This paper examines the relationship between inequality and economic growth using conflict models and computational simulations. I construct a dynamic sequential conflict model that allows us to observe the cumulative effect of inequalities in wealth and ability in a single framework. The computational simulation illustrates the dynamics of the wealth level of players. The main findings are as follows: (1) if the conflict is not intensive, then the equilibrium that achieves equal distribution of wealth is unique and stable; (2) if the conflict is intensive, the equal distribution equilibrium becomes an unstable saddle point; and furthermore (3) when the productivity of one player is lower than the other, the less productive player exploits the other through the conflict process.conflict, conflict model, inequality, war

Takeuchi, Kan

English

Research Papers in Economics

Hitotsubashi University Repository
Title
Cumulative Effect of Inequality : A Computational
Study of Conflict Models
Author(s) Takeuchi, Kan
Citation
Issue Date 2010-10
Type Technical Report
Text Version publisher
URL http://hdl.handle.net/10086/18697
RightDiscussion Paper #2010{15 1
Cumulative Eﬀect of Inequality: A Computational
Study of Conﬂict Models
Kan Takeuchi
October 28th, 2010
Abstract
This paper examines the relationship between inequality and economic growth using
conﬂict models and computational simulations. I construct a dynamic sequential conﬂict
model that allows us to observe the cumulative eﬀect of inequalities in wealth and ability
in a single framework. The computational simulation illustrates the dynamics of the
wealth level of players. The main ﬁndings are as follows: (1) if the conﬂict is not
intensive, then the equilibrium that achieves equal distribution of wealth is unique and
stable; (2) if the conﬂict is intensive, the equal distribution equilibrium becomes an
unstable saddle point; and furthermore (3) when the productivity of one player is lower
than the other, the less productive player exploits the other through the conﬂict process.
Keywords: conﬂict, conﬂict model, inequality, war.
JEL Classications: D74, D72, D63
1 Introduction
The aim of conﬂict models is to formalize the conﬂict between two (or more) groups in a
society and to analyze its general characteristics. This paper uses the conﬂict model and
computer simulations to examine the relationship between the inequality and the economic
growth.
The pioneer research on conﬂict models of Boulding (1962) was ﬁrst formalized by Bush
and Mayer (1974). They constructed a game theoretical model without assuming that
property rights are given. In their model, players need to ﬁght for their own property rights
against each other and they also can exploit the property of another player. Since the
I thank Serguey Braguinsky for introducing this framework of research to me and seminar participants
at Hitotsubashi University for helpful discussions. Takeuchi: Graduate School of Economics, Hitotsubashi
University. kan@econ.hit-u.ac.jpﬁghting activity is unproductive, if they could cooperate and keep themselves from ﬁghting,
they would achieve higher welfare. However, every one has the incentive to exploit others
and thus the cooperation cannot be equilibrium. The game was known as The Prisoner’s
Dilemma.
Skaperdas (1992) characterized three types of equilibria in a two-player conﬂict model;
both players devote all endowments to the production (full cooperation); one of them still
inputs all of the endowment to the production while the other exploits it (partial cooper-
ation); and both of them ﬁght and exploit against each other (conﬂict). The model has
two parameters. One of them is the discrepancy in the productivity and the other is the
intensity of conﬂict. The two-dimensional parameter space is divided into four regions of
diﬀerent equilibria.
The two parameters that Neary (1997) incorporates are the discrepancy in the players’
initial endowments and the aggregate level of the endowments. Neary shows that there are
four types of equilibria: full cooperation, partial cooperation, conﬂict, and banditry. In the
bandit equilibrium, the player with less endowment invests nothing in the production and
spends all for the ﬁghting eﬀort. This banditry occurs only when the aggregate level of
endowments is suﬃciently high and the discrepancy is also suﬃciently large. That is, the
hopeless poor player puts all of his endowment into the ﬁghting activity.
In these two models, there are four parameters: the inequality in the initial endowments
