Political Alternation : a suggested
interpretation∗
Pascal GAUTIER†
Raphael SOUBEYRAN‡
March 2004
Abstract
This paper proposes an explanation for the instability of parties in
power in democratic regimes. We consider a dynamic political game
with two opportunistic parties/candidates, di erent in the sets of programs they can propose. Citizens have political preferences de ned
over couples of durables public goods. We show that political alternation is more likely to happen when (i) political competition is polarized, (ii) median voter is moderate, (iii)and public goods have a low
depreciation rate. We then suggest several speci cations for parties
motivations and study how they a ect alternation. Finally, we discuss
simulated political histories.
keywords : political alternation, parties abilities and motivations,
durable public goods.
JEL classi cation :
∗
The authors thank Francis Bloch and Didier Laussel for their constant support as well
as seminars or workshops participants for helpful comments.
†
GREQAM, e.mail: pascalg [email protected], tel.0033 − (0)6 − 03 − 15 − 17 − 48
1
1
Introduction
Formal analysis of politics has a long history in economics and political
science. Since Downs [1957] and Black [1958] seminal work, political economics has became a very dynamic research area and analyzed many aspects
of democratic and/or economic life1 . Three main general topics have been
explored :
- political constraints on (governments) economic decisions.
- institutional design.
- political behaviors (politicians, parties, lobbies or voters).
Bizarrely, political alternation, that is the change of party in power after
an election, remains extremely poorly studied. Yet this phenomenon, potentially connected with all the quoted topics, has occurred in more than 50%
of elections in a majority of occidental countries since the 70th (France is a
very spectacular example with systematic alternation since 1981). Indeed,
alternation is probably one of the most robust stylized facts of political life2 !
The main ambition of this paper is to propose a stylized model of political
competition, which could (at least partially) explain why alternation is so
frequent.
First, one can observe that a priori ideological preferences (partisanship),
relying on party identity rather than on programs, don’t explain alternation
very well. If one considers the electoral competition as an opposition between
left and right and asks voters how they feel about this cleavage, many of
them seem to be ideologically attached at one of the two sides (see for instance
Mayer and Perineau [90] and [97] for the French case). Furthermore, such an
attachment is known to be quite stable. So one may expect that the party
with the largest support in a given population always wins the election.
This is not the case, even if one sometimes observes long periods of political
stability.
A rst plausible explanation for alternation is then turnout. What is crucial, the day of the election is not the ideological structure of the citizenship,
but the composition of the e ective voting population (electorate). In the
past few decades, abstention has became a very strong phenomenon, and,
if we assume that from an election to the following those who abstain are
not systematically the same3 , these modi cations in electorate might explain
changes in elections results. But alternation occurs even in countries where
1
Persson and Tabellini [2000] and Drazen [2000], for instance, o er nice surveys of the
eld.
2
One may emphasize that alternation is probably the essence of democracy. This is
certainly true, but this statement doesn’t spare us of trying to explain why it occurs!
3
Either for rational behavior or stochastic reasons.
2
voting is obligatory, which militates for other kinds of explanations4 .
Our view is that one can approach alternation in a more general framework, without regard of any change in electorate structure and/or size, from
an election to the following. The starting point of this work relies in a very
simple observation : many individuals sometimes vote for a party that is
not the one they announced as their "favorite". In a single issue election,
this could hardly happen (at least with rational and well informed agents),
because the ideological dimension translates into the (unique) political dimension. But when political competition is multi-dimensional, there is no
reason to believe that a unique candidate is viewed as better than any other
on every dimensions by a given citizen. This might be a reason for an "unfaithful" behavior.
Two related questions arise at this point : What makes a given individual
consider one party as his favorite? Why, then, does this individual sometimes
vote for an other candidate?
Our answer to the rst question is very simple, and (we believe) quite
intuitive : among the di erent policies that enter parties platforms, any
individual considers one as being more important than the others. The party
sharing the same opinion (or for which voters believe it does, or which has
o ered more of this policy in the past) will then be his favorite. This, in a
very large sense, can be assimilated to ideology.
Our answer to the second question is connected to what political scientists
call "issue voting", namely the fact that, during electoral campaigns, one
aspect of platforms emerges as a priority in a large part of society. Such an
issue has two main features : rst, the distribution of voters who consider it
as a priority is largely independent of ideological a liations. Second, it is not
systematically the same from an election to the following. Here, it is crucial to
distinguish between voters preferences, and voters priorities. We claim than
preferences are, for a large part, independent of the place and the moment,
while priorities depend upon circumstances. Imagine, for instance than you
prefer beef to chicken (but like both); if the university restaurant has been
serving beef every day for six month, you would probably enjoy some chicken.
Basically, the idea is that even with stable preferences, what individuals want
depends upon what they got in the past. Obviously, to capture this fact in
a political context, policies implemented during a legislature must have an
impact on next periods. In this work, we will then assume that policies
(public goods) are durable. Examples of such durable goods are national
4
One may suggest that, in these countries, "vote blanc" plays a similar role. We rather
believe that "vote blanc" results from quite di erent incentives than abstention. This
would probably deserve more investigation but this is not our purpose here.
3
security, public education, justice etc.
Another stylized fact is that political parties are considered as di erent
in their capacity to implement policies by the electorate. So, there exist
constraints limiting their ability to settle their platforms5 . This assumption
is consistent with one of the more salient features of empirical electoral studies
: voters believe that politicians skills are not identical. For instance, rightwing parties are often considered6 as being better than left-wing parties for
decreasing taxes or security programs. On the other side, left-wing parties
are viewed as more e cient in welfare state programs.
Interpreting these empirical results in term of e ective ability is not really
satisfactory. Political power mainly relies on bureaucracy7 , which has a great
role in implementing policies de ned by government and a large majority
of state employees keep their job even if the government is not reelected.
So there is no reason to believe that parties (once in power) have di erent
technologies . Another way to consider this question is that even if all
politicians have the same technical capacity to implement a policy, there
exist ideological or political constraints that limit or in uence their action
(lobbies, past action in o ce, ideology...). Whatever the interpretation we
give, the crucial point for our model is that voters believe that parties are
di erent and that parties know what are these beliefs.
