The Provision of Public Goods Under Alternative

Electoral Incentives
Alessandro Lizzeri and Nicola Persico

March 10, 2000

American Economic Review, forthcoming
ABSTRACT
Politicians who care about the spoils of oce may underprovide a public
good because its benets cannot be targeted to voters as easily as pork barrel
spending. We compare a winner-take-all system, where all the spoils go to the
winner, to a proportional system, where the spoils of oce are split among
candidates proportionally to their share of the vote. In a winner-take-all system
the public good is provided less often than in a proportional system when the
public good is particularly desirable. We then consider the electoral college
system and show that it is particularly subject to this ineciency.
(JEL D82, L15)
1
Democratic societies delegate to elected representatives the power to tax and to provide
public goods. A common complaint made by citizens and by the press is that a large
fraction of public spending is not devoted to genuinely useful public projects, but rather to
redistribution and pork-barrel projects. At least part of this redistribution does not seem
to be from the rich to the poor, rather it is thought to be a wasteful result of jockeying by
candidates for political advantage. Thus, it would be desirable to reduce the money tied up
in redistribution, and increase the fraction devoted to public goods.
1
The economic literature has long recognized that the political process may lead to an
inecient provision of public goods. Conventional wisdom - due to Howard R. Bowen (1943),
and presented in textbooks such as Joseph E. Stiglitz (1988) - is that the ineciency comes
from the dierence between the Samuelsonian optimum and the policy preferred by the
median voter. In this model the ineciency can manifest itself either as underprovision
or as overprovision of the public good, depending on the relative position of the median
and average voters in the population. According to this view, the source of ineciency is
voter heterogeneity; democratic provision of public goods would be ecient in a society with
identical voters. This model assumes exogenous restrictions on the policy space - such as
linear taxes - and, as a consequence, ignores a fundamental distortion in majoritarian decision
making: politicians have an incentive to tax minorities to target benets to a majority.
Candidates face a trade-o. When public money are funneled to pork-barrel projects,
fewer are available to nance public goods. The benets from the public good may be higher
on average, but they cannot be targeted to groups of voters as easily as the benets from pork
barrel projects or pure transfers. We call this a trade-o between eciency and targetability.
2
2
The goal of our analysis is to model this trade-o and to explore how alternative political
systems aect the provision of public goods and the distribution of resources across voters.
3
To highlight the basic idea, we focus on a model that is extremely simple. Two candidates
compete for election, by making a (binding) electoral promise to each voter. Candidates only
care about the outcome of the election. Voters are homogeneous: each voter will vote for
the candidate promising him the most utility. Candidates are faced with only two possible
choices: providing the public good or redistributing money. The public good gives every voter
the same utility, and is the ecient choice: social welfare is higher when the public good
is provided. Money (local public goods, pork-barrel projects), however, can be targeted
to subsets of voters. Despite this simplicity, the solution is quite rich and a number of
interesting issues emerge.
We show that targetability commands a premium, and therefore there is ineciently low
provision of public goods even in a world with identical voters. We then study how dierent
electoral incentives aect the provision of public goods and the distribution of the resources.
Our goal is to provide some insight in the dierence between political systems.
Political systems dier in (at least) two important dimensions: in the way that votes
are translated into seats in an assembly, and in the way assembly seats translate into in-
uence on policymaking. In some countries, important decisions are made by bargaining
in an assembly; in other countries, they are made by a premier elected by plurality voting.
Proportional systems are usually associated with many parties having an inuence on policy-
making, through the process of post-election bargaining; examples are countries such as Italy
and Belgium. Majoritarian systems are thought to favor the party with the highest share
3
of the vote, in the sense that more power of policy setting is conferred to that party; this
seems to be the case in countries such as Britain. We wish to collapse this complexity into
one characteristic, namely the rewards that accrue to vote shares. Vote shares are rewarded
dierently in dierent political systems, thus leading to dierent incentives for parties in the
election stage.
We focus on two extreme ways of rewarding vote shares. The rst system is a winner-
take-all system, where all the spoils of oce go to the winner. The second is a proportional
system: the spoils of oce are split among the candidates proportionally to their share of
the vote. The spoils of oce represent the benets for a party of being able to implement
its policy, and the rents from power. Since we only deal with two candidates, we do not
capture the richness of the contrast between plurality rule and proportional representation
that comes from the process of coalition formation.
4
However, in our view an important
feature of proportional systems is the opportunity to translate vote shares into commensurate
benets (political and monetary) for the politician. We show that this feature alone, even
in the absence of coalitional issues, generates important dierences between the two systems
at the electoral stage.
It seems obvious that a proportional system should generate dierent policy outcomes
than a winner-take-all system. However, in a standard Downsian model with linear taxes
the equilibrium would be the same under either system: both candidates would choose the
ideal point of the median voter. Assar Lindbeck and Jurgen Weibull (1987) also show that
the two systems lead to the same (ecient) outcome in a world where candidates engage in
pure redistribution. Similarly, absent the public good, our model yields the same prediction
4
under both systems.
The reason the two systems dier in our model reects a fundamental dierence in the
incentives they provide to candidates seeking public oce. Let us consider the case where
one candidate promises to provide the public good. Now, suppose that the opponent tries
to defeat him by redistributing money to some majority of voters. The margin of victory
of the redistributive plan against the public good declines as the public good becomes more
valuable. This is because of the reduction in the number of voters to whom a candidate
can promise benets that are worth more than the benets provided by the public good.
When candidates maximize the share of the vote, the margin of victory is important for it
determines candidates payos. Hence, in the proportional system the option of redistribut-
ing becomes less attractive as the public good becomes more valuable. In contrast, in a
winner-take-all system all that matters about a particular redistributive plan is whether it
beats the public good: the margin of victory is irrelevant. Thus, in a winner-take-all system
the attractiveness of a redistributive plan does not depend on how valuable the public good
is, as long as redistribution defeats the public good. We nd that the proportional system is
more ecient when the public good is very valuable, and the winner-take-all system is more
ecient when the public good is not very valuable. We also nd that the inequality in the
redistribution of resources increases with the value of the public good under both systems.
Another dimension of the political process that we consider involves two dierent ways
of aggregating district level votes to obtain a winner in nation-wide elections. The rst is to
elect the candidate who obtains a majority of votes in a nation-wide electoral district. The
second way is to elect the candidate who gets the majority of the votes in the majority of
5
districts. The second system will be called the electoral college since it is used in presidential
elections in the United States. We show that the electoral college generates a more unequal
