVP1 but does not
include child labor in its calculation. NVP3 uses only hired labor and does not include the cost of
using other types of labor. The instrument is the second measure of area planted collected in 2015
(see Section 3.4). Crop dummies are for maize, rice, yam, soy and groundnut.

Table 2-14: Estimates of the IR Using Operated Farm Size (IV)
Gross Value Per Hectare
Net Value Per Hectare
(1)
NVP1
NVP2
NVP3
*
Log (Area operated)
-0.861
-0.465
-0.679
-0.420
(0.454)
(0.549) (0.553) (0.547)
Crop dummies
Yes
Yes
Yes
Yes
Village dummies
Yes
Yes
Yes
Yes
***
***
***
Constant term
9.621
7.923
8.599
8.267***
(1.167)
(1.495) (1.503) (1.509)
475
366
369
406
N
R2
0.236
0.345
0.342
0.345
Standard errors are in parentheses. * p < 0.10, ** p < 0.05, *** p < 0.01
NVP1 is the net value of production that values family, communal and child labor using median
wage of agricultural activities in the respective districts. NVP2 is the same as NVP1 but does
not include child labor in its calculation. NVP3 uses only hired labor and does not include the
cost of using other types of labor. The instrument is the second measure of area planted collected
in 2015 (see Section 3.4).
Crop dummies are for maize, rice, yam, soy and groundnut

38

Table 2-15: Alternative Estimate of the IR (OLS)
Log (Gross value of production/ha)
Log (Area planted)
Gender
Education
Experience in farming
Experience squared
Hired labor days/ha
Family labor days/ha
Communal labor days/ha
Fertilizer kg/ha
Shock in past 5 year
Weedicide Dummy
Mechanization Dummy
Constant term
N
Adjusted R2

-0.238**
(0.09)
0.173
(0.26)
0.019*
(0.01)
0.018
(0.02)
-0.000
(0.00)
0.002**
(0.00)
0.003
(0.00)
0.005
(0.00)
0.003***
(0.00)
-0.204*
(0.12)
0.293*
(0.16)
0.230*
(0.14)
7.263***
(0.52)
480
0.271

This is sometimes called the production function approach.

39

Table 2-16: Estimates of the IR Using Total Factor Productivity
OLS
IV
Log(Area planted)
Crop dummies
Village dummies
Constant term
N
R2

-0.341***
(0.11)
Yes
Yes
3.878***
(0.45)
502
0.337

-1.156**
(0.48)
Yes
Yes
5.489***
(1.00)
475
0.214

Standard errors are in parentheses. * p < 0.10, ** p < 0.05, *** p < 0.01
The instrument is the second measure of area planted collected in 2015 (see Section 3.4).
Crop dummies are for maize, rice, yam, soy and groundnut.

Table 2-17: Computed Shadow Wages Versus District Median Wages
District
A
B
Median Marginal Value
District Median Wage
Product of Family Labor
(Cedis per day)
(Cedis per day)
Bibiani
6.74
15
Afram Plains South
19.98
15
Offinso North
17.37
12
Nkwanta North
9.11
10
Source: Authors’ compilation from survey data

Table 2-18: OLS Estimation of IR in Levels
Gross Value Per Hectare
Net Value Per Hectare
(1)
NVP1
NVP2
NVP3
Area planted
-80.314
-61.693
-62.406
-65.252
(119.73)
(119.45)
(119.50)
(119.59)
Crop dummies
Yes
Yes
Yes
Yes
Village dummies
Yes
Yes
Yes
Yes
Constant term
5177.758
3568.199
3620.320
3800.741
(11692.55)
(11665.48)
(11670.14)
(11679.03)
502
502
502
502
N
R2
0.139
0.138
0.138
0.138
Standard errors are in parentheses. * p < 0.10, ** p < 0.05, *** p < 0.01

40

Table 2-19: Comparing Self-reported Farm Sizes
Area Planted Category
Proportion that Over-reported
in 2015 Compared to 2014
5 hectares and less
5-20 ha
Above 20ha
N

0.267
0.483
0.723
285

Source: Author’s compilation from survey data.

41

Proportion that Underreported in 2015 Compared
to 2014
0.733
0.460
0.214
200

Table 2-20: Summary of Previous Studies
Author
Study
Range of Farm
Area
Sizes Analyzed

Sen (1962)

India

Lau and
Yotopoulos
(1971)
Bardhan (1973)
Carter (1984)

India

Bhalla & Roy
(1988)

India

Kevane (1996)

Western Sudan

India
India

Measure of
Productivity or
EfficiencyNumerator
Not available
Gross value/Value
added
Greater than and Net Profit
less than 10
acres
Not available
Gross value
Less than 1 acre Gross value
to above 12
acres
Not available
Gross value
but from 0 to
above 12 acre
from small to
above 12
Mukhammas

Gross value

42

Measure of
Productivity or
Efficiency-Denominator
Holdings

Family
Labor
Costed?
Y/N
Yes

Found
IR

Cultivable land
in acres

Yes

Yes

Cropped area
Farm size

Yes
Yes

Yes
Yes

Net cultivated
area, land
available for
cultivation
Endowment of
land own

No

IR
weakly
found

No

No

Yes

Table 2-20 (cont’d)
Barrett (1996)
Madagascar

<25 ares and
>500ares

Volume produced
minus
consumption—rice
Farm value added

No

No

Operated
holding size
Mean holding
size
Cultivated land

No

Yes

No

No

No

No
Yes but
goes
away
Yes and
No

Area cultivated

Heltberg (1998)

Pakistan

Doward (1999)

Malawi

Benjamin and
Brandt (2002)
Lamb (2003)

China

Mean is 1
hectare
<1 and >2
hectares
Mean:10.5 mu

India

Mean 2.18 acres Profit

Total cropped
area

Yes

Kimhi (2006)

Zambia

Area allocated
to maize

No

Assuncao and
Braido (2007)
Kawasaki (2010)

India

Saddle point 3
Maize output
ha
Mean: 1.78
hectares
0.08-83.87 acres Gross value

Cropped area

No

Yes

Japan

Mean:0.90 ha

Total planting
area

No

No

Output per hectare
Gross Output

Rice output kg/ha

43

Table 2-20 (cont’d)
Barrett et al
Madagascar
(2010)
Chen et al
China
(2011)
Larson et al
10 African
(2012)
countries
Carletto et al
Uganda
(2013)
Holden & Fisher Malawi
(2013)
Li et al (2013)
China
Ali & Deininger
2015
Dillon et al
(2016)

Mean: 16.24
ares
Mean: 0.07
hectare
0.41-7.28
hectares
0.01 to 600
acres
0.1-0.8 ha

Yield per unit area

Cultivated area

No

Yes

Total crop output

Farm land
cultivated
Area under crop

No

No

No

Yes

Area operated

Yes

Yes

Farm size

Yes

Yes

Farmland area

Yes

Yes

Cultivated area

Yes

No

Plot size

No

Yes

Yield kg/ha

Rwanda

0.05-2 hectare

Net agricultural
revenue
Net agricultural
revenue
Gross value/net
profit
Gross value/profit

Nigeria

Not available

Gross output value

0.3-100 mu

Source: Author’s compilation.

44

Table 2-21: OLS Results Using Sample between 1st and 99th Percentile
Log Gross Value Per Hectare
Log Net Value Per Hectare
Log (Area planted)
Main crops dummies
Village dummies
Constant term
N
R2

(1)
-0.275***
(0.08)
Yes
Yes
7.953***
(0.37)
498
0.370

NVP1 NVP2
NVP3
***
***
-0.478
-0.478
-0.209
(0.15)
(0.15)
(0.13)
Yes
Yes
Yes
Yes
Yes
Yes
***
***
8.042
8.175
7.854***
(0.67)
(0.66)
(0.61)
381
384
424
0.352
0.351
0.356

Standard errors are in parentheses * p < 0.10, ** p < 0.05, *** p < 0.01

45

0

5000

10000

Appendix B: Figures for Essay 1

5
0

20
50

100
150
area planted in hectares

Gross per hectare
NVP2 per hectare

200

NVP1 per hectare
NVP3 per hectare

Figure 2-1: Plot of Measures of Productivity and Area Planted in levels
NVP1 is the net value of production that values family, communal and child labor using median wage of agricultural
activities in the respective districts. NVP2 is the same as NVP1 but does not include child labor in its calculation.
NVP3 uses only hired labor and does not include the cost of using other types of labor.

46

10000
8000
6000
4000
0

2000

5
0

20
50

100
150
area planted in hectares

Gross value per hectare
NVP5 per hectare

200

NVP4 per hectare

Figure 2-2: Plot of Measures of Productivity and Area Planted in levels using shadow wages
NVP4 is the net value of production that values family, communal and child labor at family shadow wages. NVP5 is
the same as NVP4 but does not include child labor in its calculation.

47

6.5

7

7.5

8

Gross Value of Output per Hectare & Area Planted (in logs)

1

2

3

4

5

log of area planted in hectares

Figure 2-3: Gross Output per Hectare Against Area Planted
Family labor is value at the district median wage rate

48

6

6.5

7

7.5

8

Net Value of Output per Hectare and Area Planted (in logs)

-2

0

2
log of area planted in hectares
NVP1
NVP3

4

6

NVP2

Figure 2-4: Plot of Measures of Productivity and Area Planted in logs
NVP1 is the net value of production that values family, communal and child labor using median wage of
agricultural activities in the respective districts. NVP2 is the same as NVP1 but does not include child labor in its
calculation. NVP3 uses only hired labor and does not include the cost of using other types of labor.

.

49

.06
.04
0

.02

Density

.08

.1

Kernel density estimate

0

50

100
150
area planted in hectares

200

kernel = epanechnikov, bandwidth = 1.4190

Figure 2-5: Distribution of Area Planted Variable

50

500000
400000
300000
200000

0

100000

0

50

100
150
area planted in hectares

Figure 2-6: Sensitivity Analysis

51

200

3

ESSAY 2: JOINT LIABILITY LENDING WITH CORRELATED RISK

Introduction
The advent of micro-lending undoubtedly brought relief to many small businesses especially in
developing countries, where collateral security cannot be provided by prospective borrowers
because poverty rates are very high. It is estimated that as at 31st December 2010, more than 205
million people had been reached with loans by microfinance institutions (MFIs), (Maes & Reed,
2012). Despite backlashes received by MFIs for benefiting off the poor and despite findings that
microfinance is probably not as miraculous as we would be made to believe (Banerjee, Duflo,
Glennerster, & Kinnan, 2015), the facility remains prevalent in developing countries even in
recent times.
In addition to serving as a source of funds for existing and new businesses, microfinance
theoretically has been shown can aid some economies escape poverty traps in an occupational
choice framework (Ahlin & Jiang, 2008). Reports from the mix market11 indicate that about
3,652 MFIs are registered, and are involved in the financial intermediation, and financial services
subsector of the economies of developing countries. With free entry and free exit that exists in
the micro-lending market, successes chalked by existing MFIs have attracted potential firms both
for-profit and the initial12 not-for-profit firms that focus mainly on outreach and maximization of
borrower welfare. Particularly impressive about the performance of MFIs is the repayment rates
that have gone up considerably given that, the loans are advanced to poor people perceived to
have good projects but have neither capital nor collateral to secure loans from traditional banks.

11
12

This is a platform where registered microfinance institutions record their data and exchange information.
Originally, MFIs were mainly operated by not-for profit organizations.

52

Microfinance contracts began as a contract between a borrower and a lender; where the
individual borrower was solely responsible for his debt (individual liability). Other strategies
such as dynamic incentive; where borrowers are promised a lower future interest rate or a higher
loan amount in the future are also being used. (Karlan & Zinman, 2009) study a large MFI in
South Africa that uses the dynamic incentive strategy. Group-based joint liability lending
methods have also been used and remain common in the micro-lending markets, (de Quidt,
Fetzer, & Ghatak, 2016). However, for-profit institutions are found to use more of individual
lending while not-for-profit institutions use group-based lending methods (Cull, Demirgßç-Kunt,
& Morduch, 2009; de Quidt et al., 2016).
Joint liability lending involves borrowers forming groups to access loans. The
requirements are that, the liability is jointly held so that, successful group members are liable for
part or the entire debt of an unsuccessful group member. The success of joint liability lending in
the adverse selection setting, rests at least theoretically, (Ghatak, 1999, 2000) on the facts that
borrowers are allowed to choose their own partners or group members, and secondly on the fact
that, the loan is given on joint liability terms. The idea is that, in a village setting for instance,
where his or her neighbor knows almost everyone, borrowers have local information which is not
available to the lender and which makes it possible for safe borrowers to partner with other safe
borrowers. Thus, a joint liability lending contract offers an implicit or hidden discount to safe
borrowers and charges risky borrowers an implicit or hidden premium. Conditional on success, a
safe borrower pays less in expectation than a risky borrower does in a joint liability contract due
to having safer partners.
Although operations of MFIs are seen to be with small non-agricultural businesses, some
loans advanced to agricultural workers are also granted on joint liability basis by rural and
53

agricultural development banks. This raises concerns about the ability of joint liability lending to
help in pricing for risk, or help in improving repayment rates if project returns of borrowers are
correlated. This paper explores this scenario. As (Besley, 1994) puts it “A special feature of
agriculture which provides the income of most rural residents is the risk of income shocks. These
include weather fluctuations that affect all the producers of a particular commodity. Such shocks
affect the operation of credit markets if they create the potential for group of farmers to default at
the same time”. To be able to price for risk and improve repayments rates, lenders should be able
learn from the repayment behavior of borrowers. But as stated in (Ahlin & Waters, 2016;
Ghatak, 2000), correlated risk influences how information is revealed to the Bank. For example,
consider the case of perfect spatial correlation, then a joint liability contract for a group of two
people will provides the lender with only one independent information about a borrower. This
makes the group contract not superior to an individual liability contract.
This paper derives and characterizes the optimal lending contracts of joint liability
lending when risks are correlated under adverse selection. This problem is very important both
empirically and theoretically. Empirically, correlated risk is a pervasive reality and thus worth
exploring. Theoretically, it would be interesting to see how the correlated risk affects group
lending’s ability to positively impact credit markets. (Ahlin, 2009) delves into correlated risks. It
puts a simple structure on the correlation to understand how groups are formed and finds that
borrowers anti-diversify risk in group formation so as to lower the occurrence of having to carry
the debt burden of a group member. This anti-diversification strategy by borrowers, lays
credence to how important it is to understand further, how the optimal lending contract and
parameter space for efficiency are altered in the presence of correlation. I compare results from
this work to the efficiency outcomes under independent risks. I additionally consider a common
54

instance where the lender is faced with a pool of borrowers with a fraction having correlated
risks and the other fraction having independent risks.
I find that correlation reduces the parameter space for fully efficient lending using joint
liability contracts relative to independent risks case. The lender cannot effectively rely on the
joint liability to improve risk-pricing under correlation. This is partly because of the
monotonicity constraint, which prevents the lender from increasing the joint liability beyond the
gross interest rate, and partly because of affordability constraints. I also find that, it may be better
in some cases for lenders to serve borrowers separately, if the pool of borrowers consists of a
fraction with correlated risks and a fraction with independent risks. This may help explain
existence of specialized microfinance institutions such as agricultural banks separately from
standard microfinance institutions. Separation of these banks could stem from fear on the part of
lenders that they may have loans advanced to correlated borrowers becoming bad debts. This has
left the financing of correlated risk borrowers’ projects in the hands of governments and nongovernmental organizations in cases where collateral is not available on the part of borrowers.
(Besley, 1994; IFC, 2012; Ramana, 2004) are a few of the papers that acknowledge the
fact that covariant risk make agricultural financing unattractive to lenders. Ramana (2004) report
why MFIs do not lend to farmers citing the Gramen Bank and Unit Desa system of the Bank of
Rakyat Indonesia as banks that have focused exclusively on rural areas but not on agricultural
lending in contrast with the Bank of Agriculture and Agricultural Cooperatives (BAAC) of
Thailand which focuses exclusively on lending to agricultural workers and none to nonfarm
activities.