among players, the gap in the productivity among players, the total wealth level of the
society, and the intensity of the conﬂict. Appealing to a computer simulation model, I
incorporate all of these parameters into one framework. So, we could observe the cumulative
eﬀect of the inequalities in several aspects.
2 The Model
Suppose that there are two agents in the economy and that each of them divides his resource
endowment into two eﬀorts respectively: productive eﬀort z and “ﬁghting” eﬀort y. The
payoﬀ of agents is
π1(z1,y1,z2,y2) = p(y1,y2)F(z1,z2), (1)
π2(z1,y1,z2,y2) = (1   p(y1,y2))F(z1,z2) (2)
2where F(z1,z2) is the production and p(y1,y2) is the share of agent 1. That is, in one hand,
two agents combine their productive eﬀorts so as to generate a pool of income available
to themselves and on the other hand they determine their share by the function . In the
literature of conﬂict models, p is often called the Contest Success Function (CSF). There
are two speciﬁc forms of CSF proposed in the literature,
Hirshleifer (1991): p(y1,y2) =
exp(y1)k
exp(y1)k + exp(y2)k (3)
Tullock (1980): p(y1,y2) =
yk
1
yk
1 + yk
2
. (4)
Note that p(y1,y2) = 0.5 whenever y1 = y2. Both of these are S-shaped and the variable k
parameterizes the intensity of the conﬂict process in the sense that k propagates the gap in
the ﬁghting eﬀorts between the players. In one extreme case where k ! 0, it follows that
p ! 0.5 regardless of ﬁghting eﬀorts. On the other extreme, if k ! +1, these functions
will take on the so-called winner-take-all feature; the player who inputs more ﬁghting eﬀort
than the other will take all of the production. The maximization problem of each agent is
max
y1,z1
p(y1,y2)F(z1,z2), subject to y1 + z1  w1
max
y2,z2
(1   p(y1,y2))F(z1,z2), subject to y2 + z2  w2
where wi is the initial endowment of agent i. By specifying the functions, this model can
capture the nature of the social interactions between two groups of economic agents. For
example, they might be two ﬁrms that jointly cultivate a new market and then compete for
their shares; two employees contribute to the employer and then ask for the salary, etc. In
the following subsection, I present a plausible application of the conﬂict model.
2.1 An Application of the Conict Model
In this subsection, I present an application of the general conﬂict model. Let us assume that
there are two groups of players, h and ℓ, and their wage rates are wh and wℓ, respectively
(wh > wℓ). They divide their endowment 1 into the productive activity L and the ﬁghting
activity y. The government imposes a linear income tax and redistributes the revenue to
the two players evenly. Therefore, the consumption level of player i is
Ci = (1   t)wiLi + G, for i = h,ℓ, (5)
3where t is the tax rate and G = 0.5t(whLh+wℓLℓ). The government is partially benevolent
in the sense that it maximizes the following Cobb-Douglas social welfare function,
max
t
C
p
hC
1 p
ℓ
and the weight p of this function is not given but determined by the ﬁghting eﬀort of two
players, p = p(yh,yℓ). The optimal tax that maximizes the social welfare for a given p is
t(yh,yℓ,Lh,Lℓ) =
2whwℓ   2(whLh + wℓLℓ)p(yh,yℓ)
whLh   wℓLℓ
. (6)
By backward induction, the optimization problem of player i is
max
Li
Ci =
(
1   t(yh;yℓ;Lℓ;Lh)
)
wiLi +
1
2
t(yh;yℓ;Lℓ;Lh)
(
whLh + wℓLℓ
)
s.t. Li + yi = 1
Further calculation yields the following formulae:
max
Lh
Ch = p(1   Lh,1   Lℓ)
[
whLh + wℓLℓ
]
max
Lℓ
Ch =
{
1   p(1   Lh,1   Lℓ)
}[
whLh + wℓLℓ
]
These are exactly the same optimization problems that players solve in the general conﬂict
models. Moreover, when p is Hirshleifer’s (1991) CSF, it follows from Skaperdas (1992)
that there is a unique Nash equilibrium and p > 0.5 at the equilibrium. It implies that
the optimal tax rate is always positive and that the income is transferred from player h to
player ℓ.
3 The Dynamics
Extension of the model of Neary (1997) into multi-round game shows the cumulative eﬀect
of inequality. In the following model, two players repeatedly interact within the same
framework, but the endowments will diﬀer. As the chart illustrates below, the payoﬀ in
the current round πt
i becomes the endowment in the next round wt+1
i . In this way, we
can simulate the cumulative eﬀect of inequality. The players are assumed to play Nash
equilibrium strategy in each round independently of other rounds.
Denition (Long-run Equilibrium).
{
wt
1,wt
2
}
is a long-run equilibrium if the payoﬀs in
round t,
{
πt
1,πt
2
}
, are equal to the initial endowment
{
wt
1,wt
2
}
.
4t
t
z
z
2
1
 
