LeChatelier Effects for the Competitive Firm under Price Uncertainty.

Arthur Snow [*]

The intuition that constrained choices are less elastic as well as suboptimal is confirmed by LeChatelier's principle in economic models of optimizing behavior for environments with no uncertainty. For a competitive entrepreneurial firm facing output price uncertainty, risk preferences interact with possibilities for substituting between capital and labor in production to determine the presence or absence of LeChatelier effects for labor demanded. LeChatelier's principle holds without qualification for output supplied in the neighborhood of any long-run equilibrium with respect to both monotone likelihood ratio improvements in the price distribution and increases in risk aversion. Global LeChatelier predictions, however, are unattainable.

1. Introduction

LeChatelier effects confirm the intuition that constrained decisions are not only suboptimal but also restrained in the sense of being less elastic. Samuelson (1947) introduced this idea into economics and showed that it blends neatly with economic models of decision making under certainty, being particularly well suited to the competitive firm. As emphasized by Silberberg (1971), linearity of the objective function combines with convexity in the constraint, imposed by the production technology, to yield envelope results at the first order, and LeChatelier effects at the second, as powerful consequences of the maximization hypothesis.

The envelope results of primary interest in producer theory, Shephard's Lemma for cost functions and Hotelling's Lemma for profit functions, are equality relations reflecting the first-order stationarity of an optimum. LeChatelier effects, in contrast, are inequality relations reflecting the restrained nature of short-run responses compared with long-run, less constrained responses. Thus, the quantities of labor demanded and output supplied are more elastic with respect to wage rate and output price changes in the long run than in the short run.

These LeChatelier predictions pertain to a firm that has fully adjusted to an exogenous set of factor and output prices and thus apply only in the neighborhood of a long-run equilibrium configuration for the firm. However, they apply equally well whether capital and labor are substitute or complementary factors of production. Milgrom and Roberts (1996) have established a global LeChatelier principle under the assumption that capital and labor are always substitutes, or always complements, in production, a stipulation that is necessarily true for the infinitesimal changes contemplated in the local LeChatelier predictions.

When the firm makes decisions while facing uncertainty about output price, risk preferences influence behavior if the uncertainty cannot be fully diversified through futures contracts or the sale of ownership shares, as in the case of closely held family businesses, such as some of those engaged in farming, fishing, and independent retailing. [1] As shown in the seminal papers by Sandmo (1971) and Batra and Ullah (1974), for a risk-averse entrepreneurial firm, decreasing absolute risk aversion plays a critical role in establishing unambiguous comparative statics predictions for changes in uncertainty. Batra and Ullah (1974) find that responses to factor price changes are also influenced by substitutability between capital and labor in production.

In this paper, local LeChatelier effects are demonstrated for a competitive entrepreneurial firm's labor demanded and output supplied under output price uncertainty for own-price comparative statics effects and increases in risk aversion. The own-price LeChatelier prediction for output supplied is problematic under uncertainty with risk aversion, since the concept of a price rise must be extended to the case in which price is stochastic. Ormiston and Schlee (1993), reviewing previous literature, conclude that first-order stochastic dominance improvements are too general to yield intuitive predictions but show that the subfamily of monotone likelihood ratio improvements fits exactly with intuition. In particular, because the supply curve of a competitive firm is upward sloping under certainty, the firm's supply increases with monotone likelihood ratio improvements in the output price distribution under uncertainty, without regard to risk preferences.

In this paper, the LeChatelier prediction is established for changes in output supplied in response to monotone likelihood ratio improvements without imposing special restrictions on either technology or preferences for bearing risk. Thus, the parallel between behavioral responses to price rises under certainty and to monotone likelihood ratio improvements under uncertainty extends from first-order comparative statics to second-order LeChatelier effects. Similarly, the LeChatelier effect for output supplied applies without qualification to increases in risk aversion.

The enterpreneur's demand for labor, however, does not necessarily display LeChatelier's principle in response to increases in the wage rate. When changes in wealth influence the entrepreneur's willingness to bear risk, a rise in the wage rate induces wealth effects that tend to oppose the direct effect on labor demanded. At long-run equilibrium configurations, capital and labor tend to be substititutes in expected utility, since their marginal profits are subject to perfectly correlated risks. Thus, wealth effects influencing the demand for capital feed back on the demand for labor and, with decreasing absolute risk aversion, oppose the direct effect of an increase in the wage rate, creating the possibility that demand for labor is less elastic in the long run than in the short run. With increasing absolute risk aversion, wealth effects can lead to a rising demand curve for labor in the long run, even though the short-run demand curve is downward sloping. Thus, the LeChatelier prediction for labor demanded carries over to the environment with output price uncertainty only with constant absolute risk aversion, when wealth effects are absent.

In contrast with these local LeChatelier effects, global effects that hold away from longrun equilibria are not predicted for a competitive firm facing a stochastic output price. The uniform modularity shown to be crucial in the global context by Milgrom and Roberts (1996) is not exhibited by the expected utility criterion function. Thus, away from long-run equilibria, competitive entrepreneurial firms may not evince LeChatelier's principle.

Own-price LeChatelier effects for labor demanded and output supplied under price uncertainty are demonstrated in the following two sections. The absence of global LeChatelier predictions for environments with output price uncertainty is discussed in the subsequent section, and conclusions are presented in the final section.

