Question

Q2. Elaine is a typical student and she spends her $200 allowance on beer (X) and books (Y). Her utility function is given by U = 20 XY and PX = $2 and PY = $5.

(d) Calculate Elaine’s elasticity of demand for beer and her cross-price elasticity of demand for books with respect to the price of beer.

(e) Sketch Elaine’s demand curve for beer using at least four price-quantity demanded combinations.

Answer 1

Given

U=20XY

MUX=dU/dX=20Y

MUY=dU/dY=20X

Utility maximization requires

MUX/MUY=PX/PY

20Y/20X=PX/PY

Y/X=PX/PY

Y=(PX/PY)X

Now budget constraint is given as

I=X*PX+Y*PY

Set Y=(PX/PY)X

I=X*PX+(PX/PY)*PY*X

I=2XPX

or X=0.5I/PX (demand function of X)

and Y=(PX/PY)0.5*I/PX=0.5I/PY (demand function of Y)

d)

We have determined earlier that

X=0.5I/PX

dX/dPX=-0.5I/PX^2

Price elasticity of demand for beer (X)=(dX/dPX)*(PX/X)=(-0.5I/PX^2)*PX/(0.5I/PX)=-1

Given

We have determined earlier that

Y=0.5I/PY

dY/dPX=0

Cross price elasticity of demand of books with respect to price of beer=(dY/dPX)*(PX/Y)

Cross price elasticity of demand of books with respect to price of beer=0*(PX/Y)=0

We observe that price elasticity of demand of beer is constant at -1 and cross price elasticity of demand for books does not depend upon the price of beer in this case.

e)

X=0.5I/PX

Set I=200

X=0.5*200/PX=100/PX

or PX=100/X

X	PX
2	50.00
10	10.00
20	5.00
30	3.33
40	2.50
50	2.00
60	1.67
70	1.43
80	1.25
90	1.11