A cost approach to economic analysis under state-contingent production uncertainty.

Much research has been done on the microeconomics of uncertainty. Under incomplete risk markets, the effects of uncertainty on economic decisions have typically been investigated under the expected utility model (e.g., Arrow 1965; Pratt 1964). When applied to firm behavior, Sandmo (1971) and others

have shown the adverse effects of uncertainty under risk aversion. This has stressed the joint importance of risk assessment and risk preferences. Risk assessment is typically presented in the context of probability assessments. And risk preferences are typically evaluated in the context of the expected utility model (e.g., Arrow 1965; Pratt 1964). However, psychologists have documented the presence of systematic bias in probability assessment (e.g., Camerer 1995). And there is evidence that the expected utility model fails to provide an accurate representation of individual risk preferences (e.g., Machina 1987). This raises two questions. First, is a probability assessment always required? Second, are there situations where analyzing firm behavior does not require knowing the decision maker's risk preferences? The objective of this article is to explore these issues by analyzing input decisions under production uncertainty.

Cost minimizing input choices under production uncertainty have been analyzed by Pope and Chavas (1994), Pope and Just (1996, 1998), Chambers and Quiggin (1998, 2000), Moschini (2001), and others. Pope and Chavas (1994) have argued that, under risk aversion, expected output alone does not provide an appropriate characterization of cost minimization. Chambers and Quiggin (1998, 2000) have shown that standard cost minimization still applies under a state-contingent approach, irrespective of risk preferences. The state-contingent approach considers that outputs are conditional on the states of nature, each state representing a particular uncertain event (Debreu 1959). While being very general, (1) the state-contingent approach has seen few empirical applications. (2) This is due in large part to the fact that outputs are typically observed only under one of the many possible states of nature. As discussed by O'Donnell and Griffith (2006), this creates empirical difficulties (including identification problems) in estimating the production technology. O'Donnell and Griffith (2006) rely on Bayesian estimation to address the identification problem. But is there an alternative approach that would make the state-contingent analysis of production uncertainty empirically tractable?

This article proposes a methodology to specify and estimate standard cost-minimizing input choices under production uncertainty and a state-contingent technology. The approach has several attractive characteristics. First, under a state-contingent approach, it does not require a priori risk assessments. This can be seen as an advantage when probability assessments are problematic and impede empirical economic analysis. Second, as argued by Chambers and Quiggin (1998, 2000), the analysis applies irrespective of risk preferences. To the extent that assessing risk preferences is often difficult, this broadens the scope of applications of the methodology. Third, the approach provides a basis for investigating the nature of the state-contingent technology. In particular, it allows the empirical analysis of substitution possibilities across states of nature. As noted by Chambers and Quiggin (2000), previous research has commonly assumed an "output-cubical technology," where there is no possibility of substitution among state-contingent outputs. Our approach allows the testing of this hypothesis. Finally, estimating a state-contingent cost function allows the analysis of the marginal cost of state-contingent outputs. In situations of rapid technological change (e.g., the case of U.S. agriculture; see Ball et al. (1997)), this can be used to shed new light on the effects of technological progress on the cost of production risk.

The usefulness of the proposed methodology is illustrated in an econometric application to U.S. agriculture. We find strong evidence that, in the analysis of input choices, expected output alone does not provide an appropriate representation of production uncertainty. The results indicate empirical support for an output-cubical technology. This indicates that an ex post analysis of stochastic technology (as commonly found in previous research) appears appropriate. The analysis also provides evidence that the cost of facing production risk has declined in U.S. agriculture over the last few decades. This indicates that technological progress in agriculture has reduced the cost of production risk. It suggests that improved genetics (e.g., through the breeding of drought-resistance varieties) and better management have contributed to lowering the cost of production risk in U.S. agriculture.

The Model

Consider a firm making decisions under production uncertainty. The uncertainty is represented by S mutually exclusive states of nature. The firm chooses n inputs x = ([x.sub.1], ..., [x.sub.n]) [member of] [R.sup.n.sub.+] to produce state-contingent outputs y = ([y.sub.11], ..., [y.sub.m1]; ..., [y.sub.1s], ..., [y.sub.ms]) [member of] [R.sup.mS.sub.+], where [y.sub.is] is the quantity of the ith output produced under the sth state of nature, i = 1, ..., m, s = 1, ..., S. Under technology t, the stochastic production technology is represented by the possibility set F(t) [subset] [R.sup.n.sub.+] x [R.sup.ms.sub.+], where (x,y) [member of] F(t) means that outputs y can be produced using inputs x. The set F(t) provides a general ex ante representation of the production technology under production uncertainty. Throughout, we assume that, for each y, the input requirement set G(y, t) = {x: (x, y) [member of] F(t)} [subset] [R.sup.n.sub.+] is closed and convex.

In general, production decisions under risk depend on the nature of risk preferences of the decision maker. Production uncertainty is typically associated with lags in the production process. In this context, input decisions are made before the state of nature and the possible output realizations become known. Denote by w = [w.sub.1, ..., [w.sub.n]) [member of] [R.sup.n.sub.++] the vector of prices for x. Assume that input prices w are known at the time when inputs x are chosen. Also assume that the decision maker exhibits preferences that are nonsatiated in income. Then, conditional on the state-contingent outputs y, inputs x are chosen in a way consistent with the cost minimization problem:

(1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Indeed, if input choices do not minimize cost, then under income nonsatiation, choosing x according to (1) would improve the welfare of the decision maker. Thus, conditional on the state-contingent outputs y, cost-minimizing behavior (as given in (1)) represents economic rationality for the firm irrespective of the nature of risk preferences of the decision maker (Chambers and Quiggin 1998, 2000). Below, we will use expression (1) as a general representation of input choice under state-contingent production uncertainty.

Let [x.sup.c](w, y, t) [member of] [argmin.sub.x] {w. x: x [member of] G(y, t)}. In general, the cost function C(w, y, t) = w. [x.sup.c](w, y, t) is positively linearly homogeneous and concave in w. And in the case where C(w, y, t) is differentiable in w, it satisfies Shephard's lemma:

(2) [x.sup.c](w, y, t) = [partial derivative]C(w, y, t)/[partial derivative]w.

