This paper estimates an Almost Ideal Supply System using aggregate U.S. agricultural data. Share equations derived from an indirect production function yield elasticities that are consistent with production theory. A nested test comparing the Almost Ideal Supply System to the Translog Production Function finds little difference between the two models.Research Methods/ Statistical Methods,

Hilmer, Christiana E.

Holt, Matthew T.

English

Research Papers in Economics

THE ALMOST IDEAL SUPPLY SYSTEM AND
AGRICULTURAL PRODUCTION IN THE UNITED STATES
Christiana E. Hilmer and Matthew T. Holt
1
Selected Paper
American Agricultural Economics Association Annual Meeting
Nashville, Tennessee
August 8-11, 1999
Abstract:
This paper estimates an Almost Ideal Supply System using aggregate U.S. agricultural
data.  Share equations derived from an indirect production function yield elasticities that
are consistent with production theory.  A nested test comparing the Almost Ideal Supply
System to the Translog Production Function finds little difference between the two
models.
Copyright 1999 by Christiana E. Hilmer and Matthew T. Holt.  All rights reserved.  Readers may
make verbatim copies of the document for non-commercial purposes by any means, provided that
this copyright notice appears on all such copies.
                                                       
1 Christiana E. Hilmer is a graduate student in the Department of Agricultural and Resource
Economics and the Department of Statistics, North Carolina State University.  Matthew T. Holt
is a Professor in the Department of Agricultural and Resource Economics, North Carolina State
University.1
1.  Introduction
An important concern in applied economics is the appropriate way to model consumer
demand and firm supply.  Several methods have been proposed for modeling consumer demand.
Prominent among these are the Rotterdam model of Theil (1965), the Linear Translog model of
Christensen, Jorgenson, and Lau (1975) and the Almost Ideal Demand System (AIDS) of Deaton
and Muellbauer (1980).  Among these, the AIDS has gained widespread support in both direct
applications and extensions to more complex applications, such as the inverse demand
framework (for recent examples see Holt and Goodwin [1997], Ryan and Wales [1996], Eales
and Unnevehr [1994]). The AIDS model is appealing among the potential demand systems due
to its second order flexibility, compatibility with demand theory, and relative ease of estimation.
Given its clear intuitive appeal, it is somewhat surprising that only one attempt has been
made to apply the almost ideal framework to the firm’s supply decision (Chambers and Pope,
1994).  Previous work in production economics has focused on alternative functional forms, such
as the Translog or the Generalized Leontief.  The properties that make the framework appealing
for demand should also make it appealing for supply.  Namely, the Almost Ideal Supply System
(AISS) we propose should provide an easily estimable and sufficiently general functional form
that can be used to test the restrictions imposed in producer theory.
In this paper, we are the first to estimate an indirect production function using the AISS
model with time-series data on U.S. aggregate agricultural output.  We find estimated elasticities
that are consistent with production theory.  To assess the relative goodness of fit of the AISS to
the Translog Production Function, we use the nested test of Lewbel (1989) and find no
significant difference between the two models.  Hence our results suggest that the AISS is a
potentially appealing method for estimating modern supply theory.2
2. The Almost Ideal Supply System
The classic almost ideal demand system of Deaton and Muellbauer (1980) is derived from
the indirect utility function
(1)  (P))   g ln    -   m (ln p   =   ) m , P ( h
L
1 = k
k B -
k ￿
where h(P,m) represents indirect utility, P is a vector of consumer prices, m is income, and g(p)
is a price index defined as
(2) ￿￿ ￿ + a + a
= =
L
1 k
L
1 j
j k kj
L
1 = k
k k   0 p ln p ln B
2
1
    p ln     =   (P)   g   ln .
 To convert this almost ideal framework to supply, assume output, y, is produced by
combining capital, land, chemicals, other intermediate inputs, and labor.  Let W represent a
vector of input prices, E represent aggregate expenditure, and t represent a time trend measuring
technological change.  The almost ideal indirect production function is defined by substituting y,
W, and E for the indirect utility, price and income measures in (1) and adding t to capture
technological change.  We then have
(3) t)) E, (W,   g ln    -   E (ln w   =   ) t , E , W ( y
5
1 = k
B -
k
k ￿
where ln g(W,E,t) is defined as
(4)  ￿￿ ￿ + + + + ￿ + a + a
= =
5
1 k
5
1 j k
2
tt T k k t j k kj
5
1 = k
k k   0 t V
2
1
E ln t B w ln V  t   t V     w ln w ln B
2
1
    w ln   =   t) E, (W,   g   ln
Note that the additional terms in (4) relative to (2) are the result of adding technological change
to the production process.
Economic theory implies that production functions should be homogeneous of degree one
and symmetric in the second order parameters.  Additionaly, because we will be estimating share
equations, adding-up restrictions are needed.   These desirable properties are satisfied by3
imposing the following restrictions: ￿ ￿ ￿ ￿ = = = = = a jk kj t k jk k k B   B   and   , B V    , 0 B   , 0 B   , 1 for
all k and j.  Finally, to guarantee convergence we make the additional identifying restriction that
. 0 0 = a
Imposing the above restrictions and applying Roy’s identity to equation (3) yields a
system of share equations:
(5)    
t B 1
t)) E, g(W, ln    E (ln B tV w ln B
  =   S
T
k
i i k ik i
i -
￿ - + + + a
where Si is the cost share of the ith input and ￿ = . 1 Si
2
Next, we want to examine the relevant elasticities implicit in the AISS model.  The
elasticities are computed by converting the share equations in (5) to ordinary input demand
functions, Xi, by  i i i W / E S X = .  The subsequent uncompensated own- and cross-price
elasticities of demand are:
(6)
) t B - (1 S
  ) tV w ln B ( B
) t B - (1 S
B
1
w ln
X ln
T i
k
i k ik i i
T i
ii
i
i u
ii
￿ + + a
- + - =
¶
¶
= e
(7)      
) t B - (1 S
  ) tV w ln B ( B
) t B - (1 S
B
w ln
X ln
T i
k
j k jk j i
T i
ij
j
i u
ij
￿ + + a
- =
¶
¶
= e
and the expenditure elasticities of input demand are:
(8) 
i
i i
iE S
B
1  
E ln
X ln
- =
¶
¶
= e
Combining (6), (7), and (8) according to the Slutsky decomposition yields the compensated own-
and cross-price elasticities:
(9)   iE j
u
ij
c
ij S     e + e = e .
                                                       
