Lecture notes on common-pool externalities and
policy instruments∗
Matti Liski
May 3, 2017
Abstract
These notes present simple models to analyze the basic common-pool externalities relevant for resources that publicly owned. We identify the inefficiency and
show that it is further increased by dynamics that allow the participants choose
technologies before the common-pool situation arises. We also formalize the solutions to the externality problems. First, explicit bargaining solutions are introduced. We also discuss transaction costs and introduce the reasons that prevent
the market from achieving efficiency through bargaining. Second, we consider how
outside authorities can achieve efficiency through economic instruments such as
prices and tradable quotas. Third, we consider how uncertainty impacts the policy
choice problem.
Key words: Natural resources, pollution, common-pool problem, public goods,
commitment, market power, technology, game theory
∗
Section 2 builds on ”Games and Resources”, Bård Harstad & Matti Liski (2013), In: Shogren, J.F.,
(ed.) ”Encyclopedia of Energy, Natural Resource, and Environmental Economics”, Vol. 2, pp. 299-308
Amsterdam: Elsevier.
1
1
Introduction
The purpose of these notes is to be explicit about the nature of strategic interaction
among agents engaged in exploiting a common resource. Game theory is a natural tool
for this purpose. However, reading these notes does not require prior knowledge of
game theory, as we will go through a sequence of relatively simple strategic situations,
keeping the focus on the substance matter and on explicit solutions. Throughout the
text, references are made to resource use, activity level, output, and production. In all
cases, these terms refer to the private activity contributing to the exploitation of the
common resource. This can be, for example, private output increasing the pollution
stock that reduces the common resource such as clean air.
2
Public Resources
The publicly owned resource is measured by s ≥ 0 and the set of users is I = {1, 2, ..., n}.
Each user, i ∈ I, decides on the private action zi impacting the total resource stock. In
particular, i’s private utility is function:
U (xi , zi , s) = u (xi , zi ) − d (s) , and
X
s=
zi
(1)
(2)
i∈I
Variable xi represents other decisions made by user i, assumed to not generate any direct
externalities. It can be interpreted as a private valuation parameter or a technological
choice, for example, investments in technologies that increase the private value of using
the resource. We assume u (·) is concave in xi and that d(s) is convex.
Example 1: We can call s as the total pollution stock, and zi as emissions from agent
i (e.g., country, firm, or individual). Then, it is natural to define d(s) as the pollution
damage suffered by each agent. Assume that the damage is quadratic,
d(s) =
d X 2
zi
2 i∈I
(3)
where d > 0 is a constant. Then, the marginal damage per individual from increasing
the stock, d0 (s) = ds, is linear in the stock. To complete the example, we may assume
that the private marginal benefit is linear in zi ,
u (xi , zi ) = u(xi )zi .
(4)
We can think that u(xi ) is, for example, the monetary value of selling outputs produced.
The choice of technology xi can impact this value.
Example 2: In some situations it is reasonable to think that the marginal environmental damage is linear. Then, we would assume
X d(s) = D
zi
(5)
i∈I
2
so that d0 (s) = D > 0 is a constant for all stock levels. For example, in climate change,
this a reasonable approximation at least for a range of stock levels for greenhouse gases.
Example 3: For example, s may represent the stock of clean air rather than stock
of pollution
if zi is i’s pollution level, reducing this stock. Then, we should think s =
P
s0 − i∈I zi , so that agents extract from the resource.
The environmental
P
P harm from
pollution is then given by the convex function c ( I zi ) ≡ v (s0 ) − v (s0 − I zi ), where
v(s) is a concave function of the benefits from having a clean environment, and v (s0 ) is
simply a constant.
2.1
The Static Common Pool Problem
We refer to the first-best, denoted below by superscript F B, as the outcome preferred by
a social planner maximizing the sum of the users’ payoffs.1
Definition 1. The (interior) first-best is an allocation of {xi , zi }i∈I maximizing the
sum of utilities:
X
max
U (xi , zi , s)
{xi },{zi }
⇔
∂u (xi , zi )
=
∂zi
| {z }
private gain
i∈I
∂d (s)
n
∂s }
| {z
and
∂u (xi , zi )
= 0.
∂xi
private &public damage
In the first best, the private gain from increasing the activity level of i is balanced against
the full social loss, that is, the marginal damage that i suffers and the damage that all
others suffer. This means n times the marginal damage is full the social marginal cost.
However, in the game referred to as the problem of the commons, the variables are
not chosen by a social planner, but by the individual users. When all the users act
simultaneously, the natural equilibrium concept is Nash equlibrium. We use superscript
N E to denote the Nash Equilibrium when convenient.
Definition 2. A Nash equilibrium is a set of choices, {xi , zi }i∈I , such that, when
user i takes as given the other users’ choices, {xj , zj }j∈I\i , then {xi , zi } maximizes
private payoff U (xi , zi , s).
The static version of the game is the one where all users act simultaneously and only
once. Later, we will look at situations where agents can first choose technologies and
after this decide on the use of the common pool.
Proposition 1 Consider the static common-pool problem: (i) Each user’s choice zi is
larger than the first-best:
∂u (xi , zi )
∂d (s)
∂d (s)
=
<n
.
∂zi
∂s
∂s
1
The second-order conditions for the first best problem are satisfied when |u12 | ≤
since ∂ 2 u (xi , zi ) /∂qi2 − n2 d00 < 0 and ∂ 2 u (xi , zi ) /∂x2i < 0.
3
p
u11 (u22 − n2 d00 )
(ii) For any given zi , the equilibrium xi is socially optimal:
∂u (xi , zi )
= 0.
∂xi
Since user i does not internalize the negative externality on the other users, i extracts from the resource too much. In other words, i extracts more from the common
resource than what a social planner would prefer and as specified by the first-best. Note
that the difference between the equilibrium extraction and the first-best increases in the
number of users, n. Since choice-variable xi was assumed to create no externality, it is
optimally chosen. This fact is natural in the static setting but it might no longer hold in
dynamic settings, as we soon will learn. The following quotation describes the negative
externalities in the Nash Equilibrium:
The Tragedy of the Commons. Garrett Hardin. Science, New Series, Vol.
162, No. 3859 (Dec. 13, 1968), 1243-1248:
“The tragedy of the commons develops in this way. Picture a pasture open
to all. It is to be expected that each herdsman will try to keep as many cattle
as possible on the commons. Such an arrangement may work reasonably
satisfactorily for centuries because tribal wars, poaching, and disease keep
the numbers of both man and beast well below the carrying capacity of the
land. Finally, however, comes the day of reckoning, that is, the day when
the long-desired goal of social stability becomes a reality. At this point, the
inherent logic of the commons remorselessly generates tragedy.