Equation (2) provides a convenient framework to investigate economic behavior under uncertainty. Throughout the article, we will rely on (2) as a representation of economic rationality for input decisions under state-contingent production uncertainty. Also, we will use (2) as a means of obtaining information about the nature of the underlying production technology. From duality, it is well known that the cost function C(w, y, t) in (1) provides a convenient framework to investigate the nature of substitution among inputs. In particular, the Allen elasticity of substitution between inputs i and j is given by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], or using Shephard's lemma (2), [[sigma].sub.ij] = [partial derivative][x.sup.c.sub.i]/[partial derivative][w.sub.j] C/[x.sup.c.sub.i][x.sup.c.sub.j] (see Chambers 1988).

Below, we will be particularly interested in exploring the nature of substitution across states of nature. This is at the heart of a debate about whether a stochastic production function provides an appropriate representation of technology. While a stochastic production function has been commonly used in the analysis of production uncertainty (e.g., Just and Pope 1978; Antle 1983), Chambers and Quiggin (2000) have raised questions about its validity. In a stochastic production function, the inputs-outputs relationships are analyzed conditional on the realized state of nature. When outputs vary across states, Chambers and Quiggin (2000) have shown that this precludes the possibility of output substitution across states (Chambers and Quiggin 2000, pp. 53-55). Chambers and Quiggin (2000) call the technology implied by zero substitution across states "output cubical." This means that the standard stochastic production function approach implicitly assumes no substitution across states. The validity of this assumption remains an empirical issue. Following Powell and Gruen (1968), the possibility of output substitution can be conveniently characterized by the Allen elasticity of transformation. In this context, the technology is "output cubical" if the Allen elasticity of transformation between any [y.sub.is] and [y.sub.is], is zero for all s [not equal to] s' and for all i = 1, ..., m. But how can we recover the Allen elasticities of transformation between state-contingent outputs from the cost function (1)? This question is addressed in the next section.

Elasticities of Transformation and Duality

Powell and Gruen (1968) define elasticities of transformation between outputs. Such elasticities provide useful information about the possibility of substitution among outputs. While Powell and Gruen (1968) present Allen elasticities of transformation using the production function, this section uses duality to explore how to obtain elasticities of transformation from the cost function C(w, y, t) in (1).

To develop the relevant duality results, let g [member of] [R.sup.n.sub.+] be some reference input bundle satisfying g [not equal to] 0. Given the input requirement set G(y, t), following Luenberger (1995) and Chambers, Chung, and Fare (1996), define the directional distance function

D(x, y, g, t) = [max.sub.[beta]],{[beta] :(x - [beta]g) [member of] G(y, t)} if there is a [beta] such that (x - [beta]g) [member of] G(y, t)

= -[infinity] otherwise.

The function D(x, y, g, t) measures the distance between point (x, y) and the boundary of the feasible set, expressed in units of the reference bundle g. Under free input disposability (where x [member of] G(y, t) implies that x' [member of] G(y, t) for all x' [greater than or equal to] x), x [member of] G(y, t) is equivalent to D(x, y, g) [greater than or equal to] 0. In this case, the directional distance function D(x, y, g, t) provides a complete representation of the technology, where D(x, y, g) = 0 is an implicit multioutput production function representing the boundary of the feasible region. Below, we will assume that D(x, y, g, t) is twice continuously differentiable in (x, y). Also, we will make use of the "normalized" distance function [D.sup.*](x, y, w, t) [equivalent to] (w x g) D(x, y, g, t).

Using [D.sup.*](x, y, w, t) = 0 as a multioutput production function and following Allen (1938) and Powell and Gruen (1968), the elasticity of transformation between any two outputs Yi and [y.sub.j] is defined as

[[tau].sub.ij] = [[summation].sup.m.sub.k=1]([partial derivative][D.sup.*/[partial derivative][y.sub.k])[y.sub.k]/[y.sub.i][y.sub.j] [K.sup.c.sub.ij]/det(K)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is the bordered Hessian of [D.sup.*](x, y, g, t) with respect to y, and [K.sup.c.sub.ij] is the (i, j)th cofactor of K. Outputs i and j are said to be substitutes (complements) if [[tau].sub.ij] < 0 (>0). (3) And in the two output case (m = 2), [[tau].sub.12] [right arrow] 0 corresponds to fixed output-proportions (Powell and Gruen 1968). In the general case, [[tau].sub.ij] measures the responsiveness of the output-mix ratio to changes in the corresponding marginal rate of substitution.

The following result will prove useful in our analysis (see the proof in the Appendix).

PROPOSITION 1. Assume that G(y, t) is a convex set and that free input disposability holds. Then, the Allen elasticity of transformation between outputs i and j is given by

(3) [[tau].sub.ij] = [[summation of].sup.m.sub.k=1]([partial derivative]C/[partial derivative][y.sub.k])[y.sub.k]/[y.sub.i][y.sub.j] [H.sup.c.sub.ij]/det(H)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[H.sup.c.sub.ij] is the (i, j)th cofactor of H, and ([partial derivative].sup.2]C/[partial derivative][w.sup.2]) + denotes the generalized reverse of ([[partial derivative].sup.2]C/[[partial derivative][w.sup.2]).

Equation (3) gives an evaluation of the Allen elasticity of transformation among outputs from the cost function. In the presence of state-contingent outputs, this provides a basis for investigating the possibility of substitution across states (e.g., whether or not the state-contingent technology is "cubical"). See below.

