Problem Set III - Slutsky Decomposition and Consumer's Surplus

ECON 511 Spring 2024

Exercise 1: Income and Substitution Effect

Julia has quasi-linear utility of the formu(x, y) =f(x) +y(linear in the amount of the second good). With incomem= 600 and pricesp= 5, q= 10 Julia's optimal bundle is (x?, y?) = (20,50). When the price of the first good decreases top?= 2, Julia's new optimal bundle is (x??, y??) = (40,52).

Follow the steps below to decompose the change in quantity demanded into income and substitution effects using Slutsky's approach:

?How much income is needed for the artificial budgetm? that keeps the Slutsky real income constant?

?What is Julia's optimal bundle when faced with the artificial budgetm? ?

?Decompose the change from (x?, y?) = (20,50) to (x??, y??) = (40,52) into income and substitu- tion effects. Compute the sizes of the effects for both goods. Explain your findings.

How would your answers to part 1 change if Julia had quasi-linear utility that is linear in the amount of the first good (rather than the second), i.e.u(x, y) =x+g(y)?

How would your answers to part 1 change if Julia had a Cobb-Douglas utility function (homothetic preferences), i.e.u(x, y) =Cxayb?

1

Exercise 2: Private Value and Consumer's Surplus

Holding income and the price of good 2 fixed, Justine's demand for good 1 can be expressed as

?

??5,000?25pifp

??

Justine's income effect for the good is zero.

0 otherwise

How much are 2,500 units of the good worth to Justine? That is, find her (reservation) value of 2,500 units.

How much better off would Justine be if she faced pricep?= 50 per unit instead ofp= 80 per unit?

If Justine were offered a take-it-or-leave-it package of 1,000 units for a payment of$190,000, would she

accept?

Exercise 3: Take-It-Or-Leave-It Offers

Suppose Jingfeng has the same individual demand curve as Justine (from Exercise 2). Jingfeng is offered a take-it-or-leave-it deal of the following type:

After paying a lump-sum fee ofF= $300,000, he will be allowed to purchase as many units of the good as he desires at a price ofp= $50 per unit.1If he rejects the offer, he will not pay the fee and will not be able to purchase the good.

Does Jingfeng accept this offer?

Find the highest lump-sum fee at which Jingfeng would be willing to accept the offer. That is, call the lump-sum feeF(instead of using the numerical value$300,000) and determine the maximum number forFsuch that Jingfeng does not reject the offer.

1Jingfeng's income effect is zero, therefore, the lump sum fee does not affect his demand function. The per-unit price must be paid in addition to the lump-sum fee.

ECON 511, Spring 2024 - B.Klose 2 Problem Set III