On the Political Economy of Nonlinear Income
Taxation
Marcus Berliantyand Miguel Gouveiaz
April 17, 2024

Abstract
The literatures dealing with voting, optimal income taxation, implementation, and pure public goods are drawn on here to address the
problem of voting over income taxes to …nance a public good. In contrast with previous articles, general nonlinear income taxes that a¤ect
the labor-leisure decisions of consumers who work and vote are allowed.
Uncertainty plays an important role in that the government does notknow the true realizations of the abilities of consumers drawn from aknown distribution, but must meet the realization-dependent budget;the tax system must be robust. Even though the space of alternatives
is in…nite dimensional, conditions on primitives are found to assure existence of a majority rule equilibrium. JEL numbers: D72, D82, H21,
H41 Keywords: Voting; Income taxation; Public good; Robustness

Support provided by National Science Foundation grant SES-9007800 and the hospitality of the Whiteley Center at the University of Washington are gratefully acknowledged. The
comments of Pierre Boyer, Randy Calvert, Yan Chen, Roy Gardner, Faruk Gul, Hideo Konishi, John Ledyard, Dilip Mukherjee, Wan-hsiang Pan, Paolo Piacquadio, Michael Smart,
Jim Snyder, Max Troshkin and John Weymark are appreciated, but the authors bear full responsibility for any errors. This work was originally a chapter of M. Gouveia’s dissertation,completed at the University of Rochester.
y
Department of Economics, Washington University, MSC 1208-228-308, 1 Brookings
Drive, St. Louis, MO 63130-4899 USA. Fax: (1-314) 935-4156, e-mail: berliant@wustl.eduzUniversidade Católica Portuguesa, Lisbon, Portugal, e-mail: mig@ucp.pt

11Introduction

1.1

Background

The theory of income taxation has been an important area of study in economics. Interest in a formal theory of income taxation dates back to at least
J.S. Mill (1848), who advocated an equal sacri…ce approach to the normative
treatment of income taxes. In terms of the modern development, Musgrave(1959) argued that two basic approaches to taxation can be distinguished: thebene…t approach, which puts taxation in a Pareto e¢ ciency context; and theability to pay approach, which puts taxation in an equity context. Some ofthe early literature, such as Lindahl (1919) and Samuelson (1954, 1955), madeseminal contributions toward understanding the bene…t approach to taxationand tax systems that lead to Pareto optimal allocations. Although the importance of the problems posed by incentives and preference revelation wererecognized, scant attention was paid to solving them, perhaps due to theircomplexity and di¢ culty.
Since the in‡uential work of Mirrlees (1971), economists have been quite
concerned with incentives in the framework of income taxation. The model
proposed there postulates a government that tries to collect a given amountof revenue from the economy. For example, the level of public good provision might be …xed. Consumers have identical utility functions de…ned overconsumption and leisure, but di¤ering abilities or wage rates. The government chooses an income tax schedule that maximizes some objective, such asa utilitarian social welfare function, subject to collecting the needed revenue,
resource constraints, and incentive constraints based on the knowledge of only
the overall distribution of wages or abilities. The incentive constraints derivefrom the notion that individuals’wage levels or characteristics (such as productivity) are unknown to the government. The optimal income tax schedule
must separate individuals as well as maximize welfare and therefore is generally second best.1 The necessary conditions for welfare optimization when the
distribution of ability is bounded generally include a zero marginal tax rate
for the highest wage individual; this result only holds very locally at the topof the distribution. Intuitive and algebraic derivations of this result can befound in Seade (1977), where it is also shown that some of these necessary
1

If the government knew the type of each agent, it could impose a di¤erential head tax.
As is common in the incentives literature, one must impose a tax that accomplishes a goal
without the knowledge of the identity of each agent ex ante.

