Economic Behavior Under Uncertainty: A Joint Analysis of Risk Preferences and Technology
Author(s): Jean-Paul Chavas and Matthew T. Holt
Source: The Review of Economics and Statistics, Vol. 78, No. 2 (May, 1996), pp. 329-335
Published by: The MIT Press
Stable URL: http://www.jstor.org/stable/2109935
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ECONOMICBEHAVIORUNDER UNCERTAINTY:
A JOINT ANALYSIS OF RISK PREFERENCESAND TECHNOLOGY
Jean-PaulChavas and Matthew T. Holt*
Abstract-A methodis developedto estimatejointly risk preferencesand technology undergeneral conditions. The approachis illustratedin an application
to the analysis of U.S. corn-soybeanacreage decisions over time. The results
provide useful informationon the natureof farmers' risk preferencesand on
the influenceof priceriskandproductionriskon acreageallocationandfanners'
welfare.
tionMaximumLikelihood(FIML)estimationof the production technologytogetherwith the first orderconditionsderived from expected utility maximization. The
methodologicalapproachis developedin sectionII. An empirical applicationto acreageallocationdecisions for two
importantfield crops in the U.S., corn and soybeans,is the
I. Introduction
topic of section III. Importantly,the analysis incorporates
ONSIDERABLE researchhas focused on the influence price risk, productionrisk, and the effects of government
of risk on production decisions. On one hand, the the- price supportprogramson the decisionmaker'sunderlying
ory of the firm under uncertainty and risk aversion is now marketprice distribution.Estimationand results are diswell developed, following the work of Sandmo and others. cussed in section IV, and sectionV concludes.
C
On the other hand, there is empirical evidence that most
decision makers are risk averse and that price and/orrevenue
uncertainty has a significant influence on production decisions. Risk effects have been of special interest in agriculture
where agriculturalproducers face considerable uncertainty
due to unpredictable weather conditions and unstable agricultural markets, and where primary producers have been
found to be risk averse (e.g., Dillon and Scandizzo (1978);
Binswanger (1981)). These general conclusions and observations have stimulated considerable research into the effects of risk on aggregate commodity supply response, the
bulk of which attempts to estimate risk effects econometrically (e.g., Behrman (1968); Just (1974); Lin (1977); Chavas
and Holt (1990); Holt and Moschini (1992)). The empirical
evidence to date suggests that risk has a negative and significant effect on food supply. Although this result is presumably due to farmers' risk aversion, previous studies of aggregate supply response used a reduced form approach, an
approach that does not provide precise information on the
influence of risk preferences on production and producers'
welfare (Pope (1982)). As a result, the producers' (implicit)
cost of private risk bearing is still imprecisely known. In
general, we might expect both the underlying technology
and risk preferences to influence production decisions under
risk. Consequently, there is a need to develop an approach
that allows for joint estimation of the effects of technology
and risk preferences on production behavior and producer
welfare. Such an approach should in turn provide useful information about the role of risk and risk preferences in resource allocation.
The objective of this paper is to present a method to estimate jointly the parameters of the risk preference function
of the decision maker together with production function parameters based on observed economic behavior. We assume
the decision maker behaves in a way consistent with the
expected utility hypothesis. In this context, we propose a
model specification and estimation scheme for production
decisions under risk. Our approachis based on Full InformaReceived for publication June 8, 1992. Revision accepted for publication
August 8, 1994.
* University of Wisconsin-Madison.
II. The Model
Considera representative
agentchoosinga (n X 1) vector
x = (xl, . .. , x,)' of decision variablesunderuncertainty.
Uncertaintyis representedby a vectore of randomvariables
witha given subjectiveprobabilitydistribution.Assumethat
the decision makerhas preferencesrepresentedby the von
NeumannMorgensternutilityfunctionu(x, e, a), wherea is
a parametervectorcharacterizing
the natureof preferences.
Then,underthe expectedutilityhypothesis,economicdecisions are made as follows:
Maxx{Eu(x,e, a): g(x, /8) = 0, x
E
(1)
Rn},
where E is the expectationoperatortaken with respectto
the subjectivedistributionof e; g(x, /3) = 0 is the implicit
productionfunction which representsunderlyingtechnology; and /8is a vector of technologyparameters.
Assumethatthe optimizationproblem(1) has an interior
solution denoted by x* = argmaxx {Eu(x, e, a): g(x, /8) =
0, x E R+ }. Consider the LagrangeanL(x, A, a, /3) = Eu(x,
e, a) + A'g(x, /3), where A is a Lagrangianmultiplier. Under
differentiability,the first-orderconditionsassociatedwith
the maximizationproblem(1) are
LX(x*,A*, a, 8)=EuX(x*, e, a)+gx(x*, /3)A*0=,
LA(x*, /3)-g(x*,
8) = O,
(2a)
(2b)
where subscriptlettersdenotederivatives.'
Undersome regularityconditions,2the first orderconditions (2) providea completecharacterization
of production
behavior(e.g., Samuelson(1947)).Forexample,usingcom1 Throughoutthe paper,subscriptlettersdenote derivatives,i.e., L, = aL/Jx
= (n X 1) vector,LXX
= a Llax2 = (n X n) matrix,etc.
