Kant’s Rule of Behavior and Kant-Nash Equilibria
in Games of Contributions to Public Goods
Arghya Ghosh and Ngo Van Long
Kant’s Rule of Behavior and Kant-Nash Equilibria in
Games of Contributions to Public Goods
Arghya Ghosh and Ngo Van Long
29 March 2015
Abstract
We generalize John Roemer’s concept of Kantian Equilibrium to an environment
where not all agents are Kantian. We propose the concepts of Inclusive Kant-Nash
Equilibrium (IKNE) and Exclusive Kant-Nash Equilibrium (EKNE) and apply them
to the analysis of games of contributions to public goods. We show that social
welfare in the Inclusive Kant-Nash equilibrium is higher. In some economic environments, Kantians may be able to induce a Nashian to join their camp. Moreover,
we extend our framework in two directions. First, we propose the notion of an
Intermediate Kant-Nash equilibrium (i-KNE), which is in between the EKNE and
the IKNE. Second, we generalize the model to include dynamic considerations, and
propose the concept of Kant-Nash equilibrium in feedback strategies. We explore its
implications in some applications. In particular, we show the existence of an intermediate Kant-Nash equilibrium which results in a greater conservation of a global
environmental asset, as compared with the outcome where all agents are Nash players. The resulting social welfare is higher than in the Nash equilibrium, provided
that the public good is su¢ ciently valuable. Moreover, social welfare increases with
the Kantian population share: Given the total population, as the percentage share
of the Kantians increases, social welfare increases as a result. This is indeed good
news!
JEL-Classi…cations:C71;D62;D71.
Keywords: Kantian equilibrium; Rule of Behavior; Categorical Imperative.
1
1
Introduction
Most of economic theory is based on the assumption that individuals seek to maximize
their utility, or, more precisely, to satisfy their preferences, which can be represented
by a ‘utility function’ serving as a preference indicator. Moreover, the assumption of
utility maximization is a fundamental ingredient in non-cooperative game theory, where
the key concept is the Nash equilibrium. In a Nash equilibrium of a game, each player’s
equilibrium strategy maximizes her utility, given the utility-maximizing strategies of other
players. However, as documented below, there are economists, as well as non-economists,
who have pointed out that in many instances, individuals are guided by ‘rules of behavior’
which may not aim at, nor lead to, the maximization of a utility function. Moreover, many
people would argue that there exist circumstances under which actions ought to be guided
by moral principles, regardless of their utility consequences.
The late economist Jean-Jacques La¤ont once told the second author of this paper
that in his childhood, his family instilled in him Kant’s Rule of Behavior. For instance,
whenever he made too much noise, he would be told, “Jean-Jacques, please make less noise.
Imagine what an unpleasant place our neighborhood would be if everyone were to make
as much noise as you did.”1 Later on in life, La¤ont would write a compelling economic
paper (La¤ont, 1975) that emphasized the role of Kantian ethics, and in particular the
‘categorical imperative,’a notion that was put forward in Groundwork of the Metaphysics
of Morals (Kant, 1785). While Kant’s book o¤ers several formulations of this concept,
for our purposes it seems adequate to present it as follows: ‘Act as if the maxim of your
action were to become through your will a general natural law.’2
In his highly in‡uential book, A Theory of Justice, John Rawls (1971, pp. 253-257 and
584) also appealed to the Kantian categorical imperative in his formulation of the theory
of justice: “The principles of justice are also categorical imperatives in Kant’s sense. For
by a categorical imperative Kant understands a principle of conduct that applies to a
person in virtue of his nature as a free and equal rational being (...). To act from the
1
Many of us would recall similar lessons of moral education during our childhood. Roemer (2010, p.
18) writes: We often attempt to instill Kantian ethics in children: “Don’t litter- How would you like it if
everyone else littered as you are doing?”
2
See Bertrand Russell’s A History of Western Philosophy. As explained by Russell (1945, p. 737)
‘There are two sorts of imperative: the hypothetical imperative, which says “You must do so-and-so if
you wish to achieve such-and-such and end”; and the categorical imperative, which says that a certain
kind of action is objectively necessary, without regard to any end.’
2
principles of justice is to act from categorical imperatives in the sense that they apply to
us whatever our particular aims are.”
For the analysis of certain economic activities, John Roemer (2010) has proposed
a useful mathematical formulation of the Kantian rule of behavior. This formulation
may be brie‡y described as follows. Consider an activity that yields negative or positive
externalities, such as playing loud music, or keeping the side walk in front of your house
clean and safe. Roemer suggests that, as a Kantian, your current activity level x > 0 is
morally appropriate if and only if any scaling up or scaling down of that activity level
by a factor
6= 1 would make you worse o¤, were everyone else to scale up or down
their activity levels by the same proportion.3 Roemer (2010) proposes the concept of a
Kantian equilibrium, which he characterised as a “normative concept”(p. 17). He de…nes
a Kantian equilibrium as a con…guration of activity levels in a community where everyone
follows Kant’s rule of behavior.4 While Kantians do not judge actions on the basis of their
consequences, interestingly Roemer shows that if all agents are Kantians, the consequence
is that the resulting Kantian equilibrium is Pareto e¢ cient under some weak assumptions.
In this paper, we wish to explore further the consequences of Kantian behavior. In
particular, we consider situations where only a fraction of the community is Kantian. Can
we de…ne a state of a¤airs where in some sense, both Kantians and Non-Kantians are in
some sort of equilibrium? And what are the properties of such an equilibrium? In an
attempt to answer these questions, we propose two concepts of Kant-Nash equilibrium.
We suppose that there are people who act as Nash players, and there are people who adopt
the Kantian conduct. Our tasks are to de…ne some appropriate equilibrium concepts and
to characterise such equilibriums in some applications.
To …x ideas, we begin by considering a game of voluntary contribution to a public good.
The game takes place in a community consisting of m members, of which k members adopt Kant’s rule of behavior, and the remaining n
m
k members adopt Nash behavior.
Let K = f1; 2; 3; :::; kg denote the set of Kantian agents, and N = fk + 1; k + 2; :::; k + ng
3
Roemer proposes to use a person’s own utility level, rather than the sum of the utility levels of all
agents, as the yardstick for determining the appropriate level of her Kantian action. This echoes Rawls’s
assumption of ‘mutual disinterest’(p. 253 and 584) on the part of the contracting parties behind the veil
of ignorance: “They decide solely on the basis of what seems best calculated to further their interests so
far as they can ascertain them” (p. 584).
4
Roemer’s formulation, using utility functions, might at …rst appear to be in con‡ict with Kant’s
rejection of utilitarianism. However, recall that Roemer’s formulation does not call for utility maximization per se. In fact, every Kantian knows that the prescribed rule of behavior would not result in utility
maximization, unless everyone were Kantians.
