Question

1.Suppose a consumer consuming two commodities X and Y has the following utility function TU= X0.5 Y0.5. If price of good X and Y are 2 and 3 respectively and income constraint is Birr 60.

A. Formulate the budget equation

B. Find the MRSXY at optimum.

C. Find the quantities of good X and Y which will maximize utility

Answer 1

$P_x= 2$

$P_y= 3$

I= 60

a) Budget Eqn

2X+ 3Y= 60

b) MRS$_{xy}= \frac{Mu_x}{Mu_y}$

$Mu_x= 0.5X^{-0.5}Y^{0.5}$

$Mu_y= 0.5 X^{0.5}Y^{-0.5}$

MRS$_{xy}= \frac{ 0.5X^{-0.5}Y^{0.5}}{0.5 X^{0.5}Y^{-0.5}}$

$= \frac{ 0.5Y^{0.5}Y^{0.5}}{0.5 X^{0.5}X^{0.5}}= \frac{Y}{X}$

c) Quantities that maximize utility

$\frac{Mu_x}{Mu_y}=\frac {P_x}{P_y}$

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

3Y= 2X

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

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

Plug into the budget equation

2($\frac{3}{2}Y)+ 3Y= 60$

6Y= 60

Y$^*= 10$

$3(\frac{2}{3}X)+ 2X= 60$

4X= 60

X$^*= 15$