Discussion Papers In Economics
And Business

Endogenous Political Instability
Ryo Arawatari and Kazuo Mino
Discussion Paper 07-32-Rev.

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANEndogenous Political Instability
Ryo Arawatari and Kazuo Mino
Discussion Paper 07-32-Rev.

November 2007

この研究は「大学院経済学研究科・経済学部記念事業」
基金より援助を受けた、記して感謝する。

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANEndogenous Political Instability∗
Ryo Arawatari†and Kazuo Mino‡
November 2007

Abstract
In this paper, we construct a simple dynamic two-party electoral competition
model in which the degree of political instability is endogenously determined. We
consider the campaign contributions as stock variable which is gradually accumulated by both party’s direct investment and induced the Markov-perfect Nash equilibrium. We then examine the stability of the symmetric steady state and find thatit may be either totally stable or unstable depending on the parameter values involved in the model. We also found that under certain conditions, at least near
the symmetric steady state, there exists indeterminacy of equilibrium path: there
exist both stable and unstable paths, that is, under given levels of political assets,
both high instability political system and low instability political system can emerge
depending on expectations of political parties.
Keywords: Political assets; Dynamic political economy; Differential game; Markovperfect Nash equilibrium; Two-party model
JEL Classification: C73; D72; D78∗

We are very grateful to Junichi Itaya and Akira Yakita for their invaluable comments. We also
acknowledge the helpful comments by Takeo Hori, Shingo Ishiguro, Koichi Kawamoto and Keiichi Morimoto. Arawatari acknowledges financial support from the Japan Society for the Promotion of Science(JSPS) Research Fellowship for Young Scientists. Mino acknowledges financial support from Grant-in-Aid
for Scientific Research (No. 18530136).
†
Corresponding Author. Graduate School of Economics, Osaka University, 1-7 Machikaneyama, Toyonaka, Osaka, 560-0043, Japan. E-mail address: ege001ar@mail2.econ.osaka-u.ac.jp‡
Graduate School of Economics, Osaka University, 1-7 Machikaneyama, Toyonaka, Osaka, 560-0043,
Japan. E-mail address: mino@econ.osaka-u.ac.jp

1

Introduction

It has been argued that there is a close connection between political instability and economic performance. Casual observation suggests that the countries with poor economic
performances are often associated with unstable political systems. Several authors such as
Devereux and Wen (1998), Persson and Tabellini (1998) and Darby et al. (2004) present
the analytical frameworks that establish the negative relationship between political instability and economic growth. Such a theoretical prediction fits well to the empiricalfindings by Alesina and Perotti (1996), Barro and Sala-i-Martin (1995), Perotti (1996)
and Alesina et al. (1996)2 .
In view of the theoretical and empirical studies mentioned above, the relationshipbetween political instability and economic performance seem to be ascertained. However,the problem with these studies is that since they all treat the political instability as
an exogenously given factor, they cannot explain what is the determinant factor of thedegree of political instability. Actually, the degree of political instability differs vastly
among countries. This paper examines a theoretical model in which the degree of politicalinstability is endogenously determined. The main purpose of our study is to present a
step towards a general equilibrium consideration on the relationship between politicalinstability and economic performance.
We construct a model of electoral competition between two political parties. Following
Austen-Smith (1987), Snyder (1990), Barron (1994) and Grossman and Helpman (1996),
we assume that each party tries to increase its probability of winning by the political
campaign. The key difference between the foregoing investigations and the present paper
is that in our setting the relevant determinant of the probability of winning depends on
the accumulated stock of past spending for campaign, while the existing studies assume
that current level of campaign contributions alone determines the probability of winningof each party. Our discussion, therefore, explores the political instability in a dynamic
framework, as oppose to the foregoing studies that treat static models.
2

See also Fredriksson and Svensson (2003).

1More specifically, we set up a dynamic-game model of electoral competition between
two parties. The state variables of our dynamic game is the stocks of accumulated campaign spending (’political assets’) and the strategy of each party is the level of campaign
spending. The probability of winning is determined by the stocks of political assets as well
as by the exogenous political shocks. Each party is assumed to maximize a discounted
sum of expected profits given by the expected rent revenue minus costs for campaign.
Assuming that the political shocks are uniformly distributed, we formulate the electoralcompetition in terms of a linear-quadratic differential game and characterize the Markovperfect Nash equilibrium of the game.
In our model, the degree of political instability is represented by the probability ofwinning. We consider that the two-party system is highly unstable when the probabilityof winning of each party is close to 50%. In contrast, the political system is highly stable
if the probability of winning of one party is close to 100%. We first show that there isa symmetric steady state where both parties accumulate the same level of political asset
and the probability of winning of both parties are 50%. We then examine the stabilityof the symmetric steady sate and find that it may be either totally stable or unstabledepending on the parameter values involved in the model. It is shown that whether
the political instability prevails or not depends on the initial differences in the politicalassets of both parties. It is also revealed that under certain conditions the symmetricsteady state has a saddle point property. If this is the case, there exist a stable as wellas an unstable paths, that is, both stable and unstable political systems can emerge. In
this case, realization of one particular path is determined by the expectations of politicalparties. Such a situation is particularly interesting, because it may explain why the similartypes of economies sometimes exhibit substantially different degree of political instability.
The rest of the paper is organized as follows. Section 2 sets up the analytical framework. Section 3 characterizes the Markov-perfect Nash equilibrium of the electoral game.
This section presents our main findings and their implications. Section 4 concludes thepaper.
2