3
denote the set of Nashian agents. Let M denote the union of the two sets K and N . We
propose two distinct concepts of equilibrium, corresponding to two di¤erent interpretations of the Kantian rule of behavior. In the …rst interpretation, which we call “Inclusive
Kantian”, the words everyone else in the clause “were everyone else to scale up or down
their activity levels by the same proportion” are taken to mean “everyone else in the set
M , regardless of whether they are Kantian or Nashian.” In the second interpretation,
which we term “Exclusive Kantian”, the words “everyone else” mean “everyone else in
the set K”.
Nashian agents, on the other hand, behave in the well-known Nash fashion: Each
Nashian takes the activity levels of all other m
1 agents as given, and chooses his own
level of activity to maximize his utility. When all Kantian agents are inclusive Kantians,
we propose the concept of Inclusive Kant-Nash Equilibrium (IKNE). In the opposite case,
where all Kantian agents are exclusive Kantians, the equilibrium concept we use is the
Exclusive Kant-Nash Equilibrium (EKNE).
We compare the outcomes of the two equilibriums in our games of contributions to
public goods. Among the questions that we investigate are: (a) Is the social welfare higher
in the Inclusive Kant-Nash equilibrium? (b) For each type of Kant-Nash equilibrium, is
it gainful to Kantians to convert a Nashian into a Kantian? (c) If it is gainful, are the
marginal gains increasing or decreasing in the number of converts?
In sections 4 and 5, we extend our framework in two directions. First, we propose
the notion of an Intermediate Kant-Nash equilibrium (i-KNE), which is in between the
EKNE and the IKNE, and which include EKNE and IKNE as special cases. Second,
we generalize the model to include dynamic considerations, and propose the concept of
Kant-Nash equilibrium in feedback strategies. In Section 6, we study an application,
where the public good is the global common. We give a complete characterization of the
feedback Kant-Nash equilibrium of this game. In particular, we show the existence of an
intermediate Kant-Nash equilibrium which results in a greater conservation of a global
environmental asset, as compared with the outcome where all agents are Nash players.
The resulting social welfare is higher than in the Nash equilibrium, provided that the
public good is su¢ ciently valuable. Moreover, social welfare increases with the Kantian
population share: Given the total population, as the percentage share of the Kantians
increases, social welfare increases as a result. This is indeed good news!
4
2
A simple model of Kant-Nash contributions to a
public good
We consider a model of contributions to a local public good. To …x ideas, think of a
local community, and the public good is the degree of cleanliness of a park. There are m
agents. Let M = f1; 2; :::; mg. Agents supply e¤orts, denoted by xi , i 2 M , and the level
of the public good that is provided, denoted by Q, is equal to the sum of their e¤orts:
Q=
X
xi :
i2M
The utility function of agent i is denoted by Ui
Ui = Ui (Q; xi )
where UiQ > 0 and Uixi < 0: Agents may be heterogeneous in terms of utility functions,
and also in terms of behavior rules.
We assume there are two types of agents, Nashian agents and Kantian agents. A
Nashian agent chooses his e¤ort level xN
i to maximize his utility, taking as given the
e¤ort contributions of all other agents. A Kantian agent’s choice of e¤ort level, xK
i , is not
motivated by the maximization of utility. Rather, her behavior is dictated by a Kantian
Rule of Behavior. She knows that there are agents who are not Kantians.
Following Roemer (2010), we do not seek to determine what a Kantian ought to do
in any given environment. We restrict attention to a modest question: when ought a
Kantian be satis…ed with her current level of contribution?
For this purpose, we distinguish two alternative Kantian Rules: The Inclusive Kantian
Rule, and the Exclusive Kantian Rule.
The Inclusive Kantian Rule: Given a strictly positive con…guration of e¤ort levels
K
K
K
N
N
(xK
1 ; x2 ; :; xi ; ::; xk ; xk+1 ; ::::; xk+n ), a Kantian agent i ought to be satis…ed with her e¤ort
level xK
i if and only if the answer to the following question, Q1, is in the a¢ rmative:
Question Q1
“If I were to scale up or scale down of my e¤ort level by any non-negative factor 6= 1,
and if all other agents in the community, j 2 M fig ; were to scale up or down their
e¤ort levels by the same factor, would my utility level be (weakly) lower?”
5
While asking herself this question, the Kantian agent is not ignorant of the fact that
Nashian agents do not behave in a Kantian way. She supposes that they would, only
for the sake of determining the appropriateness of her action level. This may not sound
‘rational’from the perspective of utility maximization, but we should recall that Kantian
agents use a behavior rule that they judge to be ‘morally right’regardless of the utility
consequences.
If all Kantians adopt the Inclusive Kantian Rule, we propose the following de…nition
of an Inclusive Kant-Nash equilibrium.
De…nition D1 (Inclusive Kant-Nash equilibrium, IKNE)
n+k
N
N
K
K
K
1) A strictly positive vector (xK
1 ; x2 ; :; xi ; ::; xk ; xk+1 ; ::::; xk+n ) in R++ is called an
Inclusive Kant-Nash equilibrium if and only if
(a) for each Kantian agent i 2 K,
Ui (Q; xK
i )
for all
0, where
Q=
X
Ui ( Q; xK
i )
xK
j +
X
xN
s
s2N
j2K
and
(b) for each Nashian agent j 2 N
Uj (Q
for all real numbers x
N
+ xN
j ; xj )
j
Uj (Q
j
+ x; x)
0; where
Q
j
=
X
xK
i +
i2K
X
xN
s :
s2N fjg
2) In the special case where the set of Nashian agents is empty, the Inclusive Kant-Nash
equilibrium is called the Kantian Equilibrium.
We now turn to the Exclusive Kantian Rule.
The Exclusive Kantian Rule: Given a strictly positive con…guration of e¤ort levels
K
K
K
N
N
(xK
1 ; x2 ; :; xi ; ::; xk ; xk+1 ; ::::; xk+n ), a Kantian agent i ought to be satis…ed with her e¤ort
level xK
i if and only if the answer to the following question, Q2, is in the a¢ rmative.
6
Question Q2:
“If I were to scale up or scale down of my e¤ort level by any non-negative factor
6= 1,
and if all other Kantian agents in the community, j 2 K fig ; were to scale up or down
their e¤ort levels by the same factor, while the Nashian agents maintain their activity
levels, would my utility level be (weakly) lower?”
In this question, the Nashians are excluded from the hypothetical scaling of activity
by the same factor . Accordingly, if all Kantians adopt the Exclusive Kantian Rule, we
propose a corresponding equilibrium concept, called the Exclusive Kant-Nash equilibrium.
