Question

Given the production function
Q = AKaLpNb
where Q is the rate of output and K, L, and N represent inputs of capital, labor, and land, respectively, determine
a. The specific conditions (i.e., values of a, p\ and b) under which returns to scale would be increasing, constant, and decreasing.
b. The equation for the marginal product function for each input

Answer 1

Q = A*K^a*L^p*N^b.
a. The specific conditions (i.e., values of a, p and b) under which returns to scale would be: - increasing: a, p, b > 1,
- constant: a, p, b = 1,
- decreasing: a, p, b < 1.
b. The equations for the marginal product functions for each input are:
MPK = Q'(K) = a*A*K^(a - 1)*L^p*N^b,
MPL = Q'(L) = p*A*K^a*L^(p - 1)*N^b,
MPN = Q'(K) = b*A*K^a*L^p*N^(b - 1).