distribution of resources and less ecient provision of global public goods. The reason for
the dierence is the following. If the electorate is made up of a single district, candidates
only need to worry about fending o attacks by the other candidates on the majority of
voters. In the electoral college system candidates need to worry about the other candidate
going after 1/4 of the voters: 1/2 of the voters in 1/2 of the districts. This creates the need
to concentrate promises on smaller groups of voters and means that the relative inexibility
in the targeting of the benets from public goods must now lead to even worse ineciency.
I. Related literature
A comprehensive survey of the literature is in Torsten Persson and Guido Tabellini (1998).
Most of the literature on the ineciency of democracy focuses on decision making in leg-
islatures.
5
For instance, David Baron (1991) models the legislative process via a sequential
bargaining model. These models analyze specic, very structured legislative processes; they
do not address candidates competing in large elections. In V. V. Chari et al. (1997), policy
is determined through bargaining between a president and local representatives. This gives
rise to a common pool problem whereby an excessive number of local public projects are
nanced from general taxation (see also Barry Weingast et al.(1981)). Similarly, Torsten
Persson et al. (1997) compare congressional and parliamentary systems. In their model, the
politician choosing the level of public good provision has the option of foregoing the public
good and appropriate the money for his own district; once again, a common pool problem
6
arises. Our setup diers from these models since we focus on national candidates who do
not represent any specic district: when proposing to increase transfers to a district, our
candidates take into account the loss of another district. Thus, nationwide candidates miti-
gate the common pool problem by internalizing, in part, the costs of providing pork-barrel
projects.
The idea of the tradeo between policies with diuse benets and pork-barrel projects
has long been present in the mainstream political science literature. In one variant (see
David Mayhew (1974), pp. 52-61), legislators go for projects on which they can claim credit
with the electorate. Public goods are bad for this because voters do not know whether the
candidate was really instrumental in securing the provision of the public good, whereas it
is easy for candidates to claim credit for a local pork-barrel project. This idea relies on
incomplete information on the part of voters; one virtue of our model is that incomplete
information is not central to our arguments.
There is work comparing the eects of alternative electoral systems on the distribution
of resources, in the absence of a public good. Roger Myerson (1993) studies redistributive
politics under alternative electoral systems when there is competition among more than two
candidates. In Steven J. Brams and Michael D. Davis (1974) and James Snyder (1989),
candidates redistribute resources across electoral districts. When the environment is asym-
metric, the two systems - winner-take-all and proportional representation - yield dierent
outcomes. All these models are purely about redistribution, hence they do not yield welfare
implications.
David Austen-Smith and Jerey Banks (1988) analyze proportional representation in a
7
three-candidate model of spatial competition that integrates the electoral and legislative
processes. They obtain a sharp contrast between plurality rule with two candidates and pro-
portional representation with three candidates. Under plurality rule, both candidates adopt
the policy preferred by the median voter. Under proportional representation the equilibrium
electoral platforms are symmetrically distributed around the median. The government is
formed between the party that adopts the median position (which receives the fewest votes)
and one of the other parties. The policy outcome that emerges from the legislative process
is a compromise between the platforms of these two parties and is dierent from the median
voters preferred policy.
Additional work on the comparison between majoritarian and proportional systems has
been done by Torsten Persson and Guido Tabellini (1999). They construct a model of
redistributive politics a la Lindbeck and Weibull in which a majoritarian system generates
less public good provision than a proportional system. Persson and Tabellini also confront
this prediction with cross-country data from around 1990; they nd weak support for the
prediction that majoritarian elections are associated with less public goods.
II. The model
Our setup does not restrict the set of feasible transfers in any way except to require that
they satisfy a budget constraint. Myerson (1993) introduced this model (without the public
good) to study redistributive politics.
8
A. Economy and Agents
There are two candidates, 1 and 2. There is a continuum of consumers/voters; the set of
voters is denoted by V which can be taken as the interval [0, 1].
6
There are two goods, money
and a public good. The public good can only be produced by using all the money in the
economy.
7
Each voter has an endowment of one unit of money. The public good yields a utility
of G to each voter. Voters have no a priori preference for either candidate, and have linear
utility over goods.
8
Candidates make binding promises to each voter. A candidate can oer to provide the
public good (to all voters); alternatively, he can oer dierent taxes and transfers to dierent
voters. Because a candidates promise is only relevant if he gets elected, each voters optimal
behavior is to vote for the candidate who promises him the greatest utility.
B. Electoral Incentives
In our view, the opportunity to translate vote shares into commensurate benets for the
politician is an important feature of proportional systems relative to winner-take-all systems.
We want to understand the consequences of this phenomenon on electoral promises. In our
model, candidates are motivated to run by the prospect of spoils of oce. We discuss two
alternative systems, characterized by how these spoils are divided between candidates.
A proportional system, where the spoils are divided proportionally to the candidates
share of the vote. Thus, candidates maximize the share of the vote.
A winner-take-all system, where all the spoils go to the winner. Thus, a candidates
9
payo in the winner-take-all is 0 if his share of the vote is less than 1/2, 1 if it is greater
than 1/2, and 1/2 if the share of the vote is exactly 1/2.
C. Game
A pure strategy for a candidate species whether he chooses to oer the public good or pure
transfers (he cannot oer both, but see Alessandro Lizzeri and Nicola Persico (1998) for a
model where this is possible). In the event he chooses transfers, a pure strategy species a
promise of a transfer to each voter. Formally, a pure strategy is a function : V [1, +)
that satises the following condition: either (v) = G 1 for all v V, or
_
V
(v)dv = 0.
This is a balanced budget condition. (v) + 1 represents the utility enjoyed by voter v.
There are two stages of the game:
Stage 1 Candidates choose oers to voters simultaneously and independently.
Stage 2 Each voter v gets oers
1
(v),
2
(v) from candidates 1 and 2. After observing
the oers, voter v votes for candidate i if
i
(v) >
h
(v). If the voter gets the same oer
from both candidates, he randomizes with equal probability.
A mixed strategy in this game could in principle be a very complicated object since the
space of pure strategies is so large. However, it turns out that we only need to look at
simple distributions. In this paper we discuss the case where the oers of transfers made by
candidate i to voters are realizations of the same random variable with c.d.f. F
i
: [0, 1].
Of course, the fact that oers are realizations of the same random variable does not mean
that each voter gets the same oer. Note also that at stage 2 each voter observes his realized
promises, not random variables.
Because there are innitely many voters, F
i
will be the empirical distributions of oers
10
in the electorate; F
i
(x) is the fraction of voters who receive promises below x from candidate
i. By manipulating F
i
, candidate i is able to target transfers to sections of the populations.
In the following sections we therefore reduce the representation of a mixed strategy to the
probability of oering the public good (denote this by
i
) and the distribution of transfers
F
i
() under candidate is strategy if he chooses to oer transfers. The budget constraint
then is
_

