Chapter 17
Externalities and Public Goods
Solutions to Review Questions
1. What is the difference between a positive externality and a negative externality?
Describe an example of each.
With a negative externality, the marginal private cost of a good is less than the marginal social
cost of a good. For example, the private costs associated with driving to work on the highway
include personal time, gasoline, wear and tear on the vehicle, etc. In addition, by entering the
highway, a vehicle creates congestion that increases the time it takes all other drivers to get
where they are going. Thus, the social cost exceeds the private cost and entering the highway
creates a negative externality.
With a positive externality, the marginal private benefit is less than the marginal social benefit.
For example, when parents immunize a child they reduce the risks of the child contracting a
disease. In addition, by immunizing the child, the child is less likely to pass on certain diseases
to other people. Thus, the social benefits from immunizing exceed the private benefits and
immunization creates a positive externality.
2. Why does an otherwise competitive market with a negative externality produce more
output than would be economically efficient?
A competitive market with a negative externality produces more output than is socially optimal.
This occurs because the firms in the industry do not take into account the external costs
associated with production; they only take into account their private costs. Because they view
the cost as lower than it actually is, they produce more than would be produced if they were
forced to take into account the external costs.
3. Why does an otherwise competitive market with a positive externality produce less
output than would be economically efficient?
A competitive market with a positive externality produces less output than is socially optimal.
This occurs because consumers do not take into account the external benefits associated with
consumption; they only take into account their private benefits. Because they view the benefit as
lower than it actually is, they consume less than they would if they were forced to take into
account the external benefits.
4. When do externalities require government intervention, and when is such intervention
unlikely to be necessary?
Negative externalities may require government intervention when there is significant disparity
between the socially optimal production level of a good and the unregulated equilibrium
production level. To limit production, the government might impose taxes on production or a
production quota limiting production. If property rights are clearly defined and bargaining is
costless, the market may reach the socially efficient level of production without government
intervention.
Positive externalities may also require government intervention when there is significant
disparity between the socially optimal production level of a good and the unregulated
equilibrium production level. To encourage production, the government might provide
production subsidies. If property rights are clearly defined and bargaining is costless, the market
may reach the socially optimal level of production without government intervention.
5. How might an emissions fee lead to an efficient level of output in a market with a
negative externality?
With a negative externality, an emissions fee might lead to an efficient level of output. By
imposing a fee on production, producers are forced to take into account not only their private
costs but also the external costs (as measured by the emissions fee) of production. This has the
effect of raising the firm’s costs and reducing the firm’s production. If the level of the emissions
fee is set so that, for the last unit produced, the fee equals the external cost, this fee could lead to
an efficient level of output.
6. How might an emissions standard lead to an efficient level of output in a market with a
negative externality?
An emissions standard could lead to an efficient level of output. By setting a standard and only
selling the rights to a limited amount of emissions, the government can reduce the level of
emissions. In addition, by implementing a system whereby the rights can be traded, the
government could reduce emissions and distribute the rights so that abatement costs are as low as
possible.
7. What is the Coase Theorem, and when is it likely to be helpful in leading a market with
externalities to provide the socially efficient level of output?
The Coase Theorem states that, regardless of how property rights are assigned with an
externality, the allocation of resources will be efficient when the parties can costlessly bargain
with each other. This Theorem will be helpful in leading a market with externalities to the
socially efficient level when the cost of bargaining is low and when all parties involved can agree
on the costs and benefits associated with the externality.
8. How does a nonrival good differ from a nonexclusive good?
A nonexclusive good is one that no consumer can be prevented from consuming. A nonrival
good is one that one consumer’s consumption does not eliminate or prevent another consumer’s
consumption.
9. What is a public good? How can one determine the optimal level of provision of a public
good?
A public good is any good that is nonexclusive and nonrival. To determine the optimal level of
provision of a public good, one should determine the marginal social benefits from the public
good, which is equal to the sum of the marginal private benefits for the individual consumers,
and equate that to the marginal cost of providing the public good. Units of the good should be
provided as long as the marginal social benefit exceeds the marginal cost. This will occur up to
the point where the marginal social benefit equals the marginal cost.
10. Why does the free-rider problem make it difficult or impossible for markets to provide
public goods efficiently?
It is difficult to provide public goods efficiently when free riders exist. Free riders will consume
the good, but will pay nothing for the good, anticipating that others will pay. It may therefore
prove difficult to raise funds to finance a project with a public good, leading to an underproduction of the good, or possibly even no provision of a good with positive net benefits.
Solutions to Problems
17.1. Why is it not generally socially efficient to set an emissions standard allowing zero
pollution?
If the government were to set an emissions standard requiring zero pollution, this standard would
probably not be socially efficient. By setting the standard at zero, the government could reduce
pollution by preventing polluting industries from producing goods that society values. By setting
the standard at zero, however, the government will also eliminate the benefits to society from
production of these goods. In general, the social benefits from producing will likely exceed the
social costs up to some non-zero level of production (pollution) implying the socially efficient
level of production is non-zero.
17.2. Education is often described as a good with positive externalities. Explain how
education might generate positive external benefits. Also suggest a possible action the
government might take to induce the market for education to perform more efficiently.