As a rational being, each herdsman seeks to maximize his gain. Explicitly
or implicitly, more or less consciously, he asks, ”What is the utility to me of
adding one more animal to my herd?” This utility has one negative and one
positive component.
1) The positive component is a function of the increment of one animal. Since
the herdsman receives all the proceeds from the sale of the additional animal,
the positive utility is nearly +1.
2) The negative component is a function of the additional overgrazing created
by one more animal. Since, however, the effects of overgrazing are shared
by all the herdsmen, the negative utility for any particular decision-making
herdsman is only a fraction of 1.
Adding together the component partial utilities, the rational herdsman concludes that the only sensible course for him to pursue is to add another animal
to his herd. And another; and another... But this is the conclusion reached
by each and every rational herdsman sharing a commons. Therein is the
tragedy. Each man is locked into a system that compels him to increase his
herd without limit–in a world that is limited. Ruin is the destination toward
which all men rush, each pursuing his own best interest in a society that
believes in the freedom of the commons. Freedom in a commons brings ruin
to all.”
4
2.1.1
Working through Example 1
We work out now Example 1 above. Let us assume u(x) = u > 0 so that the marginal
gain from using the resource is a constant. The choice of xi can be thought of as made
in the past, so we consider it fixed here in this Section.
Nash equilibrium: We want to explicitly obtain the first best and Nash equilibrium
outcomes to be able to quantify the welfare loss from the externality. Consider the Nash
outcome first. Taking the emissions of the other agents as given, {zj }j∈I\i , the marginal
gain from increasing zi for agent i is
P
∂U (zi , zi + j∈I\i zj )
∂U (zi , s)
=
∂zi
∂zi
P
∂ zi + j∈I\i zj X =u−d
zi
∂zi
i∈I
X =u−d
zi
i∈I
Note that all players are symmetric, and therefore it is natural to focus on symmetric
equilibria where all players choose zi = z N E > 0 (we will look at asymmetries below).
By this, we must have for all i,
∂U (zi , s)
=0
∂zi
⇒ u = d(nz N E ) ⇒ z N E =
u
.
dn
From this, obtain the equilibrium payoff as
U (z N E , nz N E ) = uz N E −
= 1−
2
d
nz N E
2
n u2
.
2 dn
We may immediately note that U (z N E , nz N E ) is negative for all n > 2. Yet, this does
not mean that player i could do better by choosing zi = 0: this would only leave the
negative externality coming from other player’s actions but no private gain.
The first best: The first best can be obtained by choosing
5
max
X
{z1 ,..,zn }
U (zi , s) = max
X
{z1 ,..,zn }
i∈I
uzi −
i∈I
d X 2 zk
2 k∈I
⇒
P
∂ zi + j∈I\i zj X ∂U (zi , s)
=u−d
zk (1 + ... + 1)
∀i,
{z
}
|
∂zi
∂zi
k∈I
=n
X =u−d
zk n
k∈I
| {z }
=nz F B
u
= u − d nz F B n ⇒ z F B = 2
dn
Comparison & discussion: We now compare explicitly how the total payoff depends on the number of agents. First, in the Nash equilibrium of the common pool game,
we observe that
X
U
ziN E , s
= uz
i∈I
NE
d
NE 2
n
− nz
2
n u2
< 0 ⇔ n > 2.
= 1−
2 d
Thus, the total payoff goes negative quickly with the number of agents. Second, in the
first best we have
X
2 d
U ziF B , s = uz F B − nz F B
n
2
i∈I
1 u2
> 0, ∀n.
= 1−
2 dn
Careful planning thus avoids externalities and keeps the payoff positive in total. Note that
it declines with n however: it is not possible to scale up the activity with the number
of users since the total resource becomes scarce. It is important to notice that the
increase in scarcity is different from the increase in negative externalities. For example,
the illustration by Hardin above describes how a given number herdsmen overuse the
resource by increase the number of cows. However, increasing the number of herdsmen,
even if they can optimally manage the use of the commons, leads to similar problems:
the resource to be used remains limited but the number of users increases. There is less
and less of the resource per herdsman.
2.2
A Dynamic Common Pool Problem
We show next that each indvidual user would benefit if it, somehow, could commit to use
the resource more, since the other users would then find it optimal to use less. In reality,
6
user i has several opportunities to make such a commitment: any government policy set
in advance if the agents are countries, or any investment in technology, may influence
user i’s future preference, and these choices can be captured by the parameter xi . The
first-stage action xi can be anything that does not, by itself, generate externalities on
the other users. We will now refer to xi as a technology.
To make this point, we assume that agents make decision in two sequential stages:
Stage 1 Technologies xi are chosen by all i, and observed by everyone.
Stage 2 The resource use zi is chosen by all i.
With this dynamic or two-stage game, it is natural to limit attention to subgameperfect equilibria. If this definition is too quick at this point of your studies, Section 2.2.1
illustrates the concept using a simple example that captures most of substance matter.
Loosely speaking, agents can anticipate the actions that will follow in the second stage
when making their first-stage decisions. That is, we will require that the strategies at
stage two continue to constitute a Nash equilibrium in the game that is played at that
stage, for all possible first-stage actions. Since the xi s are observed before the zi s are
set, the latter choices are likely going to be functions of the set of xi -choices. That is,
zi = zi (x), where x = (x1 , ..., xn ).
Definition 3: The strategies (xi , zi (x))i∈I constitute a subgame-perfect equilibrium
if they constitute a Nash equilibrium and if also (zi (x))i∈I constitute a Nash equilibrium
at the second stage for every feasible x = (x1 , ..., xn ).
In our two-stage model, the total payoff for i from choosing xi , given x−i , is
X
zj (xi , x−i )).
u(xi , zi (xi , x−i )) − d(
j∈I
Player i can thus anticipate how its first-stage action xi influences the common-resource
use not only by i but also by every other j. We use superscript dyn to distinguish the
dynamic game from the static. There is a unique subgame-perfect equilibrium. We will
say that the technology is a strategic complement to extraction if u12 > 0 and a strategic
substitute if u12 < 0.
Proposition 2 Consider the sequential game where the xi s are chosen before the zi s. In
the subgame-perfect equilibrium,
(i) If xi is a complement to zi (i.e., if u12 > 0), then agents choose higher xi the dynamic
than in the static equilibrium. If xi is a substitute to zi (i.e., if u12 < 0), then the equilibrium levels of xi are lower .
