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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
(c) What is the best possible consumption if all are to consume the same
(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
NAME 435
(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
(c) “Lawyer’s Paradise”: Suppose that a law is passed that makes the
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
NAME 437
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?
(c) Suppose that commuters value time saved from commuting at $wper
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
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
NAME 435
(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
(c) “Lawyer’s Paradise”: Suppose that a law is passed that makes the
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
NAME 437
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?
(c) Suppose that commuters value time saved from commuting at $wper
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
