Question

Consider the production function Q = K2L , where L is labor and K is capital. a.[4] What is the Marginal Product of Capital for this production function? Is it increasing, decreasing, or constant? Briefly explain or show how you arrived at your answer. b.[4] Does this production function exhibit increasing, constant or decreasing returns to scale? Briefly explain or show how you arrived at your answer. c.[5] If the firm has capital fixed at 15 units in the short run and the firm must produce 8,000 units of the good, find the cost-minimizing quantity of Labor. If labor is paid $200 and capital is rented at $400, what is the Total Cost at this short run equilibrium? d.[10] Find the long-run cost minimizing quantities of Labor and Capital when labor is paid $200, Capital is rented at $400, and the firm must produce 8,000 units of the good. What is Total Cost at the long run equilibrium?

Answer 1

Q = K²L

(a)

MPK = Q/K = 2KL

As K increases, (2KL) increases, so MPK increases. The MPK function is increasing in K.

(b)

Doubling both inputs, new production function becomes

Q* = (2K)²L = 2²K²L = 4 x Q

Q*/Q = 4 > 2

Since doubling both inputs more than doubles the output, there are increasing returns to scale.

(c)

When K = 15 and Q = 8,000:

(15)² x L = 8,000

225L = 8,000

L = 35.56

Total cost ($) = wL + rK = 200 x 35.56 + 400 x 15 = 7,112 + 6,000 = 13,112

(d)

In long run, cost is minimized when MPL/MPK = w/r = 200/400 = 1/2

MPL = Q/L = K²

MPL/MPK = (K²) / (2KL) = K/2L = 1/2

2L = 2K

L = K

Substituting in production function with Q = 8,000:

(L)²L = 8,000

L³ = 8,000

L = 20

K = 20

Total cost ($) = 200 x 20 + 400 x 20 = 4,000 + 8,000 = 12,000