Question

1. A firm’s production function is given by Q(L,K)=5L+2k. Firm’s budget is 200,000 dollars. How much labor and capital should this firm optimally hire assuming that price of labor is 2 dollars per unit and price of capital is 5 dollars per unit? Does this production function satisfy the law of diminishing returns in the short run? What about decreasing returns to scale?

Answer 1

Q = 5L + 2K

(a)

A linear production function signifies and L and K are substitutes and isoquants are linear, touching both axes. Optimal input mix involves use of only one input.

Total cost (TC) ($) = wL + rK

200,000 = 2L + 5K

When L = 0, K = 200,000/5 = 40,000 and Q = 5 x 0 + 2 x 40,000 = 80,000

When K = 0, L = 200,000/2 = 100,000 and Q = 2 x 100,000 + 5 x 0 = 200,000

Since output is higher when L = 100,000 and K = 0, this is the optimal inpit mix.

(b)

MPL = Q/L = 5

MPK = Q/K = 2

Since MPL is independent of L, and MPK is independent of K, diminishing returns do not exist.

(c)

Doubling both inputs gives rise to the new production function:

Q1 = 5 x (2L) + 2 x (2K) = 10L + 4K = 2 x (5L + 2K) = 2 x Q

Q1/Q = 2

Since doubling both inputs exactly doubles output, there is constant returns to scale (and not decreasing returns).