The Tragedy of the Commons?
Author(s): Prajit K. Dutta and Rangarajan K. Sundaram
Source: Economic Theory, Vol. 3, No. 3 (Jul., 1993), pp. 413-426
Published by: Springer
Stable URL: http://www.jstor.org/stable/25054712
Accessed: 21/10/2010 13:56
Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at
http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless
you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you
may use content in the JSTOR archive only for your personal, non-commercial use.
Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at
http://www.jstor.org/action/showPublisher?publisherCode=springer.
Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed
page of such transmission.
JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of
content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms
of scholarship. For more information about JSTOR, please contact support@jstor.org.
Springer is collaborating with JSTOR to digitize, preserve and extend access to Economic Theory.
http://www.jstor.org
Econ. Theory 3,413-426
(1993) "^ ?
Economic
Theory 1993
?
Springer-Verlag
The tragedy of the commons?*
Prajit K. Dutta1
1
2
Department
Department
November
Received:
K. Sundaram2
and Rangarajan
of Economics,
of Economics,
22,1991;
New York, NY
10027, USA
University,
NY
of Rochester,
14627, USA
Rochester,
Columbia
University
revised
version March
16,1992
of the set of Markov-Perfect
provide a complete characterization
resource
of
games a la Levhari and
(MPE)
Equilibria
dynamic common-property
We
MPE
Mirman
find
all
of
exhibit
that
such
games
(1980).
remarkably regular
Summary.
We
nature,
however, and despite their memoryless
dynamic behavior. Surprisingly,
MPE need not result in a "tragedy of the commons",
of the
i.e., overexploitation
resource relative to the first-best solutions. We show through an example that MPE
of underexploitation
of the resource.
could, in fact, lead to the reverse phenomenon
we
are
demonstrate
Nonetheless,
that, in payoff space, MPE
always suboptimal.
1 Introduction
It is commonly
that when
several agents simultaneously
believed
exploit a
resource, the resulting externality will lead to overuse
productive common-property
or overexploitation
of the resource relative to the first-best situation, a phenomenon
the tragedy of the commons. The object of this paper is to
sense, if any, this view may be considered valid.
At least implicitly, the reasoning behind the existence of a "tragedy" involves an
to enforce
that players are not using history-dependent
strategies
assumption
is a phenomenon
"good" outcomes. In game-theoretic
terminology, overexploitation
to be associated with Markov-Perfect
of common-property
Equilibria or MPE
resource games. If players are conditioning
their equilibrium behavior solely on the
that has been
termed
in what
examine
current "state" of the game, but not on the history
*
The
first
University;
was a graduate
Sundaram
student at Cornell
when
this paper was written
was
the California
when
he was visiting
Institute
of
version
completed
to
in
are grateful
like
to many
and
would
for
their
thank,
advice,
particular,
people
version
the
that led to that state, the intuition
of
current
We
Technology.
and Karl Shell. We
Andreu Mas-Colell,
Mukul Majumdar,
Tapan Mitra, Debraj Ray, Aldo Rustichini,
at presentations
of participants
in Columbia
from the comments
also benefitted
(Fall 1988)
University
Partial support for this project was provided
and Caltech
by
(Spring 1990), and at various conferences.
the NSF
under Grant
86-06944
(principal
investigator:
Karl
Shell).
414
P. K. Dutta
and R. K. Sundaram
this reasoning appears compelling. Yet we show in this paper that, sur
prisingly, it is also false.
in a slight variant of the Levhari
In Sect. 3 of this paper, we derive an MPE
War"
Mirman
"Fish
in
which
of the resource
(1980)
example,
underexploitation
occurs from an interval of initial states. No nonconvexities
or pathological properties
the construction
is based on a simple point whose
drive this example. Rather,
behind
resource
importance may extend beyond the current context. Common-property
as
are
indeed
all
stochastic
from
games,
games,
distinguished
repeated games by the
a
state
asset - that moves
of
variable
in
this
the
stock
of
the
case,
presence
through time in response to the players' actions. The last property indicates the use
of the state as a proxy for history, albeit a limited one. In turn, this enables indirect
on history even under Markovian
behavior,
although direct con
conditioning
ditioning may be precluded.1
Details of the formalization
of these ideas may be found in Sect. 3. Here, we will
content ourselves with the observation
that this construction
appears to extend
we
shown
have
that under
similar
using
quite generally. Elsewhere,
techniques
resource
MPE
exist
in
all
games in which
symmetric common-property
exploitation
not
do
discount
the
future.2
players
If a "tragedy" (which is a positive feature of equilibrium play) need not occur
under MPE,
the question naturally arises if, normatively
speaking at least, MPE
are suboptimal. We turn to this and related questions
in Sect. 4, where we provide
an exhaustive
resource
in general common-property
characterization
of MPE
games, without any functional form restrictions.
exploitation
Briefly put, our findings are the following. Under nothing more than the usual
restrictions, all MPE exhibit remarkably well-behaved
convexity
dynamics, with
state trajectories from any initial state eventually becoming monotone
(Theorem 1).
