Moderate Voters, Polarized Parties
Richard Van Weelden∗
January 12, 2011
Abstract
I consider the welfare implications of divergent platforms in a model of repeated elections.
There are two long-lived candidates who have preferences over policy and also an opportunity
to engage in rent-seeking when in office. Candidates cannot make binding commitments before
being elected, but voters condition their re-election decision on the behavior of the candidate
in office. I consider symmetric, platform-stationary, subgame perfect equilibria with different
levels of divergence between the two candidates. I show that divergence between the candidates
can decrease the amount of rent-seeking by elected candidates, and, when utility functions are
concave, increase the welfare of the voters. I then consider how optimal divergence depends on
the distribution of voter preferences.
When utility functions are are not too concave, increasing the polarization in voter preferences will increase the range for which divergent platforms are beneficial. However, when
utility functions are sufficiently concave, increasing the polarization in voter preferences will
decrease the pressures for divergence. Consequently, greater patience will be required to support divergence, and less divergence will be optimal in equilibrium. I then consider the case of
multi-dimensional policy spaces, and show that, when utility functions are sufficiently concave,
it will be optimal to have greater divergence in the dimension in which voters are less polarized.
I discuss the implications of these results for electoral competition in the United States.
∗
Max Weber Fellow, European University Institute and Assistant Professor of Economics, University of Chicago.
Phone: +39-346-8354-093 Email: Richard.VanWeelden@eui.eu. I would like to thank Dirk Bergemann, Justin Fox,
Lucas Maestri, Massimo Morelli, Stephen Morris, Martin Osborne, Larry Samuelson, and Alyosius Siow for helpful
discussions.
1
Introduction
It has been well established that, at least in the United States, political parties are more polarized
than the voters they seek to govern.1 While there has been much work done to document the
significant, and increasing, divergence between the parties (see McCarty et al 2006 for an overview),
the voters themselves appear to be much less polarized.2 That polarized elected officials represent
a comparatively moderate populous is often seen as sign of democratic failure and evidence that
political institutions have been hijacked by relatively extreme individuals and special interests (see
Chapter 10 of Fiorina et al 2006).
In this article, I argue that polarization in candidate preferences and platforms can be beneficial,
and, in fact, may be most beneficial when voters themselves are more moderate. The argument
builds on that introduced in Van Weelden (2010), that polarization of candidates’ preferences and
platforms can create incentives for candidates to work harder to secure re-election, decreasing rentseeking. As the candidates care about the policy implemented, in addition to the rents from office,
divergence in candidate preferences and platforms allows a shirking candidate to be punished with
a policy further from their ideal point. When utility functions are concave, small changes in policy
around the median voter’s ideal point will have a much greater impact on the welfare of non-median
candidates than on the median voter. Hence, the decrease in rent-seeking behavior by the elected
candidate will more than compensate the median voter for the policy being pulled away from the
center, and divergence in candidate platforms will enhance the welfare of the median voter.3
In this paper I consider how the aggregate welfare of voters is affected by the polarization
in the policy-preferences and implemented platforms of the candidates for office. For each level of
polarization, I determine the best equilibrium which can be supported given candidates of that type,
and then compare the voters’ welfare across the equilibria for different candidates. I show that, using
a utilitarian social welfare function, divergence in candidate preferences and platforms is beneficial,
for any symmetric distribution of voter ideal points, provided the players are sufficiently patient.
This is in contrast to most of the results in the literature which find that the equilibrium with
convergence is generally welfare maximizing.4 While it is always possible, with sufficient patience,
1
As Fiorina et al (2006) describes it, “for as long as we have had data political scientists have known than political
elites are more polarized than the mass of ordinary Americans.” See also Table 2.1 in Fiorina et al (2006) for survey
data showing greater polarization in the policy preferences of the delegates in the parties’ conventions than in the
rank and file membership.
2
See Fiorina et al (2006): “[P]olarized political ideologues are a distinct minority of the American population. For
the most part Americans continue to be ambivalent, moderate, and pragmatic, in contrast to the cocksure extremists
and ideologues who dominate our political life.” (Preface to 2nd Edition)
3
Van Weelden (2010) considers a model with a single representative voter, and also shows this representative
voter can be interpreted as the median voter in a setting with a large number of voters.
4
There are, however, some papers which find divergence to be beneficial: Bernhardt et al (2009), Van Weelden
(2010). Additionally, Kedar (2005) argues that moderate voters may prefer candidates more extreme than themselves,
1
to support divergent platforms, how much patience is required will depend on the distribution
of voter preferences. While it is generally assumed that increasing divergence in voters’ ideal
points will be associated with greater divergence in candidate platforms, I show that the amount of
divergence in the socially optimal equilibrium will be increasing in the polarization of the voters only
if the players’ utility functions are not too concave. When utility functions are sufficiently concave,
as non-median voters greatly dislike having the policy pulled even further from their ideal point,
greater patience will be required to support divergent platforms when the voters are more polarized.
In fact, when utility functions are sufficiently concave, divergent platforms will be beneficial unless
the preferences of voters are too polarized. Further, when divergence is optimal, I show that, when
utility functions are sufficiently concave, the optimal degree of polarization between the candidates
will be decreasing in the polarization of the voters; that is, the more moderate the voters are, the
greater the polarization between the candidates that is desired.
Perhaps the most interesting implications of this model come from the case in which there are
multiple dimensions of policy conflict in addition to the opportunity for rent-seeking. I show that
polarization in candidates’ positions is more desirable along the dimension in which voters preferences are more polarized if, and only if, the utility functions are not too concave. Consequently,
when utility functions are sufficiently concave, it will be socially optimal to have greater divergence
in candidate platforms along the dimension in which voters are more moderate. In fact, for a range
of parameters, it will be optimal to have convergence in the dimensions in which voters are more
polarized, and divergence in the dimensions which voters are most centrist.
These results provide a new framework for analyzing party competition, and a new interpretation for the observed divergence in party platforms. Recent work on parties and political attitudes
indicates that, while voters are for the most part rather moderate, candidates and political activists
are not (Fiorina et al 2006). In addition, while voters are predominantly concerned with economic
affairs and policy (e.g. Fiorina et al 2006, McCarty et al 2006), in recent years candidates have diverged more over social policies while moving towards the center on economic policies.5 While this
polarization between candidates and parties on issues on which most voters hold moderate positions
is frequently seen as a sign of democratic failure, these results provide a more benign interpretation
of candidates polarizing over issues in which voters are centrist, and raise the possibility that such
polarization is not as harmful as is often claimed. When utility functions are concave, divergence
in candidate platforms can benefit the voters by disciplining candidates, decreasing the amount of
rent-seeking. Furthermore, when utility functions are sufficiently concave, since voters who have
knowing that the candidate will only be able to achieve part of her legislative agenda.
5
Two other explanations for convergence in certain dimensions and divergence in others are differences in how
easily messages can be targeted to certain groups (Glaeser et al 2005) and a tendency to converge with concave utility
functions and diverge with convex utility functions in probabilistic voting models (Kamada and Kojima 2010).
2
an ideal point which is relatively extreme greatly dislike having the policy pulled even further from
their ideal point, the utility loss from the policy being pulled from the center is less the fewer
extreme voters there are. Hence, from a social welfare perspective, the best outcome is to have
platforms diverge in dimensions in which voters are predominantly centrist and converge in the
dimensions in which voters’ preferences are more polarized.
Political parties who are polarized over issues on which most voters hold relatively moderate
positions is precisely the outcome which has been observed, and frequently bemoaned, in the United
States. Fiorina et al (2006) worries that “however misplaced from the standpoint of the welfare
of the larger country, an emphasis on cultural and other conflicts not of particular interest to
the the majority appears to be an integral part of contemporary electoral politics.” I compare
voter welfare between equilibria in which the candidates are polarized over the same dimension the
voters are to equilibria with polarization over the dimension in which voters are more moderate,
and show that, for a range of utility functions, polarization over issues which voters are moderate
is not “misplaced from the standpoint of ... welfare”, but rather the dimension of polarization
which maximizes welfare. In this analysis I compare only the differences in welfare for different,
exogenously specified, polarization in candidate preferences. I do not model how the candidates
are determined. One implication of these results, then, is that factors outside the model– outside
interest groups, political primaries – may actually increase aggregate welfare of the population if
they produce candidates who have policy preferences which are less representative of the electorate.
The paper is organized as follows. In section 2, I present the model and discuss the class
of equilibria which the analysis focuses on. Section 3 then presents the results for the singledimensional case. In Section 4 I show how the results extend when there are multiple dimensions
of policy conflict. In section 5 I discuss the role of the concavity assumptions and how such
assumptions relate to the preferences of voters. Section 6 then concludes. The proofs of the results
are left to the Appendix, with the Lemmas proven in the Supplemental Appendix.
2
Model
I present a simplified version of the model introduced in Van Weelden (2010). Unlike Van Weelden
(2010), which focuses on a setting with a single representative voter, I focus on how the welfare
implications of divergent platforms relates to the distribution of voter preferences. I assume there
are two infinitely-lived candidates and an infinite number of infinitely-lived voters.6 The elected
candidate chooses the policy to implement, x ∈ [−1, 1], and the amount of rent-seeking to engage
6
Whether the voters are long or short lived will not affect the analysis – within the class of equilibria considered
the voters’ best response will be to vote for the candidate they myopically prefer. For the candidates it is essential
that they care about future outcomes. In a related setup, Van Weelden (2010) shows that infinitely lived candidates
can be replaced by term-limited candidates, provided such candidates care about the outcome of future elections.
3
in, m ∈ [0, M ]. These rents could reflect outright corruption, such as taking bribes, or simply a
lack of effort – either taking time off work or not addressing difficult but important issues – by the
candidate in office. For example, the benefit to the candidate of holding office may be eroded if they
work excessively long hours, deny favors to family and friends, or take actions which antagonize
powerful lobby groups or vocal minorities. I refer to a policy-rents pair p = (x, m) as a platform.
Voters can be one of three types, v ∈ {−1, 0, 1}.7 The stage game utility, for a voter of type v,
in any period in which platform pt is implemented is
uv (pt ) = −|xt − v|λ − mt .
I will use V to refer to a specific voter, and v to refer to the voter’s type. I assume the distribution
of voters’ preferences are symmetric so that the same fraction σ ∈ (0, 12 ) of the voters are each of
the types v = 1 and v = −1. The fraction of voters who are type v = 0 is then 1 − 2σ. Note that
σ is then a measure of the polarization of voter preferences: if σ is small (close to 0) then most of
the voters are centrist; when σ is large (close to 1/2) most of the voters are at the extremes of the
political spectrum.
The candidates’ preferences are identical to the voters except that they benefit from receiving
rents if they are in office. If the candidate is not elected then their payoff is identical to a voter
with the same policy preferences. I assume that the candidates’ ideal points are symmetric about
the median voter’s ideal point, so one candidate has ideal point κ ∈ [0, 1] and the other has ideal
point −κ ∈ [−1, 0]. The utility to candidate k ∈ {−κ, κ} is then
gk (pt ) = −|xt − k|λ − mt ,
if they are not in office in that period and platform pt is implemented, and
ḡk (pt ) = −|xt − k|λ + γmt ,
if they are in office. The parameter γ ≥ 1 reflects how much the candidates in office care about
rents relative to the voters and the candidate out of office.
Note that κ then reflects the degree of polarization in candidate preferences. When κ = 0
both candidates share the policy preferences of the median voter and we have convergence in the
candidates’ preferences. When κ = 1 the candidates share the policy preferences of the polarized
voters – the bases of the parties if you will. When κ ∈ (0, 1), candidate preferences are polarized,
but less polarized than the more extreme voters. The main purpose of this analysis is to determine
the optimal degree of polarization in candidate preferences, how that affects the resulting policy
outcomes, and how the optimal polarization is affected by the distribution of voter preferences.
7
It is not important that there be only three types. What matters is that the distribution of voters is symmetric,
and that there are a positive mass of voters who hold the median position. A model with three types is used only
for simplicity, and because it allows the degree of polarization to be characterized by a single parameter.
4
The timing of the game is as follows:
1.
In each period each voter votes for a candidate, ktV ∈ {−κ, κ}. The candidate receiving
more than half the votes is elected, and, in the event that both candidates receive half the votes,
each candidate is elected with probability 1/2. Let kt denote the candidate elected in period t.
2. The elected candidate chooses platform pt = (xt , mt ) for that period.
3. The voters observe the platform pt implemented.
4.
The game is repeated with the voters deciding whether to vote for the incumbent or the
challenger.
I assume that the game is infinitely repeated with discount factor δ < 1. Since each period in
this model corresponds to an election cycle, we would expect δ to be significantly below 1. We will
then be concerned with more than just the limiting properties as δ → 1.
In each period the voters elect a candidate, kt , and the elected candidate chooses platform
pt = (xt , mt ). A history then consists of the previously elected candidates, the platforms they
implemented when in office, and the share of the vote the incumbent received, µt ∈ [1/2, 1]. That
is, a history is
ht = ((k0 , p0 , µ0 ), . . . , (kt−1 , pt−1 , µt−1 )).
I then define ht(k,µ) = (ht , k, µ) to be the history at which candidate k is elected after history ht with
t
share µ of the vote. Let H t and H(k,µ)
be the set of all t period histories ht and ht(k,µ) respectively,
t
and define H = ∪t≥0 H t and H(k,µ) = ∪t≥0 H(k,µ)
.
The strategy for each voter V is to vote for one of the candidates at every history,
sV
:
H → {−κ, κ},
sV (ht ) = ktV .
The strategy for each candidate consists of the platform they would implement, if elected by
the voters, at every history,
sk
:
H(k,µ) → [−1, 1] × [0, M ],
sk (ht(k,µ) ) = pt .
