Question

Only need answers for e,f,g,h,i

Jacob has the utility function for goods x and y asu(x, y) = ?2/3?2/3.

The price of good x is $2 each, and the price of good y is $5 each. Jacob has a total income of $120 to spend on these two goods.

(a) Write down Jacob's budget line, and show in the graph.
(b) Write down the budget exhaustion condition (BC) of Jacob’s optimization problem.(c) Write down the rationality condition (RC) of Mr. Jacob’s optimization problem.

(d) Given the two conditions from (b) and (c), calculate the optimal bundle (x*, y*) that he will choose in order to maximize his utility, and (approximately) indicate this optimal bundle in the graph.

Now, the price of good x increases to $4 each, and everything else remains the same.

(e) Given the change in the price of good x, calculate the new optimal bundle (x**, y**).

(f) How much additional income does Jacob need in order to consume at the original optimal bundle as in (d)?

(g) Suppose Jacob is a cardholder of a credit card, where the credit card company decides to compensate its members, so all of its members are able to maintain their utility level as the same as before the price changed. Write down the utilization condition (UC) of Jacob’soptimization problem after price changed.

(h) Given the condition in (g), calculate Jacob’s new optimal bundle (xc, yc) under the compensation.

(i) What is the amount that the credit card company needs to compensate Jacob? Compare the result with (f).

Answer 1

As soon as the price of good x increases the budget line shifts to leftwards as now with same income he will be able to purchase less amount of good x . The Indifference curve will now shift downwards from IC1 to IC2 as given below in the diagram :-

Answer e :-

The new optimisation bundle with utility function = U (x2,y3 )

Is U =(8x,12y)

Answer f :-

As he has curtailed a consumption of 8 units of x.

He requires (8*4) = $32 of increased income in order to return to old level

Answer g :-

Utility condition after price change = U (4x ,3y )

With the increase in price ,marginal utility of product will keep on increasing till it equates with the price of x.

Answer h :-

The new optimisation level after the compensation

(24,18)

Answer i :-

His new consumption is 24 units while the old consumption was 16 units .

The credit card company will compensate :-

4*(24-16) = 4*8 = $32