Le Chatelier Principle

Abstract

In the field of economics, the Le Chatelier principle refers to the differences in the responses of decision variables to changes in parameters when additional constraints are imposed on the system. In the context of demand theory, for example, the Le Chatelier principle is the ‘second law of demand’, that demand curves are more elastic in the long run than in the short run. In many models, additional constraints reduce the absolute response of a decision variable to a change in a parameter.

Henri Louis Le Chatelier was a French chemist born in Paris in 1850. In 1884, he offered the following observation:

Any system in stable chemical equilibrium, subjected to the influence of an external cause which tends to change either its temperature or its condensation (pressure, concentration, number of molecules in unit volume), either as a whole or in some of its parts, can only undergo such internal modifications as would, if produced alone, bring about a change of temperature or of condensation of opposite sign to that resulting from the external cause. (Oliver and Kurtz 1992)

Later writers produced a more heuristic simplification: ‘If the external conditions … are altered, the equilibrium … will tend to move in such a direction so as to oppose the change in external conditions’ (Fermi 1937, p. 111, cited in Samuelson 1949, p. 639), or even more simply: if a stress is applied to a system at equilibrium, then the system readjusts, if possible, to reduce the stress. The Le Chatelier principle is a firmly established proposition in classical thermodynamics, though its verbal statement is somewhat vague in operational content. In the field of economics, the law of demand, which states that as a price increases, ceteris paribus, consumers will decrease their consumption of that good, is in fact a direct application of the Le Chatelier principle. Consumers (or firms) mitigate the adverse effects of the price increase by utilizing less of that good or input.

Following up a suggestion by his professor and mentor E.B. Wilson at Harvard, Paul Samuelson showed that this principle was a simple application of maximizing behaviour (see especially Samuelson 1949, 1960a, 1974.) Moreover, physicists and economists – among economists, principally Samuelson – came to realize that the Le Chatelier principle was being used to describe two separate phenomena. The first referred to first-order changes in response to a change in a parameter value, such as a price. The second, which is what the Le Chatelier principle is now generally understood to mean, refers to differences in the changes as additional constraints are imposed on the system.

The General Case

First-Order Effects

The most general comparative statics model with explicit maximizing behaviour is maximize y = f(x, α) subject to g(x, α) = 0, where x(x₁, … , x_n) is a vector of decision variables, α = (α₁, … , α_m) is a vector of parameters (though for simplicity, we treat α as a scalar in the discussion below), and g(.) represents one or more constraints. Models at this level of generality, however, imply no refutable implications and are hence largely uninteresting. In particular, there are never refutable implications for parameters that enter the constraint (see, for example, Silberberg and Suen 2000). Thus we restrict the analysis to models of the form

$$ maximize\kern0.5em y=f\left(x,\alpha \right) $$

(1)

$$ subject to\ g(x)=0 $$

(2)

Since it has no effect on the analysis to follow, we consider the case of only one external constraint. Also, parameters β, which enter the constraint but which do not enter the objective function, also do not affect the analysis, and hence we suppress them in the notation. The Lagrangian for this model is L = f(x, α) + λg(x) producing the necessary first-order conditions (NFOC)

$$ {L}_i={f}_i\left(x,\alpha \right)+\uplambda {g}_i(x)=0\kern1em i=1,\dots, n $$

(3)

$$ {L}_{\uplambda}=g(x)=0 $$

(4)

Assuming the sufficient second-order conditions hold, we can in principle ‘solve’ for the n+1 explicit choice functions x = x^∗(α) and λ^∗(α). Of course, since these choice functions are the result of solving the NFOC simultaneously, each individual xi is a function of all the parameters, not just the ones which appear in Li.

