Con‡ict and Renewable Resources
1
Rafael Reuveny
School of Public and Environmental A¤airs
Indiana University
and
John W. Maxwell
Kelley School of Business
Indiana University
Full Name and Address of Corresponding Author:
Rafael Reuveny
School of Public and Environmental A¤airs
Indiana University
1309 East 10th Street
Bloomington Indiana 47405-1701
Phone: 1 812 855 3345
Fax: 1 812 855 3354
Email: rreuveny@indiana.edu
Running title: Con‡ict and Renewable Resources
1
Abstract
The economic literature on con‡ict employs a static game theoretic frame-
work developed by Jack Hirshleifer. We extend this literature by explicitly
introducing con‡ict dynamics into the model. Our speci…c application is
based on two stylized facts. First, con‡ict often arises over scarce renew-
able resources, and second those resources often lack well-de…ned and/or
enforceable property rights. Our stylized model features two rival groups,
each dependent on a single contested renewable resource. Each period, the
groups allocate their members between resource harvesting and resource
appropriation (or con‡ict) in order to maximize their income. This leads
to a complex non-linear dynamic interaction between con‡ict, the two
populations, and the resource. The system’s steady states are identi…ed
and comparative statics are computed. As developed, the model relates
most closely to con‡ict over renewable resources in primitive societies.
The system’s global dynamics are investigated in simulations calibrated
for the historical society of Easter Island. The model’s implications for
contemp orary lesser develop ed societies are examined.
Key Words: Con‡ict, Dynamics, Renewable Resources..
1 Introduction
The economic literature on con‡ict can be traced back to Malthus (1798) who argued
that con‡ict over natural resources would arise as a consequence of population growth
and environmental degradation. Contemporary studies have moved away from a focus
on natural resources. Recent advances in the literature can be traced to the seminal
game theoretic model of Hirshleifer (1989). Hirshleifer-type models, including ours,
share two central features. First, there is the lack of secure individual or group
property rights. Second, con‡ict is understood as a rational activity. Actors may
devote e¤ort to create wealth through production, and/or appropriate the wealth of
rival actors through con‡ict. A third common feature of Hirshleifer-typ e models is
that they are static. We however, o¤er a dynamic model of con‡ict concentrating
speci…cally on the dynamic interplay between con‡ict and the contested wealth.
2
The economic literature on con‡ict has abstracted from modeling the underlying
source of con‡ict and con‡ict dynamics. Although this abstraction allows authors
to claim applicability to many con‡ict situations, it weakens the ultimate predictive
power of the models, giving rise to questions that the current literature fails to ad-
dress. For example, once groups are in con‡ict, does a rise in the contested wealth
lead to a lessening or a strengthening of con‡ict? How does con‡ict a¤ect (or, how it
is a¤ected by) changes in the allocated e¤ort and/or contested wealth over time? Our
dynamic model of con‡ict allows us to gain insight into these questions. We study
con‡ict dynamics within the context of con‡ict over renewable resources. However,
3
our approach could be applied to other Hirshleifer-type models.
3
Our model features two rival groups. Each group is dependent on a single con-
tested renewable resource. Each period, the groups allocate their members between
resource harvesting and resource appropriation (or con‡ict) in order to maximize
their incomes. This leads to a complex non-linear dynamic interaction between the
two populations and the resource. The complexity arises in part because of our de-
cisions to model the disputed resource as renewable and to assume that the resource
is essential for procreation. These decisions are motivated by a desire to probe the
model’s implications for con‡ict in lesser developed societies that closely depend on
the environment for livelihood.
4
In recent years, many social scientists have argued that renewable resource scarcity
(e.g., land degradation, deforestation, …sheries depletion, food scarcity, and water
scarcity) is increasingly a factor contributing to political con‡ict.
5
In the post-1945
era, con‡ict over renewable resources has typically occurred in lesser developed coun-
tries (LDCs). For example, turmoil in Haiti has been linked to deforestation (Wallich,
1994; Homer-Dixon, 1999). Land scarcity and deforestation are said to have played
a role in the 1994 Rwandan civil war, while land pressures and hunger stimulated
the Chiapas uprising in Mexico in the early-1990s (Renner, 1996; Baechler, 1998;
Brown et a., 1999).
6
Other examples involve con‡ict over scarce water. Some ob-
servers in fact argue that future wars will be increasingly about water.
7
Con‡icts over
renewable resources have also occurred among developed countries (DCs) but with
4
lower intensity (e.g., the 1972-1973 English-Icelandic Cod War; recent US-Canada or
Canada-Spain …shing con‡icts).
8
Due to its tendency to describe speci…c episodes of con‡ict, the extant literature
on con‡ict over renewable resources in political science has generally neglected the
complex dynamic interplay between population, natural resources, and con‡ict. Our
stylized model allows to investigate this interplay, which can take relatively long peri-
ods of time to play out.
9
Thus, our model contributes to three literatures. Our chief
contribution is the dynamic extension of the static game theoretic con‡ict framework
of Hirshleifer (1989). We also extend the literature on the dynamic interplay (ab-
sent con‡ict) between population and resources by admitting con‡ict as a rational
economic activity. In this literature, the studies of Prskawetz et al. (1994), Milik
and Prskawetz (1996) and Brander and Taylor (1998) are particularly relevant to
our paper. These studies employ a similar predatory-prey setting, where man is the
predator and a renewable resource is the prey. Finally, we contribute to the litera-
ture on resource scarcity and con‡ict in political science by providing insights on the
system’s dynamic behavior while in con‡ict.
The paper proceeds as follows. In Section 2 we review the literature and discuss
our contribution. In Section 3 we develop the model. In Section 4 we consider various
modeling extensions. In Section 5 we analyze our model in the context of the Easter
Island society and evaluate its implications for other societies. Section 6 concludes.
5
2 The Ecological Competition and Con‡ict Liter-
atures
Two bodies of literature are relevant for our paper, namely the ecological competition
literature and the economic literature on con‡ict. The …rst body of literature contains
a class of dynamic models aimed at representing competition between two interacting
species that feed o¤ the same renewable resource. Arising from the works of Lotka
(1924) and Volterra (1931), these models are speci…ed as a system of equations of
motion (or, di¤erential equations) for the stocks of each of the two species and a
resource stock. Lotka and Volterra assumed that a rise in the size of either species
reduces the resource stock, whereas a rise in the size of one species reduces the size
of the other.
10
Some scholars …nd analogies between ecological competition and economic situa-
tions. However, in the Lotka-Volterra model the population numbers of one species
respond automatically to the numbers of the other species.
11
This behavior makes this
model a less attractive tool in studying human behavior. While we employ elements
of ecological competition, we add to them the notion of optimization.
Turning to the economic con‡ict literature, Hirshleifer (1989) develops a one pe-
riod game theoretic model in which con‡ict is treated as a rational activity. Studies
using this framework typically include two rival groups, modeled as unitary actors.
Hirshleifer’s initial framework has been extended in various ways including allow-
6
ing for trade among rivals, the consideration of various con‡ict interaction protocols
(e.g., Stackelberg) and non unitary actors.