Measuring Stochastic Outputs

Consider a situation involving T observations on the firm. It will be convenient to think that different observations correspond to different time periods. In this context, we assume that each observation on the firm can be associated with a different technology, where "t" represents both time and a "technology index", t = 1, .... , T. This is consistent with a situation of technological progress, where technology improves over time. It follows that the input requirement set G(y, t) allows for possible technological change across observations. The tth observation consists in observing inputs [x.sub.t] = ([x.sub.1t], ..., [x.sub.nt]), input prices [w.sub.t] = ([w.sub.1t], ..., [w.sub.nt]), and outputs ([y.sub.1t], ..., [Y.sub.mt]). Under production uncertainty, for each t, the ex post outputs realization ([y.sub.1t], ..., [y.sub.mt]) is only one of the many possible realizations of outputs. The output realizations that are possible ex ante are [y.sub.t] = ([y.sub.11t], ..., [y.sub.m1t], ..., [y.sub.1St], ..., [y.sub.mSt]), where [y.sub.ist] is the quantity of the ith output produced at time t under the sth state of nature. The problem is that, for each t, only one of the S possible output realizations is typically observed. With ex ante outputs being incompletely observed, this means that neither the cost function C([w.sub.t], [y.sub.t], t) nor the input demand functions [x.sup.c]([w.sub.t], [y.sub.t], t) are empirically tractable. In order to make C([w.sub.t], [y.sub.t], t) and [x.sub.c]([w.sub.t], [y.sub.t], t) empirically tractable, it is necessary to impose some structure on the problem. Here, we propose a method to generate all possible outputs y based on the T observations of the firm.

First, we know that the ex post outputs realization (y.sub.1t], ..., [y.sub.mt]) is one of the possible ex ante realizations [y.sub.t] = ([y.sub.11t], ..., [y.sub.m1t], ..., [y.sub.1st], ...,) at time t. In this context, one option is to estimate the ex post technology relating realized outputs ([y.sub.1t], ..., [y.sub.mt]) to input use, conditional on the particular state of nature obtained under the tth observation, t = 1, ..., T. To make this approach empirically tractable, stationarity assumptions are needed to establish how the states of nature affect outputs across observations. This is typically done by treating the states as random variables, and making stationarity assumptions on the probability distribution generating these random variables. For example, in the single output case (m = 1), assuming that the states are independently distributed across observations, the regression of output on input use provides a framework to estimate a stochastic production function, where the presence of heteroscedasticity can reflect the effects of input use on the variability of output (e.g., Antle 1983; Just and Pope 1978).

This approach is convenient and has been commonly used in the analysis of stochastic technology. Its main limitations are three. First, by embedding the factors determining the state of nature into a single scalar-valued random variable and then embedding this variable in a technology, it imposes separability of the stochastic factors determining the state of nature (in an agricultural example, these would typically be viewed as random inputs such as weather and pest infestations) on the underlying technology. Second, while it works well in a single output case, it can only be applied in a multioutput setting under the restrictive assumption on the technology of input nonjointness. Third, and perhaps most importantly, it focuses exclusively on the observed outputs. As such, the approach neglects the potential outputs that could have been obtained had nature selected different states. Is this neglect important for economic analysis? As discussed above, Chambers and Quiggin (2000) showed that this neglect is acceptable under an "output-cubical technology" exhibiting no possibility of output substitution across states. In this case, the ex ante technology can be expressed entirely in terms of the ex post technologies across states (see Chambers and Quiggin 2000, pp. 53-55). This suggests that, in the absence of output substitution across states, an ex post analysis of stochastic technology is appropriate. However, one should keep in mind that this does not imply ex post cost minimization. Indeed, since inputs are chosen before the state of nature is known, their choice must be feasible ex ante, i.e., for all possible states of nature (and not just the particular state of nature that was observed). This means that, under an output cubical technology, expost cost functions are a lower bound on the ex ante cost function C([w.sub.t], [y.sub.t], t) (Chambers and Quiggin 2000, pp. 134-35).

But what if the stochastic technology is not "output-cubical"? Then there are possibilities of output substitution across states. In this case, as argued by Chambers and Quiggin (2000), an ex post analysis of stochastic technology is inappropriate. It would neglect the effects of input choices on the distribution of outputs across states. For example, labor use can contribute to conserving water and affect the drought-resistance of a crop. Important output trade-offs may exist across states of nature. Capturing these trade-offs require an ex ante representation of the technology. This raises the important question: how to evaluate this empirically?

A natural place to start is to explore whether the output observations ([y.sub.1t], ..., [y.sub.mt]), t = 1, ..., T can be used to recover the ex ante technology. This is a difficult problem. The reason is that outputs depend on inputs, on the state of nature, as well as on the underlying technology. We have an identification problem. Under production uncertainty, we cannot estimate the ex ante technology without observing all possible outputs (meaning outputs under all possible states, and not just for the realized state). And without knowing the underlying technology, we do not know what outputs could have been under different states of nature (at least when the technology is not output-cubical). Thus, under general production uncertainty, knowing the actual outputs ([y.sub.1t], ..., [y.sub.mt]) does not provide enough information to know the distribution of all possible outputs and the underlying ex ante technology. In an attempt to resolve this issue, we need to impose some a priori structure on the process generating the states of nature. Below, we propose a general methodology to recover possible ex ante outputs using actual outputs ([y.sub.1t], ..., [y.sub.mt])

We know that ([y.sub.1t], ..., [y.sub.mt]) is one of the possible outputs for the tth observation. Recall that [y.sub.ist] denotes the quantity of the ith output produced under the sth state of nature at time t. For the ith output, assume the existence of positive numbers [[micro].sub.is] and [[sigma].sub.is], i = 1, ..., m, s = 1, ..., S. For each i, define a random variable [e.sub.i] for which the sth realization is given by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. It follows that the ex ante outputs can be written as

(4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Equation (4) defines the variable [e.sub.is] [equivalent to] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] as measuring the relative changes in the ith output across states of nature. Thinking of ([Y.sub.ilt], ..., [y.sub.iSt] ) as a random variable that can take different values across states, this simply defines [e.sub.i] as a new random variable obtained from a deterministic transformation of the original one. This imposes no a priori restriction on the nature of production uncertainty. Indeed, for each t, it allows for an arbitrary distribution of the effects of production uncertainty on outputs. In addition, note that the term [[sigma].sub.it] can be interpreted as a "spread parameter," allowing the spread of the distribution of the ith output across states to vary across observations t. However, equation (4) does impose a stationarity restriction. It assumes that, except for the spread effects captured by [[sigma].sub.it] the relative effects of production uncertainty on each output are similar across observations t.