(ii) Sequential moves worsen the common-pool problem: there is greater use of the common pool, and all payoffs are lower.
The proof of this result can be found from Harstad and Liski (2013).2 We will work
out a specific example below to illustrate the result.
2
The only difference to the current setting is that Harstad and Liski assume, instead of damage d(s),
function v(s) for the benefits of having a higher stock level, with the interpretation that the stock refers
to an unpolluted stock of the resource. See Example 3 in the beginning.
7
Complementary technologies are, for example, drilling technology, investments in polluting industries, fishing boats, or the breeding of cows for the common grassland. All
such investments are larger than what is first-best and, furthermore, they are larger than
what the users would have chosen were they not observed and reacted to by the other
users of the resource. Since every user invests too much in complementary technology,
the result is that extraction is larger, and the resource stock is smaller, than what would
be the outcome in a static setting where xi were not chosen strategically before zi .
Substitute technologies are, for example, renewable energy-sources, investments in
abatement technologies, a district’s re-education of fishermen or a gradual exit from the
extracting industry. Every such investment level is, in equilibrium, sub-optimally low
and, furthermore, it is smaller than the investment levels preferred by the users at the
extraction stage (or in a static setting where xi and zi are set simultaneously). Since
good substitutes are not invested in to a sufficiently degree, all users will extract too
much and more than they would in the similar static game.
As a simple illustration and example, suppose that each user can invest in windmills
and consume the generated renewable energy in addition to the energy from fossil fuel.
If xi measure the level of renewable energy investment, i enjoys the utility u (xi , zi ) =
b (xi + zi )−k (xi ), where b (·) is the benefit from energy consumption while k (·) is the cost
of investing in renewable energy sources. Naturally, u12 = b00 − k 00 < 0, so the technology
is, in this case, a substitute to extraction. Thus, each user invests strategically little in
windmills.
In sum, it does not matter whether the technology is a substitute or a complement
to later extraction. As long as i can set xi first, it sets xi such as to commit to more
extraction later on. The marginal direct impact on own utility is negligible, but the
consequence that other users will extract less is beneficial and of first-order magnitude
for i. When everyone act in this way, total extraction increases and utilities decline.
2.2.1
Working through examples
We work out now a modified version of Example 1 above for the dynamic case. Let
us assume that there are two agents i = 1, 2, for simplicity. In the first stage, they
can choose xi ∈ {0, 1}, where xi = 0 means “no change in technology” and xi = 1
means “new technology chosen”. The technology increases the private value of using the
common resource:
u(xi ) =
n
u
if xi = 0
u > u if xi = 1
In the second stage, the agents observe the investments made and thus if the other
player has u(0) = u or u(1) = u. Let us assume that the investment in the new technology
costs some fixed amount K > 0.
The first best: What is the first best plan in this case? That is, should there be an
investment by one or two agents in the new technology? The first observation is that, if
there is any investment, there should be only one investment:
8
u(z1 + z2 ) − d z1 + z2
| {z }
| {z }
=ẑ
2
− 2K < uẑ − dẑ 2 − K
=ẑ
where ẑ = z1 + z2 is any given resource use by two agents merged into one agent resource
use (note that we allow the planner to freely choose the production shares between the
agents; for example, agent 1 can be nominated to be only active agent). Thus, one
agent can produce the same surplus as two agents but there is only one investment; there
is not particular gain from splitting the same production into smaller pieces with the
assumed functional forms. Intuitively, in this example, it makes sense to make one agent
“efficient” by investment. This way the total benefit is maximized, and the agent who
is not active could be compensated otherwise, for example, through a side payment (we
will come back to this when discussing bargaining solutions to the problem). Suppose
now that this one investment is made. We select one agent for the investment, and ask
now much should the agent with x = 1 produce? This leads to
n
d o
u
d
FB
=
max uz − z 2 − z 2 ⇒ zx=1
z
2
2
2d
Here, there are two times the individual damage from z, but uz is only the private gain
from z. This is because, if the objective is to maximize the total payoff, we care only
about the sum of surpluses, not how it is distributed. Similarly, if there is no investment,
then x = 0 and
n
o
u
FB
2
FB
< zx=1
max uz − dz ⇒ zx=0
=
2d
FB
FB
. We can
> zx=0
Thus, investment is costly but allows one to produce more, zx=1
now ask how much more surplus the investment generates, and does this justify the
investment? It improves the total payoff, and this should be larger than the cost of
investment:
FB 2
FB
FB 2
FB
− d(zx=0
) >K
uzx=1
− d(zx=1
) − uzx=0
|
{z
}
increase in payoff due to investment
⇒
1 2
(u − u2 ) > K.
4d
This inequality must hold for the investment to make sense in the first best.
The subgame perfect equilibrium: We look now how the firms will end up investing in the equilibrium. We work backwards from the second stage to see how they use
the common resource, given the investments from the first stage. That is, take (x1 , x2 )
as given and consider first the situation where (x1 , x2 ) = (1, 0).
9
∂U (x1 = 1, z1 , s)
= 0 ⇒ u − d(z1 + z2 ) = 0
∂z1
∂U (x = 0, z2 , s)
= u − d(z1 + z2 ) < 0
⇒
∂z2
⇒ z2 = 0
The less efficient agent (i.e., with inferior technology) cannot produce since any positive
z2 would only generate negative impact on the payoff! We obtain that
u
FB
> zx=1
d
z2N E = 0.
z1N E =
It follows that the agent with the inferior technology suffers only from the externality
but does not produce and thus receives no private gain: the payoff for i = 2 is negative
and takes the value
U (x = 0, z2 = 0, z1N E ) = −
1 u2
2 d
which is exactly the external cost that 1 imposes on 2.
Suppose now both have invested so that (x1 , x2 ) = (1, 1). How much do they produce
now? We have already calculated this equilibrium outcome above (replace u by u in
Section 2.1.1):
∂U (zi , s)
=0
∂zi
⇒ u = d(nz N E ) ⇒ z N E =
u
.
dn
From this, obtain the equilibrium payoff as
U (z N E , nz N E ) = uz N E −
= 1−
2
d
nz N E
2
n u2
=0
2 dn
where the last line follows from n = 2. Thus, both players can secure zero in the second
stage payoff by investing. We need a condition for the investment cost for this equilibrium
outcome to arise: if K is low enough so that
1 u2
>K
2 d
then, by investing, the agent limits the total loss to −K (investment cost plus zero payoff
from resource use). We can check that the dominant strategy for both i = 1, 2 is to invest:
10
independently of what the other player is doing, investment makes sense individually. To
see this, take first as given x2 = 0, and consider if it is optimal to invest for agent 1
and choose x1 = 1. Clearly, investment is optimal for agent 1. Then, take as given that
x2 = 1, and note that it is again optimal to choose x1 = 1. We observe that irrespective
of 2 does, it is optimal for 1 to invest. A symmetric argument holds for agent 2, given
the choices of agent 1. Thus, it is a Nash equilibrium for both to invest.