2 Note that  ) E ln / ) t , E , W ( y ln /( ) w ln / ) t , E , W ( y ln ( E / X W S i i i i ¶ ¶ ¶ ¶ - = =4
Finally, following Chambers (1988) the Allen elasticities of substitution are
(10) j
c
ij ij S / e = s .
Turning to the effect of technology on production, the rate of technical change for the
AISS indirect production function is measured by
(11) ￿ - - - - ￿ =
¶
¶
= e
-
k
tt t k k t
k
B
k YT y
t
) t V E ln B w ln V V ( w
t ln
y ln
k
and input bias in technical change is measured by
(12)
) t B 1 (
t B
) t B 1 ( S
) t V E ln B w ln V V ( B V ( t
t ln
w ln
b
t
t
t i
k
tt t k k t i i
i
i -
+
-
￿ + + + -
=
¶
¶
= .
According to Kim (1988), if technical change is occurring YT e will be positive.  Further, if bi>0
input i is technology intensive, if bi=0 input i is technology neutral, and if bi<0 input i is
technology saving.
3.   Data Description and Empirical Results
The Almost Ideal Supply System consists of the indirect production function in (3) and
the five share equations in (5). To explore the potential usefulness of the AISS, we estimate the
above model using the familiar data on U.S. agricultural production found in Ball et. al. (1997).
These data consist of time series observation on U.S. aggregate agricultural output (Y), capital
(K), land (L), chemicals (C), other inputs (I), and labor (LB) for the period 1948-1994.
3  Note
that other inputs consist of fuels, electricity, feed, seed, and livestock purchases. Data are
aggregated using the Divisia indexing method and the base year is defined as 1987.
                                                       
