In this note, we propose a simple infinite horizon of elections with two candidates. We suppose that the government policy presents some degree of inertia, i.e. a new government cannot completely change the policy implemented by the incumbent. When the policy inertia is strong enough, no party can win the elections a consecutive infinite number of times.Policy inertia

RaphaÃ«l Soubeyran

English

Research Papers in Economics

When Inertia Generates Political Cycles 
Raphaël Soubeyran
GREQAM, University of the Mediterranean
Abstract
In this note, we propose a simple infinite horizon of elections with two candidates. We
suppose that the government policy presents some degree of inertia, i.e. a new government
cannot completely change the policy implemented by the incumbent. When the policy inertia
is strong enough, no party can win the elections a consecutive infinite number of times.
I am very grateful to Richard McLean and to an anonymous referee.
Citation: Soubeyran, Raphaël, (2006) "When Inertia Generates Political Cycles." Economics Bulletin, Vol. 4, No. 31 pp. 1-8
Submitted: August 17, 2006.  Accepted: October 2, 2006.
URL: http://economicsbulletin.vanderbilt.edu/2006/volume4/EB-06D70008A.pdf1 Introduction
In modern democracies, the alternation of political parties in power is a fre-
quent phenomenon. Why isn’t there a greater persistence of parties in power?
How can one explain the turnover of parties in government? How can one
explain political cycles? In a dynamic setting, we introduce two imperfec-
tions to the model of Downs (1957). As other scholars (see Casamatta and
De Donder 2005), I consider a programs inertia assumption. As Heckelman
(2000), I consider that parties have (diﬀerent) ﬁxed programs. The second,
we call policy inertia represents imperfections in changing the government
policy. This eﬀect has various origins in real political life. Indeed, a majority
of policies cannot be changed without any cost. In the model, I distinguish
the eﬀective policy, which is the objective state of the policy, and the gov-
ernment policy, which is the policy of the incumbent during his legislature.
We suppose that the eﬀective policy at time t is a convex combination of
the eﬀective policy at date t − 1 (past policy) and of the government policy
at time t. This assumption captures the fact that the government cannot
freely change the policy implemented in the past. I argue that this can be a
reason why similar countries with diﬀerent histories implement very diﬀerent
policies. I say that the model exhibits political cycles when no party can win
the elections a consecutive inﬁnite number of times. In this situation, parties
indeﬁnitely alternate (not necessary regularly) in power.
The main result of this note states that if the policy inertia is high enough,
political cycles appear. Furthermore, I show that the dynamic can be his-
tory dependent. Indeed, if the median voter is indiﬀerent between the two
candidates programs at an election, the future dynamic will be dramatically
aﬀected. Scholars generally explain political cycles with psychological argu-
ments (see Goertzel 2005 for a review of the American voters mood changes
literature). Schlesinger (1949, 1986, 1992) considers that the electorate is
inevitably disappointed by the party or the ideology that is in power. Klin-
berg (1952) suggests that American mood in public opinion balances between
introversion and extroversion. This could explain why domestic and foreign
concerns alternate through time and parties turnover in power. The main
explanation is certainly disappointment. The “Negativity eﬀect” theory (see
Aragones 1997 for a survey) is build on the following remark: voters’ deci-
sions are based on the incumbent’s past performance and negative pieces of
information have a greater impact on voters decisions than positive pieces of
information. In the light of the negativity eﬀect, Aragones (1997) obtains a
1result of systematic alternation of two parties implementing diﬀerent policies.
In our analysis, there is no uncertainty and electorate decisions are not based
on the government past performance, but as usually in political models, for
their preferred program at each election. I propose a simple model suggesting
that political cycles can be generated by inertia only.
2 The model
The policy setting: Two candidates L and R compete in an inﬁnite horizon
elections setting. The set of policies is the interval P = [−1,1]. Each citizen
is represented by a bliss point b αi in the set of policies and voters bliss points