For any profile of strategies, s, we can then calculate the payoffs in the repeated game for each
player. The payoff for a voter of type v ∈ {−1, 0, 1} is then
Uv (s) = (1 − δ)
∞
X
δ t uv (pt ),
t=0
and for each candidate k ∈ {−κ, κ} the payoff is
Uk (s) = (1 − δ)
∞
X
t=0
5
δ t uk (kt , pt ),
where
uk (kt , pt ) =
ḡk (pt ) if k = kt ,
gk (pt ) if k 6= kt .
In what follows, I analyze the equilibrium, and the resulting voter welfare, taking the preferences of the candidates as given. I then compare the welfare for different candidate preferences to
determine the degree of polarization that is socially optimal. Note that, as no voter can ever be
pivotal, a wide range of behavior is consistent with equilibrium. I discuss the voters’ strategies and
the class of equilibria more below, and will always assume that voters will vote for the candidate
they prefer whenever they have a strict preference for one candidate over the other.
For welfare considerations, I use a utilitarian social welfare function which weighs each type of
voter by the fraction of voters who are of that type. That is, the aggregate welfare of the voters
from a given strategy profile, s, is given by
U ∗ (s) = σU−1 (s) + (1 − 2σ)U0 (s) + σU1 (s).
Note that a utilitarian social welfare function is only one way to identify the most “efficient”
or desirable electoral outcomes. An alternative way to compare two equilibrium outcomes is to
look at whether a majority prefers one outcome to another. In Van Weelden (2010) it is shown
that the Condorcet winners among the equilibrium outcomes correspond to those which maximize
the welfare of the median voter. This will always involve non-median candidates provided the
candidates are sufficiently patient.8
2.1
Two-Dimensional Policy Space
The model can easily be extended to one with two or more dimensions of the policy space in
addition to a distaste for rent-seeking. As two dimensions are all that is necessary to illustrate
the results, I consider only the case of a two-dimensional policy space, x = (xA , xB ) ∈ [0, 1]2 . I
then allow there to be heterogeneity in the ideal points of the voters in both dimensions A and B.
To keep the model as simple as possible I assume that each voter only holds non-centrist policy
positions in one dimension, though this assumption is not necessary for the main results. There
are then five types of voters. I assume that fraction σA of the voters have ideal points x = (1, 0)
and x = (−1, 0), that fraction σB of the voters hold ideal points x = (0, 1) and x = (0, −1), and
that 1 − 2(σA + σB ) of the voters have ideal point x = (0, 0). Finally, I assume that
1
σA + σB ∈ (0, ),
2
8
In Van Weelden (2010), there a continuum of potential candidates and, rather than specify the strategy of each
individual voter, it is assumed that voters only ever elect a candidate if there is no other candidate who would
implement a platform which gives more than half the voters a higher utility. It is then shown that the resulting
outcome is equivalent to the case with a single voter. See section 5.2 of Van Weelden (2010) for details.
6
so a non-trivial fraction of the voters hold a median position in both dimensions. Notice that, when
σA < σB (σB < σA ), the voters are less (respectively more) polarized in dimension A relative to
dimension B. Without loss of generality I assume that σA ≤ σB .
As before, I assume there are two candidates, k = (kA , kB ) ∈ {−κ, κ}, where κ = (κA , κB ) ∈
[0, 1]2 . That is, the candidates have policy preferences, which may diverge, in each of the two
dimensions. Note that, by symmetry, it is without loss of generality to assume κ ∈ [0, 1]2 .
As in the one-dimensional case, candidates and voters have preferences over policy and rentseeking. That is, the stage game utilities are defined as
uv (pt ) = −(xtA − vA )λ − (xtB − vB )λ − mt ,
for the voters, and
uk (kt , pt ) =
ḡk (pt ) = −(xtA − kA )λ − (xtB − kB )λ + γmt if k = kt ,
gk (pt ) = −(xtA − kA )λ − (xtB − kB )λ − mt
if k 6= kt .
for the candidates. Histories, strategies, and payoffs in the super-game can then be defined exactly
as in the one-dimensional case, and the aggregate social welfare from a strategy profile can be
defined as
U ∗ (s) = σA U(−1,0) (s) + σB U(0,−1) (s) + (1 − 2(σA + σB ))U(0,0) (s) + σA U(1,0) (s) + σB U(0,1) (s).
2.2
Candidate Behavior
For simplicity I restrict attention to subgame perfect equilibria in pure strategies. As this is a
symmetric environment I restrict attention to symmetric equilibria in which the candidates’ strategies are stationary.9 Stationarity means that the platform the elected candidate will implement,
if elected, will be independent of the history at which they are elected, or the share of votes they
received.
Definition 1 Symmetric and Platform Stationary The candidates’ strategies are symmetric
and platform-stationary if there exists a platform p = (x, m) with x ≥ 0 and m ∈ [0, M ] such that,
for all histories ht and vote shares µ,
sκ (ht , κ, µ) = (x, m),
and
s−κ (ht , −κ, µ) = (−x, m).
9
See Van Weelden (2010) for a discussion of equilibrium selection in a similar setting.
7
Hence, if the candidates’ strategies are platform stationary, the voters will face the same choices
in every period. The voters’ can then myopically optimize without concern for how their choices
today will influence future political competition. As such, while I have assumed that all players
have the same discount factor, only the patience of the candidates will influence the equilibrium
outcomes.
2.3
Voter Behavior
I now describe the voters’ behavior. As there are a continuum of voters, the probability of each
voter being pivotal is 0. I assume that voters vote sincerely, in that if one of the two candidates
would implement a platform which gives the voter a higher payoff than the other candidate then
the voter would vote for that candidate.
Definition 2 Sincere Voting Suppose the candidates are playing symmetric, platform stationary
strategies. Then, if for voters of type v ∈ {−1, 0, 1},
uv (x, m) > uv (−x, m),
then, for all ht and all voters of type v,
sV (ht ) = κ.
Similarly, if
uv (x, m) < uv (−x, m),
then, for all ht and voters of type v,
sV (ht ) = −κ.
When the candidates’ platforms are stationary, the voters will face the same choices in every
period. That voters vote sincerely means simply that each individual will vote for the candidate
who will implement the platform more to that voter’s liking when in office. Hence, when x 6= 0, all
type 1 voters will vote for candidate κ and all type −1 voters will vote for candidate −κ. Similarly,
when the policy space is multi-dimensional, all non-median voters will have a strict preference for
one candidate (if platforms do not converge in that dimension) and so will vote for that candidate.
By symmetry, the type 0 voters will be indifferent between the two candidates. I assume that the
indifferent voters decide who to vote for based on the candidates’ performance in the last period:
they vote to re-elect the incumbent if and only if the utility received in the last period is at least
as high as the utility they would receive if the incumbent is replaced by the challenger. That is, I
assume the indifferent voters decide on the basis of retrospective voting.10
10
Fiorina (1981) introduced the concept of retrospective voting in political science. See Barro (1973), Ferejohn
(1986), Duggan (2000), and Van Weelden (2010) for formal models which use retrospective voting to provide incentives
8
Definition 3 Retrospective Voting Suppose the candidates are playing symmetric, platform stationary strategies. Then, if
uv (x, m) = uv (−x, m),
then, for all ht and all voters of type v,
sV (ht ) = kt−1 ,
if and only if
uv (pt−1 ) ≥ uv (x, m) = uv (−x, m).
I consider only subgame perfect equilibria in which the voters vote sincerely and retrospectively
and the candidates’ strategies are symmetric and platform stationary. Note that, in this complete
information environment, it is the existence of voters who are willing to condition their vote on the
incumbent’s past behavior that allows for equilibria in which candidates implement platforms other
then their ideal policy with full rent seeking when in office. Notice also that, with retrospective
voting, since the type 0 voters are pivotal, the incumbent will be re-elected forever on the equilibrium
path. This is a consequence of full information and pure strategies. In Van Weelden (2010) it is
shown that with imperfect monitoring the incumbent will be defeated with positive probability
along the equilibrium path.
I now turn to determining when divergence is beneficial. I consider first the case in which the
policy is single-dimensional and then turn to the case in which policy is multi-dimensional.
3
3.1
Results for One Dimensional Policy Space
Best equilibrium for each κ
I first consider the case in which the policy space is single-dimensional. In order to determine the
optimal degree of polarization between the candidates I first describe the best equilibrium when
the candidates’ ideal policies are κ and −κ respectively. I begin by considering the case where
κ = 0, so that the candidates and platforms converge. As the candidate cannot make binding
commitments before taking office the candidate must have an incentive to implement the specified
platform when in office in order to secure re-election. Further, as the platforms the candidates
implement are independent of history, in order for the incumbent to lose office after deviating, the
type-0 voters must be indifferent between re-electing the incumbent and electing the challenger.
Incentive constraints for the incumbent politician then immediately imply that the minimum level
to incumbent politicians. As has been noted in the previous literature (e.g. Fearon 1999) there can often be a friction
between voting based on how the candidate performed in the past and how they are expected to perform in the
future. I assume that if a voter strictly prefers the platform one candidate would implement they will always vote
for that candidate (prospective voting), but vote retrospectively if they are indifferent.
9
of rent-seeking, m0 , that can be supported in a platform-stationary subgame perfect equilibrium
with converging platforms can be determined from the equation,
ḡ0 (0, m) = (1 − δ)γM + δg0 (0, m).
(1)
Note that, as ḡ0 (0, 0) = g0 (0, 0), the amount of rent-seeking in any equilibrium must be strictly
positive.11 I now turn to describing the implemented policies and the amount of rent-seeking when
candidates’ preferences do not converge, and then proceed to describe when the voters’ welfare
is enhanced by this divergence. Consider a candidate of type κ > 0. First note that, as the
gains from rent-seeking enter linearly, the ratio of marginal utilities between the candidate and the
median voter with respect to rents is constant in the amount of rent-seeking. In equilibrium with
retrospective voting – so that voters will vote based on the utility level – the ratio of marginal
utilities over policies must be the same. So, in equilibrium, we must have
∂ḡκ (x,m)
− ∂u ∂x
0 (x,m)
∂x
=
∂ḡκ (x,m)
− ∂u ∂m
0 (x,m)
∂m
= γ,
or that the level of rent-seeking is driven to 0. Simple algebraic manipulation then reveals that, in
any symmetric and platform-stationary subgame perfect equilibrium with sincere and retrospective
voting, if m ∈ (0, M ), we must have
κ
xκ =
1
.
(2)
1 + γ λ−1
Now, as with convergence, it is straightforward to determine the minimum amount of rent-seeking,
mκ , which can be supported in equilibrium from the incentive constraints of the elected candidate,
ḡκ (xκ , m) = (1 − δ)γM + δgκ (−xκ , m).
(3)
The above equations then characterize the best equilibrium for each given level of divergence in
candidate ideal points. Substituting into the above equations we can then solve for the level of
rent-seeking in equilibrium for each κ:
mκ =
1
(1 − δ)γM
1
κ
λ
+
[1 − δ(1 + 2γ 1−λ )λ ](
1 ) .
γ+δ
γ+δ
1−λ
1+γ
(4)
1
Note that mκ is decreasing in κ if the candidates are sufficiently patient, δ > δ ≡ (1 + 2γ 1−λ )−λ ,
and increasing otherwise. Finally, when mκ is decreasing, define κ̄ to be the minimum κ for which
mκ ≤ 0. If κ̄ < 1 then on (κ̄, 1] rent-seeking will be driven to 0, and we need only look for the
minimum level of divergence which can be supported with candidates of that type. For each κ in
11
Note that we could allow candidates to receive an intrinsic benefit from holding office above and beyond the
rents they secure for themselves – i.e. we could consider the case where ḡκ (0, 0) > gκ (0, 0). The analysis presented
would go through provided ḡ0 (0, 0) < (1 − δ)ḡ0 (0, M ) + δg0 (0, 0), so that converging platforms and 0 rent-seeking
could not be supported as an equilibrium.
10
[κ̄, 1] we can then define x(κ) to be the minimum level of policy divergence necessary to support
an equilibrium with candidates of type κ and 0 rent-seeking. The first lemma shows that x(κ) has
a unique minimizer on [κ̄, 1].
Lemma 1 There exists a unique minimizer, κ∗ , of x(κ) on [κ̄, 1].
3.2
Welfare Implications of Divergence
In this subsection, I consider when moving away from the center is beneficial to the voters from a
utilitarian welfare perspective. As the electorate consists of both moderate and polarized voters,
I consider the effect on each type of voter in turn. Consider first the moderate voters (v = 0).
As shown in Van Weelden (2010), divergence will benefit such voters provided the players are
reasonably patient.12 Recalling that u0 (x, m) = −xλ − m we have
u0 (xκ , mκ ) = −
1
1
λ
1
1
(1 − δ)γM
+
(1 + γ 1−λ )−λ [δ((1 + 2γ 1−λ )λ − γ 1−λ ) − (1 + γ 1−λ )]κλ .
γ+δ
γ+δ
So the welfare of the centrist voters is increasing in κ provided the players are sufficiently patient.
How patient the players must be for divergence to benefit the centrist voters depends on how concave
utility functions are. If utility functions are concave enough, it will be possible for divergence to
benefit the centrist voters for any level of patience.
Lemma 2 There exists δ̄0 (λ, γ) ∈ (0, 1) such that the welfare of the type 0 voters is maximized
with candidates of type κ = 0 if and only if δ ≤ δ̄0 . When δ < δ̄0 , κ = 0 is the unique maximizer
of the welfare of the type-0 voters. When δ > δ̄0 the welfare of type-0 is maximized with candidates
of type
κ=
κ∗ if κ̄ < 1,
1 otherwise.
Further, δ̄0 is decreasing in λ with limλ→∞ δ̄0 = 0.
Now consider the effect of divergence on the welfare of the non-centrist voters. Obviously, if
candidate κ > 0 is elected, the welfare of a voter of type 1 and of type −1 will be quite different.