2

conditions hold for Pareto optima as well as utilitarian optima. Existence ofan optimal tax schedule for a modi…ed model was demonstrated in Kaneko(1981), and then for the classical model in Berliant and Page (2001, 2006). An
alternative view of optimal income taxation is as follows. Head taxes or lumpsum taxes are …rst best, since public goods are not explicit in the model andtherefore Lindahl taxes cannot be used. Second best are commodity taxes,
such as Ramsey taxes. Third best are income taxes, which are equivalent toa uniform marginal tax on all commodities (or expenditure).2 In our view, it
is not unreasonable to examine these third best taxes, since from a pragmatic
viewpoint, the …rst and second best taxes are infeasible.3

1.2

A Positive Political Model

How can we explain (or model) the income tax systems we observe in thereal political world? We shall attempt to answer this question with a voting
model, a positive political model, in combination with the standard income
tax model described above. As noted in the introduction of Roberts (1977),
one does not need to believe that choices are made through any particularvoting mechanism; one need only be interested in whether choices mirror theoutcomes of some voting process. Thus, what is described below is an attempt
to construct a potentially predictive model with both political and economic
content. It contains elements of the optimal income tax literature as well as
positive political theory (an excellent survey of which can be found in Calvert(1986)).
Although much of the optimal income tax literature and most of the work
cited above deals with the normative prescriptions of an optimal income tax,
there is a relatively small literature on voting over income taxes. Most of this2

In certain circumstances, nonlinear income taxes can be second best; see Laroque (2005).
Ramsey (optimal commodity) taxes typically follow the inverse elasticity rule. There
are several practical issues. First, to compute Ramsey taxes, one must estimate supply anddemand elasticities for each commodity. Second, these elasticities may vary over time, so thetaxes would vary over time. Third, there is a question regarding how …ne the classi…cationof a commodity for the purposes of taxation might be. Demand for vegetables will typicallybe more inelastic than that for potatoes and carrots, which in turn will be more inelastic
than demand for Yukon gold potatoes. As Stiglitz (2015, footnote 13) notes regarding thereasons for grouping commodities, “Partly it is administrative: the cost of having millions oftax rates, one for each precisely identi…ed commodity, would be large. Partly it is based on
information: private parties would have an incentive to try to get their products classi…ed
as one of the lower taxed products. It is costly for the government to gather the information
required to implement and enforce …ne di¤erentiations.”
3

3

literature is either restricted to consideration of only linear taxes, or does notconsider problems due to information (adverse selection and moral hazard), or
both. Examples that might …t primarily into the linear tax category which also
involve no labor disincentives on the part of agents are Foley (1967), Nakayama(1976) and Guesnerie and Oddou (1981). Aumann and Kurz (1977) use personalized lump sum taxes in a one commodity model. Hettich and Winer (1988)
present an interesting politico-economic model in which candidates seek tomaximize their political support by proposing nonlinear taxes. Work disincentives are not present in the model. Chen (2000) extends their work to themore standard optimal income tax model in the context of probabilistic voting.
Bierbrauer et al (2021) examine redistributive nonlinear income taxes with twoparties, probabilistic voting, endogenous turnout, and ethical voters who careabout the welfare of members of their party. They …nd that members of theother party might be demobilized by catering policy to them. Romer (1975),
Roberts (1977), Peck (1986), and Meltzer and Richard (1981, 1983) use linear
taxes in voting models with work disincentives. Roemer (1999) restricts toquadratic tax functions with no work disincentives but with political parties.
Perhaps the model closest in spirit to the one we propose below is in Snyder
and Kramer (1988), which uses a modi…cation of the standard (nonlinear) income tax model with a linear utility function. The modi…cation accounts foran untaxed sector, which actually is a focus of their paper. This interestingand stimulating paper considers fairness and progressivity issues, as well as
the existence of a majority equilibrium when individual preferences are single
peaked over the set of individually optimal tax schedules. (Su¢ cient conditions for single peakedness are found.) Röell (1996) considers the di¤erences
between individually optimal (or dictatorial) tax schemes and social welfare
maximizing tax schemes when there are …nitely many types of consumers. Of
particular interest are the tax schedules that are individually optimal for the
median voter type. This interesting work uses quasi-linear utility and restricts
voting to tax schedules that are optimal for some type. Brett and Weymark(2017) push this further in a continuum of types model by characterizing individually optimal tax schedules. Then they show, under conditions including
quasi-linear utility, that if the set of tax schedules is restricted to individually
optimal ones, the individually optimal tax for the median voter is a Condorcet
winner.
We propose in this paper to allow general nonlinear income taxes withwork disincentives in a voting model. The main problem encountered in4trying to …nd a majority equilibrium, as well as the reason that various setsof restrictive assumptions are used to obtain such a solution in the literature,is as follows. The set of tax schedules that are under consideration as feasible
for the economy (under any natural voting rule) is large in both number anddimension. Thus, the voting literature such as Plott (1967) or Scho…eld (1978)
tells us that it is highly unlikely that a majority rule winner will exist. Is
there a natural reduction of the number of feasible alternatives in the context
of income taxation?

