Agricultural & Applied Economics Association
Modeling Supply Response in a Multiproduct Framework
Author(s): V. Eldon Ball
Source: American Journal of Agricultural Economics, Vol. 70, No. 4 (Nov., 1988), pp. 813-825
Published by: Blackwell Publishing on behalf of the Agricultural & Applied Economics
Association
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Response
Supply
Modeling
MultiproductFramework
in
a
V. Eldon Ball
The paper models multiproduct supply response in agriculture and tests key assumptions
traditionally maintained in supply response studies. The technology is approximated by a
restricted profit function. The properties of the restricted profit function are imposed
during estimation. The hypothesis that maintains the existence of output price and
quantity indexes that satisfy the adding-up property is rejected. The existence of
individual production functions for each output is also rejected. Unless joint production
is permitted, the estimates of responsiveness of a particular commodity to changes in
own price or prices of competing outputs are likely to be considerably understated.
Key words: input nonjointness, multiproduct supply, output separability.
Statistics of income by type of farm (Somwaru) documentthe prominenceof the multiproductfarmfirm.Yet, most models of supply
response in agriculture focus on aggregate
(across commodities) supply or on own-price
response for a single commodity. Moreover,
models that do recognize multipleoutputstypically specify transformationfunctions which
impose severe a priori restrictions on the
structureof production(Vincent, Dixon, and
Powell).
The most common functional structureassumption is separability,which ensures consistent aggregationwithin these structures. It
allows decentralized decision making or,
equivalently, optimization by stages. This
opens up the possibility of multistageestimation of productiondecisions using consistent
aggregatesin the latter stages.
A further assumption is nonjointness of
production. The multiproduct firm can be
modeled as single-productfirms with inputs
allocated among the different production activities. The only constraint across the individual technologies is that the total input
utilizationnot exceed the aggregateinput endowment.
The objectives of this paper are to model
supply response in agricultureusing disaggregated output data and to test statisticallykey
V. EldonBall is an agriculturaleconomistwith the Resourcesand
TechnologyDivision, EconomicResearchService, U.S. Department of Agriculture.
The authorgratefullyacknowledgesthe contributionsof Robert
Chambersin identifyingfunctionalstructuresand of AgapiSomwaru in implementingthe econometricprocedures.
assumptions traditionally maintained in agriculturalsupply response studies. The technology in agricultureis approximatedby a restricted profit function. Estimation of the
restrictedprofitfunction model necessitates a
strategywhere both nonlinearequalityand inequality constraints can be imposed. The restrictionsused to performthe separabilityand
nonjointness tests are nonlinear equalities.
Moreover, the ability to impose nonlinearinequalityconstraintsis necessary to ensurethat
the empiricalmodel of economic behavior is
consistent with theory. If the estimated restricted profit function does not satisfy theoretical curvaturerestrictions,it is not possible
to maintain the assumption of profit-maximizing behavior and hence test for functional
structure.
The articleis organizedas follows. The next
section discusses the restrictedprofitfunction
model and presentsrestrictionsfromeconomic
theory. These restrictions on the empirical
model are imposed as part of the maintained
hypothesis. An examinationof separableand
nonjointproductionstructurescompletes this
section. In subsequent sections, the estimation procedureand data are describedand empiricalresults are presented. Some concluding
comments are offered in the final section.
The Restricted Profit Function
The technology is assumed to relate M variable inputs and outputs and N fixed inputs.
Denote the vector of variable commodities by
Copyright 1988 American Agricultural Economics Association
814 November 1988
Amer. J. Agr. Econ.
Y -(Y1,.
, is an
YM)'(Yi > O, i = 1,...,
<
=
+
...
while
I
,M, isa
1,
Yi 0,i
output
variable input) and the vector of fixed inputs
by -X -(-X,
. . . , -X)'
< ON,. Let Tbe the
set of all feasible input and output combinations. T is assumed to be a nonempty, compact, and convex set.2 In addition, the technology is assumed to exhibit constant returns
to scale, i.e., T is a cone.3 This assumption
underlies the construction of the data. Let P
(P1, ... , PM)' > OMdenote a vector of exogenous prices. Then the restricted profit function
which corresponds to T is defined as
(1)
rT(P; X) - max {P'Y: (Y; -X) e T}.
Under the assumptions made on T, 7r(P;X) is
homogenous of degree one in variable input
and output prices and in fixed input quantities,
convex in prices, and concave in quantities
(Diewert 1973). If, in addition, rT(P;X) is differentiable, it satisfies Hotelling's lemma,
(2) ar(P; X)/IPi = Yi(P; X),
i=
1, ...
Separable and Nonjoint Structures
Functional representation theorems for separable and nonjoint structures have been
proved by Lau (1972, 1978a). A restricted
profit function is weakly separable in output
prices if and only if the function can be written
as
7r(P; X) = g(h(P,,
. ., P,),
Pt+1' '
* *
' X),
PM;
where h(.) is homogenous of degree one in its
components. It is apparent from (3) that weak
separability in output prices implies the existence of an aggregate output price index,
h(P,
(4) 7T(P;X)=
gi(Pi, P+,..
. . ., PP). This in turn implies by Hotel-
>
Notation: X
O, means that each component of the
N-dimensional vector X is nonnegative, X > ONmeans that each
component is positive, and X > O means X - 0N but X * 0.