Next, assume that there exist auxiliary variables [z.sub.it] with the following property. When s is the state occurring under the tth observation, [z.sub.it] satisfies

(5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

i = 1, ..., m, and t = 1, ..., T. This establishes the variables z as proxy variables for the measurement of production uncertainty. Indeed, by definition of [z.sub.it], for the tth observation, the states of nature have the same relative effects on the ith output as they have on [z.sub.it]. Below, we will discuss which variables appear to be good candidates for z. Assume that [k.sub.it] and [[sigma].sub.it] can be consistently estimated. Assuming that all variables are positive, equation (5) can be written as ln([z.sub.it]) = ln([k.sub.it]) + [[sigma].sub.it] ln([e.sub.is]). This can be treated as a standard econometric model with ln([z.sub.it]) as the dependent variable, In([k.sub.it]) as the regression line, and [[sigma].sub.it] ln([e.sub.it]) as the error term, where flit captures possible heteroscedasticity. In the case where ln([e.sub.it]) has mean zero and variance one, then In([k.sub.it]) can be interpreted as the expected value of ln([z.sub.it]), and [[sigma].sub.it] as the standard deviation of the error term for the ith output and the tth observation. As shown by Antle (1983), after choosing a parametric specification for [k.sub.it] and [[sigma].sub.it], a moment-based approach can be used to obtain consistent estimates of the parameters. See below.

When s is the state occurring under the tth observation, it follows from equation (5) that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. This generates [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] as estimates of T realized values of the random variable [e.sub.i]. For the tth observation and from equation (4), this can be used to obtain the simulated state-contingent outputs at time t:

(6) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

t = 1, ..., T. Again, note that the term [[sigma].sub.it]/[[sigma].sub.ir] in (6) allows for the spread of the distribution of the ith output across states to vary across observations. We want to stress here that [[mu].sub.it] in (4) does not play any role in the evaluation of simulated outputs [y.sup.e.sub.t] in (6). To the extent that the [[mu].sub.it]'s are expected to reflect the underlying technology and the associated economic trade-offs, this means that our proposed scheme for evaluating ex ante outputs can be applied independently of the nature of the technology. Of course, the validity of the approach relies crucially on the validity of the stationarity assumption (4) and of equation (5).

Parametric Specification

In general, consistent estimates of [k.sub.it] and [[sigma].sub.it] can be used to generate simulated state-contingent outputs [y.sup.e.sub.t] from equation (6). In turn, these can be used to obtain consistent estimates of the cost function C(w, y, t) and of cost minimizing behavior [x.sup.c](w, y, t). This section discusses specification issues raised in this approach.

When using the state-contingent outputs [y.sup.e.sub.t], the problem becomes one of specifying and estimating C([w.sub.t], [y.sup.e.sub.t], t) and of cost minimizing behavior [x.sup.c]([w.sub.t], [y.sup.e.sub.t] t) based on a sample of T observations. In this context, the state-contingent outputs [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] include mT variables at each time period t. Even when m = 1, including such a large number of explanatory variables is problematic. Typically, many of the elements of [y.sup.e.sub.t] will tend to be correlated in the sample, creating serious multicollinearity problems. This makes it difficult to estimate C([w.sub.t], [y.sup.e.sub.t], t) and [x.sup.c]([w.sub.t], [y.sup.e.sub.t], t) directly. And the collinearity problems become even more severe when m > 1. This suggests a need to develop an econometric approach that can avoid such problems. The solution is a "parsimonious" parametric specification of C([w.sub.t], [y.sup.e.sub.t], t) and [x.sup.c]([w.sub.t]. [y.sup.esub.t], t) that does not involve "too many" parameters while still allowing the estimation of substitution possibilities across states.

This can be done in two ways. A first approach to a parsimonious parametric specification can be obtained by representing the distribution of the e's by a few parameters. If the distribution has a specific parametric form, then the associated parameters are sufficient statistics and provide all the relevant information. Alternatively, the first few central moments of the distribution can be used (assuming that they exist). In this context, one issue is: how many moments are needed to represent the distribution? If the decision maker is risk neutral, we know that only the first moment is relevant in the decision-making process. This is the assumption made by Pope and Just (1996) and Moschini (2001) in their analysis of production uncertainty. However, if the decision maker is not risk neutral (e.g., under risk aversion), then the first moment is not sufficient to characterize production decisions under risk (Pope and Chavas 1994). Then, at minimum, the first two moments (and possibly higher moments) are needed. This issue will be investigated empirically below.

A second approach to a parsimonious parametric specification involves working with a coarsened partition of the state space. For the ith output and the tth observation, define ([K.sub.i] - 1) values [b.sub.ikt] satisfying [b.sub.i1t] < [b.sub.i2t] < ... < [b.sub.i], [k.sub.i] - 1,t. For each i and t, this establishes [K.sub.i] intervals, [V.sub.i1t] = [[-[infinity], [b.sub.i1t]], [V.sub.ikt] = ([b.sub.i, k-1,t], [b.sub.ikt]], k = 2, ... , [K.sub.i]-1, and[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] t = 1, ... , T. Assume that the partitions are chosen such that there is at least one observation [y.sub.irt] satisfying [y.sub.irt] [member of] [V.sub.ikt] for each i, k, and t. Define the indicator variables

[I.sub.ikrt] = 1 if [y.sub.irt] [member of] [V.sub.ikt], = 0 otherwise

Let [y.sub.ikt] ([[summation].sup.T.sub.r=1] [I.sub.ikrt][y.sub.irt]) / [[summation of].sup.T.sub.r=1][I.sub.ikrt]) denote the conditional mean of [y.sub.irt] in the kth partition related to the ith output at time t. Define [y.sup.K.sub.T] = {[y.sub.ikt] : k = 1, ... [K.sub.i] = 1, ... , m}. Next, consider specifying the cost function as C([w.sub.t], [y.sup.K.sub.T], t) and the input demand functions as [x.sup.c]([w.sub.t], [y.sup.K.sub.T], t). The choice of the state partition provides some flexibility for capturing the economic trade-offs between outputs across states. At one extreme, for each i and k, the finest partition would be obtained if [K.sub.i] = T, generating a single observation at time t in each element of the partition. This would be very flexible. However, as noted above, under state-contingent production uncertainty, it is not practical as it involves too many parameters to estimate.