Education is a good that might generate positive external benefits. For example, when an
individual furthers her education she benefits directly in terms of higher income. In addition,
this individual, because of her increased education, might be able to develop a new technology
that benefits all of society. Thus, while the education helped the individual, by allowing the
development of the new technology (because she’s smarter!) many people benefited from her
education.
To induce the market to perform more efficiently, the government would like to entice more
individuals to further their education since education generates positive externalities. The
government could do this by providing grants or low interest student loans, for example.
17.3. a) Explain why cigarette smoking is often described as a good with negative
externalities.
b) Why might a tax on cigarettes induce the market for cigarettes to perform more
efficiently?
c) How would you evaluate a proposal to ban cigarette smoking? Would a ban on smoking
necessarily be economically efficient?
a) For one, by smoking in public, smokers force other individuals to breathe air with smoke,
known as second-hand smoke. In addition, the health problems associated with smoking force
society to pay higher health care costs to pay for smoking related illnesses, both for smokers and
for those who breathe second-hand smoke, than if no one in society smoked.
b) By imposing a tax on cigarettes the government increases the marginal private cost of
smoking and forces the individual to take into account (at least some of) the negative externality
associated with smoking. This would likely reduce the level of smoking in society, pushing the
equilibrium toward the socially efficient level of smoking.
c) A ban on smoking entirely is probably not socially efficient. To evaluate such a ban, one
would need to compare the marginal benefits with the marginal social costs. The ban would only
be socially efficient if the marginal social costs exceed the marginal benefits at a level of zero.
This would not necessarily be socially efficient because it is possible that the marginal benefit of
smoking exceeds the marginal social cost for low levels of smoking.
17.4. Consider Learning-By-Doing Exercise 17.2, with a socially efficient emissions fee.
Suppose a technological improvement shifts the marginal private cost curve down by $1. If
the government calculates the optimal fee given the new marginal private cost curve, what
will happen to the following?
a) The size of the optimal tax
b) The price consumers pay
c) The price producers receive
a) If the marginal private cost shifts down by $1, we have MPC 1 Q . With demand
P d 24 Q and marginal external cost MEC
2 Q , the social optimum occurs where
Pd
MPC MEC
24 Q (1 Q) ( 2 Q)
Q 8.33
At Q 8.33 , the price is P 15.67 .
The size of the optimal tax is the difference between the equilibrium price, 15.67, and the MPC
at the socially efficient quantity of 8.33. At this quantity the marginal private cost is
MPC 9.33 . Thus, the optimal tax is T 15.67 9.33 6.34 .
b) With this tax, consumers will pay the socially efficient price of 15.67.
c) Producers will receive the difference between the price consumers pay, 15.67, and the tax,
6.34. Thus, producers will receive 9.33.
17.5. Consider the congestion pricing problem illustrated in Figure 17.5.
a) What is the size of the deadweight loss from the negative externalities if there is no toll
imposed during the peak period?
b) Why is the optimal toll during the peak period not $3, the difference between the
marginal social cost and the marginal private cost when the traffic volume is Q5?
c) How much revenue will the toll authority collect per hour if it charges the economically
efficient toll during the peak period?
a) In Figure 17.5, the deadweight loss is area ABG. This is deadweight loss because for every
vehicle beyond the optimum, Q4, the marginal social cost exceeds the marginal benefit. Area
ABG is approximately (assuming the demand and MPC curves are nearly straight lines over this
part of the graph ) 0.5(Q5 Q4 )(8 5) 1.5(Q5 Q4 ) .
b) The socially efficient traffic volume occurs where the marginal social cost curve intersects
the marginal benefit curve. In Figure 17.5 this occurs at Q4. At Q4, the marginal benefit is $5.75
and the marginal private cost is $4.00. To achieve the social optimum the toll should be set so
that the marginal benefit equals the marginal private cost plus the toll, effectively forcing the
driver to take into account the external cost of entering the highway. At Q4, this is
$5.75 $4.00 $1.75 .
The toll is not $3.00 because the toll should be set to force the driver at Q4 to observe the
external costs imposed by entering the highway. By setting the toll at $3.00, the difference
between the MPC and MB at Q5, the toll would be set to force the driver at Q4 to observe the
external costs imposed from the driver at Q5 entering the highway. But this cost is unimportant
because at the optimum the driver at Q5 will not be on the highway. The $3.00 toll would create
a level of traffic below the social optimum.
c) If the toll authority sets a toll at the economically efficient level of $1.75, it will earn revenue
equal to the toll multiplied by the number of drivers. In this case, revenue will be $1.75Q4.
17.6. The accompanying graph (on next page) shows the demand curve for gasoline and the
supply curve for gasoline. The use of gasoline creates negative externalities, including CO2,
which is an important source of global warming. Using the graph and the table below,
identify:
• The equilibrium price and quantity of gasoline
• The producer and consumer surplus at the market equilibrium
• The cost of the externality at the free-market equilibrium
• The net social benefits arising at the free-market equilibrium
• The socially optimal price of gasoline
• The consumer and producer surplus at the social optimum
• The cost of the externality at the social optimum
• The net social benefits arising at the social optimum
• The deadweight loss due to the externality
Consumer surplus
Private producer surplus
- Cost of externality
Net social benefits
Deadweight loss
Equilibrium price
and quantity = P2
and Q2
Social optimum
price and quantity
= P1 and Q1
A+B+G+K
E+F+R+H+N
-R-H-N-G-K-M
A+B+E+F-M
M
A
B+E+F+R+H+G
-R-H-G
A+B+E+F
Zero
Difference
between social
optimum and
equilibrium
-B-G-K
B+G-N
M+N+K
M
M
17.7. The graph below shows conditions in a perfectly competitive market in which there is
some sort of externality. In this market, a consumer purchases at most one unit of the good.