55

The rest of the paper is organized as follows; the next section discusses some related
literature. Section 3.3, outlines the basic models. In sections 3.4 and 3.5, I discuss independent
risks and how I introduce correlation into the model. Optimal group lending contracts with
correlated risks are derived in sections 3.6 and 3.7. Section 3.8 expands on this theory by
dividing the pool of borrowers into correlated borrowers and uncorrelated borrowers. Section 3.9
concludes while the appendix contains proofs of propositions.

Literature Review
In the absence of collateral security from borrowers, micro-credit lenders have resorted to
strategies such as joint liability lending. They seek to price for risk and induce quick repayment
of loans. (Ghatak, 1999; Tassel, 1999), show group lending can improve efficiency of the credit
market compared to traditional individual loans. Both papers were under adverse selection
framework. A plethora of the literature on joint liability lending in the framework of adverse
selection has focused on how the method can perform better than the individual loan contracts.
The works of (Ghatak, 1999, 2000; Tassel, 1999) have all been in that direction. (Bhole &
Ogden, 2010) as well as (de Quidt et al., 2016) study individual lending versus group lending
but in a strategic default setting. They find welfare of borrowers to be higher with group lending
than individual lending under certain conditions. One of such conditions is when penalty is
allowed to be different among members of a group. Under the context of moral hazard
(Chowdhury, 2007) shows that, if loans are not advanced to group members sequentially, group
lending is not any better than individual lending13.

13

(Chowdhury, 2007; de Quidt et al., 2016) papers look at dynamic lending

56

The very early works by (Varian, 1990) and (Stiglitz, 1990) have investigated the
potency of joint liability in harnessing local information to induce high repayment rates. (Varian,
1990) proposes a model in which the bank does its own screening, and does not rely on local
information among borrowers. Banks interviewed group members and the eligibility or otherwise
of a group member determined the fate of the other partners. (Besley & Coate, 1995) have also
looked at how joint liability affects the willingness to pay on the part of borrowers. Even in the
situation where borrowers have imperfect knowledge about the project types of their group
members, (Armenda, Aghion, & Gollier, 2000) show that joint liability can lead to lower interest
rate and help overcome some credit market inefficiencies.
Much earlier, even before the seminal contributions of (Ghatak, 1999, 2000), (Besley,
1994), identifies three major things that make rural credit markets in developing countries
different from those in developed countries. These he highlights as collateral security
unavailability, covariant risk and under developed states of related institutions. Not many papers
to the best of my knowledge have looked at the optimal lending contract of Joint liability lending
when there are correlated risks. (Ahlin & Waters, 2016; Ghatak, 2000) talked about the adverse
effects of correlated risks on group lending and dynamic lending but only at the level of
conjecture.
(Ahlin & Townsend, 2007b) tested for repayment implications, using data from Thai
borrowing groups. They found that “a higher correlation of output can raise or lower repayment,
depending on the model”. The papers which discuss correlated risks in some detail as this work
seeks to do are (Ahlin, 2009; Ahlin & Townsend, 2007a; Katzur & Lensink, 2012). (Ahlin, 2009)

57

focused on matching14, which distinguishes it from this paper. (Katzur & Lensink, 2012) study
group lending with correlated project returns, which is the closest to this paper. They show that
positive correlation can improve efficiency of group lending contracts. Their results require that
the correlation between safe borrowers is sufficiently higher relative to the correlation between
risky borrowers in a two-person group. I introduce and study correlation more generally and
uniformly across risk types in this paper.

Baseline Model
3.1.1

Economic environment

The environment and baseline models follow a simple credit market and group lending
model in (Ahlin & Waters, 2016; Ghatak, 1999). Assume there is a continuum of agents that are
risk neutral and has measure one. Each agent is endowed with a unit of labor. Agents are
endowed with a project that requires one unit of capital and one unit of labor. However, agents
have no endowment of capital. Agents’ projects are known to differ in risk type 𝑝𝜖[𝑝, 1)
although there is an outside option that yields exogenously given net return of 𝑢 ≥ 0. Agents
would need to borrow a unit of capital to start their projects since they have no initial wealth.
The project of a type 𝑝 agent pays 𝑅𝑝 with probability 𝑝 and pays 0 otherwise. Assume
the riskiness of the project is private information known to the agent but not the lender. As in
(Stiglitz & Weiss, 1981), I assume that all projects have the same expected return 𝑅 = 𝑅𝑝 ∙ 𝑝 ,
∀ 𝑝𝜖[𝑝, 1). This means that risky projects pay more when an agent succeeds. I also assume

14

Ahlin (2009) found evidence of homogenous sorting by risk, and risk anti-diversifying strategy among group
members

58

limited liability, which means agents who are unsuccessful owe nothing to the lender. Output can
be verified as being either successful or failed but the lender cannot verify various shades of
success. The assumption of limited liability and costly verification of output makes debt
contracts the only feasible contracts. Borrowers who are able to repay their debts do, and so,
there are no enforcement problems. There is a single lender who is risk neutral and would be
willing to lend provided it earns an expected return of 𝜌 where 𝜌, is the opportunity cost of
capital per loan. The lender or bank knows the distribution of borrowers but not their
probabilities of success.
I assume 𝑅 > 𝜌 + 𝑢, which makes all projects have higher expected return than costs of
capital and labor invested. With social surplus strictly increasing in the number of projects
funded, fully efficient15 market would lend to all agents.
Suppose there two16 types of agents 𝑝 ∈ {𝑝𝑟 , 𝑝𝑠 } where 0 < 𝑝𝑟 < 𝑝𝑠 < 1 and 𝑅𝑠 < 𝑅𝑟 .
Let 𝜃(0, 1) be the population of risky borrowers and let the population average of any function
𝑔(𝑝) be denoted by 𝑔(𝑝) = 𝜃𝑔(𝑝𝑟 ) + (1 − 𝜃)𝑔(𝑝𝑠 ). Similarly, 𝑝 denotes the mean risk-type and
𝑝2 the mean squared-type.

15

Fully efficient lending goes hand-in-hand with maximal outreach because, from our assumptions, social surplus is
strictly increasing in the number of projects funded.
16
The model can be generalized to more than two types.

59

3.1.2

Individual Lending in a Static Environment

In this section, I review the results under individual lending and also group lending
before introducing correlation so that results can be compared. Under full information where
agents’ types are known, the lender can price for risk by charging specific interest rates such that
the following equation holds;
𝑝𝜏 𝑟𝜏 = 𝜌 ⟹ 𝑟𝜏 =

𝜌
𝑝𝜏

for each risk type 𝜏. 𝑟𝜏 (𝑟) denotes the loan amount plus interest. This is the

first best outcome which can be seen to be efficient and equitable with all surpluses accruing to
borrowers. Now, for the case where agent’s risk types are unknown to the lender,
Let 𝑝 = 𝜃𝑝𝑟 + (1 − 𝜃)𝑝𝑠 be the average success probability and 𝑟 the repayment amount. An
agent of type 𝜏 ∈ {𝑟, 𝑠}, will borrow to carry out the project if and only if,

𝑅 − 𝑝𝜏 𝑟 ≥ 𝑢̅ ⇔ 𝑟 ≤ 𝑟̂𝜏 ≡

̅
𝑅̅−𝑢
𝑝𝜏

. The first inequality says that the expected returns from the

project less the expected repayment amount should exceed the outside option’s return. The
second inequality follows from a rearrangement of the first and gives us a reservation interest
rate 𝑟̂𝜏 , above which an agent of type 𝜏 will choose the outside option instead. The reservation
interest rate is lower for the safe borrower, and thus the safe borrower is harder to attract. If safe
borrowers borrow, then so will the risky ones since safe borrowers succeed more often and repay
with a higher probability. Thus 𝑟 ≤ 𝑟̂𝑠 =

̅
𝑅̅−𝑢
𝑝𝑠

is a necessary condition for fully efficient lending.

60

A sufficient condition for both types to find it affordable to borrow is 𝑟 ≤ 𝑅𝑠 .
Now, the lender would be willing to give out a loan if 𝑝𝑟 = 𝜌 thus the break-even interest rate is
𝑟=

𝜌
𝑝̅

To attract all borrowers, we need

𝜌
𝑝̅

≤

̅
𝑅̅−𝑢
𝑝𝑠

ie 𝑁 ≥

𝑝𝑠
𝑝̅

𝑤ℎ𝑒𝑟𝑒 𝑁 =

̅
𝑅̅−𝑢
𝜌

and for affordability as

before,

𝑟 ≤ 𝑅𝑠 𝑜𝑟

𝜌
𝑝̅

≤

𝑅̅
𝑝𝑠

ie

𝑝𝑠
𝑝̅

≤ 𝐺 where 𝐺 =

𝑅̅
𝜌

As described in Ahlin and Waters (2014), 𝑁 =
market and 𝐺 =

𝑅̅
𝜌

̅
𝑅̅−𝑢
𝜌

is the net excess return to capital in this

is the gross excess return to capital in this market. Thus, efficient lending is

achieved if the net excess return to capital in this market is larger than the extent of asymmetric
information, represented by

𝑝𝑠
𝑝̅

. There is however, a second best option, which involves lending

to only risky agents. In which case there is inefficiency (in the sense that maximal outreach is not
attained) as the lender is unable to price for risk to attract safe borrowers.

Group Lending with independent risks
Here also, I review very quickly the model of joint liability lending presented in (Ghatak, 2000)
and results from (Gangopadhyay, Ghatak, & Lensink, 2005). Relying on the environment
discussed earlier, a joint liability contract requires a borrower to pay a joint liability say 𝑐, in
addition to the repayment amount 𝑟 on her own loan if the group17 member fails while she

17

We assume a group is of size two throughout this work.

61

succeeds. Here, it is assumed that agents know each other’s type but this is unknown to the
lender. (Ghatak, 2000) shows that borrowers would form groups homogenously based on risk
type in equilibrium. For a borrower of type 𝜏 ∈ {𝑟, 𝑠}, the expected payoff under homogenous
matching is given as
𝑅̅ − 𝑝𝜏 𝑟 − 𝑝𝜏 (1 − 𝑝𝜏 )𝑐 = 𝑅̅ − 𝑝𝜏 [𝑟 + (1 − 𝑝𝜏 )𝑐 ] . The effective interest rate 𝑟 +
(1 − 𝑝𝜏 )𝑐 is seen to vary positively with risk of the borrower and penalizes risky borrowers as
full information similarly does, although the lender has no information on risk types. It must be
noted that, a contract may attract only risky borrowers, both risky and safe, or none. In addition,
under any contract, the risky borrower earns more than the safe borrower (it can be seen by
examining the payoffs). Thus for group lending to achieve full efficiency, I maximize the safe
borrower’s payoff subject to lender’s zero profit, affordability and monotonicity constraints, and
assuming homogenous matching obtains.
Maximize, 𝑅̅ − 𝑝𝑠 𝑟 − 𝑝𝑠 (1 − 𝑝𝑠 )𝑐 subject to
𝑐 ≤ 𝑟, 𝑅𝑠 ≥ 𝑟 + 𝑐 and 𝑝̅ 𝑟 + ̅̅̅̅̅̅̅̅̅̅̅
𝑝(1 − 𝑝)𝑐 ≥ 𝜌
Before solving for the contracts, some key observations can be made. It can be seen that, raising
𝑐 and lowering 𝑟 along the lender’s isoprofit curve, would raise the safe borrowers payoff. This
is because the safer borrower’s indifference curve in the (𝑟, 𝑐) space has a larger slope in
magnitude relative to the bank’s iso-profit curve.18 What it means is that, a higher 𝑐 relative to 𝑟
would put more burden on the states of the world where there is a failure and thus on risky
borrowers. But since 𝑐 cannot exceed 𝑟, the best contract to attract safe borrowers is that with

18

-1/(1 − 𝑝𝑠 ) versus −𝑝̅ /̅̅̅̅̅̅̅̅̅̅̅
𝑝(1 − 𝑝)

62

𝑐 = 𝑟, where full affordability is affordable and the maximum affordable level of 𝑐 if full
liability is not affordable.
Solving the maximization problem, we have the best-for-safe contract as
𝜌

𝑐 = 𝑟 = ̅̅̅̅̅̅̅̅̅̅ assuming affordability is not an issue.
𝑝(2−𝑝)

On the other hand, when 𝑟 = 𝑐 is not affordable, the optimal contract is derived in as;

(𝑟, 𝑐) = {𝜌

̅̅̅̅̅̅̅̅̅̅𝐺
𝑝𝑠 −𝑝(1−𝑝)
𝑝̅ 𝐺 −𝑝
, 𝜌 ̅̅̅̅2𝑠 }.
̅̅̅̅
2
𝑝 𝑝
𝑝 𝑝
𝑠

𝑠

The results are thus summarized in the proposition below

Proposition 0: Under the assumptions of the model, a group contract that maximizes borrower
surplus subject to homogeneous matching, borrower limited liability, lender break even and
monotonicity achieves full efficiency, if and only if

𝑁≥{

𝐵1 −

𝐵1 −𝐵2∗
𝐶2∗ −𝐶1
𝐵2∗

[𝐺 − 𝐶1 ]

Where 𝐶1 = 𝐵1 =

𝐺 ∈ [𝐶1 , 𝐶2∗ ]
𝐺 ≥ 𝐶2∗

𝑝𝑠
𝑝̅

, 𝐵2∗ =

𝑝𝑠 (2−𝑝𝑠 )
̅̅̅̅̅̅̅̅̅̅
𝑝(2−𝑝)

2𝑝

𝑠
and 𝐶2∗ = ̅̅̅̅̅̅̅̅̅̅

𝑝(2−𝑝)

Otherwise, only risky agents borrow. This is the independent risk case results and we shall
compare it to the correlation case in the sections to follow. Essentially, we can see from the
above results that, as 𝐺 increases away from 𝐶1 , the net returns that is required for efficient
lending decreases also linearly in 𝐵1 and gets to a minimum at 𝐵2∗ . The dependence on 𝐺
illustrates the affordability consideration in deriving the contracts. We cannot go below the floor
of 𝐵2∗ because the contract cannot have 𝑐 greater than 𝑟, and indication of the limitation group
lending has in improving risk-pricing. See (Ahlin & Waters, 2016) for a full discussion and
comparison with dynamic individual lending.
63

Group Lending with Correlated Risk
I introduce correlation of risk types into the group lending model in a manner that preserves the
individual probabilities of success. Suppose there are two borrowers 𝑖 and 𝑗, with probabilities
of success 𝑝𝑖 𝑎𝑛𝑑 𝑝𝑗 respectively. The joint distribution is given in the Table 3-1.
The table presents the unique way to introduce correlation while preserving success
probabilities given 𝑝𝑖 𝑎𝑛𝑑 𝑝𝑗 . It can be verified that the columns and rows add up to equal the
individual probabilities of success. 𝜀 = 0 corresponds to the no-correlation case. I focus on
positive correlation only. (Katzur & Lensink, 2012), show that negative correlation between
risky borrowers may lead to a breakdown of positive assortative matching for some of the firstbest contracts. I maintain the assumption that borrower types are unknown to the lender. In
addition, I assume here that the lender knows project returns are correlated and knows the joint
distribution of the probabilities of success. A positive 𝜀 adds to the probabilities that both either
succeed together or fail together and subtracts from the probabilities that an agent succeeds and
the group member fails.
Although the table depicts a constant mass 𝜀 being added to or subtracted from the cells,
𝜀 may depend on the probabilities of success. There are two ways one can symmetrically
introduce correlation across all types of projects. In the first case, herein after called the
‘constant-mass case’ a constant amount of mass is added to same-outcome group events. The
probability of group members having the same outcome is increased by a constant mass relative
to the independent risk case. This also now means that the probability of group members
realizing different outcomes is lowered by the amount of the constant mass as compared to when
correlation is zero. Thus in this constant mass case, 𝜀(𝑝𝑖 , 𝑝𝑗 ) = 𝜀 > 0 for all 𝑝𝑖 𝑎𝑛𝑑 𝑝𝑗 . The

64

second case involves not adding a constant mass but scaling a mass so that, homogeneous
projects have the same correlation coefficient. I call this case the ‘constant-correlation’ case. In
this second case therefore, 𝜀(𝑝𝑖 , 𝑝𝑗 )>0 is not constant but the correlation is constant for all
𝑝𝑖 𝑎𝑛𝑑 𝑝𝑗 . Essentially both cases are doing the same thing by introducing correlation
symmetrically although the approaches differ as described above.
The result in (Katzur & Lensink, 2012) that group lending contracts may have better
efficiency outcomes, hinges on the assumption that the correlation between safe projects is
sufficiently larger than the correlation between risky projects. Thus correlation is not
symmetrically introduced in that paper. Whereas homogeneous projects have the same
correlation coefficient in our second case, it is higher for safe projects under (Katzur & Lensink,
2012).