As already noticed, there is very few literature studying political alternation. Many papers study the links between changes in power (or proximity
of elections) and changes in policies implemented8 , but they don’t try to
explain alternation itself. Several empirical studies exhibit a link between
incumbent defeat and in ation or unemployment..., but to our knowledge,
there is no attempt to rationalize these points. The closest work to ours is
probably the one by Roemer [1995]. Roemer considers a model of electoral
competition a la Wittman (i.e. with partisan politicians) with uncertainty.
More precisely, individual preferences for public and private goods depend
upon a random variable (state of the Nature) whose distribution is in uenced
by past political history. Although it is not the subject of the paper, this
model exhibits some alternation. We share with this work the very general
idea that political history has an impact on elections. Yet, we show that one
does not need any form of uncertainty to explain it.
Another related stream of literature is the one devoted to "divided government" (Alesina and Rosenthal...). As in this family of models, we assume
that voters want a relative mix of several policies, no party being able to o er
5
At least in a world of credible programs !
Even in the part of the electorate that considers itself as left-wing.
7
Laver and Shepsle [1994] for instance emphasize this point.
8
See for instance the huge literature about "political-business cycles".
6
4
them an optimal mix. Then, facing two elections (presidential and congress,
usually), voters elect candidates of di erent sensibilities in order to get a less
extreme average policy, closer to their bliss point. The contribution of our
work is to consider this idea in a multi period perspective, with only one election per period. This dynamic setting has the advantage of being applicable
to parliamentary regimes, where political power is unique.
The rest of the paper is organized as follows. Section 2 presents the
main features and assumptions of our model. Section 3 considers a oneperiod game and the nature of its equilibria. Sections 4, 5, 6 and 7 propose
re nements in political parties goals and strategies and study the dynamics
of political equilibria and alternation in o ce. In section 8, we discuss some
speci c aspects of our model. Finally, section 8 concludes.
2
Presentation of the model
2.1
Durable Public Goods
We assume there exist two durable public goods, X and Y. One can think
for instance of national defense, schooling system, public equipments, public
research, etc. At period t, any party p proposes a couple (xpt , ytp ). From a
period to the following, quantities depreciate linearly at rate δ ∈ [0, 1].
Let yts , xst be the stocks of X and Y at the beginning of period t, xit , yti
the quantities produced by incumbent at time t (that is after election), and
Xt , Yt the total quantities of goods available during the legislature t. Then:
s
yt+1
= (1 − δ)[yts + yti ] = (1 − δ)Yt
xst+1 = (1 − δ)[xst + xit ] = (1 − δ)Xt
2.2
The Political Parties
We consider a two-parties/candidates (L and R) electoral competition.
These candidates only care about being in o ce and don’t have any preferences on policies ("opportunist politicians"). Once elected, they implement
their campaign promises ("preelection politics").
Each party proposes a vector (couple) of public goods quantities, respect
to a budget constraint : at any period, incumbent is confronted to limited
amount of resources ( scal resources, for instance9 ), normalized to 1.
9
Note that, in general, literature on political competition with furniture of public good
considers that programs include a tax rate which aims to nance this good. This is not
the case here, rather, we left scal aspects in the background. The main reason for this
choice is that we don’t consider redistributive aspects of political process.
5
The crucial assumption is that voters believe candidates have di erent
abilities. More generally, as observed in many empirical studies, each party
is viewed as better than the other for the production of one of the public
goods.
Formally :
cL (x, y) = y + λL x ≤ 1
cR (x, y) = x + λR y ≤ 1
Where cL (.) and cR (.) are respectively the cost function for party L and for
party R, λL > 1 and λR > 1 measure each party relative ine ciency in
implementing one of the policies : L (resp. R) produces the good Y (resp.
X) with a lower relative cost than its competitor.
A direct consequence of previous equations is that candidates have different sets of possible programs, i.e. some programs can be proposed both
by Left-wing and Right-wing candidate, some others can be propose by L,
and some others only by R. More generally, electoral platforms must rely in
speci c subsets of <2+ , for ability or credibility reasons, that is candidates
don’t have a total latitude when determining their position10 .
Once in power, an incumbent has the possibility to "sell" part of the
stocks of public goods remaining from previous periods. Reminding that Xt
and Yt are the e ective total quantities of X and Y at period t (that is the
sum of remaining stocks and new productions), one must respect :
Xt = xst + xit ≥ 0
Yt = yts + yti ≥ 0
Note that these constraints are compatible with negative furniture of
public goods ( xit ≤ O or yti ≤ O), which would simply mean that incumbent
has destroyed part of the stock of one public good in order to have the
possibility to produce more of the other one.
Furthermore, one can notice that :
xt = Xt − xst = Xt − (1 − δ)Xt−1
yt = Yt − yts = Yt − (1 − δ)Yt−1
This leads to new constraints, in terms of total available quantities of
public goods11 :
10
X and X propose a model that describes the formation of such credibility intervals,
though in an the unidimentionnal setting.
11
We obtain BL,t and BR,t in replacing : xt by Xt −(1−δ)Xt−1 , and yt by Yt −(1−δ)Yt−1
in cL (.) and cR (.).
6
CL (Xt , Yt /xst , yts ) = Yt + λL Xt ≤ BL,t = 1 + (1 − δ)(λL Xt−1 + Yt−1 )
CR (Xt , Yt /xst , yts ) = Xt + λR Yt ≤ BR,t = 1 + (1 − δ)(Xt−1 + λR Yt−1 )
Graph X : FPP
2.3
The Electorate
Preferences of the electorate are de ned, at any period t, over couples of
available public goods quantities (Xt , Yt ) = (xst + xpt , yts + ytp ). Indeed, voters
consider what e ective available quantities of public goods will be during the
legislature, that is not only the "new productions" of X and Y.
We assume that there is a continuum of voters. Any individuals prefers
one of the two goods, but there exist a certain substitutability between
them12 . A natural way to modelize these assumptions is to consider standard
Cobb-Douglas utility functions :
U i (Xt , Yt ) = (xst + xpt )αi .(yts + ytp )1−αi = Xtαi Yt1−αi
We will say that if αi > 12 then i prefers X to Y, if αi < 21 then i prefers
Y to X, and if αi = 12 then i likes X and Y equally. In the largest possible
sense, one can consider that parameter αi captures "ideology"13 .