2. LeChatelier Effects for Labor Demanded in Response to Changes in the Wage Rate

Under conditions of certainty or uncertainty, the competitive firm is concerned only with profit, given by the relation

z(K, L, w, r, p) = pF(K, L) - rK - wL, (1)

where K and L denote the use of capital and labor inputs to a technology represented by a strictly concave production function F, with corresponding factor prices r and w, and output price p. Beginning with the case of certainty as a benchmark, let [K.sup.0](w, r, p) and [L.sup.0](w, r, p) denote the firm's capital and labor demand functions implied by maximizing long-run profit, and let [L.sup.0](K, w, r, p) denote the short-run labor demand function obtained when capital's use is fixed. The identity

[L.sup.0](w, r, p) [equivalent to] [L.sup.0]([K.sup.0](w, r, p), w, r, p) (2)

implies the following two relations:

[delta][L.sup.0]/[delta]w [equivalent to] ([delta][L.sup.0]/[delta]w) + ([delta][L.sup.0]/[delta]K)([delta][K.sup.0]/[delta]w) (3)

[delta][L.sup.0]/[delta]r [equivalent to] ([delta][L.sup.0]/[delta]K)([delta][K.sup.0]/[delta]r [equivalent to] [delta][K.sup.0]/[delta]w, (4)

where the left-hand side of Identity 4 follows from the fact that [delta][L.sup.0]/[delta]r = 0 and the right-hand side of Identity 4 expresses the reciprocity condition implied by the envelope results for the long-run profit function. Substituting for [delta][K.sup.0]/[delta]w from Identity 4 into Identity 3 yields the LeChatelier prediction for labor demanded,

[delta][L.sup.0]/[delta]w = ([delta][L.sup.0]/[delta]w) + [([delta][L.sup.0]/[delta]K).sup.2]([delta][K.sup.0]/[delta]r) [less than] [delta][L.sup.0]/[delta]w [less than] 0, (5)

where the inequalities follow from the downward slope of factor demand curves in the long run and in the short run.

Thus, the own-price LeChatelier prediction for labor demanded, which holds in the neighborhood of a long-run equilibrium configuration for the firm, is a consequence solely of optimizing behavior and does not depend on the sign of the cross-quantity effect, [delta][L.sup.0]/[delta]K = -[F.sub.KL]/[F.sub.LL] (where subscripts indicate partial derivatives). Indeed, whether capital and labor are substitutes ([F.sub.KL] [less than] 0) or complements ([F.sub.KL] [greater than] 0), the LeChatelier prediction holds because feedback from the optimal adjustment of capital always reinforces the short-run change in labor demanded.

The same reasoning does not apply under price uncertainty when the firm is risk averse. Not only must risk preferences be taken into account, but the duality relations for the profit function, including the reciprocity conditions, are unavailable when a risk averter maximizes the expected utility of profit. Indeed, as noted by Batra and Ullah (1974), long-run factor demands need not be decreasing in own prices, leading to the possibility of short-run and long-run demands having different slopes, rendering the question of comparing magnitudes of elasticities meaningless.

To derive restrictions ensuring that LeChatelier effects carry over to environments with price uncertainty, let the firm's criterion be the expected utility function

V(K, L, w, r, [delta]) = [[[integral of].sup.p].sub.p] U(z(K, L, w, r, p)) dH(p; [delta]), (6)

where U is an increasing, strictly concave function, thus exhibiting risk aversion, and H is a cumulative distribution function for the random output price p, indexed by a parameter [delta], with a support contained in the interval [p, p]. The parameter [delta] is discussed in the next section. A long-run expected utility maximum is assumed to exist with positive profit for every realization of p.

The LeChatelier effect for output-conditional labor demand with respect to the wage rate, analogous to Equation 5, carries over to an environment with uncertainty, since the firm exhibits cost-minimizing behavior even when facing a stochastic output price. However, the output supply decision is influenced by preferences for risk bearing, as shown by Sandmo (1971), and as a result, the presence of a LeChatelier effect for unconditional labor demand depends on the firm's willingness to bear risk.

Consider first the comparative statics of an increase in the wage rate. In the short run under certainty, marginal cost rises when the wage rate increases, leading to a reduction in output and demand for labor, whereas in the long run, marginal cost rises (falls) and output falls (rises) if labor is normal (inferior), so that demand for labor always declines in the long run as the wage rate rises. These predictions remain valid when the price of output is uncertain if the firm is risk neutral, since then, only the expected price is relevant, as if there were no uncertainty. However, a risk-averse firm may exhibit an additional wealth effect arising from the decline in expected profit occasioned by an increase in the wage rate. For a firm with constant absolute risk aversion (CARA), there is no wealth effect, as the firm's willingness to bear risk is unaffected by changes in wealth, so there is only the price effect, which is always negative. In contrast, a firm characterized by decreasing (increasing) absolu te risk aversion (DARA [IARA]) is less (more) willing to bear risk as wealth declines, so that the wealth effect of an increase in the wage rate reinforces (opposes) the negative short-run price effect and reinforces the negative long-run price effect if labor is normal (inferior). [2]

The following Lemma summarizes these comparative statics predictions, using [L.sup.*](K, w, r, [delta]) to denote the short-run demand for labor when output price is stochastic and [L.sup.*](w, r, [delta]) to denote the corresponding long-run demand. [3]

LEMMA 1. Under price uncertainty, the short-run demand for labor decreases whenever the wage rate increases; that is, [delta][L.sup.*]/[delta]w [less than] 0, if U exhibits CARA or DARA, whereas the long-run demand for labor decreases; that is, [delta][L.sup.*]/[delta]w [less than] 0, if U exhibits CARA or if U exhibits DARA and labor is noninferior (nonnormal) in production.

In Appendix A, the following expression is derived relating the slopes of the long-run and short-run demands for labor at a long-run equilibrium configuration,

[delta][L.sup.*]/[delta]w = [theta][delta][L.sup.*]/[delta]w - [theta]([V.sub.KL]/[V.sub.LL][V.sub.KK])L [integral of] U"([pF.sub.K] - r) dH(p; [delta]), (7)

where [theta] = [V.sub.KK][V.sub.LL]/([V.sub.KK][V.sub.LL] - [[V.sup.2].sub.KL]) [greater than] 1 given the second-order sufficient conditions. For a firm exhibiting IARA, no sign-definite prediction is possible concerning [delta][L.sup.*]/[delta]w, whereas [delta][L.sup.*]/[delta]w is unambiguously negative if labor is nonnormal, but of indeterminate sign otherwise. As a consequence, labor demand may not exhibit the LeChatelier effect when the firm's risk preferences are characterized by IARA.

For a firm exhibiting CARA, there are no wealth effects so that both [delta][L.sup.*]/[delta]w and [delta][L.sup.*]/[delta]w are unambiguously negative, and the integral on the right-hand side of Equation 7 vanishes. Because [theta] exceeds one, a firm with CARA exhibits the LeChatelier effect for labor demanded.

