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PAPERS ON AGRICULTURAL INSURANCE AND
FARM PRODUCTIVITY

By

Yanyan Liu

A DISSERTATION

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

DOCTOR OF PHILOSOPHY

Department of Agricultural Economics
Department of Economics

2006

ABSTRACT

PAPERS ON AGRICULTURAL INSURANCE AND
FARM PRODUCTIVITY

By

Yanyan Liu

This dissertation is composed of two distinct papers. The first one is a theoretical paper
on agricultural insurance pricing. The second paper studies model selection in stochastic
frontier analysis with an application to maize production in Kenya.

The first paper reviews existing agricultural insurance valuation models and provides
a new model. The new model takes explicit account of the non-diversifiable market risk
inherent in offering insurance contracts, and demonstrates how capital markets can facilitate
risk spreading and diversification. The analysis suggests that present value models may
provide appropriate insurance valuations in some circumstances, but the standard Black-
Scholes model has deficiencies for pricing agricultural insurance. Other existing methods
for pricing the market risk in agricultural insurance contracts are logically consistent and
potentially useful. However, the heterogeneous agent equilibrium model developed here is
easy to use, amenable to empirical estimation, and provides a simple and intuitive way to
value market risk in agricultural insurance contracts.

The second paper shows how to estimate the quantitative magnitude of partial effects
of exogenous firm characteristics on technical efficiency (along with their standard errors)

under a range of popular stochastic frontier model specifications. An RZ—type measure

is also derived to summarize the overall explanatory power of the exogenous factors on
firm inefficiency. The paper also applies a recently developed model selection procedure to
choose among alternative stochastic frontier specifications using data from household maize
production in Kenya. The magnitude of estimated partial effects of exogenous household
characteristics on inefficiency turns out to be very sensitive to model specification, and the
model selection procedure leads to an unambiguous choice of best model. Bootstrapping is
used to provide evidence on the size and power of the model selection procedure. The em-
pirical application also provides further evidence on how household characteristics influence

technical inefficiency in maize production in developing countries.

To my husband, Shanjun Li,
my parents, Guiying Liu and Shujun Chen,
my brother, Weiwei Chen,

and my grandparents, Honglan Li and Honglin Liu

iv

ACKNOWLEDGMENTS

I would like to thank my committee, Drs. Robert Myers (Co-Chair), Peter Schmidt (Co—
Chair), Steven Hanson, Jack Meyer, and Roy Black for providing me with valuable insights
in their respective areas of expertise. My deepest gratitude and respect go to Drs. Myers
and Schmidt for being wonderful mentors and for constant support throughout my PhD
program. I will be forever indebted to them.

My sincere thanks also go to my master’s adviser, Dr. Scott Swinton, for his genuine
support during my stay at Michigan State University, and to Dr. Roy Black who is always
willing to go out of his way to help me. My sincere thanks also extend to Dr. Steven
Hanson, who helped support my research program, and Dr. Eric Crawford, for his help and
support to my husband and me throughout our time at Michigan State. I would also like
to thank Dr. Thomas Jayne for generously providing me the data in the second paper of
my dissertation.

Thanks also to all my fellow graduate students for their friendship and encouragement.
A special mention goes to Ren Mu, Zhiying Xu, Wei Zhang, Christopher Wright, Denys
Nizalov, Lesiba Bopape, Antony Chapoto, Elliot Mghenyi, Kirimi Sindi and Gerald Nyam-
bane.

Last but not the least, I thank my husband, Shanjun Li, my parents, Guiying Liu and
Shujun Chen, my brother, Weiwei Chen, and my grandparents, Honglan Li and Honglin

Liu for their love and support.

TABLE OF CONTENTS

LIST OF TABLES ................................. viii

LIST OF FIGURES ................................ ix

1 INTRODUCTION ............................... 1
2 HOW SHOULD WE PRICE THE MARKET RISK IN AGRICUL-

TURAL INSURANCE CONTRACTS? .................. 4

2.1 Introduction .................................... 4

2.2 A Simple Insurance Model ............................ 5

2.3 Existing Approaches to Valuing Agricultural Insurance Contracts ...... 6

2.3.1 Present Value Models ........................... 6

2.3.2 Arbitrage—Based Option Pricing Models ................. 9

2.3.3 Lucas General Equilibrium Models ................... 11

2.3.4 The Chambers State-Contingent Approach ............... 13

2.4 An Alternative Valuation Model ......................... 14

2.4.1 Mutual Fund Model ........................... 15

2.4.2 Discussion ................................. 18

2.4.3 Insurance Industry Model ........................ 19

2.5 Simulation Results ................................ 21

2.6 Conclusions .................................... 25

MODEL SELECTION IN STOCHASTIC FRONTIER ANALYSIS:

MAIZE PRODUCTION IN KENYA .................... 27
3.1 Introduction .................................... 27
3.2 Stochastic Production Frontier Models ..................... 3O
3.2.l Alternative Model Specifications ..................... 32
3.3 Empirical Application .............................. 34
3.3.1 Data .................................... 35
3.3.2 Variables in the Production Frontier .................. 36
3.3.3 Exogenous Factors Affecting Efficiency ................. 38
3.4 Estimation Results from Competing Models .................. 40
3.5 Model Selection .................................. 46
3.5.1 Empirical Model Selection ........................ 46

vi

3.5.2 A Bootstrap Evaluation ......................... 47

3.6 Post-Estimation Analysis ............................. 50

3.7 Conclusion ..................................... 53
APPENDICES ................................... 54
A Derivation of Equation (2.2) ......................... 55
B Derivation of Equation (2.10) ........................ 56
C Derivation of Equation (2.13) ........................ 58
D Estimating Partial Effects of Exogenous Factors and their Standard Errors

for the General Model ............................. 59
E Results of Specifying Relevant Explanatory Variables .......... 62
BIBLIOGRAPHY ................................. 65

vii

LIST OF TABLES

2.1 Summary of the parameters in simulation model ................

3.1 Descriptive statistics for the variables in the production frontier .......
3.2 Descriptive statistics for the exogenous variables in the efficiency model . . .
3.3 Estimates for the production frontier in alternative models ..........
3.4 Estimates for the inefficiency components in alternative models ........
3.5 Correlation of efficiency estimates among alternative models .........
3.6 Partial effects of exogenous factors, evaluated at the sample mean ......
3.7 Average partial effects of EDUHIGH on E(—u,-|;r,-, zi), for the observations
within each of the four quartiles based on efficiency levels predicted in
KGMHLBC model ................................
3.8 Correlation of partial effects of EDUHIGH on E (—u,~|:r,-, 3,) among alterna-
tive models ....................................
3.9 Results of specification tests for model selection ................
3.10 Partial effects of the exogenous factors on E (—u,-|:r,-, 2i) and their 90% confi-
dence intervals based on bootstrap and the delta method in the KGMHLBC
model, evaluated at the sample mean ......................
3.11 Output elasticity with respect to inputs for local seed users and hybrid seed
users, evaluated at the sample means ......................

E.1 Specifying variables in the production frontier using OLS ...........
E.2 Tests results for specifying the exogenous factors in the efficiency component
E3 Tests results for specifying explanatory variables in the production frontier .

viii

22

38
40
43
44
44
45

47

50

50

62
62
63

2.1
2.2
2.3

3.1
3.2

LIST OF FIGURES

Insurance premiums versus interest rate .................... 24
Insurance premiums versus market risk factor ................. 24
Insurance premiums versus share of farm income in aggregate consumption . 25

Location of sample villages in Kenya ...................... 35
Kernel density estimate based on Battese and Coelli technical efficiency es-
timates ....................................... 51

ix

CHAPTER 1

INTRODUCTION

This dissertation deals with two important issues in agricultural production and risk man-
agement. The first paper is on how to price the market risk in agricultural insurance con-
tracts and the second paper identifies and quantifies the determinants of technical efficiency
in agricultural production using stochastic frontier analysis.

Agriculture is a highly risky industry. Producers are constantly faced with market price
risk as well as production risk caused by weather variations. Since its birth in 1938 in the
United States, agricultural insurance has played an increasingly significant role in risk man-
agement for farmers. In 2004, the coverage of agricultural insurance in the US was over 46
billion dollars and the premium subsidy was about 2.5 billion dollars. Although subject to
many criticisms, the contribution of agricultural insurance in promoting farmer welfare and
agricultural development is difficult to refute. Therefore, agricultural insurance programs
are being expanded both in the US. and other countries. Like all types of insurance, a cen-
tral question of agricultural insurance is how to price insurance contracts. In recent years,
traditional actuarial pricing methods have been gradually overtaken by financial valuation
models, particularly in academic research. Financial valuation models are considered supe-
rior to traditional actuarial pricing models because they take into account the role played

by diversification and portfolio management in determining the market price of the risk

embodied in insurance contracts.

The first paper on agricultural insurance has two objectives. The first is to review ex-
isting approaches to valuing agricultural insurance contracts and point out some of their
advantages and disadvantages. The second objective is to develop a new valuation model
which is more transparent about the way in which market risk is valued, and which there-
fore provides additional insights into the valuation of agricultural insurance contracts. The
new model takes explicit account of some important institutional features of agricultural
insurance contracting, such as the fact that only farmers can buy insurance and only in-
surance firms or the government offer insurance. Using simulation we also show why, and
under what conditions, alternative modeling approaches give very different answers to the
insurance valuation question.

The second paper in the dissertation is motivated by the widespread hunger problem in
Kenya. The rapidly growing population and declining growth rate of the agricultural sector
has resulted in a pressing problem of food security in Kenya. Maize is the primary staple
food in the Kenyan diet, and about 75% of farmers are engaged in maize production. Since
the area of arable land is stagnant, promoting farm productivity in the maize sub-sector
has attracted increased attention due to the significance of maize in food security. The
factors that possibly influence farm efficiency include infrastructure, education, informa—
tion technology, access to extension services, and so on. This paper aims to identify the
determinants of farm efficiency and to quantify the effects of these factors.

The methodology used is stochastic production frontier analysis allowing for specific ex-
ogenous influences on farm efficiency. The stochastic frontier model was first developed
in 1977 by two groups of researchers independently: Meeusen and van den Broeck in a
paper published in the International Economic Review, and Aigner, Lovell and Schmidt
in a paper published in Journal of Econometrics. Since then, it has been a major tool in
productivity analysis and widely used in various industries including agriculture, banking,

electric utilities, railways, and so on. Compared with traditional methods of production

function estimation, the stochastic frontier model relaxes the assumption that all producers
are successful in producing maximal possible output given their input levels. The distance
between the actual output and the maximal possible output is called technical inefficiency.
This inefficiency may exist due to explicit or implicit constraints faced by producers, or
simply because producers make mistakes in arranging their production activities. Thus,
the stochastic frontier model allows for differentiation in efficiency levels across producers.
By associating inefficiency with exogenous factors, the stochastic frontier model can be used
to identify the sources of inefficiency.

Besides addressing the empirical question of the determinants of technical efficiency in
maize production in Kenya, this paper also makes some methodological contributions to the
stochastic frontier literature. First, we provide a method for estimating the quantitative
magnitude of the partial effects of exogenous firm characteristics on firm inefficiency, show
how to put standard errors around these partial effects, and propose an Rz-type measure to
summarize the overall explanatory power of the exogenous factors on inefficiency. Second,
we show that while alternative models of the relationship between household characteristics
and technical inefficiency tend to provide the same direction of the influence of household
characteristics, the magnitudes of the partial effects on firm inefficiency are quite sensitive
to model selection. Third, we show how a recently developed model selection procedure
(Alvarez, Amsler, Orea, and Schmidt, 2006) can be used to choose among competing models.
A novel detail is that we use bootstrapping to provide evidence on the power and size of this
procedure. The model selection procedure gives an unambiguous choice of best model for
our application to Kenyan maize production. This is important because if different models
give different results, and we cannot distinguish statistically among the models, we do not
know which set of results should be used. But if we can pick a clearly best model, then the

fact that inferences and conclusions are sensitive to model selection is not such a problem.

CHAPTER 2

HOW SHOULD WE PRICE THE
MARKET RISK IN
AGRICULTURAL INSURANCE
CONTRACTS?

2. 1 Introduction

Valuation of agricultural insurance contracts has become an increasingly important issue
in recent years as the Risk Management Agency (RMA) continues to expand the range of
insurance products offered to farmers. Traditional farm-based multiple peril yield insur-
ance has been supplemented with area-based yield insurance, revenue insurance products
(both farm-based and area-based), and catastrophic risk coverage. The number of eligible
commodities has also expanded to include non-traditional and specialty crops, as well as
livestock products. Alternative approaches to valuing agricultural insurance contracts have
therefore come under increased scrutiny in both the academic literature and in the applied

construction of premium schedules for new insurance products (eg. [46]; [49]; [57]; [48];

I41]; [7]; [15])-

This paper has two objectives. The first is to review existing approaches to valuing
agricultural insurance contracts and point out some of their advantages and disadvantages.
The second objective is to develop a new valuation model which is more transparent about
the way in which market risk is valued, and which therefore provides additional insights
into the valuation of agricultural insurance contracts. The new model takes explicit account
of some important institutional features of agricultural insurance contracting, such as the
fact that only farmers can buy insurance and only insurance firms or the government offer
insurance. The new model also accounts for the presence of uninsurable background risk
which is a pervasive feature of most agricultural insurance environments. Using simulation
we also show why, and under what conditions, alternative modeling approaches give very
different answers to the insurance valuation question. The intention is to improve our
understanding of the theoretical issues underlying agricultural insurance valuation, as well
as to provide a new approach that may eventually lead to improved applied ratings of
agricultural insurance contracts.

To begin, we fix ideas by outlining a simple generic agricultural insurance contract. Next
we review existing approaches to valuing the generic contract and discuss their advantages
and disadvantages. Then our alternative model is presented, followed by simulation results
which highlight model differences and show why different valuation models can lead to such

different results. Finally there are some concluding comments.

2.2 A Simple Insurance Model

To fix ideas we focus on index-based farm revenue insurance contracts. Farmers are indexed
by 2' = 1,2, . . . ,ny and endowed with random farm income Y,- which may be correlated
across farms. There are m regional farm revenue indices Zj for j = 1, 2, . . . ,m that are

correlated with the Y,- and which form the basis for insurance contracts. If m = my and

Z j = Y,- for all 2' = j then the insurance is individual farm revenue insurance. But generally
m < fly and the indices will be based on regional or area farm revenue as opposed to
individual farm revenue. This index approach makes the insurance more consistent with
actual area-based farm revenue insurance programs and eliminates the need to account for
moral hazard or adverse selection.1

An insurance contract on Z j is a contingent claim that costs Pj (G3) and pays off V]- (0]) ==
max(Gj — Z j,0), where 03- is a guaranteed level of regional farm revenue and Pj(Gj) is

a premium schedule. We are interested in valuing Pj(G'J-) taking proper account of the

non-diversifiable market risk embodied in offering insurance contracts.