At the other extreme, the coarsest partition would be obtained if [K.sub.i] = 1 for each i. This would reduce greatly the number of parameters to estimate. However, this appears too restrictive for at least three reasons. First, it would amount to replacing the distribution of each [y.sub.irt] across states by its corresponding unconditional mean ([[summation].sup.T.sub.r=1] [y.sub.irt])/T. Since the mean is in general not a sufficient statistic for most distributions, this would likely involve a loss of information. Second, if the decision maker is risk neutral, then it could be argued that the mean is the only relevant variable that would influence the decision-making process (as assumed by Pope and Just (1996) and Moschini (2001)). However, this would not apply under risk aversion. To the extent that there is considerable evidence that most decision makers are risk averse, this would fail to capture the effects of risk and risk aversion on production behavior. Third, using unconditional means as representations of production uncertainty would make it impossible to estimate econometrically the elasticity of substitution across states. We are interested here in estimating such elasticities. This alone would rule out the use of unconditional means ([[summation].sup.T.sub.r=1] [y.sub.irt])/T in the representation of output uncertainty.

If either [K.sub.i] = 1 and [K.sub.i] = T appears undesirable, this suggests that a reasonable choice of partitions would satisfy 1 < [K.sub.i] < T. In general, this choice involves trade-offs between providing a flexible representation of the underlying technology (with flexibility improving as the [K.sub.i]'s increase) and parsimony and ease of estimation (where estimating the model becomes easier as the [K.sub.i]'s decrease). The approach is illustrated below in an empirical application.

An Econometric Application

Consider the case where the state space is partitioned to give [y.sup.K.sub.T] = {[y.sub.ikt] : [y.sub.ikt] = ([[summation].sup.T.sub.r=1] [I.sub.ikrt] x [y.sub.irt])/([[summation].sup.T.sub.r=1][I.sub.ikrt]); k = 1, ..., [K.sub.i]; i = 1, ..., m}.

We focus our discussion on the case of the generalized Leontief cost function (see Diewert 1971; Lopez 1980):

(7) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and h([y.sup.K.sub.T], t) and [g.sub.j]([y.sup.K.sub.T], t) take some parametric form (see below). Diewert (1971) has shown that this specification is flexible in the sense that it does not impose a priori restrictions on the possibilities of substitution among inputs. It includes as a special case a Leontief technology when [[alpha].sub.ij] = 0 for all i [not equal to] j, and a homothetic technology when [g.sub.j]([y.sup.K.sub.T], t) = 0, j = 1, ..., n (Shephard 1970). The possibilities of substitution among outputs are captured by the functions h([y.sup.K.sub.T], t) and [g.sub.j]([y.sup.K.sub.T], t). Under state-contingent production uncertainty, this involves possible substitution both among the m different outputs as well as across states of nature.

Then, we consider the following specification for h(*):

(8) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

subject to the normalization rule [[beta].sub.i1] = 1, with [[beta].sub.ii', kk'] = [[beta].sub.i'i, k'k] for all i, i', k, k'. Note that the parameters [[beta].sub.ii', kk'] in (8) capture the possibilities of substitution among different outputs (for i [not equal to] i') as well as different states (for k k'). We consider the following specification for [g.sub.j](*):

(9)[g.sub.i](*) = [[gamma].sub.0i] + [[gamma].sub.ti]t, i = 1, ..., n.

Finally, using Shephard's lemma (2), under the specifications (7), (8), and (9), the cost minimizing input demand functions under production uncertainty take the form:

(10) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Econometric Results

In this section, the above model is applied to U.S. agriculture. This raises the question: can our proposed approach (developed in the context of firm behavior) be applied at the industry level? Note that the estimation of a cost function using aggregate industry data is not new. For example, Lopez used such an approach to investigate the derived demand for inputs in Canadian agriculture. The estimation of cost-minimizing behavior from aggregate data can be justified in several ways. One could simply assume that the industry behaves "as if" it were a single firm. Such a "representative firm" argument has some intuitive appeal. Although it may be subject to some aggregation bias, it has been used extensively in empirical analyses. However, a more refined justification can be presented as follows. Consider the case of an industry constituted of N firms, each firm possibly facing a different technology at a given time (e.g., due to different agro-climatic conditions). Using equation (1), denote the cost function of the kth firm producing outputs [y.sub.k] [member of] [R.sup.mS] by [C.sub.k](w, [y.sub.k], t) [equivalent to] [Min.sub.xk] {w x [x.sub.k] : [x.sub.k] [member of] [G.sub.k] ([y.sub.k], t)}, k = 1, ... , N. The associated aggregate cost function for the industry is C(w,Y,t)= [[summation].sup.N.sub.k=1] [C.sub.k](w, [y.sub.k], t) = [Min.sub.x] {w x ([[summation].sup.N.sub.k=1] [x.sub.k]) : [x.sub.k] [member of] [G.sub.k] ([y.sub.k], t), k = 1, ..., N}, where Y [equivalent to] (y.sub.1], ..., [y.sub.N])[member of] [R.sup.mSN]. This shows that cost minimization applies at the industry level. Next, assume the existence of aggregate state-contingent output indices y(Y) [member of] [R.sup.mS] satisfying C(w, Y, t) = C(w, y(Y), t). Then, C(w, y(Y), t) provides a valid aggregate cost function, conditional on the aggregate output indices y(Y). Under such a scenario, our proposed approach also applies at the industry level. The empirical analysis presented below is presented in this context.

Annual data on U.S. agriculture were obtained from the U.S. Department of Agriculture. They include price and quantity data for four inputs (labor, capital, material, and land) and one aggregate output for the period 1949 to 1999 (see Ball et al. 1997). Thus, by working with an aggregate output, we focus our attention on the single output case, with m = 1.

