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

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] A. In general, [X.sup.c]([alpha]) is a restricted decision rule in situation a where [X.sup.c]: A [right arrow] P(Y). Also, from Equation lb, we define the indirect restricted objective function [f.sup.c] ([alpha]) as the function [f.sup.c]: A [right arrow] R satisfying [f.sup.c]([alpha]) = f(x.sup.c]([alpha]), [alpha]) for [alpha] [member of] A.

As discussed in the introduction, there is interest in analyzing such problems (Eqns. 1a-1b) through the use of the Lagrangean approach. Let L(x, [lambda], [alpha]) = f(x, [alpha]) [[lambda].sup.T]g(x, [alpha]) be the Lagrangean corresponding to Equations 1a and 1b, where [lambda] [member of] [R.sup.m.sub.+] is a (m x 1) vector of Lagrange multipliers associated with the in constraints g(x, [alpha]) [greater than or equal to] 0. (2) For each [alpha] [member of] A, consider the following saddle point problem associated with Equation 1a: there exist [x.sup.*]([alpha]) [member of] X and [[lambda].sup.*]([alpha]) [member of] [R.sup.m.sub.+] satisfying

L[x, [[lambda].sup.*]([alpha], [alpha]) [less than or equal to] L([x.sup.*]([alpha]), [alpha]) [less than or equal to] L([x.sup.*]([alpha]), [lambda], [alpha]), [for all]x [member of] X, [lambda] [member of] [R.sup.m.sub.+], (2a)

where [x.sup.*]([alpha]) and [[lambda].sup.*]([alpha]) correspond to a saddle point of the Lagrangean L(x, [lambda], [alpha]) under the feasible set X and L([x.sup.*]([alpha]), [[lambda].sup.*]([alpha]), [alpha]) is the unrestricted saddle value of the Lagrangean.

Similarly, for each [alpha] [member of] A, consider the following saddle point problem associated with Equation 1b: there exist [x.sup.c]([alpha]) [member of] X and [[lambda].sup.c]([alpha]) [member of] [R.sup.m.sub.+] satisfying

L(x, [lambda].sup.c]([alpha]), [alpha]) [less than or equal to] L([x.sup.c]([alpha]), [[lambda].sup.c]([alpha]), [alpha]) [less than or equal to] L([x.sup.c]([alpha]), [lambda], [alpha]), [for all]x [member of] Y, [lambda] [member of] [R.sup.m.sub.+, (2b)

where [x.sup.c]([alpha]) and [[lambda].sup.c]([alpha]) correspond to a saddle point of the Lagrangean L(x, [lambda], [alpha]) under the restricted feasible set Y and where L([x.sup.c]([alpha]), [[lambda].sup.c]([alpha]), [alpha]) is the restricted saddle value of the Lagrangean.

It is well known (see Takayama 1985, p. 74) that the existence of a saddle point always implies the following two results: [x.sup.*]([alpha]) in the saddle point problem (Eqn. 2a) is always an optimal solution to the constrained optimization problem (Eqn. 1a) and [[lambda].sup.*] [([alpha]).sup.T]g([x.sup.*]([alpha]), [alpha]) = 0 (the complementary slackness condition). The complementary slackness condition implies that the saddle value in Equation 2a and the indirect objective function from Equation 1a are identical: L([x.sup.*]([alpha]), [[lambda].sup.*]([alpha]), [alpha]) = [f.sup.*]([alpha]), [alpha] [member of] A. Similar results link the saddle point problem (Eqn. 2b) with the constrained optimization problem (Eqn. 2a). Thus, if there is a saddle point in Equation 2b, it always identifies an optimal decision [x.sup.c]. And the saddle value in Equation 2b is the indirect objective function from Equation 1b: L([x.sup.c]([alpha]), [[lambda].sup.c]([alpha]), [alpha]), = [f.sup.c]([alpha]).

Throughout this paper, we assume that a saddle point exists in both Equations 2a and 2b. Note that this is always tree, given Equations 1a and 1b, if X and Y are convex sets and if for each [alpha] [member of] A, functions f and g are concave in x and there exists a feasible point x satisfying g(x, [alpha]) > 0 (Slater's condition) (see Takayama 1985, p. 75). We focus our attention on the behavioral implications of restricting the feasible set for x from X to Y.

3. LeChntelier Principle

In this section, we investigate economic adjustments associated with a discrete change in the parameters [alpha] [member of] A and with a restriction of the feasible set from X to Y. We focus on the interaction effects between changing [alpha] and restricting the feasible set, which are at the heart of the LeChatelier principle. The analysis relies on the saddle point of the Lagrangean in Equations 2a and 2b.

First, given Y [subset] X, it is clear from Equations 1a and 1b that [f.sup.*]([alpha]) [greater than or equal to] [f.sup.c]([alpha]) for any [alpha] [member of] A. This states that the restricted indirect objective function [f.sup.c]([alpha]) has its unrestricted counterpart [f.sup.*]([alpha]) as its upper bound. Given a saddle point of the Lagrangean in Equations 2a and 2b, the saddle point value being the indirect objective function implies that

[f.sup.*]([alpha]) = L([x.sup.*]([alpha]), [[lambda].sup.*]([alpha]), [alpha]) [greater than or equal to] [f.sup.c]([alpha]) = L([x.sup.c]([alpha]), [[lambda].sup.c]([alpha]), [alpha]) (3)

for any [alpha] [member of] A. This reflects the fact that the highest possible value of the objective function cannot increase when the feasible set for x is restricted from X to Y. Although this result is intuitive, it is less clear that it contains useful information about the effects of constraints on behavior.

We present several implications of Equation 3 obtained under increasingly restrictive assumptions. In the case of a differentiable function F(y, z), we use subscript letters to denote derivatives [F.sub.y](y, z) = [partial derivative]F/[partial derivative]y, [F.sub.yz] (y,z) = [[partial derivative].sup.2]F/[partial derivative]y [partial derivative]z, [F.sub.yy](y,z) = [[partial derivative].sup.2]F/[partial derivative]y[partial derivative]z, each evaluated at (y, z). We first consider the case of two arbitrary situations [alpha] [member of] A and [alpha]' [member of] A.

PROPOSITION 1. Assume that a saddle point holds in Equations 2a and 2b for all [alpha] [member of] A; that the sets A, X, and Y are convex; that g(x, [alpha]) is differentiable in [alpha] on A for each x [member of] X; and that [f.sub.x](x, [alpha]) and [g.sub.x](x, [alpha]) exist and are differentiable in [alpha] on A for each x [member of] X. Then, for any [alpha], [alpha]' [member of] A, any [x.sup.*]([alpha]) [member of] [x.sup.*]([alpha]), and any [x.sup.c]([alpha]) [member of] [X.sup.c]([alpha]),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [x.sub.1] = [[theta].sub.1][x.sup.*]([alpha]) + (1 - [[theta].sub.1])[x.sup.c]([alpha]) and [[lambda].sub.1] = [[theta].sub.1] [[lambda].sup.*]([alpha]) + (1 - [[theta].sub.1])[[lambda].sup.c]([alpha]) for some [[theta].sub.1] [member of] [0, 1] and where [[alpha].sub.1] = [[delta].sub.1][alpha] + (1 - [[delta].sub.1])[alpha]' for some [[delta].sub.1] [member of] [0, 1].

PROOF. Let [L.sub.x](x, [lambda], [alpha]) = [f.sub.x](x, [lambda], [alpha]) + [[lambda].sup.T][g.sub.x](x, [alpha]) be the (1 X n) vector of derivatives of L(.) with respect to x, evaluated at (x, [lambda], [alpha]). Using the mean value theorem, we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

where [x.sub.1] = [[theta].sub.1][x.sup.*]([alpha]) + (1 - [[theta].sub.1])[x.sup.c]([alpha]) and [[lambda].sub.1] = [[theta].sub.1][[lambda].sup.*]([alpha]) + (1 - [[theta].sub.1])[[lambda].sup.c]([alpha]) for some [[theta].sub.1] [member of] [0, 1] and the nonnegativity condition follows from Equation 3. The differentiability of [f.sub.x](x, [alpha]) and [g.sub.x](x, [alpha]) in [alpha] implies that [L.sub.x](x, [lambda], [alpha]) is differentiable in [alpha] on A for each (x, [lambda]) [member of] X x [R.sup.m.sub.+]. Let [L.sub.[alpha]x](x, [lambda], [alpha]) be a (k x n) matrix of second cross derivatives of L(.) with respect to [alpha] and x, and let [g.sub.[alpha]](x, [alpha]) be the (m x k) matrix of derivatives of g(.) with respect to ct, each evaluated at (x, [lambda], [alpha]). From the mean value theorem applied to [H.sub.1](a) = {[L.sub.x]([x.sub.1], [[lambda].sub.1], a)[[x.sup.*]([alpha]) - [x.sup.c]([alpha])] + [g([x.sub.1], a)].sup.T][[[lambda].sup.*]([alpha]) - [[lambda].sup.c]([alpha])]}, we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [[alpha].sub.1] = [[delta].sub.1][alpha] + (1 - [[delta].sub.1])[alpha]' for some [[delta].sub.1] [member of] [0, 1]. Substituting this expression into Equation 4 completes the proof. QED.

Proposition 1 gives the general properties of the relationships between constrained behavior ([x.sup.c]([alpha]), [[lambda].sup.c]([alpha])) and unconstrained behavior ([x.sup.*]([alpha], [[lambda].sup.*]([alpha])). It holds under fairly general conditions. It holds "in the large" for any [alpha], [alpha]' [member of] A. It does not require f(x, [alpha]) or g(x, [alpha]) to be twice continuously differentiable. And it does not require the decision rule [x.sup.*]([alpha]) to be a differentiable function nor a single value mapping (thus allowing for multiple solutions). But it requires the differentiability of f(x, [alpha]) and g(x, [alpha]) in x and the differentiability of g(x, [alpha]), [f.sub.x](x, [alpha]) and [g.sub.x](x, [alpha]) in [alpha].

Next, we consider the more restrictive case where the decision roles [x.sup.*]([alpha], [x.sup.c]([alpha]), [[lambda].sup.*]([alpha]), and [[lambda].sup.c]([alpha]) are differentiable functions on A. Also, we focus on the situation where [x.sup.*]([alpha]') = [x.sup.c]([alpha]') and [[lambda].sup.*]([alpha]') = [[lambda].sup.c]([alpha]'). This latter assumption provides a formal linkage between unrestricted behavior (under X) and restricted behavior (under Y). It has been part of the classical analysis of LeChatelier principle that short-run and long-run decisions are assumed identical under situation [alpha]', but not necessarily under situation [alpha] (e.g., Samuelson 1947, 1960; Silberberg 1974; Pauwels 1979; Milgrom and Roberts 1996). The following result is then obtained as a special case of Proposition 1.