De…nition D2 (Exclusive Kant-Nash equilibrium, EKNE)
n+k
N
N
K
K
K
A strictly positive vector (xK
1 ; x2 ; :; xi ; ::; xk ; xk+1 ; ::::; xk+n ) in R++ is called an Ex-
clusive Kant-Nash equilibrium if and only if
(a) for each Kantian agent i 2 K,
Ui
X
j2K
for all
xK
j +
X
K
xN
s ; xi
s2N
!
Ui
X
xK
j +
X
s2N
j2K
K
xN
s ; xi
!
0, and
(b) for each Nashian agent j 2 N
Uj (Q
for all real number x
N
+ xN
j ; xj )
j
Uj (Q
j
+ x; x)
0; where
Q
j
=
X
xK
j +
j2K
X
xN
s :
s2N fjg
Which equilibrium would result in a greater quantity of public good? Would the utility
of a Kantian be lower in an IKNE relative to an EKNE? To explore these questions, let
us begin by considering a special linear-quadratic example.
7
3
A linear quadratic example
In this section, we assume that the utility of agent i is linear in the public good, and
quadratic in her e¤ort level:
U (Q; xi ) = AQ
1 2
x
2 i
where A > 0 is the marginal utility of the public good to the representative individual. (As
is well known, the linearity of U in Q
i
implies that the Nashian agents have a dominant
strategy; however, we will see that Kantian agents do not have a dominant strategy.)
For simplicity, in this example, we assume that all individuals have the same utility
function. We begin by considering three benchmark scenarios: the social planner solution,
the standard Nash equilibrium solution, and the Kantian equilibrium.
3.1
The planner’s solution
The planner’s solution is symmetric: all agents will be asked to provide the same amount
of e¤ort, x. The objective function of the social planner is equivalent to
max U (Q; x) = max U (mx; x)
x
x
where Q = mx. The FOC is
mUQ (mx; x) + Ux (mx; x) = 0:
(1)
This yields the familiar Samuelsonian optimality condition, that each individual’s marginal e¤ort cost is equated to the sum of marginal valuations of the public good:
xp = mA
where the superscript p indicates that this is the optimum under the planner.The quantity
of public good provided under the social planner is
Qp = mxp = Am2
8
The utility level of the representative agent is
U p = A2 m 2
3.2
0:5(Am)2 = A2
m2
2
The Nash equilibrium
This case arises if there are no Kantians, i.e., k = 0 and n = m:
Each Nashian agent i chooses his xi to maximize
A xi +
X
xj
j6=i
!
1 2
x
2 i
Since the utility is linear in the public good, this maximization problem gives the dominant
strategy
xi = A
The Nash equilibrium level of public good is
QN = Am
(2)
Obviously, the Nash equilibrium provision of the public good is below the socially e¢ cient
level. The Nash equilibrium utility level is inferior to the level achieved under the social
planner:
U
3.3
N
2
=A m
A2
= A2
2
2m 1
2
< Up
The Kantian equilibrium
This case arises if there are no Nashians, i.e., k = m. The Kantian equilibrium is identical
to the social optimum. To see this, let (xK ; xK ; ::::; xK ) > (0; 0; :::; 0) be a Kantian
equilibrium. Then it holds that
1 = arg max U (m xK ; xK )
0
9
That is, U (m xK ; xK ), considered as a function of , must attain its maximum at
The FOC with respect to
= 1.
is
mxK UQ (m xK ; xK ) + xK Ux (m xK ; xK ) = 0
Dividing by xK yields
mUQ (m xK ; xK ) + Ux (m xK ; xK ) = 0
For
= 1 to be the solution of this FOC, it is necessary that
mUQ (mxK ; xK ) + Ux (mxK ; xK ) = 0
(3)
This condition is identical to the planner’s necessary condition (1). In the linear quadratic
case, the Kantian level of public good is
QK = Am2 = Qp :
3.4
The Exclusive Kant-Nash equilibrium
Let xi = xK for all i 2 K and xj = xN for all j 2 N . Since individuals have the same
utility function, a strictly positive pair (xK ; xN ) is an exclusive Kant-Nash equilibrium
i¤: (a) the necessary condition for each Kantian (in De…nition D1) is satis…ed, i.e.,
U ( kxK + nxN ; xK )
U (kxK + nxN ; xK ) for all
(4)
0
and (b) the necessary condition for each Nashian is satis…ed, i.e.,
U (kxK + (n
1)xN + x; x)
U (kxK + nxN ; xK ) for all x
Using our speci…cation of the utility function, condition (4) becomes
( kxK + nxN )A
i.e., AkxK
1
(
2
1
( xK )2
2
(kxK + nxN )A
xK )2 reaches its maximum at
10
= 1.
1 K 2
(x )
2
0
(5)
The FOC wrt to
is
AkxK
The FOC is satis…ed at
(xK )2 = 0
= 1 i¤ AkxK
(xK )2 = 0, i.e., in the EKNE, each Kantian
contribution to the public good is
xK
E = Ak
(6)
where the subscript E refers to EKNE. Equation (6) shows that the contribution of each
Kantian increases with the number of Kantians in the population.
Turning to the Nashian agents, their FOC condition yields their contribution level
xN = A:
The Exclusive-Kant-Nash equilibrium public good level when there are k Kantians in a
game with m players is then
QE (k; m) = Ak 2 + A(m
k) = Am + A(k 2
k)
(7)
Proposition 1: The equilibrium level of the public good in the Exclusive Kant-Nash
equilibrium is greater than in the Nash equilibrium if k > 1. It is equal to the Nash
equilibrium level for k = 1 or k = 0. Starting at any k
1, the contribution of each
Kantian increases with the number of Kantians in the population.
Comparative Statics
Does an conversion of a Nashian agent into an exclusive Kantian raise the equilibrium
level of public good?
The answer is yes, as is clear from eq. (7)
What happens to the utility levels of Kantians and Nashians? How do they vary with
the number of converts?
Let uK (k; m) denote the Kantian agent’s utility level in an Exclusive-Kant-Nash equilibrium when there are k Kantians in a population of m agents. Then
uK (k; m) = A(Ak 2 + A(m
11
k))
1
(Ak)2
2
It is increasing in k for all k
1. The equilibrium utility level of a Nashian is always
higher than that of a Kantian if k > 1:
uN (k; m) = A(Ak 2 + A(m
k))
1 2
A
2
The di¤erence in their utility levels is
uN (k; m)
1
uK (k; m) = A2 (k 2
2
1)
This shows that as k increases, the utility gap widens.
Suppose by moral persuation, a Nashian is converted into a Kantian. Following the
conversion of one Nashian, the Kantian agent’s utility becomes uK (k + 1; m). It is easy
to check that the gain to each existing Kantian is positive. Thus, with the increase in the
Kantian share of the population, each Kantian contributes more to the public good, and
their utility increases.
