Question

Given utility function: U(x, y) = X1/3Y2/3, Px = 2, Py = 5 and M = 400, find:

a. The demand equation for X and Y.

b. The utility maximizing levels of X and Y.

c. The maximum utility.

d. The MRSx, y at the optimum level.

Answer 1

U=$(X^ \frac {1}{3}Y^\frac{2}{3})$

I= 400

$_x= 2$

$P_y= 5$

a)Demand functions

For utility maxmization

$\frac{\frac{1}{3}X^\frac{-2}{3}Y^\frac{2}{3}}{\frac{2}{3}X^\frac{1}{3}Y^\frac{-1}{3}}=\frac {P_x}{P_y}$

$\frac{\frac {1}{3}Y}{\frac{2}{3}X}=\frac {P_x}{P_y}$

$\frac{2}{3}P_xX=\frac{1}{3}P_yY$

X= $\frac{P_yY}{2P_x}$ 

$2P_xX=P_yY$

$Y=\frac{2P_xX}{P_y}$ 

Plug into the budget constraint

m= $P_xX+P_yY$

m=$P_x(\frac{P_yY}{2P_x})+P_yY$

2m=$3P_yY$

$Y^*= \frac{2m}{3P_y}$

m=$P_xX+P_yY$

$m=P_xX+P_Y(\frac{2P_xX}{P_y})$

m$=P_xX+2P_xX$

X*=$\frac{m}{3P_x}$

b) Utility Maximizing levels

$Y^*= \frac{2\times 400}{3\times 5}=\frac{800}{15}= 53.33$

c) Maximum utility of consuming the two goods

= 66.3=120

d) MRS

$MRS_(xy)= \frac{Mu_x}{Mu_y}= \frac{\frac{1}{3}X^\frac{-2}{3}Y^\frac{2}{3}}{\frac{2}{3}X^\frac{1}{3}Y^\frac{-1}{3}}$

$=\frac{\frac{1}{3}Y}{\frac{2}{3}X}=\frac{Y}{2X}=$

$=\frac{53.33}{66.67\times 2}=\frac{53.33}{134.34}=0.4$