2
These regularityconditions are
1. the functions g(.) and u(.) are twice continuously differentiable;
2. the matrix gx = ag(x, 8)/lx has rankone;
3. the second order sufficiency condition is satisfied:
V'LXXV< 0
for all V e
Rn
such that V ? O, gxV = 0.
These regularityconditions are assumed satisfied throughoutthe paper.
? 1996 by the Presidentand Fellows of HarvardCollege and the MassachusettsInstituteof Technology
[ 329 ]
330
THE REVIEW OF ECONOMICSAND STATISTICS
parativestatic analysis, Sandmo,Ishii, and Chavas et al.
(1988), have investigatedthe theoreticalimplicationsof the
natureof risk aversionfor productiondecisionsunderprice
risk. In the absenceof preciseinformationaboutthe nature
of riskpreferences,however,it wouldbe difficultto evaluate
the implicationsof risk for economic decisions. Consequently,thereis a need to estimatethe natureof riskpreferences (as representedby the parametersa).
Consider a partition of vector x' = (xjl, x2), where xl and
x2 have dimensions(n - 1) X 1 and(1 X 1), respectively.
Expressions(2a) and (2b) now take the form
Lxl(x*, A*, a, 83)=Euxj(x*, e, a) + gxl(x*, /)A* = O,
(3a)
Lx2(X*, A*, a, 83)=Eux2(x*g e, a)+gx2(x*,
/)A*=O,
(3b)
(3c)
LA(X*,/3)=g(x*, /3) = 0.
pendently5and identicallydistributedacross t, each with
densityfunctionf( iv, a-).Importantly,
we do not restrictthe
errortermof the technologyequationF2 to be independent
of the errortermsof the firstorderconditionsF1. Imposing
independencebetweentheerrortermof thetechnologyequation and those of the remaining(implicit)equationswould
maketheequationsystem(5) blockrecursive;theparameters
associatedwith technologyequationF2 could then be estimatedconsistentlywithoutestimatingjointlythe riskpreference parametersin Fl. Withoutthis restriction,however,
equationsystem (5) is not block recursive,in which case
recursiveestimationmethods (such as those proposedby
Antle (1984)) would be subjectto simultaneousequation
bias. In orderto avoid suchbias, the parametersin (5) must
be estimatedby a simultaneousequationmethod. By not
imposingindependence,we focus on a FIMLmethodthat
allowsus to investigateempiricallywhetheror notrecursive
methodsare appropriatein the estimationof (5).
Let J,
det[F,I/ Dxt]be the Jacobian of the transformation
Assumingthatgx2 ? 0, solving (3b) for A* and substituting givenin (5), t = 1,.. , T. Then,equation(5) is aneconomethe resultinto (3a) yields
tric modelrepresentinga systemof n implicitsimultaneous
equations.We proposeto estimatethe parametersy = (a,
FI(x*, a, /3)
Euxl(x*, e, a) - gxl(x*, /)
,/, a-)in (5) by usingMaximumLikelihood(ML)estimation
The log-likelihoodof the sampleis
techniques.
X [gx2(x*, 83)]-lEux2(x*, e, a) = 0,
F2(x*, /3)
g(x*,
/3) = 0,
T
LT(X, Y)
=
t=
or more compactly,
[log(f(F(x,, a, /3), a-)) + log I Jt I].
E
1
(6)
F(x*, a, A) = 0,
(4)
The ML estimatory* is definedto be a root of the equa-
where F(.)' = (F1(.)', F2(A)').3
tion DLT(X,
Expression(4) is a system of n implicit simultaneous
equationsthat can be solved for x* = (x*, x*). Knowing
the equationsystemin (4) is equivalentto knowingthe optimal choice functionsx*.
Thefocus hereis on estimatingthe riskpreferenceparameters a and the technologyparameters/3by using a sample
y) is nonlinear,so estimationof y can only be obtainedby
using a numericalalgorithmto maximize(6) with respect
to y. Underfairly generalconditions,the estimatory* has
desirableasymptoticproperties:it is a consistentandasymptoticallynormalestimatorof the trueparametersy (Amemiya (1985), p. 114).
of T observations on x = {xt, t = 1, 2,
...,
y)I ay
=
0. In the present case the function LT(x,
T}. In order
to use this samplein the econometricestimationof the paIII. An Application
rametervector (a, /3), a stochasticstructuremust be asIn this section we adoptthe above methodologyto the
sumed.Forpresentpurposes,we appendadditiveerrorterms
estimationof aggregateacreageresponse decisions under
to equation(4), yielding
risk.Specifically,we focus on a time-seriesanalysisof U.S.