1
xdF
i
(x) = 0. The lower support of integration is explained by the fact that voters
cannot be taxed more than their endowment.
Let S(F
i
, F
h
) denote the share of the vote of candidate h if he promises to transfer
according to F
h
and candidate i promises to transfer according to distribution F
i
. The share
of the vote of candidate h is equal to the probability that any random voter receives an oer
from h which is higher than the oer he receives from i. Thus,
S(F
i
, F
h
) =
_

1
F
i
(x)dF
h
(x)
The game among candidates is constant sum and symmetric under either assumption on
candidates objectives. Hence, in equilibrium both candidates must get 50 percent of the
votes in expectation.
D. Discussion of the Model
A few features of our model are worth discussing. First, the nature of the redistribution. In
this model private benets can be targeted to arbitrarily small subsets of the population.
Thus, the policy space is not exogenously restricted by grouping voters and then requiring
dierent voters in the same group to receive the same electoral promise. Of course, for
the logic of our results it is not necessary that candidates can literally target transfers to
11
each individual voter. We only need that candidates can discriminate among a relatively
large number of groups of voters. This is certainly possible in reality since transfers can be
dierent for voters in dierent geographical location, age group, profession, . . . .
Second, in our model each voter votes for the candidate who oers him the most. This is
in contrast with models a la Lindbeck and Weibull (1987) or models of probabilistic voting,
where increasing the electoral promise to any group delivers a continuous change in share of
the vote from that group. The potential discontinuity of the share of the vote in our model,
as one candidates promise to a voter equals and then exceeds his competitors, explains the
role for mixed strategies. Mixed strategies provide the smoothness in candidates payos
that allows us to obtain an equilibrium. This smoothness is exogenously assumed in models
of probabilistic voting.
The assumption of identical voters is useful for highlighting the dierences from the liter-
ature since we show that heterogeneity is not a driving force of our results. This assumption
has the additional benet of considerably simplifying the analysis. In Lizzeri and Persico
(1998) we allow for a limited measure of voter heterogeneity and show that the substance of
our results is unchanged.
Finally, the budget constraint. We require that transfers satisfy a budget constraint
which is determined by the amount of public resources that a candidate would be able to
allocate in case of winning the election. This is in keeping with the assumption of fully
rational and perfectly informed voters. In connection with the budget constraint note that,
with a nite population of N voters, electoral promises could not be independent across
voters, because the promises would have to sum up to the budget constraint. However, as
12
the number N goes to innity, independent promises can be constructed so that the average
deviation from the budget constraint converges to 0 with probability one. This is why we
assume an innite number of voters in our model.
9
III. Winner-Take-All vs. Proportional System
If G > 2, then under either assumption on candidates objectives the unique equilibrium
involves both candidates promising to provide the public good. To see this observe that if
candidate i oers the public good, he ties in the event candidate h also oers the public good,
and wins if candidate h oers transfers since, because of the budget constraint, candidate
h cannot oer more than G 1 > 1 to more than 50 percent of the voters. Conversely, if
candidate i chooses to oer money he gets less than 50 percent of the votes in the event that
the other candidate oers the public good.
10
Section A. describes the benchmark case where G < 1, hence provision of the public good
is dominated by a strategy of oering the same (zero) transfer to every voter.
Section B. and C. analyze the interesting parameter range, when 1 < G < 2. To develop
an intuition for the results, it is useful to consider a simple example. Suppose G = 3/2 and
assume that candidate 1 oers to provide the public good. Then candidate 2 can obtain
almost 2/3 of the votes by oering a transfer of a little more than 1/2 (hence a utility of
more than G) to a little fewer than 2/3 of the voters. These transfers are nanced by taxing
the entire endowment of the remaining voters. Suppose now that the value of the public
good is 7/4. Now candidate 2 can oer more than the value of the public good to fewer than
4/7 of the voters. The number of voters to whom candidate 2 can oer more than the value
13
of the public good declines with G although it is still sucient to win as long as G < 2. This
illustrates two fundamental aspects of the problem: First, provision of the public good is
going to be inecient since its benets are not as targetable as transfers are. Second, in the
proportional system the incentive to redistribute declines as the value of G increases, while
in the winner-take-all system the incentive to redistribute remains constant.
For values of G between 1 and 2 there is no equilibrium in pure strategies. To see this,
suppose candidate 1s strategy was
1
. If
1
(v) = G1 for all v, i.e. candidate 1 promises
each voter the public good, then candidate 2 can choose to promise more than G 1 to
more than 50 percent of the voters and obtain more than 50 percent of the votes. This
is impossible in equilibrium. Suppose then that candidate 1 chooses to oer money. Now
candidate 2 could take a set of voters V
1
with small positive measure such that
1
(v) > 1
for v V
1
, tax all their endowment and use the money to nance oers of
1
(v) + to all
other voters. The set V
1
and the can be chosen so that candidate 2 wins with a share of
the vote arbitrarily close to one hundred percent. Thus, at equilibrium both candidates will
be employing mixed strategies. We want to stress that there is a natural interpretation for
these mixed strategies: choosing F should be thought of as choosing the Lorenz curve, i.e.
the empirical distribution of transfers, in the population.
A. The Game of Pure Redistribution
Before discussing how the provision of public goods diers in the two systems it is useful
to show that the equilibrium in the game of pure redistribution is the same under the two
systems.
14
Proposition 1 (Myerson (1993)). Suppose G < 1. Then the unique equilibrium under the
winner-take-all and proportional systems involves both candidates drawing oers to all voters
from a uniform distribution on [1, 1]. Thus, for i = 1, 2
F
i
(x) =
_

