HOMEWORK 4: Strategic Interaction, Water, Climate
Change
st
Due: Monday, December 1 , 2014
50 Points
LEARNING OBJECTIVES
 Environmental and natural resource problems often arise due to strategic interactions
between economic agents.
 Environmental cooperation can be achieved in an infinite-time horizon game.
 Water resources are threatened by contamination. These contaminations can be
examined using economic criteria.
 A source of controversy is the predicted climate change response to the increasing
production of greenhouse gases (GHG).
Carefully read the question before answering. Make sure your pages are in order and stapled
together. Your work should be clear and easy to follow. When prompted, make sure that you
explain your answer completely.
QUESTIONS
1. The problem below analyses a fish stock shared between two countries. The countries
can either cooperate to sustain the fishery (participate in lower or sustainable harvests)
or the countries can choose to over harvest the fish (participate in higher/indivdual or
unstainable harvests). (http://www.worldwildlife.org/stories/illegal-fishing-puts-crabpopulations-at-risk)
a. (5 points) What is the Nash equilibrium for the one-shot game below?
Country 2
Country 1
Over Harvest (Don’t
Cooperate)
Sustainable Harvest
(Cooperate)
Over Harvest
(Don’t Cooperate)
(625,625)
(791,527)
Sustainable Harvest
(Cooperate)
(527,791)
(703, 703)
Payment to country 1
Payment to country 2
Nash equilibrium {don’t cooperate, don’t cooperate} because both countries have a
dominant strategy to over harvest.
b. (5 points) Why are the predictions of the prisoner’s dilemma so important to natural
resource management?
The prisoner’s dilemma game helps illustrate the problem with public goods and
common property resources. Answers will vary here.
c. (5 points) Assume this game is repeated infinitely. What is the required discount rate that
would ensure cooperation with the two countries if they use a tit-for-tat punishment
strategy?
To ensure cooperation, the discounted benefit of cooperating next period must be larger
than the benefit of defecting this period.
𝐻−𝐶 <
791 − 703 <
𝐶−𝐿
1+𝑟
703 − 527
1+𝑟
Solve for r
𝑟 < 1 to ensure cooperation, which is likely.
d. (5 points) Now assume the payoffs are as follows:
Country 2
Country 1
Over Harvest (Don’t
Cooperate)
Sustainable Harvest
(Cooperate)
Over Harvest
(Don’t Cooperate)
(625,625)
(791,527)
Sustainable Harvest
(Cooperate)
(527,791)
(750, 750)
Find the new discount rate that would ensure cooperation. Is cooperation more or less
likely now? Please explain the intuition behind why the discount rate changes and what
that means for the fishery?
To ensure cooperation, the discounted benefit of cooperating next period must be larger
than the benefit of defecting this period.
𝐻−𝐶 <
791 − 750 <
𝐶−𝐿
1+𝑟
750 − 527
1+𝑟
𝑟 < 4.44
A higher benefit of cooperating (750 versus 703) makes it more likely that cooperation
will occur. This means that the fishery is more likely to have sustainable harvest and not
collapse. Answers will vary.
2. The EPA has set new consumption limits of hydrochlorofluorocarbon, HCFC:
http://voices.nationalgeographic.com/2014/10/23/ozone-focus-of-epas-latestrulemaking/, with the goal of eventually phasing them out completely by 2020. HCFC is a
refrigerant that has been shown to contribute to climate change and could be considered
a transboundary pollutant. Suppose the marginal benefits and costs per unit of
refrigerant are modeled as follows to illustrate the negative externality of refrigerant
consumption:
𝑀𝑆𝐵 = 12.80 − 0.42𝑄; 𝑀𝑃𝐵 = 12.80 − 0.4𝑄
𝑀𝑆𝐶 = 𝑀𝑃𝐶 = 1.25 + 0.02𝑄
a. State the equation that represents the market externality. Give the economic
interpretation of this equation, using its specific numerical value(s).
HCFC consumption generates an externality, which in this case is MEB = –0.02Q.
This can be interpreted as follows. For every additional HCFC consumed, there is a
negative external benefit or an external cost of 2 cents per HCFC, or $20,000 on 1
HCFC. This represents damages associated with air pollution such as global warming.
b. Find the efficient equilibrium, 𝑃𝐸 and 𝑄𝐸 , for this market.
MSB = MSC
12.80 – 0.42Q = 1.25 + 0.02Q
11.55 = 0.44Q
QE = 26.25 MM
PE = $1.775
c. Find the dollar value of a per-unit refrigerant tax that would achieve the efficient
solution, and calculate the tax revenues generated to the government as a result.