are distributed over P. according to the cumulative distribution function
F. The utility function of voter i is deﬁned over the set of eﬀective policies,
ui (pt) = −|b αi − pt| where pt is the eﬀective policy at time t. We suppose that
candidates programs are ﬁxed. Candidate j proposes a policy zj. We assume
that the median voter preferred policy, b αm is such that: zL < b αm < zR.
Policy inertia: We suppose that a policy implemented at date t has
an inﬂuence on the eﬀective policy at date t + 1. Let pt+1 be the eﬀective
policy at date t + 1. We suppose this policy results from the government
policy at date t + 1 and from the eﬀective policy at date t. Let δ ∈ [0,1]
be the ”inertia” degree of past policies. Let zW(t+1) denotes the program
implemented in period t + 1 by the elected party W (t + 1) ∈ {L,R}. The
eﬀective policy at date t + 1 is then:
pt+1 = (1 − δ)z
W(t+1) + δpt,
In the case without policy inertia (δ = 0) there is no linkage between
the successive elections. In the second polar case with full policy inertia
(δ = 1), the policy is completely ﬁxed and voting has no inﬂuence on the
policy implemented. In the following, we will consider that δ < 1. Between
the two polar situations, a new government will have to face an inertia force
δpt, which can be interpreted in many ways.
Although the preferred eﬀective policy of a voter is ﬁxed, his preferred
program changes from an election to the following. To illustrate the dynamic,
let compute voter i preferred program (noted pi
t+1) at election t + 1:
p
i
t+1 =
b αi − δpt
1 − δ
,
2Then, the more the eﬀective policy of the previous period was leftist, the
more the median voter will move to the right, and the more the eﬀective
policy of the previous period was rightist, the more the median will move to
the left. This intuition underline the swing of the voters.
Policy history: We suppose that the ﬁrst election take place in period
1, the inﬂuence of period 0 depend on a degree of inertia δ0 not necessarily
equal to δ and a past policy p0. These parameters can represent diﬀerent
histories. For example δ0 = 0 could refer to a revolution preceding the ﬁrst
democratic election in t = 1 and be interpreted as the fact that past policy
is completely removed.
3 Political Cycles
We say that the set of parameters (δ, b αm,zL,zR) exhibits Political cycles if no
party can win the elections an inﬁnite consecutive number of times, formally:
Deﬁnition 1 A set of parameters (δ,p0, b αm,zL,zR) ∈]0,1[×[0,1]4 exhibits
Political cycles if and only if W (W(t) ∈ {L,R}}) does not converge when
t goes to inﬁnity.
A ﬁrst important remark, is that in situations where the median voter
is indiﬀerent between the two policies, the dynamic strongly depends on the
randomized winner. Indeed, if the median voter is indiﬀerent between the
two parties in election t, he will not be indiﬀerent in election t + 1. Let vm
t
be the utility of voter m over the set of programs at date t:
v
m
t (z) = u
i ((1 − δ)z + δpt−1),
The following result is straightforward at the light of the previous remark.
Let e pm be the eﬀective policy such that the median voter is indiﬀerent be-
tween both parties, it is deﬁned by
u
m  
(1 − δ)z
L + δe p
m
= u
m  
(1 − δ)z
R + δe p
m
.
Remark that this equation implies that zL < e pm < zR.
Now suppose that in election t − 1, pt−1 = e pm. In this situation, the
median voter is indiﬀerent between the two parties and the result of the
election t is undetermined. Suppose that L is the winner, then the policy
3implemented is zL and pt = (1 − δ)zL + δe pm < e pm. This implies that the
median voter (and then a majority of voters) will vote for party R. With the
same argument, if R wins at time t, then L wins at election t + 1. We can
conclude that:
Lemma 2 If pm
t−1 = e pm then W(t − 1) 6= W(t).
Hence, the situation where the median voter is indiﬀerent between both
parties is not stable. Now we characterize the set of parameters such that
political cycles appear:
Proposition 3 The set of parameters (δ,p0, b αm,zL,zR) exhibits Political cy-
cles if and only if 1−δ
1+δ ≤ zR−b αm
b αm−zL ≤ 1+δ
1−δ. , W converges to L if and only if
1+δ
1−δ < zR−b αm
b αm−zL, and W converges to R if and only if zR−b αm
b αm−zL < 1−δ
1+δ.
Proof. Suppose 1+δ
1−δ ≥ zR−b αm
b αm−zL ≥ 1−δ
1+δ.
step 1: Suppose that W converge to L. Then, there exists an election k such
that ∀t ≥ k, W (t) = L. This is implies ∀t ≥ k,
 pm
t+1 − zL  ≤
 zR − pm
t+1
 .
Furthermore,
pt+1 = (1 − δ)z
L + δpt,
Then,
pt+1 − z
L = δ
 