There are many such consumers, and they have different maximum willingnesses to pay.
Assume that the graph is drawn to scale.
a) What type of externality is present in this market: positive or negative?
b) What is the maximum level of social surplus that is potentially attainable in this market?
c) What is the deadweight loss that arises in a competitive equilibrium in this market?
d) Suppose a subsidy is given to producers: What is the magnitude of the subsidy per unit
that would enable this market to attain the socially efficient outcome?
For the remaining questions, please indicate whether the following government
interventions would increase social efficiency relative to the competitive equilibrium
outcome with no government intervention, decrease social efficiency, or keep it unchanged:
e) A subsidy per unit equal to 0F given to consumers who purchase the good.
f ) The government replaces private sellers and offers the good at a price of zero. (Assume
that government has no inherent cost advantage or disadvantage relative to private
producers. Assume, too, the government’s cost of production is financed by levying taxes.)
g) The government imposes a price ceiling that sets a maximum price for the good equal to
0D.
h) The government imposes a tax equal to NR on consumers who do not purchase the good.
a) The externality is positive. We can see this because the marginal social cost at any quantity is
less than the marginal private cost.
b) The maximum level of social surplus that is potentially attainable in this market is AR0. The
socially efficient outcome is at the intersection of marginal social cost and marginal social
benefit (point R), and the maximum level of social surplus is the area between the SMC and
SMB curves to the left of this point.
c) The deadweight loss that arises in a competitive equilibrium in this market is KLR. The
equilibrium point is K. The deadweight loss is the area between MSB and MSC, from the
equilibrium quantity 0M to the efficient quantity 0S.
d) The subsidy to producers that would enable this market to attain the socially efficient
outcome is 0F (or equivalently, IH, LK, RN, VU, EC, DB).
e) A subsidy equal to OF given to consumers who purchase the good increases social efficiency.
In fact, the efficient outcome is achieved. It does not matter that the subsidy is given to
consumers, not producers. A subsidy to consumers shifts the demand curve upward by 0F so that
it intersects the supply curve at N, which is the new equilibrium point. The equilibrium quantity
is the efficient quantity 0S.
f) If the government offers the good at a price of zero then social efficiency decreases. If the
good is provided at a price of zero, consumers will purchase quantity 0W. The deadweight loss is
RVW, which is the area between SMC and SMB over the range between the socially efficient
quantity 0S and 0W. This deadweight loss is larger than the deadweight loss KLR with no
government intervention, so efficiency is reduced.
g) A price ceiling that sets a maximum price for the good equal to 0D decreases social
efficiency. At this price, the consumers may wish to purchase the efficient quantity 0S, but the
producers are only willing to supply quantity 0J, and the consumers are unable to buy more than
that. The deadweight loss increases to (at least) IRG.
h) If the government imposes a tax equal to NR on consumers who do not purchase the good,
then social efficiency increases. This is a monetary incentive to purchase the good. By
purchasing a unit of the good a consumer gets two things --- the value from the good and
avoidance of the tax. Thus, a tax on consumers who do not purchase the good makes it “as if” the
demand curve is NR above the actual demand curve. The result is an equilibrium in which the
quantity is the socially efficient quantity 0S. Thus, this incentive is equally strong as the subsidy
from part e), and it has the same effect.
17.8. A competitive refining industry produces one unit of waste for each unit of refined
product. The industry disposes of the waste by releasing it into the atmosphere. The inverse
demand curve for the refined product (which is also the marginal benefit curve) is Pd = 24 Q, where Q is the quantity consumed when the price consumers pay is Pd. The inverse
supply curve (also the marginal private cost curve) for refining is MPC = 2 + Q, where
MPC is the marginal private cost when the industry produces Q units. The marginal
external cost curve is MEC = 0.5Q, where MEC is the marginal external cost when the
industry releases Q units of waste.
a) What are the equilibrium price and quantity for the refined product when there is no
correction for the externality?
b) How much of the chemical should the market supply at the social optimum?
c) How large is the deadweight loss from the externality?
d) Suppose the government imposes an emissions fee of $T per unit of emissions. How large
should the emissions fee be if the market is to produce the economically efficient amount of
the refined product?
a)
If there is no correction for the externality, the equilibrium will occur at the point where
the marginal benefit curve, P d 24 Q , intersects the marginal private cost curve,
MPC 2 Q . This occurs at
24 Q 2 Q
Q 11
At Q 11 , price is P 13 .
b)
At the social optimum marginal benefit, P d 24 Q , will equal marginal social cost,
MSC MPC MEC . This occurs where
24 Q (2 Q) 0.5Q
Q 8.80
Thus, the social optimum is to produce Q 8.80 .
c)
At the uncorrected equilibrium, the marginal social cost is MSC
Thus, the deadweight loss will be 0.5(11 8.80)(18.5 13) 6.05 .