F(x4*, a, /) = vt,
(5) corn-soybeanacreageallocationdecisionsunderboth price
and productionuncertainty,where n = 2, xl = corn acreage,
where v' = (/tj, i-t) is an n-vectorof randomvariables andx2 = soybeanacreage.We assumethataggregateacrewith meanzero.4Furthermore,
we assumethe vt'sareinde- age decisionsareconsistentwith the behaviorof a representativefarm.Also, we pay special attentionto the effects of
3The setup in (4) is analogousto using the reducedgradientmethod,a general governmentprogramson farmprices and farm income.
approachused frequentlyin the solution of nonlinearmathematicalprogramAs a specialcase of equation(1), considerthatthe objecming problems.As such, estimatesof the (optimal)LagrangemultiplierA* can
tive
functionof a representativefarmergiven by
always be recovered from the estimatedmodel structureand parameters.
4 Note that, by introducingthe errorterm v,in (5) ratherthan (3), our model
is not a "random utility model" (where the stochastic structureis assumed
consistent with the theory). Our stochastic specification has the disadvantage
of being sensitive to the equationthat is eliminatedbetween (3) and (4). But,
comparedto a randomutility specification,it has the advantageof simplifying
greatly the empirical estimationof the model.
Maxx{Eu[L7r(x,e), a]: g(x,
/)
= 0,
x
E R2+}'
(7a)
SExtending the analysis to allow for serial correlationappearsto be a good
topic for furtherresearch.
ECONOMICBEHAVIORUNDER UNCERTAINTY
331
where 1r(x,e) denotesfarmprofit,x' = (xI, x2) is the vector sign of a2. Thiswill providea convenientbasisto investigate
of acreagedecisionfor corn(xl) andsoybeans(x2),andg(x, empiricallythe DARA hypothesissuggestedby Arrow.
Ourproposedparametricspecificationfor the production
,6) = 0 is the implicitproductionfunctionwith respectto
function g(x, /3) = 0 at time t is
acreage.Profit 1r(x,e) is definedas
iT =
(7b)
q'x,
gt(Xt, /3) =
-X2t
+
whereq' = (ql, q2) = (PIYI - cl, P2Y2 - c2) denotesthe
net returnper acrefor corn(1) and soybeans(2);p' = (pl,
P2) is the vector of prices receivedby farmers;y' = ( YI,
Y2) is the vectorof yields peracre;andc' = (c1, c2) denotes
cost of productionperacreforeachcrop.At thetimeacreage
decisionsare made, the farmerdoes not know what prices
p or yields y will be realizedat harvest.As a result,p and
y aretreatedas randomvariablesthathave given subjective
Thus,the expectationoperatorE
probabilitydistributions.6
in (7a) is over the vectorof fourrandomvariablese = (Pl,
P2, YI, Y2).
To makemodel(7) empiricallytractable,we mustchoose
a parametric structure for u(17r,a) and g(x, /3), and find a
methodto evaluatethe expectationoperatorE. Considerthe
th time periodwhereprofit at time t, 7r, is definedin the
interval[L, M1].Ourproposedparametricspecificationfor
u(1r, a) at time t is
ut(7rt,
a)
tr,
=
exp(aco + alz + a2Z2 + a3t) dz,
(8)
+ (/3o + fI3[xIt + DAJ] + /2 t
33[X1t + DAt]2),
(9)
wheret denotesthe tth time period,xitis the acreageplanted
in corn (i = 1) or soybeans (i = 2), the /3's are parameters
to be estimated,and DAt is the acreage of corn diverted
Equation(9) is quadratic
undergovernmentfarmprograms.7
in xl: it is associatedwith a (strictly)concave production
function if 83(<) : 0. The time trendt in (9) reflects possible
structuralchangesin the productiontechnologyover time.
Finally,the expectationoperatorE in (4) needsto be evaluatedwithrespectto e = ( P1, P2, Y1,Y2). We note,however,
thatgovernmentprice supportprogramsinfluencethe subjective probabilityof outputpricesp received by farmers.
More specifically, a price supportprogramplaces a floor
underthe marketprice.The resultingtruncationaffectsthe
subjectiveprobabilitydistributionof pricesp. To see this,
let Pi denote the marketprice of the ith commodityin the
absence of a price supportprogram,and denote by Hi the
correspondingprice supportlevel. Then, the price received
by farmersis
JHi if Pi < Hi,
Pi
if
(10)
Pi ' Hi,
wherez is a dummyof integration,and the a's are parameters to be estimated.The time trendt in (8) accountsfor
i = 1, 2. Expression(10) indicatesthat pi is a truncated
possiblechangesin riskpreferencesover time. The specifivariablederivedfrom Pi. It has the same density
random
cation(8) has severalattractivefeatures.First,it restrictsthe
function
as Pi for Pi ' Hi, but the probabilitymass (Pi <
= exp(ao
marginalutilityof incometo be positive, )uIDh7r
to the truncationpointHi. Here,we
been
transferred
has
Hi)
+ alit + a2i7-2 + a3t) > 0. Second, it allows for riskE in (4) as follows. First,
the
to
evaluate
expectation
propose
averse as well as risk-loving behavior depending upon
of
distribution
the
(P1, P2, YI,Y2) is estimated
empirical
joint
whetherthe utility functionis concave (a 2u/I,7r2 ' 0) or
from time series data. Second, the (truncated)distribution
convex(a2u/aI7r2' 0) in 1r.Third,it includesconstantabsoof e = (PI, P2, YI, Y2) is derivedfrom (10) and from the
lute risk aversion(CARA)as a special case when (2 = 0
of (P1,P2, Y1,Y2). Third,theexpectationoperator
distribution
abso(Pratt(1964);Arrow(1971)). Indeed,the Arrow-Pratt
in
is
based on the (truncated)distribution
E
evaluated
(4)
lute riskaversioncoefficientis AR = - (a22uIhr 2)I ()uIP,7r)
the
evaluationof such a truncated
of
e.