Constant Mass Case (Case 1)
Suppose 𝜀(𝑝𝑖 , 𝑝𝑗 ) = 𝜀 > 0 for all 𝑝𝑖 𝑎𝑛𝑑 𝑝𝑗
Given the requirement that elements in the cells in the chart above, must be less than one and
greater than zero, it follows that, assuming 𝜀 ≤ 𝑝𝑟 (1 − 𝑝𝑠 ) is sufficient to ensure this. The results
on the independent risk types hinges on the homogeneous matching results derived in (Ghatak,
1999). With the introduction of correlation in project returns, I show that the homogenous
matching result still holds under case 1.

Lemma 1
65

Under the assumptions in the model, and in the presence of correlated risks, homogeneous
matching is the only equilibrium.
By examining the joint distribution, it can be seen that
Observation 1
Both types of borrowers in the presence of correlated risks pay the joint liability less often than
under independent risks. This is because they either both succeed or both fail with a higher
probability compared to independent risks case.
As mentioned already, given the assumptions in the model, a joint liability contract that
attracts safe borrowers would also attract risky ones simply because the safe borrowers succeed
more often and repay the joint liability with a higher probability. As such, to investigate the
feasibility of fully efficient lending, I zero in on what can attract the safe borrower into accepting
the contract. It follows that, if full liability is affordable, the best contract should have 𝑐 = 𝑟 and
extract the maximum possible if 𝑐 = 𝑟 is not affordable. More formally, I rely on the following
lemma to derive the condition under which fully efficient lending with borrowers having all
surpluses is attainable.

Lemma 2
The risky borrower earns more than the safe borrower in any joint liability contract with 𝑐 ≤ 𝑟.
This can be shown by comparing payoffs of the safe and risky borrowers.
This also means that a contract that attracts the safe borrower also attracts the risky borrower.

66

By Lemma 2, the contract attracts either both safe and risky borrowers or only risky borrowers or
none. To attract only risky borrowers, the lender can just set 𝑟 =

𝜌
𝑝𝑟

. To derive a necessary and

sufficient condition under which fully efficient lending is attainable, I maximize a safe
borrower’s payoff
𝑅̅ − 𝑝𝑠 𝑟 − [𝑝𝑠 (1 − 𝑝𝑠 ) − 𝜀 ]𝑐 subject to the following constraints;
1. 0 ≤ 𝑐 ≤ 𝑟

Monotonicity

2. 𝑝̅ 𝑟 + ̅̅̅̅̅̅̅̅̅̅̅
𝑝(1 − 𝑝)𝑐 − 𝜀𝑐 ≥ 𝜌

Zero profit constraint

3. 𝑟 + 𝑐 ≤ 𝑅𝑠

Limited Liability

Using the first two constraints to solve for the case where affordability is not an issue, we get
𝜌

𝑟 = 𝑐 = ̅̅̅̅̅̅̅̅̅̅

𝑝(2−𝑝)−𝜀

, granted 𝑝𝑠 𝑝𝑟 > 𝜀.19 The remaining limited liability constraint holds if

2𝑝

𝑠
𝐺 ≥ ̅̅̅̅̅̅̅̅̅̅

𝑝(2−𝑝)−𝜀

When affordability is an issue (when

𝑝𝑠
𝑝̅

2𝑝

𝑠
≤ 𝐺 < ̅̅̅̅̅̅̅̅̅̅

𝑝(2−𝑝)−𝜀

), we maximize the safe borrower’s

payoff subject to the following constraints
𝑅𝑠 ≥ 𝑟 + 𝑐 and 𝑝̅ 𝑟 + ̅̅̅̅̅̅̅̅̅̅̅
𝑝(1 − 𝑝)𝑐 − 𝜀𝑐 ≥ 𝜌.
(𝑟, 𝑐) = {𝜌

̅̅̅̅̅̅̅̅̅̅−𝜀]𝐺
𝑝𝑠 −[𝑝(1−𝑝)
𝑝̅ 𝐺 −𝑝
, 𝜌 ̅̅̅̅2 𝑠 }.
̅̅̅̅
2 +𝜀)
𝑝 (𝑝
𝑝 (𝑝 +𝜀)
𝑠

𝑠

The contract can be derived as

Clearly, monotonicity constraint is satisfied for

𝑝𝑠
𝑝̅

≤

2𝑝

𝑠
𝐺 < ̅̅̅̅̅̅̅̅̅̅

𝑝(2−𝑝)−𝜀

19

𝑝𝑠 𝑝𝑟 > 𝜀 is guaranteed by a standard assumption of 𝑝𝑠 + 𝑝𝑟 > 1 made in the literature and our earlier assumption
that 𝜀 < 𝑝𝑟 (1 − 𝑝𝑠 )

67

Proposition 1
Under the assumptions in the model, a group contract with joint liability that maximizes
borrower surplus subject to the following conditions; homogenous matching, limited liability on
borrower, lender break-even and monotonicity achieves full efficiency if and only if

𝑁≥{

𝐵1 −

𝐵1 −𝐵2
𝐶2 −𝐶1

𝐺 ∈ [𝐶1 , 𝐶2 ]

[𝐺 − 𝐶1 ]

𝐵2

Where 𝐶1 20 = 𝐵1 =

𝐺 ≥ 𝐶2
𝑝𝑠
𝑝̅

, 𝐵2 =

𝑝𝑠 (2−𝑝𝑠 )−𝜀
̅̅̅̅̅̅̅̅̅̅−𝜀
𝑝(2−𝑝)

2𝑝

𝑠
and 𝐶2 = ̅̅̅̅̅̅̅̅̅̅

𝑝(2−𝑝)−𝜀

Otherwise, only risky agents borrow.
Comparing the results here to those under independent risk, we can see that, the level of 𝑟 and 𝑐
charged when full liability is affordable are higher under correlation.
It is important to note that, under correlation, 𝑐 plays lesser role in risk-pricing as group
members are more likely to succeed or fail together. Bailouts occur less frequently under
correlation. So the extent to which the lender can rely on the joint liability component of the
contract to price for risk is curtailed in the correlation case.
The effective interest rate under this contract can be written as
𝜀

𝑟̂𝑖 = 𝑟 + [(1 − 𝑝𝑖 ) − ] 𝑐 . It can be shown that this decreases in 𝑝𝑖
𝑝𝑖

by taking the

derivative. Thus, the effective interest rate under correlation also offers a discount to safe
borrowers. I reserve the discussion of the intuition of these results until proposition 3.

20

The subscript (1) means contract is for 1 borrower.

68

Altogether, the net returns that is required for efficient lending is higher than under
independent risk case. This essentially means that correlation reduces the parameter space for
fully efficient lending and makes joint liability less efficient relative to independent risk case.

Proposition 2
The parameter space over which joint liability lending achieves full efficiency, reduces as the
correlation, represented by a constant mass of 𝜀 between homogenous projects increases.

Constant-correlation Case (Case 2)
In this section, I put some structure on 𝜀 so that it is not constant but depends on the
probabilities of success. I demonstrate that results obtained under the constant-mass case can be
replicated using the constant correlation method of symmetrically introducing correlation.
Define 𝜀(𝑝𝑖 , 𝑝𝑗 ) = 𝑣̃ ∗ min{𝑝𝑖 (1 − 𝑝𝑗 ), 𝑝𝑗 (1 − 𝑝𝑖 )}. This enables the distribution of projects of
homogeneous groups to have the same correlation coefficient. Assume 𝑣̃ ∈ [0,1]. 𝑣̃ is a fraction
of the maximum possible correlation between two projects. Thus with homogenous group, 𝑣̃ is
just the correlation coefficient. See Appendix for details.

Lemma 3
Under the constant correlation case, homogeneous matching is the only equilibrium.

69

Lemma 4
Under the constant correlation coefficient case, the risky borrower earns more than the safe
borrower does. The proof follows similarly as Lemma 1 and by noting that 0 ≤ 𝑣̃ ≤ 1.
Using Lemma 4, the contracts are similarly derived from the maximization problem but
replacing constant 𝜀 with its constant correlation form in the objective function and constraints.
When affordability is not an issue, the contract can be derived as
𝑟=𝑐=

𝜌
̅̅̅̅̅̅̅̅̅̅]
[𝑝̅ +(1−𝑣̃)𝑝(1−𝑝)

(𝑟, 𝑐) = {𝜌

and when affordability is an issue, we have the contract as

̅̅̅̅̅̅̅̅̅̅]𝐺
𝑝𝑠 −[(1−𝑣̃)𝑝(1−𝑝)
𝑝̅ 𝐺 −𝑝𝑠
̅̅̅̅̅̅̅̅̅̅] , 𝜌 𝑝𝑠 [𝑝̅ −(1−𝑣̃)𝑝(1−𝑝)
̅̅̅̅̅̅̅̅̅̅]}
𝑝𝑠 [𝑝̅ −(1−𝑣̃)𝑝(1−𝑝)

Proposition 3
Under the assumptions in the model, a joint liability contract that maximizes borrower surplus
subject to the following conditions; homogenous matching, limited liability, lender breaking
even and monotonicity achieves full efficiency if and only if

𝑁≥{

𝐵1 −

𝐵1 −𝐵2′
𝐶2′ −𝐶1

[𝐺 − 𝐶1 ]

𝐵′2

𝐺 ∈ [𝐶1 , 𝐶′2 ]
𝐺 ≥ 𝐶′2

Where
𝐶1 = 𝐵1 =

𝑝𝑠
𝑝̅

, 𝐵′2 =

[𝑝𝑠 +𝑝𝑠 (1−𝑝𝑠 )(1−𝑣̃)]
̅̅̅̅̅̅̅̅̅̅]
[𝑝̅ +(1−𝑣̃)𝑝(1−𝑝)

and 𝐶′2 =

before.

70

2𝑝𝑠
̅̅̅̅̅̅̅̅̅̅]
[𝑝̅ +(1−𝑣̃)𝑝(1−𝑝)

and 𝑣̃ is as defined

Otherwise, only risky agents borrow.
The prediction from the constant-mass correlation parameter 𝜀 is not different from the
constant correlation case. In subsequent derivations, I focus on constant-correlation case. It can
be seen that the region for fully efficient lending narrowing as correlation increases. To get an
insight on this result, I interrogate the effective interest rate in the presence of correlation. In the
face of correlation, the effective interest rate can be written as 𝑟̂𝑖 = 𝑟 + (1 − 𝑝𝑖 )(1 − 𝑣̃)𝑐.
Setting 𝑣̃ = 0 yields that of the independent risk case. When correlation is very high
(𝑣̃ approaching 1), the joint liability component becomes very small. The relationship between
the effective interest and the correlation parameter 𝑣̃ as well as 𝑝𝑖 is the same as obtained in
proposition 1.
The effective interest rate differential between safe and risky borrowers is given
by (𝑝𝑠 − 𝑝𝑟 )(1 − 𝑣̃)𝑐. Recall that the effective interest rate differential is the implicit discount
to safe borrowers and implicit premium charged to risky borrowers that makes the lender able to
improve risk-pricing and the efficiency of credit markets using group lending. Let 𝑐 ′ = (1 − 𝑣̃)𝑐
be the effective liability. 𝑐 ′ decreases as correlation 𝑣̃ increases, if 𝑐 does not change. As 𝑣̃ goes
up, we would like to raise 𝑐 to compensate for the reduction in 𝑐 ′ so that 𝑐 ′ remains the same or
unchanged. Nevertheless, 𝑐 ′ cannot be restored to its initial value because 𝑐 does not rise
enough. To show this,
First, consider the case where affordability is not an issue. The binding constraints are
𝑐 ≤ 𝑟 (Monotonicity) and 𝑝̅ 𝑟 + ̅̅̅̅̅̅̅̅̅̅̅
𝑝(1 − 𝑝)(1 − 𝑣̃)𝑐 ≥ 𝜌 (zero profit constraint).

71

As correlation 𝑣̃ increases, the effective liability 𝑐 ′ and the ̅̅̅̅̅̅̅̅̅̅̅
𝑝(1 − 𝑝)(1 − 𝑣̃)𝑐 component of the
zero profit constraint reduce if 𝑐 does not change. The lender would need to raise revenue to
restore the effective liability and hence the zero profit constraint to their initial levels. This could
be done by increasing 𝑐 sufficiently to counter the reduction in 𝑐 ′ but this cannot be achieved
without violating the binding monotonicity constraint which says 𝑐 cannot exceed 𝑟. This
suggests that 𝑟 and 𝑐 would need to be raised simultaneously. We can think about increasing 𝑟
to make way for 𝑐 to increase sufficiently by making strictly positive profits21 but this only
makes it harder to attract safe borrowers. Therefore, raising 𝑟 and 𝑐 simultaneously, without
violating both the monotonicity and zero profit constraints means 𝑐 cannot rise enough to restore
the original effective interest rate differential (or the effective liability). Unchanged profits but a
reduced implicit discount for safe borrowers makes it harder to attract safe borrowers-- thus
reducing the parameter space for fully efficient lending under joint liability.
Second, consider the case where affordability is an issue. The binding constraints are the
zero profit constraint and the affordability constraint, that is, 𝑝̅ 𝑟 + ̅̅̅̅̅̅̅̅̅̅̅
𝑝(1 − 𝑝)(1 − 𝑣̃)𝑐 ≥ 𝜌 and
𝑟 + 𝑐 ≤ 𝑅𝑠 respectively. Again, as correlation 𝑣̃ increases, ̅̅̅̅̅̅̅̅̅̅̅
𝑝(1 − 𝑝)(1 − 𝑣̃)𝑐 component of the
zero profit constraint decreases if 𝑐 does not change and the lender would need to raise revenue.
This can be done by increasing 𝑐 or 𝑟 or both to keep the lender’s zero profit constraint binding.
However, the binding affordability constraint (𝑟 + 𝑐 ≤ 𝑅𝑠 ), means 𝑐 and 𝑟 cannot increase or
decrease at the same time to achieve the intended purpose without violating the binding
affordability constraint. 𝑐 and 𝑟 must move in opposite directions (since their sum is a constant).

21

An increase in 𝑐 sufficiently would be able to restore the decline in revenue as a result of the increase in
correlation but this comes with an increase in 𝑟 which would mean there is excess revenue than needed for a binding
zero profit constraint. This then means the zero profit constraint holds with strict inequality. The lender makes
strictly positive profit from lending since every project when successful faces a higher 𝑟 and 𝑐 than those needed for
a zero profit.

72

Thus, the lender can raise revenue only by raising 𝑟 and lowering 𝑐 22. The drop in 𝑐 further
reduces the effective interest rate differential and hence the implicit discount to safe borrowers.
Again, unchanged profit but a reduced implicit discount for safe borrowers makes it harder to
attract safe borrowers—thus reducing the parameter space for fully efficient lending under joint
liability.
Figures 3-5 in the Appendix shows the relationship between the joint liability 𝑐, and
correlation 𝑣̃. It can be seen that, 𝑐 increases initially up to a point and then begins to fall. To
summarize the results, we can show the following proposition.

Proposition 4
Under the constant correlation case, the parameter space over which joint liability lending
achieves full efficiency reduces as the correlation (𝑣̃) increases.
Figure 3-1 shows the graph of the correlated risks case against independent risks case. The
correlated risks case is the broken lines. It can be seen that it lies above the independent risk case
and thus achieves full efficiency under a smaller parameter space. This is easy to see because the
efficiency parameter space for the independent risk case is larger compared to the correlated risk
case.