PROPOSITION 2. Assume that a saddle point holds in Equations 2a and 2b for all ct E A; that the sets A, X, and Y are convex; that g(x, [alpha]) is differentiable in [alpha] on A for each x [member of] X; that [f.sub.x](x, [alpha]) and [g.sub.x](x, [alpha]) exist and are differentiable in [alpha] on A for each x [member of] X; and that [x.sup.*]([alpha]), [x.sup.c]([alpha]), [[lambda].sup.*]([alpha]), and [[lambda].sup.c]([alpha]) are differentiable functions on A. Then, for [alpha]' [member of] A satisfying [x.sup.*]([alpha]') = [x.sup.c]([alpha]') and [[lambda].sup.*]([alpha]') = [[lambda].sup.c]([alpha]') and for any [alpha] [member of] A,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [alpha].sub.2] = [[delta].sub.2][alpha] + (1 - [[delta].sub.2])[alpha]' for some [[delta].sub.2] [member of] [0, 1].

PROOF: Let [x.sup.*.sub.[alpha]]([alpha]) be a (n x k) matrix of derivatives of [x.sup.*]([alpha]) with respect to [alpha], [[lambda].sup.*.sub.[alpha]]([alpha]) be the (m x k) matrix of derivatives of [[lambda].sup.*]([alpha]) with respect to [alpha]. Given [x.sup.*]([alpha]') = [x.sup.c]([alpha]') and [[lambda].sup.*]([alpha]') = [[lambda].sup.c]([alpha]') and applying the mean value theorem to [H.sub.2](a) = {[[[alpha] - [alpha]'.sup.T][L.sub.[alpha]x]([x.sub.1], [[lambda].sub.1], [alpha].sub.1])[[x.sup.*](a) - [x.sup.c](a)] + [[[alpha] - [alpha]'].sup.T][g.sub.[alpha]][([x.sub.1], [[alpha].sup.1]).sup.T][[[lambda].sup.*](a) - [[lambda].sup.c](a)]} we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [[alpha].sub.2] = [[delta].sub.2][alpha] - (1 - [[delta].sub.2])[alpha]' for some [[delta].sub.2] [member of] [0, 1]. Given [x.sup.*]([alpha]') = [x.sup.c]([alpha]') and [[lambda].sup.*]([alpha]') = [[lambda].sup.c]([alpha]'), substituting this expression into Proposition 1 proves Proposition 2. QED.

Proposition 2 is closely related to the classical LeChatelier principle found in the literature (e.g., Samuelson 1947, 1960; Silberberg 1974; Pauwels 1979; Milgrom and Roberts 1996). This can be seen through the following corollary.

COROLLARY 1 (local LeChatelier principle). Assume that a saddle point holds in Equations 2a and 2b for all [alpha] [member of] A; that the sets A, X, and Y are convex; that g(x, [alpha]) is continuous in x on X and differentiable in [alpha] on A; that ([f.sub.x] (x, [alpha]), [g.sub.x](x, [alpha])) exist and are continuous in x on X and differentiable in [alpha] on A; and that [x.sup.*]([alpha]), [x.sup.c]([alpha]), [[lambda].sup.*]([alpha]), and [[lambda].sup.c]([alpha]) are differentiable functions on A. Consider an [alpha]' [member of] A satisfying [x.sup.*]([alpha]') = [x.sup.c]([alpha]') and [[lambda].sup.*]([alpha]') = [[lambda].sup.c]([alpha]'). Then, for [alpha] [member of] A satisfying [alpha] [not equal to] [alpha]' and [alpha] [right arrow] [alpha]',

[[[alpha] - alpha]'].sup.T][L.sub.[alpha]x]([x.sub.1], [[lambda].sub.1], [[alpha].sub.1]) [x.sup.*.sub.[alpha]]([alpha].sub.2]) - [x.sup.c.sub.[alpha]]([alpha].sub.2])][[alpha] - [alpha]'] + [[alpha] - [alpha]'].sup.T][g.sub.[alpha]][([x.sub.1], [[alpha].sub.1]).sup.T][[lambda].sup.*.sub.[alpha]]([alpha].sub.2]) - [[lambda].sup.c.sub.[alpha]]([alpha].sub.2])][[alpha] - [alpha]'][greater than or equal to] 0. (5)

PROOF. By the continuity of [L.sub.x](x, [lambda], [alpha]) in (x, [lambda]) on X x [R.sup.m.sub.+] and the continuity of [x.sup.*]([alpha]) and [x.sup.c](alpha]) on A, we have [lim.sub.[alpha][right arrow][alpha], {[L.sub.x]([x.sub.1], [[lambda].sub.1], [alpha]')[[x.sup.*] ([alpha]) - [x.sup.c]([alpha])]} = [L.sub.x]([x.sup.*]([alpha]'), [[lambda].sup.*]([alpha]'), [alpha]') [[x.sup.*]([alpha]') - [x.sup.c]([alpha]')] = 0. (3) Similarly, by the continuity of g(x, [alpha]) in x on X and the continuity of [[lambda].sup.*]([alpha]) and [[lambda].sup.c]([alpha]), we have [lim.sub.[alpha][right arrow][alpha]' {g[([x.sub.1], [[alpha]').sup.T] [[[lambda].sup.*]([alpha]) - [[lambda].sup.c]([alpha])]} = g[(x([alpha]'), [[alpha]').sup.T][[lambda].sup.*]([alpha]') - [[lambda].sup.c]([alpha]')] = 0. Substituting these results into Proposition 2 completes the proof. QED.

Equation 5 is the standard LeChatelier result obtained by Samuelson (1947), Silberberg (1974), and Pauwels (1979) that established a local relationship between constrained behavior ([x.sup.c]([alpha]), [[lambda].sup.c]([alpha])) and unconstrained behavior ([x.sup.*]([alpha]), [[lambda].sup.*]([alpha])) in the neighborhood of situation [alpha]'. Equation 5 shows that in the neighborhood of [alpha]' the matrix {[L.sub.[alpha]x],([x.sub.1], [lambda].sub.1], [alpha].sub.1])[[x.sup.*.sub.[alpha]]([[alpha].sub.2]) - [x.sup.c.sub.[alpha]]([[alpha].sub.2])] + [g.sub.[alpha]] [([x.sub.1], [[alpha].sub.1]).sup.T] [[lambda].sup.*.sub.[alpha]]([[alpha].sub.2]) - [[lambda].sup.c.sub.[alpha]]([[alpha].sub.2])]} is positive semidefinite. Under what conditions would this result also apply globally (i.e., for any [alpha] [member of] A)? One way to answer this question is to evaluate [X.sup.*.sub.[alpha]]([alpha]), [x.sup.c.sub.[alpha]]([alpha]), [[lambda].sup.*.sub.[alpha]]([alpha]), and [[lambda].sup.c.sub.[alpha]]([alpha]) to check whether they satisfy Equation 5 for any [alpha] [member of] A. From comparative statics analysis, these marginal effects depend on [L.sub.xx](x, [lambda], [alpha]) = [f.sub.xx](x, [alpha]) + [lambda][g.sub.xx](x, [alpha]), on [g.sub.x](x, [alpha]), and on [L.sub.x[alpha]](x, [lambda], [alpha]) = [f.sub.x[alpha]](x, [alpha]) + [lambda][g.sub.x[alpha]](x, [alpha]). Unfortunately, these terms interact with each other in complex ways as they affect [[x.sup.*.sub.[alpha]([alpha]) - [x.sup.c.sub.[alpha]([alpha])] and [[lambda].sup.*.sub.[alpha]([alpha]) - [[lambda].sup.c.sub.[alpha]]([alpha])]. This makes it difficult to obtain general conditions for f(x, [alpha]) and g(x, [alpha]) that would make Equation 5 globally valid.

In this context, comparing Proposition 2 with Corollary 1 is instructive. First, Proposition 2 applies under slightly less restrictive conditions. Second, Corollary 1 applies only locally (i.e., in the neighborhood of [alpha]'). In contrast, Proposition 2 applies globally for any [alpha] [member of] A. As a result, Propositions 1 and 2 can be interpreted as global versions of the LeChatelier principle. And comparing Corollary 1 with Proposition 2 gives useful insights on the differences between the local LeChatelier principle and its global counterpart. It identifies the crucial role played by the expression {[L.sub.x]([X.sub.1], [[lambda].sub.1], [alpha]')[[x.sup.*]([alpha]) - [x.sup.c]([alpha])] + g[([x.sub.1], [[alpha]').sup.T][[lambda].sup.*]([alpha]) - [[lambda].sup.c]([alpha])]}. Although this expression is always equal to zero in the local LeChatelier principle, in general, it is nonzero in its global version. Comparing Corollary 1 with Proposition 2 generates the following result.

COROLLARY 2. The local LeChatelier principle in (Eqn. 5) is globally valid if

[L.sub.x]([x.sub.1], [[lambda].sub.1], [alpha]')[x.sup.*]([alpha]) - [x.sup.c]([alpha])] + g[([x.sub.1], [[alpha]').sup.T] [[lambda].sup.*]([alpha]) - [[lambda].sup.c]([alpha])] [less than or equal to] 0, for all [alpha [member of] A.

This shows that the global validity of the local LeChatelier principle rests on the sign of the expression {[L.sub.x]([x.sub.1], [[lambda].sub.1], [alpha]') [[x.sup.*]([alpha]) - [[x.sup.c]([alpha])] + g[([x.sub.1], [alpha]').sup.T][[[lambda].sup.*]([alpha]) - [[lambda].sup.c]([alpha])]} as [alpha] leaves the neighborhood of [alpha]'. It is well known that, in general, this expression can be either positive or negative, implying that the local LeChatelier principle does not always hold globally (e.g, Samuelson 1960; Milgrom and Roberts 1996; Roberts 1999). Corollary 2 establishes the general conditions for the global validity of the local LeChatelier principle. Also, it informs us about the conditions under which the local LeChatelier principle may fail to hold globally. In either case, the arguments center on how [L.sub.x]([x, [lambda], [alpha]') and g(x, [alpha]') vary with x and [lambda]. Such results are consistent with the analysis presented by Milgrom and Roberts (1996). Milgrom and Roberts (1996) have shown the role played by the sign of the second cross derivatives in [L.sub.xx] in long-run behavior (where all factors are variable) compared to the short-run behavior (where some factors are fixed). Using monotone comparative statics, they have shown that restricting globally the sign of these cross derivatives can be sufficient to generate a global version of the LeChatelier principle and that long-run behavior tends to be more price responsive than short-run behavior. Such a result implies that a failure of the local LeChatelier principle to hold globally must be associated with situations where these cross derivatives change sign as [alpha] leaves the neighborhood of [alpha]'. Examples of such situations have often been presented as illustrations that the classical LeChatelier principle holds only locally (e.g., Samuelson 1960; Milgrom and Roberts 1996; Roberts 1999).

