Question

If a consumer is consuming two commodities X and Y and his utility function U (X,Y)=2XY+6.if the price of the two commodity are 2 and 6 respectively, and the consumer has a total income of 80 birr to be spent on the two goods, a)find the utility maximizing Quantity of good X and Y. b) find the MRSxy at equilibrium. c) find the MRSyx at equilibrium.

Answer 1

$U(X,Y)= 2XY+6$

$P_x= 2$

$P_y=6$

I= 80

Budget line

2x+ 6Y= 80

a) Utility maximization Quantity

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

$\frac{2Y}{2X}=\frac{2}{6}$

12Y= 4X

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

X= 3Y

Plug in the above in the budget line

2(3Y)+ 6Y= 80

12Y= 80

Y*= 6.67

2X+ $6(\frac{1}{3}X)$= 80

4X= 80

X*= 20

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

$= \frac{2Y}{2X}= \frac{2(\frac{20}{3})}{2(20)}$

=$\frac{40}{3}\times\frac{1}{40}= 0.33$

c) $MRS_{yx}= \frac{Mu_y}{Mu_x}= \frac{2X}{2Y}$

= $\frac{20\times 20}{2\times \frac{20}{3}}$ = 3