2LetZ = (Y; -X). A set T is convex ifZ, Z' e T,0 < A
1,
implies AZ + (1 - X)Z' E T. This assumption implies that if Z is
possible, so is AZ for every number X satisfying 0 < A < 1; in other
words, that nonincreasing returns to scale prevail (Debreu, p. 41).
3 If there is free
entry into an industry, then T exhibits the
property of additivity, i.e.,Z, Z' E T implies Z + Z' E T (Debreu, p.
42). Additivity together with convexity implies that T is a cone.
i.e., Z e T, A > 0, impliesAZE T (constantreturnsto scale).
X,,
, PM;
. . , X,),
i=l1
Xj '
,M,
where Yi(P; X) is the profit-maximizing level
of output supply (input demand) if i is an output (variable input). Hence, rr(P; X) is assumed twice continuously differentiable in all
its arguments.
(3)
ling's lemma that relative output levels depend
only on output prices.
Weak separability of the profit function in
output prices implies that the underlying technology is homothetically separable in outputs.
A corresponding aggregate quantity index
exists and is homogenous of degree one in its
components. Moreover, the existence of price
and quantity indexes that are homogenous in
their components ensures that multiplying
price and quantity indexes yields the total
value of the components (Blackorby, Primont,
and Russell, pp. 206-7).
The restricted profit function is nonjoint in
inputs if and only if the function can be written
as
Xii,
j=
N.
1,.
i=l
The gi(.)'s are restricted profit functions corresponding to single-product production functions. Nonjointness in inputs implies that the
level of each output is independent of the
prices of competing outputs.
To determine the form of the restricted
profit function when (3) and (4) apply simultaneously, note that ad2r/aPiaPj= 0 (i j; i,j =
1, ... , l). Differentiating (3) yields
a27dr =
a2h + 2g ah
Oh
g
=
+
,
_ _
ah
ah2
ap,aPj
aPiaPj
dP, aPj
which implies
(5)
02g/lh2
Og/lh
-a 2h/dPiaPj
'
(ah/dPi)(9h/Pj)
The right-hand side of (5) is independent of
variable input prices and fixed inputs. Furthermore, the left-hand side can be written as
(a/ah) ln(ag/ah), which is a function of h
alone. Thus, by successive integration we obtain
g(P; X) = f(h)g,(P+1,
..,
PM;X)
+ g2(PI+1, ? -.
, PM, X),
which becomes, in order to satisfy (4),
. , PM;X)
(6) g(P; X) = g,(P,,
T h(P)
i=l
+ g2(P+, . . . ,
X).
Ball
ModelingSupplyResponse 815
= 0 (k
Finally, note that a2rr/aXjkX,,
This implies that
a2g
027,
92Ii.
aYK3XJs
2g
_
* s).
-
axj
aXjkaXjs
Differentiating (6), we obtain
a2g
1
ai2gp1
yg
1
a2
a29
(9)
a2
hi(P,)
Z
i1
dX2 dX~ 2
+
+____
92
hi(Pi) + g2(P,I+,
. . ,
pij = pji,
8jk = 8kcj
Homogeneity of degree one in prices requires
-0.
X2
aX
M
M
i = 1,
(10)
Repeated integration yields
(7) 7r(P; X) = gi(P,+,, . . . , PM)
E
The translog function is viewed as a secondorder Taylor's expansion about the unit point.
The following symmetry restrictions are imposed by the equality of cross-partial derivatives in a quadratic expansion
i=1
Aij,=
i=l
M
i=i
PM)
i=l
N
Xj + q(P+,
... , PM).
M
lpij
=
E
it =
0.
i=l
Under the assumptions on T, the restricted
profit function is homogenous of degree one in
fixed inputs. This requires
j=i
N
N
The structure of the restricted profit function
(11)
j = 1,
,
in (7) is referred to as the Gorman polar form
j=1
j=1
and meets the sufficient conditions for aggreN
gation across firms when firms maximize
=
profits (Gorman). It implies that the individual
Pij -j=i
technologies are affinely homothetic. That is,
the expansion paths are straight lines but do
not necessarily emanate from the origin. This Using Hotelling's lemma,
structure has been widely used in empirical
aln7r
PiYi S
analysis in the context of consumer demand
alnPi,
(Blackorby, Boyce, and Russell) because it
allows aggregation across households. However, in production analysis, this structure im- which applied to (8) yields
plies that variable inputs will be used at zero
M
output levels (Lopez).
(12) St = ai +
3j IlnPj
N
>
jt = 0.
j=1
j=1
The Translog Approximation
N
+
The restricted profit function (1) is approximated by the transcendental logarithmic
(translog) function with arguments, P, X, and
t, where time t indexes the level of technology
M
(8)
lnr=
N
ao+ Z a, lnP,
+
i=1
jlnX,
j=l
M M
+
f,ijnl
2I
nP
InP
i=1 j=l
N
N
j=1
k=l
+ 1/2
M
ak lInX lnXk
+
pl InPnP
lnXj
i=1
j=l
y,itl nP t
+
i=1
N
+
ijt lnXj
j=1
t + ot t + 10tt
t2.
i=
1,...,M.