12
However, each of these extensions em-
ploys a static framework. There are no equations of motion and the time trajectories
of variables are not analyzed. Several authors are aware that this is a limitation of
Hirshleifer’s approach and have called for its extension to the dynamic case.
13
We believe our study is the …rst to explicitly introduce dynamics into Hirshleifer’s
work. However, the general issue of con‡ict dynamics has been considered in prior
work. Usher (1989) develops a model in which a society moves between states of
anarchy and despotism. Yet, only population has an equation of motion, the state
of anarchy is simply assumed to be transitory due to the high costs it imposes on
actors, and the model is not solved explicitly. A few other studies employ a two-
period game theoretic model. For example, Brito and Intriligator (1985) study the
circumstances under which con‡ict over the rights to a ‡ow of a single good leads
to war. In Skaperdas and Syropoulos (1996) and Gar…nkel and Skap erdas (2000),
the factors allocated between con‡ict and production in the second period of the
game are assumed to be positively related to the payo¤s received in the …rst period.
These models are basically static, however, as the game is played only once. Recently,
Hausken (2000) has studied the mismatch between individual and group interests in
a Hirshleifer-type model. His model also is basically static, computing numerically
the one period equilibrium solution successively as does Hirshleifer (1995), which we
discuss next.
7
Hirshleifer (1995) argues that the literature only considers a one time allocation
of resources between con‡ict and production, but that his paper studies continuing
con‡ict.
14
However, he also does not specify equations of motion, employing instead
the one p eriod solution in successive iterations. Moreover, the condition he identi…es
as determining dynamic stability is not derived based on standard dynamic analysis,
but rather is the condition assuring the existence of a one period-based internal
solution, an issue to which we will return later.
15
We study the dynamic interaction between con‡ict, population and resources
in a lesser developed society. The optimization component of the model extends
Hirshleifer’s work. From the ecological competition literature, we draw the idea that
our model should have equations of motion for the rival populations and the contested
good. However, in our model, the population sizes do not respond automatically to
each other. Instead, they are a¤ected by the actors’optimal allocation choices, as in
the non-con‡ict models of Prskawetz et al. (1994), Milik and Prskawetz (1996) and
Brander and Taylor (1998).
In our model, parties …ght over wealth not only for instant grati…cation, but also
for the ability to invest their spoils in order to increase their own resource pool in the
future. This pool is then available for future productive and con‡ictive activities. In
a dynamic setting, therefore, it is necessary to link each group’s spoils to its e¤ort
pool to be allocated in subsequent periods. In our setup, the allocated e¤ort is
population. Hence, the model has di¤erential equations for the two populations, and
8
each equation includes the spoils of con‡ict of the particular group as an input.
The production of wealth (resource harvesting in our case) also may require in-
puts that cannot be easily redirected for use in con‡ict. Often, the usage rate of
these inputs has an impact on their availability in future periods. In our case, this
input is given by the renewable resource stock, which is an input into the harvesting
production process, together with labor. Thus, the model distinguishes between these
two inputs and tracks their interactions and availability over time.
Our approach could be applied to other Hirshleifer-type models, but it is not with-
out limitations. To the extent that property rights could be enforced, an economic
model of con‡ict might allow for interacting choices of optimal time path decisions,
where actors take into account future incomes. In our model, there is optimization-
based decision making, but the actors do not take into account the future conse-
quences of their chosen actions. We believe our approach is appropriate in a model
of con‡ict over resources in lesser developed societies that feature ill-de…ned or un-
enforceable property rights. We defer the development of a dynamic model featuring
foresighted actors to future research.
9
3 The Basic Mo del
This section …rst develops our model and then investigates its properties in three
respects: steady states solutions, comparative statics, and model’s dynamics.
3.1 Model Development
The model features two groups with population sizes of N
1
(t) and N
2
(t), in period
t. Each group harvests from the same resource. The groups then engage in con‡ict
over the total harvest. Population is allocated each period between harvesting e¤ort
(E) and con‡ict e¤ort (F ), in order to maximize the group’s income. Con‡ict entails
reduced harvest, but also results in the appropriation of a portion of the rival group’s
harvest. Each group fully utilizes its population. Thus, N
1
(t) = E
1
(t) + F
1
(t) and
N
2
(t) = E
2
(t) + F
2
(t).
Each group’s harvest level, H
1
(t) and H
2
(t), is given by
H
1
(t) = R (t) E
1
(t) (1)
H
2
(t) = R (t) E
2
(t) (2)
Equations (1) and (2) illustrate that the harvest depends on the resource stock (R),
the harvesting e¤ort (E
1
or E
2
), and a parameter denoting the e¢ ciency of harvesting
().
16
For now, we assume each group possesses the same harvesting e¢ ciency. We
explore the impact of di¤erences in in Section 4.
The total harvest, H (t) = H
1
(t) + H
2
(t), is contested by both groups. In
10
Hirshleifer-type mo dels, the payo¤ (the portion of the contested good won by each
group) depends on the group’s relative allocation of e¤ort to con‡ict. We de…ne P
1
(t)
and P
2
(t) as follows:
P
1
(t) =
1
F
1
(t)
m
1
F
1
(t)
m
+
2
F
2
(t)
m
(3)
P
2
(t) =
2
F
2
(t)
m
1
F
1
(t)
m
+
2
F
2
(t)
m
(4)
where,
1
and
2
denote the e¢ ciency of con‡ict e¤ort of the two groups, and m is
called the decisiveness parameter.
17
Equations (3) and (4) are typically denoted as contest success functions. These
functions have been interpreted in the con‡ict literature as either determining the
proportion of the total prize going to each side or the probability of wining the
entire prize. We adopt the former interpretation. As noted by Skap erdas (1996) and
Gar…nkel and Skaperdas (2000), many studies set m = 1 and
1
=
2
= 1. Hirshleifer
(1995) sets
1
=
2
= 1 and examines the impact of changes in m. In this section,
we set m = 1 and assume that
1
and
2
are positive. We investigate the impact of
changes in m in Section 4.
18
The income of each group (Y
1
and Y
2
) is given by the portion of the total contested
harvest it wins:
19
Y
1
(t) = P
1
(t) H (t) (5)
Y
2
(t) = P
2
(t) H (t) (6)
We now proceed to the optimization. To simplify the notation, from here on we
11
drop the time dependency of variables. It is helpful to observe that the contested
harvest may be written, using (1) and (2), as:
H = R (E
1
+ E
2
) (7)
Substituting (7), (3) and (4) into (5) and (6), we obtain each group’s income:
Y
1
=
1
F
1
1
F
1
+
2
F
2
R (E
1
+ E
2
) (8)
Y
2
=
2
F
2
1
F
1
+
2
F
2
R (E
1
+ E
2
) (9)
Each group maximizes its income by choosing how many people to allocate to
con‡ict and harvesting, subject to the constraint E
i
+ F
i
= N
i
, where i = f1; 2g.
When optimizing, the two groups are assumed to follow a Cournot-Nash type con‡ict
protocol.
