Question

Given the production function y=f(L,K)=4L^1/4K^3/4.

a) Does this function have increasing, decreasing, or constant returns to scale? explain your answer.

b)Find the factor demand functions for capital, K, and Labor, L.

Answer 1

Solution:

Production function is y = f(L, K) = 4*L^1/4*K^3/4

a) Returns to scale for any production function is identified by knowing that if all the inputs or factors of production (here, only labor and capital) are changed/increased by same factor, then by what factor does the total output changes. If output increases by same factor, we say that function exhibits constant returns to scale, if output changes by less than that factor then decreasing returns to scale and if output changes by more than that factor function exhibits increasing returns to scale.

Say, both inputs in given example, K and L increase by factor t, then new output can be found as:

y' = f(tL, tK) = 4*(tL)^1/4*(tK)^3/4

y' = 4*t^1/4+3/4*L^1/4*K^3/4 = 4t*L^1/4*K^3/4

y' = t*(4*L^1/4*K^3/4) = ty

So, output changes by same factor, t. Thus, this production function exhibits constant returns to scale.

b) Finding the factor demand functions:

Assuming that the prices of inputs are wage, w, for labor and rental rate, r, per unit of capital, we have the total cost for firm as:

TC = w*L + r*K

Optimal level of capital and labor occurs where the ratio of factor prices equals the ratio of marginal products of each input

Marginal product of labor, MPL = = 4*(1/4)*L^1/4-1*K^3/4 = (K/L)^3/4

Marginal product of capital, MPK = = 4*L^1/4*(3/4)*K^3/4-1 = 3*(L/K)^1/4

So, ratio of marginal products, MPL/MPK = [(K/L)^3/4]/[3*(L/K)^1/4] = (1/3)*(K/L)

So, at optimal, we must have MPL/MPK = w/r

So, (1/3)*(K/L) = w/r

K = 3wL/r

Production function becomes: y = 4*L^1/4*(3wL/r)^3/4

So, y = 4*L*(3w/r)^3/4

L = (y/4)*(r/3w)^3/4

So, K = (3w/r)*(y/4)*(r/3w)^3/4 = (y/4)*(3w/r)^1/4