What do we learn from this example? We see that if the investment opportunity is
not there, or if the investment cost is very high, the common pool problem is less severe:
agents have less productive technologies but also exploit the resource less, and end up
not gaining or losing surplus (zero payoff in total). If they can invest, then they are
trapped to do so and end up losing money, −2K in total. The common pool inefficiency
eats all the gains from investments but yet agents must invest since otherwise they lose
even more.
3
Solutions to the common pool problem
We consider now how the inefficiency from the externality could be removed. The most
famous solution is the one proposed by Coase.
3.1
Coasian solution
Coase proposed that if the property rights are well defined, then the market will take care
of the externality problem. Since this idea builds on the thinking that the market will
set prices for externalities, let us first modify our definition of Nash equilibrium, given
some (at this point) arbitrary payment that follows if individual i generates externalities.
Since the social cost of those externalities depends on how much the others producing
them, the externality payment depends on both zi and z−i .
Definition 2. A Nash equilibrium with externality payment p(zi , z−i ) is a set of
choices, {xi , zi }i∈I , such that, when user i takes as given the other users’ choices,
{xj , zj }j∈I\i , then {xi , zi } maximizes private payoff U (xi , zi , s) − p(zi , z−i ).
Proposition
3 Consider the static common-pool problem with p(zi , z−i ) = (n − 1)d(zi +
P
j∈I\i zj ): (i) Each user’s choice zi coincides with the first-best:
∂d (s)
∂u (xi , zi )
=n
.
∂zi
∂s
(ii) For any given zi , the equilibrium xi is socially optimal:
∂u (xi , zi )
= 0.
∂xi
The result says that if user i must pay the externality cost imposed on other agents,
then it follows, by definition, that the individual choices will be socially optimal. The
Coasian solution achieves this either by assigning property rights to those who suffer or
to those who produce the externality. In the former case, the holder of the property right
11
has the right to live without the damage from the resource use, so the agent that wants
to use it must pay for the usage.
Victims hold the rights. Going back to our Example 1 with two agents i = 1, 2,
assume that only agent 1 has some private gain from production: u(x1 , z1 ) = uz1 with
u > 0 while u(x2 , z2 ) = 0. Yet, both agents suffer from output as before: d(s) = d2 (s)2 is
the individual damage per agent. In the Nash equilibrium, choices are
u
z1N E =
d
NE
z2 = 0
since agent 1 cares only about his own damage, and agent 2 has no gain from production at
all. Now, if agent 2 owns the property right to have no damage, agent 1 must compensate
agent 2 for giving up those rights. If the compensation equals the full damage imposed
by 1 on agent 2, then we arrive at:
d
p(z1 , z2 ) = p(z1 , 0) = (z1 )2 .
2
Facing this payment, agent chooses the activity level to solve
d
∂ uz1 − (z1 )2 − p(z1 ) = 0 ⇒ u −
dz1
−
|{z}
∂z1
2
own marginal damage
dz1
|{z}
=0
marginal payment to 2
u
⇒ z1N E = z1F B =
2d
We observe that agent 1 chooses the first-best outcome, and compensates agent 2
exactly for the losses. This is what the proposition predicts. However, remember that
agent 2 owns the property right, and it is not the case that the agent suffering has to
follow exactly this pricing rule for selling her rights. The efficient sale of the rights to
pollute can be obtained in different ways. For example, agent 2 could set a unit price
u/2 so that the total payment received from the polluter is
u
p(z1 ) = z1 .
2
When facing this linear payment in z1 , agent 1 solves
∂ d
u u
uz1 − (z1 )2 − z1 = 0 ⇒ u − dz1 − = 0
∂z1
2
2
2
u
⇒ z1N E = z1F B =
2d
This gives the first best incentives as well. Why is this? Note that here agent 2 is fixing
one price for all levels z1 , and this price equals the marginal damage on agent 2 at activity
level z1N E = z1F B . For this reason, agent 1 pays a high price for all units: the price is
higher than the actual marginal damage for all z1 < z1F B . As a result, trading leaves
some extra surplus for agent 2:
d
p(z1 ) − z12
|
{z 2 }
=
u
d
u2
u2
z1 − z12 =
−
>0
2
2
4d 8d
compensation - damage
12
What do we learn from this? The Coasian solution says that it is in principle possible to
trade with rights to achieve efficiency when the property rights are well defined – it does
not tell how exactly the parties decide to transact.
Externality-generating agents hold the rights. The key proposition made by
Coase was that it does not matter for efficiency how the property rights are allocated,
as long as they are well defined. To see this, consider the other extreme where the user
of the resource has all the rights. For example, the polluter has the right to pollute, the
herdsmen have the right to use the pasture, etc. How can the market set a price for the
externality in this case? Now, the victims have to compensate the externality-generating
agents for reducing the level of the externality causing activity: the externality payment
that the producer faces must be negative, p(zi , z−i ) < 0, for choices zi < ziN E . That is, the
payment must compensate the agent for not generating the externality level associated
with the Nash equilibrium where, as we have seen, the agents ignore how much costs
they inflict on others.
Once again go back to our Example 1 with two agents i = 1, 2, and assume that
only agent 1 has some private gain from production: u(x1 , z1 ) = uz1 with u > 0 while
u(x2 , z2 ) = 0. Yet, both agents suffer from output as before: v(s) = − d2 (s)2 is the
individual damage per agent. In the Nash equilibrium, activity levels are
u
d
= 0.
z1N E =
z2N E
Now, if agent 1 owns the property right to choose whatever activity level it pleases, agent
2 must compensate agent 1 for giving up those rights. If the compensation equals the
full willingness to pay for reductions in damages, then
p(z1 , z2 ) = p(z1 , 0) =
d 2 d NE 2
z1 − (z1 )
2
|2
{z
}
d
1 u2
.
