Problem Set 9. Externalities and Public goods
EconS 526
1. A perfectly competitive market exists for wheat. The inverse demand is 𝑃𝑃 = 200 − 𝑄𝑄 where P is
the price of wheat and Q is the total quantity of wheat. The private total cost for the
unregulated market is 𝐶𝐶 = 50 + 80𝑄𝑄 + 0.5𝑄𝑄 2 . The production of wheat creates an externality
where the total external cost is𝐸𝐸 = 0.5𝑄𝑄 2 .
a. Solve for the unregulated competitive equilibrium of wheat and the socially optimal level of
wheat.
b. Derive the Pigouvian tax (per unit of output of wheat) that results in the social optimum.
c. One big company, WheatsRUs, buys out all the farmers of wheat and becomes a
monopolist. Using the same functional forms, solve for the unregulated monopoly
equilibrium.
d. Given the socially optimal level of wheat in (a), what is the optimal tax that should be placed
on the monopolist?
a. Under perfect competition, the equilibrium occurs when price equals marginal cost. Marginal
cost is 80+Q. Thus, the equilibrium occurs when,
𝑃𝑃 = 𝑀𝑀𝑀𝑀 𝑝𝑝
So Q is 60.
200 − 𝑄𝑄 = 80 + 𝑄𝑄
The socially optimal level of wheat occurs when
𝑃𝑃 = 𝑀𝑀𝑀𝑀 𝑝𝑝 + 𝑀𝑀𝑀𝑀 𝐸𝐸
So Q is 40
200 − 𝑄𝑄 = 80 + 𝑄𝑄 + 𝑄𝑄
b.
So the equation becomes
𝑃𝑃 = 𝑀𝑀𝑀𝑀 𝑝𝑝 + 𝑡𝑡
But we know we want Q=40. So now,
So t is 40.
200 − 𝑄𝑄 = 80 + 𝑄𝑄 + 𝑡𝑡
200 − 40 = 80 + 40 + 𝑡𝑡
c.
Under monopoly, the equilibrium occurs when marginal revenue equals marginal cost. Marginal revenue
is 200-2Q. Thus, the equilibrium occurs when,
𝑀𝑀𝑀𝑀 = 𝑀𝑀𝑀𝑀 𝑝𝑝
200 − 2𝑄𝑄 = 80 + 𝑄𝑄
So Q is 40.
d.
No tax is needed because the monopolist already derives the socially optimal level of wheat. Formally,
we have
𝑀𝑀𝑀𝑀 = 𝑀𝑀𝑀𝑀 𝑝𝑝 + 𝑡𝑡
But we know we want Q=40. So now,
200 − 2𝑄𝑄 = 80 + 𝑄𝑄 + 𝑡𝑡
So t is 0.
200 − 80 = 80 + 40 + 𝑡𝑡
2. Policymakers are contemplating on placing either a Pigouvian tax or a standard to regulate
sulfur dioxide emissions. Policymakers do not know the actual marginal abatement cost of firms
emitting sulfur dioxide but they do know the actual marginal damage function based on
economic estimates. What policy (tax or standard) should be chosen? Explain your answer. To
complete your explanation, (1) provide a citation of a study that estimates the form of the
marginal damage of sulfur dioxide and (2) draw a graph to prove that one regulatory instrument
is better than the other.
This is choosing a regulatory policy under uncertainty. The better policy is the one that minimizes
deadweight loss. Based on the study by Henry, Mueller and Mendelsohn (2011) in the Journal of Policy
Analysis and Management (Figure 6), the marginal damage for SO2 is flat. Therefore, we could have the
following case:
$
MAC_T
MAC_E
a
tax
c
MD
b
Std
S*
SO2
If the government estimated MAC_E but true MAC is MAC_T, then a standard will be put at Std and a
deadweight loss equal to area abc occurs. In contrast, a Pigouvian tax will lead to the socially efficient
level of emissions at S* leading to 0 deadweight loss. Therefore a tax is better in this case.
3. Anna and Bess are assigned to write a joint paper within a 24-hour period about the Pareto
optimal provision of public goods. Let tA denote the number of hours that Anna contributes to
the project and tB the number of hours that Bess contributes. The numeric grade that Anna and
Bess earn is a function, 23ln(tA + tB), of the total number of hours that they contribute to the
project. If Anna contributes tA, then she has (24- tA) hours in the day for leisure. Anna’s utility
function is 𝑈𝑈𝐴𝐴 = 23 ln(𝑡𝑡𝐴𝐴 + 𝑡𝑡𝐵𝐵 ) + ln(24 − 𝑡𝑡𝐴𝐴 ) and Bess’s utility function is 𝑈𝑈𝐵𝐵 =
23 ln(𝑡𝑡𝐴𝐴 + 𝑡𝑡𝐵𝐵 ) + ln(24 − 𝑡𝑡𝐵𝐵 ).
a. If they decide to choose hours to contribute simultaneously and independently, what is the
Nash equilibrium number of hours that each will provide?
b. What is the number of hours each should contribute to the project that maximizes the sum
of utilities?
a.
For Anna, her problem is,
max
𝑡𝑡𝐴𝐴
The FOC is
𝑈𝑈𝐴𝐴 = 23 ln(𝑡𝑡𝐴𝐴 + 𝑡𝑡𝐵𝐵 ) + ln(24 − 𝑡𝑡𝐴𝐴 )
23
1
−
=0
𝑡𝑡𝐴𝐴 + 𝑡𝑡𝐵𝐵 24 − 𝑡𝑡𝐴𝐴
Simplifying, we get the reaction function,
Bess’s reaction function is similar,
(552 − 𝑡𝑡𝐵𝐵 )
= 𝑡𝑡𝐴𝐴
24
(552 − 𝑡𝑡𝐴𝐴 )
= 𝑡𝑡𝐵𝐵
24
Substitute Bess’s reaction function into Anna’s and we get, 𝑡𝑡𝐴𝐴 = 22.08. This will be the same for Bess.
b.
The aggregate utility is now,
Or,
max
𝑡𝑡𝐴𝐴 𝑡𝑡𝐵𝐵
𝑊𝑊 = 23 ln(𝑡𝑡𝐴𝐴 + 𝑡𝑡𝐵𝐵 ) + ln(24 − 𝑡𝑡𝐴𝐴 ) + 23 ln(𝑡𝑡𝐴𝐴 + 𝑡𝑡𝐵𝐵 ) + ln(24 − 𝑡𝑡𝐵𝐵 )
The FOCs are now,
max
𝑡𝑡𝐴𝐴 𝑡𝑡𝐵𝐵
𝑊𝑊 = 46 ln(𝑡𝑡𝐴𝐴 + 𝑡𝑡𝐵𝐵 ) + ln(24 − 𝑡𝑡𝐴𝐴 ) + ln(24 − 𝑡𝑡𝐵𝐵 )
46
1
−
=0
𝑡𝑡𝐴𝐴 + 𝑡𝑡𝐵𝐵 24 − 𝑡𝑡𝐴𝐴
1
46
−
=0
𝑡𝑡𝐴𝐴 + 𝑡𝑡𝐵𝐵 24 − 𝑡𝑡𝐵𝐵
Simultaneously solving for both equations we obtain, 𝑡𝑡𝐴𝐴 = 𝑡𝑡𝐵𝐵 = 23.
Note: The problem set is due on December 10. See syllabus for penalty due to late submissions. This is a
bonus problem set that can replace the lowest problem set score that you have so far. If you decide not
to do this problem set, it will simply be dropped as your lowest problem set score.