Consider an initial equilibrium with k Kantians and n Nashian. What is the e¤ect
of a succesful conversion of a Nashian into a Kantian? The sum of utilities of k Kantian
and one potential convert is kuK (k; m) + uN (k; m). Compare this with the sum of these
k + 1 individuals after the conversion, (k + 1)uK (k + 1; m), we …nd that their net gain is
positive. This shows that it is possible to compensate the new convert so that all k + 1
agents are better o¤ after the conversion. If all this surplus is given to the new convert,
his utility will be higher than the utility he had before conversion. However, he would
gain more if he lets someone else be converted, so that he can free ride on a larger group
of Kantians. Therefore, if economic considerations alone matter, a farsighted Nashian
would not agree to be converted: if he stays out, he will reap more gain!
Proposition 2: It is possible for Kantians to induce a Nashian to join their camp.
With adequate inducement, he gains by joining, as compared with the status quo. However,
it would be better for him if he stays out and let another Nashian join the Kantian camp.
3.5
The Inclusive Kant-Nash equilibrium
Let us turn to the IKNE. A strictly positive pair (xK ; xN ) is an Inclusive Kant-Nash
equilibrium i¤, for Kantian agents,
12
U ( kxK + nxN ; xK )
U (kxK + nxN ; xK ) for all
0
and, for Nashian agents,
U (kxK + (n
1)xN + x; x)
U (kxK + nxN ; xN ) for all x
0
Using our speci…cation of the utility function, at the IKNE, we have, for Nashian
agents
xN = A
(8)
and for Inclusive Kantians,
A(kxK + nA)
(xK )2 = 0
(9)
This yields the equilibrium contributions of each Inclusive Kantian
xK
I =
p
A
k + k 2 + 4n > Ak = xK
E
2
(10)
where the subscript I denote IKNE. Thus we obtain an intuitively plausible result:
Proposition 3: When Kantians are inclusive, their contributions are greater than
under the exclusive Kant-Nash equilibrium. The Nash players contribute the same amount
in both societies.The level of the public good in an IKNE is greater the level achieved in
an EKNE.
Proposition 4: Consider an IKNE. An increase in the population share of Kantians
will lead to higher contribution by each Kantian, but each Kantian always contributes less
than she would in an ideal society where all are Kantians.
Is social welfare higher under inclusive Kant Nash equilibrium relative to exclusive
Kant Nash equilibrium? The answer is given in Proposition 5:
Proposition 5: Social welfare is higher in the Inclusive Kant-Nash equilibrium (compared to the outcome in the Exclusive Kant-Nash equilibrium), even though Kantians
would have higher utility in the Exclusive Kant-Nash equilibrium.
13
4
Generalization: Intermediate Kant-Nash Equilibrium
In the preceeding section, we considered two extreme formulations of Kantian Rule. It
seems natural to suggest an intermediate Kantian Rule. Instead of asking themselves
Question Q1 or Question Q2, they can ask:
Question Q3:
“If I were to scale up or scale down of my e¤ort level by any non-negative factor
and if all other Kantian agents in the community, j 2 K
6= 1,
fig ; were to scale up or down
their e¤ort levels by the same factor, while the Nashian agents were to scale up or down
their e¤ort levels by a factor ( ), would my utility level be (weakly) lower?”
It is important to stress that in asking this question, Kantians do not necessarily
believe that Nashians would actually scale up or down their activity levels. How should
( ) be speci…ed? It seems sensible to suppose that ( ) 6=
speci…cation would be to introduce a parameter , such that
( )=(
so that
0
too; if
> 1 then ( )
1
( )
( )=
1) + 1 where 0
1. This means that if
1; and ( )
i¤
6= 1. An operational
1
= 1 (neither scaling up nor down) then
; if
(11)
=1
< 1 (scaling down), then ( ) is such that
.
The resulting equilibrium is called the Intermediate Kant-Nash Equilibrium (i-KNE).
This speci…cation (11) has the advantage that the IKNE and the EKNE can be seen
clearly as special cases: If
= 0, Question Q3 will be identical to Question Q2 (i.e.,
the exclusive case), and if
= 1;Question Q3 will be identical to Question Q1 (i.e., the
inclusive case).
5
Extension: Kant-Nash equilibrium and the dynamics of contributions to a public good
We now turn to a model where the contributions to the public good take place over
time. The time horizon can be …nite or in…nite. As is well known, with dynamic games,
14
there is greater scope for achieving e¢ ciency if agents can use punishment strategies, such
as trigger strategies, which belong to the class of history-dependent strategies. In this
paper, to sharpen our focus, we do not consider history-dependent strategies. Instead, we
consider only the class of strategies called feedback strategies, where actions at any date
t is conditioned only on two variables, namely the observed value of the state variable at
that date, S(t), and on t itself.5
The Kantian and Nashian agents interact over T periods. Period T + 1 is called the
last period: no one can take any action in that period, though they may receive a payo¤.
The public good is a stock, denoted by St , that depreciates at a rate (1
) per period.
Agents add to the stock by investments, in periods t = 1; 2; ::; T .
Let St denote the stock at the beginning of period t. In period t, (t = 1; 2; ::; T ), agents
contribute to the stock by expending e¤orts, xi;t . During period t, they enjoy a bene…t
‡ows Gt which is a function of both the opening stock St and their aggregate contribution
during the period, Xt :
Gt = G(St ; Xt )
where
Xt =
The evolution of the stock is given by
X
xi;t
St+1 = St + Xt
The utility obtained by agent i is
Ui;t = Ui (G(St ; Xt ); xi;t )
where Ui is increasing in Gt and decreasing in xi .
Consider the last period, T + 1. Assume that agents derive pleasure from leaving a
collective bequest ST +1 . This pleasure is Bi (QT +1 ), where Bi0
0. It is discounted by
a discount factor
2 (0; 1). Then the Exclusive Kant-Nash equilibrium in the period T
game is de…ned and solved in way similar to the static game considered in the preceding
5
See Maskin and Tirole (2001), Dockner et al (2000), and Long (2010) for the concept of feedback
Nash equilibrium, or Markov-perfect Nash equilibrium. For history dependent equilibrium and trigger
strategies, see e.g. Fudenberg and Tirole (2000), and Dockner et al. (2000).
15
section. Since there is a stock ST at the beginning of the period, the period T equilibrium
actions can be expressed as functions of ST .
De…nition D3: Exclusive Kant-Nash equilibrium for the subgame in period
T
K
K
K
N
N
Given ST , a strictly positive conguration (xK
1;T ; x2;T ; :; xi;T ; ::; xk;T ; xk+1;T ; ::::; xk+n;T ) is
an exclusive Kant-Nash equilibrium for the subgame in period T if and only if
(a) For each Kantian agent i 2 K,
Ui G
X
j2K
Ui G
X
xK
j;T
+
X
s2N
xK
j;T +
j2K
for all
xN
s;T ; ST
X
!
xN
s;T ; ST
s2N
; xK
i;T
!