Substituting the ${x}_i^{\ast }$’s into the objective function yields the indirect objective function ϕ(α) = f(x^∗(α), α), the maximum value of f for given α, subject to the constraint. Since ϕ(α) is by definition a maximum value, ϕ(α) ≥ f(x, α), but ϕ(α) = f(x, α) when x = x^∗. Thus the function F(x, α) = f(x, α) − ϕ(α) has a (constrained) maximum of zero, with respect to both x and α. Thus we consider the primal-dual model

$$ maximize\kern0.5em F\left(x,\alpha \right)=f\left(x,\alpha \right)-\phi \left(\alpha \right) $$

(5)

$$ subject\kern0.5em to\kern0.5em g(x)=0 $$

(6)

where the maximization runs over x and also α. (In the latter instance, we ask, for given x_i’s, what values of the parameters would make these x_i’s the maximizing values?) The Lagrangian for this model is

$$ L=f\left(x,\alpha \right)-\phi \left(\alpha \right)+\uplambda g(x) $$

(7)

The first-order conditions with respect to x are the same as in the original model. With respect to α, the NFOC yield the famous ‘envelope theorem’

$$ {L}_{\alpha }={f}_{\alpha }-{\phi}_{\alpha }=0 $$

(8)

When α enters the constraint also, we get the envelope theorem in its most general form,

$$ {\phi}_{\alpha }={L}_{\alpha }={f}_{\alpha }+\uplambda {g}_{\alpha } $$

(8a)

Importantly, however, since we have restricted the model so that the parameters α do not enter the constraint, the primal-dual model is an unconstrained maximization in α. Hence in the α dimensions, the second-order conditions are simply

$$ {F}_{\alpha \alpha}={f}_{\alpha \alpha}-{\phi}_{\alpha \alpha}\le 0 $$

(9)

This inequality says that in the α dimensions, f is relatively more concave than ϕ. This is the fundamental geometrical property that underlies all comparative statics relationships and also the ‘second-order’ Le Chatelier relationships.

The NFOC (8) are identities when x = x^∗. That is,

$$ {\phi}_{\alpha}\left(\alpha \right)\equiv {f}_{\alpha}\left({x}^{\ast}\left(\alpha \right),\alpha \right) $$

(10)

Differentiating with respect to α,

$$ {\phi}_{\alpha \alpha}\equiv \sum \limits_1^n{f}_{\alpha i}\frac{\partial {x}_i^{\ast }}{\partial \alpha }+{f}_{\alpha \alpha} $$

(11)

Rearranging terms, using (9) and invariance to the order of differentiation,

$$ {\phi}_{\alpha \alpha}-{f}_{\alpha \alpha}\equiv \sum \limits_1^n{f}_{i\alpha}\frac{\partial {x}_i^{\ast }}{\partial \alpha}\ge 0 $$

(12)

This is the fundamental relation of comparative statics. From it, we can derive Samuelson’s famous ‘conjugate pairs’ theorem, namely, that refutable implications occur in maximization models when and only when a parameter enters one and only one first-order condition. For in that case, where say α enters only Li = 0 ,  f_jα ≡ 0 , j ≠ i, and so (12) reduces to one term:

$$ {f}_{i\alpha}\;\frac{\partial {x}_i^{\ast }}{\partial \alpha}\ge 0 $$

(13)

In this case we can say that the response of x_i is in the same direction as the disturbance to the equilibrium (or, in the case of minimization models, in the opposite direction). These relationships constitute the ‘first-order’ Le Chatelier effects. Note that these results are identical to those in models with no constraints at all, or with multiple constraints, as long as those constraints do not contain the parameter that is changing.

Second-Order Effects

Suppose now the NFOC hold at the parameter value α⁰ and consider now the imposition of an additional constraint, h(x) = 0, with the important restriction that this constraint does not change the original equilibrium, for example, a constraint holding some input fixed at the previous profit maximizing level. Then the new NFOC are solved for new explicit choice functions, ${x}_i={x}_i^s\left(\alpha \right)$, where the superscript ‘s’ stands for ‘short run’. Substituting these short run choice functions into the objective function produces a new indirect objective function, ψ(α). Since the new constraint did not disturb the equilibrium, ψ(α⁰) = ϕ(α⁰) at that point. However, since the objective function is now more constrained, for α ≠ α⁰ , ψ(α) ≤ ϕ(α). Thus the function G(α) = ψ(α) − ϕ(α) has an unconstrained maximum (of zero) at α = α⁰. The NFOC are

$$ {G}_{\alpha}\left(\upalpha \right)={\psi}_{\alpha}\left(\alpha \right)\hbox{--} {\phi}_{\alpha}\left(\alpha \right)=0 $$

(14)