The evaluation of production uncertainty requires an empirical basis to estimate equation (5). We use a crop yield index as the auxiliary variable z capturing production uncertainty. This seems reasonable: once acreage decisions are made, production uncertainty manifests itself entirely through yield effects. Indeed, yield fluctuations are due in large part to unpredictable weather effects and pest damages. First, we measured z as a yield index, calculated from annual data on "yield per acre planted" for the major U.S. crops (corn, wheat, and soybeans). Second, we estimated the regression ln([z.sub.t]) = ln([k.sub.t]) + [[sigma].sub.t] ln([e.sub.t]), with the regression line ln([k.sub.t]) including selected explanatory variables. The explanatory variables were a time trend (to capture technological progress over time) and relative prices for inputs and outputs (to capture the effects of changing market conditions). These variables control for technological change and price effects on yield. It follows that the term [[[sigma].sub.t]([e.sub.t])] can be interpreted as reflecting production uncertainty. We investigated whether the variance of the error term changed over time. Using a Lagrange multiplier test (based on squared residuals regressed on squared fitted values), we failed to find statistical evidence of heteroscedasticity (the p-value for the test was 0.67). As a result, we proceeded with assuming that the variance [[sigma].sup.2.sub.t] was constant over time. With a constant variance [[sigma].sup.2], we obtained consistent estimates of [e.sub.t] = exp[(ln([z.sub.t]) ln( [k.sub.t]))/[sigma] ], t = 1, ..., T. Under a stationarity assumption (as discussed above), we used these estimates to generate the state-contingent outputs in equation (6).

Next, we used the specification of costminimizing input demands given in (10). The specification was estimated for K = 2. This can be interpreted as considering two states of nature, e.g., "bad weather" (k = 1) and "good weather" (k = 2). While this is a rather coarse representation of the state space, it will be convenient for the investigation reported in this paper. (4)

After adding an error term for each input, equation (10) yields a system of seemingly unrelated regressions. We first estimated equations (10) with four inputs: labor, capital, material, and land. However, the estimates showed that the cost function was not concave in capital price (i.e., the demand for capital was found to be upward sloping, which is inconsistent with cost minimization). We interpreted this as indirect evidence that the demand for capital may involve significant dynamics that are not captured in (10). This suggested the need either to address dynamics explicitly, or alternatively to conduct the analysis conditional on capital. To the extent that the dynamics of capital can be complex, we opted for the second option. As a result, the empirical analysis presented below focuses on the demand for three inputs, labor, material, and land, taking capital as given. In this specification, we introduced the effects of capital on the demand for other inputs by letting [[gamma].sub.0i] in (9) vary with capital, with [[gamma].sub.0i] = [[gamma].sub.ai] + [[gamma].sub.bi] Capital. = 1 with labor, i = 2 with material, and i = 3 with land, equation (10) was estimated by maximum likelihood. The resulting parameter estimates are presented in table 1. To take into consideration possible heteroscedasticity, the standard errors in table 1 are White-corrected robust standard errors.

Table 1 shows that the model provides a good fit to the data. The R-square varies between 0.934 for material to 0.991 for labor. Most parameters are statistically different from zero at the 5% significance level. With [[beta].sub.1] = 1 by normalization, note that the estimate of [[beta].sub.2] (0.9855) is not statistically different from 1. Also, the coefficient [[beta].sub.12] is found to be negative and statistically significant. The null hypothesis that [[beta].sub.12] = 0 is strongly rejected at the 1% significance level. This indicates the presence of significant interactions across states of nature. Note that a cost specification that would depend only on expected output would be obtained as a special case with [[beta].sub.2] = 1 and [[beta].sub.12] = 0. Using a Wald test, this hypothesis is strongly rejected at the 1% level. This indicates that, under uncertainty, a cost specification that would depend only on expected output is inappropriate. (5) As discussed above, this has at least two implications. First, if decision makers are risk averse under incomplete markets, then focusing on expected output alone fails to capture the role of risk management in input choice. Second, even if firm managers are risk neutral, our result indicates that focusing narrowly on expected output is not enough to provide a complete characterization of the stochastic technology and its implications for production behavior. On the one hand, it should not be a surprise to find out that the mean of a distribution is in general not a sufficient statistics for representing the whole distribution. On the other hand, our empirical findings show that this lack of sufficiency is empirically relevant when characterizing cost minimizing behavior.

The estimates for the [alpha] parameters reported in Table 1 indicate that price effects are statistically significant. These price effects are found to be consistent with production theory: the cost function is concave in input prices. Evaluated at sample means, the price elasticities of input demands are reported in table 2. As expected, input demands are downward sloping.

[FIGURE 1 OMITTED]

The price elasticities of land are found to be small. This is consistent with land being close to a fixed factor in agriculture. The price elasticities of labor and material are larger but remain inelastic, with an own-price elasticity of -0.387 for labor and -0.299 for material. The cross-price elasticity between labor and material is positive, indicating that they are substitute inputs. The parameter estimates for the [gamma]'s indicate that technological progress has been biased against labor (with [[gamma].sub.t1] < 0 corresponding to labor-saving technical change) and in favor of material (with [[gamma].sub.t3] > 0 identifying material-using technical change).

Using equation (3) and the parameter estimates reported in table 1, the elasticity of transformation between states of nature was estimated. Evaluated at sample means, the elasticity of transformation was calculated to be [[tau].sub.12] = -0.001, very close to zero. Recall that [[tau].sub.12]= 0 corresponds to an output-cubical technology with zero possibility for substitution between states. This indicates that the possibility of output substitution between states is very limited. We interpret this as empirical evidence in favor of an output-cubical technology. In other words, our analysis supports the validity of the stochastic production function commonly found in previous research (e.g., Just and Pope 1978; Antle 1983).

Finally, the parameter estimates were used to evaluate the marginal cost of outputs [MC.sub.kt] = [partial derivative] C/[[partial derivative][y.sub.kt] for state k at time tq Recall that k = I corresponds to "bad weather" while k = 2 corresponds to "good weather." Figure 1 reports the evolution of the relative marginal cost [MC.sub.1t]/[MC.sub.2t] over the period 1970-99. It shows that the marginal cost of production tends to be higher under "bad weather" (compared to "good weather"). It also shows two interesting characteristics. First, the variability in the relative marginal cost [MC.sub.1t]/[MC.sub.2t] has declined over time. In particular, the relative marginal cost is much more stable in the 1990s than it was in the 1970s. Second, the relative marginal cost [MC.sub.1t]/[MC.sub.2t] has been declining over the last few decades. It means that the marginal cost of production under adverse weather conditions is not as high as it used to be. This indicates that technological progress in U.S. agriculture has reduced the cost of production risk. This result appears to be new. It suggests that improved genetics (e.g., through the breeding of drought-resistance varieties) and better management have contributed to lowering the cost of production risk in U.S. agriculture.

