Suppose a good is produced according to the following production function: f (E,

Question

Suppose a good is produced according to the following production function:

f (E, K) = E1/2K1/2

            so that the marginal product of labor and capital are

                                                MPE = (1/2)(K/E)1/2

MPK = (1/2)(E/K)1/2

If w = $8 and r = $4, determine the necessary conditions for the input choices, K and E to be cost-minimizing. In other words, what is the cost-minimizing ratio of K to E for this firm? What will happen to the cost-minimizing ratio if the wage stays at $8 but the price of capital rises to $10?

Explanation / Answer

Cost is minimized when MPE/MPK = w/r

MPE/MPK = [(1/2) x (K/E)1/2] / [(1/2) x (E/K)1/2] = K/E

(a) When w = $8 and r = $4,

K/E = 8/4 [Required input ratio]

(a) When w = $8 and r = $10,

K/E = 8/10 = 4/5 [Required input ratio]

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