Obviously, because of the durable nature of public goods and the existence of stocks at the beginning of every period, voters’ bliss point will be
di erent from an election to the following. The main intuition of the model
then appears clearly : political history has an in uence on economic environment, which, in turn, in uences the electorate wishes. Furthermore, because
parties are di erent, this leaves some space for possible alternation in o ce,
if incumbent does not have the technical ability to o er what median voter
requires.
12
Please note that under these speci cations, the way we consider issue voting is somehow "weak", in the sense that voters may agree to sacri ce a given amount of their favorite good if one o ers them a "big" quantity of the other public good. See Gautier and
Soubeyran (2003) for a model considering "sharp" issue voting, that is without possibility
of substitution.
13
It is interesting to notice that in our formulation, there is a superposition between
ideology and rationality. We have de ned policies in a very general way, which allows us
to imagine that, for instance, some voters economically su er from one of them (imagine
that one of the public good is "income equality"...rich voters would be disadvantaged by
such a public good). then, we don’t have to consider, as it is done in many papers, that
there is a sort of asymmetry between policies, one of them being chosen following egoistic
consideration (taxation, for instance) and the other being ideological...in the sense that it
has no economic interest (let’s say religion)!
7
Note that even if bliss points are not the same from an election to the
following, preferences do not change across time, each voter i being characterized by a constant parameter αi . But it would be formally equivalent to
consider that preferences evolve endogenously across time, if we de ne Uei (.)
as an endogenous utility function such that:
∀xst , yts , xt , yt , Uei (αie , xst , yts , xt , yt ) = U i (Xt , Yt )
This equation de nes implicitly αie as a function of xst , yts , xt , and yt . Yet,
we believe that interpreting our model via the impact of political history is
more relevant than via some changes in preferences14 .
A useful property of these utility functions is they respect the Single
Crossing Property (Gans and Smart [96]) on the global budget set. So we
know that, at any period t, a Condorcet winner will exist. Let us rst present
the de nition of Single Crossing Property we are going to use :
De nition 1 Let Ω ⊆ <2 be the set of technically feasible programs (Z),
and A ⊆]0, 1[ the set of individual heterogeneity parameters (α). If for all
Z1 , Z2 ∈ P , U αk (Z1 ) = U αk (Z2 ) has at most one solution in A, then U i (.) =
u(., αi ) satis es the Single Crossing Property.
Lemma 1 In the model, the Single Crossing Property is veri ed on Ω ⊆ <2
if and only if at most one party do not proposes a strictly positive quantity
of one good.
This lemma allows us to say that a Condorcet winner exists15 and that it
is the program preferred by the voter with median heterogeneity parameter,
αm ∈]0, 1[. Then, in the following, we can concentrate on the behavior of
this particular voter.
3
One period game and Political Equilibria
In this section, we consider a one-period game. Because of his pivotal
role, we rst describe median voter bliss-point and then specify which are the
14
Roemer introduced stochastic changes in voters preferences in a model of electoral
competition, relying on a "disappointment" argument. He obtains simulations in which
there can be political cycles and alternation. Our ambition, here is to propose an explanation to such phenomenons without any stochastic element.
15
We have assumed, for technical reasons that both candidates can not propose simultaneously nil quantities of public goods. One could like to know how this a ects our
conclusions. Indeed, such situations can not be political equilibria, so nothing would be
modi ed in the following results.
8
equilibria of the electoral competition. The main point is that, because of
di erences in technologies, only one candidate has (in general) the possibility
to propose a winning program. Furthermore, the Condorcet winner is not
the unique wining program.
3.1
Median voter’s choice
Utility during legislature t depends on stocks and on quantities produced
by new incumbent. Median voter chooses (Xt , Yt ) such as her utility is maximized. Her optimization program (M) is then :
M AX[U m = (Xt )αm (Yt )1−αm ]
r.t. (Xt , Yt ) ∈ <+ × <+
s.t. (BL): CL (Xt , Yt ) ≤ BL,t or (BR): CR (Xt , Yt ) ≤ BR,t
m
m
) be the solution of program (M) without constraint (BL)
, YR,t
Let (XR,t
m
m
and (XL,t , YL,t ) be the solution of program (M) without constraint (BR).
Then, basic calculations lead to the following results :
m
XR,t
= αm BR,t
m
YR,t
1 − αm
=
BR,t
λR
αm
BL,t
λL
= (1 − αm ) BL,t
m
XL,t
=
m
YL,t
The mainline of the reasoning will be to compare maximal utility that
median voter obtains if Left party or Right party wins the election. It will be
useful to de ne the function ∆Utm (.) as, for median voter, utility di erence
between best program R can o er to her and best program L can o er to her
at election t (given the existing stocks).
m
m
m
m
∆Utm ≡ U m XR,t
, YR,t
− U m XL,t
, YL,t
The best platform for median voter (we shall note it (Xtm , Ytm )), is then :
m
m
(XR,t
, YR,t
) if ∆Utm (.) > 0
and
m
m
(XL,t
, YL,t
) if ∆Utm (.) < 0.
9
3.2
Political equilibria
The game is sequenced as follow. First, voters and candidates observe
available stocks of both goods. Second, parties simultaneously choose their
platforms. Then elections are hold and, nally, new incumbent implements
its program.
Because of the Single Crossing Property, we know that their exist a Condorcet winner, namely the platform preferred by voter with median heterogeneity parameter (median voter). But because of di erences in technologies,
only one candidate can propose this platform in general16 . This appears
clearly on the following graph.
GRAPH X : FPP + isoquantes et bliss point de l’electeur median.
What does determine which candidate will win the election ? One can
answer in considering the sign of ∆Utm (.) :
m
m
− λαRm −1 ]
− λαRm −1 ]xst − [λ−α
∆Utm (.) ∝ [λαRm − λL−αm ]yts − [λ1−α
L
L
Recalling that the winning party will be R if ∆Utm (.) > 0, and L if
∆Utm (.) < 0, it is clear that the more important is the stock of public good
Y, and the smallest is the stock of public good X when the election takes
place, the more likely is R to win. The reason is obvious : median voter will
require "a little" of Y and "a lot" of X (which is the public good R is the more
e cient to produce). A symmetric argument describes the circumstances
under which L is more likely to be elected17 .