2 1.5(11) 18.5 .
d)
The emissions fee of $T should be set to shift the MPC curve so that it intersects the
marginal benefit curve at Q 8.80 , the socially optimal quantity. At Q 8.80 the marginal
benefit is P 15.2 and the marginal private cost is MPC 2 8.80 10.80 . Therefore, the
optimal tax is T 15.2 10.8 4.4 .
17.9. Consider a manufactured good whose production process generates pollution. The
annual demand for the good is given by Qd = 100 - 3P. The annual market supply is given
by Qs = P. In both equations, P is the price in dollars per unit. For every unit of output
produced, the industry emits one unit of pollution. The marginal damage from each unit of
pollution is given by 2Q.
a) Find the equilibrium price and quantity in a market with no government intervention.
b) At the equilibrium you computed, calculate: (i) consumer surplus; (ii) producer surplus;
(iii) total dollars of pollution damage. What are the overall social benefits in the market?
c) Find the socially optimal quantity of the good. What is the socially optimal market price?
d) At the social optimum you computed, calculate: (i) consumer surplus; (ii) producer
surplus; and (iii) total dollars of pollution damage. What are the overall social benefits in
the market?
e) Suppose an emissions fee is imposed on producers. What emissions fee would induce the
socially optimal quantity of the good?
a)
100 – 3P = P P = 25 and Q = 25.
c)
MEC = 2Q, while MPC = Q. Thus, MSC = 3Q. To find the optimal quantity, we equate
MSC to inverse demand, or 3Q = 100/3 – Q/3, or Q = 10. The socially optimal price would equal
the marginal social cost at the optimal quantity, or P = 3(10) = $30.
e)
The optimal emissions fee is equal to the difference between MSC and MPC at the
socially optimal quantity. Since MSC = 3Q and MPC = Q, and Q = 10, the optimal emissions fee
equals: 3(10) – 1(10) = $20 per unit.
Price
75
MSC = 3Q
MPC = Q
M
100/3
30
A
B
25
G
K
N
E
H
F
P = 100/3 – Q/3
R
Quantity
10
Consumer surplus
Private producer
surplus
- Cost of
externality
Net social benefits
Deadweight loss
25
Equilibrium price and Social optimum
Difference
quantity = P2 and Q2
price and quantity = between social
P1 and Q1
optimum and
equilibrium
A+B+G+K =
A=
-B-G-K =
$104.167
$1.67
-$102.50
E+F+R+H+N =
B+E+F+R+H+G = B+G-N=
$312.50
$250
-62.5
-R-H-N-G-K-M =
-R-H-G =
M+N+K=
-$625
-$100
$525
A+B+E+F-M =
A+B+E+F=
M=
-$208.33
$151.67
$360
M=
Zero
M
17.10. The demand for widgets is given by P = 60 - Q. Widgets are competitively supplied
according to the inverse supply curve (and marginal private cost) MPC = c. However, the
production of widgets releases a toxic gas into the atmosphere, creating a marginal external
cost of MEC = Q.
a) Suppose the government is considering imposing a tax of $T per unit. Find the level of
the tax, T, that ensures the socially optimal amount of widgets will be produced in a
competitive equilibrium.
b) Suppose a breakthrough in widget technology lowers the marginal private cost, c, by $1.
How will this affect the optimal tax you found in part (a)?
a)
The socially optimal level of output occurs when P = MPC + MEC, or 60 – Q = c + Q,
which implies Q = 30 – 0.5c. At this output level, P = 30 + 0.5c. The optimal tax is the
difference between price and MPC at this output level: T = 30 + 0.5c – c = 30 – 0.5c.
b)
If marginal private cost falls to MPC = c – 1, then the optimal tax becomes T = 30 –
0.5(c – 1) = 30.5 – 0.5c. That is, the optimal tax rises by $0.50. You can also see this since
ΔT/Δc = –0.5 in part (a).
17.11. The market demand for gadgets is given by Pd = 120 - Q, where Q is the quantity
consumers demand when the price they consumers pay is Pd. Gadgets are competitively
supplied according to the inverse supply curve (and marginal private cost) MPC = 2Q,
where Q is the amount suppliers will produce when they receive a price equal to MPC. The
production of gadgets releases a toxic effluent into the water supply, creating a marginal
external cost of MEC = Q. The government wants to impose a sales tax on gadgets to
correct for the externality. When producers receive a price equal to MPC, the amount
consumers must pay is (1 + t)MPC, where t is the sales tax rate. Find the level of the tax
rate that ensures the socially optimal amount of gadgets will be produced in a competitive
equilibrium.
The socially optimal output level occurs where P = MPC + MEC, or 120 – Q* = 2Q* + Q*,
implying Q = 30. With the sales tax, the equilibrium output level occurs where Pd = (1 + t)MPC,
or 120 – Q = (1 + t)(2Q). As a function of t, the equilibrium output level is then Q = 120/(3 +
2t).
The optimal sales tax level ensures that Q = Q*, or 120/(3 + 2t) = 30. Solving, we have t = 0.50.
17.12. Amityville has a competitive chocolate industry with the (inverse) supply curve Ps =
440 + Q. While the market demand for chocolate is Pd = 1200 - Q, there are external
benefits that the citizens of Amityville derive from having a chocolate odor wafting through
town. The marginal external benefit schedule is MEB = 6 - 0.05Q.
a) Without government intervention, what would be the equilibrium amount of chocolate
produced? What is the socially optimal amount of chocolate production?
b) If the government of Amityville used a subsidy of $S per unit to encourage the optimal
amount of chocolate production, what level should that subsidy be?
a)
The equilibrium level of output occurs where Pd = Ps, or 1200 – Q = 440 + Q.
