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Chapter 7: Problem 12

A computer company's cost function, which relates its average cost of production AC to its cumulative output in thousands of computers \(Q\) and its plant size in terms of thousands of computers produced per year \(q\) (within the production range of 10,000 to 50,000 computers), is given by \\[\mathrm{AC}=10-0.1 Q+0.3 q\\] a. Is there a learning-curve effect? b. Are there economies or diseconomies of scale? c. During its existence, the firm has produced a total of 40,000 computers and is producing 10,000 computers this year. Next year it plans to increase production to 12,000 computers. Will its average cost of production increase or decrease? Explain.

Short Answer

Expert verified
a. Yes, there is a learning-curve effect. b. There are diseconomies of scale. c. The average cost of production will decrease next year.

Step by step solution

01

Determine the Learning-Curve Effect

A learning-curve effect exists if cumulative output (\(Q\)) has an influence on the average cost (\(\mathrm{AC}\)). From the cost function \(\mathrm{AC}=10-0.1Q+0.3q\), it can be seen that \(Q\) does influence the average cost. Therefore, there is a learning-curve effect.
02

Determine the Scale Economies

Economies of scale exist if an increase in plant size (\(q\)) decreases the average cost, while diseconomies of scale exist if an increase in \(q\) increases the average cost. From the cost function \(\mathrm{AC}=10-0.1Q+0.3q\), a positive coefficient on \(q\) indicates that as the plant size increases, average cost also increases. Therefore, there are diseconomies of scale.
03

Predict Change in Average Cost

Using the given cost equation \(\mathrm{AC}=10-0.1Q+0.3q\) and given that \(Q=40,000\) computers have been produced in total while 10,000 computers are being produced this year \(q=10\), the average cost this year is \(\mathrm{AC}=10-0.1(40)+0.3(10)=7\) . Next year, \(q\) will be 12 and \(Q\) will be 52 (\(40+10+2\)), the predicted average cost will be \(\mathrm{AC}=10-0.1(52)+0.3(12)=5.8\). Therefore, average cost of production will decrease next year.

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Most popular questions from this chapter

A recent issue of Business Week reported the following: During the recent auto sales slump, GM, Ford, and Chrysler decided it was cheaper to sell cars to rental companies at a loss than to lay off workers. That's because closing and reopening plants is expensive, partly because the auto makers' current union contracts obligate them to pay many workers even if they're not working. When the article discusses selling cars "at a loss," is it referring to accounting profit or economic profit? How will the two differ in this case? Explain briefly.

Suppose that a paving company produces paved parking spaces \((q)\) using a fixed quantity of land \((T)\) and variable quantities of cement (C) and labor (L). The firm is currently paving 1000 parking spaces. The firm's cost of cement is \(\$ 4,000\) per acre covered, and its cost of labor is \(\$ 12 /\) hour. For the quantities of \(C\) and \(L\) that the firm has chosen, \(M P_{C}=50\) and \(M P_{L}=4\) a. Is this firm minimizing its cost of producing parking spaces? How do you know? b. If the firm is not cost-minimizing, how must it alter its choices of \(C\) and \(L\) in order to decrease cost?

You manage a plant that mass-produces engines by teams of workers using assembly machines. The technology is summarized by the production function \\[q=5 K L\\] where \(q\) is the number of engines per week, \(K\) is the number of assembly machines, and \(L\) is the number of labor teams. Each assembly machine rents for \(r=\$ 10,000\) per week, and each team costs \(w=\$ 5000\) per week. Engine costs are given by the cost of labor teams and machines, plus \(\$ 2000\) per engine for raw materials. Your plant has a fixed installation of 5 assembly machines as part of its design. a. What is the cost function for your plant-namely, how much would it cost to produce \(q\) engines? What are average and marginal costs for producing \(q\) engines? How do average costs vary with output? b. How many teams are required to produce 250 engines? What is the average cost per engine? c. You are asked to make recommendations for the design of a new production facility. What capital/ labor \((K / L)\) ratio should the new plant accommodate if it wants to minimize the total cost of producing at any level of output \(q ?\)

Suppose the long-run total cost function for an industry is given by the cubic equation \(\mathrm{TC}=\mathrm{a}+\mathrm{b} q+\mathrm{c} q^{2}+\mathrm{d} q^{3}\). Show (using calculus) that this total cost function is consistent with a U-shaped average cost curve for at least some values of \(a, b, c\), and \(d\).

The short-run cost function of a company is given by the equation \(\mathrm{TC}=200+55 q\), where \(\mathrm{TC}\) is the total cost and \(q\) is the total quantity of output, both measured in thousands. a. What is the company's fixed cost? b. If the company produced 100,000 units of goods, what would be its average variable cost? c. What would be its marginal cost of production? d. What would be its average fixed cost? e. Suppose the company borrows money and expands its factory. Its fixed cost rises by \(\$ 50,000\), but its variable cost falls to \(\$ 45,000\) per 1000 units. The cost of interest ( \(i\) ) also enters into the equation. Each 1 -point increase in the interest rate raises costs by \(\$ 3000 .\) Write the new cost equation.
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