Weingast B.R., Wittman D. The Oxford Handbook of Political Economy

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82 candidate objectives and electoral equilibrium

Duggan (2006), again abstracts away from campaigns. Politicians do make mean-

ingful choices while in oﬃce, however, as candidates must consider the information

(about preferences or abilities) their choices convey to voters. Equilibria of a simple

(stationary) form can often be shown to exist quite generally in these models.

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chapter 5

..............................................................

POLITICAL

INCOME

REDISTRIBUTION

..............................................................

john londregan

1 Introduction

.............................................................................

Douglass North has characterized government as possessing a monopoly on the

legitimate use of force. This makes the government the primary custodian of property

rights, while at the same time leaving it uniquely suited to seize and dispose of

individuals’ resources. While some resources at the disposition of the government

are allocated to the provision of non-excludable “public” goods (lighthouses are a

classic if somewhat prosaic example), or to partially excludable goods (transportation

networks are subject to congestion, but during oﬀ-peak hours they have aspects of

public goods), an important element of public policy involves the redistribution of

private consumption goods. It is to the last-mentioned class of activity, the political

redistribution of income, that this chapter is devoted. I shall take a positive rather

than a normative approach to the subject that seeks to explain why we observe the

redistributive policies that we do, rather than taking up the related problem of what

policies of redistribution should be implemented.

Income redistribution is often associated with action on the part of government,

while it is customary to think of policies that preserve the distribution of income

arising from the workings of markets and other non-governmental productive

activities as being passive ones. Yet any allocation of income requires the active

enforcement of property rights by the government, so-called “laissez-faire” policies

∗

I am grateful to seminar audience at the Juan March Institute for useful comments.

john londregan 85

rely on government action every bit as much as actively redistributive policies. The

distribution of income is always political.

In some cases we have governments that are not generally accountable for their

actions. In these environments redistribution often amounts to kleptocracy, with a

dictator, ruling clique, or royal family as the primary recipients of redistribution.

Olson (1993) recognized that even in these cases, the kleptocrat may increase the

harvest of loot if he can credibly commit to not steal everything.

When government policy is accountable to the general public matters become

much more complicated,

and it is to income redistribution within a framework of

democratic institutions that this chapter is devoted. One important set of analyses not

covered here are focused on legislative bargaining, Baron and Ferjohn (1989)being

a notable example. Those models focus on the features of legislative organization,

and use the division of a prize as a paradigm for legislative decision-making. Because

the subject of legislative organization is taken up in detail in other chapters in this

volume, I do not concentrate on that important set of models here. Likewise, I do not

focus on models of lobbyists, such as Grossman and Helpman (1994), even though

the goal of many lobbyists is to inﬂuence the political redistribution of income, as

such models are also the subject of another chapter in this volume.

This overview begins in Section 2 with a simpler but widely used class of models in

which the menu of redistributive options is artiﬁcially restricted by limiting taxes and

transfers to schemes that make post-tax income a deterministic and non-decreasing

function of pre-tax income. Because the functional form relating post-tax to pre-

tax incomes lends itself to extensions on other dimensions, for example to modeling

public goods provision when individuals can choose to relocate to other jurisdictions,

models of this class have become building blocks of the literature on redistributive

politics. Attention then turns to competitive models that allow for much greater ﬂex-

ibility in the choice of redistribution schemes, with Section 3 focusing on models that

emphasize the potential instability of redistributive politics, while Section 4 considers

models with built-in sources of stability. In Section 5 attention turns to several models

of redistributive politics that try to explain the choice of redistributive instruments

when more eﬃcient alternatives are available. A ﬁnal section concludes.

2 Competition over Restricted Income

Tax Schedules

.............................................................................

If we restrict our attention to redistributive schemes that involve a constant lump-

sum payment to everyone, coupled with a linear (Meltzer and Richard 1981),

or quadratic (Roemer 1999), tax scheme, then political competition can lead to

¹ Of course, on normative grounds virtually everyone will prefer this complexity to the crass

simplicity of a kleptocratic regime.