2

The Model

2.1

Citizens

The society consists of a continuum of citizens with a unit mass who have identicalpreferences over private consumption and public services. Each citizen receives the sameamount of endowment in each period. The instantaneous utility of a citizen is specifiedasW (c(t), g(t)) = hc (t) + H(g(t)),

h > 0,

where c(t) and g (t) are private consumption and the public spending per capita, respectively. Function H (g) is monotonically increasing, strictly concave and satisfies the Inada
conditions: limg→0 H 0 (g) = +∞ and limg→+∞ H 0 (g) = 0. We assume that citizens do nothold any financial asset, so that the private consumption is determined by their disposable
income:c(t) = (1 − τ (t))y + y0 ,where τ (t) ∈ [0, 1] denotes the rate of income tax, y (> 0) is a given, taxable endowment
and y0 (> 0) expresses a given, untaxable endowment. Thus even if τ (t) = 1, eachconsumer can maintain a positive level of consumption, y0 .

2.2

Electoral Competition

There are two political parties, indexed by j = A and B. If party j wins an election andholds the office, it can appropriate a part of tax revenue as a rent. Letting rj (t) be rent
appropriation, the budget constraint for the government office is given by

τj (t)y = gj (t) + rj (t),j = A, B,(1)

where τj (t) is the rate of income tax set by party j.
Using the government’s budget constraint, we rewrite each voter’s utility function
3

under party j’s policy as follows:

W (gj (t), rj (t)) = h(y − gj (t) − rj (t)) + hy0 + H(gj (t)).

(2)

At the time of the election, the voters base their voting decision not only on therelative welfare gain under alternative offices but also on the campaign contributions by
the parties as well as on exogenous political shocks and individual-specific ideological bias.
Specifically, voter i prefers the candidate representing party A, if the following holds:

W (gA (t), rA (t)) ≥ W (gB (t), rB (t)) + φi + Θ(t).
The above condition assumes that voter i0 s preference are affected by two additional
factors, φi and Θ (t) .
First, φi ∈ R expresses an individual-specific parameter that measures voter i’s individual ideological bias toward party B. A positive value of φi implies that voter i hasa bias in favor of party B 0 s political stance, whereas the voter prefers party A0 s ideological position if φi < 0. The voters with φi = 0 are ideologically neutral, that is,they do not have individual ideological bias. We assume that φi is distributed according to a cumulative distribution function F (·), and the distribution is symmetric, i.e.,F (−φ) = 1 − F (φ) ∀φ ∈ R and F (0) = 1/2.
Second, Θ(t) represents the average (relative) popularity of party B in period t and it
is defined as:
Θ(t) ≡ σB (t) − σA (t) − S.

(3)

Here, σj ∈ (−∞, +∞) denotes the ’political asset’ accumulated by party j through its
campaign and S denotes an exogenous shock that disturbs the voters’ decision. The
behavior of σi is discussed in the next subsection. For analytical simplicity, we assume
that S is a random variable uniformly distributed on [−s̄, s̄] where s̄ is a positive constant.
Note that in our formulation, φi is exogenously specified by voter’s preference, while Θ (t)4is an endogenous variable determined by accumulation of political assets of both parties.

2.3

Strategic Behaviors of the Political Parties

The above specification basically follows the modelling by Austen-Smith (1987), Snyder(1990), Baron (1994), and Grossman and Helpman (1996) who emphasize the campaign
contribution as an important determinant of implemented policies. The existing studies,
however, assume that the relative popularity is determined by the instantaneous campaign
contributions and thus σj is a flow variable. Additionally, the previous literature usuallyassumes that campaign activities are conducted by interest groups. In contrast to theforegoing formulations, we consider that σj is an intangible stock variable which is gradually accumulated by party j’s direct investment. Letting dj (t) be party j’s investment
level, σj changes according toσ̇j (t) = dj (t) − δσj (t) ,(4)where δ denotes the depreciation rate of σj . We assume that the rate of depreciation is
the same for both parties. In the above, dj (t) represents party j’s effort (or labor) level
to increase its political reputation. It is assumed that one unit of effort raises one unitof the political asset. Since voting habits and preferences over alternative parties usuallychange over time, our formulation may capture the voting criteria of citizens in a moreplausible way than the existing studies. Now defineσ(t) ≡ σB (t) − σA (t).

Then we may summarize two equations in (4) as:σ̇(t) = dB (t) − dA (t) − δσ(t).

(5)

Given the assumptions displayed above, citizens vote for the party whose platform,5{τj , gj , rj }, provides them with the highest utility. Recall that the distribution of φi issymmetric and F (0) = 1/2. Therefore, party A wins an election if the following holds:

F (W (gA (t), rA (t)) − W (gB (t), rB (t)) − σ(t) + S) >1
2

⇔ W (gA (t), rA (t)) − W (gB (t), rB (t)) − σ(t) + S > 0.

As a result, under our distributional assumption on S, each party’s probability of winning,

PA (t) = Pr (S > W (gB (t), rB (t)) − W (gA (t), rA (t)) + σ(t)) ,is given by






PA (t) =

1

1−






W (gB ,rB )−W (gA ,rA )+σ+s̄2s̄0if W (gB , rB ) − W (gA , rA ) + σ ≤ −s̄,
if W (gB , rB ) − W (gA , rA ) + σ ∈ (−s̄, s̄),(6)

if W (gB , rB ) − W (gA , rA ) + σ ≥ s̄.