Finally, with DARA, the wealth effect entering through the integral in Equation 7 is positive, whereas both [delta][L.sup.*]/[delta]w and [delta][L.sup.*]/[delta]w are negative if labor is normal. In this case, the firm's labor demand exhibits the LeChatelier effect if the cross-partial derivative 

[V.sub.KL] = (w/r) [integral of] U"[([pF.sub.K] - r).sup.2] dH(p; [delta]) + [F.sub.KL] [integral of] U'p dH(p; [delta]) (8)

is nonnegative; that is, if the expected utility function is supermodular with respect to capital and labor at long-run equilibrium configurations. For this to be true, labor and capital must be complements in production ([F.sub.KL] [greater than] 0). Moreover, this complementarity must be stong enough to offset the effect of risk aversion, which tends to make the factors substitutes in generating expected utility in a long-run equilibrium, since marginal profits from labor and capital are subject to perfectly correlated risks.

On the other hand, if capital and labor prove to be substitutes in expected utility ([V.sub.KL] [less than] 0), either because they are substitutes in production or insufficiently complementary to offset the effect of risk aversion, then the second term on the right-hand side of Equation 7 is positive for a firm with DARA. Hence, when capital and labor are substitutes in an expected utility function with DARA, the wealth effect of an increase in the wage rate feeds back through the long-run adjustment of capital to oppose the short-run change in labor demanded, and as a consequence, the firm's demand for labor may not exhibit the LeCatelier effect. An example is presented in Appendix B wherein a firm with constant relative risk aversion, and hence DARA, violates the LeChatelier prediction for labor demanded given a symmetric, decreasing returns to scale production function of the Cobb-Douglas form.

Thus, a definite prediction concerning the LeChatelier effect for labor demanded with respect to changes in the wage rate emerges only in the case of a firm characterized by CARA.

PROPOSITION 1. In long-run equilibrium under price uncertainty, labor demanded decreases by more in the long run than in the short run when the wage rate increases; that is, [delta][L.sup.*]/[delta]w [less than] [delta][L.sup.*]/[delta]w [less than] 0, if U exhibits CARA.

3. LeChatelier Effects for Output Supplied in Response to Changes in Uncertainty and in Risk Aversion

LeChatelier predictions can also be established with respect to changes in the probability distribution for the stochastic output price by generalizing the concept of an increase in price to account for uncertainty in the manner suggested by Ormiston and Schlee (1993). They demonstrate that the behavioral response to an increase in price under certainty is qualitatively the same as the response to a monotone likelihood ratio (MLR) improvement in the probability distribution when price is stochastic. They also note that these changes in uncertainty are behaviorally identical to reductions in risk aversion. In this section, LeChatelier predictions are established for output supplied with respect to both MLR improvements and reductions in risk aversion, thereby extending the behavioral link between improvements under certainty, MLR improvements under uncertainty, and changes in risk aversion from the first order to the second.

MLR Improvements

Formally, an MLR improvement is a first-order stochastic dominance shift from an initial cumulative probability distribution [H.sup.0](p) to a final distribution [H.sup.1](p) that satisfies the following restrictions:

[H.sup.0](p) = 1 over a nondegenerate interval ([p.sub.2], p]

[H.sup.1](p) = 0 over a nondegenerate interval [p, [p.sub.1])

[H.sup.1](p) = [[[integral of].sup.p].sub.[p.sub.i]]

Thus, MLR improvements increase, or leave unchanged, both the maximum and minimum realized values for the random price p, while ensuring that, over the common interval of support, the ratio of likelihoods is increasing with respect to p; that is, [h.sup.1](p)/[h.sup.0](p) = k(p) with [k.sup.1] [greater than] 0 for the probability densities [h.sup.1] and [h.sup.0]. As a consequence, the expected value of p increases.

MLR improvements in the output price distribution correspond to increases in the "strength" of industry demand as indicated by the parameter [delta]. A higher price, then, represents, ex post, more favorable news about the strength of demand in the sense discussed by Milgrom (198 la, b), that for any nondegenerate prior beliefs about 8, posterior beliefs conditional on a particular price stochastically dominate at the first-order posterior beliefs conditional on any lower price.

As shown by Sandmo (1971), Batra and Ullah (1974), and Epstein (1978), an additive shift in the price distribution does not necessarily lead to an increase in output supplied. Indeed, as shown by Hadar and Russell (1978) and Ormiston and Schlee (1992, 1993), the same is true of first-order stochastic dominance improvements. However, MLR improvments always lead to an increase in output supplied, as shown in the following result. [4]

LEMMA 2. Let an increase in [delta] cause a monotone likelihood ratio improvement in the probability distribution for output price, H(p; [delta]). Then output increases with increases in [delta] in both the short run and the long run; that is, [delta][y.sup.*]/[delta][delta] [greater than] 0 and [delta][y.sup.*]/[delta][delta] [greater than] 0. The long-run demand for capital increases (decreases) if capital is normal (inferior) in production.

The statement concerning short-run changes in supply represents an application of Ormiston and Schlee's (1993) Theorem 2. Specifically, since output increases whenever p increases under certainty, output increases with MLR improvements in price uncertainty. The statement concerning output supplied in the long run extends this result to the case of two decision variables in the theory of the competitive firm.

Lemma 2 contrasts with the conclusions reached by Sandmo (1971) that either CARA or DARA is sufficient for the quantity supplied by a competitive firm to increase with an additive shift of the random output price, but that with IARA, the firm's supply could decrease. In the case of MLR improvements in the random output price, no restrictions on risk preferences are needed to predict that quantity supplied increases.

By exploiting Lemma 2, the next result establishes the LeChatelier prediction for output supplied in response to MLR improvements.

PROPOSITION 2. Let an increase in [delta] cause a monotone likelihood ratio improvement in the probability distribution H(p; [delta]). Then in long-run equilibrium under price uncertainty, output supplied increases by more in the long run than in the short run when [delta] increases; that is, [[delta][y.sup.*]/[delta][delta] [greater than] [[delta][y.sup.*]/[delta][delta] [greater than] 0.

