Question

Suppose a consumer’s preference (utility function) is represented by: 0.25 0.75 U  X Y ; where U is total utility

of the consumer derived by consuming commodity X and Y. The price of X is Birr 4 while the price of Y is

Birr 2, and the consumer’s money income is Birr 1600. Then:

a. write the budget line equation and determine its slope

b. determine the marginal utilities for both commodities

c. calculate the optimal consumption bundle mathematically

d. calculate the income shares of both commodities

Answer 1

a) U(x,Y) =0.25, 0.75

X=birr 4

Y= birr2

Income, $I= 1600$

Budget line equation$I=PrX+PrY where Pr=price$

$1600=4X+2Y$

$U=X^{0.25}Y^{0.75}$

$MuX,\frac{du}{dx}=0.25X^{0.25-1}Y^{0.75}$

$MuY,\frac{du}{dy}=0.75X^{0.25}Y^{0.75-1}$

$MuX=0.25X^{-0.75}Y^{0.75}$

$MuX/PX= MuY/PY$

$\frac{0.25X^{-0.75}Y^{0.75}}{4}=$ $\frac{0.75X^{0.25}Y^{-0.25}}{2}$

Multiply both sides by 4,

$0.25X^{-0.75}Y^{0.75}=1.5X^{0.25}Y^{-0.25}$

$X^{-0.75}Y^{0.75}=6X^{0.25}Y^{-0.25}$

$X^{-0.75-0.25}=6Y^{0.25-0.75}$

$X^{-1}=6Y^{-1}$

$\frac{1}{X}=\frac{6}{Y}$

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

Substitute y/6 for x in the budget line equation

$1600=4X+2Y$

$1600=4\frac{Y}{6} +2Y$

$1600=2\frac{Y}{3}+2Y$

$1600=(2y+6y)/3$

$4800=8Y, Y=600$

$For x, 1600= 4x+2(600)$

$1600=4x+1200$

$4x=1600-1200$

$4x=400$

$x=100$

c) Optimal bundle

p1= 4, p2= 2, m(income) =1600

$MRS= -X/Y= -P1/P2 =-4/2=-2$

$X=2Y......(i)$

$2X+Y=1600....(ii)$

$5Y=1600$

$Y=1600/5= 320$

$X=2Y$

$X=2(320)=640$

$d)Income shares for X=\frac{dQx}{dQm}I/X$

$=4(1600/100)=64$

$Income shares for Y=\frac{dQy}{dQm}I/Y$

$=2(1600/600)=5.3$