The approximating function to an underlying convex function will exhibit this property
at the point of approximation. In particular,
the approximating function will have its hessian matrix positive semidefinite at the point of
approximation. Lau (1978b) has shown that
every positive semidefinite matrix has a
Cholesky factorization. The hessian of the restricted profit function can then be written as
(13)
M
N
Pj, lnXj + yiYtt,
j=1
aPapj2]
where L is a unit lower triangular matrix (Lii =
1, Lj = 0,j > i) and D is a diagonal matrix with
typical element Dii referred to as a Cholesky
value. In terms of the parameters of the translog approximation, this implies
816 November 1988
Amer. J. Agr. Econ.
311
+ al(a1
8312
+ al(C2
-
1)
122
+ a1a2
+ a2(a
P 2M
+ a2aM
12
-
.. .IM
1)
12M
+
aOIam
+
aY2Cam
(14)
+ alaM
K1IM
.*
* PM
*
Dll
L21Djl
L21Dll L21Dll + D22
LMlDll
L21LMlDl
+ LM2D22
Lau (1978b) demonstrates that a real symmetric matrix is positive semidefinite if and only if
the Cholesky values are nonnegative. To impose convexity at the point of approximation,
it is necessary to equate the algebraic expressions in the Hessian and the Cholesky representation, requiring that the Dii > 0.
Because the translog function is an approximation about a point, the hypothesis tests will
require that the hypothesis holds only at the
point of approximation. Approximate weak
separability imposes the restrictions
(15)
jk = af-ijik,
al
i,j=
1,..
&iPjs = ajPis,
,1, k= I+ 1, ...
M,
s= 1,...,N,
on the parameters of the translog function.
Linear homogeneity of the aggregator function
h(.) in output prices implies that the ratios of
output supply functions are homogenous of
degree zero in output prices. Writing this condition using Euler's theorem yields
0,
d
k=l
k=l
(97r/9P, p "
aPk. [7dr/9aPj
i : j,
i,j= ,1, 1 .
which imposes the further restrictions
(16)
T/ik
0,
i=
1...,1.
k=l
Finally, nonjointness in inputs requires that
the parameters of the translog approximation
satisfy
(17)
;ij= -aiaj,
i
* j,
i,j = 1, ....
+
aM(aM
-
1)
LMlDll
.L2,LmLDll, + LM2D22
...
L2M1Dl
+ LM2D22+ * * + DMM
The Empirical Model
The empirical model identifies five output categories including livestock, fluid milk, feed
and food grains, oilseeds, and other crops.
Variable inputs include flows from durable
equipment, land and buildings, farm-produced
durables, hired labor, energy, and non-energy
intermediate inputs. Self-employed and unpaid family labor is more accurately viewed as
a fixed input (Brown and Christensen). The
model is estimated under the maintained hypothesis of profit-maximizing behavior and assuming either weak separability in output
prices or nonjointness in input quantities or
both.
The parameter restrictions imposed by
separability and nonjointness are nonlinear
equalities. These restrictions can be algebraically embedded in the translog model. However, the curvature restrictions take the form
of nonlinear inequalities. Accordingly, to impose these restrictions a mathematical programming algorithm is employed rather than
the more common unconstrained estimation
techniques.4 The algorithm is available from
the Stanford Optimization Laboratory as a
Fortran routine called MINOS Version 5.0
(Murtagh and Saunders).
The estimator can be cast quite generally
as a system of nonlinear implicit equations
(Hazilla and Kopp)
(18) fit(Zit,
,) = ui,
i= 1 ...
,M;
t=
1,...
,T,
4 Traditional maximum likelihood estimation
represents an unconstrained optimization problem even if restrictions within and
across equations are imposed. The optimization is unconstrained
since the parameter restrictions are embedded directly into the
objective function through a reparameterization.
Ball
Modeling Supply Response
where Z is a matrix of observed data, 0 is a
vector of coefficients to be estimated, and ui, is
an error of optimization. Assuming that the
errors (ult, . .
, uMt)' are temporally inde-
pendent, each with mean zero, the same distribution, and positive definite error variance-covariance matrix 2, the Aitken-type
estimator 0 is obtained by minimizing with
respect to 0
(19) S(0) =
M
1/T
i=1
T
f
O,)' (2
fEt,(Zi,,
t=l
IM))
fit(Zit, i).
Equation(19) is minimizedwith respect to 0
given a priorconsistent estimateof E. Using 0,
a new estimate of Z is obtained based on the
inner product of estimated residuals. The estimates are iterateduntil the coefficientvector
0 and the covariance matrix E stabilize. It is
well known (Madansky) that such iteration
does not improve the asymptotic variance of
the estimator. However, when estimating a
system of equations with constraints across
one of the endogenous variables, such iteration results in estimates that are invariantto
the equation deleted under the assumed error
structure(Berndt and Savin).
To impose convexity, the optimization
problem in (19) is redefined in terms of the
lagrangianfunction,
(20) ?(Z, 0, X, X) = S(0) -
M
A
0),
Xh(Z,
i=l
where the X, are Lagrange multipliersassociated with the curvaturerestrictionshi(Z, 0).
