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462 PUBLIC GOODS (Ch. 37)
(h) With the proportional income tax scheme discussed above, what bud-
get constraint would a rich person consider in deciding how many aerobics
tax rates, how many aerobics lessons per creature would the rich favor?
(i) Calculate the utility of a rich creature under a head tax. √937.5
Under privatization. √1,250.Under a proportional income tax.
√833.3.(Hint: In each case, solve for the consumption of bread and
the consumption of aerobics lessons that a rich person gets, and plug these
into the utility function.) Now calculate the utility of each poor creature
(j) Is privatization Pareto superior to the head tax? Yes. Is a propor-
tional income tax Pareto superior to the head tax? No. Is privatization
Pareto superior to the proportional income tax? No. Explain the last
Calculus 37.8 (2) Three friends, Archie, Betty, and Veronica, are planning a
party. They disagree about how many people to invite. Each person i
has a quasilinear utility function of the form mi+ui(x)wheremiis the
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number of dollars that ihas to spend and xis the number of guests invited
to the party. Suppose that for each i,
Everyone knows the functional form of the others’ utility functions and
knows his own value of aibut does not know anyone else’s value. Let us
suppose that the actual values of aiare 20 for Archie, 40 for Betty, and
60 for Veronica.
(a) How many guests should be invited to maximize the sum of the three
(b) Suppose that the three friends decide to use the VCG mechanism to
determine the number of guests. If each plays his or her best strategy,
(c) In the VCG mechanism, if the amount of public good supplied is
x, Archie would receives a sidepayment equal to the sum of Betty’s and
Veronica’s utility for x. If Betty and Veronica play their best strategies
(without colluding) and if the amount of public good is x, this sidepay-
.
(d) In addition to receiving sidepayments, the VCG mechanism requires
that each person must pay an amount equal to the maximum possible
sum of the other two persons’ utilities. If Betty and Veronica play their
(e) If everybody plays their best strategy, what is the net amount that
(f) Suppose that the party is organized not by just three people, but by a
dormitory with 21 residents. All of these residents have utility functions
of the same form as Archie, Betty, and Veronica. Seven of them have
ai= 20, seven have ai= 40, and seven have ai= 60. In order to
maximize the sum of the residents’ utilities, how many guests should be
464 PUBLIC GOODS (Ch. 37)
seven with ai= 40 and seven with ai= 60, how many guests would have
(g) If everybody plays their best strategy in the VCG game, then after all
sidepayments and taxes are collected, how much net tax will each of the
37.9 (2) Homeowners 1, 2, and 3 live at the end of a badly deteriorated
road. Fixing the road would cost $C. The value to Homeowner 1 of fixing
the road is $3,000, the value to Homeowner 2 is $5,000, and the value to
Homeowner 3 is $8,000. Each homeowner claims that fixing the road is
not worth much to him, because each wants the others to pay the cost.
The local government suspects that the total value to these homeowners
of fixing the road is greater than $Cand has decided to require the three
homeowners to use the VCG mechanism to determine whether to fix the
road. Since the government had no idea of the individual values for fixing
the road, it decided to allocate the costs equally among the three home-
owners. Each homeowner is asked to report his value for fixing the road.
If the sum of the reported values is greater than C, the road will be fixed
and each homeowner will have to pay $C/3 and also will have to pay an
additional tax as calculated by the VCG mechanism.
(a) Suppose that C= $13,500, so that each homeowner has to pay $4,500
as his share of the cost. If homeowners report their values accurately, the
ported values is greater than $C, the government will choose to build the
road. To calculate Homeowner 3’s VCG tax, we reason as follows. If
the road is repaired, Homeowners 1 and 2 will have to pay a total of
NAME 465
Homeowner 3 be better off or worse off with the VCG outcome than if the
would the sum of the utilities of the three homeowners be higher or lower
(b) Suppose that C= $18,000 and so each homeowner will have to pay
a $6,000 share of the cost. If homeowners report their values accurately,
the sum of the reported values will be $16,000 . Since the sum
of reported values is less than $C, the government will choose not to
repair the road. Let us calculate Homeowner 1’s VCG tax. If the road
is repaired, the total amount that Homeowners 2 and 3 have to pay is
.