Equilibrium output is then Q = 380. Taking into account the positive externality, the social
optimal amount of production sets Pd + MEB = Ps, or 1200 – Q* + 60 – 0.05Q* = Q* + 440,
yielding Q* = 400.
b)
With a subsidy of $S, equilibrium occurs where Pd + S = Ps or 1200 – Q + S = 440 + Q.
To get Q = Q* = 400 the subsidy must satisfy 1200 – 400 + S = 440 + 400 or S = 40.
17.13. The only road connecting two populated islands is currently a freeway. During rush
hour, there is congestion because of the heavy traffic. The marginal external cost from
congestion rises as the amount of traffic on the road increases. At the current equilibrium,
the marginal external cost from congestion is $5 per vehicle. Would a toll charge of $5 per
vehicle lead to an economically efficient amount of traffic? If not, would you expect the
economically efficient toll to be larger than, or less than $5?
The graph below demonstrates that the optimal toll would be less than $5. Equilibrium with no
toll occurs at Q3, although the socially optimal amount of driving would occur at Q2 < Q3.
Although the marginal external cost is $5 at Q3, the socially optimal toll occurs at MEC(Q2),
which is less than $5 since MEC is upward sloping.
As an alternative interpretation, you can notice that imposing a $5 toll would cause equilibrium
to occur at Q1, where demand intersects the curve MPC + $5. By definition, at Q3 it must be true
that MSC = MPC + MEC = MPC + $5. Thus the curve MPC + $5 intersects the MSC curve at
Q3. Since MEC is upward sloping, MSC < MPC + $5 at output levels less than Q3 so the MPC +
$5 curve must intersect demand to the left of Q2.
P
Demand
MSC
MPC + $5
MPC
$5
MEC
Q1 Q2 Q3
Q
17.14. A firm can produce steel with or without a filter on its smokestack. If it produces
without a filter, the external costs on the community are $500,000 per year. If it produces
with a filter, there are no external costs on the community, and the firm will incur an
annual fixed cost of $300,000 for the filter.
a) Use the Coase Theorem to explain how costless bargaining will lead to a socially efficient
outcome, regardless of whether the property rights are owned by the community or the
producer.
b) How would your answer to part (a) change if the extra yearly fixed cost of the filter were
$600,000?
a)
If the firm installs the filter, the community benefits by $500,000 while the firm incurs a
cost of $300,000. The socially efficient outcome is for the firm to install the filter.
If the firm possesses the right to pollute, the community will have an incentive to pay the
firm some price above $300,000 (perhaps $499,999) to induce the firm to install the filter.
Afterwards, the community would be $1 better off and the firm would be $199,999 better off.
On the other hand, suppose the community possesses the right to prevent the firm from
polluting. The firm then has two choices: it could install the filter at a cost of $300,000, or it
could compensate the community for the costs of its pollution by paying them $500,001. Since
it’s clearly cheaper to install the filter, the firm will have a strong incentive to choose this
socially efficient outcome.
b)
If the filter costs $600,000, then the socially efficient outcome is for the firm to not install
the filter, since the costs exceed the benefits.
If the firm possesses the right to pollute, it will do so. Although the community is willing
to pay the firm up to $500,000 to stop polluting, that does not exceed the firm’s costs of
installing the filter.
If the community possesses the right to prevent the firm from polluting, then the firm will
have a strong incentive to compensate the community for the costs of polluting rather than
installing the filter. The community could demand that the firm either install the filter or pay it
some price less than $600,000 (say $599,999). Then the firm would be better off (by $1)
installing the filter, while the community also benefits (by $99,999) since the payment exceeds
the costs of living with the pollution.
17.15. Two farms are located next to each other. During storms, sewage from Farm 1 flows
into a stream located on Farm 2. Farm 2 relies on this stream as a source of drinking water
for its livestock, and when the stream is polluted with sewage, the livestock become sick
and die. The annual damage to Farm 2 from this form of pollution is $100,000 per year. It
is possible that Farm 1 can prevent the runoff of sewage by installing storm drains. The
cost of the storm drains is $200,000.
a) Provide an argument that the Coase Theorem holds in this situation.
b) Suppose that the damage to Farm 2 is $500,000 per year, not $100,000 per year (with the
cost of storm drains remaining fixed at $200,000). Provide an argument that the Coase
Theorem holds in this case.
a)
Suppose the property rights are assigned to Farmer 1. Farmer 2 can either pay for storm
drains at Farm 1 for $200,000 or live with the damage of $100,000. Farmer 2 will not find it
worthwhile to pay for the storm drains, and the run-off from Farm 1 will continue.
Suppose the property rights are assigned to Farmer 2. Farmer 1 can either spend $200,000 to
prevent the run-off, or can pay $100,000 in compensation for the pollution damage. Farmer 1
will find it worthwhile to pay for the damage, and the run-off will continue.