2.3 Existing Approaches to Valuing Agricultural In-
surance Contracts

We now examine several existing approaches to valuing agricultural insurance within the

context of the simple insurance model of the preceding section.

2.3.1 Present Value Models

The simplest and still most common approach to valuing agricultural insurance contracts

is to set the premium schedule equal to the present value of the expected indemnity:
02'
PJ-(Gj) = fiEIleGfll = i3 [0 (G.- — zj>f<zj>dzj (2.1)

where 6 = 1/ (1 + r) is a discount rate based on the risk-free rate of interest 7‘, E is
expectation conditional on information available when the contract is valued (sold), and
f () is the density function for the relevant insurance index. For most US. agricultural

insurance contracts administered through the RMA, premium payment is not required

 

1That is, we assume individual farms cannot influence regional revenues and that the prolmbility distri-
bution of regional revenues is known to all.

until the index is realized and claims are paid.2 In this case we would set [3 = 1 because no
discounting would be necessary.

The present value model (2.1) has the advantage of being easy to calculate under a
wide range of potential probability distributions for ZJ- because it is straightforward to
evaluate the integral in (2.1) numerically for quite flexible probability distributions. Not
surprisingly, this has led to a significant amount of research focused on what the right
probability distribution for the integration should be ([35]; [34]; [19]; [25]). The present
value model also has the advantage that the formula is independent of the returns to other
assets in the economy (other than T).

For ease of comparison with other models outlined below, we evaluate the integral in (2.1)
assuming the insurance index is lognormally distributed. Using results from the appendix

of Rubinstein ([43]) (reproduced in appendix A) this gives:

-—- - .2 9-—u-—02-
p G,N(M)_.i2+°5mv(i , )] (2.2)

j(Gj) = 3

 

0.7 03'

 

where N () is the cumulative distribution function for the standard normal, gj = ln(G]-),
and Mj = E (23-) and a]? = Var(zj) are the mean and variance of zj = ln(ZJ-), conditional on
information available when the insurance is purchased. This formula has the advantage that
it is easy to compute without resorting to numerical integration or Monte Carlo methods
because N () is already compiled and available in most computational software programs.

It should be immediately obvious that the present value model (2.1) and (2.2) values the
market risk inherent in issuing insurance contracts at zero. Put another way, the present
value model is consistent with an insurance market equilibrium in which insurers act as if
they are risk neutral and incur zero operating costs (see [42]). There are two situations
in which valuing the market risk at zero might be appropriate. The first is if the risks

being insured are fully diversifiable. For example, this might be reasonable in the case of

 

2This is a very unusual insurance feature because most insurance schemes require up—front payment
of premiums to ensure compliance. However, it is a common feature of agricultural insurance used to
encourage broader farmer participation, and therefore needs to be accounted for in agricultural insurance
valuation.

auto insurance where a large number of independent risks are being pooled across insureds
leaving negligible aggregate risks for insurers. It is generally acknowledged, however, that
agricultural revenue risks are covariate because individual contract losses have a tendency to
move up and down together. In this case, even pooling large numbers of contracts together
leaves aggregate market risk remaining (see [33]; and [17]). Of course, this still leaves the
possibility of pooling aggregate agricultural risks with other types of (uncorrelated) insured
risks via the reinsurance market. However, there are a number of reasons why reinsurance
markets may not be able to diversify all of the aggregate risk inherent in holding agricultural
insurance portfolios (see [29]; [23]; [45]; [17]).

The second situation in which valuing the market risk of agricultural insurance at zero
might be appropriate is if the insurance is being underwritten by the government which
spreads the indemnity risk across all taxpayers. If the number of taxpayers is large relative to
the size of the risk then it has been argued that the risk per individual taxpayer is negligible
and the insurance should be priced as if the market risk is zero ([6]). However, while this
argument may be reasonable for agricultural insurance in a very large developed economy
like the US, it is unlikely to hold in developing countries where a much larger proportion of
the total population is engaged in agriculture. Furthermore, even if agricultural insurance
is being underwritten by the government and spread across a large number of taxpayers,
it might still be useful to value the market risk that would have occurred without the
government underwriting, because this will provide a more complete picture of the degree
of subsidy that is inherent in the underwriting.

For all of these reasons there have been several attempts to relax the assumption of no
market risk, and begin to build more general, market-based models of agricultural insurance

valuation.

2.3.2 Arbitrage-Based Option Pricing Models

The contingent payoffs on insurance contracts are identical to the payoffs on a put option
written on the underlying insurance index Z j.3 This insight has led a number of researchers
to apply arbitrage—based option pricing models to value agricultural insurance contracts
([53]; [52]; and [49]). The advantage of the arbitrage—based approach is that valuation is
based on a fully specified equilibrium asset pricing model that implicitly includes a market
value for the risk involved in holding portfolios of insurance contracts and other assets.
Initial applications of the option pricing approach used the Black-Scholes ( [11]) formula
which, using our notation and under the usual assumption of lognormal Z j, can be expressed

as:

 

 

0 2 0 2
g'—z--r+0.50- g'—z--r—0.50-
[DJ-(0,.) zach( 3 J 3) —Z§’N( J J J), (2.3)

03' 03'
where Z]0 is the “initial value” of the index when the insurance is taken out and z? = ln(Z§-)).
While this formula does include an implicit value for the market risk from trading the
option, it suffers from two weaknesses for valuing agricultural insurance. First, equation
(2.3) assumes that the option is paid for when it is purchased but most US. agricultural
insurance does not require premium payment until maturity when the index is realized and
claims paid. Of course, this problem is trivial and can be overcome by simply compounding
the formula through to the maturity date (i.e., multiplying the formula by 1/6). A second
and more important weakness for pricing agricultural insurance is that the Black-Scholes
model prices the option (insurance) by assuming that the underlying index Zj is the price
of a tradable asset. The formula is then derived by constructing a time-varying portfolio
consisting of the underlying asset and a risk-free bond that exactly replicates the time-
varying return on the option (insurance contract). Imposing the no arbitrage condition (i.e.,

equating the returns on the portfolio and the option) under an assumption of lognormally

 

3A put option gives the owner the right, but not the obligation, to sell a unit of an underlying asset
whose price becomes Z] at maturity, at a given strike price G J- determined when the option is written. The
payoff on the option therefore becomes max(G'j -— Z 1, 0) which is equivalent to the payoff on an insurance
contract written on 2,.

distributed Z j then leads to (2.3). This approach may be reasonable for financial options
written on tradable assets such as stocks but in most agricultural insurance applications Z j
will be an index that is not the price of a tradable asset, and so the no arbitrage argument
which forms the basis of the Black-Scholes formula breaks down (see [49]; [57]; [48]).

The non-tradability of agricultural insurance indices has led more recent applications of
arbitrage based option pricing models to use a variant of the Black-Scholes model that
allows the option to be written on a non-tradable asset ([16]; [51]). The pricing formula for
options written on non-tradable assets or indices, again assuming a lognormally distributed

index, can be written using our notation as:

._ .+/\. , .2___ , g--—,u-—02-+Aa-

03' 03’

 

 

where A is the so—called “market price of risk” for the insurance contract. Again, this formula
assumes the option or insurance is paid for at the time it is taken out so, for the case of
agricultural insurance where premiums are paid at maturity, the formula would need to be
compounded to the maturity date (i.e., multiplied by 1/6). It is interesting to note that
if the market price of risk is zero (A = 0) then (2.4) gives exactly the same formula as the
present value model under lognormality (2.2). This shows that (2.4) is just a generalization
of the present value model (2.2) that incorporates a value for the market risk from offering
insurance contracts.

Nevertheless, there are two major weaknesses in using (2.4) to value agricultural insurance
contracts. First, while a value for the market risk is included explicitly, the market price of
risk A remains an undetermined coefficient. It has been suggested that A can be estimated
using an auxiliary asset pricing model such as the capital asset pricing model (CAPM)
or arbitrage option pricing theory (APT) (see [57]; and [48]). However, computing an
appropriate value for the market price of the risk embodied in the insurance index Z j using
the CAPM or APT may not be easy and, even if it can be done, why not just use the CAPM

or APT to value the insurance product directly?4 Second, the arbitrage-based formula (2.4)

 

4Of course, the reason why the CAPM or APT are not used to price the option (insurance) directly

10

for pricing an option written on a non-tradable asset is constructed by assuming that, even
though Zj is not the price of a tradable asset, there exists a portfolio of tradable assets
whose risk spans the risk in 23° (i.e. whose returns are perfectly correlated with Z 3). Put
another way, it implicitly assumes that the option (insurance contract) is a redundant
asset because its returns can be replicated using a time-varying portfolio of the spanning
portfolio and a risk—free bond.5 There are many assets, such as commodity futures and
options contracts, stocks of agribusiness and food companies, etc., that might be included
in a spanning portfolio for Z j. Yet to our knowledge there has never been any convincing
evidence presented that such a spanning portfolio exists for agricultural insurance contracts.
Indeed, all of the recent interest in developing new agricultural insurance products would

suggest that agricultural insurance contracts are not redundant assets.6

2.3.3 Lucas General Equilibrium Models

Another approach that has been used to price agricultural insurance contracts is the repre-
sentative agent general equilibrium asset pricing model of Lucas ([30]). This model prices
contingent claims (including insurance) using an equilibrium pricing kernel derived from
a dynamic optimization problem and the imposition of market clearing conditions. The

equilibrium pricing formula takes the form:

P.7(Gj) = 5E [3%l/jWfl] , (2.5)

 

is that they are viewed as having more restrictive assumptions than those underlying the arbitrage—based
models. But if these more restrictive assumptions have to be invoked to-obtain a value for the market
price of risk then they are, at least to some extent, implicitly imbedded in the arbitrage based model for
nontradables anyway.

5This assumption can be relaxed slightly by allowing the return on the spanning portfolio to track the
risk in Z )- with error, provided the tracking error is completely diversifiable using existing asset markets
[see [32]]. Nevertheless, the valuation approach is still based on the notion that there exists a “spanning
portfolio” that generate returns that are (almost) perfectly correlated with the with movements in Z j.

6Conversely, one might argue that the fact that farmers seem unwilling to buy agricultural insurance
contracts without sizable subsidies indicates that the contracts may be redundant. However, this argument
is incomplete at best because even if the insurance is redundant risk-averse farmers should be willing to
purchase actuarially fair insurance.

11

where U () is a concave von Neumann—Morgenstein utility function, (5 is a time preference
parameter, 0 is consumption in the period when Zj is realized and insurance indemnities
are paid, and C0 is consumption in the previous period when the insurance is taken out. In
equilibrium, the representative agent consumes all of the economy’s endowments (see [30]).

The Lucas model is a representative agent model and the utility function and time prefer—
ence parameter are that of the representative agent. Notice that if the representative agent
is risk neutral then (2.5) just reduces to the present value formula (2.1), with discounting
occurring at the rate of time preference parameter (5.7 But if the representative agent is risk
averse then the ratio of marginal utilities represents an adjustment factor that compensates
investors for taking on the risk of issuing the insurance contracts. This approach has been
used to price weather insurance products for agriculture (see [12] and [41]).

There are several problems with using the Lucas model (2.5) to price agricultural insur-
ance products. First, whose consumption should be used to compute the marginal utility
of consumption? Should it be aggregate consumption in the economy or consumption of
farmers or some local agricultural region?8 There is no reason why it should be farm or
regional consumption because all agents in the economy can potentially invest in firms offer-
ing agricultural insurance. And yet aggregate economy-wide consumption tends to have low
correlation with agricultural crop revenues in the US. and so applying (2.5) using aggregate
economy-wide consumption tends to just reduce to the risk neutral present value formula
(2.1). Another problem in computing the marginal utility of consumption is that a specific
form must be assumed for the utility function. Clearly, insurance pricing results may be
sensitive to which utility function and consumption variable are included in the analysis.
More importantly, while the Lucas model (2.5) does not require the existence of a Spanning

portfolio, it is a single good representative agent exchange model in which no trades of

 

7Of course, this implies that insurance premiums are paid when the insurance is taken out, rather than
when Z1 is realized and indemnities are paid. If premiums only need to be paid at the time indemnities are

paid out then the Lucas model becomes 133(0)) 2 E [fill—(QWI/(Gjfl.
8It have been done both ways(scc [12] and [41]).

12

any commodity or asset occur in equilibrium (no trades being necessary because all agents
are identical). Hence the model generates equilibrium pricing kernels for contingent claims
but the claims are redundant and never traded in equilibrium. Put another way, the Lucas
general equilibrium asset pricing model is implicitly a complete markets model and so again

prices insurance contracts assuming they are redundant assets.

2.3.4 The Chambers State-Contingent Approach

A more recent contribution to agricultural insurance pricing is the state-contingent approach
of Chambers ([15]). The Chambers model focuses on the farmer’s role as a producer rather
than as a consumer, and derives the value of an insurance contract to farmers using the
notion that farmers will never forego an opportunity to raise profit risklessly, regardless of
their risk preferences. The resulting valuation formula takes the form:

6k(a, 6)

Pjrej) = E d

W03) » (2-6)

 

 

where k(a, 6) is the producer’s cost of producing state contingent output 5 given input prices
a, 8k(a, 6) is the Gateaux derivative of k, and cf is the state contingent output price. The
basic idea is that the return on any asset held by the farmer can be replicated physically by
investing in alternative stochastic output levels, and this equivalence can be used to value
the farmer’s willingness to pay for agricultural insurance.

The Chambers model takes a novel perspective that provides some new insights into how
farmers value insurance. Furthermore, it has the advantage that the valuation formula is
independent of risk preferences (though it does depend on the form of the cost function).
Nevertheless, it is not yet a complete general equilibrium model because it only focuses
on the farmer’s decision and the farmer’s willingness to pay for insurance. Equilibrium
insurance valuation will also depend on the decisions of those offering insurance and this
side of the market is not treated in the Chambers paper. So while the Chambers approach

gives some interesting insights into farmer willingness to pay for insurance, it is not yet

13

an equilibrium valuation model in the same sense as the other valuation models discussed

above.9

2.4 An Alternative Valuation Model

Our alternative valuation model is a general equilibrium approach in the spirit of Lucas
([30]) but we allow for agent heterogeneity and uninsurable background risk. Hence, in-
surance contracts are not redundant assets and trade in insurance contracts actually takes
place. We also allow for some important institutional features of agricultural insurance,
such as that only farmers residing in a region can purchase that region’s area-based insur-
ance contract, and only insurance companies or the government offer insurance (insurance
contracts not offered or traded among individual agents).