t
t
y
y
2
1
 
t
t
p
F
 
1 1 1
2
1 1 1
1
) 1 (
- - -
- - -
- =
=
t t t
t t t
F p
F p
p
p
 
t
t
w
w
2
1
 
Round t-1  Round t  Round t+1 
1
2
1
1
+
+
t
t
w
w
 
t t t
t t t
F p
F p
) 1 ( 2
1
- =
=
p
p
 
Figure 1: The sequential structure of the game
To run a simulation model, let us suppose that the production function is a Cobb-
Douglas function and that CSF is Tullock’s. Thus, the equilibrium is always an interior.
The ﬁrst-order conditions are
∂πt
1/∂zt
1 = ptFt
1   pt
1Ft = 0. (7)
∂πt
2/∂zt
2 = (1   pt)Ft
2 + pt
2Ft = 0. (8)
where Ft
i is the partial derivative of Ft with respect to zt
i and pt
i is the partial derivative
with respect to the i-th argument. Rearranging the terms yields
(1   pt)Ft =
wt
1   zt
1
k
Ft
1. (9)
ptFt =
wt
2   zt
2
k
Ft
2. (10)
Summing up these two equations, we obtain
Ft =
1
k
(
(wt
1   zt
1)Ft
1 + (wt
2   zt
2)Ft
2
)
.
Since the production function is ﬁrst-order homogeneous, zt
1Ft
1 + zt
2Ft
2 = wt+1
1 + wt+1
2 , and
thus
wt+1
1 + wt+1
2
wt
1 + wt
2
=
σtFt
1 + (1   σt)Ft
2)
1 + k
where σt is the share of player 1’s endowment to the total endowment, σt = wt
1/(wt
1 +
wt
2). Note that the left-hand side of the equation is unity at any long-run equilibrium. A
proposition for the symmetric long-run equilibrium follows: .
5Productivity is same Productivity is diﬀerent
F = z0.5
1 z0.5
2 F = z0.52
1 z0.48
2
k = 1.45 Panel A Panel B
moderate
conﬂict
There is a unique long-run equilib-
rium at which the distribution is
equal. Moreover, any initial endow-
ment will eventually reach the long-
run equilibrium.
The unique long-run equilibrium is
located in a diﬀerent position in fa-
vor of player 2 who has lower pro-
ductivity.
k = 1.70 Panel C Panel D
intensive
conﬂict
There are 3 long-run equilibria: one
saddle point on the 45 degree line
and two sinking points.
The two of the three long-run equi-
libria in Panel C no longer exist. At
the remaining long-run equilibrium,
player 2 who has the lower produc-
tivity exploited player 1.
Table 1: The comparison of equilibria
Proposition 1. If the two players are identical, then there is a unique symmetric long-run
equilibrium, at which, ∂F
∂z1 = 1 + k.
This result implies that economic growth will end at the point where the marginal
product diminishes down to 1 + k . Note that the marginal product is decreasing in the
level of production. Thus, the intensive conﬂict indicated by large k limits the sustainable
growth of the economy. For players, the intensive conﬂict means that the marginal proﬁt
of ﬁghting eﬀort is not negligible, since the conﬂict propagates the small gap in the ﬁghting
eﬀort and results in the large gap in the distribution of the production. To ﬁnd other
long-run equilibria, I run a computer simulation in the next section.
4 The Simulation
The dynamics has already captured the inequality in the initial endowment, σt = wt
1/(wt
1+
wt
2), at each round and the level of the aggregated endowment wt
1+wt
2. Thus, the simulation
examines the eﬀect of the other two factors. Those are the discrepancy in the productivity
and the intensity of the conﬂict.
Table 1 summarizes the result of the simulations. For the gap in the productivity, I
change the parameter of the Cobb-Douglas function. To observe the eﬀect of the intensive
conﬂict, I present two cases (k = 1.45 and 1.70) for illustration.
6Figure 2 shows the trajectories of
{
wt
1,wt
2
}
for t = 1,2,.... Each arrow connects two
points from
{
wt
1,wt
2
}
to
{
wt+1
1 ,wt+1
2
}
, the wealth of players before and after the game
of round t. In Panel A (top-left), there is the unique long-run equilibrium that every
trajectory eventually reaches. That is the symmetric long-run equilibrium, and at the long-
run equilibrium the marginal product is equal to 1 + k or ∂Ft/∂zt
1. As seen in the ﬁgure,
any path of economic growth converges to the equilibrium. This feature remains unchanged
even after I introduce discrepancy in the productivity. In Panel B (top-right), in which
player 1 has the higher productivity than player 2, the long-run equilibrium is located
slightly upper-left and it is still a sinking point. It implies that player 2 exploits player 1
by investing her resource into the ﬁghting eﬀort. She does so, since the marginal proﬁt to
player 2 of ﬁghting tends to be greater than that of productive eﬀort.
Panel C shows an interesting result. Even though the productivities are identical, there
could be another asymmetric long-run equilibrium. Moreover, the symmetric long-run equi-
librium is no longer a sinking point, and the path to the symmetric long-run equilibrium
is on the knife edge. It implies that any noise or turbulence in the distribution would