However, when using a utilitarian social welfare function, given that there are the same number of
voters of each type, our concern is with the average of the two utilities. When utility functions are
sufficiently concave the average welfare of the type −1 and type 1 voters’ utility will decrease by
more than the type 0 voters’ utility as the implemented policy moves away from the center. First
note that the utility of the two types of voters are
u−1 (xκ , mκ ) = −mκ − (1 + xκ )λ ,
12
See Van Weelden (2010) for a discussion and numerical examples which demonstrate that divergence is beneficial
with patience well below 1 for reasonable parameters.
11
and
u1 (xκ , mκ ) = −mκ − (1 − xκ )λ ,
respectively. Combining the two equations we can see that the average utility of the polarized
voters is
u−1 (xκ , mκ ) + u1 (xκ , mκ )
(1 − xκ )λ + (1 + xκ )λ
= −mκ −
.
2
2
Notice that, as voters’ utilities are concave, the aggregate effect of the policy being pulled from
the center – holding the degree of rent-seeking constant – is negative, as pulling the policy away
from the median (xκ > 0) decreases the welfare of the type −1 voter more than it increases the
welfare of the type 1 voter. Holding the level of rent-seeking constant, the utility loss from policy
xκ instead of 0 is given by
(1−xκ )λ +(1+xκ )λ
2
−1. How the welfare of the polarized voters relates to the
(1−xκ )λ +(1+xκ )λ
− 1 and xλκ , which
2
< xλκ , and so the aggregate welfare
centrist voters is then determined by the relationship between
I discuss in the Appendix. When λ < 2,
(1−xκ )λ +(1+xκ )λ
2
−1
of the polarized voters will decrease by less than the centrist voters when the policy is pulled from
the center. Conversely, when λ > 2, we have
(1−xκ )λ +(1+xκ )λ
2
− 1 > xλκ , and the polarized voters are
harmed more by the policy being pulled away from the center than median voters are. Hence, it is
possible that, while divergence is beneficial to moderate voters, the aggregate welfare of polarized
voters is decreased. Due to the concavity of utility functions, polarized voters greatly dislike having
the policy pulled further from their ideal point, and so the aggregate welfare of the polarized voters
may decrease even though the welfare of the centrist voters is increased by divergence in candidate
preferences. As such, when λ > 2, greater patience will be required for the welfare of the polarized
voters to increase with divergent platforms than for the centrist voters. Nevertheless, I show that
the polarized voters will, on average, benefit as well, provided the players are sufficiently patient.
However, if utility functions are very concave, divergence will only increase the average welfare of
the polarized voters if the players are extremely patient.
Lemma 3 There exists δ̄1 (λ, γ) ∈ (0, 1) such that the welfare of the polarized candidates, u−1 (x, m)+
u1 (x, m), is maximized at κ = x = 0 if and only if δ ≤ δ̄1 . When δ < δ̄1 , κ = 0 is the unique
maximizer of the welfare of the polarized voters. When δ > δ̄1 the welfare of polarized is maximized
with candidates of a type κ > 0. When λ > 2 then the welfare is maximized with
∗
κ if κ̄ < 1,
κ=
1 otherwise.
Further, limλ→∞ δ̄1 = 1.
So we have that the aggregate welfare of the polarized candidates is maximized with convergent
candidates if and only if δ ≤ δ̄1 . Note that, when λ < 2 we will have δ̄1 < δ̄0 , and when λ > 2,
12
δ̄1 > δ̄0 . When λ = 2 the average welfare of the polarized voters will always equal the average
welfare of the centrist voters, so δ̄1 = δ̄0 . As we are interested in maximizing the social welfare
function
u0 (x, m) + σ(u−1 (x, m) + u1 (x, m)),
when δ̄1 6= δ̄0 , whether or not divergence is socially beneficial can depend on the distribution of
voter preferences. The next proposition characterizes when divergence will be beneficial for each
level of patience, and each distribution of voter ideal points, for a given degree of concavity, λ.
Proposition 1 Concavity and Convergence
1. When δ ∈ (0, min{δ̄0 , δ̄1 }), for all σ ∈ (0, 21 ), the utilitarian social welfare function is maximized with κ = 0.
2. When δ ∈ (δ̄1 , δ̄0 ), there exists a σ̄(λ, δ) ∈ (0, 12 ) such that a utilitarian social welfare function
is only maximized by setting κ = x = 0 if σ ≤ σ̄. When σ ∈ (σ̄, 12 ), welfare is maximized by
a κ > 0.
3. When δ ∈ (δ̄0 , δ̄1 ), there exists a σ̄(λ, δ) ∈ (0, 21 ) such that a utilitarian social welfare function
is only maximized by setting κ = x = 0 if σ ≥ σ̄. When σ ∈ (0, σ̄), welfare is maximized by a
κ > 0.
4. When δ ∈ (max{δ̄0 , δ̄1 }, 1), for all σ ∈ (0, 21 ), the utilitarian social welfare function is maximized with κ > 0.
The above proposition outlines the various levels of patience the players may have and the
resulting welfare implications. As there are four components I address each part in turn. The
first part of the above proposition says that, when the players are too impatient, it will always be
optimal to choose converging platforms and candidates. The reason for this is simple. While x = 0
is the welfare maximizing policy, it can still be beneficial to elect candidates who will implement
x 6= 0 as this makes it possible to discipline candidates more effectively. When δ is low, the threat
of future punishments are less effective, so it is optimal to choose centrist candidates. The final
part of the proposition says that, when the players are sufficiently patient, there exist equilibria
with non-converging platforms which outperform those with median candidates and converging
platforms regardless of the distribution of voters’ preferences.
Parts 2 and 3 of Proposition 1 show that, for an intermediate degree of patience, whether or
not divergence is beneficial will depend on the distribution of voters. When voter’s utility functions
are not too concave – that is, when λ < 2 – the average welfare of polarized voters will decrease
less than the welfare of the centrist voters when the policy is pulled away from the center, so less
patience will be needed for divergence to benefit the polarized voters (δ̄1 < δ̄0 ). Hence, when
13
δ ∈ (δ̄1 , δ̄0 ), aggregate welfare will be enhanced with polarized parties, provided there is sufficient
polarization in voters’ preferences. Conversely, when utility functions are more concave – λ > 2 –
the average welfare of the polarized voters decreases more than for the centrist voters as the policy
is pulled from the center. Hence, when δ ∈ (δ̄0 , δ̄1 ), so that there is sufficient patience for divergence
to benefit the centrist voters but not to benefit the polarized voters, polarization in candidates and
platforms will enhance aggregate welfare unless the preferences of voter are too polarized.
An alternative way to view the above proposition is to consider how much patience is required
to support divergence for each distribution of voter preferences. Holding λ fixed, for each σ, we
can define δ̄(σ) to be the minimum level of patience such that, when δ > δ̄(σ), κ = x = 0 does not
maximize a utilitarian social welfare function. Notice that δ̄(σ) ∈ [min{δ̄0 , δ̄1 }, max{δ̄0 , δ̄1 }] for all
σ. Then we have the immediate corollary which says that, when λ > 2, the necessary patience is
increasing in the degree of polarization, and when λ < 2 it is decreasing.
Corollary 1 Patience and Polarization
1. When λ > 2 the degree of patience necessary for divergence to be beneficial is increasing in
polarization:
∂ δ̄(σ)
> 0.
∂σ
2. When λ < 2 the degree of patience necessary for divergence to be beneficial is decreasing in
polarization.
∂ δ̄(σ)
< 0.
∂σ
So we see that, when λ > 2, as the voters become more polarized, greater patience is required
to support divergence. In fact, when utility functions are very concave, we can say something
stronger. Recall that by Lemma 2, no matter how impatient the players are, when utility functions
are sufficiently concave, the welfare of the centrist voters will be enhanced by candidates moving
away from the center. Similarly, by Lemma 3, no matter how patient the players are, when utility
functions are sufficiently concave, the welfare of the polarized voters will be decreased by a lack
of convergence in platforms. Hence, regardless of the level of patience, when utility functions are
sufficiently concave, divergence in candidate platforms will increase aggregate welfare when enough
voters are centrist, and decrease aggregate welfare when enough voters are polarized.
Proposition 2 High Levels of Concavity
For any δ ∈ (0, 1) there exists a λ̄(δ) ∈ (2, ∞) such that, for all λ > λ̄(δ), there exists a σ̄λ ∈ (0, 1/2)
such that κ = 0 is socially optimal if and only if σ ≥ σ̄λ .
So, we see that, for any level of patience, when utility functions are sufficiently concave, the
welfare of centrist voters will be enhanced by divergent platforms, and the welfare of the polarized
14
voters will be decreased by divergence. Consequently, the aggregate welfare of the voters will be
enhanced by polarized parties and candidates, if and only if the voters’ preferences are not too
polarized.
The results of this section, which has considered how the optimality of converging or nonconverging platforms is determined based on the level of concavity in voters utility functions (λ), the
players’ patience (δ), and the distribution of voters preferences (σ), can be summarized as follows.
Holding the degree of concavity in utility functions fixed, if the players are sufficiently patient,
divergent platforms will always be beneficial for any symmetric distribution of voter preferences.
Conversely, when players are impatient, convergence will always be optimal. For intermediate levels
of patience, whether divergence or convergence is optimal will depend on the distribution of voter
ideal points: the amount of patience necessary for divergence to be beneficial will be increasing
in voter polarization when utility functions are sufficiently concave, and decreasing when utility
functions are less concave. Alternatively, holding the amount of patience fixed, we have seen that,
if the players’ utility functions are concave enough, divergence will benefit the centrist voters, but
harm the polarized voters. Consequently, convergence will be optimal when voters are sufficiently
polarized, and divergence will be optimal when voters are sufficiently centrist.
Having now established when converging platforms will be optimal and when divergence will
be beneficial, I now turn to analyzing the optimal divergence when divergence is beneficial.
3.3
Optimal Divergence
Having now established that non-convergence in candidate preferences and platforms is welfare
enhancing, with sufficient patience, I now turn to the question of how much divergence is optimal
for different degrees of polarization in voter preferences. In this model, however, it will often be
optimal to have no rent-seeking or full polarization in candidate preferences, making it difficult to
take comparative statics. This is due to the functional form assumption that utility functions are
linear in rents. With sufficient patience, it is optimal to increase divergence until the equilibrium
rent-seeking is driven to 0, or there is maximum polarization in candidate preferences, κ = 1.
Proposition 3 Optimal Divergence
Suppose δ > max{δ̄0 , δ̄1 }. Then it is socially optimal to have κ = 1 if κ̄ ≥ 1 and κ = κ∗ ∈ [κ̄, 1] if
κ̄ < 1.
In this section, in order to consider comparative statics, I modify the model to ensure that the
optimal polarization will not involve a “corner solution” with maximum polarization or 0 rentseeking. As noted in Van Weelden (2010), divergence can still be beneficial when utility functions
are concave over rents, but the amount of rent-seeking will not be driven to 0. How much rentseeking will persist in the best equilibrium will then depend on the distribution of voter preferences.
15
In this subsection I consider a variation of the model in which voter’s utility functions are
concave over rents. That is, the voters’ utility is
uv (pt ) = −|xt − v|λ −
m1+β
t
.
1+β
When β = 0 the voters’ utility functions are exactly as in the previous subsection, but when β > 0
utility functions are concave. I then perturb the utility functions of the candidates k ∈ {−κ, κ} to
be
ḡk (pt ) = γ
m1+β
t
− |xt − k|λ ,
1+β
if the candidate is elected, and
gk (pt ) = −|xt − k|λ − mt +
γ m1+β
( t
− mt ),
δ 1+β
if they are not. While this then assumes that utility functions are convex over rents for the
candidates, and introduces additional complexity in the candidates’ utility functions, allowing the
stage game utility to depend on the level of patience for the candidate who is not elected, it
guarantees that the platform each candidate implements in office will be the same as in the case
with linear rents. This provides the simplest perturbation in which the rents will not be driven to
0 and so the degree of divergence will vary with the degree of polarization. In order to ensure that
we are not at a corner solution I assume that M is neither too big or too small – since the disutility
of rent-seeking depends on the level of rent-seeking, if M is too small it will not be beneficial to
move away from the center, and if M is too large it will be optimal to diverge until reaching max
polarization. The following proposition shows how polarization in candidates relates to polarization
in voters.
16
Proposition 4 Optimal Divergence
Suppose δ > min{δ̄0 , δ̄1 }. Then there exists β̄ > 0 such that for all β ∈ (0, β̄), there exists a
non-degenerate interval (M , M̄ ), such that for all M ∈ (M , M̄ ),
1. when λ > 2 the optimal degree of polarization in candidates is strictly decreasing in the polarization of the voters. That is, there exists σ̄(λ, δ, M, β) ∈ (0, 12 ] such that, on (0, σ̄(λ, δ, M, β)),
the best equilibrium involves less polarization,
∂κ∗ (σ)
< 0;
∂σ
and more rent-seeking,
∂x∗ (σ)
< 0,
∂σ
∂m∗ (σ)
> 0.
∂σ
2. when λ < 2 the optimal degree of polarization in candidates is strictly increasing in the polarization of the voters. That is, there exists σ̄(λ, δ, M, β) ∈ [0, 21 ) such that, on (σ̄(λ, δ, M, β), 12 ),
the best equilibrium involves more polarization,
∂κ∗ (σ)
> 0;
∂σ
and less rent-seeking,
∂x∗ (σ)
> 0,
∂σ
∂m∗ (σ)
< 0.
∂σ
So we see that, when not at a “corner solution” with full polarization or 0 rent-seeking, the
optimal degree of polarization will depend on the distribution of voter preferences. When λ > 2,
the benefits of divergence are greater for the centrist voters than for the polarized voters and so
the degree of polarization is decreasing in the fraction of polarized voters. Conversely, when λ < 2,
the benefits of polarization are greater for polarized voters, thus, the more polarized the voters the
more polarization it will be optimal to have in the candidates.