Because
analytical
= - (a1 + 2a27rt), implying that AR is a constant if and
difficult
can
be
(due to the absenceof a
quite
expectation
only if a2 = 0. This will providea basis for testingempirito use numericalMonte
form
we
closed
solution),
propose
callythe CARAassumption.Fourth,it providesa fairlyflexin
to
calculate
F1
Carlo
(4).
integration
of risk preferences.For example,given
ible representation
and
and
a numericalevaluation
Given
(9)
(8)
equations
= -2a2, it can allow for decreasing absolute risk
DARID,7r
II can be implein
E
section
in
the
method
of
presented
(4),
aversion(DARA:DAR/Ikr< 0; see Pratt)or increasingabsothe
speciMore
parametric
mented
specifically,
empirically.
lute risk aversion(IARA:DARIl$Th> 0) dependingon the
fication (8) and (9) generatesthe system of simultaneous
equations(5) that can be estimatedby using FIML tech6 Our subsequentspecificationof (homoskedastic)productionuncertaintyis
niques.This in turnis accomplishedby assumingthe error
admittedlyrestrictive;see, e.g., Justand Pope (1979) or Griffithsand Anderson
(1982) for richerspecifications.But given thatourestimationmethodis already
highly complex, the approachwe adoptherefor handlingproductionuncertainty
has the advantageof being simple to implementempirically.Investigatingmore
flexible specificationsfor productionuncertaintyis an importanttopic for further research.
7 Ourspecificationhas the advantageof being fairly flexible while remaining
simple for the purposeof estimation.As noted by a reviewer,a slight disadvantage of our productionfunction (9) is that it does not treatxl and x2 symmetrically.
332
THE REVIEW OF ECONOMICSAND STATISTICS
term v, in (5) has a joint (i.e., bivariate)normaldistribution
with mean zero and a finite variance-covariance
matrix.8
Beforeproceeding,notethatadditionala priorirestrictions
needto be imposedto renderthe systemof equationsimplied
by (8) and (9) estimable.The reasonis that, given (8) and
the fact that F1 in (4) is in implicit form, it is possible to
rescalebothparametera0 andthe standarddeviationof vI,
by some positive constantwithoutaffectingthis equation.
In otherwords,the parametersof equationF1 arenot identified. To makethe systemof equationsestimable,we impose
the identifyingrestrictionsthat the variance of vI, equal
unity.Thus,the variance-covariance
matrixof v,is specified
as follows
Var(v,)=
99',
(11)
ments for corn are addedto corn revenuein the analysis.9
All pricesarerealprices,i.e., nominalpricesdeflatedby the
consumerpriceindex (1967 = 100) as reportedby the Bureauof LaborStatistics.Averageyieldsperacreareobtained
from variousUSDA publications.
To makethe modelempiricallytractable,severalassumptions are requiredregardingthe (untruncated)distributions
of pricesandyields. For presentpurposes,we use an adaptive approachto characterizethe relevant moments (i.e.,
meanandvariance)of the untruncated
normalizedpricedistributions.Specifically,expecteduntruncatedprice at time
t is definedas the marketpricethe previousyear,plus a drift
parameter3:
E,( p,) = 3 + Pt- 1,
where
wherep, is the deflatedprice at time t, and 3 is the mean
of the first differencein P5tover the sample period. This
if [=
07]
adaptiveformulationfor determining(untruncated)
priceex02
CRl
pectationshas been employedsuccessfullyin previousresearch(e.g., Houcket al. (1976); ChavasandHolt (1990)).
O', and o-2 being parametersto be estimated.Formulation Untruncated
price varianceis definedby
(11) expressesthe variance-covariance
matrixof v, as the
3
Choleskyproductof the matrix9. This approachhas the
=
Var(p-,)
wj[pt_ -Et-j(Pt_j)],
advantageof guaranteeingthatVar(v,)is always a positive
j= 1
semi-definitematrix.Note thatimplicitin the specification
of the matrix9 is the identifyingrestrictionVar(vl,) = 1.
Also, the covariancebetweenvP,and v2,is given by E(v1,g-2,) wherethe weightswj are0.5, 0.33 and0.17. It indicatesthat
priceriskis measuredas the squareddeviationof pastprices
= o-2. Noting that o-2 = 0 is the conditionfor the system
of equations(5) to be recursive,we now have a convenient from their expected values, with declining weights. This
basisfortestingthehypothesisthattheparameters
embedded measurementis also consistentwith specificationsusedpreviously (e.g., Lin (1977); Traill (1978); Chavas and Holt
in F1 andF2 can be estimatedindependently.