_
0 for x 1
x+1
2
for 1 x 1
1 for 1 x
(1)
Proof. Let us show that this is an equilibrium. First, notice that the share of the vote from
oering the public good is F
i
(G1) < F
i
(0) = 1/2, so oering the public good is a bad
idea since it yields less than 50 percent of the votes. Suppose now F
1
satises equation 1,
and candidate 2 oers money according to a distribution F. To satisfy the budget constraint
it must be
_

1
xdF(x) = 0. We then have:
S(F
1
, F) =
_

1
F
1
(x)dF(x)

_

1
x + 1
2
dF(x) =
1
2
= S(F
1
, F
2
)
The second-to-last equality holds because of the budget constraint. Therefore deviation
cannot increase the share of the vote for candidate 2. For a proof of uniqueness see Myerson
(1993) and Alessandro Lizzeri (1997).
Notice that in this game candidates will choose to tax and redistribute even when no
public good is provided at equilibrium. This is in contrast with the standard, median voter,
models of democratic provision of public goods. This contrast is due to the fact that in our
model the redistributive tools of candidates are not restricted to linear taxes.
When voters are risk-neutral, the outcome of the political process is ecient: since G < 1
citizens value money more than they value the public good, and they do not mind the risk
15
involved in the redistribution process.
11
B. A Winner-Take-All System
Theorem 2 Suppose 1 < G < 2. Under the winner-take-all system in the unique equilibrium
both candidates oer the public good with probability (G) = 1/2 for G (1, 2). When
candidates oer money they choose transfers according to the following distribution:
F

(x) =
_

_
0 for x 1
1
2
_
x+1
2G
_
for 1 x 1 G
1
2
for 1 G x G1
1
2
_
1 +
x+1G
2G
_
for G1 x 1
1 for 1 x
(2)
When G > 2, the unique equilibrium is to oer the public good for sure: (G) = 1 for
G (2, ).
Proof. Case 1 < G < 2 : We want to show that this is an equilibrium. First, the
strategy F

is feasible since the average sum spent is

_

1
xdF

(x) = 0. Suppose candidate 2

follows the equilibrium strategy: then the share of the vote of a candidate 1 who redistributes
according to F, and meets candidate 2 who promises transfers is:
S(F

, F) =
_

1
F

(x)dF(x). (3)
When F is a best-response, it is never the case that the oers of candidate 1 fall outside
candidate 2s support: formally, F (1) = 0 and F (1) = 1. The share of the vote accruing
16
to candidate 1 (expression (3)) is
S(F

, F) =
1
2
_
_
1G
1
(
x + 1
2 G
)dF(x) +F(G1) F(1 G)
+
_
1
G1
(1 +
x + 1 G
2 G
)dF(x)
_
=
1
2
_
M
1
+M
2
2 G
+
1 G
2 G
(F(G1) F(1 G))
+ (1
G
2 G
)(1 F(G1)) +
1
2 G
_
where
M
1
:=
_
1G
1
xdF(x)
and
M
2
:=
_
1
G1
xdF(x)
are the total transfers generated by promises in the interval [1, 1 G] and [G1, 1], re-
spectively. It is obviously never a best response to promise anything in (1 G, G1) nor
is it optimal to promise G 1 to any positive measure of voters: indeed, any strategy that
promises G1 to a mass m of voters is dominated by one that is identical, except that of
the voters previously being oered G 1 are now oered 1 G, and the remaining m
are oered G1 +. Thus, we can safely restrict to checking those deviations F for which
F(G1) F(1 G) = 0. The above expression then reads
S(F

, F) =
1
2
_
M
1
+M
2
2 G
+ 2
_
G1
2 G
_
F(G1)
+
_
1
G
2 G
_
+
1
2 G
_
(4)
The problem of candidate 1 is to choose an F under the constraint that M
1
+ M
2
0
(budget constraint). First, clearly the candidate will choose to make M
1
+M
2
= 0. Second,
whenever F(G 1) <
1
2
we have S(F

, F) <
1
2
, so candidate 1 is sure to lose against
17
redistribution (but to win against the public good); whenever F(G1) >
1
2
, the candidate is
sure to win against redistribution but is sure to lose against the public good. Finally, when
F(G 1) =
1
2
the candidate ties against the public good and against redistribution. Since
= 1/2, the candidate is indierent between any F such that M
1
+M
2
= 0. Similarly, given
that candidate 2 plays the equilibrium strategy, candidate 1 is indierent between oering
the public good and oering money because the probability of victory is 1/2 in either case.
Case G > 2 : straightforward, since redistributing resources cannot give utility greater
than G to more than 50 percent of the voters.
Uniqueness: see Lizzeri and Persico (1998).
C. A Proportional System
We now discuss the system where candidates maximize the share of the vote.
Theorem 3 Suppose 1 < G < 2. Under the proportional system in the unique equilibrium
involves both candidates choosing to oer the public good with probability (G) = G 1
for G (1, 2). When candidates oer money they choose transfers according to the same
distribution as in Theorem 2.
When G > 2, the unique equilibrium is to oer the public good for sure: (G) = 1 for
G (2, ).
Proof. Case 1 < G < 2 :
Suppose that candidate 2 plays according to the equilibrium. As in the proof of Theorem
2 if candidate 1 oers money according to a distribution F and candidate 2 oer money
according to distribution F

, we can write candidate 1s share of the vote according to

18
equation (4). For the same reason as in the proof of Theorem 2, candidate 1 will chose
M
1
+M
2
= 0. Thus, we can rewrite equation (4) as follows:
S(F

, F) =
1
2(2 G)
[2(G1)F(G1) + 3 2G] (5)
If candidate 2 chooses the public good instead, candidate 1s share of the vote if he oers
money according to F is 1F(G1). Thus, given that candidate 2 chooses the public good
with probability G 1 and money with probability 2 G, candidate 1s expected share of
the vote when he oers money is:
(G1)(1 F(G1))
+(2 G)
1
2(2 G)
[2(G1)F(G1) + 3 2G] =
1
2
Thus, given that candidate 2 plays according to the equilibrium strategy candidate 1s share
of the vote is 1/2 for any distribution F. Furthermore, it is clear that given that candidate 2
plays according to F