With either property rights assignment, the outcome is the same: the run-off will continue. It is
not economically efficient to build a storm drain because the storm drain costs more than the
damage due to the run-off.
b)
Suppose the property rights are assigned to Farmer 1. Farmer 2 can either pay for storm
drains at Farm 1 for $200,000 or live with the damage of $500,000. Farmer 2 will find it
worthwhile to pay for the storm drains, and the run-off from Farm 1 will be prevented.
Suppose the property rights are assigned to Farmer 2. Farmer 1 can either spend $200,000 to
prevent the run-off, or can pay $500,000 in compensation for the pollution damage. Farmer 1
will find it worthwhile to pay for the storm drain, and the run-off will be prevented.
With either property rights assignment, the outcome is the same: the storm drain will be built,
and the run-off will be abated. It is economically efficient to build a storm drain because the
storm drain costs less than the damage due to the run-off.
17.16. Suppose a factory located next to a river discharges pollution that causes $2 million
worth of environmental damage to the residents downstream. The factory could completely
eliminate the pollution by treating the water on location at a cost of $1.6 million.
Alternatively, the residents could construct a water purification plant just upstream of
their town, at a cost of $0.8 million, which would not completely eliminate the
environmental damage to them but reduce it to $0.5 million. Under current law, the factory
must compensate the town for any environmental damage the factory causes. Bargaining
between the factory owner and the town is costless. What would the Coase Theorem imply
about the outcome of bargaining between the town and the factory owner?
There are three scenarios that could emerge: (a) status quo in which pollution damage is $2
million; (b) full elimination of pollution at a cost of $1.6 million; (c) construction of a water
purification plant near the town for $0.8, which reduces pollution damage to $0.5 million. The
Coase theorem predicts that bargaining would result in the option with the lowest total cost -damage cost plus abatement cost -- which is (c), and that furthermore, this option would emerge
irrespective of which party has the property rights.
In this case, because the town has the property rights, it could indeed force the factory to either
pay $2 million in damages or to completely eliminate the pollution at a cost of $1.6 million. This
suggests that the firm would be forced to eliminate the pollution for $1.6 million. But if
bargaining is costless, we would imagine that the factory would approach the town and say:
“Look, we have a better idea than this expensive solution. We will build you a purification plant
near your town for $0.8 million. We know that this will not fully clean up the damage, and so we
will compensate you $0.5 million for the remaining damage. We have reduced the damage by
$1.5 million and compensated you for the remaining damage of $0.5 million. You are better off,
and by the way, we are better off, too, because this solution only costs us $1.3 million, rather
than $1.6 million. This deal is win-win for both us.”
17.17. The demand for energy-efficient appliances is given by P = 100/Q, while the inverse
supply (and marginal private cost) curve is MPC = Q. By reducing demand on the
electricity network, energy-efficient appliances generate an external marginal benefit
according to MEB = eQ.
a) What is the equilibrium amount of energy-efficient appliances traded in the private
market?
b) If the socially efficient number of energy-efficient appliances is Q = 20, what is the value
of e?
c) If the government subsidized production of energy efficient appliances by $S per unit,
what level of the subsidy would induce the socially efficient level of production?
a)
Equilibrium occurs where P = MPC, or 100/Q = Q. Thus Q = 10.
b)
If the socially efficient number of appliances is Q = 20, then P + MEB = MPC at Q = 20,
or 100/20 + e*20 = 20. Solving, we get e = 0.75.
c)
With the subsidy, equilibrium occurs where P + S = MPC, or 100/Q + S = Q. Since the
efficient amount of appliances is Q = 20, the proper subsidy would solve 100/20 + S = 20 or S =
15.
17.18. The demand for air-polluting backhoes in Peoria is PD = 48 - Q. The air pollution
creates a marginal external cost according to MEC = 2 + Q. Supply of backhoes is given by
PS = 10 + cQ. If the socially efficient level of backhoes is Q* = 12, find the tax that induces
the socially efficient level of backhoes in equilibrium and the value of c.
The socially efficient quantity Q* solves PD = PS + MEC while the optimal tax T solves PD = PS
+ T. Thus, we know that the optimal tax is equal to the marginal external cost at Q* = 12, or T =
2 + Q*. Thus T = 14.
The value of c can be determined since we know that at Q* = 12, PD = PS + MEC or 48 – 12 =
10 + c*12 + 2 + 12. Solving, we get c = 1.
17.19. The town of Steeleville has three steel factories, each of which produces air pollution.
There are 10 citizens of Steeleville, each of whose marginal benefits from reducing air
pollution is represented by the curve p(Q) = 5 - Q/10, where Q is the number of units of
pollutants removed from the air. The reduction of pollution is a public good. For each of
the three sources of air pollution, the following table lists the current amount of pollution
being produced along with the constant marginal cost of reducing it.
a) On a graph, illustrate marginal benefits (“demand”) and the marginal costs (“supply”)
of reducing pollution. What is the efficient amount of pollution reduction? Which factories
should be the ones to reduce pollution, and what would the total costs of pollution
reduction be? In a private market, would any units of this public good be provided?
b) The Steeleville City Council is currently considering the following policies for reducing
pollution:
i. Requiring each factory to reduce pollution by 10 units
ii. Requiring each factory to produce only 30 units of pollution
iii. Requiring each factory to reduce pollution by one fourth
Calculate the total costs of pollution reduction associated with each policy. Compare the
total costs and amount of pollution reduction to the efficient amount you found in part (a).