!
X
+ Bi
; xK
i;T
!
xK
j;T
+
j2K
+ Bi
X
xN
s;T
+ ST
s2N
X
xK
j;T +
X
xN
s;T + ST
s2N
j2K
!
!
0, and
(b) for each Nashian agent j 2 N
for all x
Uj G X
j;T
N
+ xN
j;T ; ST ; xj;T + B
Uj (G (X
j;T
+ x; ST ) ; x) + B ( ST + X
0; where
X
j;T
X
xK
i;T +
i2K
X
ST + X
j;T
j;T
+ xN
j;T
+ x)
xN
s;T :
s2N fjg
Assume that the period T subgame has a unique Exclusive Kant-Nash equilibrium.
Then we can write the equilibrium actions as functions of ST :
xK
i;T =
K
i;T (ST ),
i2K
xN
s;T =
N
s;T (ST ),
s2N
and
(where the functional form of
K
i;T
may be di¤erent from that of
16
N
j;T ).
Then we calculate
the equilibrium utility levels of each Kantian agent in period T , which we denote by ViK ;T
K
Vi;T
(ST ) = UiK
X
G
K
j;T (ST )
+
+ Bi
N
s;T (ST ); ST
s2N
j2K
X
X
K
j;T (ST )
+
X
N
s;T (ST )
+ ST
s2N
j2K
N
For Nashians, we can de…ne Vi;T
(QT
1)
!
;
!
K
i;T (ST )
!
in a similar same way.
Then, at the beginning of period T
1, facing a given stock ST 2 , the Exclusive
Kant-Nash equilibrium in the period T 1 game can be de…ned as follows.
is
For each Kantian i 2 K, her payo¤ function for the game starting with the stock QT
Ui G
X
xK
j;T
1
+
j2K
X
K
Vi;T
X
xN
s;T
1 ; ST 1
s2N
xK
j;T 1
+
j2K
X
xN
s;T 1
+ ST
s2N
!
1
; xK
i;T
!
1
!
2
+
and for each Nashian j 2 N , the payo¤ function is
Uj (G (X
j;T 1
+ xj;T
1 ; ST 1 ) ; xj;T 1 )
where
X
j;T 1
=
X
xK
i;T
N
+ Vj;T
1+
i2K
1 (X j;T 1
X
xN
s;T
+ xj;T
1
+ ST
1)
1
s2N fjg
De…nition D3-B: Exclusive Kant-Nash equilibrium for the subgame in period
T 1
A con…guration (xK
1;T
K
K
K
N
N
1 ; x2;T 1 ; :; xi;T 1 ; ::; xk;T 1 ; xk+1;T 1 ; ::::; xk+n;T 1 )
17
is an Exclus-
ive Kant-Nash equilibrium for the subgame in period T
Ui G
X
xK
j;T
+
Ui G
xN
s;T ; ST
s2N
j2K
X
X
xK
j;T +
j2K
X
!
; xK
i;T
!
xN
s;T ; ST
s2N
!
+ Bi
!
; xK
i;T
1 if and only if, for each Kantian,
X
xK
j;T
+
xN
s;T
+ ST
s2N
j2K
+ Bi
X
X
xK
j;T +
j2K
X
!
xN
s;T + ST
s2N
!
and, for each Nashian, j 2 N
for all x
Uj G X
j;T 1
+ xN
j;T
1 ; ST 1
; xN
j;T
Uj (G (X
j;T 1
+ x; ST
1 ) ; x)
N
+ Vj;T
N
+ Vj;T
1
1 (X j;T 1
1 (X j;T 1
+ xN
j;T
+ x + ST
1
+ ST
1)
1)
0.
The whole game can be solved recursively this way. An Exclusive Kant-Nash Equilibrium in feedback strategies of the game is a strategy pro…le, which is a set of m functions,
xi (:; :), that maps the pair (S; t), for all S
0 and all t 2 f1; 2; 3; :::; T g, to the space
of action, where S is the value of the state variable observed at time t, such that at
any subgame, the actions prescribed by these strategies constitute an Exclusive Kant-Nash
equilibrium for that subgame.
If the time horizon is in…nite, the equilibrium is obtained by taking the limit T ! 1.
(An example will be provided in the next section.)
In a similar fashion, one can de…ne the Inclusive Kant-Nash Equilibrium in feedback
strategies. For example, for the subgame in period T ,
De…nition D4: Inclusive Kant-Nash equilibrium for the subgame in period
T
K
K
N
N
K
Given ST , a strictly positive conguration (xK
1;T ; x2;T ; :; xi;T ; ::; xk;T ; xk+1;T ; ::::; xk+n;T ) is
an exclusive Kant-Nash equilibrium for the subgame in period T if and only if
(a) For each Kantian agent i 2 K,
Ui G
X
j2K
Ui G
X
j2K
xK
j;T +
X
xN
s;T ; ST
s2N
xK
j;T +
X
s2N
!
; xK
i;T
xN
s;T ; ST
!
!
+ Bi
; xK
i;T
18
!
X
xK
j;T +
j2K
+ Bi
X
xN
s;T + ST
s2N
X
j2K
xK
j;T +
X
s2N
!
xN
s;T + ST
!
for all
0, and
(b) for each Nashian agent j 2 N
for all x
Uj G X
j;T
N
+ xN
j;T ; ST ; xj;T + B
Uj (G (X
j;T
+ x; ST ) ; x) + B ( ST + X
0; where
X
j;T
X
xK
i;T +
i2K
6
X
ST + X
j;T
j;T
+ xN
j;T
+ x)
xN
s;T :
s2N fjg
A dynamic game of preservation of a global common: Intermediate Kant-Nash equilibrium
We now turn to a model of dynamic contributions to a special kind of public good, called
the global common. Contributions take the form of refraining from excessive exploitation
of a global common. Think of a natural asset, called ‘the quality of the environment.’
Agents derive utility from the environmental amenities as well as from the consumption
goods. The production of the consumption goods, however, generates the emissions of
pollutants, which reduce the quality of the environment. Suppose that some agents are
Kantian and others are Nashian. We investigate the Intermediate Kant-Nash equilibrium
of this dynamic game. We apply the concept of Intermediate Kant-Nash equilibrum,
introduced in section 4, to this dynamic model.
