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Chapter 10: Problem 10

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

Expert verified
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

01

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.
02

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#
03

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: $$e_{l^{\prime}, w} + e_{k^{\prime}, v} = 0$$
04

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#
05

Find the cross-price derivatives of contingent demand functions

To find the cross-price derivatives of contingent demand functions: $$\frac{\partial l^{c}}{\partial v} = \frac{\partial k^{c}}{\partial w}$$
06

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#
07

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#
08

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.

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Most popular questions from this chapter

Suppose that a firm produces two different outputs, the quantities of which are represented by \(q_{1}\) and \(q_{2}\). In general, the firm's total costs can be represented by \(C\left(q_{1}, q_{2}\right) .\) This function exhibits economies of scope if \(C\left(q_{1}, 0\right)+C\left(0, q_{2}\right)>C\left(q_{1}, q_{2}\right)\) for all output levels of either good. a. Explain in words why this mathematical formulation implies that costs will be lower in this multiproduct firm than in two single-product firms producing each good separately. b. If the two outputs are actually the same good, we can define total output as \(q=q_{1}+q_{2}\). Suppose that in this case average cost \((=C / q)\) decreases as \(q\) increases. Show that this firm also enjoys economies of scope under the definition provided here.

Professor Smith and Professor Jones are going to produce a new introductory textbook. As true scientists, they have laid out the production function for the book as \\[ q=S^{1 / 2} J^{1 / 2} \\] where \(q=\) the number of pages in the finished book, \(S=\) the number of working hours spent by Smith, and \(J=\) the number of hours spent working by Jones. After having spent 900 hours preparing the first draft, time which he valued at \(\$ 3\) per working hour, Smith has to move on to other things and cannot contribute any more to the book. Jones, whose labor is valued at \(\$ 12\) per working hour, will revise Smith's draft to complete the book. a. How many hours will Jones have to spend to produce a finished book of 150 pages? Of 300 pages? Of 450 pages? b. What is the marginal cost of the 150 th page of the finished book? Of the 300 th page? Of the 450 th page?

Suppose that a firm's fixed proportion production function is given by \\[ q=\min (5 k, 10 l) \\] a. Calculate the firm's long-run total, average, and marginal cost functions. b. Suppose that \(k\) is fixed at 10 in the short run. Calculate the firm's short-run total, average, and marginal cost functions. c. Suppose \(v=1\) and \(w=3 .\) Calculate this firm's long-run and short-run average and marginal cost curves.

A firm producing hockey sticks has a production function given by \\[ q=2 \sqrt{k l} \\] In the short run, the firm's amount of capital equipment is fixed at \(k=100 .\) The rental rate for \(k\) is \(v=\$ 1\), and the wage rate for \(l\) is \(w=\$ 4\) a. Calculate the firm's short-run total cost curve. Calculate the short-run average cost curve. b. What is the firm's short-run marginal cost function? What are the \(S C, S A C,\) and \(S M C\) for the firm if it produces 25 hockey sticks? Fifty hockey sticks? One hundred hockey sticks? Two hundred hockey sticks? c. Graph the \(S A C\) and the \(S M C\) curves for the firm. Indicate the points found in part (b). d. Where does the \(S M C\) curve intersect the \(S A C\) curve? Explain why the \(S M C\) curve will always intersect the \(S A C\) curve at its lowest point. Suppose now that capital used for producing hockey sticks is fixed at \(\bar{k}\) in the short run. c. Calculate the firm's total costs as a function of \(q, w, v,\) and \(\bar{k}\) f. Given \(q, w,\) and \(v,\) how should the capital stock be chosen to minimize total cost? g. Use your results from part (f) to calculate the long-run total cost of hockey stick production. h. For \(w=\mathrm{s} 4, v=\$ 1,\) graph the long-run total cost curve for hockey stick production. Show that this is an envelope for the short-run curves computed in part (c) by examining values of \(\bar{k}\) of \(100,200,\) and 400

Suppose the total-cost function for a firm is given by \\[ C=q(v+2 \sqrt{v w}+w) \\] a. Use Shephard's lemma to compute the (constant output) demand function for cach input, \(k\) and \(l\). b. Use the results from part (a) to compute the underlying production function for \(q\) c. You can check the result by using results from Fxample 10.2 to show that the CES cost function with \(\sigma=0.5, \rho=-1\) generates this total-cost function.
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