Question

13. Suppose the total-cost function for a firm is given by C-qwv a) Use Shephard's lemma to compute the (constant output) demand functions for inputs l and k. b) Use your results from part (a) to calculate the underlying production function for q.

Answer 1

a) Given, total cost function is c = q w^2/3 v^1/3 the partial derivative of the total cost function with respect to input prices is the firm�s conditional input demand L = partial differential T C/partial differential w = 2/3 q w^-1/3 v^1/3 = 2/3 q (v/w)^1/3 k = partial differential T C/partial differential w = 1/3 q w^2/3 v^-2/3 = 1/3 q (w/v)^2/3 b) let a = w/v so L = 2/3 q (1/a)^1/3 (1) K = 1/3 q a^2/3 (2) solving for a from equation 1 a = (2/3 times q/L)^3 substituting a in (1) K = 1/3 times q times (2/3 times q/L)^3 times 2/3 K = 1/3 times q times 4 q^2/9 L^2 implies K = 4 q^3/27 L^2 implies q = 3 squareroot K times 27 L^2/4