Consider the Cobb-Douglas production function F(L,K) = AL?K?. Suppose that ? = 2, ? = 3, the firm has 3 units of capital and the firm's general productivity level is 20. (a) What is the firm's long-run production function? (b) What is the firm's short-run production function? (c) If the firm employs 10 workers, what are the marginal products of labor and capital? (d) If the firm employs 10 workers, what is the marginal rate of technical substitution for labor with capital? (e) Does this firm's technology exhibit increasing, decreasing or constant returns to scale?
What will be an ideal response?
(a) F(L,K) = 20L2K3
(b) F(L) = 20L233 = 180L2
(c) The marginal product of labor is given by MPL = ?AL?-1K? = (2)(20)(101)(33) = 3,600. Similarly, the marginal product of capital is given by MPK = ?AL?K?-1 = (3)(20)(102)(32) = 36,000.
(d) This question can be answered in either of two ways. First, MRTSLK = MPL/MPK, which implies that MRTSLK = 3,600/36,000 = 0.10. Secondly, for the Cobb-Douglas production function, MRTSLK = (?/?)(K/L) = (2/3)(3/10) = 0.10.
(e) Since ? + ? = 2 + 3 = 5 >1, the firm will experience increasing returns to scale.
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