Question

Below are production functions that turn capital (K) and labor (L) into output. For each of the production functions below, state and PROVE whether it is Constant/Increasing/or Decreasing Returns to scale. That is, you want to see how production changes when you increase all inputs (K,L,(M)) by a factor of α, where α > 1: (4 points each)

a) F(K,L) = K^1/3*L^1/3+2K+3

b) F(K,L) = sqr(K^3+L^3)

c) F(K,L) = (K^2/4+L^2/4)^2

d) F(K,L,M) = min(K,L)*M

Answer 1

Solution: a) This is not a homogeneous function. But analysing its each term we get that first term

K^1/3*L^1/3 follows decreaing returns to scale. Second term follows constant returns to scale. So overall F(K, L) = K^1/3*L^1/3+2K+3 follows Decreasing Returms to Scale.

b) F(K, L)= (K^3+L^3)^1/2

F(aK, aL)= a^3/2(K^3+L^3)^1/2= a^3/2F(K, L) > aF(K, L).

Hence it is an IRS production function.

c) F(K, L)= (K^2/4+L^2/4)^2

F(aK, aL)= a^2/4*2(K^2/4+L^2/4)^2= aF(K, L) = aF(K, L).

Hence it is an CRSproduction function.

d) F(K, L, M)= min(K, L)*M

F(aK, aL, aM)= a^2min(K,L)*M= a^2F(K, L) > aF(K, L).

Hence it is an IRS production function.