3 8 points Consider a firms cost minimization problem min wL

3. (8 points) Consider a firm\'s cost minimization problem. min wL+ rK subject to the production function, V +K2q L,K Here, the marginal product of labor is MPL L2 7, the marginal product of capital is MPk = K(L2-K2)--\" When answering the following questions, round to four decimal places. Find the conditional factor demands for L and K with respect to q when w = 3 and r = 4. What is the total cost function and average cost function? What type of returns to scale does this firm have? a. b. c.

Solution

(a) Cost is minimized when MPL/MPK = w/r = 3/4

MPL/MPK = L/K = 3/4

4L = 3K, therefore

K = 4L/3 and L = 3K/4

Substituting in production function,

[L² + (4L/3)²]^1/2 = q

Squaring both sides,

L² + (4L/3)²] = q²

L² + (16L²/9) = q²

25L²/9 = q²

25L² = 9q²

L² = 9q²/25

L = 3q/5 [Conditional demand for labor]

K = 4L/3 = 4 x (3q/5)/3 = 4q/5 [Conditional demand for capital]

(b) Total cost (C) = wL + rK

C = 3 x (3q/5) + 4 x (4q/5)

C = (9q/5) + (16q/5)

C = 25q/5

C = 5q [Total cost]

Average cost = C/q = 5

(c) Let us double both inputs so that new production function becomes

q1 = [(2L)² + (2K)²]^1/2 = (4L² + 4K²)^1/2 = (4)^1/2(L² + K²)^1/2 = 2 x (L² + K²)^1/2 = 2 x q

q1/q = 2

Since doubling both inputs exactly doubles output, there is constant returns to scale.