, candidate 1s share of the vote from choosing the public good is also
1/2. Thus, candidate 1 is happy to play according to the equilibrium strategy.
Case G > 2: identical to Theorem 2.
Uniqueness: see Lizzeri and Persico (1998).
In the above stylized model of proportional representation we implicitly assume that the
implemented policy is that of the winning candidate. In reality, minority parties may have
an inuence on policy and the nal outcome might depend in a complicated way on the
outcome of some post-election bargaining game. We now show that the conclusions of this
section are unchanged when we allow for more general ways of implementing policies after
an election.
19
Consider a function (s) [0, 1]. This function represents the probability that the
implemented policy will be the platform of candidate 1 when candidate 1 has a share of
the vote of s. We assume that is non decreasing in s, and that (1/2) = 1/2. Thus,
the function may be thought of as a reduced form of a bargaining game between the
two candidates.
12
Given any such , optimal behavior for voters still involves voting for the
candidate who promises them the higher utility: this increases the chance that the policy of
the favorite candidate is implemented.
13
This implies that candidates equilibrium strategies
are unchanged relative to our previous analysis, since candidates only care about shares of the
vote, and not about policy. Thus, at equilibrium the public good is oered with probability
(G) and, no matter what a candidate oers, he obtains a share of the vote of exactly 1/2.
Since (1/2) = 1/2, in equilibrium the probability that the public good is provided does not
change from the outcome of our simple model.
D. Discussion and Comparison
<INSERT FIGURE 1 ABOUT HERE>
The distribution of money across voters is the same in both systems and is illustrated
in the Figure . As the value of the public good Gincreases, this distribution becomes more
concentrated on the extremes. The probability of provision of the public good in the winner-
take-all system is independent of G for 1 < G < 2. When candidates maximize the share
of the vote the equilibrium probability of providing the public good (G) is not constant in
20
G : goes from 0 to 1 as Gincreases from 1 to 2. We discuss these features below.
The presence of the public good makes redistribution more extreme than in Theorem 1.
This is because, in order for transfers to be a protable strategy against the public good,
a majority of the voters must receive transfers that exceed the value of the public good;
the resource constraint then dictates that the remaining voters receive correspondingly lower
transfers. It is also easy to understand why no candidate oers transfers of value close to,
but less than, the value of the public good. Indeed, a candidate oering transfers of G
can win these votes only if the opponent redistributes. But then it is protable to increase
these transfers by 2; this costs marginally more, and wins over the votes also in the event
that the opponent promises the public good.
Another interesting feature of the equilibrium is that the redistribution of money is the
same in both electoral systems. To understand this fact, observe that if candidate 1 oers
transfers and candidate 2 oers the public good, the only relevant aspect of candidate 1s
redistributional strategy is the fraction of voters who receive transfers that are worth less
than the public good, i.e. F (G1). Therefore, the consideration that pins down candidate
1s redistributional strategy F (x) is that it must be a best response against a candidate 2
who redistributes, i.e. it must not be possible to change F and increase the share of the
vote of F against F

without changing F(G 1). If this were possible, there would be an

incentive to deviate from F

in both systems. Thus, the condition that pins down F is the

same under both systems.
We now discuss the intuition for why public good provision diers between winner-take-all
and proportional systems. When a candidate evaluates a deviation in the way he distributes
21
transfers, oering more than G1 to more than 50 percent of the voters is protable when
the opponent oers to provide the public good, but not when he redistributes. When an
opponent redistributes the best deviation is to downgrade the oers of G1 down to oers
of 1 G, since no voter is receiving oers in the interval [1 G, G 1]. The money saved
through this reallocation can be used to buy share of the vote. The vote share gained
by this deviation increases with G, since the higher G the greater the savings. Thus, in
the proportional system the appeal of this deviation increases with G. To discourage this
deviation, at equilibrium the probability that the opponent oers the public good must
be high enough, and increasing with G. By contrast, in the winner-take-all system the value
of this deviation is independent of G because the margin of victory is irrelevant: as soon as
a deviation guarantees more than 50 percent of the votes, it yields a payo of 1. In fact,
the value of the opposite deviation, oering more than G 1 to more than 50 percent of
the voters, is also independent of G for the same reason. In the winner-take-all system, to
discourage both kind of deviations it must be equally likely that the opponent oers the
public good and redistribution, hence = 1/2 independent of G.
This discussion suggests that what matters for our results is that, when confronted with
an opponent who promises the public good, the attractiveness of the option of redistributing
resources is dierent in the two systems.
E. Eciency and Constitutional Design
We are now in a position to compare electoral systems. When G < 1 and G > 2 equilibrium
is the same under the two systems, so we can concentrate on the case where 1 < G < 2. It
is important to distinguish between ex ante and ex post Pareto eciency. This is because
22
candidates strategies involve randomization, and thus there are two notions of allocation,
one before and one after the uncertainty is realized.
14
Before electoral promises are made,
allocations are lotteries over consumption. Ex ante Pareto eciency ranks such allocations,
taking the point of view of a voter who considers the expected utility of the outcome of
the election before receiving an electoral promise. In contrast, after the winning policy
is implemented, an allocation is a consumption vector that species how much each voter
consumes. Ex post Pareto eciency ranks such allocations, from the viewpoint of a voter
who has already received this consumption.
15
In the Proposition below we show that, under both electoral systems, while the equilib-
rium is ex post ecient, it is ex ante inecient. It is also possible to rank electoral systems
in terms of ex ante eciency.
Since each candidate is equally likely to get elected, the probability of provision of the
public good is 1/2 in the winner-take-all system and G 1 in the proportional system:
when G < 3/2 the probability of provision is higher in the winner-take-all system, when
G > 3/2 it is higher in the proportional system. Thus, risk neutral voters prefer (ex ante)
the proportional system if and only if G > 3/2.
If voters are risk averse, they care not only about the probability of public good provision,
but also about the distribution of transfers at equilibrium. However, notice that while the
probability of providing the public good may dier in the two systems, the distribution of
transfers is the same; furthermore, whenever the outcome of the two systems dier, the bet
associated to the public good has higher mean, and lower spread, than the bet associated
with transfers. Hence, risk-averse voters are ex ante unanimous in their preference for the
23
system that yields higher provision of the public good, just as in the case of risk-neutrality.
Proposition 4 (i) The equilibrium outcome is ex post ecient in both electoral systems for
any value of G.
(ii) For 1 < G < 2, the equilibrium outcome is ex ante inecient in both electoral
systems.
(iii) Suppose voters are risk averse or risk neutral. Then when G > (<) 3/2 the propor-
tional (winner-take-all) system yields higher ex ante utility for voters.
Proof. Part i. It is obvious that if G < 1 or G > 2 the equilibrium is ex post ecient.
Suppose therefore that 1 < G < 2. If ex post it turns out that the public good was provided,
this is an ex post ecient allocation because there is no distribution of transfers that gives
more utility to all voters. Suppose then that the public good was not provided. This implies
that the empirical distribution of money across the population is as specied by equation
(2). Clearly, this allocation cannot be dominated ex post by any other distribution of money.
Moreover, providing the public good requires that all the money be taken away from all
voters. Thus, the 50 percent of the voters who in equilibrium get more than G would be
worse o ex post if the public good is provided.
Part ii. Consider the winner-take-all system. To see that the equilibrium outcome is ex
ante inecient, observe that for each voter, from the ex ante perspective, the equilibrium
outcome is a lottery that puts probability (G) on the provision of the public good and
probability 1 (G) on the distribution in equation (2). Ex ante, all voters prefer the
allocation that puts probability one on the provision of the public good. The same reasoning
takes care of the proportional system.
24
Part iii. Denote with H
P
and H
W
the equilibrium distribution of utilities in the pro-
portional and winner-take-all systems, respectively. It is easy to verify that the function
D(y) =
_
y

