Question

For a firm with production function f(L,K)=√L+√K, find its cost function for arbitrary values of w and r. That is,find a formula for the cost of producing q units that includes q,and also w and r,as variables. Also find marginal and average cost,and draw a plot that shows both cost functions in the same graph.

Answer 1

Production function Q = L^0.5 + K^0.5. MPL = 0.5L^-0.5, MPK = 0.5K^-0.5. This gives MRTS = -K/L. Wage rental ratio is w/r. At optimal choice, K/L = (w/r)^2 or K = (w/r)^2L. This implies

Q = L^0.5 + (w/r)L^0.5

Q = L^0.5(1 + w/r)

This gives L = Q^2/(1 + w/r)^2

= r^2Q^2/(r + w)^2

K = w^2Q^2/(r + w)^2

Cost function C = wL + rK

C = wr^2Q^2/(r + w)^2 + rw^2Q^2/(r + w)^2

= wrQ^2(r + w)/(r + w)^2

This gives C = (wr/(w + r))Q^2

MC = 2Q*  (wr/(w + r))

AC = Q*  (wr/(w + r))

Since w and r are constant, MC and AC originate from origin and are upward sloping with MC lying above AC.