Unfortunately, there are situations where the local LeChatelier principle holds globally without the Milgrom-Roberts conditions (Milgrom and Roberts 1996, p. 177). This indicates that the Milgrom-Roberts conditions are "too strong." In addition, the condition stated in Corollary 2 is difficult to analytically assess, in general. This suggests the need for an alternative approach to the global properties of the LeChatelier principle. This is the topic of the next section.

4. Global Properties

We start with an analysis of the global behavioral implications of changing the parameters [alpha], both under the unrestricted (X) and the restricted (Y) feasible sets. We then proceed with an examination of conditions under which the local and global LeChatelier results coincide. First, consider decision-making under the unconstrained feasible set X and under situations [alpha] and [alpha]' [member of] A. By the first inequality in the saddle point problem (Eqn. 2a), we have

[f.sup.*]([alpha]') = L([x.sup.*]([alpha]'), [[lambda].sup.*]([alpha]'), [alpha]') [greater than or equal to] L(x, [[lambda].sup.*]([alpha]'), [alpha]') for all x [member of] X (6a)

or choosing x = [x.sup.*]([alpha]),

[f.sup.*]([alpha]') = L([x.sup.*]([alpha]'), [[lambda].sup.*]([alpha]'), ([alpha]') [greater than or equal to] L([x.sup.*]([alpha]), [[lambda].sup.*]([alpha]'), [alpha]') (6b)

for any [alpha], [alpha]' [member of] A. Equation 6a shows that the indirect objective function [f.sup.*]([alpha]) is an upper bound on the Lagrangean L(x, [[lambda].sup.*]([alpha]'), [alpha]') for all x [member of] X given any [alpha]' [member of] A. Indeed, the indirect objective function reflects the best that can be done. In a maximization problem, this means that it represents the highest possible value of the objective function, which must therefore be as least as large as any other alternative for x (as given by the Lagrangean L(x, [[lambda].sup.*]([alpha]'), [alpha]')). Although results from Equations 6a and 6b are quite intuitive, we show next that they contain a lot of useful information related to economic behavior.

PROPOSITION 3. Assume that a saddle point in Equation 2a holds for all [alpha] [member of] A. Then, for any [alpha], [alpha]' [member of] A,

L([x.sup.*]([alpha]'), [[lambda].sup.*] ([alpha]), [alpha]) - L([x.sup.*]([alpha]'), [[lambda].sup.*]([alpha]), [alpha]') [less than or equal to] [f.sup.*]([alpha]) - [f.sup.*] ([alpha]') [less than or equal to] L([x.sup.*]([alpha]), [[lambda].sup.*] ([alpha]'), [alpha]) - L([x.sup.*]([alpha]), [[lambda].sup.*]([alpha]'), [alpha]').

PROOF. The saddle point in Equation 2a implies that Equation 6b holds, yielding

L([x.sup.*]([alpha]), [[lambda].sup.*]([alpha]'), [alpha]') [less than or equal to] [f.sup.*]([alpha]'). (7)

Note that

[f.sup.*]([alpha]) = f([x.sup.*]([alpha]), [alpha]) [less than or equal to] f([x.sup.*]([alpha]), [alpha]) + [[lambda].sup.*] [([alpha]').sup.T] g([x.sup.*]([alpha]), [alpha]), since [[lambda].sup.*]([alpha]') [member of] [R.sup.m.sub.+] and g ([x.sup.*]([alpha]), [alpha]) [greater than or equal to] 0 by feasibility, = L([x.sup.*]([alpha]), [[lambda].sup.*] ([alpha]'), [alpha]). (8)

Summing Equations 7 and 8 gives

[f.sup.*]([lambda]) - [f.sup.*]([alpha]') [greater than or equal to] L([x.sup.*]([alpha]), [[lambda].sup.*]([alpha]'), [alpha]) - L([x.sup.*]([alpha]), [[lambda].sup.*]([alpha]'), [alpha]'). (9)

Switching [alpha] and [alpha]' in Equation 9 and multiplying by (-1) yields

L([x.sup.*]([alpha]'), [lambda]([alpha]), [alpha]) - L([x.sup.*] ([alpha]'), [[lambda].sup.*]([alpha]), [alpha]') [less than or equal to] [f.sup.*]([alpha]) - [f.sup.*]([alpha]').

This proves Proposition 3. QED.

Proposition 3 applies under general conditions. It does not require differentiability assumptions on functions f or g. It does not require the decision rule [x.sup.*]([alpha]) to be a differentiable function nor a single value mapping. And it holds "in the large" (i.e., for any [alpha], [alpha]' [member of] A). Proposition 3 provides a lower bound and an upper bound on the change in the indirect objective function [f.sup.*]([alpha]) - [f.sup.*]([alpha]'). Note, that under appropriate differentiability and continuity assumptions letting [alpha] [right arrow] [alpha]', Proposition 3 generates the classical envelope result [f.sup.*.sub.[alpha]]([alpha]') = [L.sub.[alpha]([x.sup.*]([alpha]'), [[lambda].sup.*] ([alpha]'), [alpha]'). Thus, Proposition 3 can be interpreted as a generalized envelope theorem. Finally, Proposition 3 generalizes a related analysis presented by Anderson and Takayama (1979). The bounds derived by Anderson and Takayama (1979, p. 499) (4) are

[L.sub.[alpha]]([x.sup.*]([alpha]'), [[lambda].sup.*]([alpha]), [alpha]')[[alpha] - [alpha]'] [less than or equal to] [f.sup.*]([alpha]) - [f.sup.*]([alpha]') [less than or equal to] [L.sub.[alpha]]([x.sup.*]([alpha]), [[lambda].sup.*]([alpha]'), [alpha])[[alpha] - [alpha]']

for any [alpha], [alpha]' [member of] A. Anderson and Takayama assume that L(x, [lambda], [alpha]) is convex in [alpha] on A (Anderson and Takayama 1979, p. 494). Because our Proposition 3 does not require L(x, [lambda], [alpha]) to be convex in [alpha], it applies under more general conditions than the Anderson and Takayama analysis. In addition, in situations where [alpha] (1) [not equal to] [alpha]', and L(x, [lambda], [alpha]) is strictly convex in ct, the following inequalities hold

L([x.sup.*]([alpha]'), [[lambda].sup.*]([alpha]), [alpha]), - L([x.sup.*]([alpha]'), [[lambda].sup.*]([alpha]), [alpha]') > [L.sub.[alpha]]([x.sup.*]([alpha]'), [[lambda].sup.*]([alpha]), [alpha]')[[alpha] - [alpha]']

and

L([x.sup.*]([alpha]), [[lambda].sup.*]([alpha]'), [alpha]) - L([x.sup.*]([alpha]), [[lambda].sup.*]([alpha]'), [alpha]') < [L.sub.[alpha]]([x.sup.*]([alpha]), [[lambda].sup.*]([alpha]'), [alpha])[[alpha]'-[alpha]]

implying that the bounds presented in Proposition 3 are tighter than the Anderson and Takayama bounds. In this case, the Anderson and Takayama bounds on the change in the indirect objective function, [f.sup.*]([alpha]) - [f.sup.*]([alpha]'), are "too loose." This shows that our analysis generalizes the Anderson and Takayama analysis and can give more precise results.

Results analogous to the ones presented in Proposition 3 can be obtained in the context of the restricted optimization problem (Eqn. 1b). They are presented next.

PROPOSITION 3'. Assume that a saddle point in Equation 2b holds for all [alpha] [member of] A. Then, for any [alpha], [alpha]' [member of] A,

L([x.sup.c] ([alpha]'), [[lambda].sup.c]([alpha]), [alpha]) - L([x.sup.c]([alpha]'), [[lambda].sup.c]([alpha]), [alpha]') [less than or equal to] [f.sup.c]([alpha]) - [f.sup.c]([alpha]') [less than or equal to] L ([x.sup.c]([alpha]), [[lambda].sup.c]([alpha]'), [alpha]) - L([x.sup.c]([alpha]), [[lambda].sup.c]([alpha]'), [alpha]').

The results in Propositions 3 and 3' provide a broad framework to analyze the effects of changing situations [alpha] on economic behavior. Recall that the feasible sets X and Y satisfy Y [subset] X (i.e., that Y is restricted compared to X). As indicated in Equation 3, this implies that [f.sup.c]([alpha]) [less than or equal to] [f.sup.*]([alpha]) for any [alpha] [member of] A. As in Proposition 2, we will consider the situation [alpha]' [member of] A satisfying [x.sup.*] ([alpha]) = [x.sup.c]([alpha]') and [[lambda].sup.*]([alpha]') = [[lambda].sup.c]([alpha]'). As discussed above, this formal linkage between restricted and unrestricted behavior is a basic assumption typically found in LeChatelier analysis (see Samuelson 1947, 1960; Silberberg 1974; Pauwels 1979; Milgrom and Roberts 1996). It implies that [f.sup.*]([alpha]') = [f.sup.c]([alpha]'). Combining this with Equation 3 yields

[f.sup.*]([alpha]) - [f.sup.*]([alpha]') [greater than or equal to] [f.sup.c][alpha] - [f.sup.c] ([alpha]'). (10)

Combining Equation 10 with Proposition 3 and 3' is instructive. Propositions 3 and 3' establish bounds on the change in the indirect objective function between ct and [alpha]'. And Equation 10 states that such changes tend to be larger in the unrestricted case (compared to the restricted case). This raises the question: What happens to the bounds stated in Propositions 3 as one moves from the unrestricted set X to the restricted feasible set Y? Clearly, Equation 10 implies that the upper bound in Proposition 3 (the unrestricted case) is also an upper bound to [[f.sup.c][alpha] - [f.sup.c]([alpha]')] in Proposition 3' (the restricted case). And Equation 10 implies that the lower bound in Proposition 3' (the restricted case) is also a lower bound to [[f.sup.*][alpha] - [f.sup.*]([alpha]')] in Proposition 3 (the unrestricted case). But although [f.sup.c] [alpha] - [f.sup.c] ([alpha]') [less than or equal to] [f.sup.*][alpha] - [f.sup.*]([alpha]') from Equation 10, is there a relationship between the two lower bounds stated in Propositions 3 and 3'? And is there a relationship between the two upper bounds stated in Propositions 3 and 3'? The general answer to both of these questions is no. Yet, it may be natural to expect that a reduction in the change in the indirect objective function as one moves from X to Y (given in Eqn. 10) could well be associated with a parallel shift in the associated bounds stated in Propositions 3 and 3'. In such situations, the following conditions would hold:

Condition C1 (restriction for a parallel shift in the upper bounds):

L([x.sub.c]([alpha]), [[lambda].sup.c]([alpha]'), [alpha]) - L([x.sup.c]([alpha]), [[lambda].sup.c]([alpha]'), [alpha]') [less than or equal to] L([x.sup.*]([alpha]), [[lambda].sup.*]([alpha]'), [alpha]) - L([x.sup.*]([alpha]), [[lambda].sup.*]([alpha]'), [alpha]) (C1)

for [alpha], [alpha]' [member of] A.

