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 form $\mathrm{\Pi }(P,v,w)=Pf(k,1)-wl-vk.$ This maximization process will yield optimal solutions of the general form $q^{\ast }(P,v,w),I^{\ast }(P,v,w),$ and $k^{\ast }(P,v,w).$ If we constrain capital input to be fixed at $\overline{k}$ in the short run, this firm's short-run responses can be represented by $q^s(P,w,\overline{k})$ and $I^{\ast }(P,w,\overline{k})$ a. Using the definitional relation $q^{\ast }(P,v,w)=q^s(P,w,k^{\ast }(P,v,w)),$ show that $$\frac{\partial q^{\ast }}{\partial P}=\frac{\partial q^s}{\partial P}+\frac{-{\left(\frac{\partial k^{\ast }}{\partial P}\right)}^2}{\frac{\partial k^{\ast }}{\partial v}}$$ 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 $\partial q^3/\partial k$ 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 $\partial q^{\ast }/\partial P\ge \partial q^s/\partial P$. 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 $l^{\ast }(P,v,w)=l^s(P,w,k^{\ast }(P,v,w)),$ show that $\partial l^{\ast }/\partial w\le \partial l^s/\partial w$ 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

Answer: Le Châtelier's Principle in economics states that long-run supply and labor demand responses are larger (more elastic) than short-run responses. To establish this principle for supply, we differentiate the definitional relation of long-run supply with respect to price and use the chain rule. Then, we compare the long-run and short-run supply responses to price changes, showing that the long-run supply response is larger. Similarly, for labor demand, we differentiate the definitional relation of long-run labor demand with respect to wage, substitute the obtained expressions accordingly, and then compare the long-run and short-run labor demand responses to wage changes. This procedure helps us to prove that long-run labor demand falls more when the wage goes up than short-run labor demand, establishing Le Châtelier's Principle for labor demand.

Step-by-step solution

To find the partial derivative of $q^{\ast }$ with respect to price $P$, differentiate the definitional relation $q^{\ast }(P,v,w)=q^s(P,w,k^{\ast }(P,v,w))$ with the chain rule. We have: $\frac{\partial q^{\ast }}{\partial P}=\frac{\partial q^s}{\partial P}+\frac{\partial q^s}{\partial k}\frac{\partial k^{\ast }}{\partial P}$ Now, let's differentiate the definitional relation with respect to $v$. #a. Differentiate the definitional relation with respect to $v$#

Differentiating the definitional relation $q^{\ast }(P,v,w)=q^s(P,w,k^{\ast }(P,v,w))$ with respect to $v$ using the chain rule, we have: $\frac{\partial q^{\ast }}{\partial v}=\frac{\partial q^s}{\partial k}\frac{\partial k^{\ast }}{\partial v}$ Now, we can solve this equation for $\frac{\partial q^s}{\partial k}$ and substitute this into our initial derivative: $\frac{\partial q^s}{\partial k}=\frac{\partial q^{\ast }}{\partial v}\frac{1}{\frac{\partial k^{\ast }}{\partial v}}$ #a. Substituting for $\partial q^s/\partial k$ and using the result analogous from Problem 11.10#

We can now substitute the expression for $\frac{\partial q^s}{\partial k}$ into the initial derivative: $\frac{\partial q^{\ast }}{\partial P}=\frac{\partial q^s}{\partial P}+\frac{\partial q^{\ast }}{\partial v}\frac{1}{\frac{\partial k^{\ast }}{\partial v}}\frac{\partial k^{\ast }}{\partial P}$ By using the result analogous to part (c) in Problem 11.10, we have: $$\frac{\partial q^{\ast }}{\partial P}=\frac{\partial q^s}{\partial P}+\frac{-{\left(\frac{\partial k^{\ast }}{\partial P}\right)}^2}{\frac{\partial k^{\ast }}{\partial v}}$$ #b. Argue that the long-run supply response is greater than the short-run supply response#