If the state trajectory meets a certain "asymptotic
criterion, then it
convegence"
must lead to a "tragedy" (Theorem 2). While a "tragedy" may, or may not, occur
true that in payoff space all MPE are strictly sub
it is, nonetheless
optimal (Theorem 3). Indeed, Theorem 3 is an immediate consequence of Theorem 2
and the properties of the first-best solutions.
As a final question, we examine why the many
that have been
examples
studied in the literature3 have, by and large, tended to "confirm" the presence of a
tragedy. We show that general sufficient conditions on MPE
strategies exist that
always lead to a "tragedy" (Theorem 4). Many examples in the literature satisfy at
under MPE,
least one, and typically more, of these conditions.
resource games.
Section 2 presents a general framework of common-property
of the Levhari-Mirman
Section 3 is concerned with our modification
example.
1
our work was independent, we discovered
that this idea was not altogether
original. A similar
Although
the arguments
in Fudenberg
idea motivates
and Tir?le
(1983).
2
the arguments
and Sundaram
makes
See Dutta
(1992b). The absence of discounting
considerably
simpler, since any finite number of periods are irrelevant under the long-run average criterion.
3
and Lewis (1985),
See, e.g., Lancaster
(1979), Levhari and Mirman
(1980), Easwaran
(1973), Mirman
or Reinganum
and Radner
and Stokey
(1985). See also Benhabib
(1990), who characterize
equilibria
when
players
have
linear utilities.
The
415
of the commons?
tragedy
Section 4 presents
the Appendix.
the remaining
results
listed above. All proofs
are to be found
in
2 The framework
2.1 The dynamic
resource game
we consider
The model
here is closely
related
to those in Lancaster
(1973), Levhari
and Mirman (1980),Reinganum and Stokey (1985),Amir (1987),Benhabib and
and Sundaram
Radner
(1988), Dutta
(1992a, b), and Sundaram
(1989), among
others. There is a single good, the "resource", which may be consumed or invested.
Conversion
of investment to output takes one period and is accomplished
through
a production function f: R + -? R +. In each period t of an infinite horizon, two4 agents
observe the available stock yt ^ 0 of the good, and independently and simultaneously
decide upon their consumption
denote player
plans for the period. Let aite[0,yj
i's planned period-i consumption.
If plans are collectively
feasible (au + a2t ^ y\
then they are carried out, and player i's (actual) consumption
cit equals his planned
infeasible (au + a2t > yt\ then we simply
ait. If plans are collectively
consumption
assume that each player receives half the available stock.5 The amount x,e[0,)>f]
left over after consumption
by the players forms the investment or savings in period
t and is transformed to the period-(i + 1) available stock yt+1 as yt+ x=f(xt). There
entire process now repeats itself the new stock level yt+x. Player i's utility in period
t is a function only of his own consumption
cit in the period, and is given by wf(cir).
Both players discount future utilities by the factor <5e(0,1), and attempt tomaximize
total discounted utility over the infinite duration of the game.
on the functions involved.
We make the usual neoclassical
assumptions
1. f
Assumption
is differentiable
/
is continuous,
on R++,
with
increasing
lim/'(x)
>
and strictly
concave
on R+; /(0) = 0;
= 0
l/<5, lim f'(x)
x|0
jc?oo
are continuous,
2. The functions nf:R+-^R
Assumption
concave on R+, and continuously
differentiable on R++,
increasing
i= 1,2.
and
strictly
that by the assumptions
of/, there is a unique stock level x > 0 such that
x
x
x ??x. Therefore, / maps S = [0, x] into itself,
x
x
for
and
for
^
f(x) ^
f(x) ^
loss of generality, restrict analysis to S. We also refer to S as
and we may, without
the state space of the game and to a generic element yeS as a state.
The tuple {S,fu1,u29?}
completely
specifies the dynamic resource game. We
Note
proceed
4
to define
strategies,
payoffs,
and MPE
to this game.
case purely for notational
to the two-player
simplicity.
As the reader may readily check, none of our general
of this rule is of no significance.
are instead determined
state y, allocations
if, in the event of infeasible plans (al,a2)ata
analysis changes
=
that satisfy 0^hl{y,alya2)^hl{yialia2)-\-h2(y,al,a2)
y. Indeed,
(hlth2)
by any pair of functions
For instance,
is never an equilibrium
possibility.
