Question

(a) Find the demand for y in terms of just prices using the utility function U(x, y) = [x^2 + y^2 ]^1/2 .

(b) Is y an inferior or normal good?

(c) Are x and y substitutes or complements?

(d) Let Px = 1 and Py = 2. What is the price elasticity of demand for y at equilibrium?

(e) Find the cross price elasticity of demand for y. (Hint: Use Px as your cross price.)

Answer 1

Ans A)

U(x,y)=(x^2+y^2)^(0.5)

Budget equation is

I=Px*X+Py*Y

At equilibrium bundle slope of budget line equals to slpe of indifference curve

MRS=-Px/Py

MRS=dU/dx/dU/dy

dU/dX=X/(x^2+y^2)^(0.5)
dU/dY=Y/(x^2+y^2)^(0.5)

X/Y=Px/Py

X=Px*Y/Py

USing this equality into budget equation

I=Px^2Y/Py+PyY=Y(Px^2/Py+Py)

I/(Px^2/Py+Py))=Y

I*Px/(Px^2+Py^2)=X

Ans B)

If consumption of good Y increases with income I then it is normal good otherwise inferior good

dI/dy=(Px^2/Py+Py); as Px and Py>0 dI/dY>0
hence Y is nomral good for sure

Ans C)

If inncrease in the price of good X decrease the demand of good Y then X and Y are complements otherwise substitutes

dX/dPy=I*Px*(2Py)(-0.5)/(Px^2+Py^2)^1.5<0

dX/dPy<0 hence X and Y are substitutes

Ans C)

Px=1,Py=2 then X/Y=0.5 therefore Y=2X
I=X+2(2X)=5X
0.2I=X and 0.4I=Y

Price elasticity of demand =(dX/X)(Px/dPx)=(dX/dPx)(Px/X)=

dX/dPx=I/(Px^2+Py^2)-IPx^2/(Px^2+Py^2)^1.5=I/(5)-I/5^1.5=I(0.2-0.09)=0.11I

Price elasticity of demand=0.11I*/0.2I=0.055

Ans D)

(dY/dPX)(Px/Y)=I*Py*(2Px)(-0.5)/(Px^2+Py^2)^1.5*(1/0.4I)=-(I*2/5^1.5)(1/0.4I)=-5/5^1.5=-0.45