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Week 4 – Check Your Understanding:
Chapter 7 Exercise 1, 6, 8, and 9
1. In the Deep Creek Mining Company example described in this chapter (Table 7.1), suppose again that labor is the variable input and capital is the fixed input. Specifically, assume that the firm owns a piece of equipment having a 500-bhp rating.
a. Complete the following table:
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LABOR INPUT L (NO. OF WORKERS)
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TOTAL PRODUCT TPL (= Q)
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MARGINAL PRODUCT MPL
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AVAERAGE PRODUCT APL
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1
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2
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3
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4
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5
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6
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7
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8
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9
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10
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b. Plot the (i) total product, (ii) marginal product, and (iii) average product functions.
c. Determine the boundaries of the three stages of production.
6. Consider that following short-run production function (where L = variable input, Q = output):
Q = 10L – 0.5 L2
Suppose that output can be sold for $10 per unit. Also assume that the firm can obtain as much of the variable input (L) as it needs at $20 per unit.
a. Determine the marginal revenue product function.
b. Determine the marginal factor cost function.
c. Determine the optional value of L, given that the objective is to maximize profits.
8. Based on the production function parameter estimates reported in Table 7.4:
a) Which industry (or industries) appears to exhibit decreasing returns to scale? (Ignore the issue of statistical significance.)
b) Which industry comes closest to exhibiting constant returns to scale?
c) In which industry will a given percentage increase in capital result in the largest percentage increase in output?
d) In what industry will a given percentage increase in production workers result in the largest percentage increase in output?
9. Consider the following Cobb-Douglas production function for the bus transportation system in a particular city:
Q = aLB1 FB2 KB3
Where L = labor input in worker hours
F = fuel input in gallons
K = capital input in number of buses
Q = output measured in millions of bus miles
Suppose that the parameters (a, B1, B2, and B3) of this model were estimated using annual data for the past 25 years. The following results were obtained:
a = 0.0012 B1 = 0.45 B2 = 0.20 B3 = 0.30
a) Determine the (i) labor, (ii) fuel, and (iii) capital input production elasticities.
b) Suppose that labor input (worker hours) is increased by 2 percent next year (with the other inputs held constant). Determine the approximate percentage change in output.
c) Suppose that capital input (number of buses) is decreased by 3 percent next year (when certain older buses are taken out service). Assuming that the other inputs are held constant, determine the approximate percentage change in output.
d) What type of returns to scale appears to characterize this bus transportation system? (Ignore the issue of statistical significance.)
e) Discuss some of the methodological and measurement problem one might encounter in using time-series data to estimate the parameters of this model.
Chapter 8 Exercise 2(a), 4, and 6(a)
2. Howard Bowen is a large-scale cotton farmer. The land and machinery he owns has a current market value of $4 million. Bowen owes his local bank $3 million. Last year Bowen sold $5 million worth of cotton. His variable operating costs were $4.5 million; accounting depreciation was $40,000, although the actual decline in value of Bowen’s machinery was $60,000 last year. Bowen paid himself a salary of $50,000, which is not considered part of his variable operating cost. Interest on his bank loan was $400,000. If Bowen worked for another farmer or a local manufacturer, his annual income would be about $30,000. Bowen can invest any funds that would be derived, if the farm were sold, to earn percent annually. (Ignore taxes)
a) Compute Bowen’s accounting profits.
4 From your knowledge of the relationship among the various cost functions, complete the following table.
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Q
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TC
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FC
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VC
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ATC
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AFC
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AVC
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MC
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0
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125
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10
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5
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20
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10.5
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30
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110
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40
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255
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50
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3
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60
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3
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70
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5
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80
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295
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6. The Blair Company’s three assembly plants are located in California, Georgia, and New Jersey. Previously, the company purchased a major subassembly, which becomes part of the final products, from an outside firm. Blair has decided to manufacture the subassemblies within the company and must now consider whether to rent one centrally located facility (e.g., in Missouri, where all the subassemblies would be manufactured) or to rent three separate facilities, each located near one of the assembly plants, where each facility would manufacture only the subassemblies needed for the nearby assembly plant. A single, centrally located facility, with a production capacity of 18,000 units per year, would have fixed costs of $900,000 per year and a variable cost of $250 per unit. Three separate decentralized facilities, with production capacities of 8,000, 6,000, and 4,000 units per year, would have fixed costs of $475,000, $425,000, and $400,000, respectively, and variable cost per unit of only $225 per unit, owing primarily to the reduction in shipping costs. The current production rates at the three assembly plants are 6,000, 4,500, and 3,000 units, respectively.
a) Assuming that the current production rates are maintained at the three assembly plants, which alternative should management select?
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