Condition C2 (restriction for a parallel shift in the lower bounds):

L([x.sup.c]([alpha]'), [[lambda].sup.c]([alpha]), [alpha]) - L([x.sup.c]([alpha]'), [[lambda].sup.c]([alpha]), [alpha]') [less than or equal to] L([x.sup.*]([alpha]'), [[lambda].sup.*]([alpha]), [alpha]) - L([x.sup.*]([alpha]'), [[lambda].sup.*]([alpha]), [alpha]') (C2)

for [alpha], [alpha]' [member of] A.

Condition C1 states that the upper bound in Proposition 3 is at least as large as the upper bound in Proposition 3'. Condition C2 states that the lower bound in Proposition 3 is at least as large as the lower bound in Proposition 3'. Although these conditions may not always hold, note that they do not require the differenfiability of the functions f(x, [alpha]) and g(x, [alpha]). And they apply under multiple solutions and for discrete change in [alpha]. We also consider summing the inequalities in Conditions C1 and C2. This gives

Condition C3 (restriction for a parallel shift in the average of the upper and lower bounds):

L([x.sub.c]([alpha]), [[lambda].sup.c]([alpha]'), [alpha]) - L([x.sup.c]([alpha]), [[lambda].sup.c]([alpha]'), [alpha]') + L([x.sub.c]([alpha]'), [[lambda].sup.c]([alpha]) - [alpha]) L([x.sup.c]([alpha]'), [[lambda].sup.c]([alpha]), [alpha]') [less than or equal to] L([x.sup.*]([alpha]), [[lambda].sup.*]([alpha]'), [alpha]) - L([x.sup.*]([alpha]), [[lambda].sup.*]([alpha]'), [alpha]') + L([x.sup.*]([alpha]'), [[lambda].sup.*]([alpha]), [alpha]) - L([x.sup.*]([alpha]'), [[lambda].sup.*]([alpha]), [alpha]') (C3)

for [alpha], [alpha]' [member of] A.

Clearly, Conditions C1 and C2 imply Condition C3. We show below that Condition C3 is precisely the restriction that generates global LeChatelier results. We also show that Conditions C1 and C2 have useful implications in the analysis of constrained behavior.

First, we consider the implications of Condition C1 under increasingly restrictive assumptions.

PROPOSITION 4. Assume that a saddle point holds in Equations 2a and 2b for all [alpha] [member of] A; that the sets A, X, and Y are convex; and that [f.sub.[alpha]](x, [alpha]) and [g.sub.[[alpha]](x, [alpha]) exist and are differentiable in x [member of] X for each [alpha] [member of] A. For any [alpha], [alpha]' [member of] A, consider

[[alpha]- [alpha]'].sup.t] [[L.sub.[alpha]x] ([x.sub.3], [[lambda].sub.3] [[alpha].sub.3]) [[x.sup.*] ([alpha]) - [x.sup.c] ([alpha])] + [[alpha] - [[alpha]'].sup.t] [g.sub.[alpha][([x.sub.3], [[alpha].sub.3]).sub.T] [[lambda].sup.*] ([alpha]') - [[lambda].sup.c] ([alpha]')] [greater than or equal to] 0, (C1')

where [x.sub.3] = [[theta].sub.3] [x.sup.*]([alpha]) + (1 - [[theta].sub.3])[x.sup.c]([alpha]) and [[lambda].sub.3] = [[theta].sub.3] [[lambda].sup.*]([alpha]') + (1 - [[theta].sub.3])[[lambda].sup.c]([alpha]') for some [[theta].sub.3] [member of] [0, 1] and

where [[alpha].sub.3] = [[delta].sub.3][alpha] + (1 - [[delta].sub.3])[alpha]' for some [[delta].sub.3] [member of] [0, 1] and where [[alpha].sub.3] = [[delta].sub.3][alpha] + (1 - [[delta.sub.3])[alpha]' for some [[delta].sub.3] [member of] [0, 1]. Then, for any [alpha][alpha]' [member of] A,

(a) Equation C1 implies Equation C1'.

(b) Equation C 1' is a necessary and sufficient condition for Equation C1 if [L.sub.[alpha]x] is independent of (x, [alpha]) and [[lambda].sup.*]([alpha]') = [[lambda].sup.c]([alpha]').

PROOF. The differentiability of f(x, [alpha]) and g(x, [alpha]) in [alpha] implies that L(x, [lambda], [alpha]) is differentiable in [alpha] on A. Applying the mean value theorem to the function [H.sub.3]([alpha]) = L([x.sup.*]([alpha]), [[lambda].sup.*]([alpha]')[alpha]) - L([x.sup.c]([alpha]), [[lambda].sup.c]([alpha]'), [alpha]), we obtain

L([x.sup.*]([alpha]), [[lambda].sup.*]([alpha]'), [alpha]) - L([x.sup.*]([alpha]), [[lambda].sup.*]([alpha]'), [alpha]') - L([x.sup.c]([alpha]), [[lambda].sup.c]([alpha]'), [alpha]) + L([x.sup.c]([alpha]), [[lambda].sup.c]([alpha]'), [alpha]') = [L.sub.[alpha]], ([x.sup.*]([alpha]), [[lambda].sup.*]([alpha]'), [[alpha].sub.3][[alpha] - [alpha]'] - [L.sub.[alpha]], ([x.sup.c]([alpha]), [[lambda].sup.c]([alpha]'), [[alpha].sub.3])[[alpha] - [alpha]'],

where [[alpha].sub.3] = [[delta].sub.3][alpha] + (1 - [[delta].sub.3])[alpha]' for some [[delta].sub.3] [member of] [0, 1]. Substituting this expression into Equation C1 yields

[L.sub.[alpha]] ([x.sup.*]([alpha]), [[lambda].sup.*]([alpha]'), [[alpha].sub.3])[[alpha] - [alpha]'] - [L.sub.[alpha]]([x.sup.c]([alpha]), [[lambda].sup.c]([alpha]'), [[alpha].sub.3])[[alpha] - [alpha]'] [greater than or equal to] 0. (11)

Under the differentiability of [f.sub.[alpha]](x, [alpha]) and [g.sub.[alpha]](x, [alpha]) in x, it follows that [L.sub.[alpha]](x, [lambda], [alpha]) is differentiable in (x, [lambda]) on X x [R.sup.m.sub.+] for each [alpha] [member of] A. Applying the mean value theorem to [H.sub.4](x, [lambda]) = {[L.sub.[alpha]] (x, [lambda], [[alpha].sub.3])[[alpha] - [alpha]'}, we obtain

[L.sub.[alpha]] ([x.sup.*]([alpha]), [[lambda].sup.*] ([alpha]'), [[alpha].sup.3]) [[alpha] - [alpha]'] - [L.sub.[alpha]] ([x.sup.c]([alpha]), [[lambda].sup.c]([alpha]'), [[alpha].sup.3) [[alpha] - [[alpha]']

= [[[alpha] - [alpha]'].sup.T] [L.sub.[alpha]x] ([x.sub.3], [[lambda].sub.3], [[alpha].sub.3]) [[x.sup.*]([alpha]) - [x.sup.c]([alpha])] + [[[alpha] - [alpha]'].sup.T] [g.sub.[alpha]] [([x.sub.3], [[alpha].sub.3]).sup.T] [[lambda].sup.*], ([alpha]') - [[lambda].sup.c] ([alpha]')],

where [x.sub.3] = [[theta].sub.3] [x.sup.*]([alpha]) + (1 - [[theta].sub.3]) [x.sup.c]([alpha]) and [[lambda].sub.3] = [[theta].sub.3] [[lambda].sup.*]([alpha]') + (1 - [[theta].sub.3]) [[lambda].sub.c] ([alpha]') for some [[theta].sub.3] [member of] [0, 1]. Substituting this expression into Equation 11 proves Proposition 4(a). To prove Proposition 4(b), assume that [L.sub.[alpha]x] is independent of (x, [alpha]) and [[lambda].sup.*] ([alpha]') = [[lambda].sup.c] ([alpha]'). Then,

L(x, [lambda], [alpha]) = [f.sub.1](x) + [f.sub.2]([alpha]) + [[alpha].sup.T] [f.sub.3] x + [m.summation over (j=1)] [[lambda].sub.j][[g.sub.1j](x) + [g.sub.2j]([alpha]) + [[alpha.sup.T][g.sub.3j]x],

[[lambda].sub.3] = [[lambda].sub.*]([alpha]) = [[lambda].sub.c]([alpha]'),

and either Equation C1 or C1' can be written as [[alpha] - [[alpha]'].sup.T][f.sub.3] + [[summation of].sup.m.sub.j=1] [[lambda].sup.*.sub.j]([alpha]')[g.sub.3j]] [[x.sup.*]([alpha]) - [[x.sup.c]([alpha])] [greater than or equal to] 0. QED.

Proposition 4 applies under fairly general conditions: it allows for possible multiple solutions, and it holds "in the large" (i.e., for any [alpha], [alpha]' [member of] A). We now consider a special case obtained under slightly more restrictive assumptions. In particular, we now restrict [alpha]' [member of] A such that it satisfies [x.sup.*]([alpha]') = [x.sup.c]([alpha]') and [[lambda].sup.c]([alpha]') = [[lambda].sup.c]([alpha]'). As discussed above, this latter assumption linking unrestricted and restricted behavior under situation [alpha]' is commonly found in LeChatelier analysis (see Samuelson 1947, 1960; Silberberg 1974; Pauwels 1979; Milgrom and Roberts 1996). Also, we assume that [x.sup.*]([alpha]), [x.sup.c]([alpha]), [[lambda].sup.*]([alpha]), and [[lambda].sup.c]([alpha]) are differentiable functions on A.

PROPOSITION 5. Assume that a saddle point holds in Equations 2a and 2b for all [alpha] [member of] A; that the sets A, X, and Y are convex; that [f.sub.[alpha]](x, [alpha]) and [g.sub.[alpha]](x, [alpha]) exist and are differentiable in x [member of] X for each [alpha] [member of] A; and that [x.sup.*]([alpha]), [x.sup.c]([alpha]), and [[lambda].sup.*]([alpha]), and [[lambda].sup.c]([alpha]) are differentiable functions on A. If [alpha]' [member of] A satisfies [x.sup.*]([alpha]) = [x.sup.c]([alpha]') and [[lambda].sup.*]([alpha]') = [[lambda].sup.c]([alpha]') then for [alpha] [member of] A, Condition C1' can be written as

[[[alpha] - [alpha]'].sup.T] [L.sub.[alpha]x] ([x.sub.3], [[lambda].sub.3], [[alpha].sub.3]) [[x.sup.*.sub.[alpha]] ([[alpha].sup.4]) - [[x.sup.c.sub.[alpha]]([[alpha].sub.4])] - [[alpha] - [[alpha]'] [greater than or equal to] 0. (C1")

where [[alpha].sub.4] = [[delta].sub.4][alpha] + (1 - [[delta].sub.4])[alpha]' for some [[delta].sub.4] [member of] [0, 1].

