Dr. Donna Feir
Economics 313
Problem Set 8
Public Goods
1. Two consumers – A and B – each have the utility function 𝑈 = 𝑥 2 𝑦. Good x is a public good and good
y is a private good. The price of good x = ½ and the price of good y = 1.
Each consumer is endowed with 180 units of the private good.
a. Derive the best response functions for both consumers and use these to calculate how
much of the public good each will purchase in the Nash equilibrium. Draw a diagram
illustrating this equilibrium.
𝒑𝒙
𝟐𝒚𝑨
𝟏
→
= → 𝟒𝒚𝑨 = 𝒙𝑨 + 𝒙𝑩
𝒑𝒚 𝒙𝑨 + 𝒙𝑩 𝟐
𝟏
𝟏
𝑩𝑳𝑨 : 𝒙𝑨 + 𝒚𝑨 = 𝟏𝟖𝟎 → 𝒚𝑨 = 𝟏𝟖𝟎 − 𝒙𝑨 → 𝟒𝒚𝑨 = 𝟕𝟐𝟎 − 𝟐𝒙𝑨
𝟐
𝟐
Setting the two expressions we have for 𝟒𝒚𝑨 equal to each other, we have:
𝑴𝑹𝑺𝑨 =
𝟏
𝒙𝑨 + 𝒙𝑩 = 𝟕𝟐𝟎 − 𝟐𝒙𝑨 → 𝟑𝒙𝑨 = 𝟕𝟐𝟎 − 𝒙𝑩 → 𝒙𝑨 = 𝟐𝟒𝟎 − (𝟑) 𝒙𝑩 ← 𝑩𝑹𝑨
Because the two consumers’ utility function and endowments are the same, we know that 𝑩𝑹𝑩 =
𝟏
𝟐𝟒𝟎 − (𝟑) 𝒙𝑨 .
The Nash equilibrium is where the BRs cross:
𝟏
𝟏
𝟏
𝟏
𝒙𝑨 = 𝟐𝟒𝟎 − ( ) 𝒙𝑩 = 𝟐𝟒𝟎 − ( ) (𝟐𝟒𝟎 − ( ) 𝒙𝑨 ) = 𝟏𝟔𝟎 + ( ) 𝒙𝑨
𝟑
𝟑
𝟑
𝟗
𝟖
→ ( ) 𝒙𝑨 = 𝟏𝟔𝟎 → 𝒙𝑨 = 𝟏𝟖𝟎 = 𝒙𝑩
𝟗
Dr. Donna Feir
Economics 313
b.
Calculate the efficient level of public good provision in this economy.
𝒑
Efficiency is where 𝑴𝑹𝑺𝑨 + 𝑴𝑹𝑺𝑩 = 𝒑𝒙 → 𝒙
𝒚
𝟐𝒚𝑨
𝑨 +𝒙𝑩
+𝒙
𝟐𝒚𝑩
𝑨 +𝒙𝑩
𝟏
=𝟐→
𝟏
𝟐𝒚𝑨 +𝟐𝒚𝑩
𝒙𝑨 +𝒙𝑩
𝟏
= 𝟐 → 𝟒(𝒚𝑨 + 𝒚𝑩 ) =
𝟏
(𝒙𝑨 + 𝒙𝑩 ). From the BL for each we know that 𝒚𝑨 = 𝟏𝟖𝟎 − 𝒙𝑨 and 𝒚𝑩 = 𝟏𝟖𝟎 − 𝒙𝑩 , so 𝒚𝑨 + 𝒚𝑩 =
𝟐
𝟐
𝟏
𝟐
𝟏
𝟐
𝟑𝟔𝟎 − 𝒙𝑨 − 𝒙𝑩 , so 𝟒(𝒚𝑨 + 𝒚𝑩 ) = 𝟏𝟒𝟒𝟎 − 𝟐(𝒙𝑨 + 𝒙𝑩 ).
Setting the two expressions for 𝟒(𝒚𝑨 + 𝒚𝑩 ) equal yields 𝒙𝑨 + 𝒙𝑩 = 𝟏𝟒𝟒𝟎 − 𝟐(𝒙𝑨 + 𝒙𝑩 ) →
𝟑(𝒙𝑨 + 𝒙𝑩 ) = 𝟏𝟒𝟒𝟎 → (𝒙𝑨 + 𝒙𝑩 ) = 𝟒𝟖𝟎, In the Nash equilibrium, total provision was just 360 units.
So 120 too few units are provided in the Nash equilibrium.