lead the economy to one of the asymmetric long-run equilibria. Finally, let us observe the
eﬀect of the unequal productivity. Unlike the previous change from Panel A to Panel B,
the simulation shows that the dispersion in the productivity is propagated by the conﬂict.
Through the intensive conﬂict, player 2 could exploit player 1 further. I summarize these
observations below.
Observations:
1. There is a unique symmetric long-run equilibrium when the players are identical.
(Panel A and Panel C)
2. When the conﬂict is moderate, the symmetric long-run equilibrium is the global attrac-
tor, and thus it is automatically attained regardless of initial distribution of wealth.
(Panel A)
3. The symmetric long-run equilibrium is, however, an unstable saddle point if the con-
ﬂict is intensive. (Panel C)
4. When the conﬂict is moderate, the gap in the productivity among players does not
have signiﬁcant impact on the distribution of wealth at the long-run equilibrium.
(Observe the diﬀerence between Panel A and Panel B)
7Panel A:
F = z0.5
1 z0.5
2 and k = 1.45
〮 ㄮ ㈮
〮
ㄮ
㈮
Panel B:
F = z0.52
1 z0.48
2 and k = 1.45
〮 ㄮ ㈮
〮
ㄮ
㈮
Panel C:
F = z0.5
1 z0.5
2 and k = 1.70
〮 ㄮ ㈮
〮
ㄮ
㈮
Panel D:
F = z0.52
1 z0.48
2 and k = 1.70
〮 ㄮ ㈮
〮
ㄮ
㈮
Figure 2: Simulations
Note. These ﬁgures illustrate the dynamics of sequential conﬂict games. Table 1 summarizes
the result for each panel above.
85. When the conﬂict is intensive, the gap in the productivity among players has the
impact on the signiﬁcant distribution of wealth at the long-run equilibrium. (Observe
the diﬀerence between Panel C and Panel D)
5 Concluding Remarks
I wish to conclude with two remarks. First, we should carefully interpret the “ﬁghting” eﬀort
in this model. It does not necessarily mean physical ﬁght, but it could be any form of social
interaction that aﬀects the distribution of wealth or resources such as politics, competition
among employees, competition between ﬁrms, negotiation over salary, and so forth. As
we live in the market mechanism and believe in it, we also believe that the distribution
of wealth is basically determined by skills. Indeed, it is true. The skills, however, might
include the ability of “ﬁghting,” which is characterized by the conﬂict model. Think of blue
collar and white collar workers or executives. While the blue collar workers are skillful at
making the products of a company, the executives are excellent in distributing the proﬁt
brought by the blue collar workers. Even though the executives are incapable of making any
products of the company, they take a signiﬁcant amount of the proﬁt. Then, the simulation
results appeal us, as Panel D shows that at the equilibrium in the dynamics player 2 who
is less productive takes more than 80% of the total wealth. What we believe as the market
mechanism might be part of conﬂict process.
Second, when society witnesses an increase in inequality, this tends to be attributed to
the dispersion of productivity. The simulation result, however, suggests that the change in
the nature of the conﬂict causes the change in the distribution of wealth. Suppose that the
society has been around the symmetric long-run equilibrium in Panel A. Suppose that the
society suddenly changes and the distribution process (or conﬂict) becomes more intensive.
Then, the distribution of wealth departs from the 45 degree line and it will eventually
reach either of the asymmetric long-run equilibria. Note that this could happen even if
the production function and the productivity remain unchanged. Finally, I conjecture that
these observations of the simulation can be fully or partially veriﬁed by theoretical analysis,
which remains for future work.
9References
Boulding, K.E., 1962. Conﬂict and Defense: A General Theory. Harper, New York.
Bush, W., Mayer, L., 1974. Some implications of anachy for the distribution of property.
Journal of Economic Theory 8, 400–412.
Hirshleifer, J., 1991. The technology of conﬂict as an economic activity. American Economic
Review 81, 130–134.
Neary, H., 1997. Equilibrium structure in an economic model of conﬂict. Economic Inquiry
35, 480–494.
Skaperdas, S., 1992. Cooperation, conﬂict, and power in the absence of property rights.
American Economic Review 82, 720–739.
Tullock, G., 1980. Eﬃcient reng seeking. In: Buchanan, J., Tollison, R., Tullock, G. (Eds.),
Towards a theory of the rent-seeking society, College Station, pp. 97–112.
10

Cumulative Effect of Inequality : A Computational Study of Conflict Models

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Cumulative Effect of Inequality : A Computational Study of Conflict Models

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