Note that restrictions on risk preferences are not needed to ensure that the LeChatelier prediction for output supplied carries over from the case of price increases under certainty to MLR improvements under uncertainty. This conclusion extends to the second order the result obtained by Ormiston and Schlee (1993) linking first-order behavioral responses to improvements under certainty and MLR improvements under uncertainty.

In contrast, whereas labor demanded obeys the LeChatelier prediction with respect to output price under certainty when capital and labor are normal and complementary in production, no such prediction emerges when output price is stochastic. In place of complementarity in production, the factors must be complementary in expected utility, but tend instead to be substitutes, causing wealth effects, as they feed back through the long-run adjustment of capital, to oppose the short-run change in labor demanded in response to an MLR improvement in the price distribution.

Changes in Risk Aversion

LeChatelier predictions can also be established with respect to changes in risk aversion by applying the fundamental Theorem 4 of Diamond and Sitglitz (1974). They show that, for the case of one choice variable, an increase in risk aversion causes the optimal choice to fall (rise) if the marginal value of the choice variable changes sign once from positive to negative as the random variable decreases (increases) over its domain of support.

An increase in risk aversion is represented by replacing the utility function U with the transformed function [psi](U; p), where an increase in the preference parameter p indicates that the transformation function [psi] is more concave, thereby increasing the degree of risk aversion. As an application of Diamond and Stiglitz's Theorem 4, the following Lemma establishes comparative statics predictions for output supplied with respect to increases in risk aversion that parallel those stated in Lemma 2 concerning MLR improvements.

LEMMA 3. Let an increase in p cause an increase in risk aversion. Then output supplied under price uncertainty decreases when p increases in both the long run and the short run; that is, [delta][y.sup.*]/[delta]p [less than] 0 and [delta][y.sup.*]/[delta]p [less than] 0. The long-run demand for capital decreases (increases) if capital is normal (inferior) in production.

By exploiting Lemma 3, Proposition 3 demonstrates the LeChatelier prediction for output supplied with respect to increases in risk aversion under price uncertainty.

PROPOSITION 3. Let an increase in [rho] cause an increase in risk aversion. Then in long-run equilibrium under price uncertainty, output supplied decreases by more in the long run than in the short run when [rho] increases; that is, [delta][y.sup.*]/[delta][rho] [greater than] [delta][y.sup.*]/[delta][rho] [greater than] 0.

This Proposition complements Proposition 2 concerning MLR improvements, and shows that no assumptions on risk preferences are required to ensure the LeChatelier prediction for output supplied with respect to increases in risk aversion. In addition, just as in the case of MLR improvements, so in the case of changes in risk aversion, no parallel LeChatelier prediction for labor demanded is possible.

4. Absence of Global LeChatelier Predictions

Thus far, only local LeChatelier predictions have been established. The global LeChatelier principle set forth by Milgrom and Roberts (1996) requires that the criterion function be supermodular; that is, the cross-partial derivatives involving capital, labor, and the wage rate must be nonnegative. Unfortunately, the cross derivative between capital and labor for the expected utility function V(K, L, w, r, [delta]) in the general case is given by

[V.sub.KL] = [integral of] U"(p[F.sub.k] - w) dH + [F.sub.KL] [integral of] U'p dH. (9)

In a long-run equilibrium, [F.sub.L]/[F.sub.K] = w/r and the first integral is unambiguously negative, as shown in Equation 8. Nonetheless, the expected utility function is supermodular at long-run equilibria if capital and labor are sufficiently complementary in production. However, outside the neighborhood of a long-run equilibrium, the first term in Equation 9 could dominate the second, reversing the modularity of expected utility. Because the expected utility criterion is not necessarily uniformly modular away from long-run equilibria, global LeChatelier effects cannot be predicted for labor demand.

Similarly, uniform modularity fails when the firm's criterion is formulated with output and capital as the decision variables, as in the proof of Lemma 2 in Appendix A. The sign of the cross-partial derivative for capital with respect to the probability distribution parameter [delta] depends on whether capital is above or below its long-run equilibrium level, [K.sup.*](w, r, [delta]. Hence, a global LeChatelier prediction for supply is not possible for a risk-averse competitive firm facing output price uncertainty.

5. Conclusions

LeChatelier effects involve behavioral responses at the second order, reflecting, for example, the greater local concavity of profit as a function of output price in the short run, as a consequence of maximizing behavior in the face of an additional constraint limiting the adjustment of capital. The response of a competitive firm ceteris paribus to changes in an exogenous variable under certainty are less elastic in the short run, when the use of capital is fixed, than in the long run, since feedback from the optimal adjustment of capital always reinforces short-run responses, whether capital and labor are complements or substitutes. Milgrom and Roberts (1996) show that uniform complementarity or substitutability between capital and labor ensures that feedback from optimal capital adjustments is reinforcing globally, not merely in the neighborhood of a long-run equilibrium for the firm.

Such a global LeChatelier prediction is not available for a risk-averse competitive firm facing a stochastic price for output. Risk preferences interact with technology to generate feedback effects that may, in some instances, oppose he short-run, more constrained responses to increases in the wage rate or monotone likelihood ratio improvements in the output price distribution. Local LeChatelier predictions, however, are possible at long-run equilibrium configurations for the firm. Indeed, constant absolute risk aversion is sufficient for own-price LeChatelier effects for labor demanded, since in the absence of wealth effects, risk preferences have no direct bearing on factor demands. Moreover, the LeChatelier effect for output supplied holds without qualification for monotone likelihood ratio improvements in the price distribution and increases in risk aversion. Thus, the perfect behavioral link between monotone likelihood ratio improvements and reductions in risk aversion established by Ormiston and Schlee (1993) for first-order comparative statics effects applies as well to the second-order LeChatelier effects.

(*.) Department of Economics, University of Georgia, Athens, GA 30602, USA; E-mail [e-mail]. The helpful comments and suggestions of two anonymous referees are gratefully acknowledged. Received June 1998; accepted April 1999.

(1.) An additional situation that may fit the mold of the risk-averse firm arises when a risk-averse manager's compensation is linked to profit, as in a moral hazard setting where ownership is separated from managerial control. As emphasized by Sandmo (1971), when an entrepreneurial firm has more than one owner making decisions, it must be assumed that their preferences are sufficiently similar to justify representing the firm's objective by an expected utility function. Leland (1974) and Feder, Just, and Schmitz (1980) analyzed the roles of markets for ownership shares and futures contracts in diversifying risks faced by a competitive firm.