20
Performing the optimization for group 1 yields its reaction function:
F
1
F
2
=
2
(E
1
+ E
2
)
1
F
1
+
2
F
2
: (10)
Similarly, the reaction function of group 2 is given by:
F
2
F
1
=
1
(E
1
+ E
2
)
1
F
1
+
2
F
2
: (11)
Solving (10) and (11) for F
1
and F
2
, we get:
F
1
=
p
2
(N
1
+ N
2
)
2(
p
1
+
p
2
)
(12)
F
2
=
p
1
(N
1
+ N
2
)
2(
p
1
+
p
2
)
: (13)
Substituting F
1
and F
2
in (8) and (9), we obtain the income solutions:
Y
1
=
p
1
p
1
+
p
2
R
(N
1
+ N
2
)
2
(14)
12
Y
2
=
p
2
p
1
+
p
2
R
(N
1
+ N
2
)
2
: (15)
Populations grow according to the equations
dN
i
dt
=
i
N
i
, i = f1; 2g: The pop-
ulation growth rates are given by:
1
= " +
Y
1
N
1
+ a
1
(
1
)F
1
+ b
1
(
2
)F
2
and
2
=
" +
Y
2
N
2
+ a
2
(
2
)F
2
+ b
2
(
1
)F
1
. Incorporating
1
and
2
into the population equations
of motion, we get:
dN
1
dt
= N
1
(" +
Y
1
N
1
+ a
1
(
1
)F
1
+ b
1
(
2
)F
2
) (16)
dN
2
dt
= N
2
(" +
Y
2
N
2
+ a
2
(
2
)F
2
+ b
2
(
1
)F
1
) (17)
where, " denotes the di¤erence between natural birth rate and death rate, and
Y
i
N
i
captures the positive dependence of population growth on income per capita ( >
0).
21
We assume that " < 0 implying that without the resource to consume or for
su¢ ciently low per capita income, which is tied to the resource via (14) and (15), our
resource-dependent populations will eventually decline to zero.
22
The negative terms
a
i
and b
i
model the destructive e¤ect of con‡ict on population: a
i
grows with
i
(i.e.,
as group i becomes more e¢ cient in con‡ict), and b
i
declines with
j
(j 6= i) (i.e., as
group j becomes more e¢ cient in con‡ict).
23
The growth rate of the resource is given by the di¤erence between its natural
growth and total harvesting. The natural growth of the resource is given by the
standard logistic form inside the square brackets in (18).
24
Combining the logistic
growth with the harvesting functions gives the following resource equation of motion:
dR
dt
=
r
R
1
R
K
RE
1
RE
2
(18)
13
where, r
is the rate of growth of the resource, and K
is the resource carrying
capacity. The parameters r
and K
are assumed to fall with con‡ict (F
1
+ F
2
):
r
= r + r
c
(F
1
+ F
2
) and K
= K + K
c
(F
1
+ F
2
). K and r are the intrinsic resource
carrying capacity and growth rate parameters, respectively, and r
c
and K
c
are nega-
tive coe¢ cients. Noting that E
1
= N
1
F
1
and E
2
= N
2
F
2
, Equation (18) can be
re-written as follows:
dR
dt
=
r
R
1
R
K
R
N
1
+ N
2
2
(19)
Substituting (12), (13), (14) and (15), into (16) and (17), we get a system of three
nonlinear di¤erential equations (16), (17), and (19) that describes the dynamics in
terms of R, N
1
, and N
2
.
3.2 Steady States
The model’s steady states are found by setting the time derivatives of N
1
; N
2
; and
R in (16), (17), and (19) to zero. This results in a simultaneous system of nonlinear
equations.
N
1
" +
p
1
p
1
+
p
2
R
(N
1
+ N
2
)
2N
1
!
= 0 (20)
N
2
" +
p
2
p
1
+
p
2
R
(N
1
+ N
2
)
2N
2
!
= 0 (21)
rR
1
R
K
R
N
1
+ N
2
2
= 0 (22)
To make the analysis tractable, we have abstracted from the destructive e¤ects of
con‡ict on the population and resource stocks in (20), (21) and (22), setting a
i
, b
i
,
14
r
c
and K
c
to zero. We discuss the likely impacts of these destructive e¤ects in Section
5.3.
The system of equations (20), (21) and (22) has …ve steady state solutions (four
“corner”and one ”internal”).
25
Beginning with the corner solutions, the steady state
with N
1
= 0; N
2
= 0; R = 0 depicts a situation in which both populations have
declined to zero following an exhaustion of the resource. The steady state with
N
1
= 0; N
2
= 0; R = K depicts a situation in which both populations have declined
to zero b efore the resource has been depleted. Subsequently, the resource recovers to
its carrying capacity.
26
In the next two steady states, N
1
= 0; N
2
= N
2
; and R = R
,
or N
1
= N
1
; N
2
= 0;and R = R
, respectively, where asterisks denote some positive
level. Obviously, in these four corner steady states, there is no con‡ict in the system.
The …fth, internal steady state is given by:
N
1
=
2r
(
2"
K
+ 1)
p
1
p
1
+
p
2
(23)
N
2
=
2r
(
2"
K
+ 1)
p
2
p
1
+
p
2
(24)
R =
2"
: (25)
This internal steady state features con‡ict between the two rival groups. Using (12)
and (13), the e¤ort allo cations for con‡ict and harvesting can be written as:
F
1
=
N
1
2
r
2
1
(26)
F
2
=
N
2
2
r
1
2
(27)
15
E
1
=
N
1
2
2
p
1
p
2
p
1
(28)
E
2
=
N
2
2
2
p
2
p
1
p
2
(29)
For the internal steady state to exist, the condition
2"
K
+ 1 > 0 must hold.
Otherwise, Equation (25) implies R > K, and (23) and (24) imply N
1
< 0 and
N
2
< 0. In this case, the system will collapse to one of the corner steady states.
27
3.3 Comparative Statics
The partial derivatives of equations (23)-(29) determine the impact of changes in the
exogenous variables on the internal steady state. Equations (26) and (23) imply that
in a steady state where group 2 is better at con‡ict (
2
is greater) ceteris paribus,
relative to a base case steady state, group 1 allocates less e¤ort to harvesting, while
group 2 devotes more e¤ort to harvesting (i.e.,
@E
1
@
2
< 0 and
@E
2
@
2
> 0). This is so
because the larger is
2
, the greater is the portion of the total harvest accruing to
group 2, and the smaller is the portion accruing to group 1. Thus, group 2 has a
greater incentive to harvest relative to the base case, while group 1 has less incentive
to harvest. In the new steady state, group 2 then devotes less e¤ort to con‡ict,
while group 1 devotes more e¤ort to con‡ict, relative to their base case steady state
allocations (i.e.,
@F
1
@
2
> 0 and
@F
2
@
2
< 0).
Equations (14), (15), (23) and (24) imply that a steady state featuring a greater
2
; relative to a base case, ceteris paribus, features smaller group 1 population and
income, and larger group 2 population and income. Hence, a group that is better
16
at con‡ict is able to sustains a higher income and population. But when its rival
is relatively better at con‡ict, the group’s population and income are lower. These
results are driven by the fact that a group that is better at con‡ict gains more from
it, raising its income and ultimately its population via the fertility function.