= z12 −
2
2 d
difference in agent 2 damages
Notice that agent 1 is collecting money from reducing its activity level below the Nash
equilibrium level. Facing this payment, agent 1 chooses the activity level to solve
∂ d
uz1 − (z1 )2 − p(z1 ) = 0 ⇒ u − dz1 −
∂z1
2
⇒ z1N E = z1F B
dz1
|{z}
=0
marginal compensation from 2
u
=
2d
We observe that agent 1 chooses the first-best outcome! Again, we could consider
other compensation schemes: as long as they give the right incentives to reduce the
activity at the margin, the optimum can be achieved. The main difference is that parties
end up achieving different total surpluses from the activity, depending on the pricing
schedule used. Ultimately, the chosen effective price for the externalities depends on the
bargaining power between the agents, which is an issue that we consider in detail below.
13
3.2
Transaction costs
Above, we sketched the core idea of the Coasian solution to the externality problem.
In practice, it is challenging concept since it may be difficult for the parties involved to
transact in an optimal way. These difficulties go under the concept of transaction costs,
covering several potential sources of such costs.
The hold up problem can arise when the victims have the right to avoid the external
cost. As we have seen above, those who generate the external costs must then compensate
the victims. The hold-up problem arises when some victims hold up the rest of the group
from transacting, unless a large compensation is made exactly to the agents holding up
the process. To make a stark case, suppose that all agents who suffer have been paid
compensations according to their damages, excluding one agent, denoted by j. Thus,
all agents i ∈ I\j have agreed on the compensation scheme that leads to the firstbest, provided agent j also comes on board. This agent could now ask for a different
compensation from each of the polluter:
p(z , z )
| i{z −i}
= d(zi +
X
zk ) +
k∈I\i
compensation paid by i
|
{z
Ti
|{z}
constant
}
damage caused by i on j
The first part of the compensation is what we have seen before: j is compensated for
the damage it suffers from the activity of i. The second part is a pure transfer of money:
agent j could demand the full surplus that agent i achieves from the activity through Ti .
Why is it that j could collect such extra transfers from all agents i ∈ I\j generating the
externalities? This follows because j has huge bargaining power: by refusing to sell its
rights, it denies the activity, and all agents would get zero payoff (no activity, no payoff).
The hold-up follows from the strict nature of rights; one cannot produce at all if there is
at least one agent who declines the activity.
The hold-up problems becomes a sever issue, in particular, if there is a small number
of firms who need to procure the rights from the victims. The victims have an incentive
to wait instead of striking a deal, because the terms of bargaining improve when the
victim is later in the sequence of deals between the firm and the victims.
The free rider problem can arise when the externality generating firms have the
rights. Then, the victims must get their act together and compensate the firms for
reducing the activity level. Suppose there is a meeting where all parties excluding one
victim show up. In this meeting bargaining leads to the best possible outcome for the
n − 1 agent present in the room so that they end up choosing for all i in the room:
∂d (s)
∂u (xi , zi )
= (n − 1)
.
∂zi
∂s
Notice that this is the first-best outcome when the number of agents is n − 1. This
follows because, as we have shown, those suffering can compensate those who generate
the externality to reduce the activity level and thereby reach the first best. However, there
is one agent missing from the meeting, so the outcome is not fully optimal. The agent
14
who does not show up obtains a considerable reduction in the activity level, and thus in
the negative external cost. But it does not have to compensate anyone for reductions
in the activity, which is the key to the free rider problem. When n becomes very large,
the negotiations give almost the first best outcome but no payments follow for the agent
staying out. But since all agents have incentives to stay out, the negotiations may not
happen in the first place.
Let us go back to our Example 1 with n agents i = 1, 2, .., n, and assume that
only agent 1 has some private gain from production: u(x1 , z1 ) = uz1 with u > 0 while
u(xi , zi ) = 0 for all other i. Yet, all agents suffer from output as before: v(s) = − d2 (z1 )2
is the individual damage per agent. In this case, the negotiations lead to
z1 =
u
(n − 1)d
so that the free-riding agents saves the damages
d N E 2 d 2 u2
u2
u2 1
(z1 ) − z1 =
−
=
1
−
2
2
2d 2(n − 1)2 d
2d
(n − 1)2
without contributing any resources to compensate firm 1.
Bargaining frictions are yet another source of transaction costs. These are related
to difficulties that parties might have in coming up with a bargaining protocol that
manages to obtain efficiency. These problems may be less sever if there is an outside
authority such as government with coercive powers, helping the parties to materialize
the gains from trade, for example, through a mechanism that determines actions and
contribution by each party. But, in particular, between sovereign countries, there is no
such outside authority, and problems discussed just below may arise.
There are three players a, b, c who can produce emissions reductions. When a and
b form a group, we denote that coalition by ab, and so forth. If all agents remain
separate, the payoff remains at zero. We assume that ab produce 12 for group ab. The
agent outside the group, who is c in this case, receives 9. We assume that whenever an
agent remains outside and two others form a coalition, then the outside agent receives 9.
Outside agent obtains thus some free-riding benefit: this agent does nothing but yet has
more payoff than if the two others remain separate as well. To complete the payoffs, we
assume ac gives 13, bc gives 14, and abc is of value 24.
The bargaining protocol puts a microstructure on how any one of the possible coalitions could form, including the potential transfers between the agents. One protocol is
such that agents enter a room in a particular oder, for example, a first, followed by b,
and then c arrives. Thus, in this situation a waits for b, and makes an offer to b to form
team ab. What are the options for b? First, b can stay outside and compete with a for
c when agent c arrives the room. Second, b can join a, and then they can make an offer
together to c. Which one dominates? Well, in the first case, b can offer at most the value
of bc minus the value of b alone if ac forms: 14-9=5. Similarly, a can offer c as follows:
13-9=4. We observe that b will win the competition for agent c. But b will not pay 5
but 4 to c (you can think that b can offer to pay one cent more than a to outbid a). You
see that c has no option other than to accept since c has does not have the free riding
option, as there will be no coalition if bc does not form. Thus, along this path, agent b
15
gets 14-4=10. Now, consider the decision by agent a when seeing b entering the room.
Agent a has to offer b at least 10 to join agent a. Then, ab together would have to offer
c at least 9 to prevent her from free riding. Team abc has value 24 so this leaves payoff
24-10-9=5 for agent a. Does this make sense to a? No! Agent can refuse to talk to b
when first meeting b, so this forces b to form a coalition with c, allowing a then to free
ride on this coalition. We have found the final outcome: a stays alone, bc forms, and the
payoffs are (9,10,4) to a,b,c, respectively.