Furthermore, it is important to notice that winning candidate18 has a
certain latitude when choosing its platform. Indeed there exists a full set of
winning platforms. More precisely, any feasible program such that the other
party can’t o er a better platform to median voter, is a winning program.
m
Let us then de ne Φm
L,t (resp. ΦR,t ) as the set of programs considered by
median voter as better than best program L (resp. R) can o er to her :
n
o
n
o
+
+
m
m
m
m
Φm
L,t = (X, Y ) ∈ < × < /U (X, Y ) > U (XL,t , YL,t )
+
+
m
m
m
m
Φm
R,t = (X, Y ) ∈ < × < /U (X, Y ) > U (XR,t , YR,t )
16
A very special case occurs when both L and R can o er equivalent programs (in term
of utility) to median voter. Graphically, one would observe than indi erence curve is
tangential to both production possibilities frontiers.
17
One can observe that when λR = λL = 1, ∆Utm = 0. This is logical, because
candidates then have the same technologies, and both of them can systematically propose
the platform preferred by median voter.
18
The candidate which is certain to win the election to say it in a more rigorous manner.
10
Let W (t) be a map valuated in {R,L,RL} such that :
W (t) = R means that R wins the election at time t,
W (t) = L means that L wins the election at time t,
W (t) = RL means that R and L win the election at time t with proba 12 .
Finally, let Et be the set of winning programs (with certainty) at period t.
Then the following proposition summarizes which are the winning platforms in a one-election setting. Of course, any couple of strategies such that
one of them belong to Et , is a political equilibrium, because the candidate
who is sure to loose can propose any (feasible) program.
Proposition 1 (Winning strategies) .
(i) If ∆Utm (.) > 0, then W (t) = R, Et = Φm
B 6= L,t ∩
o
n R,t
m
m
m
, YRm (t))
, YLm (t)), (XR,t
(ii) If ∆Ut (.) = 0 , then Wtm = RL and Et ≡ (XL,t
(iii) if ∆Utm (.) < 0 , then Wtm = L and Et = Φm
R,t ∩ BL,t 6= GRAPH X FPP + Isoquantes + ensemble des programmes vainqueurs
These results lead to several observations. First, because we are in a
deterministic setting, one candidate is in general certain to be elected. But
this candidate can propose many "winning programs" each of them leading to
di erent dynamics of the model. That is why, in the rest of the paper, it will
be necessary to de ne more precisely how candidates choose their announced
programs.
Second, in very speci c circumstances, median voter is indi erent between
candidates. This is not a surprise, public goods being substitutable for voters.
As it is usually done in political economics models, we assume that each
candidate then wins with proba 1/2. This event, which would be of modest
importance in a static model, becomes crucial in a dynamic one. Indeed,
because of the role of political history (via stocks of public goods), the identity
of elected candidate will dramatically a ect future priorities of voters, and
then their political choices !
4
When parties are secondary Democratic
In this section we suppose that parties are secondary "democratic" in the
sense that, among all the possible winning platforms, they propose the one
which is preferred by median voter. Such a de nition of democracy may seem
somehow surprising, but it only says that winning candidate implements the
Condorcet winner of the game.
11
4.1
The one period Game
With this "democratic" re nement, we obtain the following formal expressions for winning programs (cf. 3.1).
m
m
(i) If ∆Utm > 0, then W (t) = R and (Xt∗ , Yt∗ ) = (XR,t
, YR,t
).
m
(ii) If ∆Ut = 0, then W (t) = R, L and :
m
m
-(Xt∗ , Yt∗ ) = (XR,t
, YR,t
) with probability 1/2.
∗
∗
m
m
-(Xt , Yt ) = (XL,t , YL,t ) with probability 1/2.
m
m
).
, YL,t
(iii) if ∆Utm < 0, then W (t) = L and (Xt∗ , Yt∗ ) = (XL,t
∗
∗
With (Xt , Yt ) the platform proposed by victorious candidate.
4.2
In nite horizon Game
In this subsection, we consider our political game repeated over an in nite number of legislatures. The one period game is played at each period
and stocks result from the whole political history. We rst show that in nite
stability in power is impossible for some values of ideological parameter in
median voter’s utility function. Secondly, we study how the di erent elements of the model in uence the position of these values.
Proposition 2 (No Power stability) When parties are democratic and 0 <
δ < 1, there exist α1 <α2 belonging to ]0, 1[, such that :
if αm belongs to ]α1 , α2 [ then no party can win consecutively the elections an
in nite number of times.
Proof of Proposition 1 .
CORRIGER POUR UNICITE DE L’INTERVALLE... The mainline of this
proof is the following : we rst assume that party R wins the election from
a certain date to in nity and then show that if αm is below a certain value
(α2 ), this assumption is violated. With the same reasoning in the case of
party L, we show that L can not win the elections to in nity if αm is above
an other value (α1 ). Finally, we show that α1 is smaller than α2 .
Let hP∞ be an history in nite dimension vector of the game, such that, for
all t ≥ dP , W(t)=P. If R wins the elections an in nity of times, then :
αm
m
XR,∞
(hR
∞ , αm ) =
δ
1 − αm
m
R
YR,∞ (h∞ αm ) =
δλR
Then utility di erence will be :
12
m R
∆U∞
(h∞ , αm ) = λαRm − λL−αm
h
1−αm
δλR
i αm
−
δ
i αm
−
δλL
m
− λαRm −1
− λ1−α
L
m
− λαRm −1
λ−α
L
With the same reasoning, we obtain:
m L
(h∞ , αm ) = λαRm − λL−αm
∆U∞
h
1−αm
δ
m
− λαRm −1
− λ1−α
L
m
− λαRm −1
λ−α
L
Let us write the di erence between these two last expressions:
m R
m L
(h∞ , αm ) − ∆U
∆U∞
h
h ∞ (h∞ , αm )
i αm −1 1
1−αm
1
1−αm
m
= λαRm − λ−α
−
λ
1
−
α
−
+
λ
αm −
m
L
R
L
δ
λR
δ
αm
δλL
i
This di erence is strictly positive if αm ∈]0, 1[.