4
Multiple Dimensional Policy Space
In the previous section I considered the case in which the policy space is single dimensional. I
now consider the model with multi-dimensional policy-space as defined in Section 2.1. In order to
keep the analysis as simple as possible I assume that voters only hold non-median positions in one
dimension. This also allows for a simple representation of which dimension is more polarized.13
13
In this model, regardless of the distribution of voter preferences, there will always exist pure strategy subgame
perfect equilibria when there are only two candidates. In particular, there would always be an equilibrium in which
both candidates implement their most preferred policy with full rent seeking at every history in which they are
elected, and all voters simply vote for the candidate they prefer given that strategy. Such an equilibrium is consistent
with retrospective and sincere voting. In addition, if the assumption that all voters hold centrist preferences in at
least one dimension was replaced with the assumption than any voter who has a positive (negative) ideal point in
17
Recall that fraction σA of the voters are at either pole in one dimension and σB are at either pole
in the other dimension. The remaining 1 − 2(σA + σB ) of the voters are located in the center at
(0, 0). Note that, for each κ ∈ [0, 1]2 , we can determine the policy implemented in each dimension
as before. As in the single dimensional case, recall that divergence is more beneficial to polarized
voters than moderate voters when λ < 2 and more beneficial to moderate than polarized voters
when λ > 2. Hence, when λ < 2, it is socially optimal to have greater divergence in the dimension
in which voters are more polarized, and, when λ > 2, it is socially optimal to have divergence in
the dimension in which voters are more centrist.
Proposition 5 Dimension of Greater Polarization
Suppose δ > min{δ̄(σA ), δ̄(σB )}, σA < σB and κ̄(M/2), κ∗ (M/2) < 1,
1. if λ > 2 then it will be optimal to have greater polarization in the dimension voters are less
polarized in: xA > xB and κA > κB .
2. if λ < 2 then it will be optimal to have greater polarization in the dimension voters are more
polarized in: xA < xB and κA < κB .
So we have that, when λ > 2, it is socially optimal to have less polarization in the dimension in
which voters are more polarized, and when λ < 2 it is socially optimal to have greater polarization
in the dimension in which voters are more polarized. Note, finally, that we must have that M is
not too large – κ∗ (M/2) < 1, so that if there was an equal level of polarization in each dimension,
the most moderate policy with 0 rent-seeking would involve less than full polarization in candidate
preferences – to ensure that polarization will differ in different dimensions. When M is very large,
it may be optimal to simply have maximum polarization in both dimensions.
Now I show that, for a range of parameters, not only is it socially optimal to have greater
polarization in the dimension in which voters are more moderate, but it is socially optimal to have
convergence in the dimension in which voters are more polarized. This occurs when the opportunities for rent-seeking are modest enough that only one dimension of polarization is necessary to
dissuade elected candidates from securing rents for themselves.
one dimension has a non-negative (non-positive) ideal point in the other dimension, the set of equilibria with sincere
and retrospective voting would be completely unchanged. Banks and Duggan (2008) provide results on the existence
and uniqueness of equilibria with retrospective voting and prospective voting in a setting with multiple dimensional
policy spaces in which, unlike in this model, the candidates’ types are private information.
18
Proposition 6 Divergence and Convergence in Two-Dimensions
Assume δ > max{δ̄0 , δ̄1 }, σA < σB , and κ̄(M ), κ∗ (M ) < 1. Then,
1. if λ > 2 the best equilibrium involves divergence in dimension A and convergence is dimension
B: κA > 0 and xA > 0, and κB = xB = 0.
2. if λ < 2 the best equilibrium involves divergence in both dimensions, but greater divergence in
dimension B: 0 < κA < κB and 0 < xA < xB .
So we see that, from a social welfare perspective, when utility functions are sufficiently concave
– λ > 2 – it is optimal to have divergence in the dimension in which voters are less polarized and
convergence in the dimension in which voters are more polarized. While this may sound counterintuitive at first the reason for this result is very simple. The more polarized voters are, if utilities
are sufficiently concave, the more costly having a non-median policy implemented is for the voters –
as the voters on the extremes are most sensitive to changes in policies. Hence, by having divergence
in only the dimension in which voters are more moderate, there are fewer non-centrist voters who
are harmed by this divergence. When λ < 2, non-median policies are less harmful, on average, to
polarized voters, so we get greater polarization in the dimension in which voters are more polarized.
Unlike in the case where λ > 2, however, we get divergence in both dimensions. This is because,
when λ < 2, the aggregate welfare in one policy dimension, −σ[(1 + x)λ + (1 − x)λ ] − (1 − 2σ)xλ is
strictly concave in κλ , so the benefits from moving away from the center decrease the further away
from the center the candidates move, and thus there is always an advantage to moving a positive
distance away from the center in both dimensions, regardless of the distribution of preferences.
When λ > 2, −σ[(1 + x)λ + (1 − x)λ ] − (1 − 2σ)xλ is convex in κλ , so it is optimal to choose the
most advantageous dimension and polarize only in that dimension.
5
Discussion
The focus of this paper has been on comparing the welfare across equilibria with different degrees
of polarization in candidate preferences. While earlier work in Van Weelden (2010), has endogenized the candidates, my analysis here has focused on the normative comparison across different,
exogenously given, candidates. The aim in this analysis is then less to explain why parties locate
the way they do than to question the usual conclusions about the welfare implications of diverging
platforms. I have shown that, from a social welfare perspective, polarization in the candidates can
be welfare enhancing, whether the voters themselves are polarized or not. In fact, for a range of
utility functions, polarization is more advantageous when the voters themselves are more centrist.
In recent years political scientists (see Fiorina et al 2006, McCarty et al 2006) have observed
that the political parties have become more polarized. Further, in recent elections, parties have
19
been less polarized over economic policy and more polarized with respect to social policy (see Figure
9.5 in Fiorina et al 2006).14 This is despite the fact that income remains the best predictor of an
individual’s vote (McCarty et al 2006) and despite an apparent lack of polarized views among the
American public on social issues (Fiorina et al 2006). In the model presented, far from being a sign
of democratic failure, such an outcome – when utility functions are sufficiently concave (λ > 2)
– is simply the most efficient way to motivate candidates not to shirk their responsibilities when
in office. Divergence over the dimension in which only a small fraction of the voters hold extreme
preferences (social policy) and convergence in the dimension in which voters are more polarized
(taxation and economic policy), would, in fact, be the socially efficient way to discipline elected
officials. It could even provide a possible explanation for changes in the dimension of polarization
over time. If an increase in income inequality over time leads to increased polarization in voter
preferences over economic policy, as argued in McCarty et al (2006), my results indicate that, if
λ > 2, it would be optimal to move from parties polarized on economic issues to parties polarized
on social issues – precisely because there are so few voters who hold extreme positions on these
issues.
The key assumption driving the divergent platforms result, in this paper and in Van Weelden
(2010), is that utility functions are strictly concave (λ > 1). When utility functions are not concave,
there is no benefit to divergence for any level of patience. While concavity is usually assumed about
voter utility functions, it is not necessarily clear that welfare functions always have that shape.15
What does concavity of utility functions mean in a voting context? One interpretation of the
concavity of voters utility functions is that, with each policy issue that goes against an individual’s
liking, the more upset the voter is are about the next issue policy outcome which goes against their
preferences. That is, an individual who supports free markets, would be more angry about a new
tariff being introduced if it follows a tax increase rather than a tax cut. Such an assumption about
voter’s preferences certainly seems plausible and may be possible to test in a laboratory
In this case, the form of the optimal polarization will depend on whether or not λ > 2. Note
that λ = 2 corresponds to a quadratic loss-function. Recall that, with a quadratic loss function,
individuals care only about the expected distance from their ideal point, and the variance in policy.
As x increases, the expected policy – if each candidate is elected with probability 1/2 in the first
period – remains 0 and the variance is simply x. Hence, the average utility change for polarized
and centrist voters is the same. λ > 2 then corresponds to the case in which utilities are more
concave than the quadratic utility function – that is, increasing the variance in policy is more
14
See also Gerring (1998), who shows that the fraction of the Democratic party’s platform devoted to social and
cultural issues has increased substantially since the 1970s.
15
See Osborne (1995) for a discussion of the assumption that utility functions are concave. He concludes that “in
the absence of any convincing empirical evidence, it is not clear which of the assumptions [concave utility functions
or not] is more appropriate”.
20
harmful to a voter the further the expected policy is from the voter’s bliss point. When increasing
variance increases the disutility more the further the expected policy is from the voter’s ideal point,
polarized voters will be harmed more, on average, by changes in policy around the median voter’s
ideal point. In essence, λ > 2 means that utility functions get more concave the further the policy
is from the voter’s ideal policy. Similarly, λ < 2 corresponds to the case in which utility functions
are less concave than the quadratic. Another feature to notice about utility functions with λ > 2 is
that, calculating the third derivative of the voter’s utility function as the policy moves towards the
λ−3 > 0. Whether the third derivate of the utility
voter’s ideal point, u000
v (x) = λ(λ − 1)(λ − 2)|x − v|
function is positive or not has been extensively discussed in the macroeconomics literature – a
positive third derivative corresponding to a precautionary savings motive in life cycle consumption
(Leland 1968), which has been empirically supported (e.g. Gourinchas and Parker 2001).
6
Conclusions
In this paper I consider how polarization in candidate platforms relates to the polarization of voter
preferences. While it is often assumed that the positions of candidates closely mimic those of
the voters, this does not have to be the case. While there are many parameters for which the
optimal divergence will mirror the distribution of voter preferences, I have shown that, when utility
functions are sufficiently concave, the welfare of the voters may be higher with candidates who
have policy preferences much different than the majority of the voters: who are polarized on issues
where the voters are moderate and centrist where the voters are polarized.
This provides a new explanation for why parties are polarized when voters are not, and a
possible explanation for why parties have diverged over social issues despite a lack of evidence that
voter preferences are similarly polarized. Interestingly, for a range of preferences, divergence over
such issues is, in fact, the most efficient way to create incentives for candidates not to abuse their
position once elected: elected officials are disciplined by the threat of changes in policy which matter
greatly to them, but which are relatively unimportant to the comparatively moderate electorate.
7
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22
8
8.1
Appendix
Relationship between xλ and (1 + x)λ + (1 − x)λ
In this subsection I present some properties of xλ and (1 + x)λ + (1 − x)λ which will be useful for establishing the results of this paper. Of particular relevance is the relationship between
(1+x)λ +(1−x)λ −2
,
2
the average utility loss for voters when the policy is x rather than 0, and xλ , the
utility loss for the type 0 voters. The proofs of these properties are in the Supplemental Appendix.
Property 1
(1+x)λ +(1−x)λ −2
2
− xλ is negative and decreasing in x if λ < 2, constant in x and equal
to 0 if λ = 2, and positive and increasing in x if λ > 2.
Property 2
(1+x)λ +(1−x)λ −2
2xλ
Property 3
(1+x)λ−1 −(1−x)λ−1
2xλ−1
is decreasing in x if λ > 2 and increasing in x if λ < 2.
is greater than 1 and decreasing in x if λ > 2, and less than 1 and
increasing in x if λ < 2.
Property 4 limx→0
(1+x)λ−1 −(1−x)λ−1
2xλ−1
Property 5 limx→0
(1+x)λ +(1−x)λ −2
2xλ
8.2
= ∞ if λ > 2 and limx→0
= ∞ if λ > 2 and limx→0
(1+x)λ−1 −(1−x)λ−1
2xλ−1
(1+x)λ +(1−x)λ −2
2xλ
= 0 if λ < 2.
= 0 if λ < 2.
Proofs
Proof of Lemma 1
See Supplemental Appendix. Proof of Lemma 2
See Supplemental Appendix. Proof of Lemma 3
See Supplemental Appendix. Proof of Proposition 1
Immediate from Lemmas 2 and 3. Proof of Proposition 2
Immediate from Lemmas 2 and 3. Proof of Proposition 3
There are two cases to consider, λ ≤ 2 and λ > 2. First suppose λ ≤ 2. Then max{δ̄0 , δ̄1 } = δ̄0 .
We have established in Lemma 2 that the welfare of the type 0 voters is maximized at
(x(κ∗ ), 0) if κ̄ < 1,
∗
∗
∗
(x , m ) = x =
(x1 , m1 ) otherwise.
23
Now, since λ ≤ 2,
(1 + xκ )λ + (1 − xκ )λ − 2
− xλκ ,
2
is (at least) weakly decreasing in xκ and hence also κ. Hence if
u0 (xκ , mκ ) = −mκ − xλκ ,
is increasing for all κ ∈ [0, κ̄], then
(1 + xκ )λ + (1 − xκ )λ − 2
u−1 (xκ , mκ ) + u−1 (xκ , mκ )
= −mκ −
,
2
2
is also increasing in κ for κ ∈ [0, κ̄]. Hence the welfare of the polarized voters will be maximized at
(x(κ∗ ), 0) if κ̄ < 1,
∗
∗
∗
(x , m ) = x =
(x1 , m1 ) otherwise.
This is the same point as that which maximized the welfare of the type-0. As the social welfare
function is simply a weighted average of centrist and polarized voters’ utilities, we are done.
Now consider the case in which λ > 2. Then max{δ̄0 , δ̄1 } = δ̄1 . First note that, as δ > δ̄0 the
welfare of the type 0 voters is maximized by
(x(κ∗ ), 0) if κ̄ < 1,
∗
∗
∗
(x , m ) = x =
(x1 , m1 ) otherwise.