22

The lender cannot rather increase 𝑐 and lower 𝑟 because 𝑟 is repaid more often than 𝑐 and it requires a larger raise
in 𝑐 to make-up for a small reduction in 𝑟. This can also be seen by noting that the slope of the affordability
constraint in the (𝑟, 𝑐) space has a smaller slope in magnitude relative to that of the bank’s iso-profit curve. ie −1
versus −𝑝̅ /̅̅̅̅̅̅̅̅̅̅̅
𝑝(1 − 𝑝)(1 − 𝑣̃). Therefore, moving to a higher profit curve while staying on the affordability constraint
requires raising 𝑟 and lowering 𝑐.

73

Correlation in a Market of Mixed Borrowers
An interesting case with the introduction of correlation in the group lending strategy is a
situation, where the lender is faced with a mixed pool of borrowers. An example would be a
typical situation of a lender having to serve borrowers from a pool of farmers and a pool of small
business owners. In this section, I look at a lender who is faced with these two categories of
borrowers. It would be interesting to know whether the lender would serve these categories
separately or would pool the borrowers together in offering the contract.
I assume borrowers come from either of two pools; A and B. All borrowers from pool A
are correlated with all other borrowers in pool A. All borrowers from pool B are independent of
all other borrowers. We can think of the lender being faced with a fraction k, of borrowers
having correlated risks and a fraction (1-k) having independent risks. The correlation type
(belonging to pool A or pool B), is independent of risk type (𝑝𝑖 ). I study how this impacts the
optimal contracts, efficiency and whether or not it is better for the lender to separate borrowers
instead of pooling them.
In this section also, the bank or lender doesn’t know the risk (𝑝𝑖 ) types. The lender is
assumed to know correlation type of the borrower (A or B) but the lender still allows voluntary
group formation so that borrowers will feel buy-in.23 I use the constant-correlation case
throughout this section because it enables me plot graphs easily. In this new scenario, there can
be four types of groups in equilibrium;

23

For example, borrowers could have a disutility of having their free association restricted; sometimes people would
like to form groups on religious or social lines to enable monitoring and enforce repayment. The lender may not
want to meddle with group formation so that, borrowers would not blame their default on their inability to choose
their group members. The lender could also be interested mainly in homogeneous matching on risk type not
correlation type. We could also assume k>1/2 so that there is still some max {0, 2k-1} population of correlated
borrowers left, even if lender were to dictate matching (by diversification). The analysis then proceeds similarly
from there.

74

𝐴 𝑠𝑎𝑓𝑒 :

Safe and Safe correlated group

𝐴𝑟𝑖𝑠𝑘𝑦 : Risky and Risky correlated group
𝐵 𝑠𝑎𝑓𝑒 :

Safe and Safe uncorrelated group

𝐵𝑟𝑖𝑠𝑘𝑦 : Risky and Risky uncorrelated group
A formal proof of the matching pattern is in proposition 1 of Ahlin (2016) which I rephrase as
Lemma 5 below.
Lemma 5
Under the assumptions in the model, in equilibrium, almost every group is homogeneous in both
risk and correlation type.
It is important to note that, all two-person groups formed to apply for loans are in one of the
groups listed above. Lemma 5 rules out a case where a group A member partners with a group B
member. All such groups can be shown to have a measure zero in equilibrium.

3.1.3

Optimal Contracts

Lemma 6
The safe uncorrelated borrowers earn less than any other borrower in any joint liability contract
with 𝑐 ≤ 𝑟.

75

I show this by comparing payoffs of the groups

𝐴 𝑠𝑎𝑓𝑒 , 𝐴𝑟𝑖𝑠𝑘𝑦 , 𝐵 𝑠𝑎𝑓𝑒 and 𝐵𝑟𝑖𝑠𝑘𝑦 above. This

also means that a contract that attracts the 𝐵 𝑠𝑎𝑓𝑒 borrower also attracts borrowers in other
groups.
Safe borrowers cross-subsidize risky borrowers and in the presence of correlation, uncorrelated
borrowers also cross-subsidize correlated borrowers. To derive the optimal contract therefore, I
rely on Lemmas 5 and 6 and maximize the expected payoff of 𝐵 𝑠𝑎𝑓𝑒 group subject to our usual
constraints.
𝐵 𝑠𝑎𝑓𝑒 Group member’s payoff is 𝑅̅ − 𝑝𝑠 𝑟 − [𝑝𝑠 (1 − 𝑝𝑠 ) ]𝑐
Subject to
0 ≤ 𝑐 ≤ 𝑟, and 𝑝̅ 𝑟 + ̅̅̅̅̅̅̅̅̅̅̅
𝑝(1 − 𝑝)(1 − 𝑘𝑣̃)𝑐 ≥ 𝜌
Assume 𝑣̃ <

𝜃𝑝𝑟 (𝑝𝑠 −𝑝𝑟 )
̅̅̅̅̅̅̅̅̅̅
𝑘𝑝(1−𝑝)

. Then, when affordability is not an issue, the contract can be derived as

follows
𝑟=𝑐=

𝜌
[𝑝̅ +̅̅̅̅̅̅̅̅̅̅
𝑝(1−𝑝)(1−𝑘𝑣̃)]

Now, when affordability is an issue, (when

𝑝𝑠
𝑝̅

≤𝐺<

2𝑝𝑠
̅̅̅̅̅̅̅̅̅̅(1−𝑘𝑣̃)]
[𝑝̅ +𝑝(1−𝑝)

)

the following constraints are the relevant ones and are similar to section 3.6’s derivations.
𝑅𝑠 ≥ 𝑟 + 𝑐
𝑝̅ 𝑟 + ̅̅̅̅̅̅̅̅̅̅̅
𝑝(1 − 𝑝)(1 − 𝑘𝑣̃)𝑐 ≥ 𝜌

76

The contract is derived as (𝑟, 𝑐) = {𝜌

̅̅̅̅̅̅̅̅̅̅]𝐺
𝑝𝑠 −[(1−𝑣̃𝑘)𝑝(1−𝑝)
𝑝̅ 𝐺 −𝑝𝑠
̅̅̅̅
̅̅̅̅
2
2 +(𝑣
̅̅̅̅̅̅̅̅̅̅] , 𝜌 𝑝𝑠 [𝑝
̅̅̅̅̅̅̅̅̅̅]}
̃𝑘)𝑝(1−𝑝)
𝑝𝑠 [𝑝 +(𝑣̃𝑘)𝑝(1−𝑝)

See appendix for details.
Using Lemma 6 and results from the maximization problem, the following can be shown

Proposition 5
Under the assumptions in section 5, a joint liability contract that maximizes borrower surplus
subject to the following conditions; homogenous matching, limited liability on borrower, lender
breaking even and monotonicity achieves full efficiency for 𝑣̃ <

𝑁≥{

𝐵1 −

𝐵1 −𝐵2′′
𝐶2′′ −𝐶1

[𝐺 − 𝐶1 ]

𝐵′′2
Where 𝐶1 = 𝐵1 =

𝜃𝑝𝑟 (𝑝𝑠 −𝑝𝑟 )
̅̅̅̅̅̅̅̅̅̅
𝑘𝑝(1−𝑝)

if and only if

𝐺 ∈ [𝐶1 , 𝐶′′2 ]
𝐺 ≥ 𝐶′′2

𝑝𝑠
𝑝̅

, 𝐵"2 =

𝑝𝑠 (2−𝑝𝑠 )
̅̅̅̅̅̅̅̅̅̅
[𝑝̅ +𝑝(1−𝑝)(1−𝑘𝑣̃)]

and 𝐶"2 =

2𝑝𝑠
̅̅̅̅̅̅̅̅̅̅
[𝑝̅ +𝑝(1−𝑝)(1−𝑘𝑣̃)]

Otherwise, safe uncorrelated borrowers are excluded from the credit market.
An interesting observation from the analyses in mixed pool of borrowers’ case in section
3.8.1 is that there exists non-monotonicity in the ability to reach a larger pool of borrowers in the
presence of correlation. When correlation is zero, fully efficient lending is achieved over a larger
parameter space than when correlation is positive. One would have thought that, for a fixed
correlation between project returns, as the fraction of correlated borrowers increases from 0
towards 1, it would be easier to reach a mixed pool of borrowers than an all-correlated group
(monotonicity). It turns out that, in some instances, it is easier to reach an all-correlated pool than
a mixed pool of borrowers.
77

The non-monotonicity in the conditions for fully efficient lending with respect to 𝑘 under this
section, stems from the fact that, as 𝑘 increases, there are fewer borrowers from the uncorrelated
pool of borrowers, and it is harder to reach the few uncorrelated borrowers left. In addition, the
uncorrelated borrowers become increasingly worse off as 𝑘 increases since they cross subsidize
the correlated borrowers. In such a case, separation would do better than pooling of the mixed
pool of borrowers. When all borrowers are correlated, there are no safe, uncorrelated borrowers
that need to be reached. Consider a case where 𝑘 = 1 versus when 𝑘 = 1 − 𝛿 for some small
𝛿 > 0.
The aggregate cross-subsidy from safe borrowers to risky borrower, and from uncorrelated
borrowers to correlated borrowers would be very similar under 𝑘 = 1 and 𝑘 = 1 − 𝛿 for small 𝛿.
The problem is that while for 𝑘 = 1, there are no uncorrelated borrowers to reach, under = 1 − 𝛿
one worries about how to reach the small group of uncorrelated borrowers (𝛿) who are harder to
attract because the safe-uncorrelated borrowers are the least well-off. Although this smaller
group can be left unreached so that we have just a 𝛿 loss of efficiency, the difficulty in reaching
them illustrates how fully efficient lending discontinuously becomes easier to attain as 𝑘 reaches
1.

Proposition 6
𝜃𝑝𝑟 (𝑝𝑠 −𝑝𝑟 )
Define 𝑘̅𝑎 = ̅̅̅̅̅̅̅̅̅̅[1+(1−𝑝
𝑝(1−𝑝)

̃)]
𝑠 )(1−𝑣

𝜃𝑝𝑟 (𝑝𝑠 −𝑝𝑟 )
and 𝑘̅𝑏 = ̅̅̅̅̅̅̅̅̅̅[1−(1−𝑝
𝑝(1−𝑝)

78

. Clearly, 0 < 𝑘̅𝑎 < 𝑘̅𝑏 < 1

̃)]
𝑠 )(1−𝑣

Under assumptions in section 3.8, fully efficient lending can be achieved over a larger parameter
space (G, N), by lender pooling borrowers if 𝑘 < 𝑘̅𝑎 and by lender separating borrowers if 𝑘 >
𝑘̅𝑏

3.1.4

Graphical presentation of propositions 6

We have seen in section 3.8.1 that when the conditions for proposition 5 do not hold, or
when project net excess returns are low, the safe uncorrelated are not able to borrow and
efficiency of the lending strategy is reduced. Following this, I explore in this section, whether we
can have full efficiency by separating the banks (ie. separate into a lender that would lend to only
the borrowers in Pool A and a lender focusing exclusively on Pool B borrowers). Using the
values we use in plotting the figures, we can calculate 𝑘̅𝑎 = 0.519 and 𝑘̅𝑏 = 0.574
In Figure 3-2, we add the graph of the mixed pool of borrowers to Figure 3-1 it. For
clarity, “B2” corresponds to the independent risk case. “B2p” corresponds to the correlated risk
case, while “B2pp” corresponds to the mixed pool of borrowers’ case. The mixed pool of
borrowers’ line lies in between the other two. As we can see from the figure, when net excess
return to capital embedded in the project is low and lies between B2p and B2pp, it is better to
pool borrowers and give one contract. This would be the way to achieve full efficiency. In this
case, there is no point separating, otherwise we lose the safe correlated group.
In figure 4-3 where the fraction of correlated borrowers is 0.75, we see a situation that
calls for separation of banks into those that would serve borrowers with correlated risk and those
that would serve borrowers with independent risks for full efficiency. When net excess return to
capital embedded in the project is low and lies between B2p and B2pp, it is better to separate
79

borrowers and serve them separately. This would be the way to achieve full efficiency. In this
case, there is no point pooling borrowers, otherwise we lose the safe uncorrelated group.
The need for separation of the pool of borrowers into those with correlated risk and those
with independent risk for fully efficient lending may help explain and justify the existence of
agricultural development banks separately from banks serving self-employed borrowers. (Besley,
1994; IFC, 2012; Ramana, 2004) all acknowledge the fact that covariant risk makes agricultural
financing unattractive to microfinance firms. (Ramana, 2004) document how MFIs do not lend to
farmers citing the Gramen Bank and Unit Desa system of the Bank of Rakyat Indonesia as banks
that have focused exclusively on rural areas but not on agricultural lending as against the Bank of
Agriculture and Agricultural Cooperatives (BAAC) of Thailand which focuses exclusively on
lending to agricultural workers and none to nonfarm activities.

Conclusion
We derive and characterize the optimal lending contracts for group-based joint liability lending
with correlated risks. With the introduction of correlation, we find that the parameter space for
fully efficient lending under correlated risks is smaller compared to independent risks case.
When correlation increases, the differential effective interest rate that offers implicit discount
(charges implicit premium) to safe (risky) borrowers is reduced. This reduces the ability of the
lender to use joint liability lending to price for risk and improve efficiency of credit markets.
When full affordability is feasible, the monotonicity constraint coupled with a binding
lender zero-profit constraint, prevents the lender from increasing the joint liability component
enough to maintain effective interest rate differential that could have helped in risk-pricing. On
80

the other hand, when affordability is an issue, the affordability constraint requires that the
interest rate be increased and the joint liability component be reduced in response to greater
correlation. Reduction in the joint liability further reduces the effective differential interest rate.
unchanged profits but a reduced discount makes it difficult to attract safe borrowers. Correlation
thus reduces the effectiveness of group lending and especially when affordability is low. It can
lead to the exclusion of some potential borrowers such as the safe uncorrelated from the market
thereby reducing efficiency.
An interesting observation from the analyses is the existence of non-monotonicity in the
ability to reach a larger pool of borrowers in the presence of correlation. One would have thought
that, for a fixed correlation between project returns, as the fraction of borrowers with correlated
risk increases from 0 towards 1, it would be easier to reach a mixed pool of borrowers than an
all-correlated group. It turns out that, in some instances, it is easier to reach an all-correlated
group than a mixed group of borrowers.
The non-monotonicity stems from the fact that, as 𝑘 increases, there are fewer borrowers
from the uncorrelated pool of borrowers and it becomes harder to reach the few uncorrelated
borrowers. In such a case, separation would do better than pooling of the mixed pool of
borrowers. In addition, the uncorrelated borrowers become increasingly worse off as 𝑘 increases
since they cross subsidize the correlated borrowers. When all borrowers are correlated, there are
no safe, uncorrelated borrowers that need to be reached.
The results show that under certain conditions, such as situations where project returns
are low and the fraction of correlated borrowers is high, full efficiency requires that the banks for
correlated borrowers and those for non-correlated borrowers be separated. This may help explain

81

the existence of agricultural development banks separately from banks serving self-employed
non-agricultural borrower.