We use the area-based insurance contract and notation outlined in the beginning of the
paper but now need to add more detail to the economy. Each of the my farmers in the
economy are indexed by z' = 1, 2, ..., my and have two endowments of income—farm
income Y,- and non-farm income Wi, which could consist of wage income from off-farm
employment, returns from stock market and other investments, net borrowing etc. Both
Y, and W,- are random at the time farmers have the opportunity to buy insurance, and
their joint distribution is heterogeneous across farmers and allows the two income sources
to be correlated. There are also 72. -— ny “wage-earners” in the economy who are indexed by
2' 2 fly + 1, my + 2, . . . , n and only have endowments of non-farm income 1%, which again
is random at the time insurance decisions are made. Insurance contracts are available on
the regional revenue indexes ZJ- but there is uninsurable background risk generated by the
imperfect correlation between 23-, Y,, and Wi. We begin with a mutual fund model and
then introduce some common features of insurance industries to examine an alternative

insurance industry equilibrium.

 

9Of course, the Chambers model could be turned into an equilibrium model by making additional
assumptions regarding sellers of insurance and the trading of insurance contracts.

14

2.4.1 Mutual Fund Model

Suppose there exist m insurance contracts on regional revenue indices Z j for j = 1, 2, . . . , m
with prices Pj(Gj) and indemnities Vj(Gj) = max(Gj -— Z j, 0) where Gj is the guarantee
level defined earlier. These contracts can be traded freely and costlessly among all agents
in the economy in any continuous amount desired. Short and long sales are allowed (i.e.
all agents are allowed to offer as well as purchase insurance). The budget constraint of an

arbitrary agent 2' can then be expressed as:10

m
C,- _<_ Y,- + W,- + Zn,- [v,-(G,-) — P,»(G,-)] v2- : 1, 2, ..., n, (2.7)

j=1
where C,- is agent 2’s consumption and Xij is the amount of insurance on index Z J- purchased
(sold if negative) by agent 2'. Notice that this is essentially a mutual fund model where
optimal sharing of the insurable risks can be obtained via unconstrained and competitive
trade in insurance contracts. Also, we are assuming premium and indemnity payments both
occur at the same time (as is typical in agricultural insurance) and so they have a net effect

on income available for current consumption.

Each agent’s decision problem is to choose insurance amounts to maximize the expected
utility of consumption, E[U,'(C',-)] subject to (2.7), where each U, is an increasing and
concave von Neumann-Morgenstern utility function. Necessary conditions for a maximum

81'8le

E{U,-’(C,~)[V,~(Gj)—P,(Gj)]} =0 W: 1, 2, n and j: 1, 2, m. (2.8)

This condition is virtually identical to that obtained from Lucas ([30]) under the assump-
tion that premiums are collected at the time indemnities are paid (see footnote 7). The
only difference is that we allow for heterogeneous preferences, endowments, and probability

distributions while the Lucas model is a representative agent model.

 

10For “wage earner” agents 1' = ny + 1, fly + 2, , n, then Y.- will be zero with probability one.
11Second-order conditions are satisfied by the concavity of U,.

15

TI.
We close the model by assuming the m insurance markets all clear, 2: X ,- j = 0 Vj which

i—l
implies from (2.7) that the aggregate budget constraint is given by:
TI. n n
20,-:ZY, ZW,=>C=Y+W (2.9)
i=1 i=1 i=1

where C, Y, and W, are economy aggregate consumption, farm income, and non-farm
income respectively. A mutual fund equilibrium is a set of consumption choices, insurance
choices, and pricing functions PJ-(G'j) that satisfy (2.8) and (2.9).

To impose more structure on the model we need to make additional assumptions on
preferences and probability distributions. The most common assumptions in applying the
valuation models discussed earlier are constant relative risk aversion (CRRA) utility func-
tions and lognormally distributed random variables. So we assume that all agents have the
same CRRA utility function Ui(C,-) = Oil—a / (1 — oz) where a is the coefficient of relative
risk aversion, and that all of the C,- and Zj are joint lognormally distributed.12Using these
assumptions and results from the appendix of Rubinstein ([43]) (reproduced in appendix
B), (2.8) can be solved for Pj(Gj) and expressed as:

2
._ '+OO"’ ' . 2_ .. g'—/J'_0'-+a0"
Pj(Gj) = GjN (9] H] 2]) — eflj+0 50] 002]N( J J J U) (2.10)
Uj 0.7

 

 

where aij = Cori(c,-,zj) for c,- = ln(C,'). In equilibrium, (2.10) must hold for every 2'
because all agents are free to trade insurance contracts. Therefore, if all agents know the
joint probability distributions of each 3j and their own 0,7, then 0,-3- = 0;- must be equal for
all 2'. Then (2.10) can be expressed as:

9’—“'+aac' +0.5 2— C. g~—n-—a2.+ag¢
P,(G,)=G,-N(’ J ’)_e"J ”2 “”JN(’ J J J . (2.11)
0.1 o]-

 

 

It is interesting to note that if agents are risk neutral (a = 0), or the risk from holding

insurance contracts is completely diversifiable via the mutual fund (03C- : 0), then the market

risk is valued at zero and (2.11) simplifies to the present value formula under lognormality

 

1"This assumption requires agents to have homogeneous preferences but still allows them to have different
endowments, face different risks, and consume different amounts.

l6

(2.2).13 But if agents are risk averse and the insurance risk is not completely diversifiable,
then (2.11) gives a generalized valuation formula that incorporates a value for the non-
diversifiable market risk embodied in the insurance contracts. It is also interesting to note
that for 6 = 1 (premiums and indemnities paid at the same time) then (2.11) is exactly the
same as the arbitrage based option pricing model (2.4) for options on non-tradable indexes,
as long as Aaj = 010;. This shows that the arbitrage model (2.4) and the equilibrium based
model (2.11) are identical, except for the way in which they define and characterize the
market risk term embodied in the valuation formula.

For (2.11) to be a useful formula we need a way of characterizing the magnitude of the
market risk factor (103;. A useful property of the lognormal distribution is that if c, and z]-
are joint normal then 0021(02', 2]) = E (Ci)C0'u(c,-, 2]). Using this property, and assuming
aggregate incomes Y and W are lognormally distributed, we can apply the covariance

operator with respect to zj to the aggregate budget constraint. (2.9) to get:
E(C)a]c- = E(Y)Cov[ln(Y), zj] + E(W)Cov[ln(W), 23-]. (2.12)

A property of the lognormal distribution shown in appendix C is that Cov[ln(Y),zj] as
pyszVyCVZj and Cov[ln(W),zj] z PWZJ- CVWCVZJ. where PYZJ- and PWZJ- are corre-
lation coefficients between aggregate farm income and the insurance index, and between
aggregate non-farm income and the insurance index, respectively; and CVy, CVW, and
CV2]. are coefficients of variation for aggregate farm income, aggregate non-farm income,
and the insurance index, respectively. Substituting these expressions into (2.12) and solving
for 016- gives:

05- R: SypijCVyCI/Zj + SVVpWZj-CVWCVZj (2.13)

where Sy = E(Y)/E(C) and SW = E(W)/E(C) are the expected shares of aggregate farm

and non-farm income in aggregate consumption.

 

1("Of course, 6 = 1 in (2.11) because we assume premiums and indemnities are paid at the same time.

17

2.4.2 Discussion

Equation (2.13) provides a way of characterizing and understanding the way in which market
risk influences the valuation formula. In economies like the US, where the share of farm
income in total consumption is small, and the correlation between agricultural insurance
indices and non-farm incomes is low, then both terms in (2.13) will be small and the market
risk associated with agricultural insurance will be correspondingly small, irrespective of the
degree of risk aversion of agents. In this case, the equilibrium price of insurance contracts
will be close to actuarially fair and the present value formula (2.2) or, more generally (2.1),
may be quite appropriate. The reason this occurs is that the agricultural insurance risk is
easily diversified by spreading it across a large non-agricultural sector with incomes that
are largely uncorrelated with changes in insurance index values.

Alternatively, in developing economies (or developed economies that have a much higher
share of farm income in total consumption) then the market risk may be much higher if
aggregate farm income is highly variable and strongly correlated with the insurance index.
We would generally expect aggregate farm income to be both variable and highly correlated
with the insurance index because otherwise the insurance contract would be of little use to
farmers. Of course, the other way in which the market risk can be high is if there is a strong
correlation between non-farm incomes and the insurance index, which seems less likely to
occur in general.

The overall conclusion is that the value of the market risk factor in particular situations
and for particular insurance contracts is an empirical question and indiscriminant use of
the present value model would be unwise. Equation (2.13) provides a simple but useful
way of thinking about the factors contributing to the magnitude of the market risk. When
combined with empirical estimates and an assumption (or estimate) of the CRRA parameter
0, equation (2.13) also may lead to a reasonable estimate of the value of market risk for

particular agricultural insurance contracts in particular situations.

18

2.4.3 Insurance Industry Model

A potential criticism of the mutual fund model is that the institutional structure of actual
agricultural insurance industries does not conform to the mutual fund structure. Agricul—
tural insurance contracts can only be purchased by farmers and are only sold by insurance
companies (or government agencies). That is, there is no free and unregulated direct trade
(allowing both short and long selling) of agricultural insurance contracts among farmers
and wage earners. In this section we develop an insurance industry model that embodies
some of these key institutional features of actual insurance markets and show that the equi-
librium (and hence the insurance valuation formula) remains identical to that obtained in
the mutual fund model.

The setup and notation remain as in the mutual fund model except that now only farmers
can buy agricultural insurance, and each farmer can only buy contracts based on his or her
relevant regional revenue index. The contracts are sold by an insurance industry, which
might be a set of competitive firms or it might be a government agency. The insurance
industry acts as a broker by selling contracts to farmers and then repackaging and securi-
tizing the contracts as an asset that is then resold on financial markets. Under this setup
the farmer budget constraints become:

C,- g Y,- +W, +X,;k[Vk(Gk)— —P,,( Gk) ]+:R, 1),A, V21: 1 2 my (2.14)

j=1
where the k subscript 011 the insurance contract indicates the revenue index for the region

where farmer i is located, A,,- is the amount of the repackaged asset based on insurance
contract 3' that is purchased (sold if negative) by farmer 2', and R,- is the gross rate of return
on the repackaged asset based on insurance contract j. Similarly, the wage earners budget
constraint becomes:

C,<W,-+Z(R, A,, Vz—ny+1ny+2 . (2.15)

j: -—1
Notice that only farmers can buy insurance, and only on the revenue index for the region in

which they are located. We also impose the constraint X,-,c Z 0 (farmers can only buy not

19

sell insurance). However we allow all agents to buy or sell any perfectly divisible amount
of the repackaged asset offered by the insurance brokerage industry.

Because the insurance industry only acts as a broker and therefore bears no risk, a natural
equilibrium condition is to set insurance industry profits on each insurance contract to zero

which implies:

n n
Pj(Gj) Xe = ’ZAz'j W = 1, 2, m, and (2.16)
i=1 i=1
n n
_Vj(Gj)ZX1j 2‘ RjZAij Vj=1, 2, ..., m. (2.17)
i=1 i=1

Equation (2.16) shows that in the aggregate agents will be net short each repackaged asset
(the insurance industry collects premiums and uses the proceeds to buy the repackaged
asset from individuals, who are selling short to the industry in aggregate). Equation (2.17)
shows that when the indemnities are paid by the industry they are financed by selling the
industry’s net long position in the repackaged asset, while accounting for the equilibrium

return on investment, 12,-. Together, (2.16) and (2.17) imply:
R, = V,(G,~)/P,-(G,). (2.18)

Now we turn to the necessary conditions for individual choice of investment in the repack-
aged asset. Because all individuals can trade the asset, and short sales are allowed, the

necessary conditions for expected utility maximization include:
E{U,-’(C,-)[R,- — 1]} = 0w = 1, 2, nandj = 1, 2, m. (2.19)

But substituting (2.18) into (2.19) and multiplying through by P,(G,-) gives exactly the
same necessary condition (2.8) as in the mutual fund model. Furthermore, aggregating
the budget constraints (2.12) and (2.14), and imposing the insurance industry equilibrium
conditions (2.16) and (2.17), the aggregate budget constraint for the industry equilibrium
model simplifies to exactly the same aggregate budget constraint (2.9) as in the mutual fund

model. Together, these two results show that equilibrium in the mutual fund model and the

20

insurance industry model are formally equivalent and so, under the usual assumptions of
CRRA preferences and lognormally distributed random variables, the mutual fund insurance
valuation formulas (2.11) and (2.13) also apply in the case of the insurance industry model.
Of course, the reason this occurs is because even though risks cannot be diversified via trade
in the insurance contracts themselves, competitive trade in the repackaged asset allows the

optimal amount of insurance risk diversification and spreading to occur.

2.5 Simulation Results

It will be useful to examine just how different the results from alternative insurance pricing
models can be under various conditions. Here we address this issue using a simulation
model. We use the present value model under lognormality (2.2) as a baseline and set
6 = 1 (premiums paid at the same time as indemnities) and 0,- = 0.3.14 We also set
C,- = 6“], indicating that the guarantee level is set equal to the median value of 2,. We
then find the value of n,- that makes P(G,-) == 1 using the present value formula (2.2). In
this way, the present value baseline has been normalized to give an insurance value of one,
which makes it easy to compare the relative effects of using alternative valuation formulas.
Values and ranges for all parameters used in the simulation are provided in table 2.1.

It is conceptually difficult to apply the Black-Scholes formula (2.3) to price agricultural
insurance because it relies on the assumption that Z,- is the continuously observed price
of a tradable asset that follows a geometric Brownian motion. When applying the model
to a regional revenue index which is observed discretely, the assumption that seems to
make the most sense is to think of the underlying continuous variable as the expectation

of Z,- conditional on information available at different points in time. Then we can think

 

1“It is well known that if Z,- is lognormal then a,- is approximately equal to its coefficient of variation.
Hence, a, is a relative volatility measure and setting a,- = 0.3 indicates an approximate 30% coefficient of
variation for the uncertainty surrounding the value of the realized regional revenue index when indemnities
are paid, given information available when the insurance is taken out. The 30% figure is arbitrary but
seems a reasonable starting point for regional crop revenue indexes.