Do any of these policies create a deadweight loss?
c) Another policy option would create pollution permits, to be allocated and, if desired,
traded among the firms. If each factory is allocated tradeable permits allowing it to
produce 30 units of pollution, which factories, if any, would trade them? (Assume zero
transactions costs.) If they do trade, at what prices would the permits be traded?
d) How does your answer in part (c) relate to that in part (a)? Explain how the Coase
Theorem factors into this relationship.
a)
The market demand curve is created by adding the individual demand curves vertically:
P(Q) = 10*p(Q) = 50 – Q. Demand intersects supply at a price of $20, associated with Q = 30
units of pollution reduction. This entails factory A reducing pollution by 20 and factory B
reducing pollution by 10.
The total cost of pollution reduction would be 20*$10 + 10*$20 = $400. In a private
market, each consumer would pay no more than $5 for the first unit of pollution reduction while
no factory would reduce pollution for less than $10. Thus, no amount of this public good would
be provided in a private market.
b)
Under (i), total costs are 10*$10 + 10*$20 + 10*30 = $600.
Under (ii), A does not have to reduce pollution at all. B must reduce pollution by 10
units, and C must reduce pollution by 30 units. Thus total costs under this policy are 10*$20 +
30*$30 = $1100.
Under (iii), A reduces pollution by 20/4 = 5 units, B by 40/4 = 10 units, and C by 60/4 =
15 units. Total costs of pollution reduction under this policy are thus 5*$10 + 10*$20 + 15*$30
= $700.
While each policy achieves the efficient amount of pollution reduction (Q = 30), it does
so in a way that is more costly than the efficient allocation in part (a). In particular, the policies
lead to deadweight losses of $200, $700, and $300 respectively.
c)
With the permits, C must still reduce pollution by 30 units and B by 10 units, while A is
only creating 20 units of pollution in the first place and so does not need to reduce pollution at
all. However, Factory A could eliminate all its pollution (at a marginal cost of $10 per unit) and
sell its permits to Factory C at a price of up to $30 per unit. Thus, Factory C would buy all of
Factory A’s permits, at a price between $20 and $30 per unit (note that Factory B would
purchase some of them if the price were less than $20).
d)
By defining property rights over pollution and allowing them to be traded, the socially
efficient outcome (where A reduces pollution by 20 units and B reduces its pollution by 10 units)
is achieved, just as the Coase Theorem predicts.
17.20. A chemical producer dumps toxic waste into a river. The waste reduces the
population of fish, reducing profits for the local fishing industry by $100,000 per year. The
firm could eliminate the waste at a cost of $60,000 per year. The local fishing industry
consists of many small firms.
a) Using the Coase Theorem, explain how costless bargaining will lead to a socially efficient
outcome, regardless of whether the property rights are owned by the chemical firm or the
fishing industry.
b) Why might bargaining not be costless?
c) How would your answer to part (a) change if the waste reduces the profits for the fishing
industry by $40,000? (Assume, as before, that the firm could eliminate the waste at a cost of
$60,000 per year.)
a)
If property rights are assigned to the chemical producer, the fisherman will pay $60,000
to the firm to eliminate the toxic waste. If property rights are assigned to the fisherman, the
chemical producer will clean up the waste since this is cheaper than compensating the fisherman.
Thus, regardless of who property rights are assigned to, the toxic waste gets cleaned up because
this is less costly than the damage.
b)
Because the waste harms many fishermen, it may not be easy to organize them to bargain
about compensation. Organizing the fishermen may be costly. In addition, if the fishermen and
the firm have different perceptions regarding the costs of the externality, they might not reach an
efficient solution.
c)
If the fishermen’s profits were reduced by $40,000 rather than $100,000, then if property
rights were assigned to the chemical producer, the fishermen would not find it worthwhile to pay
for cleanup. The fishermen will receive no compensation. If property rights are assigned to the
fishermen, the chemical producer would compensate the fishermen $40,000 rather than paying
the cleanup costs. Thus, regardless of whom property rights are assigned to, the waste will not
get cleaned up. This is economically efficient because cleanup costs more than the damage.
17.21. Consider an economy with two individuals. Individual 1 has (inverse) demand curve
for a public good given by P1 = 60 - 2Q1, while individual 2 has (inverse) demand curve for
the public good given by P2 = 90 - 5Q2. The prices are measured in $ per unit. Suppose the
marginal cost of producing the public good is $10 per unit. What is the efficient level of the
public good?
The marginal social benefit curve is the vertical sum of the individual consumer’s inverse
demand curves. When we sum vertically we add prices (i.e., willingness to pay). Thus, letting Q
denote the quantity of the public good, we have:
MSB = (60 – 2Q) + (90 – 5Q) = 150 – 7Q.
Equating MSB to MC we have: 150 – 7Q = 10, or Q = 20. This is the socially efficient quantity
of the public good.
17.22. There are three consumers of a public good. The demands for the consumers are as
follows:
Consumer 1: P1 = 60 - Q
Consumer 2: P2 = 100 - Q
Consumer 3: P3 = 140 - Q
where Q measures the number of units of the good and P is the price in dollars. The
marginal cost of the public good is $180. What is the economically efficient level of
production of the good? Illustrate your answer on a clearly labeled graph.