What do we learn? The vagaries of bargaining can create frictions that prevent
the full co-operative outcome to emerge. It is essential for this outcome that there is
an externality: a can commit to free ride, given the order of moves in the bargaining.
This observation is quite general in the sense that commitment not to contribute to the
common good is often exploited, for example, by locking into dirty technologies prior to
climate change negotiations.
3.3
Policy Instruments
We consider now policies for solving the externality problems. Private parties may try to
solve them through negotiations; however, if there is a policy-maker with coercive powers,
then the problem boils down to the choice of policy instruments that the policy-maker
can use to influence private behavior. There will be three main topics:
1. Price and quantity instruments. Price instrument is the “Pigouvian tax”, and the
quantity instrument refers to the “system of tradable quotas”.
2. Uncertainty. Prices and quantities as instruments are in principle equivalent but
important differences arise when there is uncertainty regarding the private costs of
the regulations
3. Asymmetric information. When private agents have private information on the
cost of regulations such as how they value their own use of the common resources,
it becomes difficult for the planner to design regulations optimally. The optimal
design should try to make the private parties to reveal truthfully their information.
The last item is listed to remind us that uncertainty is conceptually different from asymmetric information, but we leave asymmetric information for another course.
3.3.1
Taxes: Pigouvian solution
Recall that the first-best is an allocation of {xi , zi }i∈I maximizing the sum of utilities:
X
max
U (xi , zi , s)
{xi },{zi }
⇔
(i)
(ii)
i∈I
∂U (xi , zi , s) ∂
+
∂zi
∂U (xi , zi , s)
=0
∂xi
16
P
I\i
U (xi , zi , s)
∂zi
=0
In contrast, in the private optimum, described by the Nash Equilibrium, each agent
ignores the externality imposed of the private actions on the other agents. There are two
basic policy instruments for correcting this distortion. In one, agents face a price (”tax”)
for each unit of extraction. In the other, the total quantity of extraction (”quota”) is
regulated, and when this quota is tradable between the agents, there will be a market
price for extraction. Several issues arise:
• Can we trust that the policy instruments provide correct incentives for technologies
as well (choice of xi ).
• Does the allocation of quotas matter when quantities instrument is used?
• Distortions in quota markets such as market power or trading frictions.
Consider now the private choices of extraction zi and ”technology” xi when there is
a given fixed tax τ per unit of extraction:
max U (xi , zi , s) − τ × zi
{xi },{zi }
⇔
∂U (xi , zi , s)
−τ =0
(i) :
∂zi
∂U (xi , zi , s)
(ii) :
=0
∂xi
Under standard concavity assumptions, the private activity level decreases with the
tax:
∂ 2 U (xi , zi , s)
∂ 2 U (xi , zi , s)
dzi
=
1/
dz
−
dτ
=
0
⇒
< 0.
i
∂zi2
dτ
∂zi2
We want
the tax such that it implements the socially optimal, first best,
to choose
allocation, xFi B , ziF B i∈I . We want to find τ such that zi (τ ) = ziF B . This requires that
the optimal Pigouvian tax internalizes (at the margin) the externality that i imposes on
other users of the common resource. That is, we must first find the first best activity
level, and then set the tax as follows:
∂d sF B ∂s
∂d sF B
FB
τ
= (n − 1)
= (n − 1)
∂s
∂zi
∂s
It is important to emphasize that the tax obtained this way is a constant; the formula is
∂d(sF B )
merely helping to find its level. There are n − 1 other users, and they each suffer ∂s
when the stock increases by a marginal unit.
Let us then consider agent i’s own private gain from zi to see how the agent responses
to the tax. The private gain is altered when τ F B is introduced:
(i) ⇒
∂d sF B
∂u (xi , zi )
=n
∂zi
∂s
17
The Pigouvian tax imposes the full social cost of zi . We have seen that τ has no direct
effect on xi (only through zi ). Two important lessons arise here:
1. The tax should be imposed on the action that causes the externality
2. The tax should be imposed before agents choose the technologies
In reality, these pre-conditions are often violated. For example, in Finland there is
an emissions tax on automobiles but not on the use of them. In climate change, there
are no comprehensive international policies in place but countries are making significant
investments in technologies; these technology choices will impact the future negotiations,
for example, about the appropriate tax level on emissions. Let us look at these argument
in detail next.
The tax should be imposed on the action that causes the externality. To
illustrate, suppose u (xi , zi ) is interpreted as firm i0 s conditional profit depending on the
externality causing input use zi , and technology xi . Now, a firm facing tax τ on profits
considers:
max u(xi , zi )(1 − τ ) − d(s)
{xi },{zi }
⇔
∂u(xi , zi )
(1 − τ ) − d0 (s) = 0
∂zi
∂u(xi , zi )
(ii)0 :
(1 − τ ) = 0
∂xi
(i)0 :
To make a strong case, suppose that the firm is small so that it ignores its impact on
s, v 0 (s) ≈ 0. Then,
(i)0 &(ii)0 ⇒
∂u(xi , zi )
∂u(xi , zi )
=
=0
∂zi
∂xi
The same choice as without the tax! When v 0 (s) > 0, the tax has leverage in internalizing some of the impact on others but, by (ii), the technology choices will be distorted
in general. It is thus essential that the Pigouvian tax targets directly the action that
causes the externality, not some indirect measures of the activity such as profits.
The tax should be imposed before agents choose the technologies. Recall
that when setting the optimal Pigouvian tax, the policy maker understands the technologies {xi }i∈I that are available for the agents; these determine the private valuations
and are thus in part determining the level of the tax. However, if agents can choose their
technologies before the tax is in place, they can influence the level of the tax. Effectively,
by making the technology choice, the agents are committing to a certain valuation of the
resource that then influences how the tax will be set.
The analysis of this case is similar to the subgame-perfect equilibria from Section 2.2
where the first-stage xi is chosen, and observed by everyone, before the users choose the
second-stage actions zi . Here, the second stage is replaced by the policy making stage
where the optimal Pigouvian tax, given {xi }i∈I , leads to
18
∂u (xi , zi )
∂d (s)
=n
⇒ zi = zi (xi , x−i )
∂zi
∂s
Thus, agents understand how the Pigouvian policy will dictate choices in the second stage zi (xi , x−i ) when choosing xi . This problem is almost equivalent to the one
considered in Section 2.2.
Proposition 4 Consider the sequential game where the xi s are chosen before the Pigouvian tax on zi s. In the subgame-perfect equilibrium,
(i) If xi is a complement to zi (i.e., if u12 > 0), then agents over-invest in technologies,
that is, xi becomes inefficiently large for all i. If xi is a substitute to zi (i.e., if u12 < 0),
then agents underinvest in technologies.
