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
Châtelier'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
\\[
\text { form } \Pi(P, v, w)=P f(k, l)-w l-v k . \text { This maximiza- }
\\]
tion 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 \(\bar{k}\) in the short run, this firm's short-run responses can be
represented by \(q^{s}(P, w, \bar{k})\) and \(l^{s}(P, w, \bar{k})\)
a. Using the definitional relation \(q^{*}(P, v, w)=q^{*}(P, w\)
\(\left.k^{\prime \prime}(P, v, w)\right),\) show that
$$\frac{\partial q^{*}}{\partial P}=\frac{\partial q^{s}}{\partial
P}+\frac{-\left(\frac{\partial k^{*}}{\partial P}\right)^{2}}{\frac{\partial
k^{*}}{\partial v}}$$
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 \(\partial q^{s} / \partial k\) 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 \(\partial q^{*} / \partial P
\geq\) \(\partial q^{s} / \partial P .\) This establishes Le Châtelier'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 Châtelier's
Principle applies to the effect of the wage on labor demand. That is, starting
from the definitional relation \(l^{*}(P, v, w)=l^{*}\left(P, w, k^{*}(P, v,
w)\right),\) show that \(\partial l^{*} / \partial w \leq \partial l^{5} /
\partial w,\) 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
\((w)\)