21

Table 2.1. Summary of the parameters in simulation model

 

 

 

Parameter Notation Base value Range
01 CRRA parameter 2 [0, 3]
r Risk-free interest rate .05 [0, .15]
12,- Mean of the logarithm of insurance index j 2.3
a,- Standard deviation of the logarithm of insurance index j .3
Sy Expected share of farm income in aggregate consumption 2 [.05, .70]
PYZ, Correlation coefficient between aggregate farm income .7 [.5, .9]
and the insurance index j
sz, Correlation coefficient between aggregate non-farm .1 [0, .2]

income and the insurance index 3'

 

CV2, Coefficient of variation for insurance index j .3

CVy Coefficient of variation for aggregate farm income .2 0, .4
C Vw Coefficient of variation for aggregate non-farm income .2 0, .4
a0;- Wafiet risk factor .026 [0, .166]

 

of the conditional expectation Et(Z,-) changing continuously over time as new information
becomes available about the realization of Z j. Under this interpretation it would be natural
to set Z? = E0(Z,-) where t = 0 indicates the time period when insurance is taken out. By

the properties of the lognormal distribution this would imply:
0 _ 0 _ _ 2
ln(Zj) — z,- — u, + 0.50,. (2.20)

In simulating the Black-Scholes formula (2.3) we use the same values of 9,, p,, and a,- that
were used to compute the present value formula, and then use (2.20) to obtain z? and Z9.
The only remaining unknown in the Black-Scholes formula is the risk-free rate of interest 7'.
We simulated the Black—Scholes insurance price under different interest rates ranging from
0% to 15% ( r = 0 to r' = 0.15).15

We also simulate our alternative insurance pricing formula (2.11) and (2.13) under a range
of assumptions. We begin by setting 9,, p,, and a, to the same values that were used to
compute the present value price (see table 2.1). Then we examine a range of alternative
values for the market risk factor adje- and apply (2.11). We let the coefficient of relative

risk aversion a range between 0 and 3. Then using (2.13) we let the share of farm income

in total aggregate consumption S'y range from 5% (highly developed economies) to 70%

 

15To keep the valuation consistent with the present value calculation we assume the premium is agreed
to at the time the insurance is taken out but not paid until the index has been realized and indemnities
paid. This requires multiplying (2.3) through by 1/6 = l + r.

22

(developing economies) and set SW = 1 — Sy. The correlation between aggregate farm
income and the insurance index is assumed to range from 0.5 to 0.9, and the correlation
between aggregate non-farm income and the insurance index is assumed to range from 0 to
0.2. The coefficient of variation for the insurance index is set at 30%, consistent with the
earlier assumption that 0,- = 0.3, and the coefficients of variation for farm and non-farm
incomes are assumed to range between 0 and 40%. With these assumptions, the possible
values for 00’; range from 0 to approximately 0.166 (see table 2.1).

The first simulation sets all parameters except the interest rate to their base values and
lets the interest rate range from 0% to 15%. Insurance values from the present value model,
the Black-Scholes model, and our market risk adjusted model are then graphed over alter-
native interest rates (see figure 2.5). These simulations highlight three important points.
First, as long as premiums are not paid until revenue has been realized and indemnities
paid, then the interest rate has no affect on the present value model insurance price, or on
the market risk adjusted price. Second, our market risk adjusted price adds a premium of
approximately 11% to the risk-neutral present value price at base parameter values. Third,
when the interest rate is zero the Black-Scholes value equals the present value model, but
as the interest rate rises the Black-Scholes price falls below the present value price (negative
risk premium). This happens because as long as we continue to assume Z? = E0(Z,-) then
a higher interest rate implies the insurance premium must continue to fall in order equate
returns from the insurance and the dynamically adjusted portfolio containing Z ,- and a risk
free bond. This is a nonsensical result and simply emphasizes that the Black-Scholes model
is entirely inappropriate for valuing claims on non-traded assets.

The second simulation keeps the interest rate at its base value of 5% and allows all other
parameters to vary over their entire range, which implies the market risk factor (10'; ranges
from 0 to approximately 0.166. Results are shown in figure 2.5. In this case, the Black-

Scholes price is 18% lower than the present value price, again indicating problems with the

Black-Scholes approach. The market risk adjusted price equals the present value price when

23

 

_ _ Market Risk Adjusted .. _ .. _ .Black-Sc holes
PresentVaIue

1.8 4
1.7 ~
1.5 —
1.5-
1.4 -
1.3-
1.2 ~
1.1 a ——————————————————
,. ...,

.9 - "“m...

.3- "~-..,_

 

Insurance Premium

7 --I ‘.~ .-
I ~.
’-
.6_ ~‘a‘..

.5-

 

 

I) .63 .0‘5 .69 .12 .1'5
Interest Rate

 

 

 

Figure 2.1. Insurance premiums versus interest rate

there is no market risk but may range up to 78% higher than the present value price under
extreme parameter values (see figure 2.5). This shows that market risk can have nontrivial
effects on insurance valuation, and the differential will be higher the more risky is farm
income, the more important agriculture is in the overall economy, the more risk averse are
agents, and the greater the correlation is between insurance indices and farm incomes.

For the third and final simulation we keep all parameters except the share of farm income
in aggregate consumption constant at their base values and allow the farm income share
to vary over its range from 5% to 70%. Results in figure 2.5 highlight that the relative
importance of agriculture and farming in the economy can have a sizable effect on insurance
valuation in the market risk adjusted model. While the risk premium is only around 5% in
economies where agriculture and farming are relatively unimportant, this proportion rises

to around 30% in economies that are heavily dependent on agriculture.

24

‘1-

 

_ _ Market Risk Adjusted .. _ .. _ .Black-Scholes
Present Value

1.8 -l /
1.7 -i /
1.6-1 /
1.5- /

1.4d /

1.3- /

1.2 "‘ /
1.1“ /
1-1 A

Insurance Premium

.9—
8“ .._.._.._.._.._.._.._.._......._.._.._.._.._.._.._.._
.7-1
.6—
.5d

 

 

in .d2 .64 .0'6 .da .I .12 .14 .16 .18
Market Risk Factor

 

 

 

Figure 2.2. Insurance premiums versus market risk factor

2.6 Conclusions

This paper evaluates existing methods for valuing agricultural insurance contracts and
then develops a new valuation model that offers some intuitive insights into the factors
that determine the value of market risk. Under the assumptions of CRRA preferences
and lognormal probability distributions, the new model also provides a simple, practical
way of estimating the impact of market risk for particular insurance contracts in particular
situations and environments.

Simulation results suggest that simple present value models of insurance pricing, which
assume no market risk, may be quite appropriate in large developed economies that can
spread and diversify agricultural risks over a large non-agricultural sector either through
capital markets in the case of an insurance industry equilibrium, or through the tax system

in the case of government insurance programs. In economies where the share of farm income

25

 

— — Market Risk Adjusted -. — -- — Black-Scholes
Present Value

1.8-
1.7 -
1.6-
1.5—
1.4 -
1.3- ...,-
1.2d 1"”

1.1“ __,_,.—a—'

1—l

 

Insurance Premium

 

 

L) .1'5 5 .15 .é .7’5

% Farm Income in Consumption

 

 

 

Figure 2.3. Insurance premiums versus share of farm income in aggregate consumption

in total consumption is much higher, however, the market risk may also be higher and have
a significant impact on insurance valuation.

Overall, our analysis suggests that present value models may be a reasonable approxima-
tion in some circumstances but, as is now becoming well-known, the standard Black-Scholes
model has a number of deficiencies for pricing agricultural insurance. Other methods for
characterizing the market price of risk in agricultural insurance contracts, such as arbitrage-
based pricing model for options written on non-tradable assets, the Lucas representative
agent model, and the Chambers state-contingent cost function approach are logically con-
sistent and potentially useful. However, the simple heterogeneous agent equilibrium model
developed here is easy to use, amenable to empirical estimation, and provides some use-
ful insights into the factors determining the effect of market risk on agricultural insurance

valuation.

26

CHAPTER 3

MODEL SELECTION IN
STOCHASTIC FRONTIER
ANALYSIS: MAIZE PRODUCTION
IN KENYA

3. 1 Introduction

Stochastic production frontier analysis has been widely used to study technical inefficiency
in various settings since its introduction by Aigner, Lovell, and Schmidt ([2]), and Meeusen
and van den Broeck ([31]). The approach has two components: a stochastic production
frontier serving as a benchmark against which firm inefficiency is measured, and a one-sided
error term which captures technical inefficiency and is usually assumed to be identically and
independently distributed across firms. Recent studies have generalized the one-sided error
term to allow its distribution to be heterogeneous and depend on various firm characteristics
(see [9]; [14]; [54]; [55]).

Allowing inefficiency to depend on firm characteristics allows researchers to identify the

27

determinants of inefficiency, and to suggest interventions which might improve efficiency.
However, this approach has been hampered by two problems. First, the relationship between
firm characteristics and the extent of technical inefficiency can be sensitive to the model
used to incorporate firm characteristics and little is known about how to choose among
competing models. This model uncertainty makes policy recommendations quite tenuous.
Second, existing studies have mostly focused on the directions of the influence of the firm
characteristics on technical inefficiency while generally overlooking the magnitudes of the
partial effects. This makes it diflicult to determine which type of policy intervention will
have the largest impact on inefficiency. The second problem is somewhat surprising given
that the magnitudes of the effects of explanatory variables on dependent variables are often
the focal point in other types of regression analyses.

In this paper, we make three contributions to the stochastic frontier literature. First,
we provide a method for estimating the quantitative magnitude of the partial effects of
exogenous firm characteristics on firm inefficiency, show how to put standard errors around
these partial effects, and propose an R2-type measure to summarize the overall explana—
tory power of the exogenous factors on inefficiency. Second, we examine maize production
in Kenya and show that while alternative models of the relationship between household
characteristics and technical inefficiency tend to provide the same direction of the influence
of household characteristics, the magnitudes of the partial effects on firm inefficiency are
quite sensitive to model selection. Third, we show how a recently developed model selection
procedure (Alvarez, Amsler, Orea, and Schmidt, [4], hereafter AAOS) can be used to choose
among competing models, and we use bootstrapping to provide evidence on the power of
this procedure. The model selection procedure gives an unambiguous choice of best model
for our application to Kenyan maize production. This is important because if different
models give different results, and we cannot distinguish statistically among the models, we
do not know which set of results should be used. But if we can pick a clearly best model,

then the fact that inferences and conclusions are sensitive to model selection is not such a

28

problem.

Our empirical analysis is on maize production in Kenya. The problem of hunger in
Kenya remains widespread and its economy depends heavily on agriculture with 75 percent
of Kenyans making their living from farming. Maize is the primary staple food and most
farmers are engaged in maize production. In recent years, total maize output has not kept
pace with the growing population and demand, largely due to falling land productivity:
average national maize yields have fallen from over 2 tons per hectare in the early 1980’s
to about 1.6 tons per hectare recently ([36]). The technical efficiency level of Kenya maize
production is therefore an important economic and policy issue.

The empirical analysis uses detailed household survey data from Kenya and investigates
determinants of productivity and inefficiency using stochastic frontier analysis. The vari-
ables used to explain inefficiency are related to the educational background of the household,
rural infrastructure, land tenure, credit constraints faced by the household, and farm size.
These explanatory factors go well beyond those used to study production inefficiency in
most other studies of agriculture (see [27]; [21]; [4]).

In the remainder of the paper, we first review the standard stochastic frontier production
model commonly used in the literature and then extend it to allow estimation of the magni-
tude of firm characteristic effects on technical inefficiency, put standard errors around these
estimates, and develop an R2-type measure of overall explanatory power. Next we describe
our data and variables used in the empirical analysis, followed by estimation results from
alternative model specifications. Results for the magnitude of partial effects of household
characteristics on inefliciency are quite sensitive to model specification. Then we carry out
specification tests to choose our final model. A novel approach is that we use the bootstrap
to examine the reliability of these specification tests in choosing the correct model. The
final sections contain an analysis of technical inefficiency in maize production in Kenya

based on the final model, and some concluding comments.

29

 

3.2 Stochastic Production Frontier Models

The basic setup and notation follow Wang and Schmidt (2002) and AAOS. Firms are
indexed by z' = 1, . . . , N. Let. y, be log output; :13,- be a vector of inputs; and 2, be a vector
of exogenous variables that exert influence on farm inefficiency. Let y; be the unobserved

frontier which is modeled as

31f = $25 + vi. (3.1)

where v, is distributed as N (0, 0.3) and is independent of x,- and 2,, and ,8 is a parameter
vector. The actual log output level y,- equals y: less a one-sided error, u,, whose distribution

depends on 2,. The full model is written as
I12: 313:3 + '01“ udzz'. 9). “2(3216) Z 0» (33-?)

where (9 is a vector of parameters. It is assumed that u, and v, are independent of one
another and that u,- is independent of :13,- (conditional on 2,). The model is usually imple-
mented by assuming it, is distributed as N (m, 0,2)+ with various specifications (discussed
below) used to model it, and 0,. The frontier function and the inefficiency part are gen-
erally estimated in one step using maximum likelihood estimation (MLE) to achieve both
efficiency and consistency.1
Indexing exogenous factors with k = 1, . . . , K, we take expectations conditional on x,-
and 2,, and then take partial derivatives with respect to z,- k on both sides of equation (3.2),
to get
3IE(yz'I$1, ell/321k = BIEPWIIBR Zr)l/3zrk' (33)

Here, 8[E (—u,-]:r:,-, 21)] / 32,-;c can be interpreted as the partial effect of 2,], on efficiency —u,-,

and is also the partial effect on y,. Because y,- is log output, 8[E( —u,- |:1:,, 2,1)] #32,), is the semi—

 

1Some studies use a two-step procedure where the frontier function is estimated first, and then the
inefficiency term is regressed on exogenous variables in the second step. This procedure is biased for two
reasons. The first and more obvious reason is the possible correlation between the input variables in the
frontier function and the variables in the inefficiency term. The second reason is that the inefficiency term
from the first step is measured with error and the error is correlated with the exogenous factors. See Wang
and Schmidt ([56]) for an extensive discussion and evidence from Monte Carlo experiments.

30

elasticity of output (efficiency) with respect to the exogenous factors, i.e., the percentage
change in expected output when 2,;c increases by one unit. Similarly, taking conditional

variances we have
all/(921331, Zi)l/5’Zik = 31V(ui|$t.Zi)l/3zik- (34)

So 8[V(u, [23,, 2,)] / 82, k is the partial effect of 2,;c on the variance of both the inefficiency term
u, and y.,. It can be interpreted as an estimator of the partial effect of 2,;C on production
uncertainty.

