Question

1. Given two utility functions U(x, y) = x^2/3 y^4/5 and U(x, y) = x² + y, with Px = 2, Py = 1, budget is 10 unit, show the consumer choice respectively.

Answer 1

Case 1

U(x,y)=x^2/3y^4/5

Marginal utility of X=MUX=dU(x,y)/dx=(2/3)*x^-1/3y^4/5

Marginal utility of Y=MUY=dU(x,y)/dy=(4/5)*x^2/3y^-1/5

Utility maximization requires

MUX/MUY=Px/Py

Given Px=2, Py=1

((2/3)*x^-1/3y^4/5)/((4/5)*x^2/3y^-1/5)=2/1

(5/6)*(y/x)=2

y=2*(6/5)x=2.40x

Budget constraint gives us

xPx+yPy=I

2x+y=10

Put y=2.4x

2x+2.4x=10

4.4x=10

x=10/4.4=2.2727 or 2.27 -----optimal value of x

y=2.4x=2.4*2.2727 =5.45448 or 5.45   ----optimal value of y

Case 2

U(x,y)=x²+y

Marginal utility of X=MUX=dU(x,y)/dx=2x

Marginal utility of Y=MUY=dU(x,y)/dy=1

We can see that MUX/Px=2x/2=x

MUy/y=1/1=1

So, We see that marginal utility per dollar spent in case of x is increasing. while it is constant at 1 in case of y. So, consumer should consume x only.

Budget constraint gives us

xPx+yPy=I

2x+y=10

Put y=0

2x+0=10

x=5   ----optimal value of x

y=0   ----optimal value of y