Suppose the total-cost function for a firm is given by .C=qw2/3 v1/3
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”).
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}"
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