(2.) Batra and Ullah (1974) observed that with DARA, the demand for labor is a decreasing function of the wage rate if capital and labor are complementary in production, which is sufficient, but not necessary, for labor to be a normal factor.

(3.) Proofs of results are presented in Appendix A.

(4.) Ormiston and Schlee (1993) note that an additive shift does correspond to an MLR improvement if the density function is log-concave (implying a single mode), a restriction met by normal, uniform, and exponential distributions. Athey (1996) extends the results of Ormiston and Schlee (1993) to environments with multiple decision and random variables.

References

Arrow, Kenneth J. 1965. Aspects of the theory of risk bearing. Helsinki: Yrjo Jahnsson Lectures.

Athey, Susan. 1996. Comparative statics under uncertainty: Single crossing properties and log-supermodularity. Unpublished paper, MIT.

Batra, Raveendra N., and Aman Ullah. 1974. Competitive firm and the theory of input demand under price uncertainty. Journal of Political Economy 82:537-48.

Diamond, Peter A., and Joseph E. Stiglitz. 1974. Increases in risk and in risk aversion. Journal of Economic Theory 8: 337-60.

Epstein, Larry. 1978. Production flexibility and the behavior of the competitive firm under price uncertainty. Review of Economic Studies 45:251-61.

Feder, Gershon, Richard E. Just, and Andrew Schmitz 1980. Futures market and the theory of the firm under price uncertainty, Quarterly Journal of Economics 94:317-28.

Hadar, Josef, and William Russell. 1978. Applications in economic theory and analysis. In Stochastic dominance, chapter 7, edited by G. Whitmore and M. Findlay. Lexington, MA: Lexington Books.

Leland, Hayne E. 1974. Production theory and the stock market. Bell Journal of Economics 5:125-44.

Milgrom, Paul. 1981a. Rational expectations, information acquisition, and competitive bidding. Econometrica 49:921-43.

Milgrom, Paul. 1981b. Good news and bad news: Representation theorems and applications. Bell Journal of Economics 12:380-91.

Milgrom, Paul, and John Roberts. 1996. The LeChatelier principle. American Economic Review 86:173-9.

Ormiston, Michael B., and Edward E. Schlee. 1992. Necessary conditions for comparative statics under uncertainty. Economics Letters 40:429-34.

Ormiston, Michael B., and Edward E. Schlee. 1993. Comparative statics under uncertainty for a class of economic agents. Journal of Economic Theory 61:412-22.

Samuelson, Paul A. 1947. The foundations of economic analysis. Cambridge, MA: Harvard University Press.

Sandmo, Agnar. 1971. On the theory of the competitive firm under price uncertainty. American Economic Review 61: 65-73.

Silbergerg, Eugene. 1971. The LeChatelier principle as a corollary to a generalized envelope theorem. Journal of Economic Theory 3:146-55.

Tressler, John H., and Carmen F. Menezes. 1983. Constant returns to scale and competitive equilibrium under uncertainty. Journal of Economic Theory 31:383-91.

Appendix A

In this Appendix proofs are provided for the results stated in the text. The first two Lemmas and Propositions apply to any strictly concave utility function U regardless of the degree of risk aversion.

PROOF OF LEMMA 1. The proof makes use of the following well-known fact established by Arrow (1965) and Sandmo (1971) and stated here for a general criterion function U(Z[alpha] p)) with decision variable [alpha] and random variable p. "Primes" on U denote derivatives, and subscripts denote partial derivatives for functions with multiple arguments.

Fact: If U exhibits DARA [CARA] (IARA), then

[integral of] U"(Z([alpha], p))[Z.sub.[alpha]] dH(p) [greater than] [=] ([less than] 0

whenever

[integral of] U'[Z.sub.[alpha]] dH = 0 for a function Z([alpha], p)

such that

[Z.sub.p] [greater than or equal to] 0, and [Z.sub.ap] [greater than] 0 whenever [Z.sub.[alpha]] = 0.

The choice variable a depends on the context. Intuitively, with a satisfying the first-order condition, an increase in wealth leads to an increase (decrease) in the willingness to bear risk when utility exhibits DARA (IARA) implying that the choice of [alpha] rises (falls), while the second-order sufficient condition implies that the direction of change in [alpha] has the same sign as [integral of] U" [Z.sub.[alpha]] dH.

To prove this fact, assume [Z.sub.[alpha]]([alpha], p+) = 0. With [Z.sub.p] [greater than or equal to] 0, whenever p [greater than] ([less than]) p+ one has Z [greater than] ([less than]) Z+ = Z([alpha], p+); therefore,

-U"(Z)/U'(Z) [less than] ([greater than]) - U"(Z+)/U'(Z+)

if and only if U exhibits DARA. The single crossing property assumed for [z.sub.[alpha]] ensures [Z.sub.[alpha]] [greater than] ([less than]) 0 as p [greater than] ([less than]) p+. Hence, multiplying by U' [Z.sub.[alpha]] yields

- U"(Z)[Z.sub.[alpha]] [less than] [- U"(Z+)/U'(Z+)]U'[Z.sub.[alpha]]

for all p. Multiplying by - 1 and taking the expectation of both sides yields the desired inequality,

[integral of] U"(Z([alpha], p))[Z.sub.[alpha]] dH(p) [greater than] [- U"([Z.sup.+])/U'([Z.sup.+])] [integral of] U'[Z.sub.[alpha]] dH = 0

The same proof applies mutatis mutandis to the cases of CARA and IARA.