Namely, if the divergence in political assets, σ(= σB − σA ), is small (resp. large) enoughto satisfy W (gB , rB ) − W (gA , rA ) + σ ≤ −s̄ (resp. W (gB , rB ) − W (gA , rA ) + σ ≥ s̄), thenparty A (resp. party B) always wins, regardless of the magnitude of political disturbance,
S ∈ [−s̄, s̄]. As a result, PB (= 1 − PA ) is given by






PB (t) =







0
W (gB ,rB )−W (gA ,rA )+σ+s̄2s̄1

if W (gB , rB ) − W (gA , rA ) + σ ≤ −s̄,
if W (gB , rB ) − W (gA , rA ) + σ ∈ (−s̄, s̄),(7)

if W (gB , rB ) − W (gA , rA ) + σ ≥ s̄.

We may consider that political system is stable if PA is close either to 1 or 0, while it isunstable if PA is close to 1/2.
Party j 0 s instantaneous benefit is defined as the expected utility of rent appropriation,θPj (t) rj (t) , minus the investment cost. We assume that the investment cost function is
given by (a/2) d2j (t) (a > 0). Here, θ and a are positive scale parameters that respectively
6transform rent appropriation, ri , and effort, dj , to utilities. We set the objective function
for party j as
Z ∞
Uj =
0

³
´
a
e−ρt θPj (t)rj (t) − · dj (t)2 dt, j = A, B,
2

where ρ > 0 is discount rate. Each party maximizes Uj subject to (1) and a given
initial level of σ (0) (= σB (0) − σA (0)) . The timing of events is as follows. First, bothparties announce their policy platforms, {τA , gA , rA } and {τB , gB , rB }, simultaneously.
Then an election is held, in which voters choose between the two parties. The elected
party implements the announced policy platform. We assume that the each party commits
to their policy platform in each moment. Formally seaking, our model is a differential
game between two parties in which each player’s control variables are τj , gj and dj , andthe state variable is σ (t) (≡ σB (t) − σA (t)) . In what follows, we characterize the Markovperfect Nash (feedback Nash) equilibrium of this dynamic game.

3

Political Instability

3.1

Probability of Winning

In order to derive the Markov-perfect equilibrium, we first define the value function forparty j’s dynamic optimization problem in such a way that
½Z ∞eVj (σ(t)) ≡ maxt−ρ(s−t)³¾´
a2
θPj (s)rj (s) − · dj (s) ds : subject to (4) .
2Then the Hamilton-Jacobi-Bellman (HJB) equation for party i’s optimization problem isgiven byn
ρVj (σ(t)) = max

rj ,dj ,gjoa20θPj (t)rj (t) − · dj (t) + Vj (σ(t))[dB (t) − dA (t) − δσ(t)] , j = A, B, (8)27The above equation should holds for all t ≥ 0. In the following, we abbreviate t fornotational simplicity.
Before examining the HJB equations given above, we should determine the probabilityof winning Pj . More specifically, we must know the relationship between Pj and the statevariable, σ. First, the resource constraint and feasibility conditions require that(rj , gj ) ∈ g ≡ {(rj , gj ) : rj ≥ 0, gj ≥ 0, rj + gj ≤ y}, j = A, B.

(9)

As a result, function W (gj , rj ) is expressed as
W (gj , rj ) ∈ [0, hy + H(min(g ∗ , y)) − h · min(g ∗ , y)], j = A, B,
where g ∗ satisfies the first-order condition for an interior solution that maximizes Wj (gj , rj )under a given rj , that is, h = H 0 (g ∗ ). To ensure the presence of an interior solution satisfying g ∗ < y, we make the following assumption:
Assumption 1: H 0−1 (h) < y.
Additionally, keeping in mind that

W (gB , rB ) − W (gA , rA ) = h(gA − gB + rA − rB ) + H(gB ) − H(gA ),(10)

we see that the divergence between W (gA , rA ) and W (gB , rB ) fulfills

W (gB , rB ) − W (gA , rA ) ∈ [−ψ, ψ],(11)where ψ ≡ hy + H(g ∗ ) − hg ∗ > 0. In what follows, we assume:
Assumption 2: s̄ ≥ ψ/2.
This assumption means that the variance of the stochastic term is high enough. Given
Assumptions 1 and 2, we obtain the following lemma that plays a key role in the subsequent discussion:
8

Lemma 1: Under Assumptions 1 and 2, the following results are established:(i) If σ ≤ −3s̄, then PA = 1 so that Party A always wins. The optimal levels of r andg are: rA = y, gA = 0, (rB , gB ) ∈ {(rB , gB ) : rB ≥ 0, gB ≥ 0, rB + gB ≤ y}.
(ii) If σ ∈ (−3s̄, 3s̄), it holds that PA ∈ (0, 1), rA (σ) = (3s̄−σ)/3h, rB (σ) = (3s̄+σ)/3h,
gA = gB = g ∗ .
(iii) If σ ≥ 3s̄, then PA = 0 so that party B always wins. The optimal levels of r andg are: (rA , gA ) ∈ {(rA , gA ) : rA ≥ 0, gA ≥ 0, rA + gA ≤ y}, rB = y and gB = 0.
Proof. (i) Suppose that σ ≤ −3s̄ holds. Using (10), (11) and Assumption 2, we obtain:

W (gB , rB ) − W (gA , rA ) + σ ≤ ψ + σ ≤ ψ − 3s̄ ≤ −s̄.