Suppose that the government decides to provide some of the public good. Specifically, it decides to
make up the difference between the equilibrium level of provision found in your answer to a) and the
efficient level found in your answer to b).
c. Derive the new best response functions for both consumers given the level of government
provision, and use these to calculate how much of the public good each will purchase in the
new Nash equilibrium. Draw a diagram illustrating this equilibrium. Does the government
policy ensure an efficient level of public good provision? Now suppose that the total
endowment of the private good in the economy is unchanged, but that its distribution
across the two consumers has. Specifically, assume that consumer A is now endowed with
210 units of good y and consumer B is endowed with 150 units of good y.
Dr. Donna Feir
Economics 313
Now we have total public good = xA + xB + xG, where xG = 120 (the difference between 480 and 360).
MRSA = px/py ⇒ 2yA/(xA + xB + xG) = ½ ⇒ 2yA/(xA + xB + 120) = ½ ⇒ 4yA = xA + xB + 120
BLA: yA = 210 – ½xA ⇒ 4yA = 720 - 2xA
Setting the two expressions we have for 4yA equal to each other, we have:
xA + xB + 120 = 720 - 2xA ⇒ 3xA = 600 - xB ⇒ xA = 200 – (1/3)xB
BRA
Because their endowment are equal, we know that BRB is xB = 200 – (1/3)xA
The Nash equilibrium is where the BRs cross:
xA = 200 – (1/3)xB = 200 – (1/3)(200 – (1/3)xA) = (400/3) + (1/9)xA
⇒ (8/9)xA = 400/3 ⇒ xA = 150 = xB.
Total provision is thus 420, greater than in part a), but not the efficient level. Each unit of the public
good provided by the government crowds out provision by A and B, but not one-for-one, so overall
provision increases with the government policy, but it does not reach the efficient level.
d. Once again, derive the best response functions for both consumers and use these to
calculate how much of the public good each will purchase in the Nash equilibrium. Draw a
diagram illustrating this equilibrium.
MRSA = px/py ⇒ 2yA/(xA + xB) = ½ ⇒ 4yA = xA + xB
BLA: ½xA + yA = 210 ⇒ yA = 210 – ½xA
⇒ 4yA = 840 - 2xA
Setting the two expressions we have for 4yA equal to each other, we have:
xA + xB = 840 - 2xA ⇒ 3xA = 840 - xB ⇒ xA = 280 – (1/3)xB
BRA
MRSB = px/py ⇒ 2yB/(xA + xB) = ½ ⇒ 4yB = xA + xB
BLB: ½xB + yB = 150 ⇒ yB = 150 – ½xB
⇒ 4yB = 600 - 2xB
Setting the two expressions we have for 4yB equal to each other, we have:
xA + xB = 600 - 2xB ⇒ 3xB = 600 – xA ⇒ xB = 200 – (1/3)xA
BRB
The Nash equilibrium is where the BRs cross:
xA = 280 – (1/3)xB = 280 – (1/3)(200 – (1/3)xA) = (640/3) + (1/9)xA
⇒ (8/9)xA = (640/3) ⇒ xA = 240. xB = 200 – (1/3)xA = 200 – (1/3)240 = 120.
Dr. Donna Feir
Economics 313
e. Is the total amount of the public good provided in the Nash equilibrium different in this
case, compared to part a)?
No, its 360 in either case. The aggregate amount turns out to depend only on the aggregate
endowment (and the consumer’s prefs. of course), as long as it is true that each individual is wealthy
enough to contribute positive amounts of the public good (more on this below). If we change the
distribution of the endowment, we just change the distribution of who pays for it. Here, A is richer
than B. So A ends up paying for a greater proportion of the public good than B (but note that B still
contributes).
f.
Is the efficient level of public good provision different here compared to part b)?
Again, no: this also is invariant to the distribution of the endowment.
Finally suppose again that the total endowment of the private good in the economy is unchanged, but
now its distribution is such that consumer A is now endowed with 300 units of good y and consumer B is
endowed with 60 units of good y.
g. Once again, derive the best response functions for both consumers and use these to
calculate how much of the public good each will purchase in the Nash equilibrium. Draw a
diagram illustrating this equilibrium. Also draw an indifference curve diagram for B
illustrating her utility-maximizing choice of x and y at the Nash equilibrium. Is the total
Dr. Donna Feir
Economics 313
amount of the public good provided in the Nash equilibrium different in this case, compared
to parts a) and e)?