To prove [delta][L.sup.*]/[delta]w [less than] 0 if U exhibits CARA or DARA, observe that the first-order necessary conditions for maximizing expected utility V given in Equation 6 can be written

[V.sub.L] = [integral of] U'([pF.sub.L] - w) dH = 0 (A1)

[V.sub.K] = [integral of] U'([pF.sub.K] - r) dH = 0, (A2)

implying that [F.sub.L]/[F.sub.K] = w/r in long-run equilibrium. Short-run comparative statics are determined by Equation A1 alone, so the change in labor demanded when the wage rate increases, [delta][L.sup.*]/[delta]w, has the same sign as

[V.sub.Lw] = -L [integral of] U"([pF.sub.L] - w) dH - [integral of] U' dH, (A3)

given the second-order sufficient condition. The second integral is positive, and when U exhibits CARA or DARA, the first integral is nonnegative, in which event the entire expression is negative. Hence, either CARA or DARA is sufficient for [delta][L.sup.*]/[delta]w [less than] 0.

Regarding the own-price comparative statics effects for long-run factor demands, the first-order conditions in Equations A1 and A2 imply the comparative statics relation

[delta][L.sup.*]/[delta]w = (-[V.sub.Lw][V.sub.KK] + [V.sub.Kw][V.sub.KL])/ ([V.sub.KK][V.sub.LL] - [[V.sup.2].sub.KL]). (A4)

Hence, the second-order sufficient conditions imply that demand for labor decreases (increases) as the wage rate increases, [delta][L.sup.*]/[delta]w [less than] ([greater than]) 0, if

-[V.sub.Lw][V.sub.KK] + [V.sub.Kw][V.sub.KL] (A5)

is negative (positive). Equation A2 implies

[V.sub.Kw] = -L [integral of] U"([pF.sub.k] - r) dH, (A6)

so that [V.sub.Kw] [equivalent to] 0 with CARA. In this case, [delta][L.sup.*]/[delta]w has the same sign as [V.sub.Lw], which is negative.

Finally, consider the case of DARA (IARA). Because [F.sub.L]/[F.sub.K] = w/r in long-run equilibrium, Equations A3 and A6 imply

[V.sub.Lw] = [V.sub.Kw]([F.sub.L]/[F.sub.K]) - [integral of] U' dH, (A7)

which with Equation A5 implies that the sign of [delta][L.sup.*]/[delta]w is the same as the sign of the following expression:

[V.sub.Kw][[V.sub.KL] - ([F.sub.L]/[F.sub.K])[V.sub.KK]] + [V.sub.KK] [integral of] U' dH. (A8)

Observe that

[V.sub.KK] = [integral of] U"[([pF.sub.k] - r).sup.2]dH + [F.sub.KK] [integral of] U'p dH

is negative, so the second term in Expression A8 is negative, whereas [V.sub.Kw] is negative (positive) given DARA (IARA). Hence, the proof is complete if the bracketed expression is nonnegative (nonpositive) when labor is noninferior (nonnormal). Substituting for [V.sub.KK] and for [V.sub.KL] from Equation 8 yields

[V.sub.KL] - ([F.sub.L]/[F.sub.K])[V.sub.KK] = [[F.sub.KL] - ([F.sub.L]/[F.sub.K])[F.sub.KK]] [integral of] U'p dH, (A9)

as the second-order effects involving U" are eliminated. Thus, the expression is nonnegative (nonpositive) when labor is noninferior (nonnormal). QED.

PROOF OF PROPOSITION 1. Differentiating the identity

[L.sup.*](w, r, [delta]) [equivalent to] [L.sup.*]([K.sup.*](w, r, [delta]), w, r, [delta])

with respect to the wage rate w yields

[delta][L.sup.*]/[delta]w [equivalent to] [delta][L.sup.*]/[delta]w + ([delta][L.sup.*]/[delta]K)([delta]/[K.sup.*]/[delta]w). (A10)

Let [K.sup.*](L, w, r, [delta]) denote the demand function for capital when labor's use is artificially held constant. Differentiating the identity

[K.sup.*](w, r, [delta]) [equivalent to] [K.sup.*]([L.sup.*](w, r, [delta]),w, r, [delta])

with respect to w yields

[delta][K.sup.*]/[delta]w [equivalent to] ([delta][K.sup.*]/[delta]L)([delta][L.sup.*]/[delta]w) + [delta][K.sup.*]/[delta]w. (A11)

Substituting this expression into Identity (A10) and rearranging terms yields

([delta][L.sup.*]/[delta]w)[l - ([delta][L.sup.*]/[delta]K)([delta][K.sup.*]/[delta]L)] [equivalent to] [delta][L.sup.*]/[delta]w + ([delta][L.sup.*]/[delta]K)([delta][K.sup.*]/[delta]w).

Substituting for [delta][L.sup.*]/[delta]K = -[V.sub.KL]/[V.sub.LL], [delta][K.sup.*]/[delta]L = -[V.sub.KL]/[V.sub.KK], [delta][K.sub.*]/[delta]w = -[V.sub.Kw]/[V.sub.KK], and [V.sub.Kw] from Equation A6 and solving for [delta][L.sup.*]/[delta]w yields the expression given in the text at Equation 7, and the subsequent discussion there completes the proof. QED.

PROOF OF LEMMA. 2. Let c(K, y) denote the short-run cost function, with factor prices notationally suppressed. The firm's short-run supply decision and its long-run decisions concerning output supplied and demand for capital can be characterized by maximizing the criterion function

v(K, y, [delta]) [equivalent to] [integral of] U(py - o(K, y)) dH(p; [delta]).

The first-order necessary conditions

[v.sub.K] = - [integral of] U'[dHc.sub.k] = 0 [v.sub.y] = [integral of] U'(p - [c.sub.y]) dH = 0

imply that [c.sub.k] = 0 at a long-run profit maximum for the firm. Observe that the second-order sufficient condition implies that the sign of the short-run supply response to MLR improvements, [delta][y.sup.*]/[delta][delta], is the same as the sign of [v.sub.y[delta]]. Similarly, the long-run response, [delta][y.sup.*]/[delta][delta], has the same sign as

-[v.sub.y[delta]][v.sub.KK] + [v.sub.K[delta]][v.sub.Ky] = -[v.sub.y[delta]][v.sub.KK]

since [v.sub.K[delta]] = 0 in the long run. Hence, the long-run and short-run output responses have the same sign as [v.sub.y[delta]]. The long-run change in demand for capital, [delta][K.sub.*]/[delta][delta], has the same sign as

-[v.sub.x[delta]][v.sub.yy] + [v.sub.y[delta]][v.sub.ky] = -[v.sub.y[delta]][C.sub.Ky] [integral of] U'dH.