86 political income redistribution

progressive tax schemes in which those with high incomes subsidize those with lower

incomes.

In a classic paper Meltzer and Richard (1981) set forth a simple model of income

redistribution that captures the essential trade-oﬀ between equality and output max-

imization that is intrinsic to redistributive schemes. In this model government policy

consists of a tax rate t levied on all income, and a lump-sum egalitarian transfer r .

An individual with a pre-tax income of y pays ty in taxes, and receives a transfer of r,

resulting in a post-transfer income of c:

c = y(1 −t)+r (1)

Individuals are each endowed with one unit of labor, which they can divide between

work and leisure. While they get no direct utility out of work, individuals do enjoy

leisure activity, and they enjoy goods consumption.

Individuals diﬀer in their work-

place productivity, denoted by x. If an individual with productivity level x devotes a

fraction n of her time to work, her pre-tax income will be given by x per unit of time

spent working:

y = nx (2)

In this framework, the government devotes all of tax revenues to ﬁnancing the

lump-sum transfer r , so each person’s transfer is simply their equal share of tax

revenues. Individuals recognize that they are small relative to the size of society, and

so they treat the lump sum they receive as not depending on their own personal

work eﬀort.

Thus, all individuals with the same level of productivity have the same

preferences concerning taxes and transfers, while the level of transfers, r ,isfully

determined by the tax rate t.

Meltzer and Richard show that in their set-up, an individual’s most preferred

tax rate depends on her productivity level, and that this preferred rate tends to fall

with an individual’s own labor market productivity.

However, individuals do take

the incentive eﬀects of taxation on the labor supply of others into account when

evaluating potential tax rates.

Meltzer and Richard argue that in equilibrium competition among political en-

trepreneurs will lead to a government that implements the preferred tax policy of

the voter with the median productivity level

x. If the distribution of productivity

levels is skewed rightward (so that the median voter has a below-average income)

the result will be a positive level of redistributive taxation that stops short of full

equalization of incomes. Taxation stops short of full equalization because the median

² Technically, Meltzer and Richard assume that individuals’ preferences can be represented by a

quasi-concave utility function over consumption and leisure time which is increasing in both its

arguments.

³ In Meltzer and Richard’s model individuals are part of a continuum, and each person’s contribution

totaxrevenuesiseﬀectively zero, so the impact of each person’s labor supply decision on the average

transfer is literally negligible.

⁴ Strictly, if t

∗

(x) denotes the preferred tax rate of each individual with productivity level x,thent

∗

a non-increasing function of x.

john londregan 87

voter recognizes the disincentive eﬀects of a high tax rate. As tax rates rise toward 1

individuals shift away from work and into leisure.

A crucial consideration hinges on whether the median voter works. Among tax

rates that dissuade the median voter from work, she will prefer the rate that maxi-

mizes the size of the per capita transfer. Above that rate work eﬀort falls oﬀ so rapidly

that even though people pay a higher tax rate on their remaining work hours, it is not

enough to oﬀset the decreased amount of work, and total revenues decline. In this

case, the Meltzer and Richard outcome is quite similar to the notion of a “stationary

bandit” (Olson 1993), namely a government that seeks to maximize the resources it

extracts from the economy, and which sets a tax rate below full conﬁscation only be-

cause to take everything would suﬀocate individuals’ incentives to produce. However,

among voters whose preferred tax rates are below that which would remove them

from the labor force, there is an additional impact of an increased tax rate, in that it

reduces their pre-tax incomes. So the preferred tax rate is a decreasing function of a

worker’s productivity: if the median voter works in equilibrium, then the median tax

rate will be lower than the revenue-maximizing level.

This model represented a signiﬁcant innovation over earlier models. It did not fol-

low Marx (1867) in predicting that, “false consciousness” aside, people would choose

to expropriate fully those with more wealth. Neither did it rely on the assumption of

a poorly informed electorate, about which I will have more to say later in this chapter.