(hx, h2) may also be chosen to ensure that infeasibility
=
i^j, i,j = 1,2.
whenever
+ a2\
ax + a2 > y, let ht(y, a1,a2)
a,y/[ai
5
We
restrict
The
specificity
attention
416
P. K. Dutta
2.2 Markov-Perfect
and R. K. Sundaram
in the game
equilibria
In general, a strategy for a player i specifies a (planned) consumption
level for ?after
since our interest in the sequel is solely
each possible history of the game. However,
to
inMarkovian
strategies, we avoid spurious generality, and restrict definitions
case.
no
this
Standard dynamic programming
loss in strategic
arguments show that
to a Markovian
flexibility results, since a player responding
strategy will always
have a Markovian
best-response,
if at all any best-response
A (stationary) Markovian
Definition.
that satisfies g(y)e[0,y~\
for all yeS.
strategy
exists.
for either player
is a function g:S-+S
A player's Markovian
at any state
strategy g specifies his planned consumption
yeS, as g(y), regardless of when, or how, y was reached. Let T be the set of all
Markovian
indices to distinguish
strategies. Henceforth, we will use player-specific
strategies.
A pair of Markovian
strategies (gx, g2) defines in the obvious manner, from any
?
initial state yeS, a unique history of t
th period stock levels, and period-i rewards
=
i
for
1,2.
Hence,
player
(gX9g2)
uniquely defines the total discounted
u\(gx,g2)(y)ao
for player i.Denote
this total reward by Wi(gx,g2)(y).
reward
?f= ?tu\(gx,g2)(y)
0
Definition.
The strategy g\sT
is player
l's best-response
(BR) to player 2's strategy
02erif:
Wx(gx*,g2)(y)^
A BR of player 2 to gxeT
is similarly defined.
A Markov-Perfect
>
such
that for i,j =1,2,
(91 Qi)
Definition.
Wx(gx,g2)(y)VyeS,Vgxer.
Equilibrium (MPE) is a pair of Markovian
and ?#;, gt is a BR to gy
Remark. The phrase "Markov-Perfect"
arises from the simple observation
Nash equilibria inMarkovian
strategies are also subgame-perfect.
strategies
that all
If an MPE
(gx,g2) satisfies the condition gx(y) + g2(y) ^ y for all yeS, we will
as
an
to
interior MPE. Dutta and Sundaram (1992a), and Sundaram (1989),
refer
it
=
have shown that under the additional
of "symmetry"
(ux u2), an
assumption
interior symmetric MPE
(g,g) always exists if utility functions obey the Inada
condition. Surprisingly, no one has yet (to the best of our knowledge)
established
a corresponding
result for the asymmetric game, although the examples of Levhari
and Mirman
(1973) admit such equilibria without
(1980) and Lancaster
imposing
are trivial to
player-symmetric
preferences. Of course, non-interior
equilibria
=
=
an MPE).
construct
In any
y for all yeS constitutes
(for example, gx(y)
g2(y)
event, the results of this paper hold for interior and non-interior
equilibria alike, so
this is not an important issue.
2.3 The first-best
solutions
From any initial state yeS, the set of all first-best payoffs that can be achieved are
those arising from a solution to the following problem as a varies between 0 and 1:
The
tragedy
417
of the commons?
Maximize l<*Wl(gl9g2)(y)+ (1- a)W2(gug2){y)l
(01.02)
is a well-understood
we
problem in neoclassical growth theory. Consequently,
confine ourselves to stating, without proof, its relevant characteristics.
For each ae[0,1],
there is a unique pair of Markovian
strategies (gla,g2a) that
solves this problem from each 0 < yeS. The resulting sequence of states {y*} from
for all t or y*+ ?^ yf?^ y for all
y under this solution ismonotone
(i.e., y*+i^y\^y
a
to
state
and
This
the so-called "golden rule",
converges
r),
steady
steady-state,
j?.
is independent of the choice of y and a, and is the unique solution to f(x*) = y*9
where <5/'(x*) = 1. Finally, if y # y*9 then we also have y* # y* for any i, that is {y*}
converges to y* only "asymptotically".
This
2.4 The "tragedy of the commons" defined
Fix a discount factor <5e(0,1). Since the golden-rule depends only on S (and /), but
not on a or the initial state y9 obvious definitions
of "overexploitation"
and a
and the
"tragedy" suggest themselves. Let (gl9g2) denote an arbitrary MPE,
sequence of stock levels generated from an initial state yeS under (gl9 g2) be denoted
by {yt(y)Y
Definition.
The MPE
The MPE
Definition.
from all y > 0.
(gx, g2) leads to a "tragedy"
(gug2)
from y if limsup yt(y) < y%.
t-+oo
leads to a "tragedy"
if (gi9g2)
leads to a "tragedy"
occurs from y if the sequence of states from y under
overexploitation
some
(gug2) is, beyond
point, strictly bounded below y$; and a "tragedy of the
commons" occurs if overexploitation
results from each initial state.
In words,
3 An example
This section presents an example of an interior MPE which sustains a steady-state
z lying strictly above the golden-rule. Our example is based on a modification
of
=
=
=
assume
and
who
that
Levhari and Mirman
x", ae(0,1);
ux (c) u2(c)
(1980),
f(x)
that we may take the state space S to be [0,1].
loge. Observe
Levhari and Mirman
show that there is a symmetric interior MPE
(g9g) to this
?
