Question

Firm produce output according to C.D technology Y=F(K,L)=AK^α L^(1-α). Derive short run profit maximising input demand, output supply and profit functions with this technology, ignoring firm's shutdown condition.

Answer 1

the marginal products for the Cobb–Douglas production function are: MPL = (1 – α)Y/L. MPK = αY/K. Competitive profit-maximizing firms hire labor until its marginal product equals the real wage, and hire capital until its marginal product equals the real rental rate. Using these facts and the above marginal products for the Cobb–Douglas production function, we find: W/P = MPL = (1 – α)Y/L. R/P = MPK = αY/K. Rewriting this: (W/P)L = MPL × L = (1 – α)Y. (R/P)K = MPK × K = αY. Note that the terms (W/P)L and (R/P)K are the wage bill and total return to capital, respectively.