Question

Given production function: Q=L^3/5K^1/5.

Where L is labor, K is capital, w is wage rate, and r is rental rate.

What kinds of returns to scale does your firm face?

Find cost minimizing level of L and K, and long run cost function.

Answer 1

Q = L^3/5K^1/5

(i) When both inputs are doubled, new production function becomes

Q1 = (2L)^3/5(2K)^1/5 = 2^3/52^1/5 x L^3/5K^1/5 = 2^4/5 x Q

Q1/Q = 2^4/5 < 2

Since doubling both inputs less than doubles output, there is decreasing returns to scale.

(ii) Cost is minimized when MPL/MPK = w/r

MPL = Q/L = (3/5) x K^1/5 / L^2/5

MPK = Q/K = (1/5) x L^3/5 / K^1/5

MPL/MPK = 3 x (K/L) = w/r

3K/L = w/r

K = wL/3r

Substituting in production function,

Q = L^3/5(wL/3r)^1/5

Q = L^3/5L^1/5(w/3r)^1/5

Q = L^4/5(w/3r)^1/5

L^4/5 = Q x (3r/w)^1/5

L = [Q x (3r/w)^1/5]^5/4 = Q^5/4 x (3r/w)^1/4

K = [Q^5/4 x (3r/w)^1/4] x (w/3r) = Q^5/4 x (w/3r)^3/4

Total cost = wL + rK = w x [Q^5/4 x (3r/w)^1/4] + r x [Q^5/4 x (w/3r)^3/4] = Q^5/4 x [{w x (3r/w)^1/4} + {r x (w/3r)^3/4]