Next note that, as we have established in Lemma 3, when λ > 2 and δ > δ̄1 , the aggregate welfare
of the polarized voters is maximized at
∗
∗
∗
(x , m ) = x =
(x(κ∗ ), 0) if κ̄ < 1,
(x1 , m1 ) otherwise.
This is the same point as that which maximized the welfare of the type-0 voters. As the social
welfare function is simply a weighted average of centrist and polarized voters’ utilities, we are done.
I now prove Proposition 4. To do so I first prove a couple of Lemmas. The first Lemma show
that, for each κ, the amount of divergence in candidates is the same as in the case with linear rents.
Lemma 4 Suppose δ > δ and κ ∈ (0, κ̄). Then the best equilibrium involving candidates of type κ
and −κ is determined by
κ
xκ =
1
,
1 + γ λ−1
mκ =
1
(1 − δ)M 1+β
1
κ
λ
+
[1 − δ(1 + 2γ 1−λ )λ ](
1 ) .
(γ + δ)(1 + β) γ + δ
1 + γ 1−λ
24
Proof. See Supplemental Appendix.
The second Lemma shows that, for the range of parameters considered in Proposition 4, it is
socially optimal to have divergence, but not to have so much divergence that rents are driven to 0
or the maximum polarization.
Lemma 5 For all δ > min{δ̄0 , δ̄1 }, there exists a β̄ ∈ (0, 1] such that, for all β ∈ (0, β̄), there exists
a non-degenerate interval (M , M̄ ), such that for all M ∈ (M , M̄ ),
1. if λ > 2 then there exists σ̄(λ, δ, M, β) ∈ (0, 21 ] such that the socially optimal level of divergence
is κ∗ ∈ (0, min{κ̄, 1}) when σ ∈ (0, σ̄).
2. if λ < 2 then there exists σ̄(λ, δ, M, β) ∈ [0, 12 ) such that the socially optimal level of divergence
is κ∗ ∈ (0, min{κ̄, 1}) when σ ∈ (σ̄, 21 ).
Proof. See Supplemental Appendix.
Proof of Proposition 4
By Lemmas 4 and 5 we can restrict attention to equilibria with κ ∈ (0, min{κ̄, 1}), and in which
the platform implemented in every period in equilibrium is
κ
xκ =
1
,
1 + γ λ−1
mκ =
1
(1 − δ)M 1+β
1
κ
λ
+
[1 − δ(1 + 2γ 1−λ )λ ](
1 ) .
(γ + δ)(1 + β) γ + δ
1 + γ 1−λ
I now turn to analyzing the optimal level of divergence on the interval (0, min{κ̄, 1}). Taking
derivative of w(κ) with respect to κ we get
w0 (κ) = −mβκ
∂xκ
∂mκ
+λ
[σ(1 − xκ )λ−1 − σ(1 + xκ )λ−1 − (1 − 2σ)xλ−1
κ ],
∂κ
∂κ
which is equal to 0 if and only if
−mβκ
∂mκ
∂xκ
=λ
[σ(1 + xκ )λ−1 − σ(1 − xκ )λ−1 + (1 − 2σ)xλ−1
κ ].
∂κ
∂κ
1
I first consider how mκ varies with κ. As δ > min{δ0 , δ1 } > (1 + 2γ 1−λ )−λ ,
1
1
λ
1
∂mκ
λ
1
λ
λ λ−1
=
[1−δ(1+2γ 1−λ )λ ](
=
[1−δ(1+2γ 1−λ )λ ]γ λ−1 (1+γ λ−1 )xλ−1
< 0,
1 ) κ
κ
∂κ
γ+δ
γ+δ
1 + γ 1−λ
and
1
∂ 2 mκ
λ(λ − 1)
1
λ λ−2
=
[1 − δ(1 + 2γ 1−λ )λ ](
< 0,
1 ) κ
2
∂κ
γ+δ
1−λ
1+γ
so mκ is decreasing and concave in κ.
25
I now show that, when λ > 2 the optimal amount of polarization is decreasing in σ, and when
λ < 2 it is increasing in σ. First consider the case where λ ≤ 2. Recalling that
1
∂xκ
=
1 ,
∂κ
1 + γ λ−1
the F.O.C. for welfare maximization reduces to
Cmβκ =
where C =
1
γ+δ [δ(1
σ(1 + xκ )λ−1 − σ(1 − xκ )λ−1 + (1 − 2σ)xλ−1
κ
,
xλ−1
κ
1
λ
1
+ 2γ 1−λ )λ − 1]γ λ−1 (1 + γ λ−1 )2 is constant with respect to κ.
Now note that mκ is decreasing in κ and, because λ < 2,
(1 + xκ )λ−1 − (1 − xκ )λ−1
σ(1 + xκ )λ−1 − σ(1 − xκ )λ−1 + (1 − 2σ)xλ−1
κ
= 1 − 2σ + 2σ
,
λ−1
2xλ−1
xκ
is increasing in κ. Hence there exists a unique κ∗ which satisfies w0 (κ) = 0 and the welfare of the
voters is increasing for κ < κ∗ and decreasing for κ > κ∗ . So we have that κ∗ ∈ (0, κ̄) maximizes
aggregate welfare. Finally, since for λ < 2,
(1+xκ )λ−1 −(1−xκ )λ−1
2xλ−1
κ
is less than 1, we can conclude that
∂κ∗
> 0.
∂σ
Now I turn to the case in which λ > 2. When λ > 2 there may be multiple solutions to the
FOC condition. As in the case where λ < 2 the FOC is satisfied reduces to
Cmβκ = 1 − 2σ + 2σ
(1 + xκ )λ−1 − (1 − xκ )λ−1
.
2xλ−1
Note that now both sides of the equation are decreasing in κ. Further,
(1 + xκ )λ−1 − (1 − xκ )λ−1
= ∞,
κ→0
2xλ−1
κ
lim
so w0 (κ) < 0 for κ sufficiently small. Note, however, that we have established that κ = 0 is not the
maximizer, so we must have w0 (κ) > 0 at some point. I now show that once w0 (κ) becomes positive
it will remain positive until it becomes negative at κ∗ , the point which maximizes voters’ welfare.
To see this, first note that
1 − 2σ + 2σ
(1 + xκ )λ−1 − (1 − xκ )λ−1
2xλ−1
is decreasing and convex in κ. This follows from the fact that
∂ (1 + xκ )λ−1 − (1 − xκ )λ−1
λ−1
=
[(1 − x)λ−2 − (1 + x)λ−2 ]
λ−1
∂x
2x
2xλ
and so
∂ 2 (1 + xκ )λ−1 − (1 − xκ )λ−1
λ−1
= λ+1 (2[(1 + x)λ−3 + (1 − x)λ−3 ] + λ[(1 + x)λ−3 − (1 − x)λ−3 ]) > 0.
2
λ−1
∂x
2x
2x
26
Next note that since mκ is decreasing and concave, Cmβκ is decreasing and concave for all β ∈ (0, β̄]
λ−1 −(1−x
since β̄ ≤ 1. Hence, Cmβκ and 1 − 2σ + 2σ (1+xκ )
2xλ−1
λ−1
κ)
cross at two points, the first of which
is a minimum and the second a maximum. So we have established that there exists κ∗ to maximize
the aggregate welfare of the voters, which is the greater of the two solutions to
Cmβκ = 1 − 2σ + 2σ
Now, because λ > 2 and so
(1 + xκ )λ−1 − (1 − xκ )λ−1
.
2xλ−1
(1+xκ )λ−1 −(1−xκ )λ−1
2xλ−1
> 1 we can then conclude that
∂κ∗
< 0.
∂σ
Finally note that as we have established that κ∗ is increasing in κ for λ < 2 and decreasing in
κ for λ > 2, and that m is decreasing in κ and x is increasing in κ, we are done.
I now turn to proving the results for a two dimensional policy space. I first prove the following
simple lemma.
Lemma 6 δ > min{δ̄(σA ), δ̄(σB )}. Then, if the optimal equilibrium involves κA > 0 and κB > 0,
we must have
xB
xA
=
.
κA
κB
Proof. See Supplemental Appendix.
Proof of Proposition 5
First note that, since δ > min{δ̄(σA ), δ̄(σB )}, aggregate voter welfare can be enhanced by moving
away from the center in at least one dimension. Also, since κ̄(M/2), κ∗ (M/2) < 1, it cannot be
optimal to have κA = κB = 1: there exists κ ∈ [κ̄(M/2), 1) such that there is less polarization
in policies with candidates of type (κ, κ) and (−κ, −κ) than with candidates of type (1, 1) and
(−1, −1). Hence we can conclude that we must have (0, 0) < (κA , κB ) < (1, 1).
I now show that it cannot be optimal to have κA > κB when λ < 2, or to have κA < κB when
λ > 2. To see this, recall that by Lemma 6, if κA > κB then we must have xA > xB , and if κA < κB
then we must have xA < xB . Note also that
ḡ(κA ,κB ) (xA , xB , m) ≥ (1 − δ)γM + δg(κA ,κB ) (−xA , −xB , m),
if and only if
ḡ(κB ,κA ) (xB , xA , m) ≥ (1 − δ)γM + δg(κB ,κA ) (−xB , −xA , m).
27
Hence, in order for (κA , κB ) to be welfare maximizing, we must have
(1 − 2σA − 2σB )u(0,0) (xA , xB , m) + σA [u(−1,0) (xA , xB , m) + u(1,0) (xA , xB , m)] +
σB [u(0,−1) (xA , xB , m) + u(0,1) (xA , xB , m)] ≥
(1 − 2σA − 2σB )u(0,0) (xB , xA , m) + σA [u(−1,0) (xB , xA , m) + u(1,0) (xB , xA , m)] +
σB [u(0,−1) (xB , xA , m) + u(0,1) (xB , xA , m)]
Removing constant terms and rearranging we get that this holds if and only if
(σB − σA )([(1 + xA )λ + (1 − xA )λ − 2 − 2xλA ] − [(1 + xB )λ + (1 − xB )λ − 2 − 2xλB ]) ≥ 0.
As σA < σB this holds if and only if
(1 + xA )λ + (1 − xA )λ − 2 − 2xλA ≥ (1 + xB )λ + (1 − xB )λ − 2 − 2xλB .
However, recall that, when λ < 2, (1 + x)λ + (1 − x)λ − 2 − 2xλ is decreasing in x, so this cannot
be satisfied with κA > κB . Similarly, when λ > 2, (1 + x)λ + (1 − x)λ − 2 − 2xλ is increasing in x,
so this cannot be satisfied when κA < κB . Hence we can conclude that κA ≤ κB when λ < 2 and
κA ≥ κB when λ > 2.
Finally, I show that it cannot be optimal to have κA = κB ∈ (0, 1) when λ 6= 2. Now recall
that, in the best equilibrium, the incentive constraint of the incumbent must hold with equality so
we must have
ḡ(κA ,κB ) (xA , xB , m) = (1 − δ)γM + δg(κA ,κB ) (xA , xB , m).
Holding m fixed, for each κA we can then solve for the corresponding κB which would be necessary
for the incumbent’s incentive constraint to hold with equality given that, by Lemma 5, xi = ρ−1 κi .
Solving for κB in terms of κA is then equivalent to solving for xB in terms of xA , which we can do
from the equation
−(1 − ρ)λ [xλA + xλB ] + γm = (1 − δ)γM − δ(1 + ρ)λ [xλA + xλB ] − δm.
Note that from this equation
xλA + xλB =
(1 − δ)γM − (γ + δ)m
,
δ(ρ + 1)λ − (ρ − 1)λ
and so
∂xB
xA
= −( )λ−1 .
∂xA
xB
In particular, note that, when xA = xB , ∂xB /∂xA = −1. Now, if we can show that the derivative
of the aggregate welfare is non-zero with respect to xA when λ 6= 2 and xA = xB , we could then
28
conclude that κA = κB is not a maximum. Now recall that the aggregate welfare is
(1 − 2σA − 2σB )u(0,0) (xA , xB , m) + σA [u(−1,0) (xA , xB , m) + u(1,0) (xA , xB , m)] +
σB [u(0,−1) (xA , xB , m) + u(0,1) (xA , xB , m)] =
−(1 − 2σA )xλA − (1 − 2σB )xλB − σA [(1 + xA )λ + (1 − xA )λ ] − σB [(1 + xB )λ + (1 − xB )λ ] − m.
Taking derivative with respect to xA and evaluating when xB = xA this becomes
λ(σB − σA )[(1 + xA )λ−1 − (1 − xA )λ−1 − 2xλ−1
A ],
which, as σA 6= σB and λ 6= 2, is never equal to 0. Hence we can conclude that the optimal degree
of polarization does not involve κA = κB .
We then conclude that, when λ < 2 we must have xA < xB and κA < κB , and that when λ > 2
we must have xA > xB and κA > κB .
Proof of Proposition 6
I first consider whether or not the best equilibrium will involve divergence in both dimensions when
λ 6= 2. As κ̄ < 1, and since we have established that if moving away from the center is beneficial
in either dimension it will be beneficial until rents are driven to 0 when δ > max{δ̄0 , δ̄1 }, we need
consider only candidates such that the level of rent-seeking is driven to 0. Note further that as
κ∗ < 1, the constraint κA , κB ≤ 1 cannot bind. Now suppose κA > 0 and κB > 0. Recall that by
Lemma 6 we then have
xB
xA
=
≡ ρ−1 .
κA
κB
Given this, for each κA and κB , we can then implicitly determine the level of polarization from the
equation
−(κA − xA )λ − (κB − xB )λ = (1 − δ)γM − δ[(κA + xA )λ + (κB + xB )λ ].
Note that we can decrease the number of variables by substituting in ρxA for κA and ρxB for κB ,
yielding
−(ρ − 1)λ [xλA + xλB ] = (1 − δ)γM − δ(ρ + 1)λ [xλA + xλB ],
which reduces to
xλA + xλB =
(1 − δ)γM
.
