Question

The own-price elasticities of contingent input demand for labor and capital are defined as \\[ e_{Y},_{w}=\frac{\partial l^{c}}{\partial w} \cdot \frac{w}{l^{c}}, \quad e_{k^{\prime}, v}=\frac{\partial k^{c}}{\partial v} \cdot \frac{v}{k^{\ell}} \\] a. Calculate \(e_{\mathbb{R}, \text { w }}\) and \(e_{k_{k}, v}\) for each of the cost functions shown in Example 10.2 b. Show that, in general, \(c_{r}, w+e_{L r, v}=0\) c. Show that the cross-price derivatives of contingent demand functions are equal-that is, show that \(\partial f / \partial v=\partial k^{c} / \partial w\). Use this fact to show that \(s_{1} e_{\gamma, v}=s_{k} e_{k^{\prime}}, w\) where \(s_{b} s_{k}\) are, respectively, the share of labor in total cost \((w l / C)\) and of capital in total cost \((v k / C)\) d. Use the results from parts (b) and (c) to show that \(s_{i} €_{l^{\prime}, s, s}+s_{k} c_{k^{\prime}, w}=0\) e. Interpret these various elasticity relationships in words and discuss their overall relevance to a general theory of input demand.

Short answer

Question: Calculate the own-price elasticities for labor and capital using the given cost functions and show that their sum is zero in general. Also, derive the expression showing the relationship between cross-price elasticities and the shares of labor and capital in total cost and use the results to show that the sum of weighted own-price elasticities is zero. Finally, interpret the elasticity relationships and discuss their relevance to a general theory of input demand.

Step-by-step solution

Apply the formula for own-price elasticity of labor

To calculate the own-price elasticity of demand for labor, use the given formula for \(e_{l^{\prime},w}\): $$e_{l^{\prime}, w}=\frac{\partial l^{c}}{\partial w} \cdot \frac{w}{l^{c}}$$ Plug in the values of the given cost function from Example 10.2.

Apply the formula for own-price elasticity of capital

To calculate the own-price elasticity of demand for capital, use the given formula for \(e_{k^{\prime}, v}\): $$e_{k^{\prime}, v}=\frac{\partial k^{c}}{\partial v} \cdot \frac{v}{k^{c}}$$ Plug in the values of the given cost function from Example 10.2. #b. Show that the sum of own-price elasticities is zero#

Prove the sum of own-price elasticities is zero

To show that the sum of own-price elasticities is zero, we need to prove that: $$e_{l^{\prime}, w} + e_{k^{\prime}, v} = 0$$

Use the calculated elasticities from Step 1 and Step 2

Plug in the calculated own-price elasticities of labor and capital from steps 1 and 2, and see whether their sum is zero. #c. Show that cross-price derivatives are equal and derive an expression for cross-price elasticities#

Find the cross-price derivatives of contingent demand functions

To find the cross-price derivatives of contingent demand functions: $$\frac{\partial l^{c}}{\partial v} = \frac{\partial k^{c}}{\partial w}$$

Use the cross-price derivatives to express the relationship between cross-price elasticities and shares

Using the fact that the cross-price derivatives are equal, we can show that: $$s_{l} e_{l^{c}, v}=s_{k} e_{k^{c}, w}$$ #d. Show that the sum of weighted own-price elasticities is zero#

Use the results from part b and part c

Using the results about the sum of own-price elasticities and the relationship between cross-price elasticities and shares, we can show that: $$s_{l} e_{l^{\prime}, w}+s_{k} e_{k, w}=0$$ #e. Interpret the elasticity relationships and discuss their relevance#

Interpret the elasticity relationships

Discuss the meaning of own-price elasticities, cross-price elasticities, and the relationship between own-price and cross-price elasticities in the context of input demand. Explain the importance of these relationships regarding the general theory of input demand.

Key concepts

Own-Price Elasticity

Own-price elasticity measures how the demand for a good changes when its own price changes. In terms of input demand, we look at how the demand for labor or capital shifts with price variations.

In the context of the exercise, the formula for own-price elasticity of labor is \(e_{l^{\prime}, w}=\frac{\partial l^{c}}{\partial w} \cdot \frac{w}{l^{c}}\). This can be interpreted as the percentage change in labor demand when the wage rate shifts by one percent.

Similarly, for capital, the own-price elasticity formula is \(e_{k^{\prime}, v}=\frac{\partial k^{c}}{\partial v} \cdot \frac{v}{k^{c}}\). This measures how sensitive the demand for capital is to changes in its price.

These calculations are crucial as they allow businesses to understand the responsiveness of their demand for labor and capital relative to their changing costs, helping optimize resource allocation.

Cross-Price Elasticity

Cross-price elasticity focuses on how the demand for a good changes when the price of another related good changes. For input demands, it indicates the interdependence of labor and capital.

According to the solution, it's shown that \(\frac{\partial l^{c}}{\partial v} = \frac{\partial k^{c}}{\partial w}\). This means that changes in the price of one input affect the demand for the other input equally.

The expression \(s_{l} e_{l^{c}, v}=s_{k} e_{k^{c}, w}\) uses input cost shares to show the proportional impact of these changes. It emphasizes how cost proportions (\(s_{l}\) for labor and \(s_{k}\) for capital) translate these elasticities into real-world applications.

Businesses can use cross-price elasticity to predict how adjustments in pricing strategies for one input might affect the demand for other inputs, ensuring a balanced adjustment in production factors.

Input Demand Theory

Input demand theory is the framework that explores how businesses adjust their use of different inputs like labor and capital to minimize costs and maximize output.

Within this framework, elasticities play a vital role. Understanding both own-price and cross-price elasticities allows firms to determine how sensitive their input demands are to changes in input costs.

The exercise highlights an important principle: the sum of weighted own-price elasticities equals zero (\(s_{l} e_{l^{\prime}, w}+s_{k} e_{k, w}=0\)). This implies that an increase in the cost of one input will lead to adjustments not only in that input but also potentially in others, maintaining overall efficiency in input use.

By interpreting these elasticities, firms can navigate input cost changes effectively, sustaining optimal production levels and adapting swiftly to market conditions.