?
=
=
game in linear strategies specified by g(y)
olS).
(1 a<5)/(2
ky for yeS, where k
to which all
The unique non-zero steady-state of the game under this equilibrium,
stock sequences
converge,
is given by
W
?= ?-A
The "golden
rule" j?, in this case is
tf
It is readily
golden-rule
=
(a<5)a/1-a. (2)
that y*>y?For later use, we note
=
aa/1 "a.
y* is given by lim y?
checked
that the undiscounted
418
P. K. Dutta
and R. K. Sundaram
informal comments on the procedure we employ may be useful. The idea
is one of players getting "locked" into a steady-state
underlying our construction
a discontinuity
This
in equilibrium
results
from
y'>y*.
locking
strategies at /:to
the left of y', the strategies specify consumption
levels well above those required to
maintain
over the
If either player increases his consumption
y' as a steady-state.
Some
levels at /, this results in a lower stock level, a sharp increase
proposed steady-state
in consumption
at the new stock, and consequently
rapidly declining stocks. This
an
makes deviation from the proposed
unattractive
steady-state
option when ? is
high enough.
The feasibility of this procedure
in a truncated version of the
is demonstrated
Levhari-Mirman
note
We
that
the
truncation does not affect y%which
example.
remains the unique steady-state of the first-best solutions. The reader is urged to
consult the accompanying
Fig. 1when going through this section.
To begin with, let / be given by
Since >? ? y* as ? ?1, and a(2
Consider
the truncation f(x)
game continues to have y$ as
Let
converge asymptotically.
(3)
'-(2=;)
a) < Ifor ae(0,1), we have y'>y*>
y% for all <5e(0,1).
=
y' for all x = (/)1/flt. For all <5e(0,1), this truncated
the unique steady state to which all first-best solutions
c' be half the consumption
that permits
regeneration
of/:
c'=
\1/1-a
=1 1/1
(1~a)' (4)
2{y'~{y)ll*) 2\2^~J
Now define themodified strategies (?,?) by
=
0()>) Ay,)>G[O,/)
=
+ 2c'y2,ye[y'M
l(y-y')
shows that, for any initial state y > y', these strategies ensure
Simple calculation
that the game moves
to y' in one step. We will now show that (?,?) is an MPE of
this game.6 Observe
that by construction / is a steady-state of this MPE
that lies
above the "golden-rule" y*.
take
Suppose the initial state is ye[0, y'). Given ?, neither player can unilaterally
the stock to y' in a finite number of periods. For, even with zero consumption/oreuer,
player
i faces a net production function
f(y)
=
y*(\-ky
?
sustainable
stock is [2
can have no effect
ye [0, y'), the modification
a best-response
to itself for all initial states
stock
Thus, if the game starts in [0,/),
whose maximum
6
The
obvious
modification
of the Levhari-Mirman
"a/1 "a <
<x<5]
/. Hence, for initial states
on the game, and consequently
# forms
in this segment.
levels eventually converge, under (?,?\
strategies
is another
MPE.
The
tragedy
419
of the commons?
. y=x
=
=
= Investment
of unique steady-state
(a
3/4; ?
20/21). A x
Fig. 1A, B. The Levhari-Mirman
example
of the game; x* = investment at unique steady-state
of first-best solution; x = investment at undiscounted
= investment
= Same as in
levels at the low and high steady-states
A; x and x'
golden rule. B x* and x
of the game
of the game, respectively; / = stock level at high steady-state
420
P. K. Dutta
to y?, where
approaches
the associated
consumption
the limiting value c, given by
and R. K. Sundaram
level cx? = c2?( = c?) = ?y?. As
c6
?jl,
(5)
a^-^-ar^-^l-a)
can be taken as an approximation
to c? for "high" ?.
it
is
that
for high a, c' > cb whenever \ > aa/1 "a,
and
evident
(4)
(5),
Comparing
the proposed steady-state
which is satisfied for an interval of values of a. Therefore,
a
has
level
than
the
/
strictly greater consumption
steady-state y?. If the game ever
which
enters [0, /), then under the strategy profile (0, ?) there is a finite number of periods
is
T, independent of S and the starting state in [0,/), beyond which consumption
over
For
less
than
d.
all
the
T
first
gains
uniformly strictly
sufficiently high ?,
periods
are swamped by the strictly smaller consumption
thereafter.7 For high 6, therefore,
either player is, given 0, better off also adopting ?jwhen the initial state is in [/, 1].
to itself on all of S.
