Chapter 10

In a famous article [J. Viner, "Cost Curves and Supply Curves," Zeitschrift fur Nationalokonomie 3 \(\text { (September }1931): 23-46],\) Viner criticized his draftsman who could not draw a family of \(\underline{S A C}\) curves whose points of tangency with the U-shaped \(A C\) curve were also the minimum points on each \(S A C\) curve. The draftsman protested that such a drawing was impossible to construct. Whom would you support in this debate?

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 \((=C / q)\) falls 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. 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?

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 \cdot 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. e. 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 \(w=\$ 4, v=\$ 1,\) 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 (a) by examining values of \(\bar{k}\) of \(100,200,\) and 400

An enterprising entrepreneur purchases two firms to produce widgets. Each firm produces identical products, and each has a production function given by \\[ q=\sqrt{k_{i} l_{i}}, \quad i=1,2 \\] The firms differ, however, in the amount of capital equipment each has. In particular, firm I has \(k_{1}=25\) whereas firm 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 firms? b. Given that output is optimally allocated between the two firms, 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 firms 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 firms exhibited diminishing returns to scale?

Suppose the total-cost function for a firm is given by \\[ C=q w^{2 / 3} v^{1 / 3} \\] a. Use Shephard's lemma to compute the constant output demand functions for inputs \(l\) and \(k\) b. Use your results from part (a) to calculate the underlying production function for \(q\)

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 each input, \(k\) and \(i\) 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 Example 10.2 to show that the CES cost function with \(\sigma=0.5, \rho=-1\) generates this total-cost function.

The CES production function can be generalized to permit weighting of the inputs. In the two-input case, this function is \\[ q=f(k, l)=\left[(a k)^{\rho}+(b l)^{\rho}\right]^{\gamma / \rho} \\] a. What is the total-cost function for a firm with this production function? Hint: You can, of course, work this out from scratch; easier perhaps is to use the results from Example 10.2 and reason that the price for a unit of capital input in this production function is \(v / a\) and for a unit of labor input is \(w / b\) b. If \(\gamma=1\) and \(a+b=1,\) it can be shown that this production function converges to the CobbDouglas form \(q=k^{a} l^{b}\) as \(\rho \rightarrow 0 .\) What is the total cost function for this particular version of the CES function? c. The relative labor cost share for a two-input production function is given by \(w l / v k\). Show that this share is constant for the Cobb-Douglas function in part (b). How is the relative labor share affected by the parameters \(a\) and \(b\) ? d. Calculate the relative labor cost share for the general CES function introduced above. How is that share affected by changes in \(w / v\) ? How is the direction of this effect determined by the elasticity of substitution, \(\sigma\) ? How is it affected by the sizes of the parameters \(a\) and \(b\) ?