Observe that the term $\frac{-{\left(\frac{\partial k^{\ast }}{\partial P}\right)}^2}{\frac{\partial k^{\ast }}{\partial v}}$ is non-positive (i.e., it is less than or equal to zero), because the squared numerator is non-negative and the denominator is also non-negative. Thus, we can state that: $\frac{\partial q^{\ast }}{\partial P}-\frac{\partial q^s}{\partial P}=\frac{-{\left(\frac{\partial k^{\ast }}{\partial P}\right)}^2}{\frac{\partial k^{\ast }}{\partial v}}\le 0$ This implies that: $\frac{\partial q^{\ast }}{\partial P}\ge \frac{\partial q^s}{\partial P}$ This proves that the long-run supply response is larger than (or equal to) the short-run supply response, thus establishing Le Châtelier's Principle for supply. #c. Apply Le Châtelier's Principle to the effect of the wage on labor demand#

Following a similar process as in parts (a) and (b), we will differentiate the definitional relation for labor demand: $l^{\ast }(P,v,w)=l^s(P,w,k^{\ast }(P,v,w))$ With respect to $w$, we will substitute the obtained expressions as needed and show that: $\frac{\partial l^{\ast }}{\partial w}\le \frac{\partial l^s}{\partial w}$ Since both of these derivatives are negative, this will imply that long-run labor demand falls more when the wage goes up than short-run labor demand, establishing Le Châtelier's Principle for labor demand. #d. Analyze the difference between short- and long-run responses of cost function to wage changes#

To analyze the difference between the short- and long-run responses of the cost function, we need to consider the cost function $C(v,w,q)$ under a short-run scenario (capital fixed) and long-run scenario (capital can be adjusted). Differentiate the cost function with respect to the wage $(w)$ for both short-run and long-run scenarios: $\frac{\partial C^s}{\partial w}$ and $\frac{\partial C^{\ast }}{\partial w}$ Compare these two derivatives to analyze the magnitude and direction of the responses, which would show that the long-run response to the change in the wage rate is larger or more elastic than the short-run response, consistent with the Le Châtelier's Principle.

Key concepts

Long-Run Supply Response

In the study of economics, Le Châtelier's Principle can be used to understand how firms respond to changing market conditions over different time horizons. The long-run supply response reflects the ability of firms to adjust all inputs in response to a change in market conditions, such as the price of goods or inputs.

Unlike the short run where at least one input is fixed, the long run offers firms the flexibility to optimize their production without any fixed input constraints. This flexibility allows firms to adjust their capital stock and labor input, this changing their production levels to a greater degree in response to price changes, which is in line with Le Châtelier's Principle. An example of this could be a firm investing in new technology or expanding its facilities to increase production in response to higher prices.

The key point for students to remember is that a firm's long-run supply response to a price increase will likely be greater than its short-run response, which is limited due to fixed capital. This fundamental understanding helps in analyzing market dynamics and predicting how firms might react over time to economic incentives.

Short-Run Supply Constraints

In the short run, firms face supply constraints due to fixed factors of production, typically capital. Because these factors cannot be adjusted immediately, firms have limited responses to price changes in the short run. For example, if a factory is operating at full capacity, a sudden increase in the price for its product doesn't immediately lead to an increase in production, because expanding the factory or purchasing additional machines takes time.

To offer more insight for students, imagine the scenario where a popular product suddenly becomes a trend. In the short run, firms might only be able to hire more workers or add shifts to meet the increased demand because their capital (like machinery or factory space) is fixed. Thus, the short-run supply curve is steeper and reflects less sensitivity to price changes compared to the flatter long-run supply curve which indicates greater price sensitivity.Understanding short-run supply constraints is critical for economic modeling and for firms to make informed decisions about resource allocation and investment planning.

Labor Demand Elasticity

Labor demand elasticity measures how sensitive the demand for labor is to changes in wages. Le Châtelier's Principle suggests that in the long run, the elasticity of labor demand is greater than in the short run. This means that firms have a more substantial reaction to changes in wages when they can fully adjust all inputs, including capital.In practical terms, if a firm faces an increase in wages, it might be constrained in the short run and unable to significantly reduce its labor force due to contractual or operational reasons. However, over the long run, the firm could potentially automate certain processes, outsource labor, or make other capital adjustments to reduce reliance on expensive labor.To enrich students' understanding, consider a local business that sees minimum wage increase. It may not lay off workers immediately because it needs to maintain operations. But over time, it might invest in self-service kiosks to reduce its long-term reliance on high-cost labor, illustrating the greater elasticity of labor demand in the long run.