The stock of natural asset at the beginning of period t is denoted by St . Agent i
takes action xi;t in period t. Think of xi;t as the output of the consumption good, which
generates an emission level Ei;t = "xit ; for simplicity, we suppose " = 1, so that xi;t also
is also agent i’s emission level. Let Xt denote the aggregate emissions in period t, and let
X
i;t
= Xt
xi;t . Assume that Gt , the aggregate environmental service yielded by the
natural asset in period t, is increasing in St and decreasing in Xt :
Gt = G(St ; Xt ), GS > 0 and GX < 0:
There is a natural renewal process that allows S to recover, at least to some extent, from
19
the damages caused by emissions. We suppose that
St+1 = f (St ; Xt ), fS > 0 and fX < 0:
(12)
Agent i’s utility is increasing in her consumption level xi;t , and in the environmental
service Gt .
uit = U (xit ; Gt (St ; Xt )), Ux > 0 and UG > 0:
There are m in…nitely-lived agents, of which k are Kantians and n are Nashians. To
analyse their in…nite-horizon game, we …nd it convenient to solve for a sequence of T
period horizon games, and then take the limit as T tends to in…nity. We will begin with
the one-period-horizon game, and …nd the Intermediate Kant-Nash equilibrium for that
subgame. This will allow us to analyse the two-period-horizon game, and so on.
6.1
The one-period-horizon game
To simplify notation, we denote by S the current stock level, and S 0 the next period stock
level. Then, eq. (12) gives
S 0 = f (S; X)
Consider now a game where all agents know that this is the last period where they can
take action. Agent i’s payo¤ function is
(1)
Ri = U (xi ; G(S; X
i
+ xi )) +
(S 0 )
where (:) is the ‘scrap value function’, i.e. the utility attached to the terminal stock.
(1)
Here the superscript in Ri indicates that this is the payo¤ for the one-period-horizon
game. Individuals have the same utility function U (x; G). Then we can restrict attention
to equilibriums where all Kantians will choose the same activity level, and similarly, all
Nashians behave in the same way.
(1)
Knowing the opening stock S, each Nashian chooses xN
i to maximize Ri , taking X
as given. This yields the FOC
Uxi (xN
i ; G(S; X
i
+ xN
i )) + UG GX + (
20
S 0 ) (fX )
=0
i
Each Kantian i will be in equilibrium only if
N
K
1 = arg max U (xK
i ; G(S; ( )nx + k x )) +
(S 0 )
where
S 0 = f (S; ( )nxN + k xK )
As proposed in Section 4, we use the function ( ) = (
1) + 1 where 0
1 for
the Intermediate Kant-Nash equilibrium.
Assume that there exist two values min
0 and max
1 such that for any
2 [ min ; max ] the one-period-horizon game has a unique intermediate Kant Nash equi-
librium. We denote the equilibrium actions by x(1)K (S) and x(1)N (S), where the …rst part
of superscript, (1) placed in front of K and N , indicates that this is equilibrium for the
one-period-horizon game. For convenience, de…ne
X (1) (S)
nx(1)N (S) + kx(1)K (S)
G(1) (S)
G(S; X (1) (S))
Using these notations, the one-period-horizon Kant Nash equilibrium payo¤s can be
expressed as follows:
(1)
f (S; X (1) (S)
(13)
(1)
f (S; X (1) (S)
(14)
VN (S) = U (x(1)N (S); G(1) (S)) +
and
VK (S) = U (x(1)K (S); G(1) (S)) +
Example 2: Environmental quality as a public good, with a log utility
function
We modify the model of Levhari and Mirman (1980) to allow for the enjoyment of
environmental quality. The parameter for this enjoyment is denoted by g
0. (In the
model of Levhari and Mirman (1980), g = 0 identically, and there are no Kantian agents.)
Let S 2 [0; 1] be the state variable representing a natural asset at the beginning of the
current period. The highest value that S can take is 1. Let S 0 be the value of S at the
21
beginning of the next period. We assume that
S 0 = (min f0; S
Xg) where 0 <
< 1:
Thus, if X > S, then S 0 = 0.
The level of environmental services delivered to the agents during the period is assumed
to be
G(S; X) = min f0; S
Xg
The utility function of agent i takes the form
U (xi ; G) = ln xi + g ln G where g > 0;
and the ‘scrap value function’is
(S) = ln( S), where
0:
In the one-period-horizon game, each Nashian agent j chooses xN
j to maximize
ln xN
j + g ln min 0; S
X
xN
j
j
+ ln( S 0 )
where
S 0 = min 0; S
X
xN
j
j
And each Kantian agent is in equilibrium if and only if
1 = arg max ln xK
i + g ln min(0; S
+ ln
min(0; S
n ( )xN
n ( )xN
k xK )
k xK )
To ensure the existence of an intermediate Kant-Nash equilibrium, we make the following Assumption:6
Assumption A1: 1
(m
k) > 0
Under assumption A1, there exists a unique intermediate Kant-Nash equilibrium for
6
In this model, there does not exist an inclusive Kant-Nash equilibrium.
22
the one-period-horizon game.7 The equilibrium emission levels are
x(1)N =
(m
k)(1
1
x(1)K =
(m
k
1
S for k
) + (1 + g + b)
k)
(m
1, where b
1
) + (1 + g + b)
k)(1
S
Consequently, for the one-period-horion game, the equilibrium payo¤ function of a representative Nashian is
(1)
VN (S) = (1 + g + b) ln S +
where
(1)
N
(1)
N
= ln
(m
= (g + b) ln
+
(1)
N
+ ln
1
) + (1 + g + b)
k)(1
(m
(1)
N
g+b
k)(1
) + (1 + g + b)
For Kantians, the equilibrium payo¤ function function is
(1)
VK (S) = (1 + g + b) ln S +
where
(1)
K
= ln
(1)
K
+
(1
(m k))k 1
(m k)(1
) + (1 + g + b)
(1)
K
+ ln
and
(1)
K
=
(1)
N
We observe that Nashians achieve higher payo¤s than Kantians. The di¤erence between
the payo¤ is
(1)
VN (S)
6.2
(1)
VK (S) =
(1)
N
(1)
K
= ln
1
k
(m
k)
>0
The two-period-horizon game
Now, consider the game where all agents have two periods to go. All agents know their
equilibrium payo¤s of the one-period-to-go subgame: they are given by eqs (13) and (14).
7
It is important to remark that this inequality is not satis…ed, then there does not exist an intermediate
Kant-Nash equilibrium. It follows from this remark that for the log-linear game under study, there does
not exist an Inclusive Kant-Nash equilibrium.