Summing up, ? is a best-response
a stationary MPE
We have thus demonstrated
in which there is a steady-state
lying strictly above
4 A characterization
the "golden-rule."8
of MPE
We now turn to a characterization
of the properties of MPE of general dynamic
resource games. Throughout
the rest of this paper (gX9g2) will denote an arbitrary,
but fixed, MPE; oe(0,1) will be a fixed discount factor; y will denote a non-zero
initial state in S; the sequence of states from y under (gx,g2) will be denoted
will be denoted by Vi9 i=1,2;
by {)>,()>)}; the payoff functions Wi(gx,g2):S->R
=
the
under
function"
and, finally,
(gl9g2) will be denoted
\?/9i.e., i?/(y)
"savings
m*x{09y-gx(y)-g2(y)}.
state paths - is an important one
Our first result
the monotonicity
of MPE
in the sequel. Among other things, it enables
for it simplifies analysis considerably
us to replace the "limsup" in the definition of a "tragedy" with a plain "limit".
Theorem
monotone
1. For all y>09
there is an integer T(y) such that the sequence {yt(y)}
either yt+x(y)^yt(y)
i.e.,
for t^T(y),
for t= T(y), or yt+x(y)^yt(y)for
is
t^T(y).
Remark. We
note
that the convexity
of /
is not needed
for this result.
there is a unique element
1, for each yeS,
By Theorem
set
Let
Z
the
such limit points:
denote
of
all
yt(y)-+y(y)'
Z = {yeS\y
=
y(y)
=
y(y)eS
such
that
Mmyt(y) for some yeS}.
7
An alternative, more
intuitive, way to think about this is to note that as ?j 1, players care only about
c' > c, and c is larger than
their eventual
levels for each a; by construction
(i.e., steady state) consumption
the steady-state
that results from initial states in [0,/)
for any ? < 1.
consumption
8
to those we use in this section can also be employed
to support a
We suspect that similar methods
host of other points as steady-states
of (discontinuous)
at any rate, appears roughly the same as used here.
MPE,
for instance
points
ze(y?, y*). The
intuition,
The
tragedy
421
of the commons?
For notational
convenience, we will sometimes denote a generic element of Z by z
rather than y(y). The occurrence of a "tragedy" now reduces to examining whether
z < y* for all zeZ. Our main result on this question - that, as the example of the
cannot be strenthened
is that only a partial result
previous section demonstrates,
along these lines is valid:
Theorem
2. Let zeZ.
If there is yeS
such that yt(y)^>z9 but yt(y) ^ zfor any t, then
z<y*.
2 implies, almost immediately,
is strictly suboptimal
that any MPE
Theorem
from all initial states except, possibly, yj. For, recall that from any initial state
y?^y*, the sequence of stocks in any first-best solution converges asymptotically
some initial
to the golden-rule
y%. If (gi9g2) is to generate optimal payoffs from
state y #)>*, then the sequence {yt(y)} will have to coincide with the sequence of
2 shows that this is not
states from y in some first-best solution. But Theorem
to their limits would
if
then
the
did
coincide,
convergence
sequences
possible. For,
be only asymptotic. By Theorem
must be strictly below y%9while
It immediately follows that:
Theorem
3. The MPE
except, possibly,
2, this implies that the limit of the sequence
the first-best sequence, of course, converges
(gl9g2) generates
suboptimal payoffs from
{yt(y)}
to >?.
all initial states y
y%.
As a final question, we examine general conditions on the equilibrium strategies
for this question
(0i ?g 2) under which a "tragedy" is inevitable. Part of the motivation
in
to understand why constructed
in the Introduction,
is, as explained
equilibria
We
show
that:
exhibit
the
literature
in
studied
overexploitation.
always
examples
4. Suppose (gl9g2) is interior, and satisfies one of thefollowing:
2.
g
(i) i is continuously differentiable onS9i=l9
on
is
i.
S
both
g{
increasing
(ii)
for
(iii) i//(y) is strictly increasing on S.
Then (gl9 g2) leads to a "tragedy" from all y>0.
Theorem
Appendix
The proof of the results are derived
as consequences
of various
lemmata. We
begin
by defining
S(l)
=
=
> 0}.
{$eS\ j> yt(y) for some yeS9 and \?/($)
That is, S(l) is the set of states that (i) can be "reached" from some state in S under
for the game. Clearly,
the MPE
(gl9g2)> and (ii) ensure a non-trivial continuation
S(l) is the relevant set for dynamic analysis of (gx, g2):\? $tS(\)9 then either $ cannot
be reached from any other state and is therefore irrelevant after period 1, or $ can
be reached but the continuation
from p is trivial.
Lemma
1. On S(l),
thefollowing
(i) y'<y
(ii) Fl?O<Fl(y)
(iii)?-gjt/)<y-gj{y).
conditions
are equivalent:
422
P. K. Dutta
each of these implies \j/(y)^ ^(/).
Further,
and R. K. Sundaram
If the last inequality holds strictly,
then
(i)-(iii)must hold.