δ(ρ + 1)λ − (ρ − 1)λ
Note from this equation that
∂xB
xA
= −( )λ−1 .
∂xA
xB
Now recall that the welfare of the voters is given by
−(1 − 2σA )xλA − (1 − 2σB )xλB − σA [(1 + xA )λ + (1 − xA )λ ] − σB [(1 + xB )λ + (1 − xB )λ ].
29
In addition, since we know that κA , κB < 1, if we are to have a solution with κA > 0 and κB > 0
then we must the derivative of
−(1 − 2σA )xλA − (1 − 2σB )xλB − σA [(1 + xA )λ + (1 − xA )λ ] − σB [(1 + xB )λ + (1 − xB )λ ],
with respect to xA ,
λ−1
−λ[(1−2σA )xλ−1
A +(1−2σB )xB
is equal to 0. Substituting in for
σA (1 −
∂xB
∂xB
+σA [(1+xA )λ−1 −(1−xA )λ−1 ]−σB [(1+xB )λ−1 −(1−xB )λ−1 ]
]
∂xA
∂xA
∂xB
∂xA
and dividing by xλ−1
this becomes
A
(1 + xB )λ−1 − (1 − xB )λ−1
(1 + xA )λ−1 − (1 − xA )λ−1
)
=
σ
(1
−
).
B
2xλ−1
2xλ−1
A
B
Now, when λ = 2,
(1 + xA )λ−1 − (1 − xA )λ−1
(1 + xB )λ−1 − (1 − xB )λ−1
=
= 1,
2xλ−1
2xλ−1
B
A
and we are at a maximum regardless of the dimension of polarization.
Now consider when λ < 2. Recall by Properties 1 and 2 in the previous subsection that
(1 + x)λ−1 − (1 − x)λ−1
,
2xλ−1
is less than 1 and increasing in x. Hence as σA < σB this implies that we must have xB > xA .
Finally, as
(1 + xA )λ−1 − (1 − xA )λ−1
= ∞,
xA →0
2xλ−1
A
lim
there exists a solution of this form with 0 < xA < xB . Finally, note that, as the objective function
is strictly concave, the first order conditions are also sufficient, so this is the point which maximizes
social welfare.
Now consider the case where λ > 2. Recall by Properties 1 and 2 in the previous subsection
that
(1 + x)λ−1 − (1 − x)λ−1
,
2xλ−1
is greater than 1 and decreasing in x. Hence as σA < σB this implies that we must have xB > xA .
However, by Proposition 5, the optimal divergence when λ > 2 must involve xA > xB , and so this
point is not a local maximum. Hence, we cannot have a maximum with κA > 0 and κB > 0 and
so, when λ > 2 there will be divergence in one, and only one, dimension. Finally as
(1 + x)λ + (1 − x)λ > xλ ,
when λ > 2, since σA < σB , it is socially optimal to have divergence in dimension A when λ > 2
and convergence in dimension B. 30
9
Supplemental Appendix
9.1
Proofs of the relationships between xλ and (1 + x)λ + (1 − x)λ
Property 1
(1+x)λ +(1−x)λ −2
2
− xλ is negative and decreasing in x if λ < 2, constant in x and equal
to 0 if λ = 2, and positive and increasing in x if λ > 2.
Proof. As
(1+x)λ +(1−x)λ −2
2
− xλ is equal to 0 when x = 0 for all λ we need only consider whether
it is increasing or decreasing. The derivative of
λ[
(1+x)λ +(1−x)λ −2
2
− xλ with respect to x is
(1 + x)λ−1 − (1 − x)λ−1
− xλ−1 ].
2
Note that when λ = 2 the derivative is equal to 0. Further, as x ∈ (0, 1),
(1+x)λ−1 −(1−x)λ−1
2
− xλ−1
is increasing in λ. Hence, the derivative is positive when λ > 2 and negative when λ < 2.
Property 2
(1+x)λ +(1−x)λ −2
2xλ
Proof. The derivative of
is decreasing in x if λ > 2 and increasing in x if λ < 2.
(1+x)λ +(1−x)λ −2
2xλ
with respect to x is
2λxλ [(1 + x)λ−1 − (1 − x)λ−1 ] − 2λxλ−1 [(1 + x)λ + (1 − x)λ − 2]
=
4x2λ
λ
[x(1 + x)λ−1 + x(1 − x)λ−1 − (1 + x)λ − (1 − x)λ + 2] =
2xλ+1
λ
[2 − (1 + x)λ−1 − (1 − x)λ−1 ].
2xλ+1
Notice that when x = 0, 2 − (1 + x)λ−1 − (1 − x)λ−1 = 0, and that
∂[2 − (1 + x)λ−1 − (1 − x)λ−1 ]
= (λ − 1)[(1 − x)λ−2 − (1 + x)λ−2 ],
∂x
which is positive if λ < 2, negative if λ > 2 and equal to 0 if λ = 2. As
can then conclude that
(1+x)λ +(1−x)λ −2
2xλ
λ
2xλ+1
is always positive we
is increasing if λ < 2, decreasing if λ > 2, and constant if
λ = 2.
Property 3
(1+x)λ−1 −(1−x)λ−1
2xλ−1
is greater than 1 and decreasing in x if λ > 2, and less than 1 and
increasing in x if λ < 2.
Proof. By Property 1, (1 + x)λ−1 − (1 − x)λ−1 > 2xλ−1 when λ > 2 and (1 + x)λ−1 − (1 − x)λ−1 <
2xλ−1 when λ < 2 so we must only consider the statics. The derivative of
2(λ − 1)
(1+x)λ−1 −(1−x)λ−1
2xλ−1
xλ−1 [(1 + x)λ−2 + (1 − x)λ−2 ] − xλ−2 [(1 + x)λ−1 − (1 − x)λ−1 ]
=
4x2(λ−1)
λ−1
[x(1 + x)λ−2 + x(1 − x)λ−2 − (1 + x)λ−1 + (1 − x)λ−1 ] =
2xλ
λ−1
[(1 − x)λ−2 − (1 + x)λ−2 ],
2xλ
31
is
which is negative if λ > 2 and positive if λ < 2.
Property 4 limx→0
(1+x)λ−1 −(1−x)λ−1
2xλ−1
= ∞ if λ > 2 and limx→0
(1+x)λ−1 −(1−x)λ−1
2xλ−1
= 0 if λ < 2.
Proof. Apply L’Hopital’s rule. Then we get
(1 + x)λ−2 + (1 − x)λ−2
(1 + x)λ−1 − (1 − x)λ−1
=
lim
.
x→0
x→0
2xλ−1
2xλ−2
lim
Now if λ > 2 then limx→0 2xλ−2 = 0 and limx→0 (1 + x)λ−2 + (1 − x)λ−2 = 2 so
(1 + x)λ−1 − (1 − x)λ−1
= ∞.
x→0
2xλ−1
lim
Now if λ < 2 then limx→0 2xλ−2 = ∞ and limx→0 (1 + x)λ−2 + (1 − x)λ−2 = 2 so
(1 + x)λ−1 − (1 − x)λ−1
= 0.
x→0
2xλ−1
lim
Property 5 limx→0
(1+x)λ +(1−x)λ −2
2xλ
= ∞ if λ > 2 and limx→0
(1+x)λ +(1−x)λ −2
2xλ
= 0 if λ < 2.
Proof. As both the numerator and denominator go to 0 as x → 0, this is immediate by Property
4 and L’Hopital’s rule.
9.2
Proofs of Lemmas
Proof of Lemma 1
For any κ ≥ κ̄ we can define x(κ) to be the unique solution to
(1 − δ)γM − δ(κ + x(κ))λ = −(κ − x(κ))λ .
Differentiating with respect to κ and re-arranging we have
[δ(κ + x(κ))λ−1 + (κ − x(κ))λ−1 ]x0 (κ) = (κ − x(κ))λ−1 − δ(κ + x(κ))λ−1 .
So we have that x0 (κ) = 0 if and only if
δ(κ + x(κ))λ−1 = (κ − x(κ))λ−1 ,
which implies that
1
1 − δ λ−1
1 κ.
1 + δ λ−1
Plugging this into the original expression for x(κ) we get that
x(κ) =
λ
(1 − δ)γM =
2λ (δ − δ λ−1 )
(1 + δ
32
1
λ−1
)λ
κλ .
As this equation has a unique solution,
1
∗∗
κ
1 + δ λ−1 (1 − δ)γM 1
)λ ,
=
(
1
2
δ(1 − δ λ−1 )
we see that x0 (κ) = 0 at no more than one point on [κ̄, ∞).
Next, note that taking derivative with respect to κ again we get
00
[δ(κ + x(κ))λ−1 + (κ − x(κ))λ−1 ]x (κ) = (λ − 1)[(κ − x(κ))λ−2 − δ(κ + x(κ))λ−2 ] > 0,
00
when κ = κ∗∗ . So x (κ∗∗ ) > 0 and we have a local min. Combining these two observations, we can
conclude that x(κ) is decreasing for κ < κ∗∗ and increasing for κ > κ∗∗ . Hence we know that there
exist a unique minimizer,
κ∗ = arg min x(κ).
κ∈[κ̄,1]
Notice that, if κ∗∗ ≤ κ̄ then κ∗ = κ̄. Proof of Lemma 2
We have established that the welfare of the centrist voters in the best equilibrium with candidates
of type κ ∈ [0, κ̄) and −κ is given by
u0 (xκ , mκ ) = −
1
1
λ
1
(1 − δ)γM
1
+
(1 + γ 1−λ )−λ [δ((1 + 2γ 1−λ )λ − γ 1−λ ) − (1 + γ 1−λ )]κλ .
γ+δ
γ+δ
Hence the welfare of the centrist voters is increasing provided
1
δ > δ̄0 (λ, γ) ≡
1 + γ 1−λ
1
λ
.
(1 + 2γ 1−λ )λ − γ 1−λ
Note that, when δ > δ̄0 , welfare is increasing until we either reach maximum polarization – κ = 1
– or the rent is driven to 0. If the rent is driven to 0 (κ̄ < 1) then we find the minimum level of
polarization which can support 0 rent-seeking, κ∗ . Notice that as λ > 1 and γ ≥ 1,
1
λ
1
λ
1
(1 + 2γ 1−λ )λ − γ 1−λ > 1 + 2γ 1−λ − γ 1−λ ≥ 1 + γ 1−λ > 0,
so
δ̄0 ∈ (0, 1).
I now show that when δ ≤ δ̄0 choosing κ = 0 maximizes the welfare of the type-0 voters, and is the
unique κ to maximize the welfare of the type-0 voters when δ < δ̄0 . Note that on the interval [0, κ̄]
we have established that welfare is weakly decreasing in κ for δ ≤ δ̄0 and strictly decreasing when
δ < δ̄0 , so we need only check κ ∈ (κ̄, 1], for the case in which κ̄ < 1.
Recall, from the proof of Lemma 1, that there exists κ∗∗ such that the policy is decreasing in κ
for κ < κ∗∗ and increasing for κ > κ∗∗ . Hence if we can show that κ∗∗ ≤ κ̄, then the welfare of the
33
1
type-0 voters is decreasing on [κ̄, 1]. Now note that, as x∗∗ =
we must have
1
1 − δ λ−1
1+δ
1
λ−1
<
1
1−δ λ−1
1
1+δ λ−1
κ∗∗ , in order to have κ∗∗ > κ̄
.
1
1 + γ λ−1
Re-arranging, the patience must be at least
1
δ > (1 + 2γ 1−λ )1−λ ≥ δ̄0 .
1
So, when δ ≤ δ̄0 < (1 + 2γ 1−λ )1−λ , the welfare of the type 0 voters is decreasing in κ on the interval
[κ̄, 1], and so the welfare of the type 0 voters must be maximized with candidates of type κ = 0 and
converging platforms. Further, when δ < δ̄0 the welfare of the type-0 voters is strictly decreasing
on [0, κ̄], and so κ = 0 is the unique maximizer of the type-0 voters’ welfare.
1
I now show that δ̄0 decreases in λ. First, define y = γ 1−λ . Then we have
δ̄0 =
1+y
.
(1 + 2y)λ − y λ
Note that, as y is (weakly) increasing in λ, it is sufficient to show that δ̄0 is decreasing in y and is
decreasing in λ holding y fixed. I first show that δ̄0 is decreasing in y. Taking logs and differentiating
we get
∂log δ̄0
∂y
=
1
λ
−
[2(1 + 2y)λ−1 − y λ−1 ] =
1 + y (1 + 2y)λ − y λ
1
[((1 + 2y)λ − y λ )(1 − λ) − λ((1 + 2y)λ−1 − y λ−1 )] < 0,
(1 + y)[(1 + 2y)λ − y λ ]
so δ̄0 is decreasing in y. Finally note that, holding y fixed, δ̄0 is decreasing in λ. To see this, note
that the denominator is increasing in λ,
∂[(1 + 2y)λ − y λ ]
= λ[2(1 + 2y)λ−1 − y λ−1 ] > 0,
∂λ
and the numerator is constant in λ. We can now conclude that δ̄0 is decreasing in λ as claimed.
Finally, recall that
1
0 < δ̄0 < (1 + 2γ 1−λ )1−λ .
1
1
Now, since when λ > 2, γ 1−λ ≥ γ −1 we have (1 + 2γ 1−λ )1−λ ≤ (1 + 2γ −1 )1−λ ,when λ > 2. As
(1 + 2γ −1 )1−λ clearly goes to 0 as λ → ∞, we can conclude that
lim δ̄0 = 0.
λ→∞
34
Proof of Lemma 3
I prove this result by first showing that, for all λ there exists a δ̄1 ∈ [0, 1] such that the welfare of
the polarized voters is maximized with κ = 0 if, and only if, δ ≤ δ̄. I then show that δ̄1 ∈ (0, 1),
and finally, that limλ→∞ δ̄1 = 1.