A steady state that features higher resource carrying capacity (K) or intrinsic
growth rate (r), relative to a base case, ceteris paribus, has a higher populations (see
23 and 24) and the same resource stock (see 25). To gain intuition, observe from
(19) that larger r and K imply a greater resource growth rate
dR
dt
. Hence, for any
harvesting level, the marginal return to harvesting tends to rise when r or K rise
(see 1 and 2). In turn, income also rises. Fertility then rises, which raises population
levels (see 16 and 17). In the new steady state, the higher population o¤sets the
higher resource growth rate (through harvesting), which is just enough to bring back
the resource stock to its base case level.
Equations (26) and (27) imply that the higher are r and K, the greater is the
con‡ict e¤ort. Since the steady state resource stock is unaltered by changes in r and
K, the steady state marginal return to harvesting also is unaltered (see 1 and 2).
Hence, the allo cation of e¤ort between con‡ict and harvesting in a steady state with
higher r and K is driven by the greater populations.
A steady state featuring a higher fertility parameter ceteris paribus, features a
lower resource stock (see 25) and greater populations and con‡ict e¤ort (see 23, 24, 26
and 27). Higher fertility leads to greater populations, which in turn tend to deplete
17
the resource stock. The e¤ect of a greater natural net death rate (") is naturally
opposite to that of a greater .
The comparative statics e¤ect of harvesting e¢ ciency () on con‡ict is given by:
@F
i
@
=
p
j
p
1
+
p
2
K
3
(4" K) i; j = f1; 2g; i 6= j (30)
Thus given a steady state with a relatively higher (lower) R, the greater is , the
higher (lower) is the steady state level of con‡ict.
28
From (25), R rises with " and
falls with and . Hence, when death rate is high and harvesting e¢ ciency and
fertility are low, a rise in harvesting e¢ ciency may result in more con‡ict.
3.4 Dynamics
Abstracting from the destructive e¤ects of con‡ict, our dynamic system is given by
dN
1
dt
= N
1
" +
p
1
p
1
+
p
2
(N
1
+N
2
)
2N
1
R
dN
2
dt
= N
2
" +
p
2
p
1
+
p
2
(N
1
+N
2
)
2N
2
R
dR
dt
= rR
1
R
K
(N
1
+ N
2
) R:
(31)
To the best of our knowledge, this system of nonlinear di¤erential equations does not
have an analytical solution. Two methods are typically used in such cases to learn
about the dynamics: local stability analysis and numerical simulation. We study the
dynamics via simulation.
29
In order to simulate the system, we need to chose a particular parameterization.
There are, of course, many sets of parameters from which one could choose. It is
clear that the particular outcome may only apply to the chosen set. For comparison
purposes, we set
1
and
2
to 1, as is (implicitly) the case in Hirshleifer (1995).
18
However, we also require p opulation and resource parameters. To that e¤ect, one
could cho ose a parameterization not based on real-world records.
30
Alternatively,
one could pick parameters based on real-world records. Brander and Taylor (1998),
for example, study the collapse of the Easter Island society and set their parameters
accordingly. They chose a carrying capacity K = 12; 000 units, a resource growth
rate r = 0:04 per decade, a population natural death rate " = 0:1 per decade,
a population fertility parameter = 4, a resource harvesting e¢ ciency parameter
= 0:00001, an initial population = 40, and an initial resource stock = 12; 000 units.
The story of Easter Island is interesting for our paper since it provides a natural
experiment of man-nature interaction involving con‡ict over resources in a system
that lacks well-de…ned and/or enforceable property rights. We use the parameters of
Brander and Taylor here and discuss the story of Easter Island in further detail in
Section 5.
31
Let us focus now on the model’s basic dynamics. Figure 1 presents the simulation
results for group 1’s population (N
1
), con‡ict e¤ort (F
1
), the resource stock (R),
and income (Y
1
).
32
As shown, the system cycles toward an internal steady state.
Along the dynamic path, R and Y
1
lead N
1
and F
1
. Intuitively, this is so b ecause
income a¤ects fertility, and a rise in R raises income. Note also that con‡ict is
often at its peak when the resource reaches its trough. This fact coincides with the
observed tendency of resource scarcity to promote and/or intensify con‡ict in many
less developed societies.
19
[Insert Figure 1 here]
We conclude this section with brief report of additional simulation results. Since
the general behavior of the system in these cases is similar to Figure 1, we do not
graph them.
33
When group 1 is better at con‡ict than group 2 (
1
= 1:25;
2
= 0:75),
the less con‡ict-e¤ective group allocates more e¤ort to con‡ict along each point of
the trajectory, while the more con‡ict-e¤ective group allocates less e¤ort to con‡ict.
Raising r (0:06) suppresses the ‡uctuations but raises con‡ict in the steady state.
Thus, while the system becomes less vulnerable to intensive con‡ict, there is a steady
state trade-o¤. A higher K (20; 000) makes the system less damped and also raises
con‡ict in the steady state. Finally, a higher (0:001) drives the system to the steady
state with R = K, and N
1
= N
2
= 0. In this case, both populations go to zero before
R is fully diminished. Consequently, the resource grows back to carrying capacity.
20
4 Extending the Basic Model
In the basic model, the decisiveness parameter of the contest success functions was
equal to unity, and the groups were equally e¢ cient at harvesting. In this section,
we relax these assumptions.
4.1 Changes in the Decisiveness Parameter
As in Hirshleifer (1995), the contest success functions are now given by:
34
P
i
(t) =
F
m
i
(t)
F
m
1
(t) + F
m
2
(t)
i = f1; 2g: (32)
In (32), as m grows, the marginal e¤ectiveness of group i’s …ghting e¤ort in capturing
a proportion of contested goods rises. As Hirshleifer (1995) notes, with low m, the
defensive resources have the upper hand. Hirshleifer’s main …ndings are that the
existence of the internal equilibrium requires m < 1, and a rise in m (from 0 to 1)
raises the equilibrium level of e¤ort devoted to con‡ict. He refers to the situation
with m > 1 as the breakdown of anarchy.
Hirshleifer (1995) assumes that the total allocated e¤ort (N
1
+ N
2
in our model)
are open to appropriation.
35
In other studies (e.g., Hirshleifer, 1989), he assumes, as
we do in this paper, that only the total output produced (H in our model) is open
to appropriation. In Hirshleifer (1989), the equilibrium exists for any m, or anarchy
never breaks. As we shall see, this result does not hold in our dynamic case. That is,
we can get the breakdown of anarchy when only harvested resources are contested.
21
Using (32) and assuming that the two groups have equal harvesting e¢ ciencies,
our dynamic system may be written as
dN
1
dt
= N
1
" +
(N
1
+N
2
)
2N
1
(m+1)
R
dN
2
dt
= N
2
" +
(N
1
+N
2
)
2N
2
(m+1)
R
dR
dt
= rR
1
R
K
(N
1
+N
2
)
(m+1)
R
: (33)
Similar to system (31), system (33) also has four corner steady states. The …fth
(internal) steady state is given by:
N
1
= N
2
= r(m + 1)(
(m+1)"
K
+ 1)
R =
(m+1)"
(34)
For the internal steady state to exist, the condition
(m+1)"
K
+ 1 > 0 must hold.