(c) Suppose that instead of 3 homeowners at the end of the road, there
were 30 homeowners, 10 of type 1 who valued repairing the road at $3,000,
10 of type 2 who valued repairing the road at $5,000 and ten of type 3 who
valued repairing the road at $8,000. Suppose that the cost of repairing the
road is C= $135,500. If the road is repaired, each homeowner would have
to pay a tax of $135,000/30 = $4,500. If the VCG mechanism is used
and each individual reports his true valuation, the road will be repaired,
since the sum of reported valuations will be $160,000. To calculate the
VCG tax of a type 3 consumer, we observe that the sum of the values of
Calculus 37.10 (2) (For this problem, you will want to use a calculator.) A
developer would like to build an amusement park in the midst of the
once-thriving city of Broken Axle, Michigan. In order to build this park,
he must purchase all of the land in a specific 10-acre area. If he does
not get all of this land, he cannot build the park. Nobody except the
466 PUBLIC GOODS (Ch. 37)
developer knows exactly how much he would be willing to pay for the
land. All persons other than the developer believe that the probability
that he is willing to pay at least $pfor the land is G(p)=e−p/k where
k= 200,000.
(a) Suppose that all of the land belongs to a single owner. This owner
values the land at $50,000 in its current use. He must make a take-it-or-
leave-it offer to the developer. The owner wants to make an offer pthat
maximizes his expected profit from the sale, which is (p−$50,000)G(p).
(b) Suppose that instead the land consists of 10 separate parcels with 10
different owners. Let vibe owner i’s value for his own parcel if he keeps it.
These owners have different values viand only the owner knows what it
is worth to himself. Let us assume that v=10
i=1 vi= $50,000. Suppose
that each owner isets a price pifor his land. Let us define pto be the
sum of these individual offer prices. That is, p=10
i=1 pi.If the developer
is willing to pay at least pfor the entire 10-acre parcel, then he buys it
and pays each owner i, the amount pi. In equilibrium, each owner makes
the offer that maximizes his expected profit given the offers made by the
other owners. That is, he chooses pito maximize
(pi−vi)G(p)=(pi−vi)e−pj/k
where k= $200,000. What price piwill owner iset? vi+k=
(c) The 10 separate owners recognize that if they could somehow coor-
dinate their offers, they would make much larger expected total profits.
They decide to use the VCG mechanism to do so. In this instance the
VCG mechanism works as follows. Each individual states an amount ri
that the land is worth to him. All of the land will be offered to the de-
veloper at a price pthat would maximize total profits of owners if each
owner reports his actual value, ri=vi. If the developer chooses to pur-
chase the land, then each owner will receive an equal share of the sales
price p. In addition to receiving this share if the sale is made, each owner
will pay a “tax” that depends on his response riandontheresponsesof
NAME 467
the other owners in the following way. Person iwill receive an amount
the sum of the expected profits of the other 9 owners, but must pay an
amount equal to the maximum expected profits that would be received by
the other owners if the price were chosen to maximize the expected total
profits of these other owners. With this mechanism, the best strategy
for each owner is to state his true value ri=vi. If each owner uses this
best strategy, at what price will the land be offered to the developer?
(d) Suppose that person ihas a reservation price vi=$5,900. Let us
calculate the tax that the VCG mechanism would assign to iif each
individual responds with ri=vi. If the land is offered at price pto the
developer, then expected profits of persons other than iwill be
9
10p−v∼ie−p/200,000
(e) Let us calculate total expected profit of the individual with vi= $5900
under the VCG mechanism. Whether or not there is a sale, person ipays
the amount of tax that we calculated in the previous section. With prob-
468 PUBLIC GOODS (Ch. 37)
(f) If individual jhas vj=$5,000, what is the net amount of tax that
he pays under the VCG mechanism? $0 (Hint: there is an easy
answer.)