Thus, the change in capital demand has the same (opposite) sign as [v.sub.y[delta]] if [c.sub.Ky] [less than] ([greater than]) 0. It follows that the proof is complete once [c.sub.Ky] [less than] 0 is linked with the normality of capital and [v.sub.y[delta]] is shown to be positive.

The first-order conditions for minimizing short-run cost, w - [[lambda]F.sub.L] = 0 = y - F, where the Lagrange multiplier is the marginal cost [lambda] = [c.sub.y], imply

[delta][lambda]/[delta]K = ([[lambda]F.sub.K]/[[F.sup.2].sub.L])[[F.sub.LL] - ([F.sub.L]/[F.sub.K])[F.sub.KL]].

Hence, [c.sub.Ky] is negative (positive) when capital is normal (inferior), as indicated by the sign of the term in brackets.

The proof is completed by showing that [v.sub.y[delta]] is positive in the manner of Ormiston and Schlee's (1993) proof of their theorem 2. Let [pi](K, y, p) = py - c(K, y). Note that for [p.sup.*] = [c.sub.y]([K.sub.*], [y.sub.*]), one has [delta][pi]([K.sub.*], [y.sub.*], [p.sub.*])/[delta]y = 0 and [delta][pi]([K.sub.*], [y.sub.*], p)/[delta]y [less than] ([greater than]) 0 as p [less than] ([greater than]) [p.sub.*]. Assume that H(p; [[delta].sup.1]) represents an MLR improvement of H(p; [[delta].sup.0]). The definition for MLR improvement implies

[v.sub.y]([K.sup.*], [y.sup.*], [[delta].sup.1) = [[[integral of].sup.p].sub.p] U'[[pi].sub.y] dH(p;[[delta].sup.1] = [[[integral of].sup.p2].sub.p1] U'[[pi].sub.y]K(p) dH(p; [[delta].sup.0]) + [[[integral of].sup.p].sub.p2] U'[[pi].sub.y]dH(p; [[delta].sup.1]) [greater than or equal to] [[[integral of].sup.p2].sub.p1] U'[[pi].sub.y]k(p) dH(p; [[delta].sup.0]).

where the inequality follows from [delta][pi]([K.sup.*], [y.sup.*], p)/[delta]p [greater than] 0 for p [epsilon] ([p.sub.2], p], since [p.sup.*] [less than or equal to] [p.sub.2]. If [p.sup.*] [epsilon] [p, [p.sub.1]], than

[[[integral of].sup.p2].sub.p1] U'[[pi].sub.y]k(p) dH(p)[[delta].sup.0] [greater than or equal to] 0,

since k(p) is nonnegative. It follows that [v.sub.y[delta]]([K.sub.*], [y.sub.*], [[delta].sup.0]) is positive in this case. If [p.sup.*] [epsilon] ([p.sub.1], [p.sub.2]), then

[[[integral].sup.p2].sub.p1] U'[[pi].sub.y]k(p) dH(p; [[delta].sup.0] [greater than or equal to] k([p.sub.*]){[[[integral of].sup.[p.sup.*]].sub.p1] U'[[pi].sub.y]dH(p; [[delta].sup.0]) + [[[integral of].sup.p2].sub.[p.sup.*]] U'[[pi].sub.y] dH(p; [[delta].sup.0])},

since k is increasing. The right-hand side is equal to

K([p.sup.*]) [[integral of].sup.p2].sub.p1]] U'[[pi].sub.y]dH(p;[[delta].sup.0]) = -k([p.sup.*]) [[integral of].sup.p1].sub.p] U'[[pi].sub.y] dH(p;[[delta].sup.0]) [greater than or equal to] 0,

where the equality follows from the first-order condition [v.sub.y]([K.sup.*], [y.sup.*], [[delta].sup.0]) = 0, and the inequality follows from the assumption that [p.sup.*] exceeds [p.sub.t]. Thus, [v.sub.y[delta]]([K.sup.*], [y.sup.*], [[delta].sup.0]) is again positive. QED.

PROOF OF PROPOSITION 2. Differentiating the identity

[y.sup.*] (w, r, [delta]) [equivalent to] [y.sup.*]([K.sup.*](w, r, [delta]), w, r, [delta])

with respect to the shift parameter [delta] for MLR improvements yields

[delta][y.sup.*]/[delta][delta] [equivalent to] [delta][y.sup.*]/[delta][delta] + ([delta][y.sup.*]/[delta]K)([delta][K.sup.*]/[delta][delta]).

Lemma 2 indicates that [delta][y.sup.*]/[delta][delta] and [delta][y.sup.*]/[delta][delta] are both positive. Hence, the own-price LeChatelier prediction for output is established by showing that the product in the second term is positive.

Observe that [y.sup.*] = F(K, [L.sup.*]) implies

[delta][y.sup.*]/[delta]K = [F.sub.k] + ([delta][L.sup.*]/[delta]K)[F.sub.L] = [F.sub.K] - ([V.sub.KL]/[V.sub.LL])[F.sub.L] = - [F.sub.L]/[V.sub.LL][[V.sub.KL] - ([F.sub.K]/[F.sub.L])[V.sub.LL]] = - [F.sub.L]/[V.sub.LL][[F.sub.KL] - ([F.sub.K]/[F.sub.L])[F.sub.LL]] [integral of] U'p dH.

The last term in brackets is positive (negative) when capital is normal (inferior), implying that [delta][y.sup.*]/[delta]K and [delta][K.sup.*]/[delta][delta] have the same sign, since [delta][y.sup.*]/[delta][delta] is unambiguously positive, as established in Lemma 2. It follows that the product in the second term is positive, completing the proof. QED.

The final Lemma and Proposition concern increases in the degree of risk aversion, and so exploit the transformation function [psi](U; [rho]).