Furthermore:
m R
m L
(h∞ , 0) = ∆U∞
(h∞ , 0) =
∆U∞
1
λR
−1<0
and,
m R
m L
∆U∞
(h∞ , 1) = ∆U∞
(h∞ , 1) = 1 −
1
λL
>0
m L
m R
Since ∆U∞
(h∞ , αm ) and ∆U∞
(h∞ , αm ) are continuous with respect to αm ,
they are null for speci c values of αm . Then, there exist α2 ∈]0, 1[ such that
m R
m L
m L
∆U∞
(h∞ , αm ) is zero. Then, ∆U∞
(h∞ , α2 ) > 0, since ∆U∞
(h∞ , 0) < 0 and
m L
with continuity, there exist α1 < α2 ∈]0, 1[ such that ∆U∞ (h∞ , α1 ) = 0.
m R
m L
Furthermore, if αm ∈]α1 , α2 [, ∆U∞
(h∞ , αm ) < 0 < ∆U∞
(h∞ , αm ).
This concludes the proof.
This result proves that when parties are secondary democratic, there exist
a subset of ideological parameters such that, if median voter belongs to this
subset, no party can stay in o ce for ever. We can rewrite this proposition
in the following way :
Proposition 3 (Alternation)
the in nite horizon game, if α1 < αm < α2 , then each party wins consecutively the elections a nite number of time.
Proposition 3 emphasizes that our "no power stability" result does not
say that alternation is systematic. On the contrary, very long periods of
political stability as well as more "chaotic" political histories are possible.
The following natural step is to study the features of such an "alternation
interval". More precisely, it is important to know on which part of the ideological spectrum it is, and what is the impact of parameters (δ, λR , λL ) on
13
it. The following corrolary then considers the case when candidates are symmetric in the sense that none of them has a greater technological advantage
than its competitor.
Corrolary 1
If λR = λL , then α1 <
1
2
< α2
The interpretation of this corrolary is the following : if each party has a
technical advantage in producing one of the public good, and if there exist
a "symmetry" in such advantages (that is if no party is much better in producing one good than the other party is in producing the other good), then
alternation occurs when median voter is "moderate" in her preferences. This
result ts well with some of the main features of western democracies (and of
France in particular). Indeed, in such regimes, if parties are viewed as being
di erent by the electorate, these di erences are not too sharp. In the same
time, electoral studies seem to exhibit a median position close to the center
of the ideological spectrum. Then one is not surprised, when confronting
with our model, to observe very frequent alternation.
4.3
Simulations
In this subsection, we propose several simulated political histories.
4.3.1
specialization of symmetric parties and interval of alternation
In this paragraph, we will consider the case where parties are "symmetric", that is when λL = λR . We have already proved that in this context,
α = 0, 5 belongs to the alternation interval. Following simulations emphasize
that the more specialized are the candidates (that is the higher are λL and
λR ), the largest is this interval.
αm xs0 y0s λR λL
δ
interval
0.5 0 0 1.2 1.2 0,5 [0.455,0.529]
αm
0.5
xs0
0
y0s
0
λR
2
λL
2
δ
0,5
interval
[0.35,0.65]
αm
0.5
xs0
0
y0s
0
λR
10
λL
10
δ
0,5
interval
[X,0.899]
14
The reason that explains the in uence of specialization on alternation
intervals is the following :...
4.3.2
asymmetry in technologies
We now consider a political competition in which one candidate is more
specialized than the other. Two families of observations then have to be done.
The rst one concerns the possibility of alternation all over political history.
The second, in a shorter perspective, concerns the adjustment of electoral
competition to the long term path.
αm xs0 y0s λR λL δ
.
.
.
.
.
.
period
winner
1
X
2
X
xs0
.
y0s
.
λR
.
period
winner
1
X
2
X
xs0
.
y0s
.
λR
.
period
winner
1
X
2
X
αm
.
αm
.
4.3.3
y0s
.
λR
.
period
winner
1
X
2
X
xs0
.
y0s
.
λR
.
αm
.
λL
.
3
X
λL
.
3
X
4
X
5
X
6
X
.
X
.
X
.
X
.
X
.
X
.
X
.
X
.
X
.
X
5
X
6
X
.
X
.
X
.
X
.
X
.
X
.
X
.
X
.
X
.
X
5
X
6
X
.
X
.
X
.
X
.
X
.
X
.
X
.
X
.
X
.
X
.
X
.
X
.
X
.
X
.
X
.
X
.
X
.
X
.
X
δ
.
4
X
δ
.
4
X
frequency of alternations
xs0
.
αm
.
3
X
λL
.
3
X
λL
.
δ
.
4
X
5
X
6
X
δ
.
15
period
winner
1
X
2
X
xs0
.
y0s
.
λR
.
period
winner
1
X
2
X
αm
.
4.3.4
3
X
4
X
λL
.
3
X
5
X
6
X
.
X
.
X
.
X
.
X
.
X
.
X
.
X
.
X
.
X
5
X
6
X
.
X
.
X
.
X
.
X
.
X
.
X
.
X
.
X
.
X
...
...
37
R
38
L
225
L
226
L
δ
.
4
X
a quite strange political history
αm
0.43068
t
W(t)
1
R
t
...
W(t) ...
xs0
0
2 3
L R
219
R
y0s
0
λR
2
4
R
220
L
λL
2.5
5 6
L R
221
R
δ
0.9
7
R
222
R
8 9
L R
223
L
4.3.5
asymmetry in stocks
4.3.6
various depreciation rates
5
224
R
39
R
227
R
40
L
41
R
228
L
...
...
229
R
When parties maximize their possibilities of
reelections: the incumbency advantage
In this section, we assume that, when choosing their platforms, parties
also care about what will happen in the future periods. In other words,
winning candidates try to maximize their possibilities of reelection. We will
call such politicians "Long Term Opportunists" (LTO).
5.1
Political equilibria
Clearly, given this long-term objective19 candidates will o er the victorious program including the smallest quantity of public good he has a "technical" advantage to produce, and as much as possible of the other good. In
19
"Long term" may seem an unappropriate term, but one has to remember that legislatures are usually 4 or 5 years long.