We have established that the average welfare of the polarized voters in the best equilibrium
with candidates of type κ ∈ [0, κ̄) and −κ is
(1 − xκ )λ + (1 + xκ )λ
u−1 (xκ , mκ ) + u1 (xκ , mκ )
= −mκ −
.
2
2
Substituting in for mκ and xκ gives
−
1
(1 − δ)γM
1
+
[δ(1 + 2γ 1−λ
γ+δ
γ+δ
u−1 (xκ , mκ ) + u1 (xκ , mκ )
=
2
κ
κ
1
κ
λ
λ
λ
− 1)λ ](
[(1 −
1 ) −
1 ) + (1 +
1 ) ].
2
1 + γ 1−λ
1 + γ 1−λ
1 + γ 1−λ
Now, there are two cases to consider: when λ ≤ 2 and when λ > 2. I consider the case in which
λ ≤ 2. Note first that
(1 + xκ )λ + (1 − xκ )λ − 2
− xλκ ,
2
is (at least) weakly decreasing in xκ and hence also κ. Hence since
u0 (xκ , mκ ) = −mκ − xλκ ,
is increasing for all κ ∈ [0, κ̄] when δ > δ̄0 , so
u−1 (xκ , mκ ) + u−1 (xκ , mκ )
2
(1 + xκ )λ + (1 − xκ )λ − 2
2
(1 + xκ )λ + (1 − xκ )λ − 2
= u0 (xκ , mκ ) − [
− xλκ ],
2
= −mκ −
is also increasing in κ for κ ∈ [0, κ̄]. Now define
δ̄1 = max{δ :
u−1 (x0 , m0 ) + u−1 (x0 , m0 )
u−1 (xκ , mκ ) + u−1 (xκ , mκ )
≥
for all κ ∈ [0, κ̄]}
2
2
and note that 0 ≤ δ̄1 ≤ δ̄0 < 1. In addition, by continuity, when δ = δ̄1 , for all κ ∈ [0, κ̄],
u−1 (x0 , m0 ) + u−1 (x0 , m0 )
(1 + xκ )λ + (1 − xκ )λ )
u−1 (xκ , mκ ) + u−1 (xκ , mκ )
−m0 −1 ≥ −mκ −
=
.
2
2
2
which implies that
m0 − mκ ≤
1
1
κ
(1 + xκ )λ + (1 − xκ − 2)λ )
λ
[δ(1 + 2γ 1−λ )λ − 1](
)
=
.
1
γ+δ
2
1 + γ 1−λ
As m0 − mκ is increasing in δ, when δ < δ̄1 we must have
u−1 (x0 , m0 ) + u−1 (x0 , m0 )
u−1 (xκ , mκ ) + u−1 (xκ , mκ )
<
.
2
2
35
So, when δ < δ̄1 , κ = 0 is the unique maximizer on [0, κ̄].
Finally, note that, as δ̄1 ≤ δ̄0 , if κ̄ < 1 then κ∗ = κ̄ so, when δ ≤ δ̄1 the welfare of the polarized
voters is maximized with κ = 0. We can then conclude that the welfare of the polarized voters is
maximized with κ = 0 if and only if δ ≤ δ̄1 , and that κ = 0 is the unique maximizer when δ < δ̄1 .
I now turn to the case in which λ > 2. I first show that in order to determine whether
u−1 (xκ ,mκ )+u1 (xκ ,mκ )
2
is maximized at κ = 0 or not is sufficient to evaluate the function at two
points: (x0 , m0 ) and
∗
∗
(x , m ) =
(x(κ∗ ), 0) if κ̄ < 1,
(x1 , m1 ) otherwise.
This is because, when the welfare of the polarized voters is not maximized with κ = 0, it will be
maximized with platform (x∗ , m∗ ). To see this, note that
u−1 (xκ ,mκ )+u1 (xκ ,mκ )
2
can be written in
the form
h(κ) = C + Aκλ − B[(D − κ)λ + (D + κ)λ )],
where B, D > 0, on the interval κ ∈ [0, κ̄]. Now, suppose we were to have a local extremum of h
with κ ∈ (0, κ̄). Then we have that
h0 (κ) = λ(Aκλ−1 − B[(D + κ)λ−1 − (D − κ)λ−1 ]) = 0.
Notice then that at this point, since λ > 2,
h00 (κ) = λ(λ − 1)[Aκλ−2 − B[(D + κ)λ−2 + (D + κ)λ−2 ]]
λ(λ − 1)B
[(D + κ)λ−1 − (D − κ)λ−1 − κ(D + κ)λ−2 − κ(D + κ)λ−2 ]
=
κ
λ(λ − 1)BD
[(D + κ)λ−2 − (D − κ)λ−2 ] > 0,
=
κ
making this a local minimum. As we cannot then have a local maximum with positive rent-seeking,
the maximum must either be when κ = 0, after the rents are driven to 0, or after reaching maximum
divergence. Hence we need only compare (x0 , m0 ) with (x∗ , m∗ ) where
(x(κ∗ ), 0) if κ̄ < 1,
∗
∗
(x , m ) =
(x1 , m1 ) otherwise,
to determine whether the welfare of polarized voters is maximized at κ = 0.
I now turn to showing that, if the aggregate welfare of the polarized voters is no higher with
platform (x0 , m0 ) than with (x∗ , m∗ ) for some level of patience δ 0 then it is lower for all δ > δ 0 . To
show this, it is sufficient to show that and
u−1 (x1 , m1 ) + u1 (x1 , m1 ) − u−1 (x0 , m0 ) + u1 (x0 , m0 ) > 0
and
u−1 (0, 0) + u1 (0, 0) − u−1 (x(κ∗ ), 0) − u1 (x(κ∗ ), 0)
<1
u−1 (0, 0) + u1 (0, 0) − u−1 (x0 , m0 ) − u1 (x0 , m0 )
36
for all δ > δ 0 , when these equations hold weakly when δ = δ 0 .
Consider first
u−1 (x1 , m1 ) + u1 (x1 , m1 ) − u−1 (x0 , m0 ) + u1 (x0 , m0 ) =
1
1
1
λ
2 − (1 − x1 )λ + (1 + x1 )λ +
[1 − δ(1 + 2γ 1−λ )λ ](
1 ) .
γ+δ
1−λ
1+γ
Clearly, if this is non-negative for δ 0 then, for all δ > δ 0 it is increasing, and so is positive for all
δ > δ0.
Now I turn to considering
(1 − x(κ∗ ))λ + (1 − x(κ∗ ))λ − 2
u−1 (0, 0) + u1 (0, 0) − u−1 (x(κ∗ ), 0) − u1 (x(κ∗ ), 0)
.
=
u−1 (0, 0) + u1 (0, 0) − u−1 (x0 , m0 ) − u1 (x0 , m0 )
m0
It is sufficient to show that
u−1 (0, 0) + u1 (0, 0) − u−1 (x(κ∗ ), 0) − u1 (x(κ∗ ), 0)
,
u−1 (0, 0) + u1 (0, 0) − u−1 (x0 , m0 ) − u1 (x0 , m0 )
is always decreasing whenever it is less than or equal to 1. Note that we may either have κ∗ = 1 or
κ∗ < 1. Note that for all κ ∈ [κ̄, 1] we have
∂x(κ)
−γM − (κ + x(κ))λ
=
,
∂δ
λ[(κ − x(κ))λ−1 + δ(1 + x(κ))λ−1 ]
and so
−γM − (κ∗ + x(κ∗ ))λ
∂x(κ∗ )
≤
,
∂δ
λ[(κ∗ − x(κ∗ ))λ−1 + δ(1 + x(κ∗ ))λ−1 ]
as variation in κ∗ can only decrease x(κ∗ ). Now we can calculate,
∂[(1 + x(κ∗ ))λ + (1 − x(κ∗ ))λ − 2]
∂δ
∗ + x(κ∗ ))λ
−γM
−
(κ
λ[(1 − x(κ∗ ))λ−1 + (1 + x(κ∗ ))λ−1 ]
λ[(κ∗ − x(κ∗ ))λ−1 + δ(κ∗ + x(κ∗ ))λ−1 ]
≤
<
−γM − (κ∗ + x(κ∗ ))λ ,
because
(1−x(κ∗ ))λ−1 +(1+x(κ∗ ))λ−1 > (1−x(κ∗ ))λ−1 +δ(1+x(κ∗ ))λ−1 ≥ (κ∗ −x(κ∗ ))λ−1 +δ(κ∗ +x(κ∗ ))λ−1 .
Now note that as m0 = (1 − δ)γM/(γ + δ),
∂m0
(1 − δ)γM
γM
γ+1
=−
−
=−
m0 .
2
∂δ
(γ + δ)
γ+δ
(γ + δ)(1 − δ)
So we have that the derivative of
u−1 (0,0)+u1 (0,0)−u−1 (x(κ∗ ),0)−u1 (x(κ∗ ),0)
u−1 (0,0)+u1 (0,0)−u−1 (x0 ,m0 )−u1 (x0 ,m0 )
−γM − (κ∗ + x(κ∗ ))λ +
γ+1
(γ+δ)(1−δ) [(1
m0
37
is less than
+ x(κ∗ ))λ + (1 − x(κ∗ ))λ − 2]
.
Note that this is negative whenever
(1 − δ)γM + (1 − δ)(κ∗ + x(κ∗ ))λ >
γ+1
[(1 + x(κ∗ ))λ + (1 − x(κ∗ ))λ − 2].
γ+δ
Now, recalling that by the definition of x(κ) we have
(1 − δ)γM = δ(κ∗ + x(κ∗ ))λ − (κ∗ − x(κ∗ ))λ ,
and so
(κ∗ + x(κ∗ ))λ >
Further, when
(1 − δ)γM
.
δ
u−1 (0, 0) + u1 (0, 0) − u−1 (x(κ∗ ), 0) − u1 (x(κ∗ ), 0)
≤ 1,
u−1 (0, 0) + u1 (0, 0) − u−1 (x0 , m0 ) − u1 (x0 , m0 )
we have
[(1 + x(κ∗ ))λ + (1 − x(κ∗ ))λ − 2] ≤
(1 − δ)γM
.
γ+δ
So it is sufficient to show that
γM +
(1 − δ)
γ+1
γM >
γM.
δ
(γ + δ)2
Note that this is equivalent to
(γ + δ)2
> γ + 1,
δ
which always holds when γ ≥ 1. Hence, whenever
u−1 (0, 0) + u1 (0, 0) − u−1 (x(κ∗ ), 0) − u1 (x(κ∗ ), 0)
≤ 1,
u−1 (0, 0) + u1 (0, 0) − u−1 (x0 , m0 ) − u1 (x0 , m0 )
it is always decreasing in δ. Hence, if it becomes less than 1, it can never become greater than 1
again. We can then conclude that if for some δ 0 ,
u−1 (0, 0) + u1 (0, 0) − u−1 (x(κ∗ ), 0) − u1 (x(κ∗ ), 0)
≤ 1,
u−1 (0, 0) + u1 (0, 0) − u−1 (x0 , m0 ) − u1 (x0 , m0 )
then
u−1 (0, 0) + u1 (0, 0) − u−1 (x(κ∗ ), 0) − u1 (x(κ∗ ), 0)
< 1,
u−1 (0, 0) + u1 (0, 0) − u−1 (x0 , m0 ) − u1 (x0 , m0 )
for all δ > δ 0 .
So we conclude that if κ = 0 is not a maximum for δ 0 then it is not for any δ > δ 0 . We can then
conclude that there exists δ̄1 ∈ [0, 1] such that the welfare of the voter is maximized with κ = 0 if
and only if δ ≤ δ̄1 , and that κ = 0 is the unique maximizer when δ < δ̄1 .
I now show that we must have δ̄1 < 1. To do so I must show that, when δ is sufficiently close
to 1, the aggregate welfare of the voters is higher with platform (x∗ , m∗ ) than (x0 , m0 ). First note
38
that, as limδ→1 κ̄ = 0, for δ close to 1, (x∗ , m∗ ) = (x(κ∗ ), 0). Note that, as x(κ∗ ) ≤ x(1) it is
sufficient to show that, for δ sufficiently close to 1,
u−1 (x(1), 0) + u1 (x(1), 0) = −(1 + x(1))λ − (1 − x(1))λ > −m0 − 2 = u−1 (x0 , m0 ) + u−1 (x0 , m0 ).
Next note that as
−(1 − x(1))λ = (1 − δ)γM − δ(1 + x(1))λ ,
it must be that
lim x(1) = 0,
δ→1
and, differentiating with respect to δ,
λ[(1 − x(1))λ−1 + δ(1 + x(1))λ−1 ]
∂x(1)
= −γM − (1 + x(1))λ .
∂δ
Notice that this implies that
−γM − 1
∂x(1)
=
,
δ→1 ∂δ
2λ
lim
which is finite. Now consider
(1 + x(1))λ + (1 − x(1))λ − 2
(γ + δ)[(1 + x(1))λ + (1 − x(1))λ − 2]
.
=
m0
(1 − δ)γM
As both the top and bottom go to 0 as δ → 1, using L’Hopital’s rule, and the fact that limδ→1
∂x(1)
∂δ
is finite, we can conclude that,
(1 + x(1))λ + (1 − x(1))λ − 2
=
δ→1
m0
lim
[(1 + x(1))λ + (1 − x(1))λ − 2] + (γ + δ)λ[(1 + x(1))λ−1 − (1 − x(1))λ−1 ] ∂x(1)
∂δ
= 0.
δ→1
−γM
lim
Hence, we must have δ̄1 < 1.