(ii) All choices are distorted: common-pool exploitation exceeds the first best, and payoffs
are smaller that if the tax policy is set before technologies.
• Intutition for the substitutes: when one agent can commit to a higher private value
from using the common resource, it will be allocated a larger share of the resource
when the time of setting the policy comes. Understanding this, all agents overinvest to specialize in pollution! It is therefore very important not leave the agents
waiting for the policies to come since, while waiting, they can make strategic choices
worsening the problem.
• Intuitition for the complements: when investments reduce the private valuation of
using the common resource, agents commit to a larger share by under-investment.
3.3.2
Tradable rights
Above we considered a price instrument that a policy maker could use to regulate the
externality. We now move on to consider a quantity instrument, sometimes called “tradable quotas”, or “tradable rights”. Pigouvian tax τ was chosen to reach an aggregate
target for the common resource, sF B , that was socially optimal. Now we take sF B as
given and allocate private rights for using the resource:
sF B = s1 + s2 + ... + sn
where si gives how much of the common resource i can use. Thus, if agent i just uses its
own endowment, then it can use up to zi ≤ si . When these quotas are tradable, i can
buy more from the other users and then zi − si > 0; agent i may also want to sell, and
then zi − si < 0. Let us first assume that the market for the quotas is frictionless. Let
p denote the market price that agents take as given. Just below discuss how the level of
this price determined, and what happens when agents can manipulate the price.
Agent i takes the endowment si as given, as well as the market price for quotas, and
19
chooses common resource usage zi and technology xi such that
max U (xi , zi , s) − p × (zi − si )
{xi },{zi }
⇔
∂U (xi , zi , s)
(i) :
−p=0
∂zi
∂U (xi , zi , s)
(ii) :
=0
∂xi
We observe that (i) ⇒ zi = zi (p), so that the price determines usage. This is declining
in p, similarly as zi was declining in the Pigouvian tax. Quite intuitively, the higher is
the price of emissions, for example, the lower is the emission level. With this individual
level demand for the common resource, we can find the equilibrium price, denoted by
p = pF B . It is obtained from the common idea that the aggregate demand should equal
the aggregate supply, otherwise the system of tradable rights would not at all introduce
scarcity in the market:
X
(zi (p) − si ) = 0
i∈I
⇔
X
zi (pF B ) = sF B .
i∈I
We can summarize the outcomes for tradable rights and taxes as follows:
Proposition 5 Equivalence of tradable rights and taxes
(i) equilibrium quota price equals the Pigouvian tax: pF B = τ
(ii) activity levels are socially optimal ziF B in both systems
(iii) technologies are socially optimal xFi B in both systems
Moreover, private choices are independent of how endowments {si }i∈I are allocated.
This result follows by direct verification from the agents’ optimality conditions. Note,
in particular, that each firm just looks at the market price to find its response to the
incentives, and not at all how much it received as an initial allocation si . Intuitively,
a firm receiving a generous allocation does not use more just because of the allocation,
since by using the rights extensively its marginal valuation for the rights falls; it benefits
more from selling the extra rights to other firms with higher valuations. However, this
independence of allocations result does not follow if some agents have market power: a
large firm, for example, may be able to manipulate the market price for rights. This is
an important concern since the efficiency of the system is no longer maintained under
market power.
20
Tradable quotas: market power. When agent i is large enough in the market, it
understands that the clearing price depends on its own action zi :
X
zj (p) + zi = sF B
j6=i
⇒
dp
X
zj0 (p) + dzi = 0
j6=i
⇔
X
dp
= −[
zj0 (p)]−1 > 0
dzi
j6=i
Understanding how the market price depends on agent i’s action, the situation for
agent i looks different:
max U (xi , zi , s) − p(zi ) × (zi − si )
{xi },{zi }
⇔
(i) :
∂U (xi , zi , s)
dp
− p(zi ) −
(zi − si ) = 0
∂zi
dzi
∂U (xi , zi , s)
(ii) :
=0
∂xi
From (i), we can now observe that the agent evaluates how its trading with the market
dp
impacts the price. Term p(zi ) + dz
(zi − si ) measures the marginal revenue from selling to
i
the market if the agent does not use its own endowment fully. The same term becomes
the marginal cost of buying if it needs to purchase more from the market to comply with
the regulation. As a seller, the agent tends to sell too little to raise the price, and, as a
buyer, buy too little to depress the price. Because the endowment determines on which
side of the market the agent is, the distortion depends on the initial allocations.
Proposition 6 Private choices become dependent on allocations:
(i) When first best allocations are given to all, si = ziF B for all i, there will be no trade
and {xi , zi } are socially optimal
dp
∂U (xi , zi , s)
=p+
(zi − si ) = p ⇒ zi = ziF B
∂zi
dzi
(ii)If si > ziF B , i will be a seller and use more than socially optimal:
dp
∂U (xi , zi , s)
=p+
(zi − si ) < p ⇒ zi > ziF B
∂zi
dzi
(iii)If si < ziF B , i will be a buyer and use less than socially optimal:
∂U (xi , zi , s)
dp
=p+
(zi − si ) > p ⇒ zi < ziF B
∂zi
dzi
21
As long as u1,2 6= 0, the technology choices will be distorted as well.
What do we learn? To mitigate the market power problem, we would need to allocate
the rights according to the socially optimal needs. However, this is difficult in practise,
and also raises the question of why to use tradable rights in the first place.
3.3.3
Uncertainty: choosing between quantities and prices
We move now to consider the choice between the two instruments. Just above, we observed that the choice does not matter – however, it starts to matter when there is
uncertainty about the private valuation of the regulated activity. For example, the regulator may not know if the industry under regulation will be facing a boom or downturn
once the regulation is in place. The mode of regulation, that is, the policy instrument
must be chosen before knowing this. Let xi now denote an uncertain factor that influences how agent i values usage zi . It may still be ”technology” but it is not chosen by
the agent – it may capture, for example, state of demand for the firms products.
To make the point sharply, we assume the following timeline:
1. Policy is chosen (either price instrument or tradable quotas)
2. xi is realized (for example demand for output)
Let us first see how uncertainty works its way to the activity level when the policy choice
is to set tax. For a given tax, changes in xi will lead to a changes in actions:
∂U (xi , zi , s)
−τ =0
∂zi
dzi
U1,2 > 0 ⇔
>0
dxi
For example, when the demand higher (xi higher), then emissions from polluting firms
will be higher. On the other hand, if the policy instrument is to set the total quantity
target, then z1 + ... + zn is fixed, and the total emissions cannot change. Only the price at
which quotas will be traded will change. This illustrates why the two instruments differ:
one sets the price but leaves the quantities open; the other sets the total quantities, but
leaves the price of the rights open.