Meltzer and Richard showed that partial income redistribution could be the result of

decisions by fully informed voters. The authors were interested in an empirical reg-

ularity known as “Wagner’s Law” which held that as incomes rise so does the “taste”

for redistributive policies. Meltzer and Richard’s analysis directs attention instead to

the degree of income inequality, as measured by the distance between the economy-

wide mean income and the income of the median voter, suggesting that periods of

rapid economic growth, which Kuznets (1955) predicted would be characterized by

increased inequality, would be accompanied by higher rates of taxation when voting

rights are universal.

Of course, the Meltzer and Richard framework begs a number of questions. No-

tably there is the restriction to a linear income tax scheme. One response to this is

oﬀered by Roemer (1999) (see his article in this volume) who allows tax and transfer

schemes that are quadratic functions of individuals’ pre-tax incomes, though this

added complexity (and realism) comes at the price of not allowing for production.

Restricted tax schemes have become standard building blocks from which more

complex models of income redistribution are confected. Analysts interested in the

distributive impact of some other feature of the political system such as the potential

for migration, which is discussed in detail in the chapter by David Wildasin in this

volume, have frequently appended a more sophisticated model for political decision-

making to a version of Meltzer and Richard’s linear tax scheme.

An interesting case in point is Austen-Smith (2000), who generalizes the Meltzer–

Richard model to encompass heterogeneous types of economic activity, and who

compares the eﬀects of diﬀerent electoral systems on the level of redistribution.

Austen-Smith’s model generates the prediction that proportional representation

88 political income redistribution

militates in favor of a more equal distribution of income than simple majority rule,

a prediction that seems to be borne out empirically (see for example Persson and

Ta be ll in i 2000).

3 Competition over Unrestricted

Transfer Schemes

.............................................................................

The analysis of income redistribution in this context has an interesting parallel in the

literature on industrial organization. Bertrand (1883) noted that competition among a

ﬁnite number of identical sellers possesses a fundamental tendency toward instability:

given any prevailing price each seller can capture all of the market by inﬁnitesimally

undercutting her competitors’ prices. Subsequent analyses by Sraﬀa(1926), Hotelling

(1929), and de Palma et al. (1985) identiﬁed conditions under which inﬁnitesimal price

reductions result in only inﬁnitesimal changes in market shares, conferring a degree

of stability on market outcomes.

One important family of models of income redistribution, which will be examined

in this section, possesses an instability similar to that highlighted by Bertrand (1883);

in these models redistributive schemes are highly vulnerable to being undercut by

competing alternatives that oﬀer inﬁnitesimal improvements to a voting majority.

Models in which decision are made by majority vote without restrictions on the

potential allocation of transfers

generically fail to possess equilibria for reasons

discussed in McKelvey (1976). Working within the framework of cooperative game

theory, Aumann and Kurz (1977a) show that this indeterminacy is removed if indi-

viduals can destroy their endowments rather than allow them to be distributed to

others.

In Aumann and Kurz’s model,

agents are distributed along a continuum, there

is no production, and redistribution is costless, save that each individual can, if

she chooses to, destroy her own endowment before redistribution takes place. Total

endowments are ﬁnite; utility functions (the function for type t individuals is denoted

) satisfy some fairly standard regularity conditions.

The equilibrium budget must

balance. Individuals’ ability to destroy their output constrains the beneﬁts that a

majority coalition can vote for itself. In equilibrium an individual’s allocation will

be larger the greater her initial endowment, and the less risk averse she is. The intu-

ition here is that groups of agents engage in brinksmanship, threatening to disrupt

⁵ The mechanics of his model hinge crucially on his assumptions about the objectives of the political

parties.

⁶ With no productive activities and a continuously divisible stock of a single resource this amounts to

N voters selecting a point on an N − 1 dimensional simplex.

⁷ This model is discussed in Aumann and Kutz 1977a and fully elaborated in Aumann and Kurz 1977b.

⁸ Each agent’s utility function is increasing, concave, continuously diﬀerentiable, and uniformly

bounded.

john londregan 89

coalitions and, in particular, to destroy their own output if they are treated badly in

the allocation of consumption. In such an environment, risk aversion is a liability,

limiting the set of threats that are credible, while intensifying one’s sensitivity to the

threats of others.