3 See Ball et. al. (1997) for an excellent description of the original data sources as well as the methods used to
construct the subcategory indices.5
The AISS model is estimated using a maximum likelihood routine in TSP subject to the
parameter restrictions imposed by symmetry and homogeneity.  The labor equation is dropped to
avoid the singular variance-covariance matrix that occurs because the share equations add to one.
Hence, five equations are jointly estimated, specifically, the indirect production function and
four share equations.  The resulting R
2 values are above .80 in each of the equations indicating a
fairly good fit.  However, the Durbin-Watson statistics are all below .25 suggesting the presence
of autocorrelation.  Accordingly, the model was re-estimated using the Berndt-Savin, Moschini-
Moro, and Symmetric-R autocorrelation correction methods.  The results presented below are
from the Symmetric-R specification, which had the highest log likelihood value, and provided R
2
values above .95 and Durbin-Watson statistics above two.
Compensated own- and cross-price elasticities calculated using (9) and the AISS
parameter estimates are presented in Table 1.
4   The estimated cross-price elasticities are nearly
all positive  suggesting that the five inputs are substitutes. Statistically significant exceptions
include the demand for chemicals with respect to the price of capital, the demand for land with
respect to the price of other inputs, and the demand for land with respect to the price of labor.
Estimated own price elasticities are all negative and significant, suggesting that the inputs follow
the law of factor demand.  The fact that these values are greater than negative one, suggests that
all five inputs are inelastic, with chemicals having the most elastic factor demand and land
having the most inelastic factor demand.  The compensated elasticities can be used to test
whether the indirect production function is quasi-convex. The leading eigenvalue in the
substitution matrix (calculated at the data means) is 0.9243.  To have proper curvature,
                                                       
4 Allen elasticities of substitution were also estimated.  However, these values are not reported as they contain the
same information as the compensated elasticities.6
eigenvalues of should all be negative.  Therefore, we reject the quasi-convexity of the indirect
production function.
A well known aspect of compensated elasticities is that they isolate input price effects
without considering expenditure effects that accompany changes in input prices.  To consider
both effects, we calculate the uncompensated elasticities in Table 2.  As expected, the estimated
own-price elasticities are all negative.  Turning to cross-price elasticities, the estimated
uncompensated values in Table 2 differ markedly from the estimated compensated values in
Table 1.    Most of the uncompensated elasticities are negative, suggesting that when the
expenditure effect is accounted for, the majority of inputs are complements.   The only
statistically significant exception is land, for which increases in the price of capital and
chemicals increase the demand for land.  The difference in sign between compensated and
uncompensated elasticities is consistent with previous findings by Kim (1988) and Gajanan and
Ramaiah (1996).
5
Table 3 presents the estimated expenditure elasticities in (8).  Economic theory predicts
that these elasticities should be positive, as firms should increase the amount of all inputs used in
response to increases in expenditure.  All inputs have the correct sign, except for land.  The
negative sign for land may result because land is a relatively fixed input.  The expenditure
elasticity for capital is less than one, suggesting that capital is expenditure inelastic.  Conversely,
the expenditure elasticities for chemicals, other inputs and labor are greater than one, suggesting
that those inputs are expenditure elastic. These results differ from those of Chambers and Lee
(1986) who find all expenditure elasticities are positive and close to one.
                                                       
5 Based on the sign differences, Gajanan and Ramaiah (1996) caution policy makers to consider uncompensated
rather than compensated elasticities.7
Table 4 investigates the effect of technological innovation by computing the rate of
technical change in  (11) and the input bias terms in (12).  The estimated rate of technical change
is positive and significant, implying that technology increased during the period from 1948 to
1994.  The bottom panel of Table 4 suggests that other inputs and labor were technology
intensive during that time period, capital and chemicals were technology saving, and land was
technology neutral.
As the empirical results above are generally consistent with production theory, the AISS
would seem a strong candidate for future work in production economics.  We would therefore
like some idea how the AISS compares to the popular Translog model.  Lewbel (1989) proposes
a model that nests both the AIDS and the Translog indirect utility models and allows for a simple
test between the two.  By substituting supply for demand measures and adding technology we
develop an analogous nested model for testing between the AISS and the Translog indirect
production functions.
6  A test between the alternative models does not reject the null hypothesis
of equality between the AISS and the Translog models.  Specifically, the log likelihood value for
the AISS is 913.435 while the log likelihood value for the Translog model is 909.894.  In other
words, it appears that the performance of these two models is not significantly different as a
means of estimating indirect production functions. 
4.    Conclusion
The almost ideal demand system of Deaton and Muellbauer (1980) has become one of the
most popular functional forms in empirical studies of consumer demand theory.  In fact, a recent
literature search found in excess of one hundred applications since its initial development.  In
this study we are the first to estimate an almost ideal indirect production system.  Using time
                                                       