The measures 3[E(’u,|.r,, 2,)]/r92,k and (9[V(u,].r,, 2,)] / 031k were first, proposed and used
in Wang (2002) and Wang (2003), but for a different purpose than they are used here.2
Here, we interpret 6[E(—u,|:r,,2,)]/82,k as the semi-elasticity of output with respect to
exogenous factors so that not only its Sign but also its magnitude is of economic interest.
In appendix D we provide formulas for computing estimates of 8[E(-u,|:r,, 2,)]/82,;c and
8[V(u,]:r,, 2,)]/82,k, along with their standard errors using the delta method for several
popular model specifications for n, and 0,.

It will often be useful to measure how well the vector of exogenous factors, 2, explains
inefficiency, u in a data sample. Surprisingly, this has not been addressed in the previous
literature. We suggest a statistic R3, to summarize the explanatory power of 2. To motivate

the measure, the variance of the inefficiency term 11, can be decomposed as
V011): Vz[E(Uz'Izz')l + Ezlvfuz'lzzifi» (3-5)

where VZ[E(u,|2,)] is the variance of the conditional mean function over the distribution
of 2,, and EZ[V(u,|2,-)] is the expected variance around the conditional mean of u,. The
fraction of variation in u, that is explained by 2, is Vz[E(u,|2,)]/V(u,). Thus a natural

measure of explanatory power over the sample would be

. . 2
2 £1 [Efuz'lzz'l -% 2’21 E(uilzi)]
R2 : 1 (3'6)

. . 2 .
f=1 [E(u,|z,) “R f=1 E(WI-’31)] + 231:1 Vfuilzz')

2Wang uses the two measures to show the property of monotonic/non-monotonic efficiency effect asso-

 

 

ciated with alternative model specifications.

31

where E and V indicate sample estimates of the mean and variance of u, conditional on 2,.
Letting 31 = tit/02'. R2 = ¢(Rl)l¢(Rlll—1. and R3 = 41% - Rle, where ¢(°) and (Pf)
are the density and cumulative density functions for the standard normal, then the mean

and variance of u, conditional on 2, can be expressed as

E(uilzivzi) = 01'(31+Rz) (3-7)

vrulxtz.) = 0f'(1+R3)- (3.8)

So all that remains to compute R3 is to estimate [1, and 6, for a specific model specification
(see the model specification section below) and then substitute these estimators for the
population values n, and a, in (3.7) and (3.8) to get the sample estimates of E(u,[2,) and
V(uiizi)-

Similar to R2 in an ordinary least squares regression, R: can be called the “goodness
of fit” of the efficiency component, and it can be interpreted as the fraction of the sample

variation in u that is explained by 2.

3.2.1 Alternative Model Specifications

In the original specification of stochastic frontier functions, Aigner, Lovell, and Schmidt ([2])
and Meeusen and van den Broeck ([31]) assumed an identical and independent half-normal
distribution for the one-sided error terms 11,. Subsequent studies have generalized the model
to allow for heterogeneity in the distribution of the inefficiency term while maintaining the
assumption that each u, is half normal. Kumbhakar, Ghosh, and McGuckin ([27]), Huang
and Liu ([21]), and Battese and Coelli ([9]) allow the mean of the pre-truncated normal
distribution of u, to depend on a set of exogenous factors. Reifschneider and Stevenson
(1991), Caudill and Ford ([13]), Caudill et a1. ([14]) and Hadri ([20]) allow the variance
of the pre—truncated normal distribution of u, to depend on the exogenous factors. Wang
([55]) allows both the mean and the variance of the pre—truncated distribution of u, to

depend on exogenous factors.

32

Regardless of whether we allow the mean, the variance, or both the mean and the vari-
ance of the pre-truncated normal to depend on exogenous factors, both the mean and the
variance of the truncated half normal will always depend on the exogenous factors. These
are sometimes called models of heteroscedasticity, but the fact that the mean also changes
makes this terminology potentially misleading. Whereas heteroscedasticity affects only the
efficiency of estimation in a standard linear model, in a stochastic frontier model with het-
erogeneity in the distribution of the inefficiency term, failure to model the exogenous factors
appropriately leads to biased estimation of the production frontier model, and of the level
of technical inefficiency, hence leading to poor policy conclusions (see [13]; [14]; [20]; [55]).

With different specifications available to model heterogeneity, it is unclear which should
be used in particular applied settings. The choices made in many past studies seem to be
somewhat arbitrary. However, a carefully specified model might help to increase estimation
efficiency and remove sources of potential bias and inconsistency ([55]). Moreover, there has
been little investigation on how the choice of model specification influences the estimation
results. In order to deal with the model specification problem, researchers usually do
sensitivity analysis using competing models. But if the competing models give very different
results, it is difficult to pick one and discard the others. Wang ([55]) treats this problem by
specifying a flexible model that nests most of the usual model specifications for n, and 0,.
However, a more flexible model has more parameters, which impose a higher computational
burden and reduces degrees of freedom. Given that large samples are typically difficult to
obtain in stochastic frontier models, some relevant parameters may be estimated imprecisely
in flexible model specifications.

AAOS suggest a procedure for selecting a model for the one-sided error term. First,
assume the general model of inefficiency ([55]) in which n, is distributed as N (n,,0z-2)+,
with n, =2 n - exp(2,f6) and 0, = 0,, - exp(2,’-7). This general model nests several simpler
models, many of which have been used in previous studies. In particular, the following six

models are special cases of the general model, as outline in AAOS.

33

. Scaled Stevenson model: Let 6 = 7. Then the distribution of u, becomes exp(2,’-6) -

N (u, 0.3)", which is used in Wang and Schmidt ([54]).

. KGMHLBC model: Let '7 = 0. Then the distribution of 11., becomes N (n-
exp(2z'-6),0,2,)+, which has been considered in Kumbhakar, Ghosh, and McGuckin

([27]). Huang and Liu ([21]), and Battese and Coelli ([9]).

. RSCFG—n model: Let 6 = 0. Then the distribution of 11, becomes N (p.03 -

exp(22,’-7))+-

. RSCF G model: Let [.t = 0. Then the distribution of 21., becomes exp(2z’-'y) . N (0, 0,2,)+,
which is considered in Reifschneider and Stevenson ([40]), Caudill and Ford ([13]),
and Caudill, Ford, and Gropper ([14]).

. Stevenson model: Let 6 = 7 = 0. Then the distribution of it, becomes N (n,0,2,)+,

which is the model of Stevenson ([47]).

. ALS model: Let u = 7 = 0. Then the distribution of it, becomes N (0, 0,2,)+, which is
the model of Aigner, Lovell, and Schmidt ([2]).

Among the six models, the scaled Stevenson, KGMHLBC and RSCFG-n models have the

same number of parameters. The RSCFG model is nested by the scaled Stevenson model

and the RSCFG-n model. Also notice that the Stevenson model and the ALS model do not

contain any variables (2,) that influence the distribution of inefficiency. AAOS Show how

to use likelihood ratio (LR) tests, LM tests and Wald tests to test the above restrictions,

and hence to choose a plausible model for inefficiency.

3.3 Empirical Application

The empirical application is to maize production in Kenya using detailed household survey

34

3.3.1 Data

The’data are from a rural household survey of about 1100 households planting maize in
the main season of 2003-2004 in Kenya.3 The survey was designed and implemented under
the Tegemeo Agricultural Monitoring and Policy Analysis Project, a collaboration among
'Tegemeo Institute of Egerton University, Michigan State University, and the Kenya Agricul-
tural Research Institute. Figure 3.3.1 is a map of Kenya with the round dots representing
sampled villages. These villages were chosen randomly from each of eight pre-determined

agro—economic zones and then households were sampled randomly from each selected village.

 

 

- a -- , _ - _ . \- ‘ ‘.
..t. Q @3-
Location of Sample e\ o s’ "‘9"
Villages "“~\_,<_/':>»/

Source: Suri Tavneet (2005).

 

 

 

Figure 3.1. Location of sample villages in Kenya

Field level data are available for each sampled household and some households planted
maize in more than one field. The survey includes not only detailed field production infor-
mation but also rich demographic and infrastructure characteristics of each household. The
production data for each field include size of the field, yield, labor input, fertilizer applica-
tion, and seed usage. The demographic information includes the age, gender and education
level of each household member; how far a household is from a bus stop, a motorable road,

a telephone booth, mobile phone service, and extension service; whether a household mem-

 

3See Surf ([50]) for a study of the adoption decisions of hybrid seed by maize producers in Kenya using
the same data set.

35

ber has non—farm income; whether a household receives loans; how much land a household

owns, and land tenure. Rainfall and soil quality data are also available at the village level.

3.3.2 Variables in the Production Frontier

In the production frontier part of the model, the output variable is maize yield per acre, and
the input variables are applied fertilizer nutrients, labor, maize seeds and machine usage.
Since both the output and inputs are in per acre terms, land is not explicitly included as
an input. Most of the maize fields are inter-crop fields where more than one type of crop is
planted in the same season. Because most inputs (land, fertilizer and labor) are at the field
level and cannot be separately allocated to maize production only, we generate an output

index for inter-crop fields using:

Yr: 23’1ij /P1, (39)
j

where Y, is the output index, P,- is the market price of crop j , Y,,- is the yield of crop j in
field 2', and crop 1 is maize. Fields with more than three types of crops are deleted because
we want to focus on the fields where maize is the major crop.4 Only pre-harvest labor input
(LABOR) is included because harvesting and post—harvest activities have little effect, if
any, on yield. The unit of labor is person-hours. One person-hour of labor from children
younger than 16 is transformed to 0.6 person-hours of adult labor. Nitrogen (FERTILIZER),
the most important nutrient in maize growth, is computed from fertilizer application data
according to the quantity and composition of each type of fertilizer used.5 Maize seeds
can be separated into hybrid seeds and local seeds. All fields used either hybrid seeds or
local seeds (no combinations in the same field). These seed inputs are captured by two

variables, SEED measures the amount of (hybrid or local) seed per acre applied to the

 

4637 out of the total 1718 fields are dropped.

5More than 20 types of fertilizers were applied. While some of these use nitrogen, phosphorous, and
other nutrients in various proportions, nitrogen is usually the major nutrient deficiency and many of the
major fertilizers use nitrogen and phosphorous in fixed proportions. Therefore, the level of applied nitrogen
should give a reasonably accurate measure of the impact of fertilizer on yields.

36

field, and HYBRID is a dummy variable measuring one for hybrid seeded fields and zero
otherwise. We also use a dummy variable MONO as an indicator for mono-crop fields
because these might be expected to have systematically different yields than multi—crop
fields. Tractor usage in land preparation is the only machine used for pre-harvest activities.
This is captured by a dummy variable TRACTOR with one indicating that a tractor was
used and zero otherwise.

Environmental variables are also included 011 the right hand side of the frontier production
function. Failure to control for environmental variables may cause a correlation between
some inputs and unobserved factors in the error term (for example, if a farmer makes
input decisions based on soil properties that also affect maize yield) and therefore may
bias estimates of the production frontier and inefficiency level ( [44]). In order to control
for environmental conditions, we include seven dummy variables indicating the different
agro—economic zones. Farms in the same zone share similar terrain and climate conditions.
We also include three village level variables: DRAINAGE, DRAINAGE2 and STRESS.
DRAINAGE captures the drainage property of the soil. It is a categorical variable ranging
from one to ten where one indicates the least and ten the highest drainage. DRAINAGE2
is the square of DRAINAGE. We include a quadratic term because yield is expected to
increase in DRAINAGE at lower drainage levels and decrease at higher levels. Rainfall is
a very important factor in maize production in Kenya because all of the maize fields are
rain-fed and drought is the usual cause of yield loss. We use the variable STRESS to capture
the moisture stress in maize growth. STRESS is computed as the total fraction of 20-day
periods with less than 40 millimeters of rain during the 2003-2004 main season. This is a
better measure for moisture conditions than total rainfall because total rainfall does not
reflect the distribution of rainfall over time, which is very important in maize growth.

Any observations with missing values were discarded. Because of potential measurement
errors, we also drop any observation that satisfies one of the following conditions: 1) yield

lower than 65 kg per acre or higher than 4580 kg per acre, 2) seed usage less than two kg

37

per acre or more than 20 kg per acre, and 3) labor input less than 40 person-hours per acre
or more than 2200 person-hours per acre. After these filters were applied, there are 815
fields (observations) remaining. The 815 fields were managed by 660 households. Table 3.1
summarizes the descriptive statistics for the variables included in the frontier production
function (excluding zone dummies).

Table 3.1. Descriptive statistics for the variables in the production frontier

 

 

 

Afariable Notation Mean Std. Dev. Min MEX
WIELD lVIaize yield index (kg/ acre) 1071 726 69 4410
LABOR Pre-harvest labor input (person—hour/acre) 344 271 40 2160
FERTILIZER Nitrogen fertilizer application (kg/ acre) 11 12 0 63
SEED Maize seed quantity (kg/acre) 8.5 3.3 2.5 18.8
TRACTOR If tractor used in land preparation (I=yes, O=no) 0.28 0.45 O 1
MONO If mono-crop field (I=yes, 0=no) 0.11 0.31 0 1
HYBRID If hybrid seed (I=yes, 0=no) 0.72 0.45 0 1
STRESS Moisture stress (0-1) 0.14 0.21 0 1
DRAINAGE Drainage of soil (categorical 1-10) 7.2 2.1 1 10

 

3.3.3 Exogenous Factors Affecting Efficiency

Previous studies have identified numerous factors that may limit farm productivity and
efliciency. Education is arguably an important factor and Kumbhakar, Biswas, and Bailey
([26]) find that education increases the productivity of labor and land on Utah dairy farms
while Kumbhakar, Ghosh, and McGuckin ([27]) also show that education affects produc-
tion efficiency. Huang and Kalirajan ([21]) find that average household education level is
positively correlated with technical efficiency levels for both maize and rice production in
China. Here we measure education with EDUHIGH, the highest level of education among
all household members.6 We also investigate gender effects by including a dummy variable
for female-headed households (FEMHEAD).

Physical and social infrastructure, such as road conditions, access to telephone and mobile
phone service, access to extension service, etc., have also been mentioned for their role in

rural development and farm productivity. Jacoby ([22]) examines the benefits of rural

 

6EDUHIGH may capture the effects of education on efficiency for a household better than the average
education level or the education level of the household head, in that the one who receives the highest
education can help the household head and the other household members in making production decisions.

38

roads to Nepalese farms and suggests that providing road access to markets would confer
substantial benefits through higher farm profits. Karanja, Jayne, and Strasberg ([24]) show
that distance to the nearest motorable road and access to extension services have positive
effects on maize productivity in Kenya. More developed infrastructure helps farmers to
obtain more information and thus may improve technical efficiency. Here we use three
infrastructure variables to account for these effects on efficiency—DISTBUS, distance of
the house from the nearest bus stop;7 DISTPHONE, distance of the house from the nearest
telephone or mobile phone service; and DISTEXTN, distance from the nearest extension
service office.

