Econ 525
Farnham/Gugl
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Problem Set 1
1. Consider a two-consumer economy where each consumer has identical preferences given by:
ui x i,G  ln x i  ln Gi  G j ,i 1,2, j 1,2,i  j. Individuals 1 and 2 have endowments I1 and
I2 and face budget constraints: Ii = xi + Gi. Good x is a private good while good G is a pure public
good.



a) Assuming that each consumer treats the other’s spending on the public good as fixed in
making his/her own consumption decisions, find the equilibrium level of privatelyprovided public good. Contrast this with the socially efficient level.
Consult lecture notes for derivation of best response functions. Here we report the best
response functions and the equilibrium level of the public good.
G1G2   .5(I1  G2 )
G2 G1  .5(I2  G1 )

G1*  .5(I1  .5(I2  G1* ))
.5I  .25I2 2I1  I2 * 2I2  I1
G1*  1

,G2 
.75
3
3
The efficient provision level of the public good (use Samuelson condition and see lecture notes)
is (I1 + I2)/2. The equilibrium provision is (I1 + I2)/3, hence we find underprovision of the public
good.
b) Find the equilibrium level of the privately-provided public good for the case of I1 = I2 = 10.
Show that a lump sum redistribution which changes endowments to I1 = 12, I2 = 8 does not
affect the privately-provided level of the public good.
G1 G2   .5(I1  G2 )
G2 G1  .5(I2  G1 )
with I1  I2  10
10
G1* 
 G2*
3
with I1  12,I2  8
16
4
G1*  ,G2* 
3
3
We see the public good level is the same at 20/3. Note that private consumption for each
consumers is equal to 20/3 in each case.

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c)
Suppose we start out with situation b) and the government steps in taxing both people in
the amount of 2, so that people’s after-tax wealth is 8. The government uses all its taxes
to purchase units of the public good. What is the impact of this policy on the public good
level? Is this a more efficient allocation than the one in b)? Explain.
With I1 = 12, and I2 = 4. Gg = 4, we have
G1G2   .5(8  G2  4)
G2 G1  .5(8  G1  4)
G1*  G2* 
4
.
3
20
3
The efficient level of the public good is (I1 + I2)/2 = 10. So we see that this
government intervention is not moving us closer to the efficient product mix of 10 units of the
public good and 10 units of the private good than the equilibrium contributions in b); nothing
changes compared to the situation in b). Each person’s contribution decreases by the same
amount as the taxes they pay. Both individuals are exactly as well off as before. We have
complete crowding out.
G*  G1*  G2*  Gg 

d) Starting again from equal endowments of 10 per consumer, find the maximum transfer from
person 2 to person 1 that will not alter the level of privately provided public good. Contributions
to the public good must be non-negative.
This question asks what the maximum amount is that we can subtract from 20 so that one
consumer optimally provides zero of the public good.
2I1  (20  I1 ) 20 * 2(20  I1 )  I1
 ,G2 
0
3
3
3
40
20
I1  ,I2 
3
3
G1* 
This means, that starting at Ii = 10, we can transfer 10/3 from one consumer to the other and
we’ll still get the neutrality result.

e) Suppose I1 = 14, and I2 = 6. What is the equilibrium provision of the public good? Is this a
more efficient allocation than the one in d)? Explain.

First note that by plugging in the values for income in the “equilibrium” strategies we have
28  6 22
12  14
2
"G1* "
 ,"G2* "
 0
3
3
3
3
Negative contributions are not possible, and hence Person 2 is at a
corner solution contributing zero, and person 1 is the only contributor. (This should also be clear
from d) as an income of 6 is lower than 20/3.) Person 1 provides 14/2 = 7 which is > 20/3. (BBV,
2
Theorem 5). The economy produces a product mix closer to the efficient product mix and it
would be in principle possible to make both people better off than in d). In reality, however,
person 1 is better off in e) and person 2 is worse off.
To calculate the utility of each consumer in d), I’m using a monotonic transformation of the
utility function given that makes calculation of the utility level easier. (I can do that because
utility is ordinal.) The utility function I’m using is ui x i ,Gi   x i Gi  G j , i 1,2, j 1,2,i  j.
In d) both consumers have the same utility level and it is equal to ui = 400/9 = 44.4. In e) we
have u1 = 7*7 = 49, and u2 = 7*6 = 42. The actual distribution of goods in e) does not lead to a
Pareto improvement over the allocation in d) as consumer 1 is better off but consumer 2 is worse