6 Details are available upon request.8
series data on U.S. agricultural production, we find that the AISS is a potentially beneficial
method of estimating firm supply theory.  The functional form is sufficiently general, the
estimation is computational simple, the estimated coefficients are highly significant, and the
majority of the results conform to conventional supply theory.  These findings suggest that the
AISS may be preferable to alternative methods, such as the Translog, as a means of studying
firm supply theory.
In the future we plan to extend the AISS model estimated here to examine several
important issues.  We would like to estimate a short run model that treats capital and land as
fixed.  Within the context of this short run model, we plan to take a closer look at the curvature
issue and, if needed, impose local concavity.  Current research by Pitarakis and Tridimas (1999)
allows expenditure to be endogenous in a system of demand equations.  We plan to adopt this
methodology to the supply side and treat expenditure as endogenous within the indirect
production framework.9
Table 1. Estimated Compensated Price Elasticities
Elasticity with respect to the price of
Capital Land Chemicals Other Inputs Labor
Capital -0.4213* 0.1581* -0.0063 0.0810* 0.1885*
(0.061) (0.035) (0.030) (0.039) (0.036)
Land 0.4891* -0.1330* 0.1370* -0.1486* -0.3446*
(0.054) (0.036) (0.039) (0.047) (0.059)
Chemicals -0.3768* -0.0596 -0.7046* 0.6692* 0.4718*
(0.073) (0.049) (0.072) (0.076) (0.075)
Other Inputs -0.0376 0.0087 0.0409* -0.1550* 0.1430*
(0.012) (0.013) (0.016) (0.029) (0.021)
Labor 0.0143 0.0163 -0.0009 0.1852* -0.2149*
(0.034) (0.018) (0.023) (0.041) (0.035)
Notes: Standard errors in parentheses. * denotes significance at the 5% level.10
Table 2. Estimated Uncompensated Price Elasticities
Elasticity with respect to the price of
Capital Land Chemicals Other Inputs Labor
Capital -0.4931* 0.0505 -0.0431 -0.2069* 0.0257
(0.065) (0.041) (0.033) (0.047) (0.041)
Land 0.5391* -0.0581 0.1626* 0.0518 -0.2313*
(0.064) (0.051) (0.044) (0.066) (0.074)
Chemicals -0.6056* -0.4023* -0.8219* -0.2481* -0.0469
(0.084) (0.057) (0.074) (0.086) (0.092)
Other Inputs -0.1731* -0.1942* -0.0285 -0.6980* -0.1641*
(0.026) (0.014) (0.018) (0.020) (0.027)
Labor -0.1369* -0.2102* -0.0784* -0.4209* -0.5577*
(0.045) (0.020) (0.026) (0.033) (0.046)
Notes: Standard errors in parentheses. * denotes significance at the 5% level.11
Table 3.  Estimated Expenditure Elasticities
Elasticity Estimate
Capital 0.6668*
(0.116)
Land -0.4641*
(0.200)
Chemicals 2.1247*
(0.203)
Other Inputs 1.2579*
(0.068)
Labor 1.4040*
(0.116)
Notes: Standard errors in parentheses. * denotes
significance at the 5% level.12
Table 4.  Estimated Input Biases and the Rate of Technical Change
Estimate
Rate of Technical Change 0.6875*
(0.213)
Bias
Capital -1.5283*
(0.253)
Land 0.1277
(0.160)
Chemicals -0.7589*
(0.334)
Other Inputs 0.1075*
(0.056)
Labor 1.5358*
(0.193)
Notes: Standard errors in parentheses. * denotes
significance at the 5% level.13
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A linear logarithmic expenditure system: an application to U.S. data,&quot;

A simple method for imposing local curvature in some flexible consumer demand systems,&quot; Discussion paper.

Agricultural Productivity Revisited,&quot;

An almost ideal demand system,&quot;

An Econometric Estimation of Hicksian and

Analyzing the Indirect Production Function for U.S.

Applied Production Analysis: A Dual Approach.

Constrained output maximization and US agriculture,&quot;

Elasticities of demand for food: An analysis using non-additive utility functions allowing for habit formation,&quot;

Generalized habit formation in an inverse almost ideal demand system: An application to meat expenditures in the U.S.,&quot;

THE ALMOST IDEAL SUPPLY SYSTEM AND AGRICULTURAL PRODUCTION IN THE UNITED STATES

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