(H
P
(z) H
W
(z)) dz is greater than zero for all y if and only if G < 3/2. Thus,
H
W
dominates H
P
in the sense of second order stochastic dominance if and only if G < 3/2.
Ex ante eciency appears to be a more natural concept when evaluating the eciency
of political systems. This is because the choice of a political system precedes the electoral
stage. At the stage of constitutional design, it is reasonable to believe that citizens evaluate
political systems knowing the equilibrium distribution of resources, but before receiving a
specic electoral promise.
IV. The Inefficiency of the Electoral College
In some electoral systems, candidates to national level oces, like president, are selected
on the basis of the majority of votes in a nationwide district. This is the system that was
discussed in sections B. and C.. In other systems, like the United States for presidential
elections, the selection is made on the basis of a majority of votes in a majority of districts.
This is the electoral college system. Similarly, in Britain, to form a majority government
it is only necessary to win a majority of the districts, and only a majority of the votes in
each of those districts is required. Thus, the electoral college model is also a model of the
British (Westminster) system. Here we argue that the electoral college system is even more
inecient than the systems described in the previous sections in generating incentives for
candidates to eectively provide public goods.
25
The electoral college system is characterized by a winner-take-all rule both at the district
level and at the national level. To win in our two-candidate model, a candidate must obtain
more than 50 percent of the votes in more than 50 percent of the districts. A district will be
denoted by d. We assume a continuum of measure 1 of districts.
16
All districts are identical
both in size and in the benets they receive from the public good. A strategy must now
specify, in the case that redistribution is chosen, the aggregate transfer to each district as
well as the distribution inside the district.
The electoral college system is more inecient than a nationwide winner-take-all system.
This increase in the ineciency is due to the fact that a candidate who oers to provide
the public good must now worry about his opponent going after the majority of voters in
the majority of districts. If G < 4, a candidate can oer more than G 1 to more than
50 percent of the voters in more than 50 percent of the states and zero to the rest. Such a
strategy leads to sure victory against a candidate who chooses to oer the public good with
probability one. Thus, if G < 4 there is no equilibrium where the public good is provided
with probability 1.
Theorem 5 Suppose elections are conducted under an electoral college system. Let
d
i
be
the average transfer oered by candidate i to voters in district d.
(i) Suppose G < 1. In equilibrium candidates never promise to provide the public good,
and both candidates make oers to voters according to the following process: each candidate
draws
d
i
from a uniform distribution on [1, 1]. In district d candidate i makes oers to
voters according do a uniform distribution on [1, 2
d
i
+ 1].
(ii) Suppose 1 < G < 4. The equilibrium probability of providing the public good is less
26
than or equal to 1/2.
(iii) For G > 4, the public good is provided for sure at equilibrium.
Proof. Part (i) Take a district where candidate i has dedicated
d
i
resources and candi-
date h has dedicated
d
h
resources. If
d
i
>
d
h
then, by following the equilibrium strategy,
candidate i gets strictly more than 50 percent of the votes in district i and wins the district.
This means that the candidate with more resources in the district gets all the electoral col-
lege votes in the district. If
d
i
<
d
h
, party h wins the district, and if
d
i
=
d
h
then the two
candidates split the district equally. Moreover, given
d
i
,
d
h
and given the fact that candidate
i is distributing the resources uniformly, candidate h cannot do any better than choose a
uniform. With any other distribution he would still lose if
d
i
>
d
h
, win if
d
i
<
d
h
and tie if