PROOF. Given [x.sup.*]([alpha]) = [x.sup.c]([alpha]'), applying the mean value theorem to [H.sub.4](a) = {[[alpha] - [alpha]'].sup.T] [L.sub.[alpha]x]([x.sub.3], [[lambda].sub.3], [[alpha].sub.3])[[x.sup.*](a) - [x.sup.c](a)]}, we obtain

[[[alpha] - [alpha]'].sup.T] [L.sub.[alpha]x]([x.sub.3], [[lambda].sub.3], [[alpha].sub.3])[[x.sup.*]([alpha]) - [x.sup.*]([alpha]') + [x.sup.c]([alpha]') - [x.sup.c]([alpha])] = [[[alpha] - [alpha]'].sup.T] [L.sub.[alpha]x]([x.sub.3], [[lambda].sub.3], [[alpha].sub.3])[[x.sup.*.sub.[alpha] ([[alpha].sup.4]) - [[x.sup.c.sub.[alpha]] ([[alpha].sub.4])] - [[alpha] - [alpha]'],

where [[alpha].sub.4] = [[delta].sub.4][alpha] + (1 - [[delta].sub.4])[alpha]' for some [[delta].sub.4] [member of] [0, 1]. Given [x.sup.*]([alpha]') + [x.sup.c]([alpha]') and [[lambda].sup.*], ([alpha]') = [[lambda].sup.c] ([alpha]'), substituting this result in Equation C1' gives the desired result. QED.

Proposition 5 shows that Equation CI' generates the positive semidefiniteness restrictions (Eqn. C1") involving the difference between unrestricted and restricted effects of [alpha] on [x.sup.*]([alpha]) [x.sup.c]([alpha]). Such restrictions are part of the local LeChatelier principle given in Equation 5. However, the results stated in Proposition 5 are global because they hold for any discrete change from [alpha]' to [alpha] [member of] A. Combining Propositions 4 and 5, this indicates that the parallel shift in the upper bounds given in Equation C1 plays a role in establishing the global validity of local LeChatelier results related to the optimal choices [x.sup.*]([alpha]) and [x.sup.c]([alpha]). This will be further illustrated below.

We now turn our attention to Condition C2. The implications of Condition C2 are considered next under increasingly restrictive assumptions.

PROPOSITION 6. Assume that a saddle point holds in Equations 2a and 2b for all [alpha] [member of] A; that the sets A, X, and Y are convex; and that [f.sub.[alpha]](x, [alpha]) and [g.sub.[alpha]](x, [alpha]) exist and are differentiable in x on X for each [alpha] [member of] A. For any [alpha], [alpha]' [member of] A, consider

[[alpha] - [[alpha]'].sup.T] [L.sub.[alpha]x] ([x.sub.5], [[lambda].sub.5], [[alpha].sub.5]) [[x.sup.*]([alpha]') - [x.sup.c]([alpha]')] + [[[alpha] - [alpha]'].sup.T] [g.sub.[alpha]][([x.sub.5], [[alpha].sub.5]).sup.T] [[[lambda].sup.*], ([alpha]) - [[lambda].sup.c] ([alpha])] [greater than or equal to] 0, (C2')

where [x.sub.5] = [[theta].sub.5] [x.sup.*]([alpha]') + (1 - [[theta].sub.5])[x.sup.c]([alpha]') and [[lambda].sub.5] = [[theta].sub.5] [[lambda].sup.*]([alpha]) + (1 - [[theta].sub.5]) [[lambda].sub.c] ([alpha]) for some [[theta].sub.5] [member of] [0, 1] and where [[alpha].sub.5] = [[delta].sub.5][alpha] + (1 - [[delta].sub.5])[alpha]' for some [[delta].sub.5] [member of] [0, 1]. Then, for any [alpha], [alpha]' [member of] A,

(a) Equation C2 implies Equation C2'.

(b) Equation C2' is a necessary and sufficient condition for Equation C2 if [g.sub.[alpha]] is independent of [alpha] and [[x.sup.*]([alpha]') [[x.sup.c.]([alpha]').

Proof. The differentiability of f(x, [alpha]) and g(x, [alpha]) in [alpha] implies that L(x, [lambda], [alpha]) is differentiable in [alpha] on A. Applying the mean value theorem to the function [H.sub.5](a) = L([x.sup.*], ([alpha]'), [[lambda].sup.*]([alpha]), a) - L[x.sup.c]([alpha]'), [[lambda].sup.c]([alpha]), a) we obtain we obtain

L([x.sup.*]([alpha]'), [[lambda].sup.*], ([alpha]), [alpha]) - L([x.sup.*]([alpha]'), [[lambda].sup.*], ([alpha]), [alpha]'), L([x.sup.c]([alpha]'), [[lambda].sup.c], ([alpha]), [alpha]) + L([x.sup.c]([alpha]'), [[lambda].sup.c], ([alpha]), [alpha]') = [L.sub.[alpha]]([x.sup.*]([alpha]'), [[lambda].sup.*]([alpha]), [[alpha].sub.5][[alpha] - [alpha]'] - [L.sub.[alpha]]([x.sup.c]([alpha]'), [[lambda].sup.c]([alpha]), [[alpha].sub.5][[alpha] - [alpha]'],

where [[alpha].sub.5] = [[delta].sub.5][alpha] + (1 - [[delta].sub.5])[alpha]' for some [[delta].sub.5] [member of] [0, 1]. Substituting this expression into Equation C2 yields

= [L.sub.[alpha]]([x.sup.*]([alpha]'), [[lambda].sup.*([alpha]), [[alpha].sub.5])[[alpha] - [alpha]'] - [L.sub.[alpha]]([x.sup.c]([alpha]'), [[lambda].sup.c]([alpha]), [[alpha].sub.5])[[alpha] - [alpha]'] [greater than or equal to] 0. (12)

Under the differentiability of [f.sub.[alpha]](x, [alpha]) and [g.sub.[alpha](x, [alpha])in x, it follows that [L.sub.[alpha]](x, [lambda], [alpha]) is differentiable in (x, [lambda]) on X x [R.sup.m.sub.+] for each [alpha] [member of] A. Applying the mean value theorem to [H.sub.6](x, [lambda]) = [L.sub.[alpha]]([x, [lambda], [[alpha].sub.5])[[alpha] - [alpha]']}, we obtain

[L.sub.[alpha]]([x.sup.*]([alpha]'), [[lambda].sup.*]([alpha]), [[alpha].sub.5]) [[alpha] - [alpha]'] - [L.sub.[alpha]]([x.sup.c]([alpha]'), [[lambda].sup.c]([alpha]), [[alpha].sub.5]) [[alpha] - [alpha]'] = [[[alpha] - [alpha]'].sup.T] [L.sub.[alpha]x]([x.sub.5], [[lambda].sub.5], [[alpha].sub.5]), [[alpha].sub.5])[[alpha] - [alpha]']

- L([x.sup.*]([alpha]'), [[lambda].sup.*], ([alpha]), [alpha]) - [L.sub.[alpha]]([x.sup.c]([alpha]'), [[lambda].sup.c], ([alpha]), [[alpha].sub.5])[alpha] - [alpha]'], = [L.sub.[alpha]]([[x.sup.*]([alpha]'), - [x.sup.c]([alpha]')] + [[[alpha] - [alpha]'].sup.T] [g.sub.[alpha]] [([x.sub.5], [[alpha].sup.5]).sup.T] [[lambda].sup.*([alpha]) - [[lambda].sup.c] ([alpha])],

where [x.sub.5] = [[theta].sub.5] [x.sup.*]([alpha]') + (1 - [[theta].sub.5])[x.sup.c]([alpha]') and [[lambda].sub.5] = [[theta].sub.5] [[lambda].sub.*]([alpha]) + (1 - [[theta].sub.5]) [[lambda].sup.c] ([alpha]) for some [[theta].sub.5] [member of] [0, 1]. Substituting this expression into Equation 12 proves Proposition 6(a). To prove Proposition 6(b), assume that [g.sub.[alpha]] is independent of [alpha] and [x.sup.*]([alpha]')= [x.sup.c]([alpha]'). Then, L(x, [lambda], [alpha]) = f(x, [alpha]) + [[lambda].sup.T][[g.sub.4](x) + [g.sub.5](x)[alpha]], [x.sub.5] = [x.sub.*]([alpha]') = [x.sub.c]([alpha]'), and either Equation C2 or C2' can be written as [[[alpha] - [[alpha]'].sup.T] [g.sub.5][([x.sup.*]([alpha]').sup.T] [[lambda].sup.*]([alpha]) - [[lambda].sup.c]([alpha])] [greater than or equal to] 0. QED.

Again, Proposition 6 applies under fairly general conditions (i.e., it holds "in the large" for any [alpha], [alpha]' [member of] A). We now consider the special case where we restrict [alpha]' [member of] A such that it satisfies [x.sup.*.sub.[alpha] = [x.sup.c]([alpha]') = [[Lambda].sup.c]([alpha]') as commonly assumed in LeChatelier analysis.

PROPOSITION 7. Assume that a saddle point holds in Equations 2a and 2b for all [alpha] [member of] A; that the sets A, X, and Y are convex; that [f.sub.[alpha]](x, [alpha]) and [g.sub.[alpha]](x, [alpha]) exist and are differentiable in x on X for each [alpha] [member of] A; and that [[lambda].sup.*]([alpha]) and [[lambda].sup.c]([alpha]) are differentiable functions on A. If [alpha]' [member of] A satisfies [x.sub.*]([alpha]') = [x.sup.c]([alpha]') [[lambda].sup.*]([alpha]') = [[lambda].sup.c]([alpha]'), then for any [alpha] [member of] A, Condition C2' can be expressed as

[[alpha] - [alpha]'].sup.T] [g.sub.[alpha]][([x.sub.5], [[alpha].sub.5]).sup.T][[[lambda].sup.*.sup.[alpha]]([[alpha].sub.6]) - [[lambda].sup.c.sub.[alpha]] ([[alpha].sub.6])][[alpha] - [alpha]'] [greater than or equal to] 0. (C2")

where [[alpha].sub.6] [[delta].sub.6][alpha] + (1 - [[delta].sub.6]) [alpha]' for some [[delta].sub.6] [member of] [0, 1].