By (6), PA = 1 holds if σ ≤ −3s̄. When PA = 1, HJB equations in (8) are rewritten as:noaθrA − · d2A + VA0 (σ)[dB − dA − δσ] ,rA ,dA ,gA2n aoρVB (σ) = max − · d2B + VB0 (σ)[dB − dA − δσ] .
rB ,dB ,gB
2
ρVA (σ) = max

From the resource constraint in (9), it is obvious that each party finds it optimal to set
rA = y, gA = 0 and (rB , gB ) ∈ {(rB , gB ) : rB ≥ 0, gB ≥ 0, rB + gB ≤ y}.
(ii) Suppose that σ ∈ (−3s̄, 3s̄) holds. Now let us guess that PA ∈ (0, 1) in equilibrium.
Then, using (6), (7) and (10), we write the HJB equations for each party in the following
manner:½ ·¸h(gA − gB ) + h(rA − rB ) − H(gA ) + H(gB ) + σ + s̄
ρVA (σ) = max θ 1 −· rA
rA ,dA ,gA
2s̄
o
a 2
0
(12)
− · dA + VA (σ)[dB − dA − δσ] ,2½ ·¸h(gA − gB ) + h(rA − rB ) − H(gA ) + H(gB ) + σ + s̄
ρVB (σ) = maxθ· rBrB ,dB ,gB
2s̄
o
a
(13)
− · d2B + VB0 (σ)[dB − dA − δσ] .
2
9By use of the first-order conditions for selecting gj in the left-hand sides of the HJB
equations, we obtain3 :gA = gB = g ∗ ≡ H 0−1 (h).

(14)

Therefore, each party chooses same level of public spending in this case. Note that underthe Assumption 1, it should hold that g ∗ < y. Substituting (14) into the (12) and (13),
we may rewrite the HJB equations as follows:onaρVA (σ) = max RA − · d2A + VA0 (σ)[dB − dA − δσ] ,rA ,dA2noa 20ρVB (σ) = max RB − · dB + VB (σ)[dB − dA − δσ] ,rB ,dB
2

(15)
(16)

where
·

¸
h(rA − rB ) + σ + s̄RA ≡ θ 1 −· rA ,2s̄¸·h(rA − rB ) + σ + s̄· rB .
RB ≡ θ
2s̄

(17)
(18)

Here, Rj (j = A, B) denotes the expected rent obtained by party j. Solving the first-orderconditions about rA and rB , we obtain the following policy functions:rA (σ) =1· (3s̄ − σ),
3hrB (σ) =1· (3s̄ + σ).
3h

(19)

Now, we verify that PA ∈ (0, 1) holds under above optimal policy strategies. Using (10)and the condition σ ∈ (−3s̄, 3s̄), we obtain:

W (gB , rB ) − W (gA , rA ) + σ =1· σ ∈ (−s̄, s̄).
3

Therefore, from equation (6), PA ∈ (0, 1) holds.
(iii) The proof of Lemma 1(iii) is similar to that of Lemma 1(i) and is hence omitted.
3Note that rA 6= 0 and rB 6= 0 in equilibrium. If rj = 0 the value of felicity function must be zero.
Therefore, rational parties do not choose rj = 0.

10This lemma demonstrates that if the state variable, σ, is out of the interval, (−3s̄, 3s̄),then one party always wins. In words, the political system is completely stable, when the
gap between the two party’s political assets exceeds certain limits. Conversely speaking,when the divergence between σA and σB is not too large, each party can win the election
and thus political instability remains.

3.2

Optimal Investment

In this subsection, we determine the optimal investment of each party, dj . We first consider
the region, σ ∈ (−3s̄, 3s̄), in which each party has a positive probability of winning. By
use of (17), (18) and (ii) in Lemma 1, the expected rent function of party j defined onσ ∈ (−3s̄, 3s̄) is respectively represented by·¸
σ/3 + s̄θ1−(3s̄ − σ),
RA (σ) =3h2s̄
·¸θ σ/3 + s̄RB (σ) =(3s̄ + σ).
3h2s̄As (19) shows, in our differential game each playe is not symmetric. Therefore, eventhough both parties have the identical preferences and the same cost functions, the policy
function of each party is not identical.
In view of the first-order condition for selecting the optimal dA and dB , we findadA = −VA0 (σ), adB = VB0 (σ).
Equations (15) and (16) mean that the model in this case is a linear-quadratic game withan infinite time horizon. Although it has been known that a linear-quadratic differential
game with an infinite horizon may sustain nonlinear strategies, the following discussion
focuses on the case where each political party selects a linear strategy.4 Therefore, we set

4

For the presence of nonlinear strategies in the infinite horizon, linear quadratic differential games, see
Tsutsui and Mino (1990). As for the recent development in this issue, see, for example, Rowat (2007).

11

the optimal policy functions of each party in the following manner:

dj = αj σ + βj , j = A, B.

Restricting our attention to the linear strategies reduces the generality of our analyticalconsideration. We will, however, show that even if we only examine the linear policy
functions, we may have a rich set of equilibrium outcomes of the game, and hence ourcentral message can be conveyed in the simple case of linear strategies.
The above discussion yields:
VA0 (σ) = −aαA σ − aβA ,VB0 (σ) = aαB σ + aβB .