Wei ng ast (1979) takes the specter of uncertainty about equilibrium allocations as

his point of departure. He considers an N-person legislature (with N an odd number

to preclude ties) considering N projects, each of which confers a beneﬁt of b to

adiﬀerent district, while it entails a cost c, which must be evenly shared across

all N districts. Each agent wants to maximize the well-being of her own district.

Individuals are torn between their desire to maximize their beneﬁts as a part of the

winning coalition, by restricting its membership as much as possible, and their fear

of being excluded from the winning coalition. If more projects are funded, the cost

of providing them goes up, but the probability that a randomly chosen legislator is

excluded from the beneﬁts goes down.

IfallcoalitionsofsizeM are equally likely, a legislator’s probability of having her

project funded, and receiving a beneﬁt of b,willbe

, while the cost per district

of funding M projects is

c, which she must pay whether her project is chosen or

not. Her ex ante expected payoﬀ if the winning coalition is of size M is the diﬀerence

between the expected payoﬀ and the certain cost, namely:

(b −c)(3)

Provided b > c, so that the projects’ beneﬁts exceed their costs, the value of M which

maximizes a legislator’s ex ante expected payoﬀ is M = N, which entails a universal

coalition. On the other hand, if a legislator knows she will belong to the winning

coalition her ex post payoﬀ will be:

b −

c (4)

In this case, the legislator will want to make M as small as possible, subject to the

coalition still being able to impose its will. With N odd and strict majority rule, the

value for M that achieves this is M =

N+1

, a bare majority.

What if the agents in Weingast’s model had the ability to destroy their output, as

in the Aumann and Kurz model discussed previously? In this case, the costs of the

projects undertaken by the winning coalition would have to be paid for entirely by

the winners, as the losers would destroy their output. Thus, the per capita beneﬁts

for the winning coalition would be b − c regardless of the coalition’s size, eliminating

the temptation identiﬁed in equation (4) to exclude members from the winning coali-

tion.

Weingast’s conclusion that legislators would prefer a universalistic coalition is

⁹ Riker referred to such alliances among the smallest possible decisive group as “Minimum Winning

Coalitions.”

¹⁰ Of course, these results are sensitive to the “all or nothing” nature of the projects, and the linearity

of costs.

90 political income redistribution

further strengthened if we transplant into his model the assumption of Aumann and

Kurz that one can destroy one’s output.

Another model that allows freedom in choosing the form of the tax schedule

is presented by Snyder and Kramer (1988). These authors allow for considerable

ﬂexibility in the shape of the tax schedule, but they posit a very simple model for

production. In exchange for this simplicity they obtain some remarkably strong

results about tax rates. They show that any non-linear tax schedule, where the net tax

to an individual depends on her market income y: t(y), can be dominated by a linear

one, and that steeper tax schedules can result in less equalizing redistribution than

those that rise more gently (depending on the level of lump-sum taxes or subsidies).

Given an additional condition on the elasticity of labor demand in the underground

economy, Snyder and Kramer show that the equilibrium tax policy will be equivalent

to a linear one. While this result is remarkable, it hinges crucially on unconventional

assumptions about the labor market.

While electoral competition is often modeled in terms of two candidates compet-

ing in a majoritarian system, elections frequently involve non-majoritarian electoral

rules, and multiple candidates. Meyerson (1993) shows that electoral rules have a

profound impact on the sorts of oﬀers candidates will make. He develops a common

framework for comparing various electoral conﬁgurations. Candidates compete by

oﬀering redistributive beneﬁts that are uncorrelated with the campaign promises of

other candidates. Imagine the candidates handed out lottery tickets which could be

redeemed if they were elected, while the voters care only about their transfer, so that

each voter ranks the candidates in the order of the size of the transfer each has oﬀered.