16
230
R
231
L
232
R
doing this it creates a "need" for the following period, which could favor his
reelection. Formally:
(i) If W (t) = R, party R program is:
m
∗
∗
∗ ,Y ∗ )∈E [∆U
M AX(XR,t
t
t+1 (XR,t , YR,t )]
R,t
m
∗
∗
∗∗
∗∗
∆Ut+1
is a ne, increasing with YR,t
and decreasing with XR,t
. Let (XR,t
, YR,t
)
be the solution of this program.
(ii) If W (t) = L, party R program is:
m
∗
∗
∗ ,Y ∗ )∈E [∆U
M IN(XL,t
t
t+1 (XL,t , YL,t )]
L,t
∗∗
∗∗
∗
∗
m
)
, YL,t
. Let (XL,t
and decreasing with XL,t
is a ne, increasing with YL,t
∆Ut+1
be the solution of this program.
Next proposition describes the equilibria in the LTO game.
Proposition 4 The winning strategies when candidates are LTO are :
(i) If ∆Utm > 0, then W (t) = R and
m
m
∗∗
)]
, YL,t
= Argmax[Y ∈ <+ /X + λR Y = BR,t and Utm (X, Y ) = Utm (XL,t
YR,t
∗∗
∗∗
= BR,t − λR YR,t
XR,t
(ii)If ∆Utm = 0, then W (t) = L, R and winning strategies are the same as in
the case of parties having just rst objective.
(iii)If ∆Utm < 0, then W (t) = L and
m
m
∗∗
)]
, YR,t
= Argmax[X ∈ <+ /λL X + Y = BL,t and Utm (X, Y ) = Utm (XR,t
XL,t
∗∗
∗∗
YL,t = BL,t − λL XL,t
GRAPH X FPP + Isoquantes + programme "limite" vainqueur
5.2
In nite horizon Game
Given The strategies we have described in proposition 5, the following
result emphasizes, for some values of heterogeneity parameters, the impossibility of in nite stability in power.
Proposition 5 If 0 < δ < 1 and if parties are LTO, then :
there exist α1 < α2 , such that if αm ∈]α1 , α2 [, then no party can consecutively
win the election an in nity number of times.
Ln[λR ]
Furthermore Ln[λ
∈]α1 , α2 [
R λL ]
17
This result is very similar of the one we got in the democratic context.
But it underlines the fact that, even when politicians systematically propose
the program which is the more favorable to their permanence in o ce, for
some values of median voter ideological parameter, alternation still occurs.
It is now important to discuss the in uence of parameters on this "alternation space"...still has to be done...
It is interesting to notice that winning candidates platforms converge
toward the center. Usually, the main motivation for such a convergence is
to increase vote share and/or probability of victory. Here it is not the case,
rather, parties try to create a "need" in the electorate20 .
5.3
6
Simulations
Share maximizers parties (still preliminary)
In this section, we suppose that the parties second objective is to maximize their share of votes. We will call them VSM (vote share maximizer)
candidates.
6.1
The one period Game
Proposition 6 When parties are VSM, if an equilibrium exists, then they
saturate their budget constraints.
6.2
In nite horizon Game
Proposition 7 When 0 < δ < 1 and parties are VSM, if an equilibrium
exists, then there exist α1 < α2 , such that:
If αm ∈]α1 , α2 [, then no party can consecutively win the election an in nity
of times.
Ln[λR ]
∈]α1 , α2 [
Furthermore Ln[λ
R λL ]
20
La dissolution decidee par Chirac en 1997 peut etre appreciee au regard des arguments
de cette section. Ne s’agissait-il pas d’eviter que l’election ait lieu a un moment ou la
situation sociale soit assez deterioree pour que la gauche puisse en tirer avantage...?
18
7
Rent-seeker parties (still preliminary)
8
Some further discussions
8.1
Electoral competition
la Downs and
la Wittman
One may like to know how our results are modi ed if we relax the hypothesis of o ce-motivated parties and consider an electoral competition a
la Wittman. In such a setting, candidates would have exactly the same kind
of utility functions than voters. But then, at some periods, one could observe
situations in which politicians would like...their competitor to win !
Despite its many appealing features, it seems that Wittman model does
not t very well with situations in which parties can not always propose
identical programs.
9
Conclusion
In this work, we have tried to build a model of political competition taking
into account two main features of democratic life (though rarely considered
in previous papers) : the durable nature of public goods and di erences in
parties ability to implement policies.
In this setting, we show that political alternation has to be viewed as a
natural "breathing" of democracy, in the sense it directly results from changes
in priorities of the electorate. We also discuss the importance of various
elements such as asymmetry and polarization of electoral competition.
In the same time, we have suggested, for the rst time, a very simple and
intuitive way of modeling "issue voting", as a direct consequence of voters
evaluation of the economic or social environment when elections take place,
rather than any stochastic shock in preferences.
Of course, we are aware that other reasons may explain political alternation, and that much work still has to be done, but we believe that our
paper can be a rst step in a more systematic study of this phenomenon.
Furthermore, the setting we built could be useful for studying many other
aspects of democratic life.
19
References
[1] Alesina, A. and H.Rosenthal, 1996, "A Theory of Divided Government",
Econometrica, V.64, N◦ 6, 1311-1341.
[2] Barnes, S.H. and R.Pierce, 1971, "Public Opinion and Political Preferences in France and Italy", Midwest Journal of Political Science, V.15,
N◦ 4, 643-660.
[3] Black, D., 1958,...
[4] Diermeier, D. and A.Merlo,2001, "Government Turnover in Parliamentary Democracies", Journal of Economic Theory,.............
[5] Downs, A., 1957, An Economic Theory of Democracy,...
[6] Drazen, A., 2000, Political Economy in Macroeconomics, Princeton University Press.
[7] Persson, T. and G. Tabellini, Political Economics, Explaining Economic
Policy, MIT Press.
[8] Petrocik, J.R., 1996, "Issue Ownership in Presidential Elections, with a
1980 Case Study", American Journal Of Political Science, V.40, N◦ 3,
825-850.
[9] Roemer, J.E., 1995, "Political cycles", Economics and Politics 7,1-20.
[10] Roemer, J.E., 2001, Political Competition, Harvard University Press.
[11] Usher, D., 1989, "The Dynastic Cycle and the Stationary State", The
American Economic Review 79, n◦ 5, 1031-1044.