d
i
=
d
h
. The fact that candidate h does not know
d
i
when choosing the optimal distribution
of resources in district d does not change the optimality of the uniform distribution.
Let us now consider how candidates distributed resource across districts. By preceding
argument, districts behave exactly like voters: if a candidate dedicates more resources to the
district he gets the district. Thus, choosing
d
i
from a uniform distribution on [1, 1] is part
of an equilibrium.
Part (ii) Suppose that candidate 1 was oering to provide the public good with prob-
ability greater than 1/2. Since G < 4, candidate 2 could oer more than G to more than
50 percent of the voters in more than 50 percent of the districts: this leads to an expected
payo strictly greater than 1/2, which is impossible at equilibrium.
Part (iii) Since G > 4, when your opponent follows the equilibrium strategy there is no
way to use transfers and oer more than Gto more than 1/4 of the voters, and that is the
27
minimum share of the vote needed to reach an expected payo of 1/2. So, oering money is
dominated by oering the public good.
In the previous result, we have not proved that an equilibrium exists when 1 < G < 2.
In any nite game that approximates our game, an equilibrium will exist and will exhibit
the same qualitative features presented in Theorem 5.
The logic of Theorem 5 (i) is a hierarchical version of the logic of Theorem 1. The forces
that generate a uniform distribution in a nationwide election also push toward a uniform
distribution in each state. However, it cannot be an equilibrium that all districts get the same
resources. If this were the case a candidate could deviate by targeting higher average oers
to a majority of the districts. Thus, there must be an ex-post dierence in the amount of
resources that each candidate oers to voters in dierent districts. It is then straightforward
to apply the same logic to the distribution of resources across districts. This yields a uniform
distribution on [1, 1].
When G < 1, there is a big dierence between the equilibrium in a nationwide district
election discussed in Proposition 1 and the outcome described in Theorem 5. This reects
the incentives to go after 25 percent of the voters in the electoral college system as opposed
to going after the majority in a nationwide district. The aggregate distribution of resources
is much more unequal in the electoral college system. Recall that in Proposition 1 we saw
that in a nationwide district voters consumption is distributed uniformly on [0, 2]. In the
electoral college some voters get to consume as much as 4 while a lot more voters consume
less than 1. More formally, the outcome of Theorem 5 is dominated by the nationwide
district outcome in the sense of second degree stochastic dominance;
17
thus, whenever voters
28
are risk-averse they will prefer a system with a nationwide district.
The contrast between Theorems 2 and 5 is also interesting. When G > 1, the electoral
college system delivers a probability of providing the public good which is not higher than
in the nationwide district when 1 < G < 2, and is strictly lower when 2 < G < 4.
The result in Theorem 5 is related to the literature that discusses the ineciency of
pork barrel spending in legislatures where representatives represent dierent districts. Chari
et al. (1997), for instance, show that voters in each district have an incentive to select a
representative who is strongly in favor of pork-barrel spending. This arises because of a
common pool problem where candidates in each district do not take into account the costs
to other districts. Our setup diers from these models since we focus on national candidates
who do not represent any specic district: when proposing to increase transfers to a district,
our candidates take into account the loss of another district.
V. Conclusion
We have presented a political-economic model where the provision of a public good is de-
termined by the electoral incentives of oce-seeking candidates. When candidates have the
option of redistributing resources, public goods will be underprovided relative to the ecient
outcome. This happens because benets from the public goods cannot be targeted to groups
of voters as easily as the benets from pork barrel projects or pure transfers.
In this setup, we have compared dierent electoral systems: the electoral college system
is always less ecient than a system with a nationwide district. The winner-take-all system
is less ecient than a proportional system when the public good is very desirable, and is
29
more ecient when the public good is not very desirable. In both the winner-take-all and in
the proportional system, the redistributional platforms that candidates propose become less
egalitarian as the public good becomes more desirable.
This model addresses an important feature of political competition, the trade-o between
eciency and targetability. Policies whose benets are uniform across the population are
penalized in a political equilibrium, relative to policies whose benets are (or can be made)
more concentrated.
18
Of the many simplications that make this model easy to analyze, three merit particular
emphasis. First, our model is one of oce seeking candidates and ignores the possibility that
politicians can be driven by ideological considerations. This is a very stylized representation,
but one that, we believe, captures an important incentive in electoral competition. It would
be desirable to extend this model to allow for politicians that are motivated both by ideolog-
ical considerations and the benets of oce.
19
Such an extension would allow a comparison
with citizen-candidate models where politicians care about the policy, and cannot commit
to electoral platforms.
20
The assumption that electoral promises are binding is an extreme; in reality a politician
cannot perfectly commit to honor his promises. It would be desirable to understand how
the analysis would change if commitment were only imperfect, but this requires a dynamic
analysis that is beyond the scope of the present paper.
21
Finally, an unrealistic feature of our analysis is that with positive probability some voters
are taxed all their endowment, they are fully expropriated. This is a consequence of the
assumption that candidates know voters endowments and taxes are non distortionary. If
30
either of these assumptions is relaxed, all voters would retain control of a positive amount
of resources.
We feel that the most interesting extensions of our model involve further explorations of
the political process. It would be interesting to remove the restriction to two candidates.
This would be particularly important for a comparison of proportional representation with
plurality rule, since these systems seem to be associated with dierent numbers of candidates.
What we show in this paper is that there are dierences between the two political systems
even when they have the same number of candidates (two). We expect the main features of
our analysis to carry over to the environment with more than two candidates, so long as the
proportional system allocates benets to politicians in a less extreme fashion than plurality
rule.
Such an extension would permit an analysis of issues that were mostly ignored in this
paper, namely the process of legislative bargaining and coalition formation, and its inuence
on the provision of public goods. Austen-Smith and Banks (1988) provide a benchmark
analysis of this issue in a unidimensional spatial model with strategic voters.
22
It would be
desirable to build on their analysis to extend our model in this direction. This is a matter
for future research.
31
VI. References
Austen-Smith, David and Banks, Jerey. Elections, Coalitions, and Legislative Out-
comes. American Political Science Review, June 1988, 82(2), pp. 405-22.
Baron, David. Majoritarian Incentives, Pork Barrel Programs and Procedural Control.
American Journal of Political Science, February 1991, 35(1) pp. 57-90.
Baron, David and Diermeier, Daniel. Dynamics of Parliamentary Systems: Elections,
Governments, and Parliaments. Mimeo, Northwestern University, 1998.
Besley, Timothy and Coate, Stephen (1997) An Economic Model of Representative Democ-