Land tenure is another element that affects farm performance. Secure tenure may induce
more investment (such as soil conservation) and increase farm productivity in the long run.
Place and Hazell ([38]) suggest land tenure is important to investment and productivity in
Rwanda. Puig-Junoy and Argiles ([39]) show that farms with a large proportion of rented
land have low efficiency in Spain. Here we use a dummy variable (OWNED) with one
indicating that the field is owned by the household and zero indicating the field is rented.

Financial constraints, such as limited access to credit, might also affect farm input deci—
sions and efficiency. Ali and Flinn ([3]) show that credit non-availability is positively and
significantly related to profit inefficiency for rice producers in Pakistan. Parikh, Ali, and
Shah ([37]) find that farmers with larger loans are more cost efficient in Pakistan. The
effects of financial constraints on technical efficiency seems to be unexamined to date but
may be important because the timing of input usage can be an important factor influencing
yields. So farms that face financial constraints may not be able to optimize production
because inputs were not applied at the right times. We attempt to capture this effect using
CRDCSTR (a dummy variable with one indicating the household has unsuccessfully pur-

sued credit and zero otherwise), and RNFINC (the proportion of household members that

 

7We use DISTBUS instead of how far a household is from a motorable road, because only a very small
proportion of the households in Kenya own motorable transportation tools (like tractors), and bus and
bicycles are the major transportation tools there.

39

have non-farm income).

The relationship between farm productivity and farm size has been a long-standing em-
pirical puzzle in development economics since Sen (1962) (see [10]; [8]; [28]). Empirical
results on the relationship between efficiency and farm size have been mixed. Kumbahakar,
Ghosh, and McGuckin ([27]) show that large farms are relatively more efficient both tech-
nically and allocatively. Ahmad and Bravo-Ureta ([1]) find a negative correlation between
herd size and technical efficiency, while Alvarez and Arias ([5]) find a positive relationship
between technical efficiency and size of Spanish Dairy farms. Huang and Kalirajan ([21])
show that the size of household arable land is positively related to technical efficiency in
maize, rice and wheat production in China. Parikh, Ali, and Shah ( [37]) find that cost
inefficiency increases with farm size. Hazarika and Alwang ([18]) show that cost inefficiency
in tobacco production is negatively related to tobacco plot size but unrelated to total farm
size in Malawi. Here we include farm size (TTACRES) and field size (ACRES) as measures
of the size effect. Descriptive statistics for the household survey data used to define the
exogenous factors affecting efficiency are summarized in table 3.2.

Table 3.2. Descriptive statistics for the exogenous variables in the efficiency model

 

 

 

Variable Notation Mean Std Dev Min Max
EDUHIGH # school yearSRBr memfghest educated member 12 5.5 0 24
FEMHEAD If the household head is female (I=yes, 0=no) 0.19 0.39 0 I
DISTBUS Distance to the nearest bus-stop (km) 2.4 2.4 0 20
DISTPHONE Distance to the nearest phone service (km) 0.78 1.6 0 15
DISTEXTN Distance to the nearest extension service (km) 5.2 4.5 0 33
OWNED If the field owned by the household (I=yes, 0=no) 0.86 0.35 0 l
CRDCSTR If pursued credits and was rejected (I=yes, 0=no) 0.08 0.27 0 1
RNFINC % of members that have non-farming income 0.20 0.19 0 1
TTACRES Total acres of land owned by the household 7.46 10.9 0.13 110
ACRES Acres of the field 1.46 2.01 0.03 27

 

3.4 Estimation Results from Competing Models

In this section, we report the estimation results under alternative model specifications for

the inefficiency component of the model. We use a flexible translog functional form for

FERTILIZER, LABOR, and SEED in the frontier production function. We also inter-

40

act the dummy variable for hybrid maize (HYBRID), and the variable for moisture stress
(STRESS), with FERTILIZER, LABOR, and SEED because there may be important in-
teractions between these variables. With these choices, and given the constant and seven
agro-economic zone dummy variables and 03, there are 30 parameters in the frontier pro—
duction function. Furthermore, if we use the AAOS general model, there are another 22
parameters models used to specify 11, and 0,. So the total dimension of the parameter space
is 52. Even for the simpler models, such as the scaled Stevenson model, the KGMHLBC
model, and the RSCFG model, the total parameter dimension is still large (42 parameters).
To maximize a likelihood with such a high dimension can be computationally difficult given
the complexity and non-regularity of the likelihood function. We do not want to elimi-
nate potentially important variables eat-ante, nor do we want to sacrifice the flexibility of
the general inefficiency model or use a less flexible frontier production function, such as
Cobb-Douglas.

In our analysis we follow a three-step procedure to restrict the dimensionality of the

parameter space:

1. Estimate the most general form of the production frontier using OLS and excluding
the inefficiency component. Then drop jointly insignificant variables using F tests.

This yields a reduced parameter space for the production frontier component.

2. Estimate the production frontier and the inefficiency component jointly with MLE
using the general model for n, and 0,, but using the reduced set of explanatory
variables for the production frontier obtained in step 1. Then drop jointly insignificant
exogenous factors from the inefficiency component using likelihood ratio (LR) and

Wald tests. This gives a reduced set of exogenous factors in the efficiency component.

3. Estimate the production frontier with the full set of explanatory variables and the
efficiency component with the reduced set of exogenous factors after step 2 in one

step using MLE. Then using LR and Wald tests, test the joint significance of the

41

explanatory variables dropped in the first step.

The OLS estimates in the first step are inconsistent if some of the :r,’s are correlated with
e, = v, — 11,, or if the distribution of e, is heterogeneous. In this study, we do not assume
away either of these two possibilities. Therefore, our procedure to reduce the dimension
of production frontier based on the OLS results in the first step may be inconsistent. To
overcome this weakness, we take the third step of undertaking the same exclusion restrictions
with a consistent estimator. Complete results from applying this procedure to the Kenya
maize data are available in appendix E.

Using this procedure to impose zero restrictions, and then estimating the zero-restricted
model for alternative specifications for the inefficiency part using MLE, leads to the esti-
mation results provided in tables 3.3 and 3.4. Table 3.3 contains results for the frontier
parameters and table 3.4 contains the inefficiency parameters, each under alternative model
specifications for the inefficiency part. In the frontier model the variable selection criteria
led seven parameters to be restricted to zero (second order effects for LABOR and SEED, all
interaction effects among FERTILIZER, LABOR, and SEED, as well as interaction effects
for SEED and HYBRID, and FERTILIZER and STRESS—see table 3.3). An additional
two of the zone dummy variables (not reported in table 3.3) were also restricted to zero for
a total of nine restrictions. In the inefficiency model, the variable selection criteria led to
four effects being eliminated in the general model (DISTPHONE, DISTEXTN, CRDCSTR,
and ACRES). But because these four effects enter both the mean and the variance terms
in the general model this amounts to a total of eight zero restrictions—see the first column
of table 3.4.

The parameter estimates for the frontier part of the model are very similar across alter-
native models for the inefficiency component (see table 3.3). Furthermore, Both the LR
test and Wald test reject the null hypothesis that all the exogenous factors have zero effect
on inefficiency at the 1% significance level in each of the five models (see table 3.4). Hence.

it seems clear that the exogenous factors have a statistically significant effect irrespective

42

Table 3.3. Estimates for the production frontier in alternative models

 

 

 

 

LYIELD General Scaled Stevenson KGMHLBC RSCFG-p RSCFG
TFERTILIZER .15 .020 .15 .020 .15 .020 .15 .020 .15 .020
LLABOR .33 .050 .33 .052 .33 .049 .33 .052 .33 .052
LSEED .33 .048 .32 .050 .33 .048 .32 .050 .32 .050
LFERTILIZER2 .025 (.004) .026 (. 004) .026 (. 004) .026 (.004) .026 (.0 04)
LLABOR2 0 0 0 0 0
LSEED2 0 0 0 0 0
LFERTILIZERXLLABOR 0 0 0 0 0
LFERTILIZER x LSEED 0 0 0 0 0
LLABORXLSEED 0 0 0 0 0
LFERTILIZERXHYBRID -.062 (.016) -.063 (.016 -.063 (.016) -.063 (.016) - 063 (. 016 )
LLABORXHYBRID -.16 8059) .15 [)061) -.16 8059) -.16 8.061) - .15 [)06 0)
LSEEDXHYBRID
LFERTILIZERXSTRESS 0 0 0 0 0
LLABORXSTRESS -.23 14 -.29 (014) -.26 .14] -.29 [.14 - .29 (j 4)
LSEEDXSTRESS -.29 .17 —.28 (.19) -.29 .17 -.27 .20 - 29
HYBRID .19 (.063) 20 (.059) 20 ( 063) 20 (.059) 20 (. 059)
STRESS -.38 (.18) - 36 (.18) -.39 (.18) -.36 (.18 -.37 .18
MONO -.22 (.059) - .21 (. 060) -.23 (.058) -.21 (.060) -.21.60
DRAINAGE .15 (.056) .13 (. 056) .15 (.055) .13 (. 057) .13 (.056)
DRAINAGE2 - 012 (.005) -.001 (.005) -.011 (.005) - .001 (.005) - .001 (.005)
TRACTOR 5.( 056) .15 (.051) .15 (.057) .14 (.050) .15 (.051)
Constant, Zone Dummies not reported
”,2, .16 (.023) .14 (.023) .15 (.020) .15 (.022) .13 (.021)

 

Note: LYIELD is log YIELD. LFERTILIZER, LLABOR and LSEED are defined similarly. Standard

errors are in parentheses.

of the model specification employed to model inefficiency. The Battese and Coelli efficiency
estimates are computed for each observation in all the models and their correlations across
alternative models are reported in table 3.5. The lowest correlation is 0.97. Therefore, all
five models yield similar results for the production frontier and for the rankings of ineffi-
ciency among households, consistent with previous studies (e.g. [14]).

Goodness of fit statistic for the inefficiency component, R2, are reported at the bottom
of table 3.4 for the alternative model specifications. For example, the value of R3 for the
KGMHLBC model is 0.1035, indicating that 10.35% of the sample variation in inefficiency
can be explained by the exogenous factors. Not surprisingly, the general model provides
the best fit at 12.75%.

The coefficients of the exogenous factors reported in table 3.4 are not very interesting
by themselves because they are the parameters of the pre-truncated distribution of the
inefficiency term 21,. So these parameters do not tell us how the exogenous factors affect

the distribution of 11,. In order to quantify the effects of exogenous factors, we compute

43

Table 3.4. Estimates for the inefliciency components in alternative models

 

 

 

 

 

 

LYIELD General Scaled Stevenson KGMHLBC HISCFG-p RSCFG
Variables in function 112
p -4.1(6.9) -0.30(0.36) -1.45(0.72) -0.75(0.40) 0
EDUHIGH 0.034(0.049) -0.018(0.0068) 0.053(0024) 0 0
FEMHEAD -5.3(41) 0.22(0.093) -2.3 2.0) 0 0
DISTBUS 0.368016) 0.048(0.016) -0.31 0.14) 0 0
DISTPHONE 0 0 0 0
DISTEXTN 0 0 0 0
OWNED 4.4810) 0.35()0.11) -1. 3(0. 41) 0 0
CRDCSTR 0 0 0
RNFINC 0.82[1.2) -0.36 0.19) 4.(0 73) 0 0
TTACRES 0.0018 0.045) —0.013([)0.003) 0. 024(0. 012) 0 0
ACRES 0 0 0 0
Variables in function 0,2
02 27(59) 0.42(0.13) 0.59(0.14) 0.54(0.12) 0.34(0.11)
EDUHIGH -0.0063(0.015) -0.018(0.0068) 0 -0.014(0.0048) -0.032(0.014)
FEMHEAD ~0.22 0.28) 0.22(0.093) 0 0.18(0.072) 0.41 0.17)
DISTBUS -0.014 0.044) 0.048%).016) 0 0.040(0.012) 0.087 0.030)
DISTPHONE 0 0
DISTEXTN 0 0 0 0 0
OWNED -0.061(0.46) 0.358011) 0 0.288073) 0.63%022)
CRDCSTR 0 0
RNFINC -0.14 0.36) -0.36 0.19) 0 -0.29(0.15) -0.63 0.38)
TTACRES -0.012 0.013) -0.013 0.003) 0 -0.011(0.0015) -0.020 0.014)
ACRES 0 0
firebservations 815 815 815 815 815
Log-likelihood -616.30 -623.63 -618.71 -623.42 -623.70
TR statistic 56.84 34.54 50.62 38.36 37.93
Wald statistic 26.80 18.28 29.74 77.69 27.17
1% critical value 26. 22 16. 81 16.81 16.81 16.81
R2 0.1275 0.0848 0.1035 0.0936 0. 0773

 

Nofie: Standard errors are in parentheses. The LR and Wald statistics test the null hypothesis that the

exogeneous factors have no joint influence on inefficiency.

Table 3.5. Correlation of efficiency estimates among alternative models

 

 

GeneraI Scaled Stevenson KGMHLBC RSCFG-n RSCFG

 

GeneraI 1

Sealed Stevenson 0.9793 1

KGMHLBC 0.9910 0.9848 1

RSCFG-p 0.9839 0.9986 0.9843 1

RSCFG 0.9700 0.9970 0.9833 0.9917 1

 

B[E(—u,[x,, 2,)]/02, and 8[V(u,[:r,, 2,)]/82, for each observation. The formulas for comput-

ing these measures and their standard errors for the general model are provided in appendix

D. To obtain the formulas for the nested models, we only need to impose the corresponding

restrictions on the parameters.

8

The partial effects of the exogenous factors evaluated at the sample mean are reported in

 

8Wang (2002) gives the expressions for these derivatives but not for the standard errors.

44

table 3.6 along with their standard errors. The signs of the partial effects are the same for
all the models. However, different models give quantitatively different values for the partial
effects. For example, the partial effects of TTACRES on the conditional mean of —u range
from 0.0023 to 0.0072, and these differences are large relative to the standard errors of the
estimates. So conclusions about the semi-elasticity of output with respect to farm size may
differ by a factor of more than 100%, depending on which inefficiency model is used.

