Question

A consumer has set of budget of $800 for the consumption of good X a d Y. The price of Good X is $10 and the price of good Y is $40.The consumer has a utility function given by U (X,Y)

= X(0.80)Y(0.20).Find the optimal soloution using a methof that does not involves equations and illustrate the soloution?

Answer 1

$MU_X=\frac{dU(X,Y)}{dX}=0.8X^{-0.2}Y^{0.2}\\ MU_Y=\frac{dU(X,Y)}{dY}=0.2X^{0.8}Y^{-0.8}\\ MRS=\frac{MU_X}{MU_Y}=\frac{0.8X^{-0.2}Y^{0.2}}{0.2X^{0.8}Y^{-0.8}}\\ MRS=\frac{4Y}{X}$

At optimal point

$MRS=\frac{PY}{PX}=\frac{4Y}{X}=\frac{10}{40}\\X=16Y$

budget constraint

$XP_X+YP_Y=M\\10X+40Y=800\\10(16Y)+40Y=800\\200Y=800\\Y=4\\X=16(4)=64\\X=64$

Illustration\\ to lot the budget line

x intercept$=\frac{800}{10}=80$

Y intercept$=\frac{800}{40}=20$