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Chapter 12: Problem 7

Suppose, as in Problem \(12.6,\) a firm produces hockey sticks with a production function of \(q=2 \mathrm{V} K L\). Capital stock is fixed at \(K\) in the short run. a. Calculate the firm's total costs as a function of \(q, w, v,\) and \(K\) b. Given \(q, w,\) and \(v,\) how should the capital stock be chosen to minimize total cost? c. Use your results from part (b) to calculate the long-run total cost of hockey stick production d. 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 examin ing values of \(K\) of \(100,200,\) and 400

Short Answer

Expert verified
The firm's total cost function is: \(TC(q,w,v,K) = wL + vK = w\frac{q}{2VK} + vK\) b) Given q, w, and v, what is the optimal capital stock to minimize total cost? The optimal capital stock to minimize total cost is infinite. c) What is the long-run total cost of hockey stick production? The long-run total cost of hockey stick production is 0. d) Describe and analyze the long-run total cost curve for hockey stick production and the short-run curves with K values of 100, 200, and 400. The long-run total cost curve is a horizontal line at 0 on the vertical axis. The short-run total cost curves for K values of 100, 200, and 400 form a tangential "envelope" to the long-run total cost curve, illustrating how the short-run cost curves approach the long-run cost curve as K increases.

Step by step solution

01

Calculate the firm's total costs as a function of q, w, v, and K

Firstly, we need to rewrite the production function for labor L, as we are given the values of q, K, and V. \(q = 2VKL\) \(L = \frac{q}{2VK}\) Now we can express the total costs, which consist of the cost of labor wL and the cost of capital vK. \(TC(q,w,v,K) = wL + vK = w\frac{q}{2VK} + vK\)
02

Given q, w, and v, find the optimal capital stock to minimize total cost

To find the optimal capital stock K, we need to take the derivative of the total cost function with respect to K and set it equal to 0. \(\frac{\partial TC(q,w,v,K)}{\partial K} = 0\) Now we'll solve for K to find the optimal level. \(\frac{\partial}{\partial K} (w\frac{q}{2VK} + vK) = 0\) \(\frac{-wq}{2V(K^2)} = 0\) Since w, q, and V are all non-zero, K^2 must be infinity, and K must be infinity.
03

Calculate the long-run total cost of hockey stick production

We've found that the optimal K is infinite, so we now need to find the minimum long-run total cost. To do this, we will find the minimum of the total cost function for any given level of q, assuming that V and w are fixed (v = \(1 and w = \)4). We know that: \(TC(q,w,v,K) = wL + vK = 4\frac{q}{2V(\infty)} + 1(\infty) = 0\) Therefore, the long-run total cost is 0.
04

Graph the long-run total cost curve for hockey stick production and analyze short-run curves

Since the long-run total cost is always 0 regardless of the quantity of hockey sticks produced, the long-run total cost curve is a horizontal line at the 0 value on the vertical axis. For the short-run curves with K values of 100, 200, and 400, we can calculate the total cost using the equation from Step 1. \(TC(q,w,v,K) = wL + vK = 4\frac{q}{2V(K)} + 1(K)\) For K = 100: \(TC(q,4,1,100) = 4\frac{q}{2V(100)} + 1(100)\) For K = 200: \(TC(q,4,1,200) = 4\frac{q}{2V(200)} + 1(200)\) For K = 400: \(TC(q,4,1,400) = 4\frac{q}{2V(400)} + 1(400)\) We can then plot these on the same graph with the long-run total cost curve. These will form a tangential "envelope" to the long-run total cost curve, illustrating how the short-run cost curves approach the long-run cost curve as K increases.

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Most popular questions from this chapter

Suppose the total cost function for a firm is given by \\[ T C-q w^{23} v^{13} \\] a. Use Shephard's lemma (footnote 8 ) 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 that a firm produces two different outputs, the quantities of which are represented by \(q^{\wedge}\) and \(q_{2},\) In general, the firm's total costs can be represented by \(T C\left\\{q_{n} q_{2}\right)\) This function \right. exhibits economies of scope if \(T C\left(q_{w} 0\right)+T C\left(0, q_{2}\right)>T C\left(q_{w} q_{2}\right)\) 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 \(q=<,+q_{2}\) Suppose that in this case average cost \((=T C / q)\) falls as \(q\) increases. Show that this firm also enjoys economies of scope under the definition provided here.

Suppose that a firm's fixed proportion production function is given by \\[ q=\min (5 K, 10 \mathrm{L}) \\] and that the rental rates for capital and labor are given by \(v=1, w-3\) a. Calculate the firm's long-run total, average, and marginal cost curves. b. Suppose that Xis fixed at 10 in the short run. Calculate the firm's short- run total, aver age, and marginal cost curves, What is the marginal cost of the 10 th unit? The 50 th unit? The 100 th unit?

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?

A firm producing hockey sticks has a production function given by In the short run, the firm's amount of capital equipment is fixed at \(K=100\). The rental rate for \(\mathrm{AT}\) 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 \(S T C, S A T C,\) 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 \(S A T C\) and the \(S M C\) curves for the firm. Indicate the points found in part (b). d. Where does the SMC curve intersect the SATC curve? Fxplain why the SMC curve will al ways intersect the \(S A T C\) curve at its lowest point.
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