LB A firm uses only two inputs L and K. which have prices PL and PK Its producti
ID: 1104662 • Letter: L
Question
LB A firm uses only two inputs L and K. which have prices PL and PK Its production function s q = 12-La-Kus (3) a. Does the firm have increasing, constant or decreasing returns to scale? Explain how you know. (4) b. Derive the marginal product of K (remember you can leave it nsinplified) (3) c. Assume now that q = L-Ka. Using this, solve for K (as a function of L and q). (14) c. Assume PL-1, P,-8, and q 2. Substitute these values into the cost expression and derive using calculus how much L it employs to minimize costs. Points for (explaining) each step and you may leave your answer as a fraction. From this, find how much capital and the cost of producing 2 units.Explanation / Answer
a) Multiply each factor with a
new q = 12(aL)^(1/2)(aK)^(1/3)
= 12a^(1/2 + 1/3)L^(1/2)K^(1/3)
= a^(5/6) x 12L^(1/2)K^(1/3)
Since new q is less than old q because a^(5/6) < a, there are decreasing returns to scale
b) MPK = 12 x (1/3) L^(1/2) K^(-2/3) = 4L^(1/2) K^(-2/3)
c) q = LK^(1/2)
q/L = K^(1/2)
K* = (q/L)^2
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