PROOF. Given [[lambda].sup.*]([alpha]') = [[lambda].sup.c]([alpha]') applying the mean value theorem to [H.sub.4](a) = {[[[alpha] - [alpha]'].sup.T][g.sub.[alpha]]([x.sub.5], [[alpha].sub.5]) [[[lambda].sup.*](a) - [[lambda].sup.c](a)]}, we obtain

[[[alpha] - [alpha]'].sup.T] [g.sub.[alpha]]([x.sub.5], [[alpha].sub.5])[[[lambda].sup.*]([alpha]) - [[lambda].sup.*]([alpha]') + [[lambda].sup.c]([alpha]') - [[lambda].sup.c]([alpha])] = [[[alpha] - [alpha]'].sup.T][g.sub.[alpha]]([x.sub.5], [[alpha].sub.5]) [[lambda].sup.*.sub.[alpha]] ([alpha].sub.6]) - [[lambda].sup.c.sub.[alpha]]([alpha].sub.6])][[alpha] - [alpha]'],

where [[alpha].sub.6] = [[delta].sub.6][alpha] + (1 - [[delta].sub.6])[alpha]' for some [[delta].sub.6] [member of] [0, 1]. Given [x.sup.*]([alpha]') = [x.sup.c]([alpha]') = [x.sub.5] and [[lambda].sup.*]([alpha]') = [[lambda].sup.c]([alpha]'), substituting this expression into Equation C2' gives the desired result. QED.

Proposition 7 gives another version of a global LeChatelier result. It shows that Equation C2' generates the positive semidefiniteness restrictions in Equation C2" involving the difference between unrestricted and restricted effects of [alpha] on [[lambda].sup.*]([alpha]) and [[lambda].sup.c]([alpha]). Again, such restrictions are part of the local LeChatelier principle stated in Equation 5. However, the results stated in Proposition 7 are global: they hold for any discrete change from [alpha]' to [alpha] [member of] A. Combining Propositions 6 and 7, this indicates that the parallel shift in the lower bounds given in Equation C2 plays a role in establishing the global validity of local LeChatelier results related to the Lagrange multipliers [[lambda].sup.*]([alpha]) and [[lambda].sup.c]([alpha]). This will be further illustrated below.

Finally, we consider Condition C3. Recall that Condition C3 is the sum of Conditions C1 and C2. It follows that the implications of Condition C3 follow directly from Propositions 4-7. These implications are presented next under increasingly restrictive assumptions. Combining Propositions 4 and 6, we obtain Proposition 8.

PROPOSITION 8. Assume that a saddle point holds in Equations 2a and 2b for all [alpha] [member of] A; that the sets A, X, and Y are convex; and that [f.sub.[alpha]](x, [alpha]) and [g.sub.[alpha]](x, [alpha]) exist and are differentiable in x [member of] X for each [alpha] [member of] A. For any [alpha], [alpha]' [member of] A, consider

[[[alpha] - [alpha]'].sup.T] [L.sub.[alpha]x] ([x.sub.3], [[lambda].sub.3], [[alpha].sub.3]) [[x.sup.*]([alpha]) - [x.sup.c]([alpha])] + [[[alpha] - [alpha]'].sup.T] [g.sub.[alpha]][([x.sub.3], [[alpha].sub.3]).sup.T] [[[lambda].sup.*]([alpha]') - [[lambda].sup.c]([alpha]')] + [[[alpha] - [alpha]'].sup.T] [L.sub.[alpha]x] ([x.sub.5], [[lambda].sub.5], [[alpha].sub.5]) [[x.sup.*]([alpha]') - [x.sup.c]([alpha]')] + [[[alpha] - [alpha]'].sup.T] [g.sub.[alpha]] [([x.sub.5], [[alpha].sub.5]).sup.T] [[[lambda].sup.*]([alpha]) - [[lambda].sup.c]([alpha])] [greater than or equal to] 0. (C3')

Then, for any [alpha], [alpha]' [member of] A,

(a) Equation C3 implies Equation C3'.

(b) Equation C3' is a necessary and sufficient condition for Equation C3 if [L.sub.[alpha]x] is independent of (x, [alpha]), [g.sub.[alpha]] is independent of [alpha], [[lambda].sup.*]([alpha]') = [[lambda].sup.c]([alpha]'), and [x.sup.*]([alpha]') = [x.sup.c]([alpha]').

Again, Proposition 8 applies under fairly general conditions: it does not require that the functions f or g be differentiable in x, it allows for possible multiple solutions, and it holds "in the large" (i.e., for any [alpha], [alpha]' [member of] A). We now consider the special case where [alpha]' [member of] A satisfies [x.sup.*]([alpha]') = [x.sup.c]([alpha]') and [[lambda].sup.*]([alpha]') = [[lambda].sup.c]([alpha]') (as commonly assumed in LeChatelier analysis). Combining Propositions 5 and 7, we obtain Proposition 9.

Proposition 9. Assume that a saddle point holds in Equations 2a and 2b for all [alpha] [member of] A; that the sets A, X, and Y are convex; that [f.sub.[alpha]](x, [alpha]) and [g.sub.[alpha]](x, [alpha]) exist and are differentiable in x [member of] X for each [alpha] [member of] A; and that [x.sup.*]([alpha]), [x.sup.c]([alpha]), [[lambda].sup.*]([alpha]) and [[lambda].sup.c]([alpha]) are differentiable functions on A. If [alpha]' [member of] A satisfies [x.sup.*]([alpha]') = [x.sup.c]([alpha]') and [[lambda].sup.*]([alpha]') and [[lambda].sup.c]([alpha]'), then for any [alpha] [member of] A, Condition C3' can be written as

[[alpha] - [alpha]'].sup.T] [L.sub.[alpha]x] ([x.sub.3], [[lambda].sub.3], [[alpha].sub.3]) [[x.sup.*.sub.[alpha]]([alpha].sub.4]) - [x.sup.c.sub.[alpha]]([alpha].sub.4])] [[alpha] - [alpha]'] + [[[alpha] - [alpha]'].sup.T] [g.sub.[alpha]] [([x.sub.5], [[alpha].sub.5]).sup.T] [[[lambda].sup.*.sub.[alpha]]([alpha].sub.6]) - [[lambda].sup.c.sub.[alpha]]([[alpha].sub.6])][[alpha] - [alpha]'] [greater than or equal to] 0. (C3")

Proposition 9 shows that Equation C3' implies positive semidefiniteness restrictions involving the difference between unrestricted and restricted effects of [alpha] on [x.sup.*]([alpha]) and [x.sup.c]([alpha]) and [[lambda].sup.*]([alpha]) and [[lambda].sup.c]([alpha]). Such restrictions are in fact the local LeChatelier principle stated in Equation 5. However, the results stated in Proposition 9 are global because they hold for any discrete change from [alpha]' to [alpha] [member of] A. Combining Propositions 8 and 9 yields Proposition 10.

Proposition 10 (the main result). Interpret Equation C3" as the global generalization of the local LeChatelier principle (Eqn. 5). Then, for any [alpha] [member of] A and any [alpha]' [member of] A satisfying [x.sup.*]([alpha]') = [x.sup.c]([alpha]') and [[lambda].sup.*]([alpha]') = [[lambda].sup.c]([alpha]').

(a) Condition C3 implies that the local LeChatelier principle is globally valid.

(b) If [L.sub.[alpha]x] is independent of (x, [alpha]) and [g.sub.[alpha]] is independent of [alpha], then Condition C3 is necessary and sufficient for the global validity of the LeChatelier principle.

Proposition 10 establishes the close linkages between Condition C3 and the global validity of LeChatelier results. From Proposition 10(a), the parallel shift in the average of the upper and lower bounds given in Condition C3 implies that the local LeChatelier results hold globally. And from Proposition 10(b), this same condition is necessary and sufficient for the global validity of the LeChatelier principle when [L.sub.[alpha]x]. is independent of (x, [alpha]) and [g.sub.[alpha]] is independent of [alpha].

Finally, it can be easily verified that when [x.sup.*]([alpha]') = [x.sup.c][alpha]' and [[lambda].sup.*]([alpha]') = [[lambda].sup.c]([alpha]'), Condition C1 always holds if [L.sub.[alpha]x]. = 0, and Condition C2 always holds if [g.sub.[alpha]], = 0. Because Conditions C1 and C2 imply Condition C3, Proposition 10 generates Proposition 11.

Proposition 11. For any [alpha] [member of] A and [alpha]' [member of] A satisfying [x.sup.*]([alpha]') = [x.sup.c]([alpha]') and [[lambda].sup.*]([alpha]') = [[lambda].sup.c]([alpha]'), (a) If [g.sub.[alpha]] = 0, then,

* Condition C2 always holds.

* Condition C1 implies that the local LeChatelier result (Eqn. 5) is globally valid.

* If [L.sub.[alpha]x] is independent of (x, [alpha]), then Condition C1 is necessary and sufficient for the global validity of the local LeChatelier result (Eqn. 5).

(b) If [L.sub.[alpha]x] = 0, then,

* Condition C1 always holds.

* Condition C2 implies that the local LeChatelier result (Eqn. 5) is globally valid.

* If [g.sub.[alpha]] is independent of [alpha], then Condition C2 is necessary and sufficient for the global validity of the local LeChatelier result (Eqn. 5).

Propositions 10 and 11 summarize the relationships between the parallel shift in bounds stated in Conditions C1, C2, and C3 and the global validity of the local LeChatelier principle (valid in the neighborhood of [alpha]' [member of] A) for any discrete change from [alpha]' to [alpha] [member of] A. The usefulness of these relationships is illustrated next in selected examples.

5. Applications

Properties of Lagrange Multipliers

Consider the case where f = f(x) and g(x, [alpha]) = [alpha] + h(x) [greater than or equal to] 0 and where [alpha] [member of] A [subset] [R.sup.m]. Then, any change in the vector [alpha] affects how binding the m constraints are. The associated Lagrangean is L(x, [alpha], [lambda]) = f(x) + [[lambda].sup.T] [[alpha] + h(x)], [lambda] [member of] [R.sup.m.sub.+].

Applying Propositions 3 and 3', we obtain: [[[lambda].sup.*]([alpha]).sup.T][[alpha] - [alpha]'] [less than or equal to] [f.sup.*]([alpha]) - [f.sup.*]([alpha]') [less than or equal to] [[[lambda].sup.*]([alpha]').sup.T][[alpha]- [alpha]'] and [[[lambda].sup.c]([alpha]).sup.T][[alpha] - [alpha]'] [less than or equal to] [f.sup.c]([alpha]) - [f.sup.c]([alpha]') [less than or equal to] [[[lambda].sup.c]([alpha]').sup.T][[alpha] - [alpha]'] for any [alpha], [alpha]' [member of] A. These results are quite general: they do not require differentiability assumptions or unique solutions, and they are global. They reflect the classical result that Lagrange multipliers can be interpreted as "marginal values" or "shadow values" of the constraints. They also imply [[lambda].sup.*]([alpha]) - [[[lambda].sup.*]([alpha]').sup.T] [[alpha] - [alpha]'] [less than or equal to] 0 and [[lambda].sup.c]([alpha])- [[[lambda].sup.c]([alpha]').sup.T] [[alpha] - [alpha]'] [less than or equal to] 0. This means that the mappings [[lambda].sup.*]([alpha]) and [[lambda].sup.c]([alpha]) are nonincreasing in [alpha]; relaxing the constraints (through an increase in [alpha]) tends to reduce their shadow values.