Using the above relations and the HJB equations, we can expresses αj and βj by the given
parameter values. The following lemma presents our findings:
Lemma 2: Suppose that the policy functions of dA and dB are linear in σ, i.e., dj =αj σ +βj , j = A, B. Then, there exists two sets of equilibrium policy functions whose
coefficients are respectively given byq1
αA = αA≡−(ρ + 2δ) +q(ρ + 2δ)2 −

4θ
3ahs̄

6

βA = βB = β 1 ≡,1αB = α B≡(ρ + 2δ) −4θ(ρ + 2δ)2 − 3ahs̄

6

,

θ
,
1
3ah(ρ + δ − αB
)

and

2
≡
αA = αAq4θ−(ρ + 2δ) − (ρ + 2δ)2 − 3ahs̄

βA = βB = β 2 ≡62≡, αB = αBq4θ(ρ + 2δ) + (ρ + 2δ)2 − 3ahs̄
6

,

θ
.
2
3ah(ρ + δ − αB)

Proof. See Appendix.
At this moment, it is indeterminate which set of parameter values is selected. We
12discuss this issue in the next subsection. In the following, we impose two restrictions on
the parameter values. First, αA and αB given above should be real numbers, and hence
we assume:4θ
Assumption 3: s̄ > 3ah(ρ+2δ)2.

This assumption implies that the variance of the stochastic term should be high enoughto ensures that the αA and αB must be real numbers.
Second, when we obtain a set of interior solutions of dj (t) , σ (t) should be in someinterval. More specifically, we find that dj (t) = αji σ(t) + βji ≥ 0 for any set of coefficients,¡ i i¢
αj , βj (j = A, B and i = 1, 2) , if σ (t) satisfies
σ (t) ∈ [−σ0 , σ0 ] ,

where

σ0 =

12
h
i,
√
ah (ρ + 2δ) (5ρ + 4δ) + 2 (2ρ + δ) D − D

D ≡ (ρ + 2δ)2 −

4θ
.
3ahs̄

Therefore, if −3s̄ < −σ0 < σ0 < 3s̄, at least one of the parties obtains a corner solution(dj (t) = 0), even though both party have a positive probability of winning. To avoid
analytical complexity, we make the following assumption:2(3−θ)
Assumption 4: s̄ ≤ 9ah(δ+2ρ)[ρ(1−θ)+δ(2−θ)].

This assumption ensures that σ0 > 3s̄. Given this restriction, each parties’ investment
level is strictly positive for all σ (t) ∈ (−3s̄, 3s̄) in which Pj > 0 (j = A, B). We now
consider the cases of σ ≤ −3s̄ and σ ≥ 3s̄. The optimal investment in those intervals issummarized by the following lemma:
Lemma 3: Suppose that each party takes the equilibrium policy, {rj , gj }, j = A, B,
characterized in Lemma 1.
13

(i) If σ ≤ −3s̄, then the optimal levels of investment are dA = dB = 0.
(ii)If σ ≥ 3s̄, them the optimal levels of investment are dA = dB = 0.
Proof. See Appendix.
Figure 1 displays the both parties’ optimal investment levels. When σ ∈ (−3s̄, 3s̄),there exist two sets of equilibrium policy functions. If σ ∈/ (−3s̄, 3s̄), both parties cease toinvest. This reduces the divergence between the political asset of each party and weakens
the situation in which one party dominates the other in the electoral competition.
[Figure 1 is around here.]In the strategic equilibrium with αj = αj1 (j = A, B), both parties’ investment choices
are less sensitive to a change in σ than the case where they select the strategies with
αj = αj2 (j = A, B). Suppose that σ ∈ (−3s̄, 3s̄) and that σ increases. Then, in responseto the rise in σ, party A lowers its investment spending, while party B raises it (see Fig.1).
1
If party B selects the insensitive strategy, αB = αB
, the increase in party B 0 s investment

is small so that party A’s incentive to reduce its investment level is relatively weak. As a1
result, party A also chooses the less sensitive strategy, αA = αA
. Conversely, suppose that1
σ increases and party A takes insensitive strategy, αA = αA
. In this case a rise in σ yields

a relatively small decline in party B 0 s investment. Hence, responding to the small change
1in the opposite party’s investment, party B selects the insensitive strategy, αB = αB
, as1
1well. Consequently, αA = αAand αB = αB
constitute one of the Nash equilibrium. In

the rest of this paper, we refer to the Nash equilibrium with αj = αj1 (j = A, B) as theequilibrium with insensitive strategies.
Alternatively, in the strategic equilibrium with αj = αj2 (j = A, B), the optimal
investment of both parties are more sensitive than in the case of αj = αj1 (j = A, B).
Intuitive implication of this Nash equilibrium is similar to the case of insensitive strategies.
We will refer to the Nash equilibrium with αj = αj2 (j = A, B) as the equilibrium withsensitive strategies.
14

3.3

Patterns of Dynamics

We can now examine equilibrium dynamics generated by our political game. Lemma 2
and 3 mean that in the Markov-perfect Nash equilibrium the dynamic equation (5) isexpressed as







σ̇(t) =−δσ(t)

if σ(t) ≤ −3s̄,(αB − αA − δ)σ(t) if − 3s̄ < σ(t) < 3s̄, ,−δσ(t)

if σ(t) ≥ 3s̄,2
2
1
1given in Lemma 2.
, αB = αB
or αA = αA
, α B = αB
where αA = αA

It is obvious that when the initial value of σ is either less than −3s̄ or larger than 3s̄,the dynamic behavior of σ is stable. For example, if the initial level of σ is less than −3s̄
so that σA (0) > σB (0) + 3s̄, then σ (t) continues increasing. Since in this situation bothparties cease to invest, the divergence between σA and σB continues reducing at a rateof depreciation, δ. In this sense, if the initial value of σ is out of (−3s̄, 3s̄) , the politicalsystem is stable.
As for the case of σ ∈ (−3s̄, 3s̄) , the dynamic system has a symmetric steady state
where σ = 0. Lemma 2 shows that there are two dynamic equations: σ̇(t) = (αB − αA −δ)σ(t) is expressed as either
"
#
r
σ (t)
4θ
σ̇ (t) =(ρ − δ) − (ρ + 2δ)2 −
3
3ahs̄
or