23
Then, given the opening stock S, Nashian agent i chooses the current period emssions
level xN
i to maximize
(2)
(1)
Ri = U (xN
i ; G(S; X
i
+ xi )) + VN (S 0 )
And Kantians will be in equilibrium if and only if
(1)
1 = arg max U ( xK ; G(S; n ( )xN + k xK )) + VK (S 0 )
Assume that there exists a unique equilibrium for the current actions. Then we denote
the equilibrium actions by x(2)K (S) and x(2)N (S), where the superscript indicates that
this is equilibrium action pro…le when there are two periods to go. Let us de…ne de…ne
X (2) (S) = nx(2)N (S) + kx(2)K (S)
G(2) (S) = G(S; X (2) (S))
Then the two-period-horizon Kant Nash equilibrium payo¤s are
(2)
(1)
f (S; X (2) (S)
(2)
(1)
f (S; X (2) (S)
VN (S) = U (x(2)N (S); G(2) (S)) + VN
VK (S) = U (x(2)K (S); G(2) (S)) + VK
Example 2: Environmental quality as a public good, with a log utility
function (continued)
For T = 2, the equilibrium emissions are
x(2)N =
(m
k)(1
and
x(2)K =
1
S for k
) + (1 + g)(1 + b) + b2
1
(m
k
k)
1
x(2)N
The equilibrium payo¤ functions are as follows. For each Nashian,
(2)
(2)
(2)
VN (S) = ((1 + g)(1 + b) + b2 ) ln S + AN + BN +
24
2
ln
where
(2)
(2)
N
AN =
+
(1)
N ,
with
(2)
N
ln
(m
1
) + (1 + g)(1 + b) + b2
k)(1
and
(2)
BN
with
(2)
N
=
(2)
N
(1)
N ,
+
((1 + g)(1 + b) + b2
1) ln
(m
(1 + g)(1 + b) + b2 1
k)(1
) + (1 + g)(1 + b) + b2
For each Kantian,
(2)
(2)
(2)
2
VK (S) = ((1 + g)(1 + b) + b2 ) ln x + AK + BK +
ln
where
(2)
AK =
(2)
K
= ln
(1
k)(1
(m
(2)
K
+
(1)
K ,
(2)
(2)
BK = BN , and
(m k))k 1
) + (1 + g)(1 + b) + b2
and
Thus
(2)
VN (S)
6.3
(2)
(2)
VK (S) = AN
(2)
AK = (1 + ) ln
1
(2)
K
k
(m
=
(2)
N
k)
The q-period-horizon game
Then, given the opening stock S, Nashian agent i chooses the current period emissions
level xN
i to maximize
(q)
Ri = U (xN
i ; G(S; X
(q 1)
i
+ xi )) + VN
(S 0 )
And Kantians will be in equilibrium if and only if
(q 1)
1 = arg max U ( xK ; G(S; n ( )xN + k xK )) + VK
(S 0 )
Assume there exists a unique equilibrium for the current actions. We denote the equilibrium actions by x(q)K (S) and x(q)N (S), where the superscript indicates that this is
25
equilibrium action pro…le when there are two periods to go. Next, we de…ne
X (q) (S) = nx(q)N (S) + kx(q)K (S)
G(q) (S) = G(S; X (q) (S))
The q-period-horizon Kant Nash equilibrium payo¤s are as follows:
(q)
(q 1)
f (S; X (q) (S)
(q)
(q 1)
f (S; X (q) (S)
VN (S) = U (x(q)N (S); G(q) (S)) + VN
VK (S) = U (x(q)K (S); G(q) (S)) + VK
Example 2: Environmental quality as a public good, with a log utility function
(continued)
For T = q, we have
x(q)N =
(m
1
S for k
Pq 1 s
+ bq
) + (1 + g)
s=0 b
k)(1
1
x(q)K =
(m
k
k)
1
x(q)N
The value functions are
(q)
VN (S)
=
s
!
(q 1)
N
+
q 1
X
(1 + g)
b
s=0
(q)
AN
with
(q)
N
(q)
N
=
ln
+
(m
+b
(q)
with
(q)
N
=
(q)
N
+
(1 + g)
(q 1)
+
N
q 1
X
s=0
::: +
!
bs
!
(q)
+ :::
q
q 1 (1)
N ,
1
Pq 1 s
+ bq
) + (1 + g)
s=0 b
k)(1
q 1 (1)
N ,
+ bq
(q)
ln x + AN + BN +
2 (q 2)
N
and
BN
q
!
1 ln
26
(1 + g)
(m
k)(1
Pq
1 s
s=0 b
!
+ bq 1
Pq 1 s
) + (1 + g)
+ bq
s=0 b
!
The di¤erence between the Nashian payo¤ and the Kantian payo¤ is
(q)
(q)
VN (S)
VK (S) = (1 +
2
+
q 1
+ ::: +
) ln
1
k
(m
k)
Note that the di¤erence is independent of S.
6.4
The in…nite-horizon problem
Taking the limit as q tends to in…nity, we obtain the equilibrium strategies of Nashian
and Kantian players for the in…nite horizon problem.
Example 2: Environmental quality as a public good, with a log utility
function (continued)
We obtain time-independent emissions strategies
xN =
1
xK =
(m
(m
k
1 b
S
)(1 b) + (1 + g)
k)(1
k)
(m
1 b
S
)(1 b) + (1 + g)
k)(1
The value function of the representative Nashian is
VN (S) = (1 + g)
+
1
1
1
1
b
ln S +
g+b
1 b
ln
1
ln
1
(m
(m
k)(1
k)(1
1 b
)(1 b) + (1 + g)
g+b
)(1 b) + (1 + g)
and the value function of the representative Kantian is
VK (S) = VN (S)
Along the equilibrium path, St+1 = (St
St+1 =
(m
k)(1
1
1
ln
1
k
(m
k)
Xt ) , implying
g+b
)(1 b) + (1 + g)
27
St
The steady state level of the stock is
S =
(m
g+b
)(1 b) + (1 + g)
k)(1
1
<1
The steady state is stable: starting at any positive S0 , the stock will converge to S .
We obtain the following results.
Proposition 6: For the dynamic model under study, the intermediate Kant-Nash
equilibrium in feedback strategies display the following properties.
Fact 1: VN (S) is increasing in
(in the Kantians’degree of inclusiveness) and in k
(the population share of Kantians).
(1)
Fact 2: A su¢ cient condition for VK (S) to an increase in k is m 1 0.
Fact 3 Assume that m 1 > 0, and that k is su¢ ciently large such that 1 (m k) >
0. Then as k increases from k to k + 1 or higher values, the gap between VN (S)
VK (S)
become smaller.
Fact 4: (Good News) Regardless of the sign of
then an increase in k will increase social welfare.
m
1; if g is su¢ ciently great,
Fact 5: The steady state environmental quality increases in k.
Fact 6: The steady state environmental quality of the Intermediate KNE is greater
than that of the EKNE (where = 0):
Finally, let us compare the in…nite horizon Kant-Nash equilibrium with the in…nitehorizon Nash equilibrium, where all m agents are Nashians. We call this the ‘Pure
Nashian’ case, and use the subscript or superscript P N to denote the pure Nashian
Values.
Fact 7: The pure Nash steady state stock level is smaller than the Exclusive Kant-Nash
steady state level if k
7
2.
Concluding Remarks
We have formulated the concepts of Inclusive Kant-Nash equilibrium, and Exclusive KantNash equilibrium, and showed how these concepts may shed light on games of contributions to a public good when not all agents are Nashian. In a linear-quadratic example,
we were able to compare the outcomes under the two alternative equilibrium concepts.