pt − z
L
,
Since δ ∈]0,1[, lim
t→+∞
 
pt+1 − zL
= 0. Finally, lim
t→+∞pt+1 = zL. Now, we must
have that the median voter (strictly, because of the Lemma) prefers L to R:
u
i  
z
L
− u
i  
z
R
= −
   
b αm − zL
1 − δ
    +
   z
R − b α
m − δ
b αm − zL
1 − δ
    > 0,
Since b αm > zL, this is equivalent to:
   z
R − b α
m − δ
b αm − zL
1 − δ
    >
b αm − zL
1 − δ
, (1)
This inequality is equivalent to:
zR − b αm
b αm − zL >
1 + δ
1 − δ
if
zR − b αm
b αm − zL >
δ
1 − δ
,
zR − b αm
b αm − zL < −1 if
zR − b αm
b αm − zL ≤
δ
1 − δ
,
4The second case cannot hold because zR−b αm
b αm−zL > 0, then (1) is equivalent to:
zR − b αm
b αm − zL >
1 + δ
1 − δ
,
Contradiction.
step 2: Suppose that W converge to R. Then, there exists an election k
such that ∀t ≥ k, W (t) = R. This implies ∀t ≥ k,

zR − pm
t+1

 ≤

pm
t+1 − zL
.
Furthermore,
pt+1 = (1 − δ)z
R + δpt,
Then,
pt+1 − z
R = δ
 
pt − z
R
,
Since δ ∈]0,1[, lim
t→+∞
 
pt+1 − zR
= 0. Finally, lim
t→+∞
pt+1 = zR. Now, we must
have that the median voter prefers R to L (strictly, because of the Lemma).
u
i  
z
R
− u
i  
z
L
= −
   
zR − b αm
1 − δ
  
 +
  
b α
m − z
L − δ
zR − b αm
1 − δ
  
 > 0,
Since zR > b αm is equivalent to:
zR − b αm
1 − δ
<
  
b α
m − z
L − δ
zR − b αm
1 − δ
 
 ,
Equivalent to:
zR − b αm
b αm − zL <
1 − δ
1 + δ
if
zR − b αm
b αm − zL <
1 − δ
δ
,
zR − b αm
b αm − zL < −1 if
zR − b αm
b αm − zL ≥
1 − δ
δ
,
The second case cannot hold because zR−b αm
b αm−zL > 0, then this is equivalent to:
zR − b αm
b αm − zL <
1 − δ
1 + δ
,
Contradiction.
Now, suppose 1+δ
1−δ < zR−b αm
b αm−zL, this is equivalent to b αm <
zR(1−δ)+zL(1+δ)
2 .
Furthermore,
pt = (1 − δ)
t X
k=1
δ
t−kz
W(k) + δ
tp0,
5Then,
p
m
t+1 ∼
t→+∞
b αm
1 − δ
−
t X
k=1
δ
t+1−kz
W(k),
Thus,
p
m
t+1 <
t→+∞
zR + zL (1+δ)
1−δ
2
−
t X
k=1
δ
t+1−kz
W(k) <
zR + zL
2
,
Then W (t) converges to L.
Finally, suppose zR−b αm
b αm−zL < 1−δ
1+δ, this is equivalent to b αm >
zR(1+δ)+zL(1−δ)
2 .
Then,
p
m
t+1 >
t→+∞
zR + zL
2
+
δ
1 − δ
z
R −
t X
k=1
δ
t+1−kz
W(k),
Furthermore, ∀k, zW(k) ≤ zR, then
t X
k=1
δ
t+1−kz
W(k) ≤
t→+∞
δ
1 − δ
z
R,
Thus,
p
m
t+1 >
t→+∞
zR + zL
2
,
Then W (t) converges to R.
Finally, when the policy inertia degree is high enough

|zR+zL−2b αm|
zR−zL ≤ δ

,
parties alternate in power. Indeed, if party L is the incumbent, he will
necessary loose one future election. Then R wins the power and he will
necessary loose one future election, and so on...
We have claimed that the dynamic can take two diﬀerent paths when the
median voter is indiﬀerent between L and R. Indeed, suppose the median
voter is indiﬀerent in election t, formally,
 pm
t − zL  =
 pm
t − zR . Since zR 6=
zL, we have pm
t = zL+zR
2 . Now we compare the case where L wins the election
to the case where R wins the election.
If L wins the election, pt = b αm + (1 − δ) zL−zR
2 . If R wins the election
pt = b αm + (1 − δ) zR−zL
2 . Then the dynamic can change dramatically.
64 Conclusion
We have proposed a simple inﬁnite horizon dynamic model of elections where
candidates have ﬁxed programs and policies present some degree of inertia.
We have shown that inertia can generate political cycles.
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[3] Downs, A. (1957) An Economic Theory of Democracy, New York: Harper.
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An Economic Theory of Democracy,

Cycles in the Mass Psychology of American Electoral Behavior”, forthcoming in

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On the inﬂuence of extreme parties in electoral competition with policy-motivated candidates”

Paths to the Present,

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When Inertia Generates Political Cycles

http://www.accessecon.com/pubs/EB/2006/Volume4/EB-06D70008A.pdf

When Inertia Generates Political Cycles

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