Question

Aconsumer consuming two commodities x and y has the following utility functions U= xy + 2z. If the price of the two commodities are 4 and 2 respectively and his/her budget is birr 60.

Find the quantities of good x and y which with miximize utility

Find the MRSxy at optimum

Answer 1

A.

U=XY+2X

budget line

I=P_xX+P_yY , where P_x=4,P_y=2 ,Hence

I=4X+2Y

$\frac{MUx}{MUy}$

At maximum Utility

$\frac{MUx}{MUy}=\frac{Px}{Py}$

MU_x=$\frac{du}{dx}$ XY+2X

=Y+2

MU_y=$\frac{du}{dy}$ XY+2X

=X

Now

$\frac{MUx}{MUy}=\frac{Px}{Py}$

$\frac{Y+2}{X}=\frac{4}{2}$

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

Y+2=2X

Y=2X-2 ,substitute this to budget line:

60=4X+2Y

60=4X+2$\times$ (2X-2)

60=4X+4X-4

64=8x

x=8

Y=2X-2

Y=(2$\times8$ )-2

Y=16-2

Y=14

At maximum utility X=8,Y=14

B).

MRS_xy=$\frac{MUx}{MUy }$

=$\frac{Y+2}{X}$ ,where Y=14,X=8

=$\frac{14+2}{8}$

=$\frac{16}{8}$

=2

Hence, MRS_xy=2