"

σ̇ (t) =σ (t)(ρ − δ) +3r(ρ + 2δ)2 −Now suppose that ρ < δ. Then, we can obtain [(ρ − δ) −
and

"

1
(ρ − δ) +3r(ρ + 2δ)2 −

#

4θ
.
3ahs̄
p

(21)

(ρ + 2δ)2 − 4θ/3ahs̄]/3 < 0#
4θ  < 0 if s̄ < 4θ/9ahδ(δ + 2ρ),
3ahs̄  > 0 if s̄ > 4θ/9ahδ(δ + 2ρ).

15

(20)

.

Conversely, if ρ > δ, it holds that [(ρ − δ) +"1
(ρ − δ) −3r(ρ + 2δ)2 −p(ρ + 2δ)2 − 4θ/3ahs̄]/3 > 0 and#
4θ  > 0 if s̄ < 4θ/9ahδ(δ + 2ρ),
3ahs̄  < 0 if s̄ > 4θ/9ahδ(δ + 2ρ).

.

Therefore, dynamic equations of (21) and (20) yield the following proposition, which is
the main finding of this paper:
Proposition 1: Suppose that the state variable, σ, is in the range of σ ∈ (−3s̄, 3s̄)and Assumption 2, 3 and 4 hold. Then we obtain three patterns of dynamics listed
below:(i) If s̄ < 4θ/9ahδ(δ + 2ρ) and ρ < δ, then regardless of the equilibrium strategies takenby the parties, σ (t) converges to zero.
(ii) If s̄ < 4θ/9ahδ(δ + 2ρ) and ρ > δ, then regardless of the equilibrium strategies takenby the parties, σ (t) diverges from zero.
(iii) If s̄ > 4θ/9ahδ(δ + 2ρ), there are two possible trajectories of σ (t) , one of whichconverges to the symmetric steady state, while the other diverges from it.
Before discussing Proposition 1, let us make the following assumption.
Assumption 5: θ < 1.
This assumption ensures that we can have all the cases in Proposition 1 under the
restrictions on parameter values given by Assumptions 3 and 4.5
Now, we discuss each case of Proposition 1. First, consider case (i). In this case,even though we have restricted our attention to the linear policy functions, there are dual
equilibrium strategies under which σ converges to zero: see Figure 2(a). Which equilibrium path is actually taken depends on the expectations coordination among both parties
5

More precisely, θ < 1 ensures that the graph of s̄ = (3 − θ)2 /9ah(δ + 2ρ) [ρ(1 − θ) + δ(2 − θ)] is above
the graph of s̄ = 4θ/3ah(ρ + 2δ)2 at least for ρ = 0 and ρ = δ.

16(and the public). Since we have not introduced a specific mechanism of expectations
coordination, the actual trajectory may fluctuate between the two paths in the presenceof extrinsic uncertainty that disturbs expectations of agents. However, regardless of theselection of equilibrium path, the absolute value of σ continues converging to zero. Noticethat in the symmetric steady state where σ = 0, it holds that PA = PB = 1/2, i.e., wehave the highest degree of political instability. Thus case (i) always produces the highest
political instability in the steady-state equilibrium.
[Figure 2 is around here.]
In Proposition 1 (ii), there is no converging path towards the highest political instability (σ = 0) . Regardless of the selection of strategies, the unstable behavior of σ leads thedominance of one party unless the initial value of σ happens to equal zero. Figure 2 (b)depicts case (ii). Proposition 1 (iii) means that dynamic equation (21) is unstable, while11(20) is stable. Namely, the symmetric steady state is stable for αA = αA
, αB = αB, and22
it is unstable for αA = αA
, α B = αB(see Fig.2(c)). If both parties select the second set ofstrategies, then σ diverges from zero. On this equilibrium path, the destiny of the politicalsystem is that party A dominates if the initial value of σ is negative (σB (0) < σA (0)) . The
result is opposite if the initial value of σ is positive (σB (0) > σA (0)). In those situations,the political system maintains complete stability. In contrast, if the first set of strategies
is selected, then the political instability continues rising. Again, if expectations change,
the temporary equilibrium may fluctuate between the stable and unstable trajectories.
If the initial value of σ (t) is out of (−3s̄, 3s̄) , the dynamic behavior of the modelis extremely simple. Since in this region dj (t) = 0, σ (t) changes according to σ̇ (t) =−δσ (t) . Hence, if σ (0) > 3s̄ (resp. σ (0) < −3s̄), then σ (t) reaches 3s̄ (resp. −3s) in a
finite time.