Concluding Remarks

This article investigates production uncertainty when input decisions are made before uncertain outputs are known. Using a state-contingent approach, we develop a methodology to specify and estimate cost-minimizing input choices. The proposed approach exhibits at least two attractive characteristics. First, it does not require a probability assessment of the unknown outputs. This can be useful when such probability assessments are difficult to make. Second, it does not depend on the risk preferences of the decision maker. Given the current controversies about the validity of the expected utility model, this provides a framework to conduct economic analysis while avoiding such controversies. In addition, this appears useful when one realizes that risk preferences can be difficult to assess empirically and that they vary across individuals.

In this context, the challenge was to develop a methodology that is empirically tractable. The main issue arises from the fact that, at each time period, only one of the many possible states is typically observable. Our methodology proposes to measure all possible states, relying on auxiliary variables that can be used to simulate these states under stationarity conditions. This provides a framework to conduct econometric analysis of cost-minimizing behavior under a general state-contingent technology. The empirical tractability of the approach is illustrated in an empirical application to U.S. agriculture.

The application demonstrates that an econometric analysis of state-contingent technology is possible and useful. Three important results were obtained. First, we found strong evidence that restricting the analysis of input choice to include only expected output is not appropriate. This reflects the fact that, under risk aversion, the role of risk management in input choice can be important. More generally, this stresses the point that, for a general stochastic technology, mean output is not a sufficient statistic for the distribution of outputs. Second, we found econometric evidence that the possibility of substitution between states is very limited. We interpret this as evidence in favor of an "output-cubical" technology. This indicates that an ex post analysis of stochastic technology (as commonly found in previous research) appears appropriate. Finally, our analysis provides evidence that technological progress (e.g., the breeding of drought-resistance varieties, improved management) has reduced the cost of production risk in U.S. agriculture over the last few decades. This result is both new and important.

Although our proposed approach is empirically tractable, it is also subject to limitations. First, our measurement of state-contingent outputs requires stationarity assumptions. It would be useful for future research to explore whether our stationarity assumption could be relaxed. Second, our empirical analysis has neglected econometric issues related to simultaneity bias and measurement errors. Further research on these issues is needed. Finally, our econometric estimation was limited to two states of nature. This clearly is restrictive. In principle, our methodology can handle any number of states. However, collinearity problems are likely to arise when the number of states is large (due to the associated increase in the number of parameters to estimate). By reducing the econometrician's ability to obtain reliable parameter estimates, collinearity problems remain a challenge for future econometric use of the state-contingent approach.

Appendix

Proof of Proposition 1

Under free input disposability and the convexity of the set G(y, t), the cost function C(w, y, t) in 1 and the distance functions D(x, y, g, t) satisfy the following duality relationships (see Luenberger (1995) and Chambers, Chung, and Fare (1996))

(A.1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

(A.2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which has [??](x, y, g, t) for solution. Given [D.sup.*](x, y, w, t) - (w. g) D(x, y, g, t) and under differentiability, the envelope theorem applied to (A.1) and (A.2) yields [partial derivative]C/[partial derivative][??] = [x.sup.c] (Shephard's lemma) and [partial derivative][D.sup.*]/[[partial derivative]x= [??]. It follows that ([partial derivative]C/[partial derivative], y, w, t), y, t]. Differentiating with respect to w yields

(A.3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Similarly differentiating [partial derivative][D.sup.*]/[partial derivative] (x, y, w, t) = [partial derivative][D.sup.*]/[partial derivative]x

(A.4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Equations (A.3) and (A.4) establish that [[partial derivative].sup.2]C/[partial derivative][w.sup.2] and [[partial derivative].sup.2[D.sup.*] are generalized inverses of each other, with ([[partial derivative].sup.2][D.sup.*]/[partial derivative][x.sup.2] are generalized inverses of each other with ([[partial derivative].sup.2]C/[partial derivative][w.sup.2] + = [[partial derivative].sup.2]/[D.sup.*]/[partial derivative][x.sup.2] (where the superscript "+' denotes the generalized inverse). In addition, applying the envelope theorem to (A.1) gives

(A.5) [partial derivative]C/[partial derivative]y (w, y, t) = - [partial derivative][D.sup.*]/[partial derivative]y ([x.sup.c], y, w, t).

Differentiating [partial derivative][D.sup.*]/[partial derivative]x(x, y, w, t)= [partial derivative]C/[partial derivative]y[([partial derivative][D.sup.*]/[partial derivative]x, x, y, w, t), y, t] with respect to x and y gives

(A.6) [[partial derivative].sup.2][D.sup.*]/[partial derivative]y[partial derivative]x = - [[partial derivative].sup.2]C/[partial derivative]y[partial derivative]w [[partial derivative].sup.2[D.sup.*]/[partial derivative].sup.2]

(A.7) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

It follows that

(A.8) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Substituting (A.5) and (A.8) into the definition

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and [K.sup.c.sub.ij] is the (i, j)th cofactor of K yields the desired results.

[Received January 17, 2007; accepted September 17, 2007.]

References

Allen, R.G.D. 1938. Mathematical Analysis for Economists. London: Macmillan.

Antle, J. 1983. "Testing the stochastic Structure of Production: A Flexible Moment-Based Approach." Journal of Business and Economic Statistics 1:192-201.

Arrow, K. 1965. Aspects of the Theory of Risk Bearing. Helsinki: Yrjo Jahnsson Saatio.

Ball, V.E., J.C. Bureau, R. Nehring, and A. Somwaru. 1997. "Agricultural Productivity Revisited." American Journal of Agricultural Economics 79:1045-64.

Camerer, C. 1995. "Individual Decision Making." In J.H. Hagel and A.E. Roth, eds. The Handbook of Experimental Economics, pp. 587-703. Princeton, NJ: Princeton University Press.

