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Chapter 10: Problem 2

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?

Short Answer

Expert verified
Answer: By calculating Jones' working hours for different book lengths (150, 300, and 450 pages) and evaluating the marginal costs for the 150th, 300th, and 450th pages, we find the following: - For a 150-page book, Professor Jones has to spend \(J_{150} = \frac{150^2}{900}\) hours, and the marginal cost of the 150th page is \(MC_{150} = \$12 \cdot \frac{2*150}{900}\). - For a 300-page book, Professor Jones has to spend \(J_{300} = \frac{300^2}{900}\) hours, and the marginal cost of the 300th page is \(MC_{300} = \$12 \cdot \frac{2*300}{900}\). - For a 450-page book, Professor Jones has to spend \(J_{450} = \frac{450^2}{900}\) hours, and the marginal cost of the 450th page is \(MC_{450} = \$12 \cdot \frac{2*450}{900}\).

Step by step solution

01

Replace S in the production function

Since we know \(S=900\), replace \(S\) in the production function: $$q=900^{1 / 2} J^{1 / 2}$$ ##Step 2: Solve for J##
02

Calculate J for different values of q

Re-arrange for J: $$J=\frac{q^2}{900}$$ Plug in the desired number of pages \(q\) and solve for \(J\): For \(q=150\): $$J_{150}=\frac{150^2}{900}$$ For \(q=300\): $$J_{300}=\frac{300^2}{900}$$ For \(q=450\): $$J_{450}=\frac{450^2}{900}$$ ##Step 3: Calculate Marginal Costs##
03

Differentiate to obtain marginal costs

The marginal cost is the additional cost for producing one additional page. To find this, we need to differentiate the total cost function with respect to the number of pages produced \(q\). The total cost is given by the sum of costs for Smith and Jones: $$C(q)=\$3\cdot S + \$12\cdot J(q)$$ Since \(S\) is constant at 900 hours, we differentiate with respect to \(q\) to get the marginal cost: $$MC(q)=\frac{dC}{dq} = \$12 \cdot \frac{dJ}{dq}$$ To find the marginal cost of the 150th, 300th, and 450th page, we need to evaluate the above expression at \(q=150\), \(q=300\), and \(q=450\) ##Step 4: Evaluate Marginal Costs##
04

Calculate marginal costs at q values

First, we find \(\frac{dJ}{dq}\). From the expression for \(J\) derived in step 2: $$\frac{dJ}{dq} = \frac{2q}{900}$$ Now, evaluate the marginal cost expression at the different values of \(q\): For \(q=150\): $$MC_{150}=\$12 \cdot \frac{2*150}{900}$$ For \(q=300\): $$MC_{300}=\$12 \cdot \frac{2*300}{900}$$ For \(q=450\): $$MC_{450}=\$12 \cdot \frac{2*450}{900}$$

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Most popular questions from this chapter

An enterprising entrepreneur purchases two factories to produce widgets. Each factory produces identical products, and each has a production function given by \\[ q=\sqrt{k_{i} l_{i}}, \quad i=1,2 \\] The factories differ, however, in the amount of capital equipment each has, In particular, factory 1 has \(k_{1}=25,\) whereas factory 2 has \(k_{2}=100 .\) Rental rates for \(k\) and \(l\) are given by \(w=v=\$ 1\) a. If the entrepreneur wishes to minimize short-run total costs of widget production, how should output be allocated between the two factories? b. Given that output is optimally allocated between the two factories, calculate the short-run total, average, and marginal cost curves. What is the marginal cost of the 100 th widget? The 125 th widget? The 200 th widget? c. How should the entrepreneur allocate widget production between the two factories in the long run? Calculate the long-run total, average, and marginal cost curves for widget production. d. How would your answer to part (c) change if both factories exhibited diminishing returns to scale?

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.

Many empirical studies of costs report an alternative definition of the elasticity of substitution between inputs. This alternative definition was first proposed by \(\mathrm{R}\). G. \(\mathrm{D}\). Allen in the 1930 s and further clarified by H. Uzawa in the 1960 s. This definition builds directly on the production function-based elasticity of substitution defined in footnote 6 of Chapter \(9: A_{i j}=C_{i j} C / C_{i} C_{j}\) where the subscripts indicate partial differentiation with respect to various input prices. Clearly, the Allen definition is symmetric. a. Show that \(A_{i j}=e_{x_{i}, \ldots, n} / s_{j},\) where \(s_{j}\) is the share of input \(j\) in total cost. b. Show that the elasticity of \(s_{i}\) with respect to the price of input \(j\) is related to the Allen elasticity by \(e_{s_{1}, p_{1}}=s_{j}\left(A_{j}-1\right)\) c. Show that, with only two inputs, \(A_{k l}=1\) for the Cobb-Douglas case and \(A_{k l}=\sigma\) for the CFS case. d. Read Blackorby and Russell (1989: "Will the Real Elasticity of Substitution Please Stand Up?") to see why the Morishima definition is preferred for most purposes.

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.

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.
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