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Answers to Back-of-the-Chapter Problems
1. Maximizing average output is typically non-optimal. First, we should
emphasize that maximizing total output and maximizing average output
are two different things. For instance, in Table 5.2, the firm’s maximum
2. The production function, Q = 10L - .5L2 + 24K - K2, has marginal
products: MPL = 10 - L and MPK = 24 - 2K. Both marginal products
3. a. The marginal products for labor and capital are given by: MPL = 10 - L
and MPK = 24 - 2K. For L equal to K (in the range 0 to 10), capital's
b. Setting the price of each input equal to its marginal revenue product
4. a. Labor’s MPL is dQ/dL = 1 - L/400. Setting MRPL = wage implies
b. With P = $50, MRPL becomes 50 - L/8. The new solution is L* = 240
labor hours and Q = 168 dresses. With the price increase, optimal
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c. The 25% increase in productivity implies: MPL = (1.25)(1 – L/400).
5. The law of diminishing returns states that an input’s marginal product
declines as one increases its use past some point (holding other inputs
6. a. In the short run, the restaurant should hire more wait staff (and possibly
chefs) and squeeze in some extra tables so as to produce the maximum
b. In the longer run, the restaurant would be wise to increase its entire
scale of operation. Perhaps, it could expand its space in the new
b. The cost of the 68-60 mix is: ($.10)(68) + ($.07)(60) = $11.00 per day.
c. For a 200-pound steer, the cheapest mix is 56-70. Given constant
returns to scale, feeding a 250-pound steer would require (250/200) =
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8. In all likelihood, Chrysler's move to 24-hour production was prompted by
the high capital cost of building new factories combined with a slowdown
9. a. Production of steel by electric furnace has the lowest average cost per
10. Here is a graphical explanation. The firm's initial (optimal) input mix
occurs where the lowest isocost line is tangent to its isoquant. If labor’s
11. a. The grade improvements offered by extra hours of studying finance are
b. The “first” hour should be devoted to finance (an 8-point increase), the
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c. This allocation is optimal. Devoting her first 5 hours to finance and
economics offers the greatest point opportunities. Then, devoting 2
12. The optimal input condition is: MPK/MPL = PK/PL. The inputs’ respective
marginal products are: MPK = ßLαKß-1 and MPL = αLα-1Kß. Therefore,
13. a. For N1 = 16 and N2 = 24, the average catch at the first lake is Q1/N1 =
[(10)(16) - .1(16)2]/16 = 8.4 fish, and the average catch at the second
b. Movement between lakes will cease when all individuals obtain the
same average catch. Equating the average catches at the lakes implies
c. The commissioner seeks to maximize Q1 + Q2 subject to N1 + N2 = 40.
The optimum solution to this constrained maximization problem
implies that the marginal catch of the last fisher should be equal across
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Spreadsheet Problems
S1 a. The optimal quantity of labor is L* = 50. To find this solution by
hand, vary the labor hours in Cell C5 until MRPL = MCL (compare
b. In the long run, the firm should use L* = 64.64 and K* = 39.64 to
produce 175 units (the output found in part a) at minimum total cost.
c. After cutting its inputs by half, we have L* = 32.32 and K* = 19.82
implying an output of 105.8 units. Output has fallen by less than 50 percent
S2 a. (We assume that the student has created a new spreadsheet for this
problem based on the format of the example on page 222 of the text.)
In the short run with a fixed price P = $10 and capital at K = 9, the
b. The firm seeks to produce 180 units at minimum cost. Using the
spreadsheet optimizer, minimize cost in cell I9 by changing cells C5
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c. If PL falls to 18, we use the optimizer to find the new least-cost input
S3. a. Using the spreadsheet optimizer, the objective is to maximize cell E12
by changing cells C8, D8, and D10, subject to cell F12 being greater
or equal to the value of 32 in cell F14. According to the solution,
b. With the cost of aluminum only 10 percent greater than steel, the new
optimized solution finds that aluminum in trucks should increase to a
c. At a fleet-wide average of 36 MPG (and aluminum 30 percent more
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