3.4

Discussion

Figure 3 depicts the set of parameters (s̄, ρ) classified according to the each case of Proposition 1. This figure clearly shows that if ρ < δ, then case (i) holds under relatively small
17

levels of s̄ and ρ. Namely, the symmetric steady state (σ = 0) is stable under both sensitive and insensitive strategies. Conversely, as stated in case (ii) of Proposition 1, if ρ > δ,then the system is totally unstable under a smaller level of s̄ and ρ. If s̄ and ρ are large¡¢
enough to satisfy s̄ > 4θ/9ahδ(δ +2ρ), the sensitive strategies αj = αj2 yield an unstable
path diverging from the symmetric steady state, while the path with insensitive strategies¡¢
αj = αj1 is stable. This last case corresponds to (iii) in Proposition 1.
[Figure 3 is around here.]
In order to obtain intuitive implications of the graphical analysis shown above, let us
inspect the investment behavior of each party out of the symmetric steady state. First,
consider the case of σ (t) ∈ (−3s̄, 3s̄) in which both party conduct positive investment.
In this situation, the stability of σ (t) simply depends on the sign ofdσ̇ (t)
i
i
= αB
− αA
− δ (i = 1, 2).
dσ (t)(22)

By use of Lemma 2, we can show the following:
2
1
2
1
∂(αB
∂(αB
− αA
− αA
− δ)
− δ)
< 0,
> 0,
∂ρ
∂ρ
1
1
2
2
∂(αB
− αA
∂(αB
− δ)
− αA
− δ)
< 0,
> 0,
∂s̄
∂s̄

(23)
(24)

i
i
where αA< 0 and αB> 0 (i = 1, 2). Inequalities in (23) mean that a rise in discount rate,

ρ, makes the insensitive strategies (i.e., αj = αj1 : j = A, B) less sensitive, while it makes
the sensitive strategies (αj = αj2 : j = A, B) more sensitive. An increase in the time
discount rate, ρ, means that both parties become more myopic, implying that they have
weaker incentive to invest. As a consequence, investment behaviors of both parties becomeless sensitive to change in σ. On the other hand, the myopic parties have deeper concern
with the probability of winning and the level utility in the near future, which makes themmore sensitive to change in σ. As shown by (23), the first effect dominates the second in
the equilibrium with insensitive strategies (αj = αj1 j = A, B) . In contrast, the second18

effect dominates in the equilibrium with sensitive strategies (αj = αj2 j = A, B).
In a similar vein, inequalities in (24) reveal that a higher variance of political shock,s̄, makes the insensitive strategies (i.e., αj = αj1 : j = A, B) less sensitive, while it makes
the sensitive strategies (αj = αj2 : j = A, B) more sensitive. Since a rise in s̄ implies thatboth parties are more likely to be hit by bigger political shocks, it reduces the expected
gain of investment. As a consequence, the investment behaviors of both parties becomeless sensitive to a change in σ. At the same time, a higher s̄ increases each party’s incentive
to accumulate a larger stock of political asset in order to keep an advantageous position
over the opposite party. This effect makes investment of both parties more sensitive to achange in σ. Inequalities in (23) show that in the equilibrium with insensitive strategies(αj = αj1 j = A, B), the first effect dominates the second. The opposite result holds inthe equilibrium with sensitive strategies (αj = αj2 j = A, B).
Finally, combining Lemma 1 with Proposition 1, we can examine the global dynamics
of our model. If we assume that s̄ < 4θ/9ahδ(δ + 2ρ) and ρ < δ, the interior solutions arestable. Therefore, σ (t) ultimately converges to the symmetric steady state where σ = 0.
In this case, the dynamic system satisfies global stability, which implies that the destinyof political system is totally unstable because Pj = 1/2 in the steady state. In contrast,
if s̄ < 4θ/9ahδ(δ + 2ρ) and ρ > δ, then the interior solutions are unstable. Hence, σ (t)converges to 3s̄ (resp. −3s̄), if the initial value of σ is higher (resp. lower) than zero.
In this situation, the destiny of the political system depends on the history (the initialvalue of the state variable) alone. In the steady state, political system becomes stable inthe sense that one party always holds the office. If s̄ > 4θ/9ahδ(δ + 2ρ), there are one
stable and one unstable path for σ (t) ∈ (−3s̄, 3s̄) . Thus the ultimate state to which thepolitical system converges depends on expectations of agents: if both parties select αj1 ,the symmetric steady state is globally stable. In contrast, if the parties choose a2j , thenthe realized steady-state value of σ will be either 3s̄ or −3s̄, depending on the initial levelof σ (t) . This case is particularly interesting, because it may explain why the similar typesof economies sometimes exhibit substantially different degree of political instability.
19

4Conclusion

The existing literature on the relationship between political instability and economic
performances have assumed that the degree of political instability is exogenously given.
In this paper, we have constructed a simple, dynamic two-party electoral competition
model in which the degree of political instability is endogenously determined. The key
assumption of our model is that the campaign contributions can be accumulated as apolitical asset which is gradually increased by the party’s investment. We formulate theelectoral competition as a differential game and characterized the Markov-perfect Nashequilibrium. By examining the stability of the symmetric steady-state, we have found thatit may be either totally stable or unstable depending on the parameter values involvedin the model. We have also revealed that under certain conditions the symmetric steadystate exhibits a saddle point property: there exist both stable and unstable paths. In thiscase, the path the political system actually follows depends on expectations formation of
agents.
To focus on endogenous determination of political instability, we have assumed away
the production side of the economy. When discussing the effects of political instability on
economic growth and development, we should integrate our model with an appropriate
framework of growth economics. Our next task is to conduct such an extension of ourstudy.

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Austen-Smith, D.: Interest Groups, Campaign Contributions and Probabilistic Voting.
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Economics Review. 42, 1635–51 (1998)

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1482 (1998)

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nomic Dynamics and Control, 31, 31, 79-3202 (2007).

