Question

suppose a consumer consuming two comodities X and Y has the following utility function U=X^0.4Y^0.6. if the price of good X and Y are 2 and 3 respectively and the income constraint is birr 50. find the quantities of X anf Y which maximize utility

Answer 1

$U(X,Y) = X^{0.4}Y^{0.6}$

Budget line

$50 = X+Y \\ P(X)=2 \\ P(Y) = 3$

At optimum : $\frac{MU(X)}{P(X)}= \frac{MU(Y)}{P(Y)}$

$MU(Y) = \frac{ΔU}{ΔX} = 0.4X^{-0.6}Y^{0.6} \\ MU(y) = \frac{ΔU}{ΔY} = 0.6X^{0.4}Y^{-0.4} \\ \frac{0.4X^{-0.6}Y^{0.6}}{2} = \frac{0.6X^{0.4}Y^{-0.4}}{3} \\ 0.2X^{-0.6}Y^{0.6}=0.2X^{0.4}Y^{-0.4} \\ X=Y$

Putting X=Y in budget line

$50 = Y+Y \\ 50 = 2Y \\ Y = \frac{50}{2} = 25 \\ X= 25 \\ U(X,Y)_{max} = 25^{0.4} \times 25^{0.6} = 25$