In contrast to Hirshleifer (1995), m > 1 need not ensure the breakdown of anarchy
(recall that " < 0). Our breakdown of anarchy condition also depends on resource and
population parameters. In addition to large m, ine¢ cient harvesting, low fertility, a
high death rate, and a low carrying capacity all lead to the breakdown of anarchy via
system collapse.
We now examine the impact of m, assuming anarchy do es not break. The compar-
ative statics of population with respect to m in (34) are ambiguous, ceteris paribus.
Since the optimal allocation of e¤ort for con‡ict is given by F
1
= F
2
=
m
m+1
N
1
+N
2
2
,
the comparative statics of con‡ict allocations also are ambiguous, which is a markedly
di¤erent result from Hirshleifer (1995). In that paper, a rise in m raises the con‡ict
allocation. In our paper, the e¤ect of m on con‡ict is ambiguous, depending on the
population and resource parameters. We may gain insight by inspecting the e¤ect of
22
m on the relative con‡ict allocation
F
1
N
1
=
m
m+1
. This ratio is growing with m as
in Hirshleifer (1995). However, since N
1
and N
2
are exogenous in Hirshleifer (1995),
that model does not exhibit the ambiguity of our dynamic case, where N
1
and N
2
are
endogenous.
The dynamic system (33) is structurally similar to (31). The only di¤erences
being the replacement of
p
i
=
p
1
+
p
2
in the …rst two di¤erential equations
by 1= (m + 1), and the introduction of 1= (m + 1) in the third di¤erential equation.
Thus, the dynamics of the two systems will be qualitatively similar.
4.2 Di¤erences in Harvesting E¢ ciency
In section 3.1, we saw that raising the con‡ict e¢ ciency of a group relative to its
rival raises its income and lowers the e¤ort it devotes to con‡ict. Since total e¤ort is
devoted to con‡ict or harvesting, a natural question to ask is what happens when the
harvesting e¢ ciency of one group is raised relative to its rival? In order to answer this
question, we rework the basic model under the assumption of di¤erences in harvesting
e¢ ciency.
Letting
i
denote the harvesting e¢ ciency of group i, the internal steady state is
given by:
N
1
= 2r
p
1
2
1
p
1
2
+
2
p
2
1
1 +
2"
K
p
1
2
+
p
2
1
1
p
1
2
+
2
p
2
1
N
1
= 2r
p
2
1
1
p
1
2
+
2
p
2
1
1 +
2"
K
p
1
2
+
p
2
1
1
p
1
2
+
2
p
2
1
R =
2"
p
1
2
+
p
2
1
1
p
1
2
+
2
p
2
1
: (35)
23
The steady state con‡ict e¤orts are:
36
F
1
=
N
1
2
r
2
1
"
1
p
1
2
+
2
p
2
1
1
p
2
2
+
2
p
1
1
#
; (36)
F
2
=
N
2
2
r
1
2
"
1
p
1
2
+
2
p
2
1
1
p
2
2
+
2
p
1
1
#
: (37)
Next, we study the comparative statics e¤ects of
1
and
2
on con‡ict. It is clear
from (36) and (37) that the e¤ects of
1
and
2
on con‡ict depend on their e¤ects on
N
1
and N
2
. In order to examine these impacts, we rewrite the …rst equation in (35)
as
N
1
= 2r
"
p
1
2
1
p
1
2
+
2
p
2
1
#
+
2"
K
"
1
2
+
p
1
2
1
2
1
p
1
2
+
2
p
2
1
2
#!
(38)
Each of the expressions written in square brackets in (38) is decreasing in
1
and
2
. Recalling that " < 0, we see that the impact of an increase in
1
or
2
on N
1
is
ambiguous. The same is true for N
2
. It follows from (36) and (37) that the e¤ect of
changes in the harvesting e¢ ciencies on F
1
and F
2
also are ambiguous. These results
are similar to those we derived in the basic model and occur for the same reason,
speci…cally, they are driven by the size of the resource stock. When the resource
stock is high, increases in the harvesting e¢ ciency raises con‡ict. The opposite will
be true when the resource stock is low.
24
5 Implications for Historical and Contemporary
Con‡ict
In our simulation we have employed parameters for the historical society of Easter
Island. In Figure 1, the society exhibited a brief ‡owering and then declined to a
dismal state with low population, resources and income. In this section, we compare
our results with historical accounts of Easter Island. Scholars such as Tainter (1988),
Ponting (1991) and Bahn and Flenley (1992) argue that Easter Island is just one of
several examples of historical societal collapse precipitated by con‡ict over degrading
resources. Following our discussion of Easter Island, we discuss other so cieties that
experienced a similar history. These scholars and others argue that contemporary
societies face similar risks (though possibly of a weaker strength), particularly in
LDCs, where societies are closely dependent on the natural environment. We end
this section with a discussion of our model’s implications for LDCs.
5.1 Easter Island
Many years ago abundant forests and a society thrived on Easter Island. By the
time Europeans arrived on the island in the early 18th century, they found land
without trees and with a small population living in poverty and con‡ict. Many
scholars have puzzled over this story. Recently, several studies have explained the
collapse by thinking about the island in the spirit of Malthus (1798). That is, the
human population overexploited the island’s resources, leading to its own decline.
37
25
Brander and Taylor (1998) have modeled the Malthusian interpretation of the Easter
Island story, while ignoring con‡ict over resources. Our model o¤ers the possibility
to investigate the history of Easter Island based on the assumption that there was
con‡ict over natural resources on the island.
The foundation of our model captures the general setting of the Easter Island so-
ciety. A considerable literature argues there was ample con‡ict over natural resources
between well organized clans on Easter Island (e.g., see Ponting, 1991; Keegan, 1993;
and Lee, 2000). As noted by Ponting (1991), the clans were each led by a dominant
chief, supporting our modeling of groups as unitary actors.
38
Naturally, con‡ict in
primitive societies such as Easter Island was labor intensive, as we have assumed.
Our simplifying assumption that con‡ict does not kill people or damage the resource
is appropriate for Easter Island. Anthropologists who have studied ancient Easter
Island skulls found evidence of injuries, but not life threatening injuries, indicating
con‡ict, but not fatal wounds. The loser in the con‡ict often lost his property but
not his life (Lee, 2000).
39
Also, it is likely that Easter Island’s society did not de-
velop e¢ cient property rights institutions.
40
And, as noted by Brander and Taylor
(1998), being a primitive society, the assumptions that population rises with income
and actors maximize current incomes are reasonable for Easter Island.
Turning to the simulation, period 0 in Figure 1 corresponds to year 400-700 AD,
which is the time range settlers are said to have arrived on Easter Island.
41
Con‡ict
intensi…es harvesting in the beginning of the simulation. The population then rises,
26
and the resource declines. As a result, population also declines. The population peaks
at 14,000 around period 50, and declines to around 2,000 around p eriod 130 (year
1700). In Brander and Taylor’s (1998) paper, population peaks at around 10,000
people 25 periods later and then declines to 3,800 around period 130. In our paper,
the resource reaches a minimum of around 3,000 units around period 80, where as in
Brander and Taylor’s paper it reaches a minimum of around 5,000 units 25 periods
later.