Given the parametricspecification(9)-(1 1) and the nor- (1990)). Yield expectationsare obtainedby firstregressing
mality of v,, the system of equations(5) can then be esti- actualyields on a trendvariable.The resultingprediction
matedby usinga nonlinearalgorithmto maximizelog-likeli- was taken as expectedyield, and the varianceof the error
hood function(6), with the MonteCarlointegrationscheme term as a measureof yield risk. The combinedresults of
used to evaluateE being nested within the ML estimation these proceduresgeneratedexpected values and variances
for pricesandyields.'0 Finally,correlationsbetweeneachof
algorithm.
the four randomvariableswere estimatedusing the sample
deviationof each randomvariablefrom its expectation.1
IV. Estimation and Results
The first-orderconditionsF1 in (4) requirethe evaluation
This sectionreportsthe applicationof the modelto corn- of EuXiandEU,2, whereexpectationsaretakenwith respect
soybeanacreagedecisionsin the UnitedStates.The analysis to the (truncated)quadravariate
distributionof four correcovers the period 1954-1985 and is based on an updating lated randomvariables:two (truncated)prices Pi and P2,
of the dataset reportedpreviouslyin Gallagher(1978).Vari- and two yields Yi andY2. As discussedabove, we employ
ablesxl andx2 denoteacreageplantedto cornandsoybeans,
respectively;prices are U.S. average prices received by
9 The effective supportprices for corn and soybeans, and the effective diverfarmers;and costs of productionper acre (cl and c2) are sion paymentsfor corn were constructedby Houck et al. (1976) and Gallagher
those reportedby Gallagherand, for lateryears, in USDA (1978).
10Note thatour expectationformulationsare all adaptivein nature.Although
sources. Following Houck et al. and Gallagher,effective such
formulationshave been commonly used in previous research,they are in
supportprices(H) areusedto quantifytheeffectsof govern- general not consistent with the rationalexpectation hypothesis. For example,
mentprice supportprograms.Also, effective diversionpay- the influence of governmentprogramson the dynamics of price expectations
U
8 Assuming thaterrortermsarejoint normallydistributedis standardpractice
in applied econometric analyses; however, such an assumptioncould clearly
be restrictive.Investigatingalternativespecifications for the joint distribution
of z.,thus appearsto be a good topic for futureresearch.
is likely more complex than our adaptiveformulations.Also, a more thorough
investigationof the natureof expectations and their rationalitywould require
modelingthe "demandside" as well. Investigatingsuch issues is left for further
research.
1l The correlationcoefficient betweenany two randomvariableswas assumed
constant throughoutthe sample period.
ECONOMICBEHAVIORUNDER UNCERTAINTY
TABLE
l. -PARAMETER
TABLE 2.-RISK
ESTIMATES
Estimate
ao
a3
4.0270
- 16.7331
4.5666
0.2035
0.2545
2.4755
0.9574
0.0380
/3o
- 2.7456
0.4870
6.6745
0.0185
- 3.7638
1.0867
0.0010
0.6112
0.0417
- 0.0189
0.0066
a2
/33
0,
U2
Note: log likelihood
ATTITUDE ESTIMATES
Estimate
Asymptotic
Standard
Error
12.171
1.710
6.070
0.887
Asymptotic
Standard Error
Parameter
a,
333
Arrow-PrattAbsolute Risk Aversion
AR = - (a2U/aT2)1(aUlaT)
Relative Risk Aversion
RR
=
-
T(a2U/a
T2)1(aUla
Downside Risk Aversion
DR = (a3u1aT3)1(aT)
T)
157.270
43.424
Note: The estimates are evaluated at the 1967 expected profit E(1T)=4.99 billion dollars.
0.0120
217.10.
premium R defined to be the sure amount of money satisfying: Eu('nr)= u(E(17r)- R). The risk premium R, which
numerical methods to evaluate the expectation operator measures the willingness to pay of a decision maker to elimiembedded in Fl. More specifically, F1 was calculated by nate the risk associated with X7by replacing it with its exusing Monte Carlo integration techniques, where 2000 pected value E(17r), has been central to the theory of risk
points12 were generated assuming that the vector (PI, P2, Yi, aversion. It can be approximated as
Y2) is drawn from a quadravariatenormal distribution. The
Monte Carlo integration routine was nested within a FIML
R = AR M2/2-DR
(12)
M3/6
algorithm in the maximization of equation (6).'3
The resulting maximum likelihood parameter estimates
are presented in table 1. Most of the estimated parameters where Mi = E{[7T E(1T)]i} is the ithcentral moment of 1T
are significant at the 5% level. The parameter a2 is found (see Pratt (1964); Menezes et al. (1980)), M2 > 0 being
to be positive and significantly different from zero. This the variance of ir, and M3 measuring the skewness of the
implies that the null hypothesis of CARA preferences is re- distribution of ir. From (12), the AR coefficient is a local
jected by the sample evidence. With JARlD1T= - 2a2 < 0, measure of risk aversion: aversion to risk implies AR > 0;
our results provide statistical evidence that risk preferences and larger values of AR imply stronger aversion to risk (see
Arrow (1971); Pratt (1964)). In addition, Pratt has shown
exhibit decreasing absolute risk aversion. The parameter/3
is negative and significantly different from zero. This indi- that JRI/E(1T) = sign(DAR/D-g)for all ir, implying that the
cates that the productionfunction is strictly concave. Finally, properties of AR provide useful information on the effect of
the estimated value for covariance parametero2 is negative, expected profit on the willingness to pay for risk R. Similar
but not significantly different from zero. This result indicates comments apply to the coefficient RR = irAR: it is a local
there is not strong evidence against recursivity of equation measure of risk aversion reflecting the effect of a proportional risk on relative risk aversion as measured by RI 7T(see
system (5).
