11.11 Le Chateliers Principle Because firms have greater flexibility in the long run, their reactions to price...

11.11 Le Chaˆtelier’s Principle Because firms have greater flexibility in the long run, their reactions to price changes may be greater in the long run than in the short run. Paul Samuelson was perhaps the first economist to recognize that such reactions were analogous to a principle from physical chemistry termed the Le Chaˆtelier’s Principle. The basic idea of the principle is that any disturbance to an equilibrium

(such as that caused by a price change) will not only have a direct effect but may also set off feedback effects that enhance the response. In this problem we look at a few examples. Consider a price-taking firm that chooses its inputs to maximize a profit function of the form P(P, v, w) ¼ Pf(k, l) % wl % vk. This maximization process will yield optimal solutions of the general form q&

(P, v, w), l

&

(P, v, w), and k&

(P, v, w). If we constrain capital input to be fixed at k in the short run, this firm’s short-run responses can be represented by qs

ðP, w, kÞ and l s

ðP, w, kÞ.

a. Using the definitional relation q&

(P, v, w) ¼ qs

(P, w, k&

(P, v, w)), show that

@q&

@P ¼ @qs

@P þ

% @k&

@P

" #2

@k&

@v

:

Do this in three steps. First, differentiate the definitional relation with respect to P using the chain rule. Next, differentiate the definitional relation with respect to v (again using the chain rule), and use the result to substitute for @qs

=@k in the initial derivative. Finally, substitute a result analogous to part

(c) of Problem 11.10 to give the displayed equation.

b. Use the result from part

(a) to argue that @q&=@P + @qs

=@P. This establishes Le Chaˆtelier’s Principle for supply: Long-run supply responses are larger than (constrained) short-run supply responses.

c. Using similar methods as in parts

(a) and (b), prove that Le Chaˆtelier’s Principle applies to the effect of the wage on labor demand. That is, starting from the definitional relation l

&

(P, v, w) ¼ l s

(P, w, k&

(P, v, w)), show that @l

&

=@w , @l s

=@w, implying that long-run labor demand falls more when wage goes up than short-run labor demand (note that both of these derivatives are negative).

d. Develop your own analysis of the difference between the short- and long-run responses of the firm’s cost function

[C (v, w, q)] to a change in the wage (w).