To achieve efficiency, the tax should equal the MEB at QE, or when Q = 26.25
 Tax = – 0.02(26.25) = $0.525 per gallon
Tax revenues = (Per Unit Tax) * QE = 0.525 * 26.25 million =
$13,781,250.
d. (5 points) Explain how side payments can be used to achieve cooperation among
countries with transboundary pollution (covered in the strategic behavior section).
Side payments can be used among jurisdictions to achieve a cooperative equilibrium in a prison
dilemma game. For example, suppose Country A produces a good that has a negative impact
on County B. Country B can then provide a payment to Country A to act “coopertatively. The
side payment will have to be large enough so that Country A has a profit equal to their “don’t
cooperate” outcome. This payment also has small enough so that country B doesn’t have a
lower outcome than the “don’t cooperate” outcome.
3. Assume for simplicity that there are two identified point sources discharging agriculture
wastes into a local water body in Florida:
http://www.washingtonpost.com/national/health-science/in-florida-a-water-pollutionwarning-that-glows-at-night/2014/10/26/402cb636-5bba-11e4-8264deed989ae9a2_story.html. Currently, each source releases 30 units of pollution, for a
total 60 units. To improve water quality, suppose that the government sets an aggregate
abatement standard of 30 units. The two polluters’ abatement cost functions are
Point source 1: 𝑇𝐴𝐶1 = 10 + 𝐴12 ; 𝑀𝐴𝐶1 = 2𝐴1
Point source 2: 𝑇𝐴𝐶2 = 20 + 2𝐴22 ; 𝑀𝐴𝐶2 = 4𝐴2
a. (5 points) Suppose the government allocates the abatement responsibility equally
across the two point sources so that each must abate 15 units of pollution. Explain
why this abatement allocation does not yield a cost-effective solution. Support your
answer numerically.
Algebraically, if A1 = A2 =15, then MAC1 = 2A1 = 30 and MAC2 = 4A2 = 60. Hence, the
uniform standard approach is not cost-effective, since the resulting MACs are not equal.
b. (5 points) What cost condition is required for the government’s abatement allocation
to be cost effective?
For the abatement allocation to be cost-effective, the two polluters’ MACs must be equal.
Graphically, the cost savings can be illustrated by comparing the combined total
abatement costs (TAC) under the uniform approach with the combined TAC where the
MACs are equal. When A1 = A2 =15, the sum of the total abatement costs (TAC) for
each firm is represented as the sum of the areas under each MAC curve up to that point.
This area is larger than the area under the two MACs at the point where the two MACs
intersect. This excess proves that cost savings can be achieved by seeking the costeffective solution.
[Students might wish to solve for the cost-effective solution by setting the two MACs
equal, subject to the constraint that A1 + A2 = 30. They should find that the solution
occurs where A1 = 20 and A2 =10.]
c. (5 points) Suppose that instead of using an abatement standard, the government
institutes a pollution (or effluent) fee of $40 per unit of pollution. How many units of
pollution would each point source abate? Is the $40 fee a cost-effective strategy for
meeting the 30-unit abatement standard? Explain.
If the effluent fee is $40 per unit of abatement, Point Source 1 will abate 20 units of
effluent and Point Source 2 will abate 10 units of effluent. This is found by setting each
MAC function equal to the MEF of $40, and solving for A, as follows:
MAC1 = 2A1 = $40, or A1 = 20
MAC2 = 4A2 = $40, or A2 = 10
At this solution, the standard of 30 units of abatement is satisfied, since A1 + A2 = 30.
Also, we know that the cost-effective solution is achieved because MAC1 = 40 when A1
= 20, and MAC2 = 40 when A2 = 10.
d. (5 points) What are the main types and sources of water pollution? Why might the
source of the problem matter for policy design?
 From industry, in the form of point-source discharges of heavy metals, organic
wastes and other pollutants
 From sewage treatment works or direct sewage outfalls
 As leachate from landfill sites
 As pathogens washing off from fields in which cattle are kept
 As run-off of oils and solvents from city streets
 As run-off of fertilizers and pesticides, and soil erosion from farmland and forests
 As accidental (unplanned) spillages, e.g. oil tankers.
Reasons why the source of pollutions matter for policy will vary. However, non-point
pollutants are harder to correct using policy. Also, pollutants non-uniformity of emissions
also makes it difficult to determine optimal abatement levels.