Proof (ii)=>(i) Suppose fory, y'eS(l) we had V^y)> V^y')but y < y'.Then / could
not be in S(l): for, player iwould prefer to go to y which can be achieved through
a unilateral increase in consumption
over the level required to reach / (thus gaining
in immediate
and
in the
with
reward),
V^y) > V^y'), i does
strictly better
continuation
also.
i's optimal actions at these
(i) => (iii) Suppose y, y' eS( 1) and y > y'. By hypothesis,
?
states do not terminate
set of
If / ?
the game.
g}(y') ^ y
gj(y), then the
at
at
is
in
that
Thus
the
actions
contained
continuation
from
y
non-terminating
y'.
as
as
to
at
least
is
the
But
continuation
from
reach
rather
than
y.
y'
y'
(for i)
good
a
or
a
current
current
in
unilateral
increase
y requires only
greater
by i,
consumption
reward. This implies that y$S(\), a contradiction.
actions at y contains those at y', and by
(iii)=>(ii) The set of non-terminating
sets
these
the
contain
hypothesis,
optimal actions.
from a standard argument
in intertemporal
This
follows
(m)=>il/(y)^\j/(y')
of
that
the
allocation
strict
ut. See, e.g., Sundaram
theory
concavity
exploits
converse
Lemma
The
is
the
for
established
details.
(1989,
II.4)
by reversing
argument.
QED
Proof of Theorem
Define
T=
1
inf{i|>^f(jv)
=
follows, we suppress
0}. In what
the argument
y.
for all t^ 1. Suppose for some t, we had
(?) T= oo. By hypothesis
yteS(l)
=
all
then
then by Lemma
for
+
1,
yt i?yt
yt
fe^O. If yt+i>y9
clearly yt+k
>
turn
in
the
that
monotonicity
off,
i?/(yt+J ^ \l/(yt) or, by
Vi(yt+1)
^?(.Vr) implying
that yt+2 ^ yt+1. Thus j;t+1^}'(=>)'f+2^};f+1.An
analogous argument establishes
for
that whenever yt+1 ^ yt, we must have yt+2 = yt+i' Therefore, yt ismonotone
t^ 1, and T(y) = 1 suffices.
Case
Case (ii) T is finite. In this case, yt(y) = 0 for all t^ T, so that T(y) = T obviously
satisfies Theorem
1. QED
Proof of Theorem
2
For ease of notation,
Lemma
2. 3ci9c2
we suppress
such that c?=
the dependence
on the initial state y.
lim gi(yt).
f-?oo
Proof. For convenience, we prove the lemma for the case yt > yt+x > y, yt[y. The
other case yt+i < yt < y, yt]y9 is handled analogously.
Suppose for some i, there
were subsequences
t
of
that
such
tk91?
lim gAyJ = at >bt=
Pick a subsequence
lim g^yj.
?f(k)of t such that i/(k)> tk for all k. For all i, yteS(l)9
and for
The
of the commons?
tragedy
423
each k
yt,w<ytk
Therefore,
yt,(k)-g?yt,(k))<ytk-gAytk)
by Lemma
?
or ?biS
limits as k-> oo now yields
- lim
lim
y
S y
gi(ytk)
k-?oo gi(ytak))
k-*oo
1.Taking
at; a contradiction
establishing
the lemma.
QED
Now definefa :S- S byfa( y) = #?(y) for >>
/ ? fa( y) = c{.Recall thatf(x) = y, and
note
?
that
y
?
cx
c2
=
x.
Define
=
W?(*) "<(* ^ *) + &*i(/W ?j(f(x)) x)
W\(x)
=
ut(yt
-
-
?j(yt)
x) + ?Wi(/(x)
-
-
^.(/(x))
xt+J,
are taken to be such that the implied consumptions
are
the domains
since x and xt are respectively
in the
(The domains are non-empty
non-negative.
domains of W? and W\.) The two objectives are nothing but the returns to two period
deviations which leave the evolution of the game unchanged after these two periods.
We now prove,
where
is T such that (i) x, (respectively x) is a feasible deviation at y
t
(respectively yt), ^ T, and (ii)W?(x) ?>Wf(xf) (respectively, W\(xt) ^ W\(x))91 ^ T.
=
=
1,2, whenever y > 0. (if y
0, Theorem 2
Proof We begin by showing that c?> 0, i
is trivial, so we ignore this case.) Since player i always has the option of making
the
?
in
it
for
feasible consumption
follows
that
all
r,
f,
yt
period
gj(yt)
Lemma
3. There
Vt(yt) = ut(yt
-
gj(yt)) + iM|(0)/[l
-
?].
^uf(xf) + aM0)/[l-<5].
1, {xj is a monotone
sequence for t^ 1, and since y > 0, x > 0. Let
By Lemma
0 = inf xt = min {xx, x}. Then, 0 > 0, so we have for all i,
?
r?fo) ^ M0) + ?Wf(0)/[1 ?] > ?W?(0)/[1 <5].