Table 3.6. Partial effects of exogenous factors, evaluated at the sample mean

 

 

 

 

General Scaled Stevenson KGMHLBC RSCFG-p RSCF G
Partial effects on E(—u,|a:,, 2,)
EDUHIGH .0080(.0044) .0079 .0012) .0052(.0044) .0080(.0008I) .0081 .0029)
FEMHEAD -.12[.11) -.10 .051) -.14(.058) -.11 .049) -.11 .052)
DISTBUS -.037 .025) -.021 .0038) -.037(.016) -.022 .0028) -.022 .0083)
OWNED -.19(.074) -.14 .047) -.I7(.052) -.14 .042) -.14 .058)
RNFINC .19[.12) .16[.039) .13[.11) .17[.028) .16(.090)
TTACRES .0075 .0021) .0058 .00067) .0023 .0015) .0061 .00040) .0049(.0023)
Partial effects on V(u,[:1:,,2,)
EDUHIGH -.0042(.0020) -.0045(.0015) -.0024(.0020) -.0044(.0012) -.0045(.0016)
FEMHEAD .035 .058 .064(.037) .066(.026) .063(.034) .065(.038)
DISTBUS .016 .013 .012(.0055) .017(.0072) .012(.0049) .012(.0057)
OWNED .083 .040 .070(.029) .078(.021) .068(.026) .071(.035)
RNFINC -.097[.062) -.091 .048) -0.061(.050) -.091(.043) -.088[.051)
TTACRES -.0046 .0016) -.0033 .0011) -.0011(.00070) -.0033(.00083) -.0028 .0014)

 

Note: Standard errors are in parentheses.

Table 3.7 reports the average partial effects of EDUHIGH on E (—u,[:r,, 2,) for alternative
model specifications over observations within each of the four quartiles of the efficiency
levels.9 The KGMHLBC model shows an increasing trend of the partial effect of education
on efficiency levels from low to high quartiles, while the scaled Stevenson model, RSCFG-[u
model and RSCFG model suggest a decreasing trend. So using the KGMHLBC model we
would conclude that the households with lower efficiency levels would not benefit as much
from increased education as the ones with higher efficiency levels. However, an opposite
conclusion would follow if we use the scaled Stevenson model, the RSCFG-u model or the
RSCF G model.10

Table 3.8 reports the correlations of partial effects of EDUHIGH on E (—u,|:r,, 2,) among

 

9The quartiles were computed using the KGMHLBC model.
10 Similar patterns are observed for the other exogenous factors but these results are not reported to
conserve space.

45

Table 3.7. Average partial effects of EDUHIGH on E (—u, [27,, 2,), for the observations within
each of the four quartiles based on efficiency levels predicted in KGMHLBC model

 

 

General Scaled Stevenson KGMHLBC TISCFG-n RSCFCT

 

0—25% percentile 0.0067 0.0092 0.0039 0.0092 0.0092
25-50% percentile 0.0074 0.0085 0.0052 0.0085 0.0085
50-75% percentile 0.0078 0.0080 0.0059 0.0081 0.0081
75-100% percentile 0.0079 0.0069 0.0072 0.0070 0.0071

 

alternative models. Most correlations are very low and some are even negative.11 This
further confirms that different models yield rather different partial effects. Therefore, if we
are only interested in the signs of the yield semi—elasticities with respect to exogenous factors,
model specification is not important. However, if we are interested in the magnitudes of
the yield semi-elasticities, it is important to choose the appropriate model specification.

Table 3.8. Correlation of partial effects of EDUHIGH on E(—u,[r,, 2,) among alternative
models

 

 

General Scaled Stevenson KGMHLBC RSCFG-p RSCFG
General 1

 

Scaled Stevenson -0.3910 1

KGMLBC 0.7811 —0.7899 1

RSCFG-n -0.3716 0.9991 -0.7861 1

RSCFG -0.4140 0.9882 -0.8047 0.9970 1

 

3.5 Model Selection

In this section, we apply the procedure proposed by AAOS to select an appropriate model
for our empirical application. A bootstrap analysis then follows to evaluate the performance

of the model selection procedure.

3.5.1 Empirical Model Selection

We start with the general model, and then use LR tests to find simpler models that the

data do not reject. Estimation of the general model yields a log-likelihood value of -616.30.

 

11 Similar patterns are observed for the other exogenous factors but these results are not reported to
conserve space.

46

Table 3.9 reports the log-likelihood values for the six restricted models nested in the general
model. Taking the general model as the unrestricted model, we then test the restrictions
that would reduce the general model to simpler specifications. LR test statistics with Chi-

squared critical values are listed in table 3.9 and provide the following results:

0 We can reject the scaled Stevenson model (6 = 'y), RSCFG-p. model (6 = 0), and
RSCFG model (a = 0) at the 5% significance level.

0 We fail to reject the KGMHLBC model (7 = 0) at any reasonable significance level.

0 We can reject the Stevenson model (6 = '7 = 0) and ALS model (,11 = "y = O) at any

reasonable significance level.

Because both the Stevenson model and ALS model are rejected, we conclude that the
exogenous factors do affect efficiency. Among RSCF G, RSCFG-11, and scaled Stevenson
models, the RSCFG model is preferred because we fail to reject the RSCFG model at any
reasonable significance level using the RSCFG—p model or the scaled Stevenson model as
the unrestricted model. Moreover, among all the models, the KGMHLBC model is most
preferred because it is the only one that we can accept at any reasonable significance level.
Therefore, we select the KGMHLBC model as our final model.

Table 3.9. Results of specification tests for model selection

 

 

Scaled Stevenson KGMHLBC RSCFG-p RSCFG Stevenson ALS

 

 

log-likelihood -623.63 -618.71 -623.42 -623.70 -64I.44 -642.04
LR statistics 14.66 4.82 14.24 14.80 50.28 51.48
# restrictions 6 6 6 7 12 13
1% c.v. 16.81 16.81 16.81 18.48 26.22 27.69
5% c.v. 12.59 12.59 12.59 14.07 21.03 22.36
10% c.v. 10.64 10.64 10.64 12.02 18.55 19.81

 

The value of log-likelihood for the general model is -616.30.

3.5.2 A Bootstrap Evaluation

The model selection procedure proposed by AAOS leads to one clearly preferred model, the

KGMHLBC model, among the set of competing models. However, it is also relevant to ask

47

about the reliability of the model selection criterion, which is a question of the size and
power properties of the LR tests. We investigate this question using the bootstrap. That
is, we generate data via the bootstrap assuming that the KGMHLBC model is correct, and
then we see how reliably the model selection procedure picks the KGMHLBC model. So
far as we are aware this approach has not been used previously in the literature. It is useful
because we are using the bootstrap to evaluate the probability with which the actual model

selection procedure will pick the correct model.

The KGMHLBC model is written as
y, = 2726 + v, — 11,, where u, N N[n - exp(2[6), 0,2,]+ and ’0, ~ N(0,0,2,). (3.10)
We take the following steps to conduct the parametric bootstrap:

1. Using the actual sample data {(y,,:r,, 2,)}?21, estimate the KGMLBC model using
MLE to get 6 = {3,5,11,03,03}. These results are provided in tables 3.3 and 3.4.

2. Next generate pseudo—data sets based on the parameter estimates from step 1. That
is, for z' = 1,. . . ,n, draw 11.: from N[11- exp(2[6),0,2,]+, '0: from N(0,03), and then
*

compute y: = mgfi + 12;" — u, .

3. Based on the pseudo—data {31: , 2,, 2,};1 generated in step 2, estimate all seven ineffi-
ciency models using MLE. Take the log-likelihood value (11* ) and parameter estimates

(0*) in each of the models, denoted as C“ = {(165, 63‘) 3:1, where j indexes the differ-

ent models.
4. Repeat steps 2 and 3 1000 times to obtain % = {cg},1,2010.

We use the log—likelihood statistics in Q to conduct the AAOS specification tests for each
pseudo-data set, taking the general model as the unrestricted model and conduct LR tests

at the 5% significance level. The results are:

48

0 We reject the true model in 5.7% of the pseudo-data sets, the scaled Stevenson model
in 75% of the pseudo-data sets, the RSCFG—n in 78% of the pseudo-data sets, and
the RSCF G in 75% of the pseudo-data sets.

0 We reject both the Stevenson model and the ALS model in 99.9% of the pseudo-data
sets. That is, in only one of the 1000 data sets, we would wrongly conclude that the

set of exogenous factors do not affect efficiency.

0 We accept the true model and reject all of the other models in 66.0% of the pseudo-
data sets. We reject the true model and accept an alternative one at the same time

in only 0.4% of the data sets.

0 In 28.4% of the pseudo-data sets, we simultaneously accept the true model and at
least one of the alternative models. And we reject all of the models simultaneously in

5.3% of the data sets.

These results suggest that the AAOS model selection criteria do a good job of discrimi—
nating between models. If the KGMHLBC model is correct, the model selection procedure
will reject it with small probability (6%), and will pick it unambiguously with relatively
high probability (66%).

The bootstrap results also can be used to generate confidence intervals for any of our
original estimates. These confidence intervals may be more accurate in finite samples than
those generated by first order asymptotic approximations such as the delta method. For
example, we can use the parameter estimates of the KGMHLBC model in Q to compute
the partial effects for every observation in each pseudo-data set. Confidence intervals then
follow directly from the set of .% estimates. For example, given 1000 pseudo-data sets a
90% confidence interval for a parameter ranges from the 50th to the 950th largest values
of the bootstrap estimates of that parameter. This is called the “percentile bootstrap”.
Table 3.10 reports 90% percentile bootstrap confidence intervals for the partial effects in

the KGMHLBC model, evaluated at the sample mean. For purposes of comparison, it also

49

gives the 90% confidence intervals based on the delta method (i.e. using the standard errors
computed as in appendix D and reported in table 3.6). The confidence intervals given by
bootstrap and the delta method are not very different. This confirms the reliability of the
delta method.

Table 3.10. Partial effects of the exogenous factors on E(-u,[:1:,, 2,) and their 90% confidence
intervals based on bootstrap and the delta method in the KGMHLBC model, evaluated at
the sample mean

 

 

EDUHIGH FEMHEAD DISTBUS OWNED RNFINC TTACRES
.0052 -.I4 -.U37 -.19 .13 .0023

Bootstrap (00047, .011) [-.22,-.048[ (-.058,-.0078) -.2s,-.035[ [011,30] (.00011, .0053)

Delta M. (-0020, .012) -.24,-.045 (-.063,-.011) -.26,-.084 -.051, .31 (-.0017, .0048)

3.6 Post-Estimation Analysis

Post-estimation analysis is based on the results of our selected KGMHLBC model. Table
3.11 reports output elasticity estimates for local seed users and hybrid seed users calculated
at their respective sample means with their standard errors in parentheses.12 The sum
of the output elasticities with respect to FERTILIZER, LABOR, and SEED is less than 1
(0.80 for local seed users and 0.74 for hybrid seed users). However, this is expected and
does not mean the technology is decreasing returns to scale because we are holding land
constant (production is measured as yield per acre). Results show that output elasticities
with respect to FERTILIZER and SEED are higher for hybrid seed users than local seed
users, but the output elasticity with respect to LABOR is higher for local seed users.

Table 3.11. Output elasticity with respect to inputs for local seed users and hybrid seed
users, evaluated at the sample means

 

 

 

Inputs Local seed users Hybrid seed users
FERTILIZER 0.209 (.00076) 0.224 .0011
LABOR 0.300 [.0027] 0.177 .0063
SEED 0.293 .0032 0.336 .0026

 

Note: Standard errors are in parentheses.

 

’2 The means of FERTILIZER, LABOR, and SEED are computed after taking logarithms.

50

Figure 3.6 plots the density of the Battese and Coelli technical efficiency estimates. The
minimum efficiency level is 18% and the maximum is 98%. The mean of technical efficiency

is 71%, while the mode is around 80%. The distribution is left skewed.

 

Densrty

 

 

I I I l I

.4 .6
tech efficiency index of E(exp(-u)|e)

 

 

 

Figure 3.2. Kernel density estimate based on Battese and Coelli technical efficiency esti-
mates

R2 suggests that about 10% of the sample variation in inefficiency can be explained by the
set of exogenous factors (see bottom of table 3.4). From table 3.6, EDUHIGH, RNFINC
and TTACRES all have positive partial effects on the mean and negative effects on the
variance of efficiency. FEMHEAD, DISTBUS, and OWNED all have negative effects on
the mean and positive effects on the variance of efficiency. Therefore, an average household
tends to have a higher efficiency level and a lower uncertainty on efficiency if it has a higher
education level, more off—farm income, or larger farm size. Alternatively, it tends to have
a lower efficiency level and higher uncertainty of efficiency if it has a female head, or is far

from a bus-stop.

51

These results are mostly consistent with a priori reasoning and the previous literature.
The effects of education, credit constraints, farm size and infrastructure on efficiency have
been discussed extensively in the previous literature. The effect of female head could be due
to the fact that females are subject to social discrimination in Kenya. There are generally
two situations in which a female can become the head of a household. One is that she
is a single mother, and the other is that her husband is dead. Females do not have the
same inheritance rights as males in rural Kenya. A widow cannot obtain full rights to the
land left by her husband and has to give away a certain proportion of the harvest to her
husband’s brothers. This may reduce the incentive to work intensively.

A surprising result is that farmers tend to be more efficient in rented fields than in their
own fields. There are possible two reasons: 1) a fixed rent has to be paid at planting time,
which provides more incentives for farmers who work in a rented field than in their own
fields; 2) farmers rent fields that they know are productive. To the extent the second reason
is a factor, the variable OWNED might capture the unobserved land quality not included
as a covariate in the production frontier.

As explained earlier, not only the directions but the values of the partial effects on
E(—u,[:r,,2,) are of economic interest. According to the KGMHLBC model (see table
3.10), one more school year would increase yield per acre by a little over half a percent for
an average household, ceteris paribus. Being one kilometer closer to public transportation
would increase yield per acre by 3.7 percent. An increase of one acre in farm size would raise
yield per acre by less than one third of a percent. If the proportion of household members
who receive off-farm income increases by 10 percent, yield per acre would increase by 1.3
percent. However, using the same amount and the same quality of inputs, a household with
a female head tends to produce 14 percent less maize than a household with a male head,
and farmers tend to produce 17 percent more maize working in rented fields than in their

own fields.

52

3. 7 Conclusion

This paper makes three contributions to the stochastic frontier literature. First, we provide
formulas to compute the partial effects of exogenous farm characteristics on output levels
and their standard errors for alternative model specifications. We also develop an R2-type
measure that shows the explanatory power of the exogenous factors that affect inefficiency.

Second, we examine the effects of model selection on inferences about firm inefliciency
by applying several popular model specifications for the effects of firm characteristics on
firm efficiency. The application is to Kenyan maize production data and we find that
different specifications provide similar efficiency rankings of households and predict the
same directions for partial effects of exogenous factors. However, the magnitudes of these
estimated partial effects are rather different across model specifications. This finding calls
for more attention to model selection in empirical stochastic frontier analysis.