Pratt
(1964); Menezes and Hanson (1970)). Note that the
The implications of the results for risk preferences can be
RR
coefficient
can also be written as the elasticity RR
summarized in terms of the following risk aversion mea-g)IJ
ln(ir) which has the advantage of being
D
ln(D
ulI
sures:
"unit free." Finally from (12), the DR coefficient is a local
1. the Arrow-Prattabsolute risk aversion coefficient AR = measure of preferences toward skewness of the distribution
- (a 2U/aVT2)/( Ula VT);
of ir. For example, "skewness to the left" means higher
2. the relative risk aversion coefficient RR = 7T AR = downside risk and a negative value for M3. Thus expression
- ira 2UI/aJ2)I(a uIa T);14
(12) indicates that aversion to downside risk implies that
3. and the downside risk aversion coefficient DR = aJ3uIaJ1T3
> 0 and DR > 0 (see Menezes et al. (1980)).
To i3U/te
3)r ( cUtstt).
Evaluated at the 1967 expected profit, the estimates of
To interpretthese coefficients, consider the Arrow-Prattrisk AR, RR and DR are presented in table 2 along with their
standard errors.'5 The AR and RR coefficients are positive
and
significantly different from zero. This provides statisti12
This choice of 2000 was one of the largestnumberof points feasible given
cal
evidence
that farmers are risk averse. The large estimate
the availablecomputerresources(a 486 computerwith 8 Megabytesof RAM).
The results reportedbelow were found to be fairly insensitive to the number of RR (6.07) suggests a fairly strong degree of risk averof points used in the Monte Carlo integration.
sion.'6 The DR coefficient is positive and significantly dif13 In order to guaranteemeaningfulconvergence results, the same pseudo-
random variables were used as part of the Monte Carlo integrationat each
iteration in the maximization of the likelihood function (6). The maximum
likelihoodalgorithmused was MAXLIKin the GAUSS programminglanguage.
14 Menezes and Hanson (1970) distinguishbetweenrelativerisk aversionand
partialrelative risk aversion. Since we do not differentiatebetween profit and
terminalwealth, this distinctionvanishes in our analysis. Exploringthis issue
empirically appearsto be a good topic for furtherresearch.
15 The
standarderrorsare calculated using the delta method.
Previous estimates of the relative risk aversion coefficient in agriculture
have varied from 0 to over 7.5, with a median estimate around 1 (see Arrow
(1971); Binswanger(1981)). Thus, our estimateof RR appearsto be high. This
result suggests a strongdegree of risk aversion by U.S. corn-soybeanfarmers.
16
THE REVIEW OF ECONOMICSAND STATISTICS
334
TABLE 3.-EVOLUTION
Year
Expected Profit E(iT)
(10 billion 1967 dollars)
Absolute Risk Aversion Coefficient
(AR = -(a 2U/aJ12)/(a Ula1T)
Relative Risk Aversion Coefficient
(RR = - T(a2U/a1T2)1(aUla1T)
Downside Risk Aversion Coefficient
(DR = (a3u/ajT3)/(aJU/1x)
OF RISK ATTITUDES OVER TIME
1960
1965
1970
1975
1980
1985
0.0888
0.3939
0.4642
1.4463
0.8472
0.6108
15.922
13.136
12.494
3.523
8.995
11.154
1.414
5.174
5.800
5.095
7.621
6.813
262.649
181.679
165.228
21.547
90.041
133.550
Note: The estimates are evaluated at the expected profit of the correspondingyear.
ferentfrom zero. This indicatesthat farmersare averse to
downside risk."7
The evolutionof risk preferencesover the sampleperiod
is presentedin table 3 for selected years. It indicatesthat
the absoluterisk aversion (AR) coefficient has decreased
between 1960 and 1975, and then increased afterwards.
Table3 also showsthattherelativeriskaversion(RR)coefficienthas in generalincreasedover time. Finally,it indicates
somefairlylargechangesin the downsideriskaversion(DR)
coefficient over time: first a sharpdecline up to the mid1970s followed by a large increasein the 1980s.
Ourresultscan provideuseful insightson productionbehavior.Solving numericallythe estimatedequations(4) for
the decisionvariablesgeneratesthe optimalacreageallocation. This was done numericallyusing a Gauss-Seidelalgorithm.The optimalacreagedecisionsfor cornandsoybeans
were then calculatedunderalternativeconditionsin order
to evaluatethe implicationsof the model for economicadjustments.More specifically,using 1967 as the base year,
we solved the first orderconditionsby changingselected
parametersby 1%.The resultingoptimalacreagewas then
used to calculatevarious elasticities of acreageresponse.