If &&)
-
0> then for all e > 0, there is 7(e)
(A.1)
such that for ?^ TOO,
K^XMeVCi-?].
a
-a],
(A.l) implies that there is rj>0 such that for all r, Vt{yt)^wf(^)/[l
This proves that Cj, c2 > 0.
contradiction.
?
Since yt9 gj(yt) converge to y, c, as ?- oo, it is now immediate that yt ? gj(yt)
?
?
x > 0 and j> c,
x, > 0 for t sufficiently large. This proves (i).
Observe
in
that,
fact, xt is a feasible deviation at yt for t sufficiently large, since
But
in period
xt is optimal
By hypothesis,
- oo establishes
t
as
the
second
limits
Taking
xt-+x.
i, so W](xf)^ W\(xT) for all such t.
inequality in (ii). The first inequality
424
P. K. Dutta
in a similar fashion:
may be established
have
ifwe had Wi(xt) > W{(x)
and R. K. Sundaram
then we must
also
0;O>t) *t) + aut(yt + ! gj(yt +1)-xt+1)
> Ui(yt *t+1)
xt) + ?M,(yt+1
ty(y.+i)
g?{yt)
"i^t
-
the optimality
since yt, 0/O\), xt->j>, c,-, x, respectively, as t-> oo. This contradicts
to g}) in period t. QED
of xt (in i's best-response
=
=
1,2. We shall show that
Next, let fcj [?i &(}>,)]/[}>
yj, for i
?
=
Lemma 4. <5/'(x) (1
1,2.
limsup k\) ^ 1,;
From Lemma
Proo/.
By the Mean
3,
?
?
?
?
there is y'te(y,
x? y,
x) such that
g?(yt)
g?{yt)
x)
u?(yt g?(yt)
xt)
_
9j(yt)
Value Theorem
t*i(yt
?
x,
there is ?|e(/(x)
Also,
x
)- xt +1) and x,e(x,xt)
-Cj-xt+1,yl+1-g?(yt+1
JM)Wi-3j(>Wi)-*,+i)-?
' '
L
L *r * J
*,-*
(/(*)-?j-*,+i)l
Cj
xt-x
y,+i~y
So for all large t, ?f'(xt_ t)(l - *})? ?'?(yp'W
But y[, ?j -* y
')
x as t-* oo, so the RHS goes
J
to 1 as t -* oo. Taking
<5/'(x) 1 lim supfc' ^ 1.(A.2)
k})^
For
the case yt\y, we use from Lemma
Identical
Lemma
as above
arguments
5. /'(x)(l
-
3 the fact that for all large i,
then yield (A.2).
? lim
lim sup k\
sup k'2)^
t
t
QED
1.
=
LetH{y,)
yt+1-yrThen,atheTH{yt)<0,H(yt)-*0,orH{yt)>0,H{yt)^0.
?
?
=
0, we have, in either case,
y
cl?c2)
Defining H(y) =f(y
Proof.
^-"^P
yt-y
such that
for all,
limits
The
tragedy
425
of the commons?
or, for all t:
yt~$
f(yt-gi(y?)-Qiiyt))-f(y-?i-?2)
Therefore,
rewriting
this inequality, we obtain
if(xt)-fix)\/
and taking
Write
k{
=
gx(yt)-cx
g2(yt)-c2\
yt-y
yt-y
xt-x A
\
<0
yt-y
yt-y
limits as t -
oo yields
lim sup k\. By Lemmas
t
the lemma.
^ {
)
QED
4 and 5, the following
inequalities
hold
?ff(x)(\-kx)^f\x)(\-kx-k2)
?f(m-k2)^ff(x)(l-kx-k2).
can hold simultaneously
only if kt > 0 for some iwhich implies that Sf'(x) >
therefore x < x* by the concavity of/, proving Theorem 2. QED
These
Proof
of Theorem
1,
4
Suppose g{ isC1 on S. Let y > 0. By hypothesis ^(y) > 0. Let P=f(\?f(y)) > 0. Since
g i is a best-response
to gi9 so ^(y) solves
max {uj(y- gt(y)- x) + ?Uj(f(x) - gt(f(x)) - <A(j>))}.
X
Since
this is an interior maximum,
the first-order
ufjigj(y)i=Mjtijmri'Hyn
Clearly
We
(A.3) holds
now prove
=
Hy) W\
conditions
a mi
=
1,2, for all y > 0 where j)=/#()>)).
for;
that i// is strictly increasing on S. Suppose
Then, j)=/(^(y))=/(^(/))
=
f,
imply
(A-3)
for y9 y'eS we had
so the RHS's of (A.3) coincide.
=
this implies y = y'. Therefore, ^
Since \j/(y)= ^(/),
Therefore, g^y)
gj(y')9j =1,2.
on
S. Since g{ is differentiable
for both i, so is ^, therefore, ^' ^ 0 or
is one-to-one
=
> 0 for y > 0, so ^'(y) = 0 for all y = 0. Since
^ ^ 0 on S. Since ^(0) 0 and \?/(y)
so i?/ is strictly increasing on 5.