We have also extended the model to allow for Intermediate Kant-Nash Equilibrium, from
28
which we can obtain the IKNE and EKNE as limiting cases. Finally, we considered
Kant-Nash games where the stock of the public good evolves over time. We found in an
example that social welfare increases with the Kantian population share: Given the total
population, as the percentage share of the Kantians increases, social welfare increases as
a result. This is indeed good news!
Appendix
Proof of Proposition 4
Let k = m where 0
Kantians,
1, so that n = (1
= 1, and each would contribute xK ( m; (1
K
where
)m. In a world where all are
2 (0; 1), we can show that x ( m; (1
xK ( ) =
A
2
m+
p
The derivative of xK with respect to
Am
dxK
=
1+
d
2
2
2 m2
)m;
)m; ) < mA, and it is increasing in
+ 4(1
)m
:
)m < mA
is positive for all
m2 + 4(1
= 1) = mA. In a world
1=2
2 (0; 1):
( m
2) > 0
This expression has positive sign i¤ the following inequality holds:
2
m2 + 4(1
)m
1=2
>2
m
If m > 2 then the above inequality is clearly satis…ed. If m < 1; then 4m
4m
4
4 since m
4 m>
2:Therefore the above inequality is always satis…ed.
Proof of Proposition 5
Under both equilibrium, the contribution of a Nashian is always xN = A. Kantians
contribute more under the inclusive Kant-Nash equilibrium in comparison to what they
contribute under the exclusive Kant-Nash equilibrium. Are they better o¤ under the
inclusive Kant-Nash equilibrium?
Under the E-K-N, contribution by a Kantian is
xK
E (k; m) = Ak
29
and total contribution is
QE = (Ak)k + An
Welfare is
2
k 2 + (m
uK
E (k; m) = A
A2
(k)2
2
k)
Under the IKNE, contribution by a Kantian is
xK
I = A
k+
p
k 2 + 4n
2
!
> Ak if n > 0
and total contribution is
QI = A k
k+
p
!
+n
!
1
2
k 2 + 4n
2
!
Thus
2
uK
I (k; m) = A
k
Then, uK
I (k; m)
k
=
k+
p
k 2 + 4(m
2
k)
!
+ (m
k)
A
k+
p
k 2 + 4(m
2
k)
!!2
uK
E (k; m) is proportional to
k+
p
k 2 + 4n
2
1 p 2
k k + 4n
4
1
n
2
!
+n
!
1
2
k+
p
k 2 + 4n
2
!2
k2 + n
1
(k)2
2
1 2
k
4
The RHS is negative i¤
p
k k 2 + 4n < 2n + k 2
i¤
k 2 (k 2 + 4n) < 4n2 + k 4 + 4nk 2
which is always true. Thus, in an I-K-N equilibrium, Kantians are worse o¤. The total
30
loss to all k Kantians are
1 p 2
k k + 4n
4
1
n
2
1 2
k k
4
Nashians are better o¤,
UEN (k; n)
UIN (k; n)
Their gains are proportional to
k
k+
p
k 2 + 4n
2
k+
p
k+
p
=k
!
+n
!
(k 2 + n)
!
k2
k 2 + 4n
2
The total gains to all Nashians are
n k
k 2 + 4n
2
!
k2
!
Are the (across-regime) gains to Nashians more than compensate for the losses to all
Kantians?
n k
=
k+
1
k 2n
4
!
k 2 + 4n
2
p
k k 2 + 4n
p
k2
!
+
1 p 2
k k + 4n
4
1
n
2
p
2n k 2 + 4n + 2kn + k 2
The RHS is positive i¤
2n
i¤
p
k k 2 + 4n
p
2n k 2 + 4n + 2kn + k 2 < 0
p
2n + 2kn + k 2 < (2n + k) k 2 + 4n
i¤
2n + 2kn + k 2
2
< (2n + k)2 (k 2 + 4n)
31
1 2
k k
4
i¤
(2n + k)2 (k 2 + 4n)
i¤ 4n2 (2k + 4n
2n + 2kn + k 2
2
>0
1) > 0, which holds always!
Proof of Fact 2
(1)
The derivative of VK (S) with respect to k is
@VK (S)
@VN (S)
=
@k
@k
=
1+g
1
(m
@
1
@k 1
ln
k
(m
1
(1
)
k)(1
)(1 b) + (1 + g)
1
1
k)
1
k
1
(m
k)
which is positive i¤
[(1
)(1 + g) (1
(m
k)) k] + ( m
1) [(m
The …rst term is strictly positive always since 1
second term is non-negative if m 1 0.
(m
k)(1
)(1
b) + (1 + g)]
k) > 0 by Assumption A1. The
Proof of Fact 4
SW = (m
= (m
= mVN
k)VN + kVK
k)VN + k VN
k
1
ln
1
1
1
k
(m
ln
1
k
(m
k)
k)
Then
dSW
dk
dVN (S)
k
1
k
1
m
+
ln
2
dk
1
1
1
(m k)
(1
(m k))
1
k
1
m
1+g
= m
+
1
(m k)(1
)(1 b) + (1 + g) 1
(1
(m k))2
1
k
ln
> 0 if g is su¢ ciently large.
1
1
(m k)
= m
Proof of Fact 7 The pure Nashian equilibrium emission strategy in the q–period
32
horizon game is
xP N (q) =
1
S
Pq 1 s
q
1) + (1 + g)
b
+
b
s=0
(m
and the value function is
V P N (q) (S) =
q 1
X
(1 + g)
bs
s=0
!
+ bq
!
(q)
(q)
ln S + AP N + BP N +
q
where
(q)
(q)
PN
AP N =
(q)
PN
with
ln
(q 1)
PN
+
(q)
with
(q)
PN
(q)
PN
+
(1 + g)
(m
(q 1)
2 (q 2)
PN +
! PN
q 1
X
s
q
b
+b
s=0
+ +::: + ,
!
1 ln
(m
Taking the limit as q ! 1; we obtain
xP N =
VP N (S) = (1 + g)
+
1
1
(m
1
1
b
ln
q 1 (1)
PN
(m
!
Pq
1 s
s=0 b
+ bq 1
Pq 1 s
+ bq
1) + (1 + g)
s=0 b
(1 + g)
1
1
(m
ln
(m
1 b
1)(1 b) + (1 + g)
g+b
1)(1 b) + (1 + g)
The stock will converge to its steady state level, given by
SP N =
q 1 (1)
PN,
1 b
S
1)(1 b) + (1 + g)
ln S +
g+b
1 b
+ :::
1
Pq 1 s
1) + (1 + g)
+ bq
s=0 b
and
BP N =
2 (q 2)
PN
+
g+b
1)(1 b) + (1 + g)
33
1
!
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