Question

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. Smith values his labor as \(\$ 3\) per working hour. He has spent 900 hours preparing the first draft. 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?12.1 In a famous article [J. Viner, "Cost Curves and Supply Curves," Zeilschríl fur Nationalokonomie \(3 \text { (September } 1931 \text { ): } 23-46]\), Viner criticized his draftsman who could not draw a family of \(S A T C\) curves whose points of tangency with the U-shaped \(A C\) curve were also the minimum points on each \(S A T C\) curve. The draftsman protested that such a drawing was impossible to construct. Whom would you support in this debate?

Short answer

Answer: The marginal costs are approximately \$1,500 for the 150th page, \$4,500 for the 300th page, and \$7,500 for the 450th page.

Step-by-step solution

Part (a): Hours spent by Jones for different book lengths

Use the production function \(q=S^{1 / 2} J^{1 / 2}\) where \(q\) is the number of pages, \(S\) is the hours spent by Smith and \(J\) is the hours spent by Jones. Smith has spent 900 hours (\(S = 900\)) on the first draft. We will solve for \(J\) for different lengths of the book: 150 pages, 300 pages, and 450 pages. For a 150-page book: \\[150 = \sqrt{900} \sqrt{J}\\] \\[\Rightarrow J=\left(\frac{150}{\sqrt{900}}\right)^{2} \approx 125\]\\ Jones has to spend approximately 125 hours on the 150-page book. For a 300-page book: \\[300 = \sqrt{900} \sqrt{J}\\] \\[\Rightarrow J=\left(\frac{300}{\sqrt{900}}\right)^{2} \approx 500\]\\ Jones has to spend approximately 500 hours on the 300-page book. For a 450-page book: \\[450 = \sqrt{900} \sqrt{J}\\] \\[\Rightarrow J=\left(\frac{450}{\sqrt{900}}\right)^{2} \approx 1125\]\\ Jones has to spend approximately 1125 hours on the 450-page book.

Part (b): Marginal cost of the 150th, 300th, and 450th pages

To find the marginal cost, first calculate the marginal cost for each hour of work by Jones, and then calculate the marginal cost for the given page numbers. Jones values his labor at \$12 per hour. The marginal cost for each hour worked by Jones is therefore equal to \$12. For the 150th page, we have spent about 125 additional hours on the book. Marginal cost of the 150th page \(=\) 125 hours \(\times\) \$12 \(\approx\) \$1,500. For the 300th page, we have spent about 375 additional hours on the book (500 hours for 300 pages - 125 hours for 150 pages). Marginal cost of the 300th page \(=\) 375 hours \(\times\) \$12 \(\approx\) \$4,500. For the 450th page, we have spent about 625 additional hours on the book (1125 hours for 450 pages - 500 hours for 300 pages). Marginal cost of the 450th page \(=\) 625 hours \(\times\) \$12 \(\approx\) \$7,500.