82

APPENDICES

83

Appendix A: Tables and Figures for Essay 2

Table 3-1: Joint Output Distribution in the Presence of Correlation
𝑗
Succeeds (𝑝𝑗 )
𝑗

Fails (1-𝑝𝑗 )

𝑖

succeeds (𝑝𝑖 )

𝑝𝑖 𝑝𝑗 + 𝜀

𝑝𝑖 (1 − 𝑝𝑗 ) − 𝜀

𝑖

Fails (1- 𝑝𝑖 )

(1 − 𝑝𝑖 )𝑝𝑗 − 𝜀

(1-𝑝𝑖 )(1 − 𝑝𝑗 ) + 𝜀

Figure 3-1: Correlated Borrowers

84

Figure 3-2: Mixed Pool of Borrowers for Lower k

85

Figure 3-3: Mixed Pool of Borrowers for Higher k

86

Figure 3-4: K=0.61 Where Mixed Line Crosses Correlated Line to Lie Above it)

87

0.56

0.54

0.52

0.50
0.48

0.46

0.44

0.2

0.4

0.6

0.8

1.0 V

Figure 3-5: Plot of Joint Liability against Correlation among borrowers (C against V)
This is plotted by setting G=2, 𝜌 = 1, 𝑝𝑠 = 0.9, 𝑝𝑠 = 0.6, 𝜃 = 0.5
v<0.09(When Full Liability is Affordable) v>0.09(When Full Liability is Not Affordable)

88

Appendix B: Proofs of Lemmas and Propositions in Essay 2

Proof of Lemma 1 and Lemma 3
To show that homogeneous matching obtains, is efficient and is the only equilibrium, I compare
the sum of group payoff from homogenous matching to group payoff from heterogeneous
matching. This condition ensures that homogeneous matching is stable even in the presence of
side contracting.
Consider two groups each of size two. I begin with the proof of Lemma 3. Lemma 1 follows
similarly.
The group payoff of a safe borrower for partnering with another safe borrower,
𝑅̅ − 𝑝𝑠 𝑟 − [𝑝𝑠 (1 − 𝑝𝑠 ) − 𝜀 24𝑠𝑠 ]𝑐 + 𝑅̅ − 𝑝𝑠 𝑟 − [𝑝𝑠 (1 − 𝑝𝑠 ) − 𝜀𝑠𝑠 ]𝑐

(1)

The group payoff of a risky borrower pairing with another risky borrower,
𝑅̅ − 𝑝𝑟 𝑟 − [𝑝𝑟 (1 − 𝑝𝑟 ) − 𝜀𝑟𝑟 ]𝑐+ 𝑅̅ − 𝑝𝑟 𝑟 − [𝑝𝑟 (1 − 𝑝𝑟 ) − 𝜀𝑟𝑟 ]𝑐

(2)

a non-homogenous group’s payoff (1st agent safe , second risky) is
𝑅̅ − 𝑝𝑠 𝑟 − [𝑝𝑠 (1 − 𝑝𝑟 ) − 𝜀𝑠𝑟 ] 𝑐 + 𝑅̅ − 𝑝𝑟 𝑟 − [𝑝𝑟 (1 − 𝑝𝑠 ) − 𝜀𝑠𝑟 ]𝑐

(3)

(1st agent risky and second agent safe)
𝑅̅ − 𝑝𝑟 𝑟 − [𝑝𝑟 (1 − 𝑝𝑠 ) − 𝜀𝑟𝑠 ]𝑐 + 𝑅̅ − 𝑝𝑠 𝑟 − [𝑝𝑠 (1 − 𝑝𝑟 ) − 𝜀𝑟𝑠 ]𝑐
(1) + (2) Yields,
24

𝜀𝑠𝑠 represents the 𝜀 when a safe agent partners with another safe agent

89

(4)

4𝑅̅ − 2𝑝𝑠 𝑟 − 2𝑝𝑟 𝑟 − 2𝑝𝑠 (1 − 𝑝𝑠 )𝑐 − 2𝑝𝑟 (1 − 𝑝𝑟 )𝑐 + 2𝜀𝑠𝑠 𝑐 + 2𝜀𝑟𝑟 𝑐

(5)

And (3) + (4) yields
4𝑅̅ − 2𝑝𝑠 𝑟 − 2𝑝𝑟 𝑟 − 2𝑝𝑠 (1 − 𝑝𝑟 )𝑐 − 2𝑝𝑟 (1 − 𝑝𝑠 )𝑐 + 2𝜀𝑠𝑟 𝑐 + 2𝜀𝑟𝑠 𝑐
(6). this implies that we need
−2𝑝𝑠 (1 − 𝑝𝑠 )𝑐 − 2𝑝𝑟 (1 − 𝑝𝑟 )𝑐 +

2𝜀𝑠𝑠 𝑐

+

2𝜀𝑟𝑟 𝑐 ≥ −2𝑝𝑠 (1 − 𝑝𝑟 )𝑐 − 2𝑝𝑟 (1 − 𝑝𝑠 )𝑐 +

2𝜀𝑠𝑟 𝑐 + 2𝜀𝑟𝑠 𝑐
After some algebra and using the assumption about the form of the correlation parameter 𝜀 in
this second case, we can have that, (𝑝𝑠 − 𝑝𝑟 )2 − Σ ≥ 0
⇔

where Σ = 𝜀𝑠𝑟 + 𝜀𝑟𝑠 − 𝜀𝑠𝑠 − 𝜀𝑟𝑟 .

Σ ≤ 𝑣̃𝑝𝑠 (1 − 𝑝𝑟 ) + 𝑣̃𝑝𝑟 (1 − 𝑝𝑠 ) − 𝑣̃𝑝𝑠 (1 − 𝑝𝑠 ) − 𝑣̃𝑝𝑟 (1 − 𝑝𝑟 )

=𝑣̃(𝑝𝑠 − 𝑝𝑟 )2 .Thus,

homogenous matching obtains. Proof of Lemma 1 follows similarly by substituting 𝜀 in place of
𝜀𝑠𝑟 , 𝜀𝑟𝑠 , 𝜀𝑠𝑠 , 𝜀𝑟𝑟 .
Proof of Lemma 2
This is true because 𝑅̅ − 𝑝𝑠 𝑟 − [𝑝𝑠 (1 − 𝑝𝑠 ) − 𝜀 ]𝑐 < 𝑅̅ − 𝑝𝑟 𝑟 − [𝑝𝑟 (1 − 𝑝𝑟 ) − 𝜀 ]𝑐
⇔ (𝑝𝑠 − 𝑝𝑟 )(𝑝𝑠 + 𝑝𝑟 − 1)𝑐 < (𝑝𝑠 − 𝑝𝑟 )𝑟 Which is true since (𝑝𝑠 + 𝑝𝑟 − 1) < 1 and 𝑐 ≤ 𝑟
Proof of Proposition 1
Max

𝑌̅ = 𝑅̅ − 𝑝𝑠 𝑟 − [𝑝𝑠 (1 − 𝑝𝑠 ) − 𝜀 ]𝑐 subject to
0 ≤ 𝑐 ≤ 𝑟, Monotonicity

⌊𝑚⌋

𝑝̅ 𝑟 + ̅̅̅̅̅̅̅̅̅̅̅
𝑝(1 − 𝑝)𝑐 − 𝜀𝑐 ≥ 𝜌

Lender break even or zero profit constraint ⌊𝜇⌋

Where m and 𝜇 are the Lagrange multipliers respectively
90

The first order conditions are
⌊𝑟⌋: −𝑝𝑠 + 𝜇𝑝̅ + 𝑚 = 0
̅̅̅̅̅̅̅̅̅̅̅
⌊𝑐⌋: −𝑝𝑠 (1 − 𝑝𝑠 ) + 𝜀 + 𝜇𝑝(1
− 𝑝) − 𝜀𝜇 − 𝑚 = 0
Solve to get
𝜇=

𝑝𝑠 (2−𝑝𝑠 )−𝜀
̅̅̅̅̅̅̅̅̅̅
𝑝(2−𝑝)−𝜀

> 0 and 𝑚 =

have the solutions as

𝜃(𝑝𝑠 −𝑝𝑟 )(𝑝𝑠 𝑝𝑟 −𝜀)
̅̅̅̅̅̅̅̅̅̅
𝑝(2−𝑝)−𝜀

> 0 if we assume 𝑝𝑠 𝑝𝑟 > 𝜀 in which case we

𝜌

𝑟 = 𝑐 = ̅̅̅̅̅̅̅̅̅̅

𝑝(2−𝑝)−𝜀

The remaining affordability constraint is satisfied if
𝜌

𝑅𝑠 ≥ 𝑟 + 𝑐 ⇔ 𝑅𝑠 ≥ 2𝑟 = 2 ̅̅̅̅̅̅̅̅̅̅

𝑝(2−𝑝)−𝜀

𝜌

⇔ 𝑅𝑠 𝑝𝑠 ≥ 2𝑝𝑠 ̅̅̅̅̅̅̅̅̅̅

𝑝(2−𝑝)−𝜀

⇔

𝑅̅
𝜌

2𝑝

𝑠
≥ ̅̅̅̅̅̅̅̅̅̅

𝑝(2−𝑝)−𝜀

2𝑝

𝑠
⟹ 𝐺 ≥ ̅̅̅̅̅̅̅̅̅̅

𝑝(2−𝑝)−𝜀

To attract the safe type, we need the expected payoff from the project to exceed the outside
option. The condition is the inequality below
𝑅̅ − 𝑝𝑠 𝑟 − [𝑝𝑠 (1 − 𝑝𝑠 ) − 𝜀 ]𝑟 ≥ 𝑢̅
⟹𝑁≥

Now,

𝑝𝑠 (2−𝑝𝑠 )−𝜀
̅̅̅̅̅̅̅̅̅̅−𝜀
𝑝(2−𝑝)

when

affordability

is

an

issue,

(when

𝑝𝑠
𝑝̅

The problem is
Max 𝑌̅ = 𝑅̅ − 𝑝𝑠 𝑟 − [𝑝𝑠 (1 − 𝑝𝑠 ) − 𝜀 ]𝑐 subject to the following constraints
𝑅𝑠 ≥ 𝑟 + 𝑐

⌊𝜆⌋
91

2𝑝

𝑠
≤ 𝐺 < ̅̅̅̅̅̅̅̅̅̅

𝑝(2−𝑝)−𝜀

)

𝑝̅ 𝑟 + ̅̅̅̅̅̅̅̅̅̅̅
𝑝(1 − 𝑝)𝑐 − 𝜀𝑐 ≥ 𝜌

⌊𝜇⌋

⌊𝑟⌋: −𝑝𝑠 − 𝜆 + 𝜇𝑝̅ = 0
̅̅̅̅̅̅̅̅̅̅̅
⌊𝑐⌋: −[𝑝𝑠 (1 − 𝑝𝑠 ) − 𝜀] − 𝜆 + 𝜇𝑝(1
− 𝑝) − 𝜀𝜇 = 0
Solve to obtain the Lagrange multipliers as

𝜇=

𝑝𝑠2
̅̅̅̅
2 +𝜀)
(𝑝

> 0 And 𝜆 =

𝜃(𝑝𝑠 −𝑝𝑟 )(𝑝𝑠 𝑝𝑟 −𝜀)
̅̅̅̅
2 +𝜀)
(𝑝

> 0 as before, if we assume 𝑝𝑠 𝑝𝑟 > 𝜀,

The contract can be derived as

(𝑟, 𝑐) = {𝜌

̅̅̅̅̅̅̅̅̅̅−𝜀]𝐺
𝑝𝑠 −[𝑝(1−𝑝)
𝑝̅ 𝐺 −𝑝
, 𝜌 ̅̅̅̅2 𝑠 }
̅̅̅̅
2 +𝜀)
𝑝 (𝑝
𝑝 (𝑝 +𝜀)
𝑠

𝑠

To attract safe borrowers, we can show that we need the following condition,

𝑁≥

𝑝𝑠2 +𝜀
̅̅̅̅
2 +𝜀)
(𝑝

−[

𝜃(𝑝𝑠 −𝑝𝑟 )(𝑝𝑠 𝑝𝑟 )
]𝐺
̅̅̅̅
2 +𝜀)
𝑝 (𝑝
𝑠

Constant correlation coefficient
𝜀(𝑝𝑖 , 𝑝𝑗 ) = 𝑣̃ ∗ min{𝑝𝑖 (1 − 𝑝𝑗 ), 𝑝𝑗 (1 − 𝑝𝑖 )}
Let 𝑡̃ be the correlation coefficient between two projects with success probabilities 𝑝𝑖 and 𝑝𝑗 .
By definition, 𝑡̃ =

𝐶𝑜𝑣 (𝑌𝑖 ,𝑌𝑗 )
𝜎𝑖 𝜎𝑗

where 𝑌𝑖 , 𝑌𝑗 are the Bernoulli random variables associated with the

project returns and 𝜎𝑖 and 𝜎𝑗 are the standard deviations.

92

𝑡̃ =

𝐶𝑜𝑣 (𝑌𝑖 ,𝑌𝑗 )
𝜎𝑖 𝜎𝑗

=

𝐸(𝑌𝑖 𝑌𝑗 )−𝐸(𝑌𝑖 )𝐸( 𝑌𝑗 )
√𝑝𝑖 (1−𝑝𝑖 )√𝑝𝑗 (1−𝑝𝑗 )

(𝑝𝑖 𝑝𝑗 + 𝜀−𝑝𝑖 𝑝𝑗 )𝑌𝑖 𝑌𝑗

=

𝜀

=

𝑌𝑖 √𝑝𝑖 (1−𝑝𝑖 ) 𝑌𝑗 √𝑝𝑗 (1−𝑝𝑗 )

=

√𝑝𝑖 (1−𝑝𝑖 )√𝑝𝑗 (1−𝑝𝑗 )

𝑣̃𝑝𝑖 (1−𝑝𝑗 )
√𝑝𝑖 (1−𝑝𝑖 )√𝑝𝑗 (1−𝑝𝑗 )

The last equality comes from using 𝜀=𝑣̃𝑝𝑖 (1 − 𝑝𝑗 ) (assuming 𝑝𝑖 ≤ 𝑝𝑗 without loss of generality.
Hence for homogeneous projects (𝑝𝑖 = 𝑝𝑗 ) we have
𝑡̃ = 𝑣̃. Hence 𝑣̃ is just the correlation coefficient.
For non-homogeneous projects,

⟺ 𝑡̃ =

𝑣̃𝑝𝑖 (1−𝑝𝑗 )
√𝑝𝑖 (1−𝑝𝑖 )√𝑝𝑖 (1−𝑝𝑗 )

⟺ 𝑡̃ =

𝑣̃𝑝𝑖 (1−𝑝𝑗 )

=

𝑣̃𝑝𝑖 (1−𝑝𝑗 )
√𝑝𝑗 (1−𝑝𝑖 )√𝑝𝑖 (1−𝑝𝑗 )

√𝑝𝑖 (1−𝑝𝑖 )√𝑝𝑗 (1−𝑝𝑗 )

=

𝑣̃ √𝑝𝑖 (1−𝑝𝑗 )

so that,

√𝑝𝑗 (1−𝑝𝑖 )

𝑡̃ is written generally as 𝑡̃ = 𝑣̃𝑣̅ where 𝑣̅ = 𝑚𝑖𝑛 {

√𝑝𝑖 (1−𝑝𝑗 )
√𝑝𝑗 (1−𝑝𝑖 )

,

√𝑝𝑗 (1−𝑝𝑖 )

} is the maximum

√𝑝𝑖 (1−𝑝𝑗 )

correlation possible between two projects.
Thus 𝑣̃, is a fraction of the maximum correlation possible between the two project returns. In
particular, for homogenous groups for which 𝑝𝑖 = 𝑝𝑗 , the maximum correlation possible is 1
hence 𝑣̃ is the correlation coefficient as a straight forward calculation shows.

93

Proof of Proposition 2
Consider for instance the condition 𝑁 ≥

Let 𝑀 =

𝑝𝑠 (2−𝑝𝑠 )−𝜀
.
̅̅̅̅̅̅̅̅̅̅
𝑝(2−𝑝)−𝜀

𝜀 yields,

𝑑𝑀
𝑑𝜀

Secondly,

=

𝑑𝐶2
𝑑𝜀

which is obtained under full affordability.

Differentiating M, the right hand side of the above inequality with respect to

𝜃(𝑝𝑠 −𝑝𝑟 )(2−𝑝𝑠 −𝑝𝑟 )
̅̅̅̅̅̅̅̅̅̅−𝜀]2
[𝑝(1−𝑝)

=

𝑝𝑠 (2−𝑝𝑠 )−𝜀
̅̅̅̅̅̅̅̅̅̅−𝜀
𝑝(2−𝑝)

2𝑝𝑠
2
̅̅̅̅̅̅̅̅̅̅
[𝑝(1−𝑝)−𝜀]

>0

which means that, M increases in 𝜀 and

>0

Thus, the parameter space for fully efficient lending reduces as correlation increases. The
boundary of the parameter space comprises of a horizontal part and a negatively sloped part
which starts from (𝐶1 , 𝐵1 ) and gets to its floor at (𝐶2 , 𝑀). The entire boundary of the efficiency
parameter space rotates upward with the introduction of correlation.
Proof of Proposition 3
Max 𝑅̅ − 𝑝𝑠 𝑟 − [𝑝𝑠 (1 − 𝑝𝑠 ) − 𝑣̃𝑝𝑠 (1 − 𝑝𝑠 ) ]𝑐 subject to
0 ≤ 𝑐 ≤ 𝑟,

𝑅𝑠 ≥ 𝑟 + 𝑐 , and 𝑝̅ 𝑟 + ̅̅̅̅̅̅̅̅̅̅̅
𝑝(1 − 𝑝)(1 − 𝑣̃)𝑐 ≥ 𝜌. The rest follows as in proof of

proposition 1.
Proposition 4
Consider for instance the condition 𝑁 ≥

[𝑝𝑠 +𝑝𝑠 (1−𝑝𝑠 )(1−𝑣̃)]
̅̅̅̅̅̅̅̅̅̅] ,
[𝑝̅ +(1−𝑣̃)𝑝(1−𝑝)

affordability.

