A global analysis of constrained behavior: the LeChatelier principle 'in the large'.

• Author: Chavas. Jean-Paul

1. Introduction

   Samuelson (1947) first introduced the LeChatelier principle in economics. The principle has its simplest interpretation in the theory of the firm. It states that output supply or variable input demand functions are more price elastic in the long run (where all inputs are chosen) than in their short-run counterpart (where capital input is treated as fixed). This result has strong intuitive appeal because it indicates that putting restrictions on production choices tends to reduce the firm's ability to adjust to changing market conditions. This has generated much interest in refining the nature and implications of the LeChatelier principle in resource allocation (e.g., Samuelson 1947, 1960; Silberberg 1974; Pauwels 1979; Milgrom and Roberts 1996; Roberts 1999). The leading method of analysis has relied on comparative statics analysis based on applying the implicit function theorem to a system of equations that represents economic behavior (e.g., Samuelson 1947; Silberberg 1974; Pauwels 1979; Hatta 1980). It has generated useful insights into economic behavior and the adjustments of resource allocation under changing economic conditions. However, there are several limitations in the traditional approach to LeChatelier principle analysis. First, it assumes the presence of a unique equilibrium. This appears restrictive because it excludes situations of multiple equilibria. Second, it relies on differentiability assumptions. Although such assumptions often seem reasonable, they are not suitable in a number of situations. For example, they are ill-equipped to analyze switching between alternative behavioral regimes, such as those generated by the presence of "kinks" in an agent's objective function. Third, the use of the implicit function theorem gives results that are "local" and valid only in the neighborhood of a point. Actual changes often involve "large changes." It is well known that "local" LeChatelier results do not necessarily hold "globally" (Samuelson 1960; Milgrom and Roberts 1996). This raises the question: Under what conditions would the "local" LeChatelier principle apply "in the large"?

   In an effort to conduct economic analysis "in the large," several approaches have been explored, all avoiding a reliance on the implicit function theorem. The monotone comparative statics approach has been proposed to investigate the global qualitative properties of behavioral functions (e.g., Milgrom and Shannon 1994; Topkis 1995). This approach uses a supermodularity assumption to establish monotonicity properties of optimal decisions (Milgrom and Shannon 1994). Milgrom and Roberts (1996) have shown that under supermodularity assumptions some global LeChatelier results can be obtained. However, although supermodularity appears to be a reasonable assumption in some contexts (e.g., firm behavior, see Topkis 1995), it does not apply in other contexts (e.g., consumer behavior; see Sundaram 1996, p. 261). This suggests the need to examine an alternative approach.

   Another way to address the problem is to rely on the Lagrangean approach to constrained optimization analysis. This is the same starting point as the traditional local derivation of the LeChatelier principle (Samuelson 1947, 1960; Silberberg 1974; Pauwels 1979). In this context, the issue is how to use the Lagrangean approach to obtain global results, possibly without requiring differentiability or the existence of a unique solution. This approach appears attractive on several grounds. First, under some regularity conditions, (1) solving a constrained optimization problem is equivalent to finding a saddle point to the associated Lagrangean (Takayama 1985, p. 75). Second, the Lagrangean approach is at the heart of economic analysis. For example, consumption analysis is typically developed in this context. And the theory of value, Pareto optimality, and competitive market equilibrium can all be formulated and analyzed using the Lagrangean approach (e.g., Luenberger 1994).

   This paper relies on the Lagrangean approach to develop LeChatelier-Samuelson results "in the large." It extends previous research in several ways. First, our analysis applies under less restrictive assumptions. It does not require differentiability, and it allows for multiple solutions. Also, we consider the behavioral effects of generic changes in the feasible set. This is more general than the typical LeChatelier situation, which focuses on restricting the fixed factors to be constant in the short run. This greater generality appears relevant to a number of situations, including the analysis of government regulations (which restrict the feasible set) or of technological progress (which expands the feasible set). Second, our approach generates new and more precise results than previous analyses. For example, we derive new conditions under which local LeChatelier results apply globally. Third, our results are simple, yet they apply under general conditions. They provide a powerful way of analyzing the effects of restricting the feasible set on economic behavior "in the large." They rely on restrictions placed on the upper bounds and lower bounds of the change in the indirect objective function between the restricted and the unrestricted situations. We show that under general conditions a parallel shift in the upper bounds implies the global validity of the LeChatelier principle with respect to optimal choices. And that in constrained optimization problems a parallel shift in the lower bound implies the global validity of the LeChatelier principle with respect to the Lagrange multipliers (measuring the shadow value of the constraints). The approach is illustrated in several applications, showing how our analysis generates useful insights into the effects of a changing feasible set on global economic adjustments. In particular, new LeChatelier results are obtained for consumption behavior.