1.3

The Role of Uncertainty and Feasibility

The answer appears to be yes. The (optimal) income tax model has a natural
uncertainty structure that has yet to be exploited in the voting context. As in
the classical optimal income tax model, all worker/consumers have the samewell-behaved utility function, but there is a nonatomic distribution of wages
or abilities. In standard models, such as the Mirrlees model or its modern
descendants, the distribution of consumers by type is known by all and theaggregate revenue requirement is …xed at a scalar; it is 0 in models of pureredistribution. (This applies whether the number of consumer/workers is …niteor a continuum.) Suppose that a …nite sample is drawn from this nonatomic
distribution.4 The …nite sample will be the true economy, and the revenue
requirement imposed by the government can depend on the draw. In fact
this dependence is just a natural extension of the standard optimal income
tax model. In that model, the amount of revenue to be raised (the revenue
requirement in our terminology) is a …xed parameter, something that makes
perfect sense since the population in the economy and the distribution of thecharacteristics of that population are both …xed, and thus we can take public
expenditures also as …xed. But consider now the optimal tax problem for thecases when the characteristics of the actual population are unknown. That isexactly what happens when we consider that the true population is a …nite
draw from a given distribution. In such circumstances, it is not reasonable to…x the revenue requirement at some exogenously given target level, but instead
the revenue requirement should be a function of the population characteristics.
It is possible to derive the aggregate revenue requirement from primitives indi¤erent ways. In our analysis below, the revenue requirements for a particulardraw will be derived from the Pareto e¢ cient level of public good provision for
4This assumption is similar to the one used in Bierbrauer (2011), though the purpose ofthat work is entirely di¤erent from ours.

5that draw, leading to intrinsic variation in revenue requirements across draws.5
We shall make assumptions on primitives implying that the Pareto e¢ cientlevel of public good is unique but generally di¤erent for each draw.
Why do we focus on the Pareto e¢ cient level for a given draw? Given
utility functions that are separable in the public good, ex post recontracting
on the level of public good provision will not be bene…cial.
It seems natural for us to require that any proposed tax system must be
feasible (in terms of the revenue it raises) for any draw, as no player (including
the government) knows the realization of the draw before a tax is imposed. In
other words, the consumers do not transmit information, such as their labor
income, to the government prior to the imposition of the income tax. For
example, an abstract government planner might not know precisely the top
ability of individuals in the economy, and therefore might not be able to follow
optimal income tax rules to give the top ability individual a marginal rate ofzero, as described above. The key implication of using …nite draws as the true
economies is that requiring ex post feasibility of any proposable tax system forany draw narrows down the set of alternatives, which we call the feasible set,
to a manageable number (even a singleton in some cases).6
To be clear, the assumption is that the government must commit to a setof feasible tax systems (where a tax system maps income to tax liability) before knowing the realization of the draw of abilities from the distribution ofabilities, this set of feasible tax systems cannot depend on the draw, and it
must raise su¢ cient draw-dependent revenue no matter the draw. Voting thenchooses among the feasible tax systems, and the government must implement
this choice.7 All agents are forward looking, and for example each worker cananticipate their labor supply for any given tax function. Finally, consumers
supply labor and their types are revealed through their choice of labor supply. If we allowed the set of feasible tax systems to depend on the draw ofabilities, we would be back in the situation the rest of the literature has foundunsolvable, since in general any tax system can be defeated by a majority fora given draw. In other words, if the government doesn’t have to commit and5