We note that ψ_α(α) = ϕ_α(α) = f_α using the same analysis leading to Eq. 8, since α appears in neither constraint. The second-order conditions are

$$ {G}_{\alpha \alpha}\left(\alpha \right)={\psi}_{\alpha \alpha}\left(\alpha \right)-{\phi}_{\alpha \alpha}\left(\alpha \right)\le 0 $$

(15)

That is, the more constrained indirect objective function ψ(α) is tangent to ϕ (α) at α = α⁰, but it is relatively more concave, or less convex. Using ψ_α (α) ≡ ϕ_α(α) ≡ f_α expressed as identities, and proceeding as in Eqs. (10) through (12), inequality (15) yields the general second-order Le Chatelier effects:

$$ \sum \limits_1^n{f}_{i\alpha}\left(\frac{\partial {x}_i^{\ast }}{\partial x}-\frac{\partial {x}_i^s}{\partial x}\right)\ge 0 $$

(16)

In the empirically important case where α enters only the ith first-order condition, this summation reduces to one term, producing

$$ {f}_{i\alpha}\frac{\partial {x}_i^{\ast }}{\partial \alpha}\ge {f}_{i\alpha}\;\frac{\partial {x}_i^s}{\partial \alpha } $$

(17)

Thus $\partial {x}_i^{\ast }/\partial \alpha \ge \partial {x}_i^s/\partial \alpha \ge 0$ when f_iα > 0, and $\partial {x}_i^{\ast }/\partial \alpha \le \partial {x}_i^s/\partial \alpha \le 0$ when f_iα < 0. In either case, $\left|\partial {x}_i^{\ast }/\partial \alpha \right|\ge \left|\partial {x}_i^s/\partial \alpha \right|$.

Examples

Profit Maximization

Consider the profit-maximization model maximize π = f(x, w, p) = pθ(x₁…, x_n) – ∑ w_ix_i. Each parameter w_i enters only the ith NFOC, and ${f}_{x_i{w}_i}=-1$, so that (13) yields the negative slope property ∂x_i/∂w_i ≤ 0.

Moreover, (17) yields, in addition, for any additional constraint (not involving w_i) imposed on the initial equilibrium,

$$ \frac{\partial {x}_i^{\ast }}{\partial {w}_i}\le \frac{\partial {x}_i^s}{\partial {w}_i}\le 0 $$

(18)

The ‘long-run’ factor demand functions are more elastic than any short-run factor demands defined as above.

In the case where the additional constraint is simply ${x}_n={x}_n^0$, an analysis based on ‘conditional demands’ (Pollak 1969) is available. If we substitute this constraint directly into the objective function, the ‘short-run’ demand functions are ${x}_i={x}_i^s\left({w}_1,\dots {w}_{n-1},\kern0.5em p,\kern0.5em {x}_n^0\right)$. These functions are related to the long-run demands by the identity

$$ {\displaystyle \begin{array}{l}{x}_i^{\ast}\left({w}_1,\dots, {w}_n,\kern0.5em p\right)\hfill \\ {}\times \equiv {x}_i^s\left({w}_1,\dots {w}_{n-1},\kern0.5em p,\kern0.5em {x}_n^{\ast}\left({w}_1,\dots, {w}_n,\kern0.5em p\right)\right)\hfill \end{array}} $$

(19)

Differentiating both sides of this identity with respect to w_i and w_n,

$$ \frac{\partial {x}_i^{\ast }}{\partial {w}_i}\equiv \frac{\partial {x}_i^s}{\partial {w}_i}+\frac{\partial {x}_i^s}{\partial {x}_n^0}\frac{\partial {x}_n^{\ast }}{\partial {w}_i} $$

(20)

$$ \frac{\partial {x}_i^{\ast }}{\partial {w}_n}\equiv \frac{\partial {x}_i^s}{\partial {x}_n^0}\frac{\partial {x}_n^{\ast }}{\partial {w}_n} $$

(21)

Substituting (21) into (20) and using a well-known reciprocity condition yields

$$ \frac{\partial {x}_i^{\ast }}{\partial {w}_i}\equiv \frac{\partial {x}_i^s}{\partial {w}_i}+\frac{{\left(\partial {x}_i^{\ast }/\partial {w}_n\right)}^2}{\partial {x}_n^{\ast }/\partial {w}_n} $$

(22)

Since the last term in (22) is negative, we get the Le Chatelier result (18).

