Question

Given the utility function:

U = X ^3/4 . Y^1/4

Find out the optimal quantities of the two commodities using Lagrange method and simplified procedure, if it is given that price of X is Rs.6 and price of Y is Rs.3 and income is equal to Rs.120.

Answer 1

The quantities of the two commodities are optimal when:

$MUx/Px = MUy/Py$

$MUx = U'(X) = 0.75×(Y/X)^{1/4}$

$MUy = U'(Y) = 0.25×(X/Y)^{3/4}$

$0.75×(Y/X)^{1/4}/6 = 0.25×(X/Y)^{3/4}/3,$

3Y = X,

6×3Y + 3Y = 120,

Y = 5.71,

X = 17.14.