Our interest here focuses on the relationship between [[lambda].sup.*]([alpha]) and [[lambda].sup.c]([alpha]). We want to analyze how restricting the feasible set from X to Y can affect the shadow values of the constraints g(x, [alpha]) [greater than or equal to] 0 when [alpha]' satisfies [[lambda].sup.*]([alpha]') = [[lambda].sup.c]([alpha]'). Under appropriate differentiability assumptions, Equation 5 gives the well-known local LeChatelier result

[[[alpha] - [alpha]'].sup.T] [[lambda].sup.*.sub.[alpha]]([alpha].sub.2]) - [[lambda].sup.c.sub.[alpha]]([alpha].sub.2])][[alpha] - [alpha]'] [greater than or equal to] 0 (13)

for a in the neighborhood of [alpha]' [member of] A. This shows that locally (i.e., in the neighborhood of [alpha]') [[lambda].sup.*.sub.[alpha]]([alpha].sub.2]) tends to exceed [[lambda].sup.c.sub.[alpha]]([alpha].sub.2]) by a positive semidefinite matrix. Because the mappings [[lambda].sup.*]([alpha]) and [[lambda].sup.c]([alpha]) are each nonincreasing in [alpha], this means that restricting the feasible set from X to Y tends to increase the negative effect of [alpha] on the shadow value of the constraints g(x, [alpha]) [greater than or equal to] 0. In other words, restricting choices makes the shadow value of the constraints more sensitive to changing scarcity levels.

Under what condition is this local result globally valid? Note that the Lagrangean L(x, [alpha], [lambda]) = f(x) + [[lambda].sup.T][[alpha] + h(x)] implies that [L.sub.[alpha]x] = 0 and that [g.sub.[alpha]] is independent of [alpha]. From Proposition 11(b), it follows that Condition C1 is always satisfied and that Condition C2 is necessary and sufficient for the global validity of the local LeChatelier principle. When [x.sup.*]([alpha]') = [x.sup.c]([alpha]') and [[lambda].sup.*]([alpha]') = [[lambda].sup.c]([alpha]'), Conditions C2 and C2' are equivalent (from Proposition 6[b]), and Condition C2' can be written as [[[lambda].sup.*]([alpha]) - [[lambda].sup.*]([alpha]')].sup.T][[[alpha] - [alpha]'].sup.3] [greater than or equal to] [[[lambda].sup.c]([alpha]) - [[lamda].sup.c]([alpha]')].sup.T][[alpha] - [alpha]'] for any [alpha] [member of] A. Under differentiability assumptions (as stated in Proposition 7), this can be written as Condition C2" [[[alpha] - [alpha]'].sup.T][[[lambda].sup.*.sub.[alpha]]([[alpha].sub.6]) - [[lambda].sup.c.sub.[alpha]]([[alpha].sub.6])] [[alpha] - [alpha]'] [greater than or equal to] 0 for any [alpha] [member of] A.

But this is just a restatement of the local LeChatelier result (Eqn. 13), except that it holds "in the large" for any discrete change in [alpha] (where [alpha] [member of] A is not necessarily in the neighborhood of [alpha]'). Thus the parallel shift in the lower bounds given in Condition C2 is a necessary and sufficient condition for the local LeChatelier principle (Eqn. 13) to hold globally. This gives useful information on the properties of the Lagrange multipliers (measuring the shadow value of the constraints).

Production Theory

Consider a competitive firm involved in the choice of a (n x 1) netput vector x [member of] X where outputs are defined to be positive and inputs to be negative. Denote the prices for x by [alpha] [member of] A [subset] [R.sup.n], where [alpha] is a (n x 1) vector of parameters. In this context, firm profit is [[alpha].sup.T]x. Let {x:g(x) [greater than or equal to] 0, x [member of] X} denote the feasible set for the firm. Then, the firm decisions are represented by the profit maximization problem (5)

[f.sup.*]([alpha]) = [max over x] {[[alpha].sup.T]x:g(x) [greater than or equal to] 0, x [member of] X},

which has for solution the supply-demand choices [x.sup.*]([alpha]) [member of] [X.sup.*]([alpha]). The indirect profit function is [f.sup.*]([alpha]) = [[alpha].sup.T] [x.sup.*]([alpha]). The associated Lagrangean is L(x, [alpha], [lambda]) = [[alpha].sup.T] x + [[lambda].sup.T]g(x) [lambda] [member of] [R.sup.m.sub.+].

Applying Propositions 3 and 3', we obtain the classical results [x.sup.*][([alpha').sup.T][[alpha] - [alpha]'] [less than or equal to] (f.sup.*)([alpha]) - [f.sup.*)([alpha]') [less than or equal to] [x.sup.*][([alpha]).sup.T][[alpha] - [alpha]'] and [x.sup.c]([[alpha]').sup.T][[alpha] - [alpha]'] [less than or equal to] [f.sup.c]([alpha]) - [f.sup.c]([alpha]') [less than or equal to] [x.sup.c] [([alpha]).sup.T][[alpha] - [alpha]'] for any [alpha], [alpha]' [member of] A. This gives a generalized Hotelling's lemma, stating that profit maximizing netput choices are the marginal values of the indirect profit function with respect to [alpha]. It provides the basis for nonparametric production analysis (see Hanoch and Rothschild 1972; Varian 1982). It also implies [[x.sup.*]([alpha]) - [[x.sup.*]([alpha]')].sup.T] [[alpha] - [alpha]'] [greater than or equal to] 0 and [[x.sup.c]([alpha]) - [x.sup.c][([alpha]')].sup.T][[alpha] - [alpha]'] [greater than or equal to] 0. This means that the mappings [x.sup.*]([alpha]) and [x.sup.c]([alpha]) are each nondecreasing in [alpha] (i.e., that profit maximizing netput choice mappings tend to be upward sloping with respect to the corresponding prices [alpha]). These well-known results are quite general: they do not require differentiability assumptions or unique solutions, and they are global.

Our interest here focuses on the relationship between [x.sup.*]([alpha]) and [x.sup.c]([alpha]). We want to analyze how restricting the feasible set from X to Y can affect profit maximizing choices when [alpha]' is chosen such that [x.sup.*]([alpha]') = [x.sup.c]([alpha]'). Under appropriate differentiability assumptions, Equation 5 gives the local LeChatelier result

[[[alpha] - [alpha]'].sup.T][[x.sup.*.sub.[alpha]]([[alpha].sub.2]) - [x.sup.c.sub.[alpha]] ([[alpha].sub.2])[[alpha] - [alpha]'] [greater than or equal to] 0 (14)

for any [alpha] in the neighborhood of [alpha]'[??]A. This shows that locally (i.e., in the neighborhood of [alpha]') [x.sup.*.sub.[alpha]]([[alpha].sub.2]) tends to exceed [x.sup.c.sub.[alpha]]([[alpha].sub.2]) by a positive semidefinite matrix. Since the mappings [x.sup.*]([alpha]) and [x.sup.c]([alpha]) are each nondecreasing in [alpha], this means that restricting the feasible set from X to Y tends to decrease the positive effect of [alpha] on profit maximizing choices. In other words, restricting choices makes profit maximizing choices less responsive to changing market prices.

Under what condition is this result globally valid? Note that the Lagrangean L(x, [lambda], [alpha]) = [[alpha].sup.T]x + [[lambda].sup.T]g(x) implies that [L.sub.[alpha]x] is independent of ([alpha], x) and g[alpha] = 0. From Proposition 11(a), it follows that Condition C2 always holds and that Condition C1 is a necessary and sufficient condition for the global validity of the local LeChatelier result. When [x.sup.*]([alpha]') = [x.sup.c]([alpha]') and [[lambda].sup.*]([alpha]') = [[lambda].sup.c]]([alpha]'), Conditions C1 and C1' are equivalent (from Proposition 4[b]), and Condition C1' can be written as [[[alpha] - [alpha]'].sup.T][[x.sup.*]([alpha]) - [x.sup.*]([alpha]')] [greater than or equal to] [[[alpha] - [alpha]'].sup.T][[x.sup.c]([alpha]) - [x.sup.c]([alpha]')]. Under differentiability assumptions (as stated in Proposition 5), this can be expressed as Condition C1":

[[[alpha] - [alpha]'].sup.T][[x.sup.*.sub.[alpha]]([[alpha].sub.4]) - [[x.sup.c].sub.[alpha]]([[alpha].sub.4])][[alpha] - [alpha]'] [greater than or equal to] 0 for any [alpha], [alpha]' [member of] A.

But this is a restatement of the local LeChatelier result (Eqn. 14), except that it holds "in the large" (i.e., for any discrete change in [alpha], where [alpha] [member of] A is not necessarily in the neighborhood of [alpha]'). Thus the parallel shift in the upper bounds given in Condition C1 is a necessary and sufficient condition for the local LeChatelier principle (Eqn. 14) to hold globally. This gives useful information on the properties of the decision rules [x.sup.*]([alpha]) - [x.sup.c]([alpha])].

Consumption Theory

Consider a household with preferences represented by the utility function f(x), x [member of] X [subset] [R.sup.n] being a (n x 1) vector of consumer goods. With m = 1, the household is facing the budget constraint g(x, [alpha]) = a - [p.sup.T]x [greater than or equal to] 0 where [alpha] = (a,p) [member of] A, a [member of] [R.sub.++] is household income and where p [member of] [R.sup.m.sub.+] is a (n x 1) price vector for x. Under the feasible set X, let consumption decisions be made in a way consistent with the utility maximization problem 

[f.sup.*]([alpha]) = f([x.sup.*]([alpha])) = [max over x] {f(x):a - [p.sup.T]x [greater than or equal to] 0, x [member of] X},

where [alpha] = (a, p) [member of] A, [x.sup.*]([alpha]) is the utility maximizing (Marshallian) household choice, and [f.sub.*]([alpha]) is the indirect utility function. The associated Lagrangean is L(x, [alpha], [lambda]) = f(x) + [lambda][a - [p.sup.T]x], [lambda] [member of] [R.sub.+].