Cost (Expenditure) Minimization

The cost functions in production theory are derived from the model, minimize C = ∑ w_ix_i subject to f(x₁, … , x_n) = y, where y is now a parameter, that is, it is an arbitrary fixed level of output. This model is directly related to the profit maximization model. Write the profit maximization model as maximize py – ∑ w_ix_i subject to f(x₁, … , x_n) = y. When output y is a variable, this model is the profit- maximization model. If y is parametric, it is the constrained cost minimization model. Thus we see that the cost minimization model is the profit maximization model with an added constraint. Denoting the factor demands derived from cost minimization as ${x}_i={x}_i^y\left({w}_1,\kern0.5em {w}_n,\kern0.5em y\right)$, we apply (13) and (17) to derive $\partial {x}_i^{\ast }/\partial {w}_i\le \partial {x}_i^y/\partial {w}_i\le 0$. The profit maximizing factor demand function, which incorporates an output effect, is always more elastic with respect to its own price than the constant output factor demand functions, regardless of whether the output effect is positive or negative. We can also show by this method that, if another constraint is imposed on the factors, these cost-minimizing demand functions become less elastic. When the additional constraint takes on the form of holding some factor fixed, as in the above profit-maximization model, a similar conditional demand process is available (see Silberberg and Suen 2000).

Marginal Cost Functions

Many – perhaps most – important economic models incorporate a constraint of the form g(x₁, … , x_n) = k. The cost minimization model is an example; so are the various two-factor two-good models in which endowment levels are fixed. The Lagrangian for the cost minimization model is L = ∑ w_ix_i + λ(y − f(x₁, … , x_n)). The indirect objective function is the cost function C = C^∗(w₁, … , w_n,  y). The envelope theorem (8a) identifies λ^∗(w₁, … , w_n,  y) as the marginal cost function: ${C}_y^{\ast }={\lambda}^{\ast }$. We know from the above comparative statics discussion that cost minimization does not imply a sign for the slope of the marginal cost function, that is, ∂λ^∗/∂y/0 → ∂λ^∗/∂λ^∗/∂y ≷ 0. Nonetheless, we can still derive a Le Chatelier result for the marginal cost function.

Adding a new constraint h(x) = 0 to the cost minimization model consistent with the original equilibrium produces a new ‘short run’ cost function C^s( w₁, …w_n,  y). Since this is more constrained than C^∗, it must be the case that C^∗ ≤ C^s, but the two are equal at the original equilibrium. Thus the function F = C^∗ − C^s has an unconstrained maximum (of zero) with respect to all the parameters, and in particular, y. Thus ${F}_y={C}_y^{\ast }-{C}_y^s=0$ and ${F}_{yy}={C}_{yy}^{\ast }-{C}_{yy}^s\le 0.$ But this latter inequality is ∂λ^∗/∂y ≤ ∂λ^s/∂y. That is, the long-run marginal cost function either falls faster or rises slower than the short-run marginal cost function. This is the mathematical foundation for the famous article by Viner (1932) and his draftsman Wong that started it all.

Extensions

The Le Chatelier principle is a local result. Even with the usual sufficient second-order conditions, if some price changes by a finite amount, it is not an implication of the model that the long-run effects are absolutely larger than the short-run effects.

However, Milgrom and Roberts (1996) showed, using lattice theory, that, for example, for the profit-maximizing firm model, if all the cross-partials of the production function are everywhere non-negative, the Le Chatelier results hold in the large. A few years later, Suen, Silberberg and Tseng (2000) provided an easier proof of this result, showing also that the global Le Chatelier result held when the factors of production and the fixed factor do not switch from being substitutes to being complements (or vice versa) over the relevant price range.

Samuelson (1960a) analysed Le Chatelier phenomena for equilibrium systems not resulting from an explicit maximization hypothesis, using the ‘well-known’ theorem of reciprocal determinants of Jacobi. (I used to joke to my classes that the theorem was well-known to Jacobi and to Samuelson.) Lady and Quirk (2004) have analysed non-maximizing systems using a theory of cycles in determinants; they prove the Le Chatelier principle applies to systems identified by Morishima (1952), which allows substitutes and complements.

See Also

• Comparative Statics

• Envelope Theorem