42
It is hard to compare the resource in the model to the real world. As both our
model and the one of Brander and Taylor are stylized, the resource represents an
ecological complex consisting of soil, …sh species, forestry, water, etc. Nonetheless,
we can discuss the population trajectory. The available information on Easter Island
is based on archeological inquiries. The estimated maximum population ranges from
7,000 to 20,000, whereas the timing of this maximum is in the range of 1100 to 1500
AD.
43
When Easter Island was discovered in the 18th century, the Dutch Admiral
Rogeveen estimated there were 3,000 people on the island and the British Captain
Cook estimated there were 2,000 people.
Based purely on simulation results, the model of Brander and Taylor is plausi-
ble. However, their model does not include con‡ict on Easter Island, which is well
documented in the literature. Our simulation results suggest that the inclusion of
con‡ict is consistent with historical and anthropological accounts of the Island’s soci-
ety. Thus, our model also is a plausible description of the main social forces op erating
27
on Easter Island.
5.2 Other Historical Societies
While the story of Easter Island is likely the most famous, several examples of societal
collapse precipitated by con‡ict over degrading renewable resources exist. Weiskel
(1989, p.104) notes that each of these societies exhibited “gradual emergence, brief
‡owering and rapid collapse of civilization,”accompanied by con‡icts driven by the
desire to control arable land or other essential renewable resources. In this section,
we brie‡y discuss the cases of the Sumerian and the Maya civilizations.
44
The Sumerian society, which arose in the fertile valley of the twin rivers, the Tigris
and Euphrates, is generally accepted to be the world’s …rst literate society, having
attained this status by about 3000 BC. The society was comprised of a number of
cities that were often in con‡ict over the land separating them. The land was valuable
because of the innovation of irrigation. With irrigation, the Sumerian society moved
from subsistence farming towards cash crops, traded within the society and with
non-Sumerians for such things as metals and manufactured go ods.
Because of the ability to create wealth through cash crops, the Sumerians began
to overexploit the land via almost constant irrigation. Traditional agricultural tech-
niques such as crop shifting and allowing lands to lie fallow were abandoned. The
constant irrigation eventually led to a complete salinization of a vast majority of the
crop lands. The early stages of decline saw the loss of cash crops, which weakened the
28
society materially, while later stages saw the loss of essential harvests. The Sumerian
society saw increases in the death rate and con‡ict over resources and decline in in-
come and fertility. In a weakened state, the society was conquered in 2370 BC by the
Akkadian empire.
The story of the Sumerians illustrates one implication of our model: advancements
in production (in this case irrigation) need not improve the long-term prosp ects of
society. While there is little doubt that irrigation increased the short-term wealth of
the Sumerians, this wealth allowed the population to grow, which in turn led to the
over-exploitation of the resource and con‡ict over the degraded resource base.
The Maya story provides yet another example of the forces we model. Early the-
ories of the Mayan society, which dates from 2500 BC and was located in southern-
North and Central America, were at o dds with our model. Historians once thought
that this great civilization was peaceful. The Mayans also were thought to have prac-
ticed environmentally friendly agricultural techniques. As a result of this thinking,
historians were at a loss to explain the collapse of the Mayan civilization. Much
like Easter Island, the civilization went into decline long before European contact.
When the ‘lost cities’(so-named because the ancient pyramid temples were lost to
the encroaching jungle, having been abandoned for generations) were discovered by
American archeologists in the late 1830s, descendants of the Maya had no knowledge
of them.
Subsequent research of the Mayan culture has changed scholarly thinking. Mainly
29
due to the translation of the Mayan script, historians now know that far from being
a peaceful culture, the Mayan society was comprised of cities that were almost con-
tinually at con‡ict over arable land. As with the Sumerians, Mayan land increased in
value as agricultural innovation allowed the society to move beyond the subsistence
level. As with Easter Island, a major factor in the Mayan decline appears to be
deforestation and subsequent soil erosion, which occurred as large amounts of land
were cleared for agricultural purposes.
5.3 Contemporary LDCs
While we believe our model captures the underlying tendencies inherent in many
LDCs, the model’s implications should b e considered carefully. This is so, due to the
potentially mitigating e¤ects that non resource-based sectors, demographic transition,
property rights institutions, foreign aid and trade, and technological innovation might
have on resource dependent societies, all of which are not included in our model. As
noted by Reuveny and Decker (2000), while these e¤ects may not have been signi…cant
on Easter Island, they could be more signi…cant in LDCs.
Dep endence on the environment for livelihoo d is more prevalent in LDCs than
DCs. The build-up of non-resource based sectors might alleviate the pressures that
LDCs place on natural resources. However, since this is a costly and lengthy process,
con‡ict similar to that we have modeled may b e plausible for some LDCs in the future
and, as noted, according to some scholars is beginning to emerge.
30
Our model ignores the theory of demographic transition. According to this theory,
when income p er capita is low, population growth rises with income. As income per
capita rises above some threshold, population growth declines with income (Heerink,
1994). This theory is not without critics, but it is accepted by many scholars.
45
Demographic transition implies that economic growth may mitigate pressures on the
environment in LDCs. However, this approach also entails a cost. Economic growth
increases pollution, resource depletion, and often results in deforestation. Moreover,
several authors also argue that the biosphere cannot sustain the DCs’current per
capita income for all countries.
46
Similar to all Hirshleifer-type models, this paper assumes the absence of well-
developed and enforced system of property rights. With such institutions in place,
the model’s basic structure becomes less applicable as basis for analysis. However,
property right institutions are generally de…ned and enforced less rigorously in LDCs
than in DCs.
47
This does not mean that e¢ cient institutions cannot arise in LDCs.
Ostrom (1990) observes cases in which such institutions arose in poor societies, as
well as cases where they did not. Hence, the emergence of e¢ cient property rights
institutions in LDCs cannot be taken for granted and may require intervention from
DCs or international organizations.
We also have ignored the role of foreign aid and trade. Of course, resource scarcity
may be alleviated by foreign aid. However, we believe our simple model may b e of
value in gaining insight into the underlying tendencies of the system without aid.
31
As for trade, if a natural resource dependent economy has a comparative advantage
in a non resource-based sector, trade prompts the allocation of more labor to this
sector, reducing harvesting and raising social welfare. However, LDCs typically have
comparative advantage in their resource-based sectors. In this case, trade stimulates
resource harvesting. Over time, the resource gets overexploited and social welfare
declines relative to autarky.
48
Consequently, the rising resource scarcity may induce
con‡ict along lines suggested in the introduction.
The model has a Malthusian spirit. The typical argument made against the
Malthusian prediction is that it does not consider technological innovation. As ar-
gued by Homer-Dixon (1999), con‡ict may reduce society’s ability to innovate, to
begin with.
49
Ignoring this point for the moment, in our setup innovation could, for
example, raise the harvesting e¢ ciency, the carrying capacity, the resource growth
rate, and reduce the death rate. According to our model, raising the resource growth
or carrying capacity would not have made societies less con‡ictive. Additionally a
rise in harvesting e¢ ciency and a reduction in the death rate might lead to systemic
collapse. In other words, technological innovation may not be the panacea to Malthu-
sian con‡ict. At the same time, con‡icts over natural resources are currently more
frequent and intense in LDCs than in DCs. Hence, it is possible that once resources
become plentiful, actors’behavior changes so that con‡ict is no longer considered a
rational option to begin with.