350
300
Economically efficient
level of production at
Q=40
MSB
Price
250
200
MC
150
100
50
D1
0
0
50
D2
100
D3
150
200
Quantity
The economically efficient level of output occurs where MSB
three consumers are in the market we have
MC . Since this occurs where all
(60 Q) (100 Q) (140 Q) 180
3Q 120
Q 40
17.23. Suppose that the good described in Problem 17.22 is not provided at all because of
the free rider problem. What is the size of the deadweight loss arising from this market
failure?
If the good is not provided at all, the deadweight loss would be the area under the demand curve
(the MSB curve) and above the marginal cost curve, or 0.5(300 – 180)(40) = 2400. This is a
deadweight loss because it measures the potential net economic benefits that would disappear if
the good were not offered.
17.24. In Problem 17.22, how would your answer change if the marginal cost of the public
good is $60? What if the marginal cost is $350?
350
300
MSB
Price
250
Economically efficient
level of production at
Q=90
200
150
100
MC
50
D1
0
0
D2
50
100
D3
150
200
Quantity
When marginal cost is MC 60 , the MSB and MC curves intersect at a level of output at
which only consumers 2 and 3 have a positive willingness to pay. This implies the efficient level
of production will occur where
(100 Q) (140 Q) 60
2Q 180
Q 90
If the marginal cost was $350, the marginal cost would exceed the marginal social benefits at all
levels of output. Therefore, at a marginal cost of $350, the economically efficient level of output
would be zero.
17.25. A small town in Florida is considering hiring an orchestra to play in the park during
the year. The music from the orchestra is nonrival and nonexclusive. A careful study of the
town’s music tastes reveals two types of individuals: music lovers and intense music lovers.
If forced to pay for an outdoor concert, the demand curve for music lovers would be Q1 =
100 - (1/20)P1, where Q1 is the number of concerts that would be attended and P1 is the
price per (hypothetical) ticket (in dollars) to the concert. The demand curve for intense
music lovers would be Q2 = 200 - (1/10)P2. Assuming the marginal cost of a concert is
$2800, what is the efficient number of concerts to offer each year?
To find the equation of the MSB curve, we need to add the inverse demand curves. Thus, as first
step, we determine the inverse demand curves of each type of individual:
Music lovers: Q1 = 100 – (1/20)P1
P1 = 2000 – 20Q1.
Intense music lovers: Q2 = 200 – (1/10)P2
P2 = 2000 – 10Q2.
Letting Q denote the (common) quantity of orchestra performances, the MSB is determined as
follows:
MSB = 100(2000 – 20Q) + 50(2000 – 10Q) = 300,000 – 2500Q.
Since the marginal cost of a concert is $250,000, the socially efficient number of performances
is:
300,000 – 2500Q = 250,000
Q = 20.
17.26. Some observers have argued that the Internet is overused in times of network
congestion.
a) Do you think the Internet serves as common property? Are people ever denied access to
the Internet?
b) Draw a graph illustrating why the amount of traffic is higher than the efficient level
during a period of peak demand when there is congestion. Let your graph reflect the
following characteristics of the Internet:
i. At low traffic levels, there is no congestion, with marginal private cost equal to marginal
external cost.
ii. However, at higher usage levels, marginal external costs are positive, and the marginal
external cost increases as traffic grows.
c) On your graph explain how a tax might be used to improve economic efficiency in the
use of the Internet during a period of congestion.
d) As an alternative to a tax, one could simply deny access to additional users once the
economically efficient volume of traffic is on the Internet. Why might an optimal tax be
more efficient than denying access?
a)
The Internet can be viewed as common property because virtually anyone has access to
it. In practice, people are sometimes denied access, particularly when the congestion is great and
consumers cannot connect to it.
b)
The graph might be very similar to Figure 17.5.
Price
MSC = MPC + MEC
D2
MPC
D1
Pb
MEC
Pa
Q2
Volume of
interconnections
to the internet
When the demand for connections to the internet is D1, there is no congestion. However, when
the demand is high at D2, congestion creates a positive marginal external cost.
c)
When the demand is large, a tax equal to (Pb – Pa) would lead users to demand the
efficient number of connections Q2.
d)
A tax would ensure that users who value connections the most would be able to connect.
If access is denied to some users, some users with a higher value for an interconnection might be
unable to connect, while other users with a lower value for a connection might be able to go
online. This would not be economically efficient because the scarce resource (connections)
would not necessarily be allocated to consumers who value connection the most.
17.27. There are two types of citizens in Pulmonia. The first type has an inelastic demand
for public broadcasting at Q = 8 hours per day; however, they are willing to pay only up to
$30 per hour for each hour up to Q = 8. The second type demands public broadcasting
according to P = 60 - 3Q.
a) Suppose the marginal cost of public broadcasting is MC = 15. What is the economically
efficient level of public broadcasting? Hint: it will help if you draw a careful sketch of the
demand curve of each type of citizen.
b) Repeat part (a) for MC = 45.
a)
Marginal social benefits are given by P = 90 – 3Q for Q < 8 and P = 60 – 3Q for Q > 8.
In the graph below, marginal social benefits are the curve ABCD. At MC = 15, the efficient
amount of production is Q = 15.
P
$90
$60
A
B
MC = 45
$30
C
8 10
b)
15
MC = 15
D
20
Q
At MC = 45, the efficient amount of production is Q = 8.