^' = 0 and ^ is one-to-one,
So (i) is a sufficient condition for (iii). Although
(i)may now be proved directly
of \j/,we give a unified proof for (i) and (iii) that only
using the differentiability
uses (iii).
Note that if \?tis strictly increasing on S, then we must have y>y'=>
\?f(y)'> \?f(y')9
so that yt(y) > yt(y')9 and y(y) _ y(y'). Further, it is clear that if an initial state y is
not itself a steady state, then yt(y) must be a strictly monotone
sequence, so finite
to show that y(y) < y*
of yt(y) to a limit is ruled out. Therefore,
time convergence
for all yeS9 it now suffices to show that there is no steady state y > y*.
=
Recall
So let z* be the sup over all steady-states,
i.e., z*
sup{z|z =f(\//(z))}.
we must have
then for ye(z*,x],
of S. If z*<x,
that x is the upper-endpoint
to y(y) only
But since yt(y) can converge
for all t, so y(y)^z*.
yt(y)e(z*,x]9
>
we
This
3.2.
Theorem
must
have
also
implies that
y(y)
by
y*
asymptotically,
426
and R. K. Sundaram
P. K. Dutta
= x. But this is absurd since we must now have
y* > z*. The only other case is z*
= O for i=
1,2. Clearly, either player could deviate unilterally and assure
0f(z*)
himself of positive consumption
in at least one period. This completes the proof for
(i)and (iii).
The argument for hypothesis
(ii) is in two steps. Let y be any steady state of the
is interior, there is a left neighborhood
of {?/(y)which
game. Since the equilibrium
is a feasible deviation at y9 i.e., x in (\?/(y)? e9 \j/(y)) for some e > 0, such that x ^ 0
?
? x and
?
?
and implied consumptions:
y
f(x)
\j/(y)9are also non
gj(y)
gj(f(x))
negative. The last claim uses the fact that g}(f(x)) ^ g}(y).
_ x)= yt ] y.
The second step is almost identical to Lemma 4.We have xt | \?/(y)9
f(xt
=
=
Write, as before, k\
1,2.
[cf &()>,)]/[)>
yt\ i
?
?
?
?
Repeated use of the Mean Value Theorem, yields r\\?(y
Cj x.y
Cj x^J
?
?
?
?
and X\e(y
x) such that
c, x, yt
gj(yt)
(A.4)
Sf'^W-k^uWIu'W
-?
-? c' ^ c?. Hence,
r?\ ch and k\
taking limits in (A.4)
Sf'mi-hmsapknzi.
The
H(y,)
remainder
=
yt+l-y,
of the proof
to get
f'(x)(
and hence
the theorem.
is identical
to that of Lemma
4, using
the function
1- lim supk?- limsup it,2 ^ 1
)
QED
References
existence
games of resource extraction:
Amir, R.: Sequential
Discussion
Paper No. 825,1987
Working
of productive
R.: Joint exploitation
J., Radner,
Benhabib,
of Nash
asset:
Foundation
Cowles
equilibrium
a game-theoretic
Econ.
approach.
Theory 2,155-190(1992)
Dutta,
P.K.,
theorems
Dutta,
R.K.:
Sundaram,
P.K.,
M.,
How
in Markov-perfect
underconsumption
Easwaran,
existence
in a class of stochastic
games:
Equilibrium
Econ. Theory
models.
2,197-214
(1992)
and
can strategic models
be? Nonexistence,
different
chaos,
R. K.: Markovian
Sundaram,
and undiscounted
for discounted
Lewis,
T.: Appropriability
J. Econ. Theory,
equilibria.
of
the extraction
and
forthcoming
a common
property
resource.
to deter mobility.
J. Econ.
Econ?mica 51, 393-400 (1985)
Fudenberg,
D.,
Tir?le,
J.: Capital
as commitment:
strategic
investment
Theory 31, 227-256 (1983)
Lancaster,
K.: The
dynamic
L.: The
Levhari, D., Mirman,
J. Econ. 11,322-334(1980)
models
L.: Dynamic
Mirman,
theory
Reinganum,
period
Sundaram,
in mathematical
J.F., Stokey,
of commitment
R.: Perfect
153-177(1989)
J. Pol. Econ. 81,1098-1109
(1973)
inefficiency of capitalism.
a dynamic Cournot-Nash
great fish war: an example using
In: Liu, P.T., Sutinen
of fishing: a heuristic
approach.
1979
economics,
pp. 39-73. New York: Decker
resource:
of a natural
extraction
N.L.: Oligopolistic
in dynamic
equilibrium
games. Int. Econ. Rev. 26,161-174
in a class of symmetric
dynamic
Bell
solution.
J.G. (eds.) Control
importance
of
the
(1985)
games.
J. Econ.
Theory
47,