Theseelasticitiesarereportedin table4. In general,the estimatesappearreasonable.For example,the acreageelastici17 This
is an expected result since DARA preferences necessarily implies
au/'u w3 > 0, i.e., aversion to down-side risk (Pratt (1964); Menezes et al.
(1980)).
TABLE 4.-ELASTICITY
ESTIMATES
Corn
Acreage
Elasticity of with respect to
Expected corn price E(p1)
Expected soybean price E(p2)
Expected corn yield E(y1)
Expected soybean yield E(Y2)
Corn price variance Var(PI)
Soybean price variance Var(P2)
Corn yield variance Var(yl)
Soybean yield variance Var(y2)
Correlationcorn price PI, soybean price P2
Correlationcorn price PI, corn yield Yi
Correlationcorn price PI, soybean yield Y2
Correlationsoybean price P2, corn yield y
Correlationsoybean price P2, soybean yield Y2
Correlationcorn yield yl, soybean yield Y2
Note: The elasticities are evaluated at the 1967 data point.
(xi)
Soybean
Acreage
(X2)
0.2490
-0.1210
0.2925
-0.1210
- 0.0335
-0.00003
-0.0879
0.0024
- 0.0037
- 0.0086
-0.00001
- 0.0011
-0.0012
-0.0013
-0.2183
0.1035
- 0.2572
0.1035
0.0288
0.00003
0.0753
-0.0021
0.0032
0.0074
0.00001
0.0009
0.0010
0.0011
TABLE 5.-EVOLUTION
OF RISK PREMIUM R, OVER TIME
Year
1960
1965
1970
1975
1980
1985
Risk PremiumR,
(billion 1967 dollars)
0.782
2.081
0.593
0.219
0.238
0.814
ties with respectto expectedpricescomparefavorablywith
those reportedelsewhere (e.g., Gallagher(1978); Lee and
Helmberger(1985);Tegeneet al. (1988)). Althoughalways
inelastic, the own price elasticities are positive while the
cross priceelasticitiesarenegative.The elasticitieswith respect to variancesare in general small. These results are
generallyconsistentwith those obtainedpreviouslyby Chavas andHolt (1990). The elasticitiesreportedin table4 provide moredetailon acreageresponsethanfoundin any previous research.For example,measuringthe separateeffects
of yield varianceand price varianceon acreagedecisions
appearsuseful.Increasingyield riskis foundto havea negative own effect anda positivecross effect on acreage.Also,
the influenceof correlationsbetweenpricesand/oryields on
acreagedecisionsis reportedin table 4.
Finally, in the absence of risk markets,our model can
provide a basis for evaluatingthe welfare implicationsof
risk for producers.FollowingPratt,the implicitcost of risk
at time t can be measuredby the riskpremiumR, defined
implicitlyas
E,u[1T(x,, e,), a] = u[E,'i(x,, e,)
-
R, a],
(13)
whereE, denotestheexpectationoperatorbasedon theinformation availableat time t. Given the specificationof the
utilityfunctiongiven in equation(8) and the estimateof a
reportedin table 1, equation(13) can be solved for R,. This
solutionwas obtainednumericallyusing Gaussianquadratureto calculatethe integralin (8) anda gradientmethodto
solve equation(13) forR,. Theresultingestimatesof therisk
premiumare reportedin table 5 for selected years. They
show that the cost of privaterisk bearingin corn-soybean
productioncan be large:the risk premiumR, varies during
the sample period between $200 million and $2 billion.
These resultsindicatethatrisk andrisk aversioncan have a
producers.
largenegativeeffecton thewelfareof agricultural
V. Conclusion
This paperhas presenteda methodfor estimatingjointly
technology and risk preferenceparameters.Our approach
ECONOMICBEHAVIOR UNDER UNCERTAINTY
relies on FIMLestimationof all the parametersof the productionfunctionandof the firstorderconditionsassociated
with the maximizationof expectedutility. The method is
appliedto U.S. aggregatecornandsoybeanacreageresponse
decisions,incorporating
bothpricerisk andproductionrisk.
Particularattentionis given to the effects of government
price supportprogramsthat truncatethe price distribution
and thus reduce price risk. The empirical analysis relies
heavily on numericalmethods:for evaluatingthe expectation in the first orderconditions;for maximizingthe log
likelihoodfunction;for solvingthe firstorderconditionsfor
optimalacreagedecisions;and for estimatingthe implicit
cost of risk. The obviousadvantageof our approachis that
the methods we employ do not requirethe existence of
closedformsolutions,andthuscanhandledifferentassumptions aboutthe form of the utilityfunctionor the natureof
the uncertaintyfacingthe decisionmaker.Forexample,our
analysishandlesa quadravariate
truncatedrisk distribution
(involvingtwo truncatedprices and two yields). This indicatesthatourproposedmethodis flexibleandappearspromisingin thejointanalysisof technologyandriskpreferences.
