978-0393123982 Chapter 35 Solution Manual Part 2

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NAME 433

Suppose that each consumer owns 1 unit of the consumption good and

consumes it.

(b) Suppose that each consumer consumes only 3/4 of a unit. Will all

(d) Suppose that everybody around the lake is consuming 1 unit. Can

any two people make themselves both better off either by redistributing

(f) How large is the smallest group that could cooperate to benefit all its

35.7 (0) Jim and Tammy are partners in Business and in Life. As

is all too common in this imperfect world, each has a little habit that

annoys the other. Jim’s habit, we will call activity X, and Tammy’s

habit, activity Y.Letxbe the amount of activity Xthat Jim pursues

and ybe the amount of activity Ythat Tammy pursues. Due to a series

of unfortunate reverses, Jim and Tammy have a total of only $1,000,000

a year to spend. Jim’s utility function is UJ=cJ+ 500 ln x−10y,where

cJis the money he spends per year on goods other than his habit, xis

the number of units of activity Xthat he consumes per year, and yis the

number of units of activity Ythat Tammy consumes per year. Tammy’s

utility function is UT=cT+ 500 ln y−10x,wherecTis the amount of

money she spends on goods other than activity Y,yis the number of

units of activity Ythat she consumes, and xis the number of units of

activity Xthat Jim consumes. Activity Xcosts $20 per unit. Activity

Ycosts $100 per unit.

(a) Suppose that Jim has a right to half their joint income and Tammy

has a right to the other half. Suppose further that they make no bargains

with each other about how much activity Xand Ythey will consume.

How much of activity Xwill Jim choose to consume? 25 units.

How much of activity Ywill Tammy consume? 5 units.

434 EXTERNALITIES (Ch. 35)

(b) Because Jim and Tammy have quasilinear utility functions, their util-

ity possibility frontier includes a straight line segment. Furthermore, this

segment can be found by maximizing the sum of their utilities. Notice

that

UJ(cJ,x,y)+UT(cT,x,y)

But we know from the family budget constraint that cJ+cT=1,000,000−

20x−100y. Therefore we can write

Let us now choose xand yso as to maximize UJ(cJ,x,y)+UT(cT,x,y).

Setting the partial derivatives with respect to xand yequal to zero, we

find the maximum where x=16.67 and y=4.54 .Ifweplug

these numbers into the equation UJ(cJ,x,y)+UT(cT,x,y)=1,000,000+

500 ln x+500 ln y−30x−110y, we find that the utility possibility frontier is

a calculator or a log table to find this answer.) Along this frontier, the

total expenditure on the annoying habits Xand Yby Jim and Tammy is

way of dividing this expenditure corresponds to a different point on the

utility possibility frontier. The slope of the utility possibility frontier

constructed in this way is -1.

35.8 (0) An airport is located next to a large tract of land owned by a

housing developer. The developer would like to build houses on this land,

but noise from the airport reduces the value of the land. The more planes

that ﬂy, the lower is the amount of profits that the developer makes. Let

Xbe the number of planes that ﬂy per day and let Ybe the number of

houses that the developer builds. The airport’s total profits are 48X−X2,

and the developer’s total profits are 60Y−Y2−XY . Let us consider the

outcome under various assumptions about institutional rules and about

bargaining between the airport and the developer.

(a) “Free to Choose with No Bargaining”: Suppose that no bargains can

be struck between the airport and the developer and that each can decide

on its own level of activity. No matter how many houses the developer

builds, the number of planes per day that maximizes profits for the airport

is 24. Given that the airport is landing this number of planes, the

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(b) “Strict Prohibition”: Suppose that a local ordinance makes it illegal

to land planes at the airport because they impose an externality on the

developer. Then no planes will ﬂy. The developer will build 30

airport liable for all damages to the developer’s property values. Since the

developer’s profits are 60Y−Y2−XY and his profits would be 60Y−Y2

if no planes were ﬂown, the total amount of damages awarded to the

developer will be XY . Therefore if the airport ﬂies Xplanes and the

developer builds Yhouses, then the airport’s profits after it has paid

damages will be 48X−X2−XY . The developer’s profits including the

amount he receives in payment of damages will be 60Y−Y2−XY +XY =

60Y−Y2. To maximize his net profits, the developer will choose to build

30 houses no matter how many planes are ﬂown. To maximize its

profits, net of damages, the airport will choose to land 9planes.