MRSA = px/py  2yA/(xA + xB) = ½  4yA = xA + xB
BLA: ½xA + yA = 300  yA = 300 – ½xA
 4yA = 1200 - 2xA
Setting the two expressions we have for 4yA equal to each other, we have:
xA + xB = 1200 - 2xA  3xA = 1200 - xB  xA = 400 – (1/3)xB  BRA
MRSB = px/py  2yB/(xA + xB) = ½  4yB = xA + xB
BLB: ½xB + yB = 60  yB = 60 – ½xB
 4yB = 240 - 2xB
Setting the two expressions we have for 4yB equal to each other, we have:
xA + xB = 240 - 2xB  3xB = 240 – xA  xB = 80 – (1/3)xA  BRB
The Nash equilibrium is where the BRs cross. This time we will substitute A’s BR into B’s (you’ll see
why in a minute):
xB = 80 – (1/3)xA = 80 – (1/3)(400 – (1/3)xB) = (-160/3) + (1/9)xB
 (8/9)xB = (-160/3)  xB = -60. But negative contributions to the public good are not
possible, so xB must equal zero. When xB = 0, xA = 400.
Dr. Donna Feir
Economics 313
Think carefully about this diagram (it might look a little complicated at first, but it is very similar to the
IC diagram we looked at in class). The first budget line is B’s budget line if A were to contribute no
units of the public good. From B’s BR, we know that in that situation, B would contribute 80 units,
From B’s BL, if xB = 80, then yB = 20. This consumption bundle is represented by point U in the diagram.
We also know that if A then begins providing units of the public good, B will decrease his contribution.
The second budget line is drawn for a level of contribution by A of 120 units (this is an arbitrarily
chosen level). From B’s BR, we know that if xA = 120, xB = 40, and so xA + xB = 160. From B’s BL, if xB =
40, yB = 40. This consumption bundle is represented by point V in the diagram. The third budget line is
drawn for xA = 240. This is NOT arbitrarily chosen: it is the level of xA that just drives xB to zero. When
xB = 0, yB = 60. This consumption bundle is represented by point W in the diagram. Any further
increase in A’s contributions will not change B’s behavior. He will still contribute no units of the public
Dr. Donna Feir
Economics 313
good, and hence will consume all 60 units of his endowment of y. Thus when xA = 400, as it is in
equilibrium here, the consumer will be at point Z on the diagram. Note that this point is NOT
characterized by a tangency between B’s BL and IC.
h. Is the total amount of the public good provided in the Nash equilibrium different in this
case, compared to parts a) and e)?
Yes. Now we have 400 units provided. A provides more, in part because she is richer, but also in part
because she can no longer free-ride on B’s provision. This latter effect causes overall contributions to
be higher.
i.
Is the efficient level of public good provision different here compared to parts b) and f)?
No again. The efficient level of contribution is determined by the total endowment and the
preferences. The distribution of the endowment does not change the efficient quantity. Note that in
this case, we actually get closer to efficiency (but not all the way there). This is because we have partly
solved part of the free-rider problem: B still free-rides on A, but A does not free-ride on B (as B
doesn’t contribute anything).
2. Four roommates are trying to decide how much money to spend on a new couch, which will be a
public good in their household (assume that the couch seats at least four people). They each have up to
$500 to contribute to the purchase of the couch, and each has the utility function 𝑈 = 20𝐶 1/2 + 𝑀,
where C = the TOTAL amount of money spent on the couch, and M = the amount of money each has left
over from their $500, once they have contributed to the couch purchase.
a. How much money in TOTAL will the household spend on the couch in the Nash equilibrium?
(Hint: the quasi-linear utility functions makes this simple, in terms of the number of math
steps you need to take).
For any consumer, MRS = 10C-½. The price ratio equals one here (each good is dollars spent on
something), so each of the four consumers sets their MRS = price ratio  10C-½ = 1  C-½ = 1/10  C½
= 10  C = 100. So the Nash equilibrium involves+ buying a $100 couch.
b. If each roommate pays one quarter of the amount you calculated in part a), how much
money will each have leftover after contributing his or her share to the couch purchase?
What level of utility will each achieve?
If each contributes $25 to the couch, each will have $475 left over. For each, U = 20C ½ + M = 20(100)½ +
475 = 675.
c. Does the household spend the efficient amount on the couch? If not, what is the efficient
amount to spend?