racy. Quarterly Journal of Economics, February 1997, 112(1), pp. 85-114.
Besley, Timothy and Coate, Stephen Sources of Ineciency in a Representative Democ-
racy: a Dynamic Analysis. American Economic Review, March 1998, 88(1), pp.
139-56.
Bowen, Howard R. The Interpretation of Voting in the Allocation of Economic Resources.
Quarterly Journal of Economics, November 1943, 58(1), pp. 27-48.
Brams, Stephen J. and Davis Michael E. The 3/2s Rule in Presidential Campaigning.
American Political Science Review, March 1974, 68(1), pp. 113-34.
Buchanan James and Tullock Gordon. The Calculus of Consent. Ann Arbor: University
of Michigan Press, 1962.
Chari, V.V. Jones, Larry E. and Marimon, Ramon. The Economics of Split-ticket Voting
in Representative Democracies. American Economic Review, December 1997, 87(5),
32
pp. 957-76.
Coate, Stephen and Morris, Stephen. On the Form of Transfers to Special Interests.
Journal of Political Economy, December 1995, 103(6), pp.1210-35.
Lindbeck, Assar and Weibull, Jurgen. Balanced-Budget Redistribution as the Outcome of
Political Competition. Public Choice 1987 52, pp. 273-297.
Lizzeri, Alessandro. Budget Decit and Redistributive Politics. Mimeo, Princeton Uni-
versity, 1997, Review of Economic Studies, forthcoming.
Lizzeri, Alessandro and Persico, Nicola. The Provision of Public Goods Under Alternative
Electoral Incentives. CARESS Working Paper No. 98-08, University of Pennsylvania,
1998.
Mayhew, David. Congress: The Electoral Connection. New Haven: Yale University Press,
1974.
Myerson, Roger. Incentives to Cultivate Favored Minorities Under Alternative Electoral
Systems. American Political Science Review, December 1993, 87 (4), pp. 856-69.
Osborne, Martin and Slivinski, Al. A Model of Political Competition with Citizen-
Candidates. Quarterly Journal of Economics, February 1996, 111(1), pp. 65-96.
Persson, Torsten and Tabellini, Guido. Federal Fiscal Constitutions: Risk Sharing and
Redistribution. Journal of Political Economy, February 1996a, 104(5) pp. 978-1009.
. Federal Fiscal Constitutions: Risk Sharing and Moral Hazard. Econo-
metrica, May 1996b, 64(3), pp. 623-646.
33
. Political Economics and Public Finance. Mimeo 1998, forthcoming in the
Handbook of Public Economics.
. Alfred Marshall Lecture The Size and Scope of Government: Comparative
Politics with Rational Politicians. European Economic Review April 1999, 43(4-6),
pp. 699-735.
Persson, Torsten; Roland, Gerard and Tabellini, Guido. Comparative Politics and Public
Finance. Mimeo, Institute for International Economic Studies, Stockholm, 1997.
Poole, Keith and Rosenthal, Howard. Are legislators ideologues or the agents of con-
stituents? European Economic Review, April 1996, 40(3-5), pp.707-17.
Snyder, James. Election Goals and the Allocation of Campaign Resources. Econometrica,
May 1989, 57(3), pp. 637-60.
Stiglitz, Joseph E. Economics of the Public Sector. New York: Norton 1988 (2nd ed.).
Weingast, Barry; Shepsle, Kenneth and Johnsen, Christopher. The Political Economy of
Benets and Costs: A Neoclassical Approach To Distributive Politics. Journal of
Political Economy, August 1981, 89(4), pp. 642-64.
34
Footnotes
* Lizzeri: Department of Economics, Princeton University, Fisher Hall, Princeton, NJ
08544, Email lizzeri@princeton.edu, URL http://www.princeton.edu/lizzeri.; Persico: De-
partment of Economics, University of Pennsylvania, 3718 Locust Walk, Philadelphia, PA
19104, Email persico@ssc.upenn.edu, URL http://www.ssc.upenn.edu/persico. We wish to
thank Andrew Postlewaite, Tim Besley, and two anonymous referees for valuable comments.
1. Here we identify pork-barrel projects (or local public goods) with redistribution, not
public goods. This is because the benets from pork-barrel projects are concentrated
on few voters. Indeed, pork-barrel projects are widely understood as a means of (non-
monetary) redistribution disguised as public goods. What characterizes public goods in
our model is that their benets cannot be easily targeted to subsets of the population.
2. See Section E. for a discussion of the notion of eciency used in the paper.
3. The informal idea that distortions will arise out of majoritarian decision making if
policies with concentrated benets coexist with policies whose benets are diuse, can
be attributed to James Buchanan and Gordon Tollock (1962).
4. In the conclusion we discuss these issues further.
5. An exception is Stephen Coate and Stephen Morris (1995), who focus on the fact that
politicians have private information on the eects of government policy.
6. This is a convenient assumption because it will allow us to invoke the law of large
numbers on a number of occasions. This is meant to be an approximation for a game
with a large (but nite) number of voters.
35
7. Alessandro Lizzeri and Nicola Persico (1998) relax this assumption and show that the
analysis is robust.
8. This assumption is made solely to simplify the notation and presentation of the results;
all the results hold for any increasing utility function. Thus, we will feel free to interpret
the results for the case in which voters are risk averse.
9. See Myerson (1993), footnote 1, for further discussion.
10. Consider an election with n candidates. Let all candidates oer the public good, and
assume that at this strategy combination each candidate receives a share of the vote
of 1/n. Then, it is only when G is greater than n that this is an equilibrium. When
1 < G < n the equilibrium is going to be inecient. This suggests that the ineciency
is increasing in the number of candidates. However, a full analysis of the case of n
candidates is not an easy extension of the two candidate model.
11. For a discussion of the appropriate notion of eciency see Section E.. With risk
neutrality and G < 1, the equilibrium outcome is ecient according to all denitions
discussed in that section.
12. Because of our assumption of discrete investment in the public good, it is appropriate
to model the compromise between two parties as a probability rather than, say, a
coecient in the convex combination between the two policies. A policy outcome that
is a convex combination between public good and redistribution would be inconsistent
with that assumption.
13. More precisely, this behavior is the limit of optimal behavior in nite elections.
36
14. Timothy Besley and Stephen Coate (1998) discuss a model where it is important to
distinguish between alternative notions of eciency in a dynamic setting.
15. Another possible notion of eciency is surplus maximization. Notice that in our model
voters are treated equally ex ante by a candidates strategy. For such strategies, surplus
maximization coincides with ex ante Pareto eciency.
16. Just as in the previous analysis this assumption is made for convenience, to avoid
the complications of meeting the budget constraint with a stochastic strategy. Things
would be approximately the same when there are a large number (e.g., 50) of districts.
17. A random variable A dominates a random variable B is the sense of second order
stochastic dominance i
B A +, with E (|A) = 0.
That is, B is equal, in distribution, to A plus a noise term. This denition is equivalent
to the one presented in the proof of Proposition 4.
18. The analysis can be extended to allow for voters who have heterogeneous valuations
for the public good (see Lizzeri and Persico (1998)). In this case the benets from the
public good are less uniform across the population, and this makes the public good
more likely to be provided than when voters are homogeneous. This is in contrast with
the conclusions from the textbook median-voter model, where public goods provision
becomes more ecient as the population becomes more homogeneous.
19. For an empirical discussion of the importance of ideology in legislatures see for instance
Keith Poole and Howard Rosenthal (1996).
37
20. See Besley and Coate (1997) and (1998), and Martin Osborne and Al Slivinski (1996).
21. See Persson et al. (1997) for a recent dynamic analysis of policy making that does not
assume commitment on the part of politicians.
22. David Baron and Daniel Diermeier (1998) provide a more recent analysis of similar
issues. They focus on the dynamics of parliamentary systems.
38
TITLE FOR FIGURE 1: Redistribution of resources at equilibrium
39