# Below are production functions that turn capital (K) and labor (L) into output. For each of...

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

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.

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