If 0 is a solution to the optimizationproblem
posed in (20), then it can be shown (In
triligator,pp. 49-56) that the Lagrangemulti
pliers must satisfy
(21) O?(Z, 0, E, x)/0o =
M
aS(o)/ao - E X,ah,(Z, O)/ao = 0,
i=l
,i - 0,
Xihi(Z, 6) = 0,
-hi(Z, 6) < 0.
These Kuhn-Tuckernecessary conditions are
thus satisfied when the gradientof the lagrangian function equals zero, the complementary
slackness condition holds, and the inequality
817
constraints related to curvatureare nonnegative.
The tests of hypothesis on the structureof
productionare based on the likelihood ratio,
X, which is the ratio of the maximumof the
likelihoodfunction underthe null and alternative hypothesis. The statistic, -2 log X, is distributed asymptotically as chi-square with
degrees of freedom equal to the number of
restrictions imposed. It is importantto note
that the asymptotic distributionsof the statistics for tests of nonlinearequality restrictions
are independentof the imposition of inequality restrictions associated with convexity
(Rothenberg,p. 50).
The Data
The data cover the period 1948-79. Torqvist
price indexes are constructedfor the five output and seven inputcategories. Diewert (1976)
has shown that the Torqvist index will be
exact if the underlying aggregatorfunctions
are homogenous translog. Implicit quantity
indexes are obtained as ratios of value to the
Tornqvist price index. This is because the
Torqvist index satisfies the weak factor reversal test only approximately.
The output series are defined as the quantities marketed (including unredeemed CommodityCreditCorporationloans) plus changes
in farmer-owned inventories and quantities
consumedby farmhouseholds. The indexes in
each category are based on value to the producer; direct payments to producers under
government programs were included in the
value of production.
Data developed by Gollop and Jorgenson
are used to construct a measure of the labor
input. They disaggregated labor input and
laborcost into cells cross-classifiedby the two
sexes, eight age groups, five educational
groups, two employment classes (hired and
self-employed), and ten occupationalgroups.
No existing household or establishment survey, includingthe recently expanded Current
PopulationSurvey, is designed to provide annual dataon the distributionof workersamong
the 1,600 cells. However, existing surveys do
provide marginal totals cross-classified by
two, three, and sometimesfour characteristics
of labor input. These marginaldistributions,
availablefor each year 1948to 1979,provided
the basis for the estimates of labor input and
818 November 1988
labor cost. Extensive use was made of the
suitably generalized biproportional matrix
method.
The value of labor services equals the value
of labor payments plus the imputed value of
self-employed and unpaid family labor. The
imputedwage is set equal to the mean wage of
hired farm workers with the same occupational and demographiccharacteristics.
The capital input data are derived from information on investment and the outlay on
capital services. There are twelve investment
series used to calculate capital stocks. The
perpetual inventory method (Jorgenson) is
used and the service lives are those of Bulletin
F. Rentalprices for each asset are constructed
taking account of variations in effective tax
rates and rates of return, depreciation, and
capital gains. The value of capital services is
computed as the product of the rental price
and the quantity of capital at the end of the
preceding period. These data are controlled
to industry totals in the national income and
productaccounts. Using these data, Tornqvist
price and implicit quantity indexes are constructedfor durableequipment,land and service buildings,and farm-produceddurables.A
more detailed discussion of the procedures
used in constructing these capital price and
quantity series is found in Ball.
Data on energy flows in agriculturehave
been developed by Jack Fawcett Associates.
These data are used to construct Tornqvist
price and implicit quantity indexes of petroleum fuels, naturalgas, and electricity. Nonenergy intermediateinputs include chemical
fertilizers, agriculturalpesticides, feed, seed,
and purchased services. Price and implicit
quantityindexes are reportedin tables 1 and 2.
Total profit and profit shares for the variable
commodities are reported in table 3.
EmpiricalResults
Equations for profit shares were estimated
using inquality-constrainedmaximum likelihood methods. The parameterestimates for
the most generalmodel are reportedin table 4
together with their estimated standarderrors.
The value of the likelihoodfunction,as well as
the pseudo-R-squared(Baxterand Cragg),are
reported at the bottom of the table.
Test statistics for hypotheses on the structure of productionare reportedin table 5. The
hypothesis of weak separability in output
Amer. J. Agr. Econ.
prices is rejected at the 1%level. Hence, we
reject the existence of price and quantityindexes that satisfy the adding-upproperty.
The hypothesis that the technology is nonjoint in inputsis also rejected.The rejectionof
this hypothesis is consistent with the observation of multiproductfarms. This result is of
particularinterestto researcherswho have attempted to model productionof an individual
commodityin isolationof other productionactivitieswhich may coexist on the farm.In addition, it suggests careful considerationof policies which may be directedat a single output.
Specifically,this result suggests that such policies may be expected to affect all production
decisions, not simplythose made with respect
to the particularcommodityfor which the policy is targeted.
For completeness, the hypotheses of output
separabilityand input nonjointnessare tested
simultaneously.The results reportedin table 5
lead to the rejectionof the null hypothesisthat
each output is produced according to an
affinely homothetic productionfunction.
Table 6 reports gross elasticities of supply
and demand for the maintainedmodel. Estimation of this model subject to theoretical
curvaturerestrictions ensures positive (negative) own-elasticities of supply (demand).