94

which is obtained under full

Let 𝑀′ =

[𝑝𝑠 +𝑝𝑠 (1−𝑝𝑠 )(1−𝑣̃)]
̅̅̅̅̅̅̅̅̅̅]
[𝑝̅ +(1−𝑣̃)𝑝(1−𝑝)

respect to 𝑣̃ yields,

Secondly,

𝑑𝐶′2
𝑑𝑣̃

=

𝑑𝑀′
𝑑𝑣̃

=

. Differentiating 𝑀′ the right hand side of the above inequality with
𝜃𝑝𝑟 𝑝𝑠 (𝑝𝑠 −𝑝𝑟 )
̅̅̅̅̅̅̅̅̅̅]2
[𝑝̅ +(1−𝑣̃)𝑝(1−𝑝)

2𝑝𝑠 ̅̅̅̅̅̅̅̅̅̅
𝑝(1−𝑝)
̅̅̅̅̅̅̅̅̅̅]2
[𝑝̅ +(1−𝑣̃)𝑝(1−𝑝)

> 0 which means that, 𝑀′ increases in 𝑣̃

>0

Thus, the parameter space for fully efficient lending reduces as correlation increases.
As in proof of proposition 2, the boundary of the parameter space comprises of a horizontal part
and a negatively sloped part which starts from (𝐶1 , 𝐵1 ) and gets to its floor at (𝐶2′ , 𝑀′ ). The
entire boundary of the efficiency parameter space rotates upward with the introduction of
correlation.
Lemma 6
We show this by comparing payoffs of the groups

𝐴 𝑠𝑎𝑓𝑒 , 𝐴𝑟𝑖𝑠𝑘𝑦 , 𝐵 𝑠𝑎𝑓𝑒 and 𝐵𝑟𝑖𝑠𝑘𝑦 . The

payoffs are as follows
𝐴 𝑠𝑎𝑓𝑒 : 𝑅̅ − 𝑝𝑠 𝑟 − [𝑝𝑠 (1 − 𝑝𝑠 )(1 − 𝑣̃) ]𝑐
𝐴𝑟𝑖𝑠𝑘𝑦 : 𝑅̅ − 𝑝𝑟 𝑟 − [𝑝𝑟 (1 − 𝑝𝑟 )(1 − 𝑣̃) ]𝑐
𝐵 𝑠𝑎𝑓𝑒 : 𝑅̅ − 𝑝𝑠 𝑟 − [𝑝𝑠 (1 − 𝑝𝑠 ) ]𝑐
𝐵𝑟𝑖𝑠𝑘𝑦 : 𝑅̅ − 𝑝𝑟 𝑟 − [𝑝𝑟 (1 − 𝑝𝑟 ) ]𝑐
By inspection, 𝐴 𝑠𝑎𝑓𝑒 > 𝐵 𝑠𝑎𝑓𝑒 and 𝐴𝑟𝑖𝑠𝑘𝑦 > 𝐵𝑟𝑖𝑠𝑘𝑦 since 𝑣̃ < 1. Also, 𝐵𝑟𝑖𝑠𝑘𝑦 > 𝐵 𝑠𝑎𝑓𝑒 hence
𝐵 𝑠𝑎𝑓𝑒 is the least well off in the group.

95

Proposition 5
𝑅̅ − 𝑝𝑠 𝑟 − [𝑝𝑠 (1 − 𝑝𝑠 ) ]𝑐

Maximize
Subject to

0 ≤ 𝑐 ≤ 𝑟, and 𝑝̅ 𝑟 + ̅̅̅̅̅̅̅̅̅̅̅
𝑝(1 − 𝑝)(1 − 𝑘𝑣̅ )𝑐 ≥ 𝜌

(When full affordability is feasible)

⌊𝑟⌋: −𝑝𝑠 + 𝜇𝑝̅ + 𝜆 = 0
̅̅̅̅̅̅̅̅̅̅̅
⌊𝑐⌋: −𝑝𝑠 (1 − 𝑝𝑠 ) + 𝜇𝑝(1
− 𝑝)(1 − 𝑘𝑣̅ ) − 𝜆 = 0
Solve to get; 𝜇 =

𝑝𝑠 (2−𝑝𝑠 )
̅̅̅̅̅̅̅̅̅̅(1−𝑘𝑣̃)]
[𝑝̅ +𝑝(1−𝑝)

> 0 and 𝜆 = 𝑝𝑠 − 𝜇𝑝̅ > 0 if 𝑣̃ <

𝜃𝑝𝑟 (𝑝𝑠 −𝑝𝑟 )
̅̅̅̅̅̅̅̅̅̅
𝑘𝑝(1−𝑝)

The contract is then derived as
𝑟=𝑐=

𝜌
̅̅̅̅̅̅̅̅̅̅
[𝑝̅ +𝑝(1−𝑝)(1−𝑘𝑣̃)]

Assuming 𝑣̃ <

𝜃𝑝𝑟 (𝑝𝑠 −𝑝𝑟 )
̅̅̅̅̅̅̅̅̅̅
𝑘𝑝(1−𝑝)

The remaining affordability constraint is satisfied as before if
𝐺≥

2𝑝𝑠

. To attract safe uncorrelated borrowers, we need

̅̅̅̅̅̅̅̅̅̅(1−𝑘𝑣̃)]
[𝑝̅ +𝑝(1−𝑝)

𝑅̅ − 𝑝𝑠 𝑟 − [𝑝𝑠 (1 − 𝑝𝑠 ) ]𝑐 > 𝑢̅
That is 𝑁 ≥

𝑝𝑠 (2−𝑝𝑠 )
[𝑝̅ +̅̅̅̅̅̅̅̅̅̅
𝑝(1−𝑝)(1−𝑘𝑣̃)]

When affordability is an issue, (when

𝑝𝑠
𝑝̅

≤𝐺<

2𝑝𝑠
̅̅̅̅̅̅̅̅̅̅(1−𝑘𝑣̃)]
[𝑝̅ +𝑝(1−𝑝)

)

We have the following constraints similar to section 3.6’s derivations.
𝑅𝑠 ≥ 𝑟 + 𝑐
96

𝑝̅ 𝑟 + ̅̅̅̅̅̅̅̅̅̅̅
𝑝(1 − 𝑝)(1 − 𝑘𝑣̃)𝑐 ≥ 𝜌

Solve first order conditions to get

𝜇=

The contract is derived as; (𝑟, 𝑐) = {𝜌

𝑝𝑠2
̅̅̅̅̅̅̅̅̅̅
[𝑝̅ −𝑝(1−𝑝)(1−𝑘𝑣̃)]

> 0 and 𝜆 = 𝜇𝑝̅ − 𝑝𝑠 > 0

̅̅̅̅̅̅̅̅̅̅]𝐺
𝑝𝑠 −[(1−𝑣̃𝑘)𝑝(1−𝑝)
𝑝̅ 𝐺 −𝑝𝑠
̅̅̅̅
̅̅̅̅
2
2 +(𝑣
̅̅̅̅̅̅̅̅̅̅] , 𝜌 𝑝𝑠 [𝑝
̅̅̅̅̅̅̅̅̅̅]}
̃𝑘)𝑝(1−𝑝)
𝑝𝑠 [𝑝 +(𝑣̃𝑘)𝑝(1−𝑝)

and to attract the

safe uncorrelated borrowers, we need

𝑁≥

𝑝𝑠2
̅̅̅̅
2 +(𝑣
̅̅̅̅̅̅̅̅̅̅)
̃𝑘)𝑝(1−𝑝)
(𝑝

−[

𝜃(𝑝𝑠 −𝑝𝑟 )𝑝𝑟 −𝑣̃𝑘𝑝(1−𝑝)
̅̅̅̅
2 +(𝑣
̅̅̅̅̅̅̅̅̅̅) ] 𝐺
̃𝑘)𝑝(1−𝑝)
(𝑝

Propositions 6
𝜃𝑝𝑟 (𝑝𝑠 −𝑝𝑟 )
Define 𝑘̅𝑎 = ̅̅̅̅̅̅̅̅̅̅[1+(1−𝑝
𝑝(1−𝑝)

𝜃𝑝𝑟 (𝑝𝑠 −𝑝𝑟 )
̅𝑏 =
and
𝑘
̅̅̅̅̅̅̅̅̅̅
[1−(1−𝑝
̃)]

𝑠 )(1−𝑣

. Clearly, 0 < 𝑘̅𝑎 < 𝑘̅𝑏 .

̃)]
𝑠 )(1−𝑣

𝑝(1−𝑝)

To show 𝑘̅𝑎 < 𝑘̅𝑏 < 1, it suffices to show 𝑘̅𝑏 < 1.
𝜃𝑝𝑟 (𝑝𝑠 −𝑝𝑟 )
𝑘̅𝑏 < 1 ⇔ ̅̅̅̅̅̅̅̅̅̅[1−(1−𝑝
𝑝(1−𝑝)

̃)]
𝑠 )(1−𝑣

< 1 ⟺ 𝜃𝑝𝑟 (𝑝𝑠 − 𝑝𝑟 ) < ̅̅̅̅̅̅̅̅̅̅̅
𝑝(1 − 𝑝)[1 − (1 − 𝑝𝑠 )(1 − 𝑣̃)]

̅̅̅2
⇔ −𝑣̃𝑝̅ < (1 − 𝑣̃)𝑝
The cutoff values 𝑘̅𝑎 and 𝑘̅𝑏 that we have in propositions 6 are obtained by comparing the 𝐵2
levels (for 𝑘̅𝑎 ) and the slopes of the negatively sloped portion of the efficient lending parameter
space boundary of the all-correlated, and that of the mixed-borrowers contracts (for 𝑘̅𝑏 ). Thus for
the cutoff value of 𝑘 (for 𝑘̅𝑎 ), we solve
value of 𝑘 ( for 𝑘̅𝑏 ), we solve

𝑝𝑠 (2−𝑝𝑠 )
̅̅̅̅̅̅̅̅̅̅(1−𝑘𝑣̃)]
[𝑝̅ +𝑝(1−𝑝)

𝑝𝑠
𝑝𝑠 (2−𝑝𝑠 )
− ̅̅̅̅̅̅̅̅̅̅̅
̅ [𝑝
𝑝
̅ +𝑝(1−𝑝)(1−𝑘𝑣
̃)]
2𝑝𝑠
𝑝
− ̅𝑠
̅̅̅̅̅̅̅̅̅̅̅(1−𝑘𝑣
̅ +𝑝(1−𝑝)
̃)] 𝑝
[𝑝

<

that the functions intersect once.

97

<

[𝑝𝑠 +𝑝𝑠 (1−𝑝𝑠 )(1−𝑣̃)]
̅̅̅̅̅̅̅̅̅̅] ,
[𝑝̅ +(1−𝑣̃)𝑝(1−𝑝)

̃)]
𝑝𝑠 [𝑝𝑠 +𝑝𝑠 (1−𝑝𝑠 )(1−𝑣
−
̅̅̅̅̅̅̅̅̅̅̅]
̅
𝑝
̅ +(1−𝑣
̃)𝑝(1−𝑝)
[𝑝
2𝑝𝑠
𝑝
− ̅𝑠
̅̅̅̅̅̅̅̅̅̅̅(1−𝑣
̅ +𝑝(1−𝑝)
̃)] 𝑝
[𝑝

and for the cutoff

. For 𝑘𝜖(𝑘̅𝑎 , 𝑘̅𝑏 ), it is clear

We use the figures in section 3.9 to illustrate how these values are derived using the graphs in
Figures 3-2, 3-3 and 3-4.
The displayed graphs in Figures 3-2, 3-3 and 3-4 are for 𝑘 =0.25,

0.75 , 0.61

respectively, it can be seen from the graphs that, first of all, the lower bound of the gross excess
return (G) line for the Mixed borrowers line (M), is smaller than that of the “All correlated” line
(C). This means, we can have only 3 scenarios as 𝑘 increases. And the three scenarios are
depicted in the graphs. In figure 3-2, the efficient lending parameter space boundary of C lies
completely above that of M and the reverse of this is the case in figure 3-3. In figure 3-4, the line
for M crosses that of C and later lies above of it. We can see that, at low values of 𝑘, C lies
above and as k increases, M rises, crosses initially and is steeper than C. Then with further
increases in 𝑘, M lies above of C and is less steep. Thus, C lying above M is sufficient condition
for pooling of borrowers. M lying above is not sufficient to call for separation of borrowers
because, we can have the case where k=0.61 for instance, and separation is not fully justified.
However, as 𝑘 rises and M becomes less steep, we see M falling above C and remains there.
Thus the condition that the slope of M is less steep compared to the slope of C is sufficient
condition to call for separation. This ends the proof.

98

4

ESSAY 3: PREDICTORS OF THE CHOICE OF RURAL NONFARM ACTIVITY IN
TANZANIA

Introduction
Rural nonfarm economies (RNFEs) have been found to be a good source of income- booster for
rural dwellers in many countries and have contributed to the reduction of poverty. (FAO, 2008)
reports average nonfarm income shares in developing countries to be about “42% for Africa,
40% for Latin America and 32% for Asia”. Nonfarm income share of household income is
between 21 and 23% for Tanzania (Haggblade, Hazell, & Brown, 1989). The non-agricultural
participation rate is estimated to be about 75% in Ghana and 93% in Malawi (B. Davis et al.,
2010). More recently (Ackah, 2013; Owusu, Abdulai, & Abdul-Rahman, 2011) used data from
northern Ghana and found positive effects of nonfarm activities on income and food security
statuses of households. Participation in RNFEs, does not only provide a diversified source of
income to shield households against any negative shocks, they have also provided farmers, the
means to purchase inputs and or reinvest in their farms, although there is mixed findings in the
literature as to the use of income by farmers from nonfarm activities (Reardon, Crawford, &
Kelly, 1994).
Though a plethora of the literature establishes a connection between participating in rural
nonfarm activities and welfare or poverty statuses of rural households,

(Bezu, Barrett, &

Holden, 2012; Davis, 2004), there are barriers to entry so not all rural dwellers can participate.
While the barriers may differ depending on the activity, the high-yielding nonfarm activities
seem to be an option for the few rural dwellers who can afford or are wealthy. (Reardon, 1997),
and (Barrett, Reardon, & Webb, 2001) report that higher income households tend to have a

99

greater share of their income from nonfarm sector. This suggests that, higher income households
may be able to take advantage of the RNFE while low income households may not.
In this paper, we focus on the predictors of participation in the RNFE, and predictors of
choice between rural wage employment and self-employment conditional on participation.
Recent debates in development economics, on the relative desirability of wage employment and
self-employment options, motivate this paper. This debate is whether poor citizens are just
frustrated wage earners, or they are just frustrated entrepreneurs. It is important to note that selfemployed rural dwellers are not necessarily entrepreneurs especially if they engage in the RNFE
as a way of coping with risk temporarily unlike entrepreneurs who take risks and have plans for
expansion.
Wage employments tend to pay more than self-employment in rural areas and some
previous papers have established that, poor people opt for self-employment because there really
isn’t a choice for them. See for example (Contreras, Gillmore, & Puentes, 2017; Fields, 2014).
Contreras et al found double selection. In that, some workers choose themselves to be selfemployed while some are forced into self-employment because they couldn’t get access to wage
employments. As (Fields, 2014) notes, more research is needed to ascertain why households
choose their type of employments. Understanding why households choose the type of
employment is very vital to efforts aimed at eradicating extreme poverty in developing countries.
In my effort to learn some of the predictors of households’ choice of self or wage
employment upon entering the nonfarm economy, I restrict attention to three categories of
households. The first category has households that did not engage in any nonfarm activity. The
second category comprises of households that engaged in some nonfarm activity consisting of