2. The Model

   Consider an agent choosing a (n x 1) vector x of decision variables in an economic environment represented by a (k x 1) vector [alpha] of parameters, where x [member of] X [subset] [R.sup.n], and [alpha] [member of] A [subset] [R.sup.k]. X, A, and R represent real numbers. X is the set of feasible numbers for x, A is the set of feasible numbers for [alpha], and R is the "real line." The agent has preferences represented by the objective function f(x, [alpha]), where f: X x A [right arrow] R, and faces a set of in constraints g(x, [alpha]) [greater than or equal to] 0, where g: X x A [right arrow] [R.sup.m]. Assume that the agent makes decisions in a way consistent with the maximization problem

   [x.sup.*]([alpha]), [member of] [x.sup.*]([alpha]) = [argmax.sub.x] {f(x, [alpha]):g(x, [alpha]) [greather than or equal to] 0, x [member of] X}, (1a)

   where [x.sup.*]([alpha]) is the set of optimal solutions to the optimization problem (Eqn. la) under situation [alpha] [member of] A and where [x.sup.*]([alpha]) is an element of that set and [argmax.sub.x] represents the value of the decision variables that maximize the objective function (subject to feasibility constraints). Throughout this paper, we assume that the set [x.sup.*]([alpha]) is nonempty for any a [member of] A. In general [x.sup.*]([alpha]) is a decision rule expressing which decisions x are optimal in situation [alpha]. If a unique decision x is associated with each [alpha], then [x.sup.*]: A [right arrow] X is a single value mapping and [x.sup.*]([alpha]) is the optimal decision function. More generally, we allow for the possibility of multiple solutions where [x.sup.*]: A [right arrow] P(X) is a correspondence, P(X) being the power set of X. In the context of Equation la, we define the indirect objective function [f.sup.*]([alpha]) as the function [f.sup.*]: A [right arrow] R satisfying [f.sup.*]([alpha]) = f{[x.sup.*]([alpha]), [alpha]} for [alpha] [member of] A, as the direct objective function f(x, [alpha]) evaluated at optimum [x.sup.*]([alpha]) [member of] [x.sup.*]([alpha]) under situation [alpha].

   Equation 1a corresponds to a situation where the feasible set for x is X. We also consider some alternative feasible set Y satisfying Y [subset] X, where Y is a restricted subset of X. This means that some options available in the choice of x under the feasible set X are no longer available under set Y. It includes, as a special case, the situation where x is partitioned into two subsets x = ([x.sup.a], [x.sup.b]) and Y = {([x.sup.a], [x.sup.b]):([x.sup.a], [x.sup.b]) [member of] X and [x.sup.a] = [x.sup.*]([alpha]')} for some [alpha] [member of] A. This corresponds to the typical LeChatelier situation where X is the long-run feasible set and Y is the short-run feasible set with [x.sup.a] being treated as a fixed factor (e.g., Samuelson 1946; Silberberg 1974; Pauwels 1979; Milgrom and Roberts 1996). Choosing [x.sup.a] = [x.sup.a*]([alpha]') guarantees that Y [subset] X. However, it should be kept in mind that our analysis covers much more general situations because it allows for various ways of changing the feasible set. For example, it also covers the case of government regulations that impose partial restrictions on x. As such, our analysis of the LeChatelier principle has implications for the economics of regulation. It is also relevant in analyzing the effects of technological progress that expands the feasible set.

   In a fashion parallel to Equation 1a, we assume that when facing the restricted set Y that the agent makes decisions according to the maximization problem

   [x.sup.c]([alpha]) [member of] [X.sup.c]([alpha]) = [argmax.sub.x]{f(x, [alpha]):g(x, [alpha]) [greater than or equal to] 0, x [member of] Y}, (1b)

   where Y [subset] X, [X.sup.C]([alpha]) is the set of optimal restricted solutions to the optimization problem (Eqn. lb) under situation [alpha] [member of] A and where [x.sup.c]([alpha]) is an element of that set. Again, we assume that the set [X.sup.c]([alpha]) is nonempty for any [alpha] [member of]...