Question

Suppose the total-cost function for a firm is given by .C=qw^2/3 v^1/3

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 (q as a function of “k” and “l”).

Answer 1

Given the total-cost function for a firm is given by .$C=qw^{2/3 }v^{1/3}$

The partial derivative of the total cost function with respect to input prices is the firm's conditional input demand.

$L=\frac{\delta C}{\delta w}=\frac{2}{3}qw^\frac{-1}{3}v^\frac{1}{3}=\frac{2}{3}q(\frac{v}{w})^\frac{1}{3}$ ....1

$K=\frac{\delta C}{\delta v}=\frac{1}{3}qw^\frac{2}{3}v^\frac{-2}{3}=\frac{1}{3}q(\frac{w}{v})^\frac{2}{3}...2$

b)

Let a$=\frac{w}{v}$

L=$\frac{2}{3}q(\frac{1}{a})^\frac{1}{3}$

K=$\frac{1}{3}q(a)^\frac{1}{3}$

Solving for a;

a=$(\frac{2}{3}\times\frac{q}{L})^3$

Substituting in equation 2;

K$=\frac{1}{3}q(\frac{2}{3}\times\frac{q}{L})^2$

K=$\frac{4Q^3}{27L^2}$

q=$(\frac{K\times27L^2}{4})^\frac{1}{3}$