Answer to Question #269193 in Microeconomics for zack

Question #269193

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”).

1
Expert's answer
2021-11-22T10:04:17-0500

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}"



Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!

Comments

No comments. Be the first!

Leave a comment

LATEST TUTORIALS
New on Blog
APPROVED BY CLIENTS