Applying Propositions 3 and 3', we obtain [[lambda].sup.*]([alpha])[a - a' - [[p - p'].sup.T][x.sup.*]([alpha]')] [less than or equal to] [f.sup.*]([alpha]) - [f.sup.*]([alpha]') [less than or equal to] [[lambda].sup.*]([alpha]')[a - a' - [[p - p'].sup.T][x.sup.*]([alpha])] (15a)

and

[[lambda].sup.c]([alpha])[a - a' - [[p - p'].sup.T][x.sup.c]([alpha]')] [less than or equal to] [f.sup.c]([alpha] - [f.sup.c]([alpha]') [less than or equal to] [[lambda].sup.c]([alpha]')[a - a' - [[p - p'].sup.T][x.sup.c]([alpha])] (15b)

for any [alpha], [alpha]' [member of] A. Each expression is a generalized Roy's identity, indicating that the marginal value of the indirect utility function with respect to price is the marginal utility of income ([lambda]) multiplied by the Marshallian demand. Under nonsatiation (where the marginal utility of income is strictly positive), this provides the basis for nonparametric consumption analysis (see Afriat 1967; Varian 1984). These well-known results are quite general: they do not require differentiability assumptions or unique solutions, and they are global.

Our interest here focuses on the relationship between [x.sup.*]([alpha]) and [x.sup.c]([alpha]). We want to analyze how restricting the feasible set from X to Y affects Marshallian choices when [alpha]' satisfies [x.sup.*]([alpha]') = [x.sup.c]([alpha]') and [lambda].sup.*]([alpha]') = [lambda].sup.c]([alpha]'). Under appropriate differentiability assumptions, Equation 5 gives the local LeChatelier result:

- [[lambda].sup.*]([alpha]') [[p - p'].sup.T] [[x.sup.*.sub.[alpha]]([[alpha].sub.2]) - [x.sup.c.sub.[alpha]]([alpha].sub.2][[alpha] - [alpha]'] + [a - a' - [[p - p'].sup.T][x.sup.*] ([alpha]')][[lambda].sup.*.sub.[alpha]]([[alpha].sub.2] - [[lambda].sup.c.sub.[alpha]]([[alpha].sub.2)] [[alpha] - [alpha]'] [greater than or equal to] 0 (16)

for [alpha] = (a, p) in the neighborhood of [alpha]'[member of] A. Note that this imposes joint restrictions on Marshallian behavior [[x.sup.*.sub.[alpha]]([alpha]) - [[x.sup.c.sub.[alpha]]([alpha])] and on [[lambda].sup.*.sub.[alpha]]([alpha]) - [[lambda].sup.c.sub.[alpha]]([alpha]).

Next we examine the global validity of the local LeChatelier principle (Eqn. 16). Note that the Lagrangean L(x, [alpha], [lambda]) = f(x) + [lambda][a - [p.sup.T]x] satisfies [L.sub.[alpha]x] = [[0 over [lambda][l.sub.n]]] (which is independent of ([alpha], x) and g[alpha] = [1, - [x.sup.T], which is independent of [alpha]). It follows from Proposition 10(b) that Condition C3 is necessary and sufficient for the global validity of the local LeChatelier result (Eqn. 16). When [x.sup.*]([alpha]') = [x.sup.c]([alpha]') and [[lambda].sup.*]([alpha]') = [lambda].sup.c]([alpha]'), Conditions C3 and C3' are equivalent (from Proposition 8[b]), and under differentiability assumptions (as stated in Proposition 9), Condition C3 can be written as Condition C3":

- [[lambda].sup.*]([alpha]') [[p - p'].sup.T] [[x.sup.*.sub.[alpha]]([[alpha].sub.4]) - [x.sup.c.sub.[alpha]]([alpha].sub.4])] [[alpha] - [alpha]'] + [a - a' - [(p - p').sup.T][x.sup.*] ([alpha]')] [[lambda].sub.[alpha].sup.*] ([[alpha].sub.6]) - [[lambda].sup.c.sub.[alpha]]([[alpha].sub.6])] [[alpha] - [alpha]'] [greater than or equal to] 0. (17)

This is just a restatement of the local LeChatelier result (Eqn. 16), except that it holds "in the large" (i.e., for any discrete change in [alpha], where [alpha] is not necessarily in the neighborhood of [alpha]'). Thus, in this case, the parallel shift in the average bounds given in Condition C3 is a necessary and sufficient condition for the local LeChatelier principle (Eqn. 16) to hold globally.

Unfortunately, Equations 16 or 17 involve both [[x.sup.*.sub.[alpha]]([alpha]) - [x.sup.c.sub.[alpha]]([alpha)] and [[lambda].sup.*.sub.[alpha]]([alpha]) - [[lambda].sup.c.sub.[alpha]]([alpha])]. As such, these LeChatelier results appear of limited value in economic analysis. Indeed, it would be more useful to be able to assess separately the terms [[x.sup.*.sub.[alpha]]([alpha]) - [x.sup.c.sub.[alpha]]([alpha])] and [[lambda].sup.*.sub.[alpha]]([alpha]) - [[lambda].sup.c.sub.[alpha]]([alpha])]. This can be done using Conditions C1 and C2. First, consider the implications of Condition C1 for Marshallian behavior. When [[lambda].sup.c]([alpha]') = [[lambda].sup.*]([alpha]') > 0 (under nonsatiation) and [x.sup.c]([alpha]') = [x.sup.*]([alpha]'), Conditions C1 and C1' are equivalent (from Proposition 4[b] and Condition C1') and can be written as

[[p - p'].sup.T][[x.sup.*]([alpha]) - (x.sup.*]([alpha]')] [less than or equal to] [[p - p'].sup.T][[x.sup.c]([alpha]) - [[x.sup.c]([alpha]')]. (18a)

Under differentiability assumptions (as stated in Proposition 5), this can be expressed as Condition C1":

[[p - p'].sup.T][[x.sup.*.sub.[alpha]]([[alpha].sub.4]) - [x.sup.c.sub.[alpha]]([alpha].sub.4])][[alpha] - [alpha]'] [less than or equal to 0 (18b)

for any [alpha] = (a, p) [member of] A. In the case of a discrete change in [alpha], this means that, under Condition C1, Equation 18a or 18b give a version of the LeChatelier principle applying globally to Marshallian effects [[x.sup.*]([alpha]) - [x.sup.c]([alpha])]. This appears to be a new result. When a = a', this implies as a special case [[p - p'].sup.T][[x.sup.*.sub.p]([[alpha].sub.4]) - [x.sup.c.sub.p]([[alpha].sub.4])][p - p'] [less than or equal to] 0 for any p, p'. It means that under Condition C1, restricting the feasible set from X to Y tends to reduce the magnitude of Marshallian price adjustments. This result holds "in the large" for any discrete change in p (where p is not necessarily in the neighborhood of p'). It shows that the parallel shift in the upper bounds given in Condition C1 implies the global validity of a version of the LeChatelier principle related to Marshallian effects [[x.sup.*]([alpha]) - [x.sup.c]([alpha])].

Second, consider Condition C2. When [x.sup.*]([alpha]') = [x.sup.c]([alpha]') and [[lambda.sup.*]([alpha]') = [[lambda].sup.c]([alpha]'), Conditions C2 and C2' are equivalent (from Proposition 6[b]), and Condition C2' can be expressed as

[a - a' - [[p - p'].sup.T][x.sup.*]([alpha]')]{[[lambda].sup.*]([alpha]) - [[lambda].sup.*]([alpha]')] - [[[lambda].sup.c]([alpha]) - [[lambda].sup.c]([alpha]')]} [greater than or less than] 0. (19a)

Under differentiability assumptions (as stated in Proposition 7), this can be written as Condition C2":

[a - a' - [[p - p'].sup.T][x.sup.*]([alpha]')][[lambda].sup.*.sub.[alpha]]([[alpha].sub.6]) - [[lambda].sup.c.sub.[alpha]] ([[alpha].sub.6])][[alpha] - [alpha]'] [greater than or equal to] 0 (19b)

for any [alpha] = (a, p) [member of] A. In the case of a discrete change in a, this means that under Condition C2, Equation 19a or 19b gives a version of the LeChatelier principle that applies globally to the marginal utilities [[[lambda].sup.*]([alpha]) - [[lambda].sup.c.]([alpha])]. The term [a - a' - [[p - p'].sup.T][x.sup.*]([alpha]')] can be interpreted as measuring the change in real income between situations [alpha]' and [alpha]. It follows from Equation 19a that under Condition C2 any change from [alpha]' to [alpha] generating an increase (decrease) in real income implies that [[[lambda].sup.c]([alpha]) - [[lambda].sup.c]([alpha]')] [less than or equal to] ([greater than or equal to])[[[lambda].sup.*]([alpha]) - [[lambda].sup.c]([alpha]')]. Intuitively, this states that the marginal utility of income (as measured by [lambda]) tends to increase more (or to decrease less) in the unconstrained choice (compared to the constrained choice). Again, this appears to be a new result. It is another global version of the LeChatelier principle linking the parallel shift in the lower bound given in Condition C2 to the properties of the Lagrange multiplier as it adjusts to changes in the feasible set.

We would like to thank two anonymous reviewers for their useful comments on an earlier draft of the paper.

Received September 2003; accepted June 2004.

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(1) The regularity conditions involve appropriate convexity conditions and a constraint qualification (Takayama 1985, p. 75). The constraint qualification is often taken to be Slater's condition, stating that there must exist a feasible point at which all constraints are not binding.

(2) The superscript "T" is used to denote "transpose": [[lambda].sup.T] is the transpose of [lambda].

(3) For a function h mapping U [subset] [R.sup.r] into V [subset] [R.sup.s], q = [lim.sub.y[right arrow]p](y) denotes the limit of the function h at p, with q [member of] V. By definition, it satisfies the following property: for every [epsilon] > 0, there exists a [delta] > 0 such that [parallel]h(y)--q[parallel] < [epsilon] for all points y [member of] U for which 0 < [parallel]y - P[parallel] < [delta], where [parallel].[parallel] denotes the Euclidian distance.

(4) Note that although the Anderson and Takayama bounds coincide with those presented in Proposition 3 when L(X, [lambda], [alpha]) is linear in a, our results differ from the Anderson and Takayama analysis whenever L(x, [lambda], [alpha]) is nonlinear in [alpha].

(5) Note that the same optimization structure arises in the context of Hicksian consumer behavior. To see that, let m = 1 and g(x) = v(x) - u where x is (n x 1) vector of consumption goods, v(x) is the consumer utility function, and u is a given utility level. Then, Hicksian expenditure minimization can be written as

[min.sub.x] {[[alpha].sup.T]x:g(x) [greater than or equal to] 0, x [member of] X} = [-max.sub.x] {[-[alpha].sup.T] x:g(x) [greater than or equal to] 0, x [member of] X}

where [alpha] is a (n x 1) vector of market prices for x. It follows that, except for a change in sign, all the results presented in this section can be easily adapted to the analysis of Hicksian consumer behavior. The case of Marshallian consumer behaviour is discussed below.

Jean-Paul Chavas, Department of Agricultural and Applied Economics, University of Wisconsin, Taylor Hall, Madison, WI 53706, USA; E-mail [e-mail].