It has now been established that, for both the cases in which λ ≤ 2 and λ > 2, that there exists
δ̄1 ∈ [0, 1) such that the aggregate welfare of the polarized voters is maximized with κ = 0 if and
only if δ ≤ δ̄1 . I now show that δ̄1 > 0. To see this, recall that the level of rent-seeking is decreasing
1
in κ when δ < δ = (1 + 2γ 1−λ )−λ . As the aggregate welfare of the polarized voters is decreasing in
both m and x, we can conclude that we must have δ̄1 > δ > 0.
Hence we can conclude that there exists δ̄1 ∈ (0, 1) such that the aggregate welfare of the
polarized voters is maximized with κ = 0 if and only if δ ≤ δ̄1 .
I conclude by showing that
lim δ̄1 = 1.
λ→∞
Recall that, when λ > 2, to determine whether or not κ = 0 maximizes the welfare of the polarized
voters it is sufficient to compare the utility from (x0 , m0 ) with
(x(κ∗ ), 0) if κ̄ < 1,
∗
∗
(x , m ) =
(x1 , m1 ) otherwise.
39
Hence it is sufficient to show that
u−1 (x∗ , m∗ ) + u1 (x∗ , m∗ )
u−1 (x0 , m0 ) + u1 (x0 , m0 )
< lim
= −m0 − 1.
λ→∞
λ→∞
2
2
lim
To show this, I start by showing that
lim (x∗ , m∗ ) = lim (x(1), 0).
λ→∞
λ→∞
First note that as
κ
lim xκ = lim
λ→∞
λ→∞
1+γ
1
λ−1
=
κ
,
2
lim (1 − x1 )λ = 0,
λ→∞
and
lim (1 + x1 )λ = ∞.
λ→∞
Recalling mκ solves
−(κ − xκ )λ + γmκ = (1 − δ)γM − δ(κ + xκ )λ − δmκ ,
we can conclude that lim→∞ κ̄ < 1. Hence, we have limλ→∞ (x∗ , m∗ ) = limλ→∞ (x(κ∗ ), 0). Now
recall that, from the proof of Lemma 1, when κ∗ > κ̄, then κ∗ = min{κ∗∗ , 1}, where
1
κ∗∗ =
1
As
1+δ λ−1
2
goes to 1 as λ → ∞ and
1 + δ λ−1 (1 − δ)γM 1
(
)λ .
1
2
δ(1 − δ λ−1 )
(1−δ)γM
1
is greater than 1 for λ large, this implies that
δ(1−δ λ−1 )
lim κ∗ = 1,
λ→∞
and so we can conclude that
lim (x∗ , m∗ ) = lim (x(1), 0).
λ→∞
λ→∞
I have then established that, in order to show that limλ→∞ δ̄1 = 1 we need only show that, for all
δ ∈ (0, 1),
lim
λ→∞
u−1 (x(1), 0) + u1 (x(1), 0)
< −m0 − 1.
2
I now turn to showing that this equation holds. Recall that x(1) is the solution to
−(1 − x(1))λ = (1 − δ)γM − δ(1 + x(1))λ .
Notice that (1 − δ)γM is constant with respect to λ, so, since 1 − x(1) ∈ (0, 1),
lim δ(1 + x(1))λ = (1 − δ)γM + lim (1 − x(1))λ
λ→∞
λ→∞
40
is finite. Therefore we must have that
lim x(1) = 0.
λ→∞
Now define
L∗ ≡ lim (1 + x(1))λ .
λ→∞
I now show that
lim (1 − x(1))λ =
λ→∞
1
.
L∗
To see this, note that, taking logs and applying L’Hopital’s rule,
λ log(1 + x(1))
−(1 − x(1))
= lim
= −1.
λ→∞ λ log(1 − x(1))
λ→∞ 1 + x(1)
lim
Hence,
lim λ log(1 − x(1)) = − lim λ log(1 + x(1)) = − log L∗
λ→∞
λ→∞
So we can conclude that
lim (1 − x(1))λ = lim (1 + x(1))−λ =
λ→∞
λ→∞
1
.
L∗
Now, substituting in for the limits, we have
δL∗ −
1
= (1 − δ)γM = m0 (γ + δ).
L∗
Next, recall that
u−1 (x(1), 0) + u1 (x(1), 0)
< −m0 − 1,
2
if and only if
2m0 + 2 − u−1 (x(1), 0) + u1 (x(1), 0) = 2m0 + 2 − (1 + x(1))λ − (1 − x(1))λ < 0.
Taking limit and substituting in for m0 we get that limλ→∞
u−1 (x(1),0)+u1 (x(1),0)
2
< −m0 − 1 if and
only if
2+
2δ ∗
2 1
1
L −
− L∗ − ∗ < 0.
∗
γ+δ
γ+δL
L
This is equivalent to
(γ − δ)L∗ + (γ + δ + 2)
1
> 2(γ + δ).
L∗
Removing 1/L∗ from this equation then gives
(γ − δ)L∗ + (γ + δ + 2)δL∗ − (γ + δ + 2)(1 − δ)γM > 2(γ + δ),
and dividing by γ + δ gives
(1 + δ)L∗ − (γ + δ + 2)m0 > 2.
41
In order to show that this holds, I need verify only that it holds for M = 0, and that (1 + δ)L∗ −
(γ + δ + 2)m0 is increasing in M. Recall that L∗ is the solution to
δL∗ −
1
= (1 − δ)γM.
L∗
1
Hence, when M = 0 we have L∗ = δ − 2 . Further we have that
∂L∗
(1 − δ)γ
>
.
∂M
δ
Hence we can conclude that
∂(1 + δ)L∗ − (γ + δ + 2)m0
1+δ γ+δ+2
> (1 − δ)γ(
−
) > 0,
∂M
δ
γ+δ
as γ > δ, and so it is sufficient to show that (1 + δ)L∗ − (γ + δ + 2)m0 > 2 when M = 0. As
1
L∗ = δ − 2 and m0 = 0 when M = 0, and, since for all δ ∈ (0, 1),
1+δ
1
> 2, we can then conclude
δ2
that
(1 + δ)L∗ − (γ + δ + 2)m0 > 2,
in fact holds. Hence, we can conclude that
lim
λ→∞
u−1 (x(1), 0) + u1 (x(1), 0)
u−1 (x0 , m0 ) + u1 (x0 , m0 )
< −m0 − 1 = lim
,
λ→∞
2
2
for all δ. Therefore we can conclude that
lim δ̄1 = 1.
λ→∞
I have then shown that, for all λ > 1, there exists a δ̄1 ∈ (0, 1) such that the welfare of the polarized
voters is maximized with κ = 0 if and only if δ ≤ δ̄1 , and that limλ→∞ δ̄1 = 1. Further we have
that κ = 0 is the unique maximizer when δ < δ̄1 , and when λ > 2, the unique maximizer is
∗
κ if κ̄ < 1,
κ=
1 otherwise.
Proof of Lemma 4
Recalling that, as the centrist voters vote based on the utility they receive the elected candidate
must implement a platform which satisfies
∂ḡκ (x,m)
− ∂u ∂x
0 (x,m)
∂x
∂ḡ (x,m)
κ
(κ − x)λ−1
γmβ
∂m
=
=
−
=
,
∂u0 (x,m)
xλ−1
mβ
∂m
in the best equilibrium with type κ and −κ, provided the level of rent-seeking is non-0. Therefore
we have
κ
x=
1
1 + γ λ−1
42
,
as in the case with linear rents. Hence the level of rent-seeking in the best equilibrium can be
determined from the equation
ḡκ (xκ , m) = (1 − δ)γ
M 1+β
+ gκ (xκ , m),
1+β
which reveals that the level of rent seeking will be
mκ =
1
1
κ
(1 − δ)M 1+β
λ
+
[1 − δ(1 + 2γ 1−λ )λ ](
1 ) .
(γ + δ)(1 + β) γ + δ
1 + γ 1−λ
Proof of Lemma 5
As δ > min{δ̄0 , δ̄1 } > δ, for each M there exits a κ̄(M ) such that the amount of rent-seeking is
driven to 0 if the degree of polarization is at least κ̄(M ). I now show that there exists a β1 > 0
such that, for all β ∈ (0, β1 ), there exists M 0 < κ̄−1 (1) such that, for M > M 0 it will be optimal
to move away from the center, for an appropriate distribution of voter preferences. Recall that the
aggregate welfare of the population when the candidates are type κ ∈ [0, κ̄] and −κ is
w(κ) = σu−1 (xκ , mκ ) + (1 − 2σ)u0 (xκ , mκ ) + σu1 (xκ , mκ ) =
−
m1+β
κ
− σ[(1 + xκ )λ + (1 − xκ )λ ] − (1 − 2σ)xλκ .
1+β
There are two cases to consider: λ ≤ 2 and λ > 2. Consider first the case in which λ ≤ 2. Note
that average welfare of the type −1 and 1 voters is higher with candidate of type κ than type 0 if
and only if
m1+β
− m1+β
(1 + xκ )λ + (1 − xκ )λ − 2
κ
0
>
.
1+β
2
Now, when M = κ̄−1 (1), we can compare κ = 0 with κ = κ̄ = 1 which becomes the equation
(1 + x1 )λ + (1 − x1 )λ − 2
m1+β
0
>
.
1+β
2
Now as δ > δ̄1 ,
m1+β
(1 − δ)γM
(1 + x1 )λ + (1 − x1 )λ − 2
0
=
>
,
β→0 1 + β
γ+δ
2
lim
so there exists a β1 > 0 such that, for all β ∈ (0, β1 ),
m1+β
(1 + x1 )λ + (1 − x1 )λ − 2
0
>
,
1+β
2
holds, and so κ = 1 provides higher utility to the polarized voters than than κ = 0 when M =
m1+β
−m1+β
κ
0
is increasing in M we can conclude that,
1+β
−1
κ̄ (1) such that the average welfare of the of the type
κ̄−1 (1). Finally, as
0
exists an M <
0
maximized by setting κ = 0 if and only if M > M .
43
for all β ∈ (0, β1 ) there
−1 and 1 voters is not
Therefore, for all β ∈ (0, β1 ), there exists M 0 < κ̄−1 (1) such that for all M > M 0 there exists
σ̄(λ, δ, M, β) ∈ [0, 12 ) such that, for all σ ∈ (σ̄(λ, δ, M, β), 12 ), w(κ) is not maximized with κ = 0.
Now consider the case in which λ > 2. Note that welfare of the type 0 voters is higher with
candidate of type κ if and only if
m1+β
− m1+β
κ
0
> xλκ .
1+β
Now, when M = κ̄−1 (1), we can compare κ = 0 with κ = κ̄ = 1 which becomes the equation
m1+β
0
> xλ1 .
1+β
Now as δ > δ̄0 ,
m1+β
(1 − δ)γM
0
=
> xλ1 ,
β→0 1 + β
γ+δ
lim
so there exists a β1 > 0 such that, for all β ∈ (0, β1 ),
m1+β
0
> xλ1 ,
1+β
holds, and so κ = 1 provides higher utility to the centrist voters than than κ = 0 when M = κ̄−1 (1).
Finally, as
0
M <
m1+β
−m1+β
κ
0
1+β
κ̄−1 (1)
is increasing in M we can conclude that, for all β ∈ (0, β1 ) there exists an
such that the welfare of the of the type 0 voters is not maximized by setting κ = 0
if and only if M > M 0 . Therefore, for all β ∈ (0, β1 ), there exists M 0 such that for all M > M 0
there exists σ̄(λ, δ, M, β) ∈ (0, 21 ] such that, for all σ ∈ (0, σ̄(λ, δ, M, β)), w(κ) is not maximized
with κ = 0.
We have now established that there exists M 0 < κ̄−1 (1) such that, for any M > M 0 , such that
κ = 0 does not maximize w(κ), for an appropriate distribution of voter preferences.
Now I show that, there exists a M̄ > κ̄−1 (1) > M 0 such that, for all M ∈ (M 0 , M̄ ), w(κ)
is maximized on (0, κ̄] by κ∗ < min{κ̄, 1}. Recall that, as δ > δ, mκ is decreasing in δ and
mκ̄ = 0. Consequently, since mβκ → 0 as mκ → 0, there exist M̄ > κ̄−1 (1) such that, there exists
κ0 < min{1, κ̄} such that, for all κ ∈ (κ0 , κ̄], w0 (κ) < 0. Hence the optimal level of divergence on the
interval [0, κ̄] must be in [0, κ̄) for all M < M̄ . We can then conclude that, for all M ∈ (M 0 , M̄ ),w(κ)
0
is maximized on (0, κ̄] at some point κ∗ ≤ κ < min{κ̄, 1}.
Now we have established the maximum on [0, κ̄] is κ∗ ∈ (0, κ∗ ). We need to now establish that
the welfare is never higher with κ ∈ [κ̄, 1]. Note that, since w(κ∗ ) > w(κ̄), if κ̄ is sufficiently close
00
to 1, then w(κ∗ ) > w(κ) for all κ ∈ [κ̄, 1]. As such, there exists M < κ̄−1 (1) < M̄ , such that no
00
κ ∈ (κ̄, 1] can give higher utility than κ∗ when M > M . Hence, defining M = max{M 0 , M 00 } <
κ̄−1 (1) < M̄ we have that, for all M ∈ (M , M̄ ), the optimal level of divergence, κ∗ ∈ (0, min{κ̄, 1}).
Finally, defining β̄ = min{β1 , 1} we are done. 44
Proof of Lemma 6
Suppose κA , κB > 0. From the assumption of retrospective voting we must have that the candidate
chooses platform ((xA , xB ), m) to maximize
gκ (xA , xB , m)) = −(κA − xA )λ − (κB − xB )λ + γm,
subject to the constraints m ≥ 0 and
u0 (xA , xB , m) = −xλA − xλB − m ≥ ū.
Note that this implies
(κB − xB )λ−1
(κA − xA )λ−1
=
≥ γ,
xλ−1
xλ−1
A
B
and so
xA
xB
=
≡ ρ−1 .
κA
κB
45