Calculus 35.9 (1) This problem continues the story of the airport and the devel-

oper from the previous problem.

(a) “Merger”: Suppose that the housing developer purchases the airport.

To maximize joint profits, it should build 24 houses and let

12 planes land. Combined profit is now 1008 . Explain why

each of the institutional rules proposed in the previous problem fails to

achieve an efficient outcome and hence has lower combined profits. In

(a), the airport does not bear the cost of

damages from landing planes. In (b), planes

436 EXTERNALITIES (Ch. 35)

(b) “Dealing”: Suppose that the airport and the developer remain in-

dependent. If the original situation was one of “free to choose,” could

the developer increase his net profits by bribing the airport to cut back

one ﬂight per day if the developer has to pay for all of the airport’s lost

profits? Yes. The developer decides to get the airport to reduce its

ﬂights by paying for all lost profits coming from the reduction of ﬂights.

To maximize his own net profits, how many ﬂights per day should he pay

the airport to eliminate? 12.

35.10 (1) Every morning, 6,000 commuters must travel from East Potato

to West Potato. Commuters all try to minimize the time it takes to get to

work. There are two ways to make the trip. One way is to drive straight

across town, through the heart of Middle Potato. The other way is to take

the Beltline Freeway that circles the Potatoes. The Beltline Freeway is

entirely uncongested, but the drive is roundabout and it takes 45 minutes

to get from East Potato to West Potato by this means. The road through

Middle Potato is much shorter, and if it were un-congested, it would take

only 20 minutes to travel from East Potato to West Potato by this means.

But this road can get congested. In fact, if the number of commuters who

use this road is N, then the number of minutes that it takes to drive from

East Potato to West Potato through Middle Potato is 20 + N/100.

(a) Assuming that no tolls are charged for using either road, in equilib-

rium how many commuters will use the road through Middle Potato?

(b) Suppose that a social planner controlled access to the road through

Middle Potato and set the number of persons permitted to travel this

way so as to minimize the total number of person-minutes per day spent

by commuters travelling from East Potato to West Potato. Write an

expression for the total number of person-minutes per day spent by

commuters travelling from East Potato to West Potato as a function of

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the number Nof commuters permitted to travel on the Middle Potato

road. N(20 + N

muters who drove through Middle Potato to get to work? 32.5

minutes. What would be the total number of person-minutes per

day spent by commuters travelling from East Potato to West Potato?

minute and that the Greater Potato metropolitan government charges a

toll for using the Middle Potato road and divides the revenue from this

toll equally among all 6,000 commuters. If the government chooses the

toll in such a way as to minimize the total amount of time that people

spend commuting from East Potato to West Potato, how high should it

set the toll? $12.5w.How much revenue will it collect per day from

this toll? $15,625w.Show that with this policy every commuter is

better off than he or she was without the tolls and evaluate the gain per

consumer in dollars. Before the toll was in place,

all commuters spent 45 minutes travelling

who travel through Middle Potato are

indifferent between spending 45 minutes

travelling on the Beltline and paying the

438 EXTERNALITIES (Ch. 35)

35.11 (2) Suppose that the Greater Potato metropolitan government

rejects the idea of imposing traffic tolls and decides instead to rebuild the

Middle Potato highway so as to double its capacity. With the doubled

capacity, the amount of time it takes to travel from East Potato to West

Potato on the Middle Potato highway is given by 20 + N/200, where

Nis the number of commuters who use the Middle Potato highway. In

the new equilibrium, with expanded capacity and no tolls, how many

commuters will use the Middle Potato highway? 5,000 How long