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Chapter 7: Problem 8

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 \(q ?\)

Short Answer

Expert verified
This problem involves the calculation of a cost function, the average cost and the marginal costs based on a specific production function. It also involves determining an optimal capital-labor ratio for a new production facility and estimating the number of labor teams required to produce 250 engines, along with the average cost per engine.

Step by step solution

01

Calculate Cost Function

First, calculate the cost function, C(q), which should provide the total cost to produce \(q\) engines. We will estimate this cost function based on given machine and labor costs plus the cost of the raw materials. Since the plant has 5 machines, which costs $10,000 each per week, \(K = 5\) and \(rK = \$10,000 * 5\). Then, calculate the Labor Cost replacing \(L\) from the production function \(q=5KL\) by \(L = \dfrac{q}{5K}\) and multiplying it by the labor cost per team (w), we have: \(wL = \$5,000 * \dfrac{q}{5K}\). Finally, add the raw material cost per engine to get the cost function: \(C(q) = rK + wL + \$2,000*q\).
02

Calculate Average and Marginal Costs

Next, calculate the average cost (AC) by dividing the cost function by the quantity \(q\) (i.e., \(AC(q) = C(q) / q\)). Marginal cost (MC), which is the cost of producing one additional unit, is the derivative of the cost function (i.e., \(MC(q) = dC(q) / dq\)).
03

Determine Labor Needs for 250 Engines

Substitute \(q = 250\) into the production function to determine the number of labor teams needed to produce 250 engines. We can get an equation as \(250 = 5*K*L\). Here \(K = 5\), therefore, substitute \(K\) and solve the equation to get the value of \(L\). Calculate the average cost per engine by substituting \(q = 250\) into the average cost function.
04

Recommendation for New Production Facility

Here, we need to derive a general \(K/L\) ratio that would minimize the total cost for any level of output \(q\). This requires differentiating the cost function with respect to \(L\) and setting the result to zero, then solving for the \(L\). This will give the optimal capital-labor ratio \(K^*/L^*\).

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Most popular questions from this chapter

Suppose the economy takes a downturn, and that labor costs fall by 50 percent and are expected to stay at that level for a long time. Show graphically how this change in the relative price of labor and capital affects the firm's expansion path.

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 \mathrm{Q}+0.3 \mathrm{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.

The cost of flying a passenger plane from point \(A\) to point \(B\) is \(\$ 50,000\). The airline flies this route four times per day at \(7 \mathrm{AM}, 10 \mathrm{AM}, 1 \mathrm{PM}\), and \(4 \mathrm{PM}\). The first and last flights are filled to capacity with 240 people. The second and third flights are only half full. Find the average cost per passenger for each flight. Suppose the airline hires you as a marketing consultant and wants to know which type of customer it should try to attract-the off-peak customer (the middle two flights) or the rush-hour customer (the first and last flights). What advice would you offer?

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.

A chair manufacturer hires its assembly-line labor for \(\$ 30\) an hour and calculates that the rental cost of its machinery is \(\$ 15\) per hour. Suppose that a chair can be produced using 4 hours of labor or machinery in any combination. If the firm is currently using 3 hours of labor for each hour of machine time, is it minimizing its costs of production? If so, why? If not, how can it improve the situation? Graphically illustrate the isoquant and the two isocost lines for the current combination of labor and capital and for the optimal combination of labor and capital.
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