100

only wage employment. The third category comprises of households that engaged in some
nonfarm activity consisting of only self-employment. Many previous works have only compared
participating households to non-participating ones making this paper different. In addition, I use
panel data to mitigate bias due to unobserved household characteristics. Many previous papers
have used cross-sectional data. I enrich the literature by providing newer evidence on the
hypothesized relationships between key variables and participation in the RNFE.
The results suggest wealth and land assets are key predictors of participation in the
RNFE. They also predict self-employment over rural wage employment, conditional on
participation. I find that, 1 standard deviation increase in non-agricultural wealth index, predicts
an average increase in the likelihood of participating in rural self-employment by approximately
16%. This result is about 5% for likelihood of participating in rural wage employment. 1
standard deviation increase in non-agricultural wealth index, predicts an average increase in the
likelihood of choosing self-employment over wage employment by approximately 5%. This
latter result is conditional on participation. Households appear to age out of self-employment or
that, rural self-employment appear to be engaged in by younger households.
There are always endogeneity issues to grapple with when studying participation in the
RNFE. That is, do wealthy households tend to participate or there is a reverse causality in the
sense that, households rather participate and later become wealthy. Households can be pulled
into the RNFE for asset accumulation purposes and households can also be pushed into the
RNFE as a way of coping with economic hardships. (Dimova & Sen, 2010) found that,
households in the Kagera region of Tanzania, rather diversify income sources in order to
accumulate assets and not for survival purposes. I find significant Fixed Effects results between
participation decision and wealth index variables suggesting that, reverse causality could be of
101

less concern. This could be so because it is not likely that within three years of the panel data,
households have been able to participate in the RNFE and accumulated enough wealth that can
provide enough within household variation needed for Fixed Effect estimations.
The RNFE needs to be studied in more details because aside being a source of asset
accumulation, it provides a suitable platform for empowering the poor and vulnerable. In
addition, rural dwellers are increasingly making efforts themselves to engage in nonfarm
activities to increase their incomes (Winters et al., 2009). Governments are able to help the rural
poor to overcome some of the barriers to entry in order to improve the livelihoods of poor
households (Barrett et al., 2001; Delgado & Siamwalla, 1997), if we know the predictors and the
barriers. Tanzania’s emerging economy makes its rural economy an area of interest especially
when recent studies have shown that rural dwellers adopt diversification as a pathway out of
poverty, (De Weerdt, 2010).
The rest of the paper is organized as follows. Section 4.2 gives a description of the data
used in the analysis. The estimation methods are presented in section 4.3. The results and
discussions are in section 4.4. Section 4.5 has the conclusion.

Data Description and Patterns of Rural Income Diversification in Tanzania
The data used in this paper is the Tanzanian rural income generating activities data obtained
from the FAO database. The FAO constructs this rural income generating activities data (RIGA)
using the LSMS data made available by the World Bank. I use the first three waves collected in
2008/2009, 2010/2011 and 2012/2013. The survey was designed to be nationally, urban/rural and
agro-ecologically representative. The households were clustered into 409 enumeration areas,
102

which comprised of 2063 rural and 1,202 urban households in the first round. I use only the rural
sample of the data that form a balanced panel. The World Bank database has the documentation
on the details of the sampling methods and procedures.
Table 4-1 describes the pattern of income diversification and some demographic
characteristics in the data set. The average household size in the three waves of the data is about
5 persons per household. These households on average also have about 25% of them headed by a
female. The size of the labor force is approximately less than three people per household.
Households in the rural areas do not appear to have land abundantly available because the
average land size owned by the households is about 2 hectares. In terms of pattern of income
diversification, about 22% of households had at least one of their members participate in
agricultural wage activities in the first wave. This fraction became almost 30% by 2012/2013 in
the third wave.
Non-agricultural wage on the other hand hovered around 23% in the second and third
waves of the data, which indicates that non-agricultural wage employment, and diversification is
still popular in rural Tanzania. In total, the percentage of households participating in the
agricultural sector in general fell from almost 99% in 2008/2009 to 92% in the 2012/2013 wave.
The pattern is similar with participation in livestock production.
Agricultural consisted about 70% of household income in the first wave and about 62% in
the third wave. Non-agricultural share of household income stood at about 30% in the first wave
and increased to about 38% in the third wave. These provide evidence that households in rural
Tanzania are diversifying their sources of income. The balanced panel of rural households
obtained for this analysis is 4500 and the standard deviations of our key variables namely nonagricultural wealth index, agricultural wealth index and land owned in hectares, from this 4500

103

balanced panel are 1, 0.95 and 2.446 respectively. Details of descriptive statistics for the
subsample of the waves used in the regressions are contained in tables 4-2, 4-3 and 4-4. The final
data used in this study vary depending on which regression is being run but only balanced panel
is used throughout.

Estimating Participation in RNFE
Because the participation in RNFE is a binary variable, I model the decision to participate in
nonfarm wage or self-employment using the linear probability Fixed Effect (LFE) and Correlated
Random Effects (CRE) models. A natural place to start in terms of predicting the probability of a
household participating in nonfarm wage or self-employment activities is to consider the linear
probability model with Fixed Effects. This model has the potential of providing good estimates
in addition to it not requiring a distributional assumption on the unobserved heterogeneity term
conditional on observables (Wooldridge, 2010). Specifically, it could give estimates that are free
of bias stemming from household-specific unobserved variables that are fixed between panel
waves, such as ability or underlying wealth. I report results from Correlated Random Effects and
those from Linear Fixed Effects probability models for comparison purposes.
More formally, we define the likelihood to participate in the RNFE following (Wooldridge,
2010) as follows
𝑦𝑖𝑡∗ = 𝑋𝑖𝑡 𝛽 + 𝑐𝑖 + 𝑒𝑖𝑡 , 𝑦𝑖𝑡 = 1[𝑦𝑖𝑡∗ > 0]

(1)

where 𝑦𝑖𝑡∗ is a latent variable that captures the marginal value of time to the household in the
RNFE. 𝑦𝑖𝑡 is the decision to participate or not in the RNFE. Then, the response probability for
the linear probability model is

104

𝑝𝑟𝑜𝑏(𝑦𝑖𝑡 = 1|𝑋𝑖𝑡 , 𝑐𝑖 ) = 𝑋𝑖𝑡 𝛽 + 𝑐𝑖 ,

(12)

The Probit model of this, takes the form
𝑝𝑟𝑜𝑏(𝑦𝑖𝑡 = 1|𝑋𝑖𝑡 , 𝑐𝑖 ) = Φ(𝑋𝑖𝑡 𝛽 + 𝑐𝑖 ) where 𝑋𝑖𝑡 in the models above is a vector of covariates
and 𝑐𝑖 is the unobserved heterogeneity term. The Correlated Random Effects model which
allows for dependence between 𝑐𝑖 , and 𝑋𝑖𝑡 takes the form
𝑝𝑟𝑜𝑏(𝑦𝑖𝑡 = 1|𝑋𝑖𝑡 , 𝑐𝑖 ) = Φ(𝑋𝑖𝑡 𝛽 + 𝑐𝑖 ) , 𝑎𝑖 /𝑋𝑖 ~ Normal (0,𝜎𝑎2 )
𝑐

25

𝑖

= 𝜓 + 𝑋̅𝑖 𝜉 + 𝑎𝑖 .

The full model can be written as follows
𝑝𝑟𝑜𝑏(𝑦𝑖𝑡 = 1|𝑋𝑖𝑡 , 𝑐𝑖 ) = Φ(𝜓𝑎 + 𝑋𝑖𝑡 𝛽𝑎 + 𝑋̅𝑖 𝜉𝑎 + 𝑐𝑖 ).
𝜓𝑎 , 𝛽𝑎 𝑎𝑛𝑑 𝜉𝑎 are obtained by multiplying 𝜓, 𝛽, 𝑎𝑛𝑑 𝜉 by (1 + 𝜎𝑎2 )−1/2 . In this case,
𝜓𝑎 , 𝛽𝑎 𝑎𝑛𝑑 𝜉𝑎 are estimated consistently in a pooled Probit of 𝑦𝑖𝑡 on 𝑋𝑖𝑡 , 𝑋̅𝑖 . I test for the joint
significance of the coefficients of 𝑋̅𝑖 as is normally done and find that they are jointly significant
which means CRE use is justified.

Results and Discussions
Tables 4-2, 4-3 and 4-4 show the summary statistics of covariates used in the regressions. The
tables contain information on per capita household expenditures among the three household
categories considered in this paper. The three categories as a reminder are households that did

25

I use the Mundlak version. The general form has just 𝑋𝑖𝑡 and not their averages.

105

not engage in any nonfarm activity, households that engaged in some nonfarm activity consisting
of only wage employment, and households that engaged in some nonfarm activity consisting of
only self-employment. For simplicity, I refer to the first two categories of household henceforth
as wage employment and self-employment households respectively.
From these tables, we see that the mean per capita expenditures are higher for nonagricultural wage employees in the balanced panel used. The mean per capita expenditure for the
rural self-employed category is also higher in all survey rounds (for the balanced panel sample
used) than for the category of households that did not participating in any non-agricultural
activity.
The expected sign of covariates in predicting participation in rural nonfarm economy or
in predicting what type of activity the household would choose when they participate, depends
on whether the household is diversifying for accumulation of assets or income growth, or the
household is diversifying income sources as a way of coping with risk. Some of the covariates
can have either positive or negative expected signs depending on the motive for diversification.
For example, if households engage in the RNFE as a way of coping with risks, then we would
expect households to not participate in the RNFE when they have enough assets (that is a
negative relationship between assets and participation decision) and vice versa.
Agricultural Wealth, Non-agricultural Wealth and Landholdings
While agricultural wealth makes it easy for households to expand their agriculture and may
probably not engage in nonfarm activities, non-agricultural wealth plays an important role in
predicting the participation decision of households in the RNFE. Once these households choose
to participate, the wealth status of the household can also predict whether they self-finance or can
106

access a loan and engage in self-employment or opt for rural wage employment. As mentioned in
the introduction, households may be pushed into the RNFE as a way of getting means to survive
and households could also be pulled into the RNFE as a way of accumulating assets. Availability
of land to a household is expected to aid the household see an enhancement in their agricultural
activities relative to households without land. Land however can assist households to participate
in non-agricultural rural self-employment by serving as a source of collateral security. Thus,
households with a good amount of land could have flexibility or ease when it comes to
participating in the RNFE. Conditional on participating in the RNFE, these households can also
choose self-employment because they can finance while those with little or no land may
participate in the RNFE as wage earners if credit markets are imperfect.
(Taylor & Yunez-Naude, 2000) found land variable and farm engagements to be
positively related using data from Mexico. They also found a negative relationship between land
and rural wage employment participation. In addition, (Winters et al., 2009) used data covering
15 countries and found that availability of land makes it easy for households to engage in
agricultural activities to improve their welfare. (Ackah, 2013) finds land endowment to be
negatively correlated with RNFE participation.
Table 4-5 shows the results of predictors of participation in rural non-agricultural wage
employment. I compare rural dwellers who participated in rural wage employment, with the
group that did not participate in any non-agricultural employment. In table 4-6, I predict
participation in the RNFE as a self-employed household. Here, I compare rural dwellers who
participated in self-employment, against the group that did not participate in any non-agricultural
employment. Then in Table 4-7, I show the predictors of choice between nonfarm selfemployment and rural wage employment conditional on participating in the RNFE. This table
107

enables us to learn about what type of rural nonfarm activity household choose to undertake once
they opt to participate in the RNFE. All the households in tables 4-5 to 4-7 that participated in
the RNFE may or may not have engaged in some agricultural activities.
While both Tables 4-5 and 4-6 show a positive relationship between non-agricultural
wealth index and participation in rural wage employment, or rural self-employment, they both
also show a negative relationship between agricultural wealth index and participation in rural
wage employment and in rural self-employment. The relationship between participating in
nonfarm self-employment and wealth, shown in Tables 4-5 and 4-6, appear to be in favor of the
“pull factors” literature. The “pull factors” literature says the RNFE or activities may rather pull
the wealthy or those with the means to participate in them (Haggblade, Hazell, & Reardon,
2010), as against the push factors argument, (Bardhan & Udry, 1999) which says households are
pushed to diversify as a way of coping with risk.
The Linear Fixed Effects estimates are highly significant for the non-agricultural wealth
index variable in Tables 4-5 and 4-6. Since Fixed Effects estimation uses within household
variations only, it must be that there is significant within household variations in the data and
thus gives credence to the estimates obtained from the Correlated Random Effects. The
significance of the Linear Fixed Effects estimates may also suggest that reverse causality should
be of less concern. It is less likely that households within three years of the survey rounds have
accumulated enough assets as a result of being in the RNFE such that there is within household
variation sufficient to yield significant Fixed Effect estimates. Thus the pull factors literature is
supported here too.

108

The amount of land the household owns is significant and negative in the rural wageemployment equation (Table 4-5) but not significant in the rural self-employment regression in
Table 4-6. The wealth variables may have absorbed the effects of the land variable in Table 4-6.
The results on the land variable carry the expected signs in both Tables 4-5 and 4-6. The Pooled
CRE results in Table 4-5 suggests that, 1 standard deviation increase in non-agricultural wealth
index, predicts an average increase in the likelihood of participating in rural wage-employment
by approximately 5%. On the other hand, 1 standard deviation increase in agricultural wealth
index, predicts an average decrease in the likelihood of participating in rural wage employment
by approximately 3%. Looking at Table 4-6, we can see that, 1 standard deviation increase in
non-agricultural wealth index, predicts an average increase in the likelihood of participating in
rural self-employment by approximately 16%. On the other hand, 1 standard deviation increase
in agricultural wealth index, predicts an average decrease in the likelihood of participating in
rural wage employment by approximately 3%. Altogether, the results seem consistent with the
“pull factors” side of the RNFE participation literature.
Table 4-7 shows that, for households that participate in the RNFE, non-agricultural
wealth is positively associated with choosing to be in rural self-employment relative to rural
wage employment. This is not surprising because we would expect households with enough nonagricultural wealth to self-finance and thus are likely to choose self-employment over wage
employment once they decide to participate in the RNFE. This result is also not so consistent
with reverse causality since wage earners earn more than self-employed households on average
(Tables 4-2, 4-3 and 4-4). From Table 4-7, we interpret the CRE results to mean, 1 standard
deviation increase in non-agricultural wealth index, predicts an average increase in the likelihood
of choosing rural self-employment over wage employment by approximately 5%.
109

Still on Table 4-7, the land variable is significantly and positively associated with
participating households choosing self-employment over rural wage employment. 1 standard
deviation increase in hectares of land owned by households, predicts an average increase in the
likelihood of choosing rural self-employment over wage employment by approximately 11%.
This could suggest that, households with more land choose to do self-employment over
wage employment because they can leverage their land assets to be able to self-finance their
business. This is consistent with the argument that some rural nonfarm opportunities may be the
sole preserve of asset owners since credit markets are imperfect in developing countries. The
Linear Fixed Effects results are highly significant indicating a significant variation within
households in terms of land holdings and that the result holds within households.
In sum, wealth or assets do not only predict participation, they predict whether
households would choose self-employment over wage employment at least in this data. In other
words, wealth or assets are key predictors of occupational choice by households conditional on
participating in the RNFE. It appears that the assets allow households may have the flexibility
and the wherewithal required to venture into some of the rural nonfarm activities. Households
that are able to self-finance would do so and those that cannot but have collateral securities
would be able to access funds to be able to finance their projects.
Demographic Characteristics and other Covariates
The demographics of a household can also predict participation in the RNFE. For example, the
household head’s gender can influence the ownership and distribution of resources, which in turn
influences the productivity and income levels of the household. Similarly, the household head’s
marital status and education level can influence the allocation of resources in the household.
110

Education is expected to make a household more eligible to participate in the RNFE.
Conditioned on participation, education is likely to make households choose high-paying wage
employment than self-employment in the rural areas. Some previous studies like (Lanjouw,
Quizon, & Sparrow, 2001) found that education aids participation in non-agricultural wage
employment and even self-employment in Tanzania. They also find proximity to infrastructure
as key determinant of nonfarm income of the peri-urban dweller in Tanzania. (Ackah, 2013)
finds human capital to be positively related to nonfarm participation.
From our results in Tables 4-5 and 4-6, average education level in the household is
significant with a positive sign. That is, higher education can predict participation in wage
employment and self-employment but weakly significant in the self-employment equation as
expected from the discussion above. The age of the head of the household variable has a negative
association with participation in rural self-employment but no significant relationship with
participation in rural wage employment as shown in Tables 4-5 and 4-6. The summary statistics
in Tables 4-2, 4-3 and 4-4 show that in all survey rounds, age of no-non-agricultural employment
category of rural dwellers is a bit higher than those in the self-employment, and the rural wageemployment categories. This suggests or predicts that, older people are less inclined to go into
rural self-employment relative to agricultural activities. This could be because such old people
would have garnered so much experience in farming so that, they are unwilling to move to the
RNFE. This may also help us to understand that younger people are increasingly getting
interested in the RNFE and are leaving the farm sector for the older people to undertake.
In Table 4-7, where we examine the choice of occupation upon participating in the
RNFE, the age of the household head variable has a zero and insignificant coefficient. This

111

suggests that age may not matter in what a household chooses to participate in conditional on
participation.
Married household head variable has a negative and insignificant association with
participation in rural wage employment and a negative and significant relationship with rural
self-employment in Tables 4-5 and 4-6. This could also be because, married household heads
tend to have enough hands to help on their farms and thus are less inclined to leave agriculture
entirely.
In Table 4-7, we see that for the households that participate in the RNFE, the likelihood
of choosing self-employment over wage employment is lower for household heads that are
married. This could be perhaps because these married heads of households think about food
security and how to provide for the family and thus would like to opt for an activity that provides
quick and guaranteed source of income.
The household size variable has a positive and significant relationship with participating
in rural wage and self-employment equations as shown in Tables 4-5 and 4-6. The bigger the size
of the household, the more likely it is that a member is qualified to or is able to participate in the
RNFE. This result is expected.
Although in Tables 4-5 and 4-6, we see no significant results in relation to distance from
the nearest market and participation in nonfarm employment, the summary statistics show that
those who are farther from the nearest market are those who did not engage in any nonagricultural employment for income. In Table 4-7 the result shows there is no relationship
between distances of plot to the nearest market and the choice of rural nonfarm activity.