Finally, in the analytical solution to our model we have abstracted from the po-
32
tentially destructive e¤ects of con‡ict on the population and resource base. Some
con‡icts over resources in LDCs are not intense enough to signi…cantly a¤ect the re-
source base or the overall death rate. In terms of our model, relatively low intensity
scarcity-induced con‡icts could be, and some say already are, a steady state outcome
in LDCs. However, some con‡icts in Africa, for example, are said to have already
registered a negative e¤ect on p opulation forecasts and the environment. In extreme
cases, aided by technological innovation in …ghting, the e¤ect of con‡ict could be so
destructive as to eventually drive the system into one of its corner steady states.
33
6 Conclusions
We have developed a dynamic model of con‡ict based on Hirshleifer’s initial static
game theoretic framework. To our knowledge, this is the …rst model in the economic
literature on con‡ict that makes Hirshleifer’s framework dynamic. We have employed
the model to study con‡ict over renewable resources in historical and present day less
developed societies.
Our model has …ve steady states. Four steady states exhibit no con‡ict because
either one or both groups are extinct. The condition for the breakdown of anarchy in
our model is more complicated than in Hirshleifer (1995). In our case, the breakdown
depends on parameters of the resource and population, not only on the decisiveness
parameter. We focused on a …fth steady state that features con‡ict. The compara-
tive statics reveal that changes enhancing the resource stock or the population raise
con‡ict. A rise in the con‡ict e¢ ciency of one group relative to the other raises the
group’s income and reduces its con‡ict e¤ort. A rise in the model’s decisiveness para-
meter generates an ambiguous e¤ect on con‡ict, which also di¤ers from Hirshleifer’s
(1995) static model. Finally, the e¤ect of raising harvesting e¢ ciency on con‡ict is
positive when the resource stock is high.
Turning to the dynamics, our results generally accord with the stories of historical
societies that exhibited a relatively brief ‡owering, followed by decay, all the while
exhibiting con‡ict over the resource base. Finally, we have discussed the model’s
34
implications for contemporary LDCs, paying particular attention to the limitations
resulting from our modeling approach.
We have employed a relatively simple framework. Several research extensions,
therefore, are worth pursuing. For example, the agents in the model maximize their
current incomes. While we …nd this assumption appropriate in our case, it would
be interesting to intro duce foresight into the model. Second, given our focus on less
developed societies, we ignored demographic transition. Incorporating demographic
transition into the model is an interesting extension. It would also be interesting to
add more goods and factors of production. These features are expected to remove
pressure from the resource, but we believe that the resource-population ‡uctuations
will not disappear in their presence. Third, in the solution we have ignored the de-
structive e¤ects of con‡ict. It would be interesting, although mathematically compli-
cated, to relax this assumption. However, as long as the destructive e¤ects of con‡ict
are not so strong as to result in system collapse, we suspect that this extension would
not change the nature of our …ndings.
In the end, while the model’s trajectory is consistent with the spirit of the his-
tory of several ancient societies, contemporary LDCs di¤er, of course, from these
cases. That said, we believe our …ndings serve as a warning of what the future look
like should societies choose to …ght over renewable resources instead of devising the
appropriate institutions to control their exploitation.
35
Rafael Reuveny is Assistant Professor of Political Economy and Public Policy
at the School of Public and Environmental A¤airs, Indiana University. In addition
to con‡ict over scarce natural resources, his research interests include the relation-
ship between trade, con‡ict and democracy, the leadership long-cycle approach to
international p olitical economy, and the determinants of sustainable development.
John W. Maxwell is Associate Professor of Business Economics and Public Policy
at the Kelley School of Business, Indiana University. In addition to con‡ict over scarce
natural resources his research interests include the political economy of environmental
regulation with special interest in voluntary environmental agreements.
36
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44
Notes
1
We would like to thank two anonymous reviewers and the editor of The Journal
of Con‡ict Resolution for their many comments and suggestions.
2
The next section reviews the relevant literature, placing our model within it.
3
While we focus on renewable resources, as noted by Hirshleifer (1995) and Neary
(1997), Hirshleifer’s setup could be used to analyze many situations, including gang
warfare, criminal-victim interactions, labor disputes, legal disputes, and animal ter-
itoriality disputes. Using a static model, Neary (1997) motivates his general-forms
Hirshleifer-type model by refering to Homer-Dixon’s (1994) review of real world con-
‡icts over renewable resources.
4
Con‡ict over non-renewable resources also exists. For reasons of tractability,
however, our model has only one resource. The resource is meant to capture the entire
basket of resources on which a primitive society depends. Given this dependence it
is appropriate to model the resources as renewable, as most life-giving resources are
renewable.
5
Scholars suggest four channels through which this tends to happen: a decline in
economic performance, ethnic clashes due to population migration, a weakening of
political institutions, and a general exacerbation of existing so cio-economic-political
cleavages. For reviews of theories and case studies, see Myers (1993), Baechler (1998)
45
and Homer-Dixon (1999).
6
This list is by no mean exhaustive. For example, social strife in post-1945 Philip-
pines has been linked to deforestation and land degradation leading to population
displacements (Hawes, 1990). Durham (1979) describes how land scarcity caused
migration from El Salvador to Honduras, resulting in competition between immi-
grants and natives over land, leading to a war in 1969. Subsistence crisis in Peru
led to the Luminoso rebellion (McClintock, 1984). Since the 1970s, cropland and
food scarcities have aggravated ethnic con‡ict along the border between India and
Bangladesh (Ashok, 1996). Since the 1980s, there has been a rise in piracy directed
at …shing boats in LDCs (Oceans and the Law of the Sea, 1998), and land pressures
have stimulated squatters on ranches in Brazil (The Economist, April 13, 1996).
7
See, e.g., press release by The World Bank’s Vice President Ismail Serageldin
(1995). Contemporary cases include the 1989 Mauritania-Senegal con‡ict, the on-
going Arab-Israeli con‡ict, the mid-1980s South Africa-Lesotho con‡ict, the ongoing
Syrian-Turkish con‡ict (Myers, 1993; Homer-Dixon, 1999), and the 1990s social strife
in China’s Ningxia province (Pomfret, 1998).
8
Choucri and North (1975) argue that land pressures in Europe caused World
War I, and Westing (1986) lists wars from 1914 to 1982 involving DCs and natural
resources.
9
Maxwell and Reuveny (2000) study the dynamics of con‡ict over renewable re-
46
sources, but they do not model the con‡ict decision.
10
Dep ending on their parameters, the Lotka-Volterra trajectories oscillate over
time, converging to one out of …ve steady states. In one steady state, the two popu-
lations and the resource coexist. In the second and third, only one population exists,
respectively. In the fourth, both populations vanish, and in the …fth steady state the
population and resource vanish. See, e.g., MacArthur (1972) and Slobodkin (1980).
11
For studies that …nd analogies between ecological competition and economic sit-
uations see, e.g., Jacquemin (1987) and Hirshleifer (1977).