The empiricalresultsfor corn and soybeanacreageindicate thatcorn-soybeanfarmersarerisk averseandthatthey
exhibitdecreasingabsoluterisk aversionanddownsiderisk
aversion.The analysisalso providesuseful informationon
theinfluenceof riskon acreagedecisionsandon thefarmers'
implicitcost of privaterisk bearing.This suggeststhatthe
proposedmethodcan help refine the economic analysisof
productionbehaviorunderuncertainty.It is hoped that it
will helpstimulatefurtherresearchon governmentfarmprogramsandtheirinfluenceon the efficiencyof riskallocation
in agriculture.
REFERENCES
Antle, J. M., "EconometricEstimationof Producers'Risk Attitudes," American Journal of AgriculturalEconomics 66 (1984), 342-350.
Amemiya, Takeshi, AdvancedEconometrics(Cambridge:HarvardUniversity
Press, 1985).
Arrow, KennethJ., Essays in the Theoryof Risk Bearing (Amsterdam:NorthHolland Publishing Co., 1971).
Arrow, KennethJ., L. Hurwicz and H. Uzawa, "ConstraintQualificationsin
MaximizationProblems," Naval Research Logistics Quarterly8 (June
1961), 175-191.
Behrman,JereR., SupplyResponsein UnderdevelopedAgriculture(New York:
North-HollandPublishing Co., 1968).
335
Binswanger, H. P., "Attitudes Toward Risk: Theoretical Implications of an
Experiment in Rural India," The Economic Journal 91 (1981),
867-890.
Chavas,Jean-Paul,R. D. Pope, and H. Leathers,"Competitive IndustryEquilibrium under Uncertainty and Free Entry," Economic Inquiry 26
(1988), 331-344.
Chavas, Jean-Paul, and Matthew T. Holt, "Acreage Decisions under Risk:
The Case of Corn and Soybeans," American Journal of Agricultural
Economics 72 (1990), 529-538.
Dillon, J., and P. Scandizzo, "Risk Attitudesof SubsistenceFarmersin Northern Brazil: A Sampling Approach,"AmericanJournal of Agricultural
Economics 60 (1978), 425-435.
Gallagher,P., "The Effectiveness of Price SupportPolicy: Some Evidence for
U.S. Corn Acreage Response," Agricultural Economic Research 30
(1978), 8-14.
Griffiths, W. E., and J. R. Anderson, "Using Time-Series and Cross-Section
Data to Estimate a ProductionFunction with Positive and Negative
Marginal Risk," Journal of the American Statistical Association 77
(1982), 529-536.
Holt, Matthew T., and G. Moschini, "Alternative Measures of Risk in Commodity Supply Models: An Analysis of Sow FarrowingDecisions in
the United States," Journal of Agriculturaland Resource Economics
17 (1992), 1-12.
Houck, J. P., M. E. Abel, M. E. Ryan, P. W. Gallagher,R. G. Hoffman, and
J. B. Penn, Analyzing the Impact of GovernmentPrograms on Crop
Acreage (Washington,D.C.: U.S. Departmentof AgricultureTechnology Bulletin 1548, Aug. 1976).
Ishii, Y., "On the Theory of the Competitive Firm under Price Uncertainty:
Note," AmericanEconomic Review 67 (1977), 768-769.
Just,R. E., "An Investigationof the Importanceof Risk on Farmer'sDecisions,
" AmericanJournal of AgriculturalEconomics 56 (1974), 14-25.
Just, R. E., and R. D. Pope, "Production Function Estimation and Related
Risk Considerations,"AmericanJournal ofAgriculturalEconomics 61
(1979), 277-284.
Lee, D. R., andP. G. Heimberger,"EstimatingSupplyResponsein the Presence
of Farm Programs,"AmericanJournal of AgriculturalEconomics 67
(1985), 193-203.
Lin, W., "MeasuringAggregate Supply Response Under Instability," American Journal of AgriculturalEconomics 59 (1977), 903-907.
Menezes, C., and D. L. Hanson, "On the Theory of Risk Aversion," International Economic Review 11 (1970), 481-487.
Menezes, C., C. Geiss, andJ. Tressler,"IncreasingDownside Risk," American
Economic Review 70 (1980), 921-932.
Pope, Rulon D., "EmpiricalEstimationand Use of Risk Preferences:An Appraisal of EstimationMethods that Use Economic Decisions," American Journal of AgriculturalEconomics 64 (1982), 376-383.
Pratt, J. W., "Risk Aversion in the Small and in the Large," Econometrica
32 (1964), 122-136.
Samuelson,Paul,FoundationsofEconomicAnalysis (Cambridge:HarvardUniversity Press, 1947).
Sandmo,A., "On the Theory of the CompeitiveFirm underPrice Uncertainty,
" AmericanEconomic Review 61 (1971), 65-73.
Tegene, A., W. E. Huffman and J. A. Miranowsky, "Dynamic Corn Supply
Functions:A Model with Explicit Optimization,"AmericanJournal of
AgriculturalEconomics 70 (1988), 103-111.
Traill, B., "Risk Variablesin EconometricSupply Models," Journal of Agricultural Economics 29 (1978), 53-61.