off.
The allocation in e) is more efficient because in principle, we could take away 1/2 unit of the
private good from person 1 and give it to person 2, so that both have a utility level of 7*6.5 =
45.5 > 44.4.


f) Suppose we start out with situation d) and the government steps in taxing both people in the
amount of 2, so that person 1 has after tax wealth of 12, and person 2 has after tax wealth of 4.
The govt. uses all its taxes to purchase units of the public good. What is the impact of this policy
on the public good level. Is this a more efficient allocation than the one in d) and/or e)? Explain.
With I1 = 12, and I2 = 4. Gg = 4, we have
G1 G2   .5(12  G2  4)
G2 G1   .5(4  G1  4)  0
G1*  .5(12  4)  4.
G*  G1*  Gg  8

The efficient level of the public good is (I1 + I2)/2 = 10. So we see that this government
intervention is moving us closer to the efficient product mix of 10 units of the public good and
10 units of the private good.
To calculate the utility of each consumer in c), I’m using a monotonic transformation of the
utility function given that makes calculation of the utility level easier. (I can do that because
utility is ordinal.) The utility function I’m using is ui x i ,G  x i Gi  G j , i 1,2, j 1,2,i  j.
In d) both consumers have the same utility level and it is equal to ui = 400/9 = 44.4. In d) we
have u1 = 7*7 = 49, and u2 = 7*6 = 42. In e) we have u1 = 8*8=64 and u2 = 8*4 = 32. The actual
distribution of goods in f) does not lead to a Pareto improvement over the allocation in d) or e) as
worse off.
consumer 1 is better off but consumer 2 is
The allocation in f) is more efficient than the one in d) or e) because in principle, we could take
away (for example) 1.75 units of the private good from person 1 and give it to person 2 in f), so
that u1 = 8*6.25 = 50 and u2 = 8*5.75 = 46; both consumers would then end up with a higher
utility level than the one they experience in d) or e).

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
g) Keeping the amount of wealth equal to 20 in this economy, show that the level of public good
provided decreases as we add two new consumers with the same preferences and as each
consumer receives an equal endowment of 5. (Hint, if all consumers are identical, the Nash
equilibrium will be characterized by identical equilibrium contributions.)
G1 G2  G3  G4   .5(I1  G2  G3  G4 )
G2 G1  G2  G4   .5(I2  G1  G2  G4 )
etc.
Look for G1*  G2*  G3*  G4* ,Ii  5
Gi*  .5(Ii  Gi*  Gi*  Gi* )
.5Ii
20
Gi* 
 1,  Gi*  4 
1  1.5
3


2. Suppose in an N-person economy, everyone has a quasilinear utility function,
uxi,G  vG  xi , where G is the level of the public good, and xi is the level of the private
good consumed by consumer i; vG is concave and twice continuously differentiable. Let wi be
individual i's initial endowment for the private good, and ti be her tax share. Assume a linear
technology for which the marginal rate of transformation equals 1.

a) Assume an interior solution. Characterize the socially optimal amount of public good
provision.
 v' G  1
nv' G  1  v' G   1/n

b) Assume an interior solution. Characterize the voluntary contributions equilibria and compare
the result with the first-best.
max v(Gi  
j i
G j )  x i   w i  x i  Gi 
FOCs
wrt x i :   1

wrt Gi : v' Gi  
j i

Gj 1
This implies that there exists a symmetric Nash Equilibrium with everybody contributing the
same amount.