PROOF OF LEMMA 3. The proof of Lemma 2 mutatis mutandis based on the criterion v(K, y, p) = [integral of] [psi](U(py - [delta](K, y)); [rho]) dH(p) with short-run cost function c(K, y), applies here with [rho] replacing [delta] to show that [delta][y.sup.*]/[delta][rho] and [delta][y.sup.*]/[delta][rho] have the same sign as [v.sub.yp], and that [delta][K.sup.*]/[delta][rho] also has the same (opposite) sign as [v.sub.yp] if capital is normal (inferior) in production.

To establish [v.sub.yp] [less than] 0 and complete the proof, differentiate the first-order necessary condition

[v.sub.y] = [integral of] [[psi].sub.U]U'(p - [c.sub.y]) dH = 0

with respect to the risk-aversion index [rho] to obtain

[v.sub.yp] = [integral of] ([[psi].sub.Up]/[[psi].sub.U])[[psi].sub.U]U'(p - [c.sub.y]) dH = [integral of] [([[psi].sub.Up]/[[psi].sub.U]) - ([[[psi].sup.+].sub.Up]/[[[psi].sup.+].sub.U])][[psi].sub.U]U'(p - [c.sub.y]) dH,

where [[[psi].sup.+].sub.Up]/[[[psi].sup.+].sub.U])is evaluated at the price p+ such that U'(p - [c.sub.y]) is positive (negative) when p is above (below) p+. Because increases in [rho] induce increases in risk aversion, [[delta].sup.2] In [[psi].sub.U]/[[delta]p[delta]U is negative, as discussed by Diamond and Stiglitz (1974), implying that the term in brackets is negative (positive) when p is above (below) p+. It follows that [v.sub.yp] is negative. QED.

PROOF OF PROPOSITION 3. The proof of this Proposition follows that of Proposition 2, with [rho] replacing [delta] and Lemma 3 replacing Lemma 2. QED.

Appendix B

In this Appendix, an example is presented in which a competitive firm facing output price uncertainty does not exhibit the LeChatelier effect for labor demanded. It follows from Equation A10 that the LeChatelier effect is absent if the inequality

([delta][L.sup.*]/[delta]K)([delta][K.sup.*]/[delta]w) [greater than] 0

holds at a long-run equilibrium. After substituting for [delta][L.sup.*]/[delta]K = - [V.sub.KL]/[V.sub.LL] and for [delta][K.sup.*]/[delta]w from Equation A11, and substituting for [delta][K.sup.*]/[delta]L = - [V.sub.KL]/[V.sub.KK] and [delta][K.sup.*]/[delta]w = - [V.sub.Kw]/[V.sub.KK] and rearranging terms, while recalling the second-order sufficient conditions, the desired inequality becomes

[delta][L.sup.*]/[delta]w [greater than] - [V.sub.Kw]/[V.sub.KL].

Substituting for [delta][L.sup.*]/[delta]w from Equation A4 yields

[V.sub.Lw] [greater than] [V.sub.Kw] [V.sub.LL]/[V.sub.KL]

after rearranging terms. Now substituting for [V.sub.Lw] from Equation A7, using

[V.sub.LL] -- ([F.sub.L]/[F.sub.K])[V.sub.KL] = [[F.sub.LL] - ([F.sub.L]/[F.sub.K])[F.sub.KL] [integral of] U'p dH,

which is obtained in the same manner as the expression in Equation A9, and using

[integral of] U' dH = ([F.sub.L]/w) [integral of] U'p dH,

which is obtained from the first-order condition in Equation A1, yields

-[F.sub.L]/w [greater than] [[F.sub.LL - ([F.sub.L]/[F.sub.K])[F.sub.KL]]([V.sub.Kw]/[V.sub.KL]).

It will be shown in the example that [V.sub.KL] is negative. Thus, substituting for [V.sub.Kw] from Equation A6 and for [V.sub.KL] from Equation 8 yields

[integral of] U"(P - M)[[F.sub.L](p - M) + w[eta]] dH + (F.sub.KL]/[F.sub.K])[[integral of] U'p dH + wL [integral of] U"(p - M) dH][greater than] 0, (A12)

where M = r/[F.sub.K] denotes long-run marginal cost and [eta] = -[LF.sub.LL]/[F.sub.L] is (minus) the elasticity of the marginal product of labor with respect to labor usage.

With DARA, the second integral within brackets at A12 is positive. Thus, when capital and labor are complements in production ([F.sub.KL] [greater than] 0), the second term in the sum at Equation A12 is positive with DARA. Hence, the desired inequality holds if the first integral is also positive.

In the example, an expected utility maximum is assumed to occur on the unit isoquant with wage and interest rates both equal to one. The production function has the symmetric, generalized Cobb-Douglas form F(K, L) [(KL).sup.[alpha]] with decreasing returns to scale, so that [alpha] is less than one half. Tressler and Menezes (1983) show that production with constant returns so scale is inconsistent with free entry by risk-averse entrepreneurs. Under these assumptions [F.sub.L] = [alpha] and z = p -- A, where A denotes average cost. The utility function U(z)= In (z) has constant relative risk aversion equal to one, and so exhibits DARA with U" = -1/[Z.sup.2]. Under these assumptions, the first integral in Equation A12 is positive if the expression

[integral of] [(p - M)/(p - A)][[alpha](p - M) + [eta]]/ (p - A) dH (A13)

is negative. The first-order condition in Equation Al implies that

[integral of] [(p - M)/(p - A)] dH = 0,

so the expression in Equation A13 has the same sign as the covariance between [(p - M)]/(p - A) and [[alpha](p - M) + [eta]]/(p - A). Because the former is increasing with p, the covariance is negative, and Equation A13 has the desired negative sign, if the latter is decreasing with p, which requires [alpha](M -A) [greater than][eta]. It is easy to check that this inequality holds, since the assumed conditions on production imply M = 1/[alpha], A = 2 and [eta] = 1 - [alpha]. Finally, to verify that [V.sub.KL] is negative, note that [F.sub.KL]/[F.sub.k] = [alpha] under the assumed conditions, and Equation 8 can be written as

[V.sub.KL]/[F.sub.K] = [integral of] [(p - M)/(p - A)] ([-(p - M)/(p - A)] + [alpha]p/(p - M)] dH,

which is negative, since the covariance between the two terms in braces is negative as the first is increasing with P, whereas the second is decreasing.