As an alternative, the variation in revenue requirements can be seen as variation in …scalpressure on the government; see Heathcote and Tsujiyama (2021) for discussion.
6
Pierre Boyer has pointed out that the cost of the public good could be unknown in
addition to or instead of the abilities of the individuals in the economy. This would represent
an aggregate risk to the economy, in contrast with individual abilities, that are idiosyncraticrisks. In accordance with the assumptions concerning asymmetric information in the optimal
tax literature, we stick to unknown individual abilities on the part of the tax designer.
7
We shall be much more precise about the voting stage of this game in section 3.1.

6

can propose a state or draw contingent set of feasible taxes, we have the same
situation as if there is no uncertainty and a …nite number of worker/consumers
with given types, so there generally will be no Condorcet winner in any given
state.
Our model …ts naturally into the literature in public …nance on robustness,examining allocation when agents have ambiguous beliefs about the state of
the world. Recent innovations include an application to public goods by
Kocherlakota and Song (2019), that analyzes e¢ ciency in the context of adiscrete decision about whether or not to produce a public good in a static
environment; and Lensman and Troshkin (forthcoming) that examines optimal
allocative policy in a dynamic environment. Robustness has been used in thecontext of voting as well; see Berliant and Konishi (2005).
A signi…cant di¤erence between the work here and the balance of the literature is how robustness enters the model. In the literature just cited, agents
have multiple priors over the state of the world and are ambiguity averse. In
contrast, we shall assume that the government must generate su¢ cient revenue independent of the draw of agent types. One way to envision this is
to assume that the government is ambiguity averse and each element of theset of distributions it considers possible assigns a draw probability 1. Noticethat in this case, ambiguity enters into a government constraint rather than aconsumer’s objective function.
In sum, given government concerns about robustness, our setup guarantees
that a majority rule equilibrium exists and the government budget constraint
will be met.
Our arguments apply to …nite numbers of agents. The model has a discontinuity when one goes from a …nite to an in…nite number of agents. In thislatter case there is no uncertainty about the composition of the draw, so we do
not have a continuum of ex post feasibility restrictions, one for each possible
draw. Instead we have only that the revenue constraint needs to be satis…ed
for the known population. Thus, for our purposes, even a little uncertainty
is su¢ cient, and it is possible to view perfect certainty about the draw as aknife-edge case.8 Moreover, there are further conceptual issues pertaining tomodels with a pure public good and a continuum of consumers; see Berliant
and Rothstein (2000).
In relation to the literature that deals with voting over linear taxes, ourmodel of voting over nonlinear taxes will not yield a linear tax as a solu8

We are indebted to Jim Snyder for some of these thoughts.