[12] Wittman, D., 1973, "Parties as Utility Maximizers", American Political
Science Review 67, 490-498.
[13] Wittman, D., 1977, "Candidate motivation : A synthesis of alternative
theories", American Political Science Review 77, 142-157.
20
10
10.1
ANNEX
proofs
Proof of lemma 1:
Suppose the reverse.
u(Z2 , αk ), Z1 , Z2 ∈ Ω.
Let αa 6= αb be the solutions of u(Z1 , αk ) =
Let Z1 = (X1 , Y1 ) and Z2 = (X2 , Y2 ).
X1 Y2 αa
Then, [ X
] =
2 Y1
Y2
Y1
1 Y2 αb
= [X
] and so, αa = αb .
X2 Y1
Proof of proposition 1:
We make the proof of (i) and (ii). Proof of (iii) is the same as proof of
(i).
Let rstly prove part (i):
m
m
), then W (t) = R and Et 6= . Now,
, YR,t
If ∆Utm > 0 and R plays (XR,t
suppose R choose a program which does not belong to Et . By de nition,
this program does not belong to BR,t . Then the only possible case is that it
does not belong to ΦL,t . But, since U is strictly increasing with respect to X
and Y, there exists a program such that is feasible by party L that defeats
party’s R program. This contradicts W (t) = R. Now prove part (ii):
m
m
m
m
) they
, YL,t
) and (XR,t
, YR,t
If ∆Utm = 0. If R and L respectively plays (XL,t
give the same utility to median voter. If one party do not, then he gives less
utility to median voter and lose the election.
Proof of proposition 2:
Let de ne the history of the game h. At period t+1, the history of the
game, ht+1 , is a dimension t vector. Each element of ht+1 is a value taken in
the election results set W = {R, L, RL}.
Then, ht+1 = (W (1), ..., W (t)) ∈ W t .
21
Let hPt+1 ∈ W t , such that W(t)=P, P=R,L. Then:
αm
−αm
m
m
m
m
m
∆Utm (hLt+1 )−∆Ut+1
(hR
)(YL,t
−YR,t
)−(λ1−α
−λαRm −1 )(XL,t
−
L
t+1 ) = (λR −λL
m
XR,t ) > 0
m
Furthermore, if αm = 0, then ∆Utm (hLt+1 ) and ∆Ut+1
(hR
t+1 ) are negative, and,
R
m
m L
if αm = 1, ∆Ut (ht+1 ) and ∆Ut+1 (ht+1 ) are positive. Since ∆Utm (hLt+1 ) and
m
(hR
∆Ut+1
t+1 ) are continuous functions of αm , there exist at least one pair
(αa,t , αb,t ), αa,t < αb,t such that, if αm ∈]αa,t , αb,t [, then ∆Utm (hLt+1 ) is posim
(hR
tive and ∆Ut+1
t+1 ) is negative.
Proof of proposition 4:
The proof of this proposition is straightforward. Suppose party R wins
the elections from a date t to in nity. By proposition 3, that is false, then
there exists a date T>t, such that L wins the election. Now, suppose that
party L wins the elections from T to unity, proposition 3 says that is false,
then there exists a date d > T , such that party R win the election, and so
on... Repeating this reasoning to in nity ends the proof.
Proof of proposition 5:
Proof of proposition 6:
We have to show that the following statements can be false for the same
values of αm :
(H1) There exist dR such that for all t ≥ dR , W (t) = R
(H2) There exist dL such that for all t ≥ dL , W (t) = L
Suppose (H1) is true. Then:
∗∗
∗∗
∗∗
∗∗
∀t ≥ dR + 1, XR,t
+ λR YR,t
= 1 + (1 − δ)[XR,t−1
+ λR YR,t−1
]
∗∗
∗∗
Then, this budget converges (because δ < 1) toward XR,∞
+ λR YR,∞
= 1/δ
which is nite (because 0 < δ).
∗∗
Furthermore, XR,t
≥ δ[λλRRλ−1
+
L −1]
22
where the right hand term is the "abcisse" of the intersection point of possibility production frontiers and is an in nitesimal strictly positive number.
Let write maximum level of utility party L can o er to the median voter:
m
m
U m (XL,∞
, YL,∞
)=
αm
λL
αm
λL
=
≥
α m
αm
λL
α m
∗∗
∗∗
(1 − αm )1−αm [λL XR,∞
+ YR,∞
]
(1 − αm )1−αm 1
∗∗
]
[ + (λL λR − 1)XR,∞
λR
δ
α m
1
(1 − αm )1−αm [ + e]
δ
Furthermore, U
It follows:
m
αm
∆U∞
≤ αm
m
m
m
(XR,∞
, YR,∞
)
=
αm
αm
1 − αm
λR
1−αm
1
δ
(1 − αm )1−αm αm −1
m
[λR
− λ−α
(1 + ee)]
L
δ
Then it su ces that αm <
pose (H2) is true.Then:
Ln[λR (1+e
e)]
Ln[λR λL ]
for (H1) to be contradicted. Now, sup-
∗∗
∗∗
∗∗
∗∗
∀t ≥ dL + 1, λL XL,t
+ YR,t
= 1 + (1 − δ)[λL XL,t−1
+ YR,t−1
]
∗∗
∗∗
Then, this budget converge (because δ < 1) toward λL XL,∞
+ YL,∞
= 1/δ
which is nite (because 0 < δ).
∗∗
≥ δ[λλRLλ−1
+ 0
Furthermore, YL,t
L −1]
where the right hand term is the "ordonnØe" of the intersection point of possibility production frontiers and 0 is an in nitesimal strictly positive number.
Let write maximum level of utility party R can o er to the median voter:
U
m
m
m
(XR,∞
, YR,∞
)
=
αm
αm
≥
αm
αm
=
αm
αm
1 − αm
λR
1−αm
∗∗
∗∗
[XL,∞
+ λR YL,∞
]
1 − αm
λR
1−αm
1 1
∗∗
[ + (λL λR − 1)YL,∞
]
λL δ
1 − αm
λR
1−αm
1
[ + e0 ]
δ
m
m
Furthermore, under (H2), U m (XL,∞
, YL,∞
) = ( αλm
)αm (1 − αm )1−αm
L
23
1
δ
It follows:
m
∆U∞
≤
αm (1
αm
− αm )1−αm αm −1
m
]
[λR (1 + ee0 ) − λ−α
L
δ
Ln
Then it su ces that αm >
su ces that
Ln
λR
λR
(1+)
e
e
Ln[λR λL ]
for (H2) to be contradicted. Finally, it
(1+)
e
e
Ln[λR λL ]
< αm <
Ln[λR (1+e
e)]
Ln[λR λL ]
for (H1) and (H2) to be contradicted.
