Question

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 \((w)\)

Short answer

Based on the step by step solution, we can conclude that in the short run, when wage goes up, labor demand will fall, but in the long run, labor demand falls by even more because the firm can adjust capital input as well. Additionally, the response in the cost function to a change in wage will be larger in the long run as the firm can adjust capital input to minimize costs, which is not possible in the short run. This result demonstrates the application of Le Châtelier's Principle in the analysis of the effects of price changes on a firm's input decisions in both the short run and long run.

Step-by-step solution

Differentiate Definitional Relations with Respect to Price (P) and Substitution Cost (v)

Differentiate the given relationship of \(q^*(P, v, w) = q^s(P, w, k^*(P, v, w))\) with respect to P using the chain rule: \(\frac{\partial q^*}{\partial P} = \frac{\partial q^s}{\partial P} + \frac{\partial q^s}{\partial k}\frac{\partial k^*}{\partial P}\) Now, differentiate the given relationship with respect to v using the chain rule: \(\frac{\partial q^*}{\partial v} = \frac{\partial q^s}{\partial k}\frac{\partial k^*}{\partial v}\) Next, re-write the expression we derived for \(\frac{\partial q^*}{\partial P}\) by substituting the result we derived for \(\frac{\partial q^s}{\partial k}\) from the expression for \(\frac{\partial q^*}{\partial v}\).

Find the Relation between Long-run and Short-run Supply Responses

Solve the expression for \(\frac{\partial q^s}{\partial k}\) from \(\frac{\partial q^*}{\partial v}\): \(\frac{\partial q^s}{\partial k} = \frac{\frac{\partial q^*}{\partial v}}{\frac{\partial k^*}{\partial v}}\) Replace the expression for \(\frac{\partial q^s}{\partial k}\) in the expression for \(\frac{\partial q^*}{\partial P}\): \(\frac{\partial q^*}{\partial P} = \frac{\partial q^s}{\partial P} + \frac{\frac{\partial q^*}{\partial v}}{\frac{\partial k^*}{\partial v}}\frac{\partial k^*}{\partial P}\) Simplifying the above equation, we get: \(\frac{\partial q^*}{\partial P} = \frac{\partial q^s}{\partial P} + \frac{-\left(\frac{\partial k^*}{\partial P}\right)^{2}}{\frac{\partial k^*}{\partial v}}\)

Prove Le Châtelier's Principle for Labor Demand

Now, differentiating the relation \(l^*(P, v, w) = l^*(P, w, k^*(P, v, w))\) with respect to w: \(\frac{\partial l^*}{\partial w} = \frac{\partial l^s}{\partial w} + \frac{\partial l^s}{\partial k}\frac{\partial k^*}{\partial w}\) Differentiate the relation with respect to v: \(\frac{\partial l^*}{\partial v} = \frac{\partial l^s}{\partial k}\frac{\partial k^*}{\partial v}\) Solve the expression for \(\frac{\partial l^s}{\partial k}\) from \(\frac{\partial l^*}{\partial v}\): \(\frac{\partial l^s}{\partial k} = \frac{\frac{\partial l^*}{\partial v}}{\frac{\partial k^*}{\partial v}}\) Replace the expression for \(\frac{\partial l^s}{\partial k}\) in the expression for \(\frac{\partial l^*}{\partial w}\): \(\frac{\partial l^*}{\partial w} = \frac{\partial l^s}{\partial w} + \frac{\frac{\partial l^*}{\partial v}}{\frac{\partial k^*}{\partial v}}\frac{\partial k^*}{\partial w}\) After simplifying, we get \(\frac{\partial l^*}{\partial w} \leq \frac{\partial l^5}{\partial w}\) Which implies that long-run labor demand falls more when wage goes up than short-run labor demand.

Analyze the Difference Between Short-and Long-run Responses of Cost Function

In the short run, the capital input \(k\) is fixed, so the cost function \(C(v, w, q)\) will only respond to changes in wage, \(w\). In the long run, however, the capital input can also be adjusted. Therefore, the feedback effect and additional flexibility in the long run from adjusting capital input might lead to a larger response to changes in wage. For example, if wage increases, the firm may respond by adjusting capital input to minimize costs, which could lead to a larger change in the cost function compared to the short run where such adjustment is not possible.