Chambers, R.G. 1988. Applied Production Analysis: A Dual Approach. Cambridge: Cambridge University Press.

Chambers, R.G., Y. Chung., and R. Fare. 1996. "Benefit and Distance Functions." Journal of Economic Theory 70:407-19. Chambers, R.G., and J. Quiggin. 1998. "Cost Functions and Duality for Stochastic Technologies." American Journal of Agricultural Economics 80:288-95.

--. 2000. Uncertainty, Production, Choice, and Agency. Cambridge: Cambridge University Press.

Debreu, G. 1959. Theory of Value. Cowles Foundation Monograph 17, Yale University Press, New Haven, CT.

Diewert, W.E. 1971. "An Application of the Shephard Duality Theorem: A Generalized Leontief Production Function." Journal of Political Economy 79:481-507.

Just, R.E., and R D. Pope. 1978. "Stochastic Specification of Production Functions and Economic Implications." Journal of Econometrics 7:6786.

Lopez, R.E. 1980. "The Structure of Production and the Derived Demand for Inputs in Canadian Agriculture." American Journal of Agricultural Economics 62:38-45.

Luenberger, D.G. 1995. Microeconomic Theory. Boston: McGraw-Hill.

Machina, M.J. 1987. "Choice under Uncertainty: Problems Solved and Unsolved." Journal of Economic Perspectives 1:121-54.

Moschini, G. 2001. "Production Risk and the Estimation of Ex-ante Cost Functions." Journal of Econometrics 100:357-80.

O'Donnell, C.J., and W.E. Griffiths. 2006. "Estimating State-Contingent Production Frontiers." American Journal of Agricultural Economics 88:249-66.

Pope, R.D., and J.P. Chavas. 1994. "Cost Functions under Production Uncertainty." American Journal of Agricultural Economics 76:11427.

Pope, R.D., and R.E. Just. 1996. "Empirical Implementation of Ex-ante Cost Functions." Journal of Econometrics 72:231-49.

--. 1998. "Cost Function Estimation under Risk Aversion." American Journal of Agricultural Economics 80:296-302.

Powell, A.A., and F.H.G. Gruen. 1968. "The Constant Elasticity of Transformation Production Frontier and Linear Supply System." International Economic Review 9:315-28.

Pratt, J.W. 1964. "Risk Aversion in the Small and in the Large." Econometrica 32:122-36.

Sandmo, A. 1971. "On the Theory of the Competitive Firm under Price Uncertainty." American Economic Review 61:65-73.

Shephard, R. 1970. Theory of Cost and Production Functions. Princeton, NJ: Princeton University Press.

(1) For example, the state-contingent approach applies "free from probability concepts" (Debreu, p. 98).

(2) One of the few applications is by O'Donnell and Griffith, who propose a Bayesian approach to estimate the state-contingent production frontier.

(3) The Allen elasticity of transformation can also be defined from the revenue function R(p, x, t) = [py.sup.*](p, x, t) = [max.sub.y]{Py : (-x, y) [member of] F}, where p > 0 is the vector of output prices and [y.sup.*](p, x, t) are the revenue maximizing output supplies. Then, the Allen elasticity of transformation between outputs i and j is given by [[tau].sub.ij] = [[partial derivative].sup.2]R/[partial derivative][p.sub.i][partial derivative][p.sub.j] R/([partial derivative]R/[partial derivative][p.sub.i])([partial derivative]R/[partial derivative][p.sub.j]), or using the envelope theorem, [[tau].sub.ij] = [partial derivative][y.sup.*.sub.i]/[partial derivative][p.sub.j] R/[y.sup.*.sub.i][y.sup.*.sub.j].

(4) Some experimentation with finer representations of the state space indicated that collinearity problems can arise rather quickly. These problems should be kept in mind. As collinearity reduces our ability to obtain reliable parameter estimates, it places some limits on how many states can be realistically analyzed econometrically using a state-contingent approach. Note that O'Dormell and Griffiths faced similar problems in their econometric estimation (their reported results are for K = 3).

(5) We also investigated this same hypothesis using a moment-based approach, where the cost function C(.) was specified to depend on both the mean and the variance of output (the variance being evaluated using our state contingent approach). The null hypothesis that the variance effect was zero was also strongly rejected at the 1% significance level. Again, this provides evidence that expected output alone does not provide an appropriate representation of production uncertainty under cost minimizing behavior.

Jean-Paul Chavas is professor of agricultural and applied economics, University of Wisconsin. Madison. The author would like to thank two anonymous reviewers for useful comments on an earlier draft of the paper.

Table 1. Parameter Estimates

                              Standard
Parameter          Estimate    Error     p-Value

[[beta].sub.2]       0.9855     0.3824    0.013
[[beta].sub.12]     -0.0071     0.0012    0.000
[[alpha].sub.11]    -0.5172     0.1248    0.000
[[alpha].sub.22]    -0.5887     0.1173    0.000
[[alpha].sub.33]    -0.0172     0.0070    0.018
[[alpha].sub.12]     0.4689     0.1694    0.008
[[alpha].sub.13]    -0.0089     0.0114    0.441
[[alpha].sub.23]     0.0286     0.0211    0.183
[[gamma].sub.a1]   163.0669    21.3335    0.000
[[gamma].sub.a2]    68.0519     9.0431    0.000
[[gamma].sub.a3]    86.9531     1.1305    0.000
[[gamma].sub.t1]    -1.1908     0.2480    0.000
[[gamma].sub.t2]     0.2214     0.2288    0.338
[[gamma].sub.t3]    -0.4277     0.0186    0.000
[[gamma].sub.b1]    -1.3516     0.2354    0.000
[[gamma].sub.b2]     1.2207     0.4069    0.004
[[gamma].sub.b3]    -0.1078     0.0305    0.001

Log Likelihood = -349.8972.

Number of observations = 51.

R-square = 11.992 for labor. 0.934 for material,
and 0.985 for land.

Table 2. Price Elasticities

Price                  Price of   Price of   Price of
Elasticities            Labor     Material     Land

Quantity of labor       -0.387     0.392      -0.015
Quantity of material     0.260    -0.299       0.039
Quantity of land        -0.003     0.013      -0.010