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Proof of Lemma 2.
Substituting (19) into (17) and (18), we find that
·

¸
1 2
1
s̄
RA (σ) = θσ − σ+
,
18hs̄
3h
2h
·
¸
1 2
1
s̄
RB (σ) = θσ + σ+
.
18hs̄
3h
2h

(A.1)
(A.2)

Using (A.1), (A.2), (12) and (13), we obtain following envelope conditions:
·

¸
1
1
00
+ VA (σ) [dB (σ) − dA (σ) − δσ] + VA0 (σ) [d0B (σ) − δ],
σ−
9hs̄
3h
·
¸
1
1
00
0
ρVB (σ) = θσ++ VB (σ) [dB (σ) − dA (σ) − δσ] + VB0 (σ) [−d0A (σ) − δ].
9hs̄
3hρVA0 (σ) = θ

(A.3)
(A.4)

Suppose that the policy functions are linear in σ : dj = αj σ + βj , j = A, B. Then it holdsthat
VA0 (σ) = −aαA σ − aβA ,VA00 (σ) = −aαA ,VB0 (σ) = aαB σ + aβB ,VB00 (σ) = aαB .

Substituting these expressions into (A.3) and (A.4) gives·¸θ−ρaαA −+ aαA (αB − αA − δ) + aαA (αB − δ) σ
9hs̄
¸
·
θ
+
− ρaβA + aαA (βB − βA ) + aβA (αB − δ) = 0,3h¸·θ− aαB (αB − αA − δ) − aαB (−αA − δ) σρaαB −9hs̄·¸θ+ ρaβB −− aαB (βB − βA ) − aβB (−αA − δ) = 0.
3h

23

(A.5)

(A.6)

If both (A.5) and (A.6) hold for any feasible σ, we should have−ρaαA −

θ+ aαA (αB − αA − δ) + aαA (αB − δ) = 0,
9hs̄

(A.7)

θ
− ρaβA + aαA (βB − βA ) + aβA (αB − δ) = 0,3hθρaαB −− aαB (αB − αA − δ) − aαB (−αA − δ) = 0,9hs̄θρaβB −− aαB (βB − βA ) − aβB (−αA − δ) = 0.
3h

(A.8)
(A.9)
(A.10)

Combining (A.7) and (A.9), we obtain:2.
(ρ + 2δ)(−αA ) − (−αA )2 = (ρ + 2δ)αB − αB

The above means thatαA = −αB ,(A.11)

In view of (A.11), (A.7) and (A.9) present two sets of solutions:
q

q
αA =

−(ρ + 2δ) +4θ(ρ + 2δ)2 − 3ahs̄

6, αB =

4θ(ρ + 2δ)2 − 3ahs̄(ρ + 2δ) −

6

,

and
q
αA =

−(ρ + 2δ) −q(ρ + 2δ)2 −

4θ
3ahs̄

6

, αB =(ρ + 2δ) +4θ(ρ + 2δ)2 − 3ahs̄

6.

Next, consider βA and βB . Combining (A.8) and (A.10) and using (A.11), we obtain:√
ρ+ D· (βB − βA ) = 0,2which implies βA = βB . As a result, using (A.8) and (A.10), βA and βB are given by:

βA = βB =θ.
3ah(ρ + δ − αB )24

¤

Proof of Lemma 3.
(i) Suppose that σ ≤ −3s̄. Using Lemma 1, HJB equations of each party is given by:
n
o
a 2
0
ρVA (σ) = max θy − · dA + VA (σ)[dB − dA − δσ] ,
dA
2
o
n a
ρVB (σ) = max − · d2B + VB0 (σ)[dB − dA − δσ] .
dB
2Kuhn-Tucker conditions are given by:−adA − VA0 (σ) ≤ 0, dA ≥ 0,

(B.1)

−adB + VB0 (σ) ≤ 0, dB ≥ 0.

(B.2)

dA = dB = 0,

(B.3)

Now let us assume thatare optimal policy functions. Substituting (B.3) into HJB equations, we obtain following
differential equations:θy δVA0 (σ)
−
· σ,
ρ
ρ
δV 0 (σ)
VB (σ) = − B· σ.
ρVA (σ) =Solving these differential equations, we obtain:

VA (σ) =θy,ρVB (σ) = 0.

25

(B.4)
(B.5)

Then we obtain:
VA0 (σ) = VB0 (σ) = 0.
Therefore, the strategies given by (B.3) fulfill all of Kuhn-Tucker conditions of (B.1) and(B.2). As a result, we conclude that the strategies given by (B.3) are optimal policy.
(ii) The proof of Lemma 3(ii) is similar to that of Lemma 3(i) and is hence omitted.
¤

26

Figure 1: The figures displays the both parties’ investment level.

274µ(a) The case of s¹ < 9ah±(±+2½)
and ½ < ±4µand ½ > ±(b) The case of s¹ < 9ah±(±+2½)4µ(c) The case of s¹ > 9ah±(±+2½)

Figure 2: The figures display the dynamics of σ. The panel (a) illustrates the caseof ρ < δ and s̄ < 4θ/9ahδ(δ + 2ρ); the panel (b) illustrates the case of ρ > δ ands̄ < 4θ/9ahδ(δ + 2ρ); the panel (c) illustrates the case of s̄ > 4θ/9ahδ(δ + 2ρ).

28

(3¡µ)29ah± 2 (2¡µ)

4µ9ah± 2(3¡µ)227ah± 2 (3¡2µ)4µ12ah±22(3¡µ)
s¹ = 9ah(±+2½)[½(1¡µ)+±(2¡µ)]4µs¹ = 9ah±(±+2½)4µ27ah±2
Ã
2

s¹ =

4µ
3ah(½+2±)2

Figure 3: The figures display the set of parameters (s̄, ρ) classified according to the eachcase of Proposition 1.

29