Sakai defines a "normal"technology as satisfying the followingconditions:(a) the marginal
cost of an output tends to increase when the
quantities of other outputs decrease or when
the prices of inputs increase, and (b) the marginalrevenueof an inputincreaseswhen quantities of other inputs increase or when output
prices increase. These conditions on the technology are sufficientto show that the outputs
jointly produced are never gross substitutes,
nor are the inputs employed, and that the
input-output relations are not regressive.
These resultsare extremelyimportantbecause
they imply a set of inequality restrictions on
the entire matrixof gross elasticities. All elasticity estimates satisfy the normalcase restrictions when evaluated at the point of approximation.
The own-elasticitiesof supply are generally
less than unity; only the supply functions for
livestock and "other crops" are price elastic.
The gross complementarityof outputs suggests that an increase in the price of a particular output would result in increased production of all outputs.This would occur only if the
increase in input usage resultingfrom an output price increase shifted the product trans-
Table 1.
Price Indexes of Outputs and Inputs, U.S. Agriculture 1948-79
Year
P1
P2
P3
P4
P5
P6
P7
P8
P9
1948
1949
1950
1951
1952
1953
1954
1955
1956
1957
1958
1959
1960
1961
1962
1963
1964
1965
1966
1967
1968
1969
1970
1971
1972
1973
1974
1975
1976
1977
1978
1979
0.7558
0.6496
0.6575
0.7851
0.6795
0.6262
0.5807
0.5259
0.5005
0.5482
0.6260
0.5609
0.5551
0.5485
0.5635
0.5340
0.5049
0.5674
0.6429
0.5812
0.6034
0.6899
0.6976
0.6659
0.7882
1.1098
0.9551
1.0030
1.0145
1.0000
1.2617
1.4711
0.4995
0.4119
0.4069
0.4744
0.5017
0.4512
0.4150
0.4185
0.4311
0.4379
0.4304
0.4335
0.4383
0.4394
0.4266
0.4279
0.4330
0.4409
0.4993
0.5198
0.5432
0.5689
0.5916
0.6081
0.6280
0.7374
0.8592
0.9016
0.9938
1.0000
1.0898
1.2328
0.7850
0.7325
0.7035
0.6899
0.6943
0.6818
0.6895
0.6539
0.6471
0.6297
0.6162
0.5794
0.5707
0.6755
0.7703
0.7476
0.7563
0.7274
0.8182
0.6747
0.7020
0.7283
0.7551
0.6883
0.7817
1.2044
1.5285
1.3044
1.1642
1.0000
1.1491
1.2504
0.4439
0.3640
0.3997
0.4290
0.4387
0.4010
0.4004
0.3392
0.3550
0.3193
0.3042
0.2995
0.3059
0.3722
0.3489
0.3779
0.3677
0.3886
0.4349
0.3828
0.3692
0.3506
0.4000
0.4295
0.4865
0.7767
0.9821
0.7522
0.8066
1.0000
1.0156
1.0569
0.4627
0.4267
0.4737
0.4780
0.4992
0.4641
0.4527
0.4568
0.4683
0.4562
0.4635
0.4650
0.4777
0.4669
0.4776
0.4845
0.4861
0.5135
0.5354
0.6027
0.5651
0.5486
0.5797
0.6180
0.6573
0.8679
0.9635
0.8786
0.9192
1.0000
1.0317
1.1067
0.3036
0.1476
0.2797
0.3687
0.2937
0.2455
0.3009
0.2799
0.2850
0.3788
0.3916
0.3431
0.3739
0.4599
0.4425
0.4574
0.4220
0.5154
0.5742
0.5140
0.5089
0.5304
0.5631
0.5742
0.6297
0.8163
0.8381
0.8153
0.7619
1.0000
1.1882
1.4397
0.1828
0.0599
0.1580
0.2829
0.1209
0.1187
0.1086
0.1727
0.1128
0.1317
0.2120
0.1632
0.1329
0.2413
0.3667
0.4327
0.2835
0.5257
0.6554
0.4748
0.4425
0.7948
0.6430
0.7660
1.1184
2.1688
1.5153
0.6528
0.8064
1.0000
1.1229
1.8406
0.4741
0.3634
0.3679
0.1875
0.4517
0.4542
0.3686
0.2472
0.1817
0.1892
0.2123
0.0950
0.4433
0.2991
0.3037
0.2957
0.3590
0.3990
0.3303
0.4815
0.5328
0.2763
0.4271
0.4833
0.5529
0.7932
1.1115
1.9672
1.3438
1.0000
1.4083
1.3307
0.192
0.194
0.182
0.203
0.208
0.209
0.210
0.212
0.231
0.251
0.261
0.271
0.285
0.297
0.305
0.316
0.350
0.377
0.409
0.444
0.482
0.491
0.558
0.581
0.607
0.697
0.762
0.798
0.884
1.000
1.098
1.205
Note:P1 is livestock,P2 is fluidmilk,P3 is feed andfood grains,P4 is oilseeds,P5 is othercrops,P6 is durableequipment,P7 is reales
labor,Pi0 is energy, and Pil is other purchasedinputs; WI is self-employedlabor.
Table 2.