G1*  G2*  ...  Gn*
v' nGi*  1

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Since G *  nGi* we can see that there is severe underprovision in the Nash equilibrium
(remember v’(G) is decreasing in G and hence the G* such that v’(G*) =1 is lower than the G
such that v’(G)=1/n ).
Note, however, that there exist asymmetric Nash equilibria as well. For any given contributions
of the n-1 consumers, consumer i will optimally respond by contributing an amount such that the
total amount of contributions satisfies v’(G*) =1. To see this more clearly, take 2 individuals and
suppose the amount of G satisfying v’(G) =1 is G = 10. Person i’s best response function is given
implicitly by
v' Gi*  G*j  1 This means, if person j contributes 3 units, person i will contribute 7 units, but if
person i contributes 7 units, person j’s best response is to contribute 3 units. Hence one person
contributing 3 units and the other contributing 7 units is also a Nash equilibrium. So we see that
this game has multiple Nash equilibria in which G1* 0,10,G2* 0,10,G1*  G2*  10



Hint: If you are unsure of the equilibrium strategies, you can set n = 2 and choose a specific
function for v(G). This allows you to draw the best response functions and you find out that they

both have a slope equal to -1 whenever the best response of a person is Gi* 0.
c) What happens as n gets larger?
As n gets larger the total amount contributed in the Nash equilibrium stays the same -v’(G*) =1 does not depend on n – but the optimal amount of G increases with n, since G
should be provided according to v’(G)=1/n).
d) Suppose the government is taxing everybody in the same amount. Can the government make
everybody better off by taxing them more than their equilibrium contribution? What is true about
private contributions in this case?
Consider the case of the symmetric Nash equilibrium. Yes, the govt can make everybody better
off by taxing them in the same amount and at an amount higher than their eqm contribution. In
this case, v’(G*) < 1 and hence nobody contributes voluntarily to the public good. However, they
are all better off. To see this, we can use the indirect utility function of a person. This function is
given by v(G) + wi – G/n. Starting with the symmetric NE without govt intervention we have
v(G*) + wi – G*/n. Taking the derivative with respect to G and taking into account that each
person is taxed t = 1/n for each additional unit of the public good, we find that the indirect utility
goes up by v’(G*) – 1/n at G =G*. This amount is positive because we are starting at a G such
that v’(G) = 1. So the government should increase the tax liability of each person until v’(G)
=1/n.
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3) Consider the following table.
Quantity
1
2
3
4
Alice
a1
a2
a3
a4
Brenda
b1
b2
b3
b4
Chip
c1
c2
c3
c4
SNetB
a1+b1+c1
a2+b2+c2
a3+b3+c3
a4+b4+c4
Person
Show that no matter what the others report, nobody has an incentive to tell a lie if the GrovesClarke mechanism is used.
It should be clear that without loss of generality, we can just focus on one person given all the
other persons’ reports. Let this person be Alice.
Note that Alice can control the amount of public good provided given the other people’s
reports and suppose she prefers a quantity different from the one that would arise if she would
tell the truth. That is, al>ak, for all k not equal to l. It must be the case that without Alice’ report
some other quantity than l is the outcome. Let this quantity be m. Then it must be true that bm +
cm > bl + cl. If m is also the group choice with Alice’ report, she does not pay any taxes when she
reveals truthfully and she is al - am worse off than with al. If Alice’s truthful report changes the
group choice from n to m she pays bn + cn – (bm + cm) in addition to being al - am worse off.
What happens if she changes her report so that l is chosen?
We consider 2 cases: If m would be chosen without her, she pays a tax of bm + cm – (bl + cl).
But if m was also the group choice we know that am + bm + cm > al + bl + cl or equivalently bm +
cm – (bl + cl) > al - am. So she is worse off than reporting the truth.
If some other quantity n would be chosen without her, she would still pay the same taxes as
before going to m and then additional taxes for going from m to l. Again lying ultimately lowers
her welfare by more than al - am +[bn + cn – (bm + cm)].
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