Question

Given the utility function:

U = X 3/4 . Y1/4

Estimate the demand functions of commodity X and commodity Y using Lagrange method, if it is given that price of X is Px and price of Y is Py and Income is M.

Answer 1

$L=X*3/4*Y*1/4+\lambda(M-PxX-PyY)$

$\frac{\delta L}{\delta X}=3/16*Y-\lambda Px=0$

$\frac{\delta L}{\delta Y}=3/16*X-\lambda Py=0$

$\frac{\delta L}{\delta \lambda}=M-PxX-PyY=0$

$\frac{3/16*Y}{\lambda}=Px$

$\frac{3/16*X}{\lambda}=Py$

$M-\frac{3/16*Y}{\lambda}X-\frac{3/16*X}{\lambda}Y=0$

$M-\frac{3/8*YX}{\lambda}=0$

$\lambda*M=3/8*Y*X$

Commodities X and Y are substitutes does not matter their ratio you will get maximum utility