Implicit Quantity Indexes of Outputs and Inputs, U.S. Agriculture
1948-79
Year
Y1
Y2
Y3
Y4
Y5
-Y6
-Y7
- Y8
1948
1949
1950
1951
1952
1953
1954
1955
1956
1957
1958
1959
1960
1961
1962
1963
1964
1965
1966
1967
1968
1969
1970
1971
1972
1973
1974
1975
1976
1977
1978
1979
0.5558
0.5898
0.6004
0.6574
0.6338
0.6495
0.6821
0.7126
0.7394
0.7124
0.7191
0.7616
0.7654
0.7855
0.7879
0.8250
0.8687
0.8582
0.8880
0.9196
0.9232
0.9246
0.9671
0.9990
0.9723
0.9593
0.9845
0.9665
1.0063
1.0000
0.9859
0.9931
0.8648
0.8888
0.8883
0.8807
0.8849
0.9275
0.9405
0.9491
0.9681
0.9714
0.9648
0.9581
0.9688
0.9973
1.0047
0.9988
1.0154
0.9962
0.9637
0.9566
0.9471
0.9409
0.9493
0.9620
0.9746
0.9384
0.9410
0.9401
0.9794
1.0000
0.9910
1.0081
0.4678
0.2558
0.3579
0.3092
0.3966
0.3570
0.3889
0.4070
0.4100
0.4200
0.5133
0.5247
0.5613
0.4818
0.4705
0.5466
0.4782
0.6425
0.5865
0.7525
0.6627
0.7242
0.6706
0.8672
0.8675
0.9631
0.8335
1.0429
1.0101
1.0000
1.0100
1.1518
0.2601
0.2337
0.2618
0.2620
0.2541
0.2509
0.2791
0.3156
0.3445
0.4786
0.3989
0.3954
0.4176
0.4477
0.4701
0.5049
0.4919
0.5556
0.5629
0.6064
0.6743
0.7286
0.8005
0.7605
0.8142
0.9286
0.9101
0.9890
0.8321
1.0000
1.0596
1.2819
0.6872
0.8272
0.6288
0.7035
0.7331
0.7701
0.7909
0.8018
0.7286
0.7736
0.7539
0.8197
0.8077
0.8343
0.8462
0.8496
0.8019
0.8870
0.8224
0.7276
0.8246
0.9017
0.8227
0.8620
0.9592
0.9864
0.9209
0.9052
0.8945
1.0000
0.9897
1.0950
0.3661
0.4689
0.5364
0.6000
0.6488
0.6656
0.6967
0.7031
0.7094
0.6946
0.6843
0.6925
0.7051
0.6941
0.6875
0.6932
0.7071
0.7254
0.7505
0.7820
0.8163
0.8275
0.8309
0.8375
0.8394
0.8562
0.9101
0.9473
0.9676
1.0000
1.0218
1.0551
1.1582
1.1554
1.1070
1.1386
1.2877
0.8960
0.9150
0.9314
0.9403
0.9425
0.9509
0.9829
1.0035
1.0103
1.0179
1.0230
1.0323
1.0368
1.0396
1.0422
1.0473
1.0502
1.0481
1.0481
1.0458
1.0430
1.0479
1.0515
0.9954
1.0000
1.0080
1.0213
0.7869
0.9267
0.7369
0.7600
0.7964
0.8255
0.8155
0.8207
0.8130
0.8135
0.7762
0.8007
0.8140
0.8919
0.8519
0.8662
0.8782
0.8645
0.8696
0.8553
0.8657
0.8650
0.8727
0.8746
0.8975
0.9233
0.9524
1.0060
1.0397
1.0000
1.0144
1.0092
- Y
1.8
1.7
1.8
1.7
1.6
1.5
1.4
1.4
1.3
1.3
1.3
1.3
1.3
1.3
1.3
1.3
1.1
1.1
1.0
0.99
0.94
0.99
0.92
0.90
0.90
0.93
0.99
1.0
1.0
1.0
1.0
1.0
Note: Yl is livestock,Y2is fluidmilk, Y3is feed andfood grains,Y4is oilseeds, Y5is othercrops, Y6is durableequipment,Y7is reale
labor, Y10is energy, and Yll is other purchasedinputs;X1 is self-employedlabor.
Table 3.