12
Works in this area include, among others, Hirshleifer (1991, 1995), Skaperdas
(1992, 1996), Grossman and Kim (1995), Skaperdas and Syropoulos (1996), Neary
(1997), Anderton et al. (1999), Gar…nkel and Skaperdas (2000) and Hausken (2000).
13
See e.g., Skaperdas (1992), Grossman and Kim (1995) and Hirshleifer (1995).
14
See Hirshleifer (1995:29).
15
Hirshleifer acknowledges that his “steady state assumption rules out issues in-
volving timing, such as arms races, economic growth or (on a smaller time scale)
signaling resolve through successive escalation (1995: 47).” (Italics are original).
16
This technology was proposed by Schaefer (1957), and is popular in the resource
literature (e.g., Clark, 1990: Chapter 1; Brander and Taylor, 1998). Expressions (1)
and (2) assume that each group’s harvest is independent of the harvest of the rival
47
group. While this assumption is likely to hold when the resource is in abundance,
when the resource is scarce each group’s harvest may impose a negative externality
on its rival’s harvest. While our assumption has been made principally for analytical
tractability, it is worth observing that the marginal return to harvesting e¤ort, S,
falls as the resource declines.
17
The parameters
1
,
2
, and m are positive (Hirshleifer, 1989, 1991, 1995).
18
In some papers, including his 1989 paper, Hirshleifer sets m = 1. While many
Hirshleifer-type models are based on (3) and (4), some studies specify these equations
as general forms. For example, in Neary (1997) P
1
=
f(F
1
)
f(F
1
)+f(F
2
)
, where f is twice
continuously di¤erentiable. We employ a speci…c form to be able to compare to
Hirshleifer work, and since we investigate the dynamics in numerical simulations.
19
This assumption is conceptually equivalent to assuming that each group tries to
consume its own harvest, but that the harvest also is subject to appropriation by the
rival group.
20
That is, each group takes the e¤ort allocation of its rival as given when choosing
its own allocation.
21
This assumption is used in many studies (e.g., Prskawetz et al., 1994; Milik and
Prskawetz, 1996; Brander and Taylor, 1998) and is supported empirically in LDCs
(Heerink, 1994). In many DCs, fertility seems to decline with income. We return to
this topic in Section 5. An alternative interpretation, which may well also apply to
48
DCs, is that the resource is essential for procreation (e.g., when food declines, fertility
declines).
22
For a similar assumption in a model without con‡ict see, e.g., Brander and Taylor
(1998).
23
One could assume that " and di¤er across groups. This would complicate the
model without adding much insight, as there is no a priori reason to assume that the
rival groups di¤er in these respects.
24
The logistic form applies to renewable resources, which are our focus (see, e.g.,
Clark, 1990: 10). The model can be applied to non renewable resources by setting
r
= 0. In this case, one may want to introduce a term for resource discovery.
25
This steady state con…guration is typical in the ecological competition literature
(see Section 2).
26
In Section 5, we provide human historical examples of societies going extinct due
to environmental degradation precipitated by con‡ict over renewable resources.
27
When the groups are equal in every respect, the steady states S = K; R
1
= R
2
= 0
or S = 0; R
1
= R
2
= 0, could b e attained. When the groups di¤er, the system also
could collapse to one of the steady states with only one group. For that to occur, the
rate of population growth of one of the groups needs to be always negative due to a
too low income (e.g., because its con‡ict e¢ ciency is too low).
49
28
The sign of (30) is positive if in steady state S > K=2, and negative if S < K=2
(see 25).
29
A local stability analysis involves …nding the system’s eigenvalues around each
steady state. This method is not tractable here since the system’s characteristic
equation (which determines the eigenvalues) is cubic. Since the system is of order 3,
the phase diagram approach also is not tractable. For studies that employ dynamic
numerical simulation see, e.g. Prskawetz et al. (1994), Milik and Prskawetz (1996)
and Brander and Taylor (1998).
30
In this case, the goal is mainly to demonstrate the mathematical properties of
the dynamic system simulated, as in, e.g., Prskawetz, et al. (1994) and Milik and
Prskawetz (1996).
31
As noted by Brander and Taylor, the estimated initial population for the island
ranges from around 20 to 100. Our simulation results are virtually the same for
di¤erent initial populations within this range.
32
In this symmetric case, the values for group 2 are identical. Income is plotted at
ten times its actual level to better visualize it.
33
In each case, the parameters not mentioned are kept as in Figure 1.
34
To make the analysis tractable and focus attention on m, we assume
1
=
2
= 1
and keep our earlier assumption regarding the non-destructive aspects of con‡ict,
50
both of which are as in Hirshleifer (1995).
35
This assumption would imply in our model that human beings can be forced to
…ght against their own group or harvest for the other group, both of which we do not
…nd appealing.
36
It is worth noting that the term in square brackets in (36) and (37) equals unity
when
1
=
2
, and we then recover equations (26) and (27). Similarly, equations (35)
collapse to equations (23), (24) and (25), respectively, under the same condition.
37
See, e.g., Ponting (1991), Bahn and Flenley (1992), Van Tilberg (1994), Gowdy
(1998), Brander and Taylor (1998), Brown and Flavin (1999), and Reuveny and
Decker (2000).
38
Extending our model to include more than two clans does not require changing
its structure, but will make it less tractable.
39
Con‡ict of this type also is sometimes observed in other primitive societies. For
example, see Keegan’s (1993) account of the African Zulus.
40
See Ponting (1991), Van Tilberg (1994), Brander and Taylor (1998) and Luter-
bacher (2001). It is suggested that the islanders did not develop e¢ cient institutions
to deal with the degradation also because the island’s trees grew slowly and people
did not grasp the nature of the slow change taking place.
41
The date the island was …rst settled varies across studies. Brander and Taylor,
51
for example, use the date 400 AD, Gowdy (1998) and Bahn and Flenley (1992) use
700 AD, and Brown and Flavin (1999) use 500 AD.
42
The model becomes less applicable in the early 1800s, when the island is no longer
a closed system.
43
See Ponting (1991), Bahn and Flenley (1992), Van Tilberg (1994) and Brander
and Taylor (1998).
44
Our discussion of the Sumerian and Maya cultures is based on Ponting (1991),
and our discussion of the Zulu culture is based on Keegan (1993). Interested readers
are directed to these sources for further details of the rise and fall of these societies,
and others that met a similar fate.
45
For critics, see, e.g., Abernethy (1993) and Dilworth (1994).
46
See Cohen (1995) for a detailed review of many studies demonstrating this claim.
47
In fact, according to the 2001 Heritage Foundation’s Index of Economic Freedom
(www.heritage.org/index/), none of the 17 countries we have mentioned in the in-
troduction as experiencing resource con‡icts are ranked in the top 30 worldwide in
regard to property rights, and only 1 county is ranked in the top 50 (Turkey). Out of
the 17, the majority are LDCs. We would like to thank an anonymous reviewer for
bringing up this point.
48
Bee (1987), Brown (1995), and Brander and Taylor (1997) provide empirical ex-
52
amples of the detrimental e¤ect of international trade on resource-dependent economies.
49
This point is controversial. For example, Simon (1996) and Boserup (1981) argue
that these same adverse forces generate more innovation, since necessity is the mother
of invention.
53