Proof of proposition 7:
ft be the voter indi erent between R program and L program type.
Let α
∗
∗
∗
∗
) be an equilibrium of the game with only parties
, YL,t
), (XL,t
, YR,t
Let ((XR,t
ft is such that:
rst objective. Then α
∗ αm ∗ (1−αm )
∗ αm ∗ (1−αm )
) YR,t
= (XR,t
) YR,t
(XR,t
It follows:
ft =
α
Y∗
Ln Y L,t
∗
R,t
Y ∗ X∗
R,t
Ln YL,t
∗ X∗
R,t L,t
∗
∗
∗
∗
.
6= XL,t
and YR,t
6= YL,t
By de nition of E(t), we know that XR,t
Since there is a continuum of voters, party R share of votes is decreasing
X∗
Y∗
e and party L share is increasing with α
e if XR,t
with α
> YR,t
(case A), party R
∗
L
L,t
e and party L share is decreasing with α
e if
share is increasing with α
∗
XR,t
∗
XL,t
<
∗
YR,t
∗
YL,t
(case B). Then, depending on their relative programs, parties catches votes
from left or right side of the electorate. Party which proposes relatively
more of good Y obtains votes from the left side of the electorate, and party
which proposes relatively more of X catch the votes from the right side of
the electorate (voters with high value αi ). These di erent cases have to be
considered (for the clarity of the exposition, we do not write indice t in this
proof):
24
Suppose ∆Utm > 0 then, with proposition ???, we know W (t) = R, and
∗
∗
(XR,t
, YR,t
) ∈ E(t).
Case A:
∗
∗
YR,t
XR,t
>
∗
∗
XL,t
YL,t
In this case, party R program is equivalent to the following program:

Ln

∗ ,Y ∗ ) α
e=
M IN(XR,t
R,t 
Ln
∗
YL,t
∗
YR,t

∗ X∗
YL,t
R,t
∗ X∗
YR,t
L,t



∗
∗
s.t. (XR,t
, YR,t
) ∈ E(t)
And party L program is:


Y∗
L,t
Y∗
R,t
X∗ Y ∗
R,t L,t
X∗ Y ∗
L,t R,t
Ln

∗ ,Y ∗ ) α
e=
M AX(XL,t
L,t 
Ln



∗
∗
s.t. (XL,t
, YL,t
) ∈ BL,t
e we obtain:
Di erentiating α,
∂α
e
∗
∂XR,t
∗
∗
∝ YR,t
− YL,t
∂α
e
∗
∂YR,t
∗
∗
∝ XL,t
− XR,t
∂α
e
∗
∂XL,t
∗
∗
∝ YL,t
− YR,t
∂α
e
∗
∂YL,t
∗
∗
∝ XR,t
− XL,t
Since E(t) 6= , these programs have no interior solution. Then, there are
three possibilities.
∗
∗
∗
∗
∗
∗
∗
≥ YL,t
and XL,t
< XR,t
, then YL,t
is maximal. Since (XR,t
, YR,t
)∈
Suppose YR,t
∗
∗
BR,t , YL,t > YR,t , which is contradictory.
∗
∗
∗
∗
∗
∗
Suppose YR,t
> YL,t
and XL,t
≥ XR,t
, then XL,t
= 0. This implies XL,t
<
∗
XR,t , which is contradictory.
∗
∗
∗
∗
Then, if a solution exists, it is such that YR,t
< YL,t
and XL,t
≤ XR,t
.
Case B:
∗
∗
XR,t
YR,t
<
∗
∗
XL,t
YL,t
In this case, party R program is equivalent to the following program:
25

Ln

∗ ,Y ∗ ) α
e=
M AX(XR,t
R,t 
Ln
∗
YL,t
∗
YR,t
∗ Y∗
XR,t
L,t
∗ Y∗
XL,t
R,t




∗
∗
) ∈ E(t)
, YR,t
s.t. (XR,t
And party L program is:


∗ ,Y ∗ ) α
e=
M IN(XL,t
L,t 

Y∗
Ln Y L,t
∗
R,t
XR Y ∗
Ln X ∗ YL,t
∗
L,t R,t



∗
∗
) ∈ BL,t
, YL,t
s.t. (XL,t
As in case A, we have to consider three sub cases:
∗
∗
∗
∗
∗
∗
)∈
, YR,t
= 0. Since (XR,t
). Then XL,t
and XL > XR,t
≤ YL,t
Suppose (YR,t
∗
∗
E(t), then XR,t > XL,t . This contradicts the hypothesis.
∗
∗
∗
∗
∗
e = 1. Playing
= 0 and α
, then YL,t
≤ XR,t
and XL,t
> YL,t
Suppose YR,t
∗
∗
e less than 1. Then, it is not an equilibrium.
XL,t , YL,t > 0 would make α
∗
∗
∗
∗
e > 1. This is a dominated stratSuppose YR,t > YL,t and XL,t > XR,t
, then α
∗
∗
∗
∗
∗
∗
egy of party L, if he plays (XL,t , YL,t
) such that YR,t
6= YL,t
and XL,t
≤ XR,t
,
e < 1.
then α
∗∗
∗∗
∗∗
∗∗
.
≤ XR,t
and XL,t
< YL,t
Finally, if a solution exists, it is such that YR,t
Then budget constraints are saturated.
Similar arguments would prove the same result in the case where ∆Utm < 0.
Finally, in the case where ∆Utm = 0, it is straightforward, there exist a unique
equilibrium of the game with only rst objective, then it is again the equilibrium of the game with a second objective. Furthermore, at this equilibrium,
budget constraints are saturated.
Proof of proposition 8:
See the proof of proposition 6.
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