Total Variable Profit and Profit Shares of Outputs and Variable Inputs, U.S. A
Year
Total
Variable
Profit
S1
S2
S3
S4
S5
-S6
-S7
1948
1949
1950
1951
1952
1953
1954
1955
1956
1957
1958
1959
1960
1961
1962
1963
1964
1965
1966
1967
1968
1969
1970
1971
1972
1973
1974
1975
1976
1977
1978
1979
14,883
13,215
11,865
8,892
11,016
16,511
14,740
12,991
10,703
9,210
12,652
16,214
12,191
13,265
9,871
9,900
13,223
13,637
13,324
8,960
10,192
13,707
13,529
12,979
9,003
10,625
14,483
13,295
13,773
9,195
10,536
15,989
0.8352
0.7691
0.9581
1.1199
0.9420
0.9053
0.8869
0.8535
0.8256
0.8673
1.0262
0.9178
1.059
1.1911
1.3308
1.4549
1.4415
1.5671
1.8998
1.7169
1.6651
1.8491
1.8788
1.8682
2.0586
2.4897
2.1040
2.0926
2.0857
1.8506
2.2294
2.6662
0.3446
0.2949
0.3520
0.3637
0.389
0.3737
0.3507
0.3630
0.3735
0.3791
0.3798
0.3580
0.4249
0.4860
0.5155
0.5663
0.5798
0.5671
0.6424
0.6410
0.6170
0.6235
0.6275
0.6592
0.6597
0.6493
0.7259
0.7342
0.7978
0.7425
0.7766
0.9100
0.4094
0.2109
0.3427
0.2595
0.3377
0.3038
0.3367
0.3399
0.3318
0.3294
0.4044
0.3662
0.4479
0.5045
0.6093
0.7567
0.6666
0.8434
0.8955
0.9147
0.7798
0.8588
0.8117
0.9400
1.0215
1.5211
1.5987
1.6468
1.3472
1.0378
1.1664
1.4738
0.0960
0.0714
0.1062
0.1020
0.1020
0.0937
0.1047
0.1020
0.1141
0.1420
0.1157
0.1065
0.1333
0.1927
0.2057
0.2636
0.2487
0.2907
0.3408
0.3120
0.3113
0.3102
0.3731
0.3837
0.4452
0.7057
0.8367
0.6719
0.5736
0.7742
0.8068
1.0344
0.6413
0.7186
0.7333
0.7401
0.8118
0.8069
0.8133
0.8462
0.7720
0.7949
0.8081
0.8306
0.9760
1.0924
1.2288
1.3789
1.2996
1.4867
1.4862
1.4290
1.4126
1.4567
1.3472
1.5174
1.7179
2.0308
2.0139
1.7415
1.7039
1.8771
1.8562
2.2432
0.1076
0.0676
0.1773
0.2336
0.2029
0.1770
0.2285
0.2182
0.2195
0.2844
0.2973
0.2484
0.3201
0.4295
0.4439
0.5096
0.4774
0.5857
0.6981
0.6286
0.6044
0.6202
0.6242
0.6574
0.6911
0.7956
0.8308
0.8116
0.7332
0.9008
1.0591
1.3494
0.1694
0.0559
0.1709
0.2812
0.1370
0.0953
0.0895
0.1474
0.0952
0.1109
0.1850
0.1386
0.1339
0.2712
0.4502
0.5883
0.3871
0.7058
0.9124
0.6397
0.5573
0.9750
0.7552
0.9072
1.2642
2.1286
1.4298
0.5962
0.6599
0.7446
0.8162
1.3804
-S
0.2
0.1
0.1
0.0
0.2
0.2
0.1
0.1
0.0
0.0
0.1
0.0
0.2
0.2
0.2
0.2
0.2
0.3
0.2
0.3
0.3
0.1
0.2
0.3
0.3
0.4
0.6
1.1
0.7
0.5
0.7
0.6
S7 is realesta
Note:S1 is livestock,S2 is fluidmilk,S3 isfeedandfoodgrains,S4 isoilseeds, S5is othercrops,S6 is durableequipment,
S10 is energy, and 511 is other purchasedinputs.
Amer. J. Agr. Econ.
822 November 1988
Table 4.
Parameter Estimates for the Translog Restricted Profit Function
Estimated
Value
Parameter
1.6932
0.6609
0.8135
0.6052
1.5703
-0.6651
-0.5389
0.4699
-0.4372
-0.3299
1.9017
0.6708
-0.2820
0.5712
-0.3499
-0.9451
0.2218
0.4467
0.1702
0.0296
0.0746
0.5348
0.6482
-0.1383
-0.0851
0.2628
0.0721
0.1453
0.0958
-0.0770
-0.0525
-0.0635
0.8327
-0.1580
-0.5072
0.3846
0.0931
-0.0002
0.1059
0.1335
-0.1749
0.5003
-0.3312
0.0880
a,
a2
a3
a4
a5
a.
a7
a.
ag
all
/311
/316
1814
1816
/317
181
/319
/322
/323
/324
/325
/326
/327
/328
/329
/3210
/3211
/333
/335
/336
/383
/339
/310
/3311
/344
/345
/346
in L
?2
-
Standard
Error
0.0935
0.0294
0.1122
0.0544
0.0939
0.0627
0.1205
0.0558
0.0234
0.0162
0.1097
0.2217
0.0805
0.2028
0. 1049
0.1807
0.1291
0.1232
0.0929
0.0621
0.0383
0.2432
0.0923
0.0814
0.0502
0.1078
0.0564
0.0366
0.0328
0.0589
0.0328
0. 1339
0.3338
0. 1232
0.2064
0.1401
0.1491
0. 1233
0.0675
0.0408
0.3273
0.0844
0.1229
0.0831
Parameter
347
/48
/349
/3410
/3411
/355
P56
1357
/358
/859
/3510
/3511
/366
/367
/368
/369
/361
/3611
P;7
P,7
P,7
/3710
/3711
P,8
/389
18810
/3011
P,9
/3910
3911
/31010
/31011
/31111
Yit
72t
73t
74Y
Ys5
76t
Y7t
'Y8t
Yot
Yiot
Ylit
Estimated
Value
Standard
Error
0.1190
0.0364
0.0479
0.0056
0.1271
0.8473
0.0824
0.4120
0.2369
-0.0543
0.1744
0.3475
0.2623
-0.2306
-0.1608
0.0040
-0.0059
-0.1933
-0.5151
0.0821
0.0999
0.0669
-0.3857
-0.1451
-0.0918
-0.0522
-0.1714
0.0275
0.0216
0.0865
-0.1285
-0.1035
-0.0033
0.0024
-0.0019
0.0068
1.9E-05
-0.0099
0.0087
0.0219
0.0066
-0.0018
-0.0019
-0.0178
0.0692
0.0550
0.0440
0.0249
0.1489
0.3211
0.1363
0.1211
0.0860
0.1006
0.0482
0.2944
0.1183
0.0819
0.0697
0.0502
0.0321
0.1661
0.1694
0.0783
0.0287
0.0211
0.1381
0.0778
0.0278
0.0197
0.1186
0.0563
0.0247
0.1063
0.0173
0.0589
0.4471
0.0143
0.0048
0.0169
0.0080
0.0139
0.0099
0.0166
0.0091
0.0040
0.0027
0.0171
513.7525
= 0.9851
Note: I is livestock, 2 is fluid milk, 3 is feed and food grains, 4 is oilseeds, 5 is other crops, 6 is durable equipment, 7 is real estate, 8 is
farm-produceddurables,9 is hiredlabor, 10 is energy, and II is other purchasedinputs.
