An aggregate production function F(K,L) gives you how much (aggre-gate) output Y

Question

An aggregate production function F(K,L) gives you how much (aggre-gate) output Y can be produced using (aggregate) inputs K and L, i.e. Y = F (K,L). For each of the following production functions (a) Determine whether they have constant returns to scale. (b) Can you write the per worker production function y = f(k)? That is, can you find an expression for output per worker y in terms of capital per worker k only?

F(K,L)= 8K+L F(K,L)=3K^2+2L^2 F(K,L)=2K^0.6 + 5L^0.6   F(K,L)=K^4/L^2 F(K,L)=K^3/L^2

F(K,L)=K^1/4 L^3/4

Explanation / Answer

a)

F(K,L)= 8K+L now increasing K,L x times F = 8Kx + Lx = x (8K+L) so function also increased x time so it has contant returns to scale and

F(K,L)=3K^2+2L^2 now increasing K,L x times F = 3(Kx)^2+2(Lx)^2 = x^2 (3K^2+2L^2) so function increased x^2 time so it has increasing returns to scale

F(K,L)=2K^0.6 + 5L^0.6 now increasing K,L x times F = 2(Kx)^0.6 + 5(Lx)^0.6 = x^0.6 (2K^0.6 + 5L^0.6) so function increased x^0.6 time so it has decreasing returns to scale   

F(K,L)=K^4/L^2 now increasing K,L x times F = (Kx)^4/(Lx)^2 = x^2 (K^4/L^2) so function increased x^2 time so it has increasing returns to scale

F(K,L)=K^3/L^2 now increasing K,L x times F = (Kx)^3 / (Lx)^2 = x (K^3/L^2) so function also increased x time so it has contant returns to scale

F(K,L)=K^1/4 L^3/4 now increasing K,L x times F = (Kx)^1/4 (Lx)^3/4 = x^1/3 (K^1/4 L^3/4) so function increased x^1/3 time so it has decreasing returns to scale   

b) to convert it into per worker we have to divide function by L i.e. f = F(K,L)/Land k = K/L only three function can be converted in f=(k) form

F(K,L)= 8K+L f = 8(K/L) + 1 = 8k + 1   

F(K,L)=K^3/L^2 so f = K^3/L*L^2 = (K/L)^3 = k^3

F(K,L)=K^1/4 L^3/4 now f = (K^1/4 L^3/4)/L = K^1/4/L^-/4 = (K/L)^1/4 = k^1/4

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