Both instruments will lead to some inefficiencies because the instrument is not adjusted to the changes in xi . The question is now: which one of the two instruments
deviates less from what would be optimal ex post, that is, after the realization of the uncertainty? The answer to this question determines if the policy maker should use prices
or quantities. To answer the question, we assume that there is only one representative
agent choosing z, facing one of the policy instruments. We assume that U (.) is such that
the marginal private value is linear in z, and additively increasing in x:
pd = z ∗ − a(z − z ∗ − x)
where pd denotes private marginal value of z when x is given. Note that this equation is
an inverse demand for emissions; it is a downward sloping relationship between quantities
22
and prices that experiences shifts when x changes. Thus, x is a demand shifter, with
high x being a state of boom. The terms z ∗ > 0 and a > 0 are some given parameters.
They capture the level of the valuation (that is, z ∗ ), and the slope (that is, a) giving
how quickly the valuation declines in z. Note that here we drop index i since there is
one representative agent. Assume that that the policy maker has some expectations for
the level of the demand, x. These are denoted by E{x}, and we assume that E{x} = 0,
which means that the policy maker expects no systematic change in x, for example, that
the demand systematically increases. Also assume that d(.) is such that the social cost
of extraction is also linear and increasing in z:
ps = z ∗ + b(z − z ∗ )
where ps is the social marginal value of z, independent of x, with slope b > 0.
How should we choose activity z if we could do that after observing x? Just equate
the private demand price and the social cost, pd = ps :
Proposition 7 (first best) The socially optimal activity level is
a
x
zF B = z∗ +
a+b
You see that when x = 0 (no uncertainty), then z F B = z ∗ . However, we cannot
observe x at the time of policy making: we are restricted to second-best policy. What
would be this second-best policy if one is restricted to use quantity regulation? We let
SB denote second best, and state first the following observation:
Proposition 8 (second-best: quantity policy) Suppose that quantity z must be chosen
before observing x. Then, the optimal quantity is
z SB = z ∗ .
Proof: The expected loss from setting z is:
Z z
a+b
(z − z ∗ )2
E
(pd (k) − ps (k))dk = −
2
∗
z
which is minimized by setting z SB = z ∗ .
If, instead of quantity, the policy is to set the price on the activity. Note that when
facing tax τ per unit of z, the private response is to choose z such that
τ = pd = z ∗ − a(z − z ∗ − x).
This is so because if the private value exceeds (or falls short) τ it would be optimal to
demand more (or less) of the resourced. Note that the policy maker can now choose the
expected activity level by choosing the tax: we can set τ so that E{z(τ )} = z SB .
Proposition 9 (second-best: price policy) Optimal tax τ per unit of z is
τ ∗ = z∗
⇒
E{z(τ )} = z SB
23
What do we learn? It is optimal to choose the level of the instrument so that the
expected activity level is the same in both cases. But ex post, the activity levels will be
off in both cases. To see this, recall that once the uncertainty is realized, x 6= 0, and it
so deviates from the expected value x = 0 that was used in planning at the outset; thus,
the policy will lead to an outcome that deviates from the first best. The quantity policy
will lead to fixed final emissions, denoted by z Q , that will be off the first best by this
much
a
x
zF B − zQ =
a+b
while the price policy induced final quantity, denoted z τ , will deviate in the other direction
zF B − zτ = −
b
x.
a+b
To make the choice between the instruments, we need to compare the resulting losses
from these deviations. Let ∆Q and ∆τ denote the expected loss from deviations z F BE −z Q
and z F BE − z τ , respectively
Proposition 10 (Uncertainty: choosing between quantities and prices) The optimal policy depends only on the slopes of the marginal private valuation and the marginal social
costs:
∆Q < ∆τ ⇔ b > a
Thus, if and only if the slope of the marginal social cost is greater than the slope of the
private benefit, the quantity instrument is better.
Now we can draw some conclusions from this exercise:
• price instrument makes sense in climate change: the social cost arises from changes
in stocks ⇒ b is low.
• Suppose uncertainty can take two values, x ∈ [xL , xH ]. The quantity instrument
can be supplemented with prices to achieve first best! Regulator can sell more
rights in state x = xH , and buy back permits in the low state x = xL . Difficult to
implement if uncertainty has a richer structure but gains in general to be achieved
through this ”hybrid” price-quantity scheme.
• When private value of emissions and the social cost are correlated, then the comparison of policies gets more complicated.
4
Further Reading
The Coasian solution to the externality problem (1960) can be found from most textbooks
but the exposition in Friedman (2000), including the discussion on transaction costs, is
outstanding. Beccherle & Tirole (2011) and Harstad (2012) provide the climate policy
implications of the dynamic aspects in common-pool problems. Liski & Montero (2011)
focus on market power in tradable right systems.
24
Beccherle, Julien and Tirole, Jean (2011): ”Regional Initiatives and the Cost of Delaying
Binding Climate Change Agreements,” Journal of Public Economics, Volume 95,
Issues 11–12, Pages 1339-1348.
Buchholz, Wolfgang and Konrad, Kai (1994): ”Global Environmental Problems and the
Strategic Choice of Technology,” Journal of Economics 60 (3): 299-321.
Coase, Ronald H. (1960): ”The Problem of Social Cost,” Journal of Law and Economics
3: 1-44.
Friedman, D, ” Law’s order ( 2000). Princeton University Press.
Fudenberg, Drew and Tirole, Jean (1991): Game Theory. MIT Press.
Gerlagh, Reyer, and Liski, Matti (2011): ”Strategic resource dependence,” Journal of
Economic Theory, Volume 146, Issue 2, Pages 699-727.
Harstad, Bård (2012): ”Climate Contracts: A Game of Emissions, Investments, Negotiations, and Renegotiations,” Review of Economic Studies, forthcoming.
Harstad Bård & Matti Liski (2013)”Games and Resources”, , In: Shogren, J.F., (ed.)
”Encyclopedia of Energy, Natural Resource, and Environmental Economics”, Vol.
2, pp. 299-308 Amsterdam: Elsevier.
Liski, Matti, and Montero, Juan-Pablo (2011): ”Market power in an exhaustible resource market: The case of storable pollution permits,” The Economic Journal,
121: 116-144.
25