Third, we apply the specification tests recently proposed by Alvarez, Amsler, Orea, and
Schmidt ([4]) to choose between alternative model specifications for the Kenyan maize data.
In our application these tests yield an unambiguous choice of best model, and an analysis of
the model choice procedure using the bootstrap indicates that the model choice procedure
is reliable. To our knowledge, bootstrapping has not been used previously to examine the
size and power of these model selection criteria.

The empirical application uses the preferred model to identify factors that limit technical
efficiency in maize production in Kenya, and quantify their partial effects on maize yields.
We examine the effects of education, female head of household, distance from a bus stop,
land owned or rented, extent of off-farm income, and farm size on the level of efficiency.
Approximately 10% of the variation in efficiency levels is accounted for by these household
characteristics, and while education, non-farm income, and farm size increase technical
efficiency, female-headed households, distance from a bus stop, and land being owned rather

than rented all decrease it.

53

APPENDICES

54

APPENDIX A

Derivation of Equation (2.2)

By definition of lognormality, Z,- can be written as Z,- = cu}. +0112 where x,- is standard

normal, p,- = E(2,), and for 2,- : ln(Z,-). Then equation (2.1) can be written as:

 

(gj-ujl/Uj . . . 1 _ 2. 2
Pj(Gj) = fi/ (C¥J'-€“’+"in)-2—e $J/ dl‘j
-00 1r
[ (IT—Vl/U' _2 (g°-ir)/0' , ,, 1 _2
= fl Gj/ J J J I e :rJ/2dxj_/ J J JepJ-Jrgjxj e xJ/2dxj
b --00 V277 —oc V271
P gj—uj 1.40.502 /(g,—u,)/o, 1 —(:r-—0')2/2
= 0N —— — J J —— J J d2:-
f’ , f a, i e -0. «276 ,
_ -— ,- . 2 -— --02-
= fl GjN (g, [l])_euj+0.50jN(gj H] J):[,
L Uj Uj

 

where N () is the cumulative distribution function for the standard normal and g,- = ln(G,-).

55

APPENDIX B

Derivation of Equation (2.10)

Equation (2.8) can be written using the CRRA assumption as:
E {Ci—“[max(G,- — 2,, 0) — 13,1} : 0 (3.1)
or

[000 /OG’ 0,“ Z,-)f(Z,-,C',-)dZ,- dC,-= P,,.—E(0 01) (13.2)

Using the same notation as in A above and defining C, = euci‘l’acifci with yo, = E [ln(C,)]
and 02, = Var[ln(C,)] where :0,- and :06, are bivariate standard normal with correlation

coefficient p,,-, then (8.2) can be expressed:

/: /:ULJP1 e‘"(llc,'+0a dang-HQ- euj+0j$ j)f(x,,xc,)dx,dg;c, =P,“E[e (um-+00, 3:0,),
(13.3)
or
M 00
/ Uj (Gj—e#j+ajmj)f(xj)d$j/ e-OWUU‘ffo’flxct | 23,-)dscc, = PjE[e—a(#ci+acixci)]'
‘°° m (13.4)

Also, from the properties of bivariate normal distributions, we have:

 

f(ivci I 5173') = 2 expff-Ta' - Pij-Tj)2/12(1— Pfjfif (13-5)

56

Substituting (85) into (8.4) and doing some algebra, we get

“L19. H' 2 2 2
a- . . . — -— -- - ,- ,I' — .. . . . .
/ _] (Gj _ eIIJ-l-OJ‘T] )6 ”(‘02 purioarj-i-O 0(1 1)”)(1 0C7'f(.rj)d23j = PjE[e—a(nm+0c,$c,)]
-—oo

(B.6)
Furthermore, we know that f(x,) = fiexpwgfl) and E{exp[—a(nc, + 0C,xc,)]} =

exp(—a/rc, + 050202,). Substituting these expressions into (B.6) gives

92 ”2' 2
/ 0] (Gj _ ejIj-f-szrj)e—(xj+p,jrmc,) /2d.’13j = Pj (B.7)

00

Of

 

g' — H. #I'+0.502--—ap- -0-0 - g- — pg
Pj = GjN( J 0- j '— ap,ch,) —6 J J I] J GIN (JG—“l + Opija'ci -~ 0,) (8.8)
3 J

Finally, notice that p,,-0,0C, = Cov(c,, 2,) E 0,,- so that (8.8) becomes (2.10).

57

APPENDIX C

Derivation of Equation (2.13)

Equation (2.13) can be easily derived from (2.12) if we show Cov(y, 2,) a: pyzj CVyCVZj
and Cov(w,2,) z pWZjCVWCI/Zj, where y = ln(Y), 2,- : ln(Zj) and w = ln(W). To
begin, note that:

Cove/,2) E(YZ-)—E(Y)E(Z~) E(YZ-)
pYZjCVYCVZj=E(1/)E(ZJ,)= E(Y)E(Z,~) J =E(Y)E(JZ,)

 

— 1. (Cl)
Then from the properties of bivariate normal distributions, we have

E(YZJ') = E[exp(y + 2,)]
= exp [E(y) + E(2,-) + 0.5Var(y) + 0.5Var(2,) + Cov(y, 2,)]
= exp [E(y) + 0.5Var(y)] x exp [E(Zj) + O.5Var(2,~)] X exP [001431, 2,)]

= E(Y)E(Z,-) exp [Cov(y, 2,)]
Substituting this result into (B5), and rearranging gives:
Cov(y, Zj) = In (I + pyszVyCI/Zj) z PYZjCVYCVZj- (0.2)

Cov(w, 2,) a: PWZ,CVWCVZ, can be shown similarly.

58

APPENDIX D

Estimating Partial Effects of
Exogenous Factors and their

Standard Errors for the General

hdodel

Assume there are K exogenous factors (K 1 continuous variables and K2 = K — K1 dummy
variables). We deal with the continuous variables first. Let 2,0 be the K1 dimensional vector
of the continuous variables. We derive the partial effects of 2,0 on the mean and variance of

efficiency via differentiation as

5E(-Ui|$i. 211/34? = 7601(3133 - R2) - 56011310 + R3) (D-l)

0V(u,[$,, .20/(92,-C : 270012“ '1' R3 '1' R4) — 6602:2124, (D2)

where n, = n . exp(2z’-6), 0, = 0,, - exp(2£7), (SC and 7C are the coefficient vectors associated

with 2,6, R1, R2, and R3 are as defined in the text, and R4 = R1(R2 + R1R3 + 2R2R3).
Next we derive the variances of the partial effects of 2?. Let 6’ = (6' 7’), and 9(6) =

0[E(—u,[:r,, 2,)]/62,C, and h(6) = 6[V(u,|z,, 2,)]/62,C, where both 9(6) and 11(6) are K1 x 1

59

dimensional vectors. Following the delta method,

mare—gran —: Nro.(ag(9))0(a~f’“))'[.

 

 

66’ (96’

L

 

 

 

\/7_1[h(6)— 11(9)] —. 1v F0, (we) 0 (8h(6))l[ .

66’ 66’

We derive Bg(6)/86’, 09(6)/37’, 0h(6)/86' and 0h(6)/87’ as

‘99—“?
66’
09(9)
87’
(9)1(6)
86’
6h(6)
87’

 

 

 

-Uz'(7C

017023-31 - R2 - RlR41+ 02113031133 - R2) + 0156435.

0,2 [7C2

2i + D)R1(1+ R3) - 01(56 - 761435.

;(R6 — 2R4) — 602,126 — R40] ,

where D = [1K1 0K1 x K2] is a K1 x K dimensional matrix, and

86

R5

Rs

0.; a) = [97(0) 97(0)] and 05(0) = [116) 5(0)

7] are K1 x 2K dimensional matrices, which
69 6

= R1(1+ R3) — R134.

= R4 + R1(2R1R3 + 2R1R§ — R1R4 — 2R2R4).

(967 6

(0.3)

(0.4)

(0.5)
(D.6)

(0.7)

._ 0,2 [7cz§(4 + 4R3 + 4R4 — R6) + 6622036 — 2R4) + 2(1+ R3 + R4)D](,D.8)

(0.9)

(0.10)

depend on the model parameters 6 and 7. We can get the estimates of $7) and 3%? by

substituting the estimates of 6 and 7 into the above formulas. The variances of the partial

effects can be estimated by substituting the estimate of

69(6
66

variance-covariance matrix of 6 into the formulas (D3) and (D4).

as well as the estimate of the

Next we compute partial effects of dummy variables. Let 2,,C be the dummy of concern.

The partial effect of 2,,c on E(-u,[:r,, 2,) and V(u,|:r,, 2,) are

E(—Ui[$i,2i,2ik =1)— E(—’U,i[.’lfi, Zia Z’ik = O)

= l-UilRl + R2)llz,k=1 - I—szRl + R2)Ilz,k=0

= V(u,[2:,,2,,2,k = I) - V(u,[a:,, Z,, 2,}, = 0)

= [030 + R3)llz,k:1-[0f(1+ 12010,:

60

(0.11)

(0.12)

Similarly, following the delta method, we have

We then have 6d(6)/86’, (9d(6)/67', 6r(6)/86', and 67(6)/87’ as follows

aawyas’
5d(9)/37'
67(6)/66’
0r(6)/87'

fild(9)—d(0)l —+ N 0.(
l.

filr(9)-r(6)l —: N 0.(

 

l-UiRilRl + R3)Ziliz,k=1 - {-0219de + R3)Zillz,k=0

l—Ui(R2 - R1R3)Zil|z,k:1 - l-Ur(R2 - R133)zil|z,k:0

8d(6)
66’

67(6)
86’

 

 

M
M

l—02'2R4szlzikT—l - I-UER4ZfIIZ,k=O

[(2 + 2R3 + R4)02'22fliz,k=1 - [(2 + 2R3 + R4)0i2z2"llz,k=0

8d(6)
596’

67(6)
36’

 

 

)..

.l

 

)1

(0.13)

(0.14)

(0.15)
(0.16)
(0.17)

(D.18)

(26%;) = [$63) 91%)] and 8g]? = [5%) [$2] are 1x 2K dimensional matrices. The variances

of the partial effects for Zik can be estimated similarly as for the continuous variables

described earlier.

61

APPENDIX E

Results of Specifying Relevant

Explanatory Variables

Table E.1 reports the first-step OLS estimates for the original production frontier and the
OLS estimates after we drop nine jointly insignificant variables. Standard errors are in
parentheses. Robust standard errors using the Huber—White sandwich estimator of vari-
ance for households as clusters are used to model heteroscedasticity and autocorrelation
among fields planted by the same households. Nine variables are dropped based on the F
test [F(9,659)=0.25, P value=0.986]. Only STRESS, DRAINAGE and DRAINAGE2 are
individually insignificant at the 10% significance level in the remaining explanatory vari-
ables. However, STRESS is marginal [P value=0.112]. DRAINAGE and its squared term
are jointly significant both in the original model [F(2,659)=4.57, P value=0.0107] and in
the specified model [F(2,659)=4.72, P value:0.0092]. Therefore, we keep these variables in
the production frontier component.

Table E.2 reports the second-step results of the LR and Wald tests for the general model,
the scaled Stevenson model, the KGMHLBC model, the RSCFG-n model, and the RSCFG
model respectively. Our unrestricted models are the ones with the full set of exogenous

factors. The restricted models have the reduced set of exogenous factors (EDUHIGH,

62

Table E.1. Specifying variables in the production frontier using OLS

 

 

 

 

Ln(outp1fl Original Model Specified Model
LnN 0.15 0.024 0.15 (0.023)
LnLabor 0.36 0.072 .34 (0.060)
LnSeed 0.25 (0.11) .32 0.059
LnN2 0.025 (0.0049) 0.025 (0.0047)
LnLabor2 -0.00002 (0.036) 0
LnSeed2 -0.065 (0.099) 0
LnN anLabor -0.0023 (0.013) 0
LnN anSeed -0.00031 (0.022) 0
LnLaboranSeed -0.45 (0.079) 0
LnN xHybrid -0.058 (0.019) -0.061 (0.020)
LnLaborxHybrid -0.19 (0.089) --.17 (0.071)
LnSeedxHybrid 0.094 (0.15) 0
LnN xStress 0.026 (0.046) 0
LnLaborxStress -0.31 0.16 -0.25 0.14
LnSeedetress -0.39 0.25 -0.43 0.25
Hybrid 0.24 (0.069) 0.23 (0.069)
Stress —0.34 (0.30) -0.33 (0.21)
Mono -0.23 (0.070) -0.23 (0.070)
Drainage 0.10 (0.068) 0.10 (0.066)
Drainagez -0007 (0.0065) -0007 (0.0063)
Tractor 0.19 (0.065) 0.19 (0.057)
Zones Dummies not reported
7}- Observations 815 815
R—squared 0.5236 0.5220

 

FEMHEAD, DISTBUS, OWNED, RNF INC, and TTACRES). Both the LR test and the

Wald test fail to reject the null hypothesis that the four exogenous factors we dropped
(DISTPHONE, DISTEXTN, CRDCSTR, and ACRES) all equal zero at any reasonable

significance level for each of the models.

Table E.2. Tests results for specifying the exogenous factors in the efficiency component

 

 

 

 

General Scaled Stevenson KGMHLBC RSCFG-p RSCFG
Log-likelihood unrestricted) -613.02 -622.67 —616.13 -622.31 -623.05
Log-likelihood restricted) -616.30 -623.63 -618.71 -623.42 -623.70
LR statistics 6.56 1.92 5.16 2.22 1.30
Wald statistics 7.03 2.14 4.62 3.35 2.09
# restrictions 8 4 4 4 4
10% critical value 13.36 7.78 7.78 7.78 7.78

 

Table E.3 reports the third-step results of the LR and Wald tests for the alternative

model specifications. The unrestricted models have the full set of explanatory variables

in the production frontier while the restricted models have the reduced set of explanatory

variables. Both the LR test and the Wald test fail to reject the null hypothesis that the

63

nine explanatory variables we dropped from the production frontier model in the first step
all equal zero at any reasonable significance level for each of the models. In the end, we

keep the reduced set of variables from the first step and the second step in our subsequent

analysis.

Table E.3. Tests results for specifying explanatory variables in the production frontier

 

 

GeneraNEcaled Stevenson KGMHLBC RSCFG-[1 RSCFG

 

Log-likelihood unrestricted) -614.89 -622.11 -617.21 -621.85 -622.16
Log-likelihood restricted) -616.30 -623.63 -618.71 -623.42 -623.70
LR statistics 2.82 3.04 3.00 3.14 3.08
Wald statistics 2.95 3.12 3.04 3.22 3.11

 

64

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68

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