Myerson considers a stylized setting with a very large number of voters

each of

whom casts a “sincere” vote that reﬂects his actual preference ranking, and a ﬁxed

amount of money that the winning candidate can distribute if she is victorious. A

key feature of Myerson’s model is that the “lottery tickets” handed out randomly by

candidates cannot be shown to anyone else. Thus, Myerson’s framework precludes

candidates targeting speciﬁc groups, and it rules out their making oﬀers contingent

on the oﬀer a voter has from the candidate’s competitors. This assumption makes the

model tractable, but at a signiﬁcant cost in terms of realism.

Meyerson obtains analytical results for systems involving a “single non-transferable

vote,” which includes the “simple plurality” rule widely used in the United States,

as well as for “Borda Counts,” “negative plurality voting,” and for multiple non-

cumulative votes. He also presents simulation results for “approval voting.”

A sense of how the electoral system aﬀects the transfers politicians oﬀer is conveyed

by Meyerson’s analysis of distributive politics under a simple plurality rule, and under

negative approval voting.

The simple plurality rule is probably the system most familiar to the reader: each

voter casts a ballot for one candidate, and the candidate receiving the most votes

is elected. Under negative approval voting, each voter casts one “negative ballot,”

and the candidate receiving the fewest negative ballots is elected. With only two

¹¹ His voters are modeled as a continuum.

john londregan 91

candidates, the two systems are eﬀectively identical: a vote for “candidate A” under

simple plurality rule has the same eﬀect as a negative vote for “candidate B” under

negative approval voting.

With either voting system, the equilibrium for a two-candidate election will not

involve the candidates making matching oﬀers to each voter. Why not? Suppose that

one candidate was foolish enough to do so: oﬀering an equal amount to each voter.

His opponent could then exclude a small set of voters from beneﬁts while she used the

savings to slightly outbid him for the remaining voters, thereby winning the election.

In equilibrium each candidate will make a range of unequal oﬀers. In the case of a

two-candidate election with either simple plurality rule or negative approval voting,

each candidate’s oﬀers will be uniformly distributed on the interval [0, 2], with some

voters getting nothing, while the most highly bribed receive as much as twice the

average oﬀer. If the other candidate is using a uniform distribution for oﬀers, then

the increased probability of outbidding the other candidate for a given voter that

results from a small increase in the oﬀer to that voter is just oﬀset by the increased

cost. When both candidates employ this strategy we are in equilibrium as neither has

an incentive to change her behavior. If we normalize the mean transfer to equal 1,

then the standard deviation of the distribution of transfers oﬀered by each candidate

under either simple plurality rule or negative approval voting will equal

√

,or

about 0.58.

Now suppose that instead of two candidates competing in an election, there are

three. In this setting the simple plurality rule and negative approval voting no longer

coincide. First let’s see what happens with the simple plurality rule. The uniform

distribution on the interval [0, 2] over oﬀers is no longer going to be a symmetric

equilibrium. If the other two candidates used such a distribution, then the third

contender could promise two and a half times the mean oﬀer of 1 to 40 per cent

of the voters, and nothing to the remainder, and expect to receive 40 per cent of

the vote, thereby defeating each of his opponents who would divide the remaining

votes evenly between themselves. This pressure to make unequal oﬀers carries over

to the equilibrium, as barely a third of the electorate is needed to prevail. In the

symmetric equilibrium each candidate makes generous oﬀers to a minority of voters

while making much smaller oﬀers to a majority of the electorate. Relative to the mean

oﬀer of 1, candidates’ oﬀer distributions will be skewed

with a minimum oﬀer of 0,

a maximum of 3, and a standard deviation of approximately 0.89. With the simple

plurality rule in elections with three or more contenders, candidates need friends

more than they fear enemies—with sincere voting, a candidate who is everyone’s

second choice receives no votes under a simple plurality rule.

With negative approval voting in a three-candidate election, the incentives run in

the opposite direction. Suppose that two of the candidates oﬀered a uniform distribu-

tion (on the interval [0, 2]) of payments to voters. The third candidate could win the

election by oﬀering a more equal distribution! For example, by oﬀering everyone the

¹² In the symmetric equilibrium the probability density function f (x) for each candidate’s oﬀers will

be f (x)=

√

−

for x on the interval [0, 3], and zero otherwise.