Table 5.
Chi-Square Statistics for Hypothesis Tests
Hypothesis
Output separability
Input nonjointness
Affinely homothetic
production functions
Critical Value
Calculated
Value
Degrees
of
Freedom
0.05
0.01
68.08
92.35
29
10
42.56
18.31
49.59
23.21
164.69
39
54.58
62.53
Table 6.
Output Supply and Input Demand Elasticities
Elasticity with Respect to Price of
Commodity
Livestock
Fluid milk
Grains
Oilseeds
Other crops
Durable
equipment
Real estate
Farm produced
durables
Hired labor
Energy
Other purchased
inputs
Grains
Oilseeds
Other
Crops
Durable
Equipment
Real
Estate
FarmProduced
Durables
0.494
0.642
0.491
0.502
0.493
0.476
0.604
0.838
0.552
0.491
0.399
0.477
0.411
0.432
0.394
1.012
1.173
0.947
1.023
1.110
-0.534
-0.556
-0.192
-0.519
-0.613
-0.275
-0.319
-0.425
-0.342
-0.277
-0.369
-0.325
-0.470
-0.409
-0.319
1.359
0.864
0.552
0.391
0.235
0.641
0.473
0.384
1.446
0.806
-1.271
-0.237
-0.192
-0.584
-0.228
-0.622
1.331
1.625
1.467
0.457
0.837
0.820
0.814
0.571
0.409
0.528
0.496
0.500
1.066
1.694
1.042
-0.323
-0.674
-0.647
-0.713
-0.310
-0.336
-1.162
-0.260
-0.312
1.412
0.694
0.906
0.538
1.388
-0.564
-0.336
-0.379
Livestock
Fluid
Milk
1.089
1.266
0.991
1.115
1.091
824 November 1988
formation frontier outward sufficiently to
allow absolute increases in all outputs. The
magnitude of the input demand elasticities
with respect to outputprices suggests that this
may, indeed, be the case.
The input demand functions are generally
price elastic. Of particularinterest is the evidence that the demandfor hired farm labor is
elastic. This has importantimplicationsfor the
analysis of the effects of farm labor unionization. The estimates suggest that returns to
hired farm labor and the level of farm employmentmay decrease dramaticallyas a consequence of increasing effective wages associated with unionization.
An increase in the hired labor wage rate
would result in absolute reductionsin all outputs as well as induce changes in the composition of output. Finally, the gross complementarity of the inputs suggests that the reduction
in outputwould be accompaniedby reductions
in the demand for all factors of production.
ConcludingComments
The objective of this paper has been to model
supply response in agriculturewithout imposing separabilityor nonjointnessas part of the
maintainedhypothesis. First, the implications
of these assumptions for the form of the restricted profit function were examined. Both
special forms of the technology were rejected.
The rejection of separabilityrestrictions suggests that consistent aggregationof the outputs is not possible. The existence of individual productionfunctions for each output was
also ruled out. Unless joint productionis permitted, the resultingestimates of responsiveness of a particularcommodity to changes in
own price and prices of competingoutputsare
likely to be considerably understated.
The tests of restrictionson the structureof
productionwere carried out under the maintained hypothesis of theoretical consistency.
That is, the propertiesof the restricted profit
function were imposed during estimation.
While it is known that the imposition of inequality constraints does not affect the
Cramer-Raolower bound for the variance of
the estimator (Rothenberg, p. 50), it is still
importantto impose the curvaturerestrictions
implied by economic theory.
The derivationof the separabilityand nonjointness restrictionsrests upon the properties
of a well-behaved restricted profit function.
Amer. J. Agr. Econ.
More specifically, these properties allow for
a dual interpretationof the technology. If
these properties are not present, the duality
theorems do not apply; consequently, the hypothesis tests have no economic interpretation. Thus, the tests of the null hypothesis of
functional structureis actually a joint test of
functional structure/theoretical consistency
against the alternative of theoretical consistency. Unless the data yield empirical estimates underthe null and alternativehypotheses that are theoretically consistent without
constraints,the constraintsmust be imposed.
[Received November 1987; final revision
received March 1988.]
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