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Unformatted text preview: Chapter 5: Answers to Questions and Problems 1. a. When K = 16 and L = 16 , ( 29 ( 29 0.75 0.25 16 16 16 Q = = . Thus, AP L = Q/L = 16/16 = 1. When K = 16 and L = 81 , ( 29 ( 29 ( 29 ( 29 0.75 0.25 16 81 8 3 24 Q = = = . Thus, AP L = 24/81 = 8/27 . b. The marginal product of labor is ( 29 3 4 2 L MP L = . When L = 16 , ( 29 3 4 2 16 1/ 4 L MP = = . When L = 81 , ( 29 3 4 2 81 2 / 27 L MP = = . Thus, as the number of units of labor hired increases, the marginal product of labor decreases ( 29 ( 29 16 1/ 4 2 / 27 81 L L MP MP = = , holding the level of capital fixed. c. We must equate the value marginal product of labor equal to the wage and solve for L. Here, ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 3/ 4 3/ 4 $100 2 200 L L VMP P MP L L = = = . Setting this equal to the wage of $25 gives ( 29 3/ 4 200 25 L = . Solving for L , the optimal quantity of labor is L = 16. 2. See Table 51. (1) (2) (3) (4) (5) (6) (7) Capital Labor Output Marginal Product of Capital Average Product of Capital Average Product of Labor Value Marginal Product of Capital K L Q MP K AP K AP L VMP K 20 1 20 50 50 50 2.50 100 2 20 150 100 75 7.50 200 3 20 300 150 100 15 300 4 20 400 100 100 20 200 5 20 450 50 90 22.50 100 6 20 475 25 79.17 23.75 50 7 20 475 67.86 23.75 8 20 45025 56.25 22.5050 9 20 40050 44.44 20100 10 20 300100 30 15200 11 20 150150 13.64 7.50300 Table 51 a. Labor is the fixed input while capital is the variable input. b. Fixed costs are 20($15) = $300. c. To produce 475 units in the leastcost manner requires 6 units of capital, which cost $75 each. Thus, variable costs are ($75)(6) = $450. d. Using the VMP K = r rule, K = 5 maximizes profits. 51 e. The maximum profits are $2(450) $15(20) $75(5) $225 = . f. There are increasing marginal returns when K is between 0 and 3. g. There are decreasing marginal returns when K is between 3 and 11. h. There are negative marginal returns when K is greater than 7. 3. The law of diminishing marginal returns is the decline in marginal productivity experienced when input usage increases, holding all other inputs constant. In contrast, the law of diminishing marginal rate of technical substitution is a property of a production function stating that as less of one input is used, increasing amounts of another input must be employed to produce the same level of output. 4. a. FC = 50 b. ( 29 ( 29 ( 29 ( 29 2 3 10 25 10 30 10 5 10 $8,250 VC = + + = ....
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 Summer '10
 HOLLAND

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