112

Availability of electricity at the dwelling place of the household variable is significant in
the wage-employment regression in Table 4-5 and carries a positive sign. It also is significant
and negative in the self-employment regression in Table 4-6. Electricity expansion, availability
or coverage in the area may be taking place based on some criteria not observed in the data.
In Table 4-7 having electricity at the dwelling place has a negative relationship with the
likelihood of choosing rural self-employment over rural wage employment. This suggests that
wage employment may be lucrative than self-employment as portrayed by the summary statistics
in Tables 4-2, 4-3 and 4-4 or that wage employees have electricity provided for them by their
employers. It could also be that infrastructure presence is correlated with wage employment
opportunities. Proximity to infrastructure in general has been found in (Winters, Davis, & Corral,
2002) to be a predictor of participation in the RNFE in Mexico.

Conclusion
I study the predictors of participation in rural wage, and rural self-employment. I focus on three
categories of households and these are, households that did not engage in any nonfarm activity,
households that engaged in some nonfarm activity consisting of only wage employment,
households that engaged in some nonfarm activity consisting of only self-employment. I do not
include households that participated in both rural wage and self-employment in our analysis.
Aside studying the predictors of participating in the rural nonfarm economy, I also look at
predictors of the choice between rural self-employment and wage employment conditional on
participation.

113

Most previous studies have just compared participants in the rural nonfarm economy to
non-participants. I enrich the literature by providing newer evidence on the relationships between
key variables and households’ decisions to partake in the rural nonfarm economy. The
relationship between participating in nonfarm wage, self-employment and wealth, appear to be in
favor of the “pull factors” literature. The “pull factors” literature says the rural nonfarm activities
may rather pull the wealthy or those capable of venturing into it for asset accumulation purposes
(Haggblade et al., 2010), against the push factors argument, (Bardhan & Udry, 1999) which says
households are pushed to diversify as a way of coping with risk. Aside the wealth index
variables, the amount of land available to households plays a significant role in a household’s
decision to go into the rural nonfarm economy.
Putting all together, we can learn that wealth and land assets predict household’s
engagement in agriculture and their continuous stay or their ability to move to nonfarm activities.
The type of rural nonfarm activity they choose is also related to the availability of assets, thereby
siding with the “pull factors” argument about why households engage in income diversification.
Some significant Fixed Effects results suggest that, reverse causality in the form of households
gaining wealth after joining the rural nonfarm economy, could be of less concern.

114

APPENDIX

115

Appendix: Tables for Essay 3

Table 4-1: Rural Tanzania Income Generating Activities
Survey Round:
2008/2009
Variables
Household size
Age of head
Edu of head
Female head (%)
Married head (%)
Household labor force
Years of edu of labor force
Per capita consumption exp
Ag. wage emp income
Non-ag wage emp income
Self-employment income
Total household income
% Ag. wage participation
%non-Ag. wage
participation
% Ag. participation

Sample size
2055
2055
2025
2055
2055
2055
2053
2055
2055
2055
2055
2055
2055
2055

Mean or %
5.43
47.22
4.49
24.39
66.5
2.52
3.92
33501.73
22931.38
105820.2
215522
768844.9
21.77
14.67

2010/2011
2012/2013
Means (TZS) or Percentages (%)
Sample Size Mean or %
Sample size
2576
5.48
3144
2575
47.60
3143
2563
4.67
3127
2489
25.61
3144
2576
55.05
3144
2576
2.59
3144
2576
3.40
2980
2576
517302.5
3144
2622
43942.35
3209
2622
245225.9
3209
2622
347306.4
3209
2622
1984343
3209
2622
27.77
3209
2622
22.56
3209

2055

98.51

2622

116

94.322

3209

Mean or %
5.26
47.00
4.89
25.79
60.58
2.51
5.71
708717
103343.7
379923.4
481059.7
1763192
29.87
22.82
92.22

Table 4-1 (cont’d)
Agricultural(%)
On farm(%)
Non ag (%)
Nonfarm (%)
Ag wage
Non ag wage
Land owned(ha)
Household owns
dwellings

Share of Income Generating Activity
2050
70.22
2602
66.60
2050
65.94
2602
61.24
2050
29.78
2602
33.39
2050
19.24
2602
24.97
2050
4.28
2602
5.37
2050
6.52
2602
10.16
2055
1.59
2576
1.83
2055
91.89
2576
86.68

Survey weights applied.

117

3194
3194
3194
3194
3194
3194
3144
3144

61.70
54.08
38.30
25.86
7.63
10.92
1.81
84.64

Table 4-2: Means and Percentages (2008 Survey Data)
Wage
Employment
(N=139)
-0.15
Ag wealth index
0.55
Wealth index
43.89
Age of household head
0.76
Married household head
5.43
Household size
5.53
Average education of household
0.90
Land owned in hectares
0.12
Dwelling has electricity
Distance to nearest government school 0.33
4.24
Distance to nearest market
44664.80
Per capita expenditure
803921.10
Non ag wage employment income
0.00
Self-employment income
44562.59
Transfer income
611.51
Other income

SelfEmployment
(N=342)

No Non-Ag
Employment
(N=724)

Full
Sample
(N=1205)

-0.04
0.25
44.90
0.65
5.34
4.50
1.66
0.04
0.19
4.04
36570.19
0.00
717001.20
44732.28
0.01

-0.04
-0.27
49.82
0.65
5.27
3.41
1.47
0.02
0.10
3.80
30407.00
0.00
0.00
56120.44
546.96

-0.06
-0.03
47.74
0.66
5.31
3.96
1.46
0.03
0.15
3.91
33800.89
92734.46
203497.40
51555.05
399.17

Source: Author’s computation from survey data. Means are in Tanzanian Shillings.
per capita expenditure is the total expenditure of the household divided by the household size.
Wealth indexes are computed by the FAO using a broad range of assets owned by the household. A principal
component analysis is used in computing the indexes. The assets include television, radio, refrigerator, flooring of
the house, tractor, plough, etc (for agricultural wealth indexes) See (Wealth Index Mapping in the horn of AfricaFAO 2011)

118

Table 4-3: Means and Percentages (2010 Survey Data)
Wage
Employment
(N=159)
-0.22
Ag wealth index
0.06
Wealth index
43.11
Age of household head
0.51
Married household head
4.84
Household size
3.91
Average education of household
1.24
Land owned in hectares
0.11
Dwelling has electricity
Distance to nearest government school 0.26
2.97
Distance to nearest market
687218.40
Per capita expenditure
996955.50
Non ag wage employment income
0.00
Self-employment income
47185.64
Transfer income
5157.23
Other Income

SelfEmployment
(N=369)

No Non-Ag
Employment
(N=677)

Full
Sample
(N=1205)

-0.04
0.01
44.61
0.53
5.40
3.25
2.21
0.04
0.19
2.75
521153.30
0.00
841496.60
47641.84
12555.56

0.03
-0.29
50.68
0.59
4.98
2.87
2.01
0.03
0.18
2.67
441931.10
0.00
0.00
47994.34
1556.87

-0.03
-0.15
47.82
0.56
5.09
3.12
1.97
0.04
0.20
2.73
498556.60
131548.50
257686.50
47779.69
5400.00

Source: Author’s computation from survey data. Means are in Tanzanian Shillings.
Per capita expenditure is the total expenditure of the household divided by the household size.
Wealth indexes are computed by the FAO using a broad range of assets owned by the household. A principal
component analysis is used in computing the indexes. The assets include television, radio, refrigerator, flooring of
the house, tractor, plough, etc. (for agricultural wealth indexes) See (Wealth Index Mapping in the horn of AfricaFAO 2011)

119

Table 4-4: Means and Percentages (2012 Survey Data)

Ag wealth index
Wealth index
Age of household head
Married household head
Household size
Average education of household
Land owned in hectares
Dwelling has electricity
Distance to nearest government school
Distance to nearest market
Per capita expenditure
Non Ag wage employment income
Self-employment income
Transfer income
Other income

Wage
Employment
(N=137)

SelfEmployment
(N=363)

No Non-Ag
Employment
(N=705)

Full
Sample
(N=1205)

-0.16
0.21
43.70
0.68
5.18
4.40
1.40
0.17
0.06
2.45
795294.10
1824722.00
0.00
74595.13
51576.64

0.05
0.10
45.31
0.61
5.77
3.53
2.34
0.08
0.05
1.64
684298.30
0.00
1237116.00
67399.09
86457.30

-0.04
-0.27
51.76
0.62
5.09
3.00
2.16
0.03
0.08
2.27
562346.20
0.00
0.00
101726.70
14936.17

0.03
-0.10
48.90
0.63
5.31
3.32
2.13
0.06
0.07
2.10
625568.10
207458.00
372674.60
88301.00
40647.30

Source: Author’s computation from survey data. Means are in Tanzanian Shillings.
per capita expenditure is the total expenditure of the household divided by the household size.
Wealth indexes are computed by the FAO using a broad range of assets owned by the household. A principal
component analysis is used in computing the indexes. The assets include television, radio, refrigerator, flooring of
the house, tractor, plough, etc (for agricultural wealth indexes) See (Wealth Index Mapping in the horn of AfricaFAO 2011)

120

Table 4-5: Predictors of participation in Non-Ag Wage Employment
(1)
Linear Fixed
Effects
0.043**
(0.021)
-0.026
(0.019)
-0.001
(0.001)
-0.009
(0.027)
0.007
(0.005)
0.033***
(0.006)
-0.014**
(0.006)
0.154*
(0.080)
0.003
(0.012)
0.000
(0.001)
Yes

Non-agricultural wealth index
Agricultural wealth index
Age of household head
Head of household married
Household size
Average years of education in household
Land owned in hectares
Dwelling has electricity
KM from community to nearest government
primary school
KM from plot to nearest market
Year dummy (2008 base year)
Time average of covariates
Constant

No
0.084
(0.075)
1464
0.105

Observations
R2

(2)
CRE
Probit
Pooled
0.053***
(0.015)
-0.029*
(0.016)
-0.001
(0.001)
-0.018
(0.020)
0.011**
(0.005)
0.028***
(0.006)
-0.027***
(0.007)
0.102**
(0.049)
0.000
(0.014)
0.000
(0.001)
Yes

(3)
CRE
Probit
Joint
0.050***
(0.013)
-0.029**
(0.014)
-0.001
(0.001)
-0.016
(0.020)
0.011**
(0.005)
0.028***
(0.006)
-0.027***
(0.008)
0.104**
(0.049)
0.001
(0.016)
0.000
(0.001)
Yes

Yes

Yes

1464

1464

Standard errors are in parentheses and are clustered at the household. Households with Non-agricultural wage
employment are compared to those with no non-agricultural employment
*
p < 0.10, ** p < 0.05, *** p < 0.01

121

Table 4-6: Predictors of Participation in Non-Ag Self Employment
(1)
Linear Fixed
Effects
0.167***
(0.021)
-0.035**
(0.015)
-0.006***
(0.001)
-0.053**
(0.025)
0.014***
(0.005)
0.009
(0.006)
0.004
(0.005)
-0.155*
(0.079)
0.009
(0.011)
-0.001
(0.001)
Yes
No
0.529***
(0.067)
2589
0.103

Non-agricultural wealth index
Agricultural wealth index
Age head of household
Head of household married
Household size
Average years of education in household
Land owned in hectares
Dwelling has electricity
KM from community to nearest government
primary school
KM from plot to nearest market
Year dummy (2008 base year)
Time average of covariates
Constant
Observations
R2

(2)
CRE
Probit
Pooled
0.163***
(0.021)
-0.031***
(0.011)
-0.006***
(0.001)
-0.076***
(0.020)
0.016***
(0.005)
0.010
(0.006)
0.003
(0.005)
-0.118*
(0.065)
0.009
(0.010)
-0.001
(0.001)
Yes
Yes

(3)
CRE
Probit
Joint
0.163***
(0.015)
-0.032***
(0.011)
-0.006***
(0.001)
-0.073***
(0.019)
0.016***
(0.005)
0.010*
(0.006)
0.003
(0.005)
-0.130**
(0.058)
0.009
(0.012)
-0.001
(0.001)
Yes
Yes

2589

2589

Standard errors are in parentheses and are clustered at the household. Households with Non-agricultural selfemployment are compared to those with no non-agricultural employment
*
p < 0.10, ** p < 0.05, *** p < 0.01

122

Table 4-7: Predictors of Choice of Non-Agricultural Self-Employment
(1)
Linear Fixed
Effects
0.099***
(0.038)
-0.001
(0.043)
-0.000
(0.002)
-0.070
(0.063)
-0.006
(0.014)
-0.007
(0.016)
0.023
(0.015)
-0.359**
(0.139)
-0.044**
(0.018)
0.000
(0.002)
Yes
No
0.864***
(0.173)
372
0.113

Non-agricultural wealth index
Agricultural wealth index
Age of household head
Head of household married
Household size
Average years of education in household
Land owned in hectares
Dwelling has electricity
KM from community to nearest government
primary school
KM from plot to nearest market
Year dummy (2008 base year)
Time average of covariates
Constant
Observations
R2

(2)
CRE
Probit
Pooled
0.054*
(0.029)
-0.045
(0.037)
-0.000
(0.002)
-0.089*
(0.050)
0.007
(0.016)
-0.004
(0.015)
0.047**
(0.023)
-0.212**
(0.102)
-0.040**
(0.020)
0.000
(0.002)
Yes
Yes

(3)
CRE
Probit
Joint
0.064***
(0.025)
-0.031
(0.039)
-0.000
(0.002)
-0.082*
(0.047)
0.005
(0.013)
-0.005
(0.012)
0.043*
(0.023)
-0.234**
(0.092)
-0.039*
(0.021)
0.000
(0.002)
Yes
Yes

372

372

Standard errors are in parentheses and are clustered at the household. hh denotes household. Households with Nonagricultural wage employment are compared to those with non-agricultural self employment. Households may have
some agricultural activities but not both wage and self-employment.
*
p < 0.10, ** p < 0.05, *** p < 0.01

123

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