LeChatelier Effects for the Competitive Firm under Price Uncertainty.

• Author: Snow, Arthur

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...