Question

Bridgit’s utility function is U(x1, x2)= x1 + ln x2 x1 - stamps

x2 - beer

Bridgit’s budget p1 x1 + p2 x2 = m

p1 – price of stamps

p2 – price of beer

m – Bridgit’s budget

a) What is Bridgit’s demand for beer and stamps?

b) Is it true that Bridgit would spend every dollar in additional income on stamps?

c) What happens to demand when Bridgit’s income changes (i.e. find the income elasticity)?

d) What happens to demand when p1 and p2 increase (i.e. find the price elasticities)?

Answer 1

Given-

U(x1, x2)= x1 + ln x2 ...........equation (1)

where x1 - stamps; x2 - beer

Bridgit’s budget : p1 x1 + p2 x2 = m ......equation (2) where

p1 – price of stamps

p2 – price of beer

m – Bridgit’s budget

a) The optimal demand functions of x1 and c2 can be found by setting the Marginal Rate Of Substitution (MRS) equal to the price ratio of two goods.

At equilibrium,

Slope Of Indifference curve = Slope Of Budget Line MRS = MUx1/MUx2 = p1/P2

Calculate MRS

MUx1 = dU/dx1 = 1

MUx2 = dU/dx2 = 1/x2

MRS (x1,x2) = 1/(1/x2) = x2

So, MRS (x1,x2)= x2

Settijg the Equilbrium condition -

x2 = p1/p2

or p1 = x2.p2 ...........(3)

x2 = p1/p2.....Demand function for x2

Substituting the value of p1 in equation (2)

p1x1 + p1 = m

p1(x1 + 1) = m

x1 + 1 = m/p1

x1 = (m/p1 ) - 1 ....Demand function for x1

b)

Yes, it is true that Bridgit would spend every dollar in additional income on stamps as the demand of beer is income independent . For a given price ratio, the demand for beer does not depend on the level of his income . Thus, every additional dollar will be spend on stamps, given the prices of beer and stamp.

C)

Income Elasticity Of Demand = %change in quantity demanded / % change in income

For beer , income Elasticity is zero as the quantity demanded doesn’t not depend on the income level of the consumer and therefore changes in income will have no effect on Bridgit Demand of beer, given the prices of both goods.

For stamps, with an increase in income the quantity demanded of stamps increases and vice versa.

D)

Price Elasticity Of Demand = % change in quantity demanded/ % change in prices

When p1 and p2 increases —

The Demand Of beer will decrease or increase or will remain same will depend on the proportion of price increases. If the increase in p1 is greater than that of p2 then demand for beer will rise and vice Versa.

The demand for stamps will decrease due to fall in price (p1)