For efficiency, we want the SUM of the MRSs to equal the price ratio. There are four individuals, each
with identical utility functions, so the sum of the MRSs is just four times an individual’s MRS. So the
Dr. Donna Feir
Economics 313
sum of the MRSs = 4  10C-½ = 40C-½. Setting this equal to the price ratio, we have  40C-½ = 1  C-½ =
1/40  C½ = 40  C = 1600. So efficiency involves buying a $1600 couch.
d. Suppose the roommates agree to spend the efficient amount on the couch, and also agree
that they will split the cost equally among them. How much money will each have leftover
after contributing his or her share? What level of utility will each achieve?
If each contributes $400 to the couch, each will have $100 left over. For each, U = 20C½ + M =
20(1600)½ + 100 = 900.
e. The utility that each achieves at the efficient level of couch spending exceeds the utility at
the equilibrium level of couch spending. Explain carefully why – if each consumer is
maximizing his or her utility, which is fully described by the utility function provided – no
individual has an incentive to stick to the agreement that they reached.
If each roommate is maximizing their utility, then they will be on their BR function. At the Nash
equilibrium (a $100 couch), they are ALL on their BR functions. So we know that it is not a BR for any
individual to agree to contribute $400, even if all the others agree. Each would have an incentive to
cheat on the agreement (think about cheating as perhaps claiming to not have the $400 once the
roommates are all at the furniture store looking at couches).
f.
Are there reasons why you might think that in reality public goods problems at the
household level such as this one can be solved, and efficiency can be achieved?
Many household goods and services can be thought of as public goods (as long as there are at least
two individuals in the household). Think about your decision to clean the bathroom in your house or
apartment. Have you ever thought: hmmm, perhaps if I wait another day or two, my roommate will
clean the bathroom instead of me? If so, then you are attempting to free ride on your roommate’s
provision of a household service (cleanliness), which is a public good. As a result, the house probably
gets dirtier. How might we (or do we, in practice) solve these problems? Often through forms of social
sanctions or punishments. Your roommate might stop talking to you, or worse, if you refuse to do
your share of household tasks. This might be true in our couch example as well. So what is missing
from our model here that might capture this? There is nothing in the utility function that describes the
(utility) cost to each roommate that might be associated with such sanctions or punishments. If we
incorporated this, we might be able to achieve efficiency as an equilibrium outcome (or at least get
closer to efficiency).
3. Two consumers – A and B – each have the utility function U = 3x + 5y. Good x is a public good and
good y is a private good. The price of good x equals the price of good y and each consumer is endowed
with 100 units of the private good.
a. How much of the public good will each consumer provide in the Nash equilibrium?
Each individual’s MRS = 3/5 < px/py  neither will want to contribute any units of the public good.
Dr. Donna Feir
Economics 313
Why not? Each is willing to pay up to 3/5 of a unit of the private good in order to get one unit of the
public good, but each unit of the public good costs 1 unit of the private good. So each contributes
nothing and so no units of the public good are provided in the Nash equilibrium.
b. Is the equilibrium level of provision efficient? If not, what is the efficient level of public good
provision? Now suppose that instead consumer B’s utility function is U = x + 5y, but
everything else in the problem remains unchanged.
MRSA + MRSB = 6/5 > 1 = px/py  in aggregate the consumers are always wiling to pay more for a unit
of the public good than the a units of the public good costs. Which means it will be efficient to spend
all the economy’s endowment on the public good.
c. How much of the public good will each consumer provide in the new Nash equilibrium?
Each individual’s MRS is still < px/py  neither will want to contribute any units of the public good, so
no units of the public good are provided in the Nash equilibrium.
d. Is the equilibrium level of provision efficient? If not, what is the efficient level of public good
provision? Explain why it is that your answer here is different from that in part b).
Now MRSA + MRSB = 4/5 < 1 = px/py  in even in aggregate the consumers are not wiling to pay
enough for units of the public good to justify the expenditure. So in fact, it is efficient to spend
nothing on the public good, which is exactly what the consumers do in the Nash equilibrium.
Comment on this question: I like questions like this one, not because I think they describe reality (we
don’t have linear utility functions, at least for most goods), but because they test student
understanding of the public good model without lots of math. If you have simply memorized math
steps (such as, Nash equilibrium is where MRS = price ratio and efficiency is where sum of MRSs =
price ratio) then you will get stuck on this question as soon as you try to write down 3/5 = 1. To solve
this problem you need to understand that the MRS represents the marginal benefit of the public good,
measured in units of the private good, which also tells us the consumer’s maximum willingness to pay
for an additional unit of the public good. You also need to understand that the price ratio is the
marginal cost of the public good, measured in units of the private good. Once you’ve understood
these things, this problem is trivially simple. Understand the economics; don’t memorize the math.