7

tion without very extreme assumptions. This will be explained in section 6
below. Moreover, our second order condition for incentive compatibility will
generally be much weaker than those used in the literature on linear taxes;
compare our assumptions below with the Hierarchical Adherence assumptionof Roberts (1977). As noted by L’Ollivier and Rochet (1983), these second
order conditions are generally not addressed in the optimal income taxation
literature, though they ought to be addressed there. In what follows, we employ the results contained in Berliant and Gouveia (2001) and more generally
in Berliant and Page (1996) to be sure that the second order conditions forincentive compatibility hold in our model.
At this point, it is important to remark on the anatomy of our analysis.
We shall introduce two models: an endowment economy, where there is no
choice of labor supply, where each consumer knows only their own endowment
and the prior distribution from which endowments are drawn, and where alltaxes are lump sum; and an optimal income tax economy, where each consumer
knows only their own productivity and the prior distribution of types, where
the government knows only the prior distribution from which agents’ types
are drawn, where taxes induce distortions in labor supply, and where incentive
constraints on labor supply must be satis…ed. Although results on existenceof Condorcet winners for the endowment economy may be of independentinterest, our primary objective is to apply these results to the income tax
model with distortions as follows. The method for accomplishing this is to use
a result on implementation of lump sum taxes from the endowment economy,
the …rst model, in terms of a tax on labor income in the optimal income tax
economy, the second model. Under assumptions we shall specify, this result
implements the lump sum tax system in the sense that each consumer facing
a labor-leisure choice ends up paying exactly the tax speci…ed by the lumpsum tax system; notice that income taxes are an indirect mechanism. In theliterature on optimal income taxation, this is called the “Taxation Principle.”Moreover, we show that characteristics of the lump sum tax systems, such as
single crossing, are inherited by the taxes implementing them in the frameworkwith incentives.
To prove our main result, we combine the restriction on tax systems, as
outlined above, with assumptions on the utility function and the cost function for public goods. In particular, we assume that the utility function isquasi-linear in consumption and additively separable in the consumption commodity, labor supply, and the public good level; moreover, the public good8

subutility is multiplicative in type. Due to quasi-linearity and separabilityof the utility function, the Pareto e¢ cient level of public good, and thus aggregate revenue requirements, will be unique for each draw. Furthermore,
the aggregate revenue requirements will be concave in the draw, where theworst case for the government budget occurs when the draw consists of identical individuals.9 The tangent to the aggregate revenue requirement function
where all individuals in a draw are identical (a one dimensional domain for thefunction) represents a minimal feasible individual revenue requirement. These
are single crossing, and their implementations in terms of income taxes or net
income schedules are also single crossing. Among these, the tax system most
preferred by the median voter is a majority rule equilibrium, since induced
preferences over net income schedules are single crossing, as is standard in the
optimal income tax literature.
The structure of the paper is as follows. First, we introduce our frameworkand notation in section 2. In section 3 our main result on voting over bothpublic goods and income taxes is stated. Section 4 contains a discussion of
the techniques we use in the proofs, whereas section 5 contains two examples
of interest. Finally, section 6 contains conclusions and suggestions for further
research. The appendix contains proofs of most results.

2

The Model

2.1

Basic Notation and De…nitions

We shall develop an initial model of an endowment economy as a tool. Although it might be of independent interest, our primary purpose is to apply
this model and the results we obtain to the standard optimal income tax modelin the succeeding sections.
There is a single consumption good c and consumers’preferences are identical and given by the utility function v(c) = c, with c 2 R+ . A consumer’s endowment, which is also her type, is described by w 2 [w; w], where [w; w] R+ .
In this section the endowment can also be seen as pre-tax income or, following classical terminology in Public Finance, the ability to pay of each agent.
9

We have succeeded in proving analogous results when the aggregate draw-dependent
revenue requirements function is convex rather than concave, so the worst case scenario forthe government budget is when the draw consists of a given type and the extreme type mostunlike it. However, we were unable to derive this kind of aggregate revenue requirementsfunction from primitives.

9References to measure are to Lebesgue measure on [w; w].
The distribution of consumers’endowments has a measurable density f (w),where f (w) > 0 a:s:
Let k be a positive integer and let A [w; w]k , the collection of all possible
draws of k individuals from the distribution with density f .10 Formally, adraw is an element (w1 ; w2 ; :::; wk ) 2 A.
In order to be able to determine what any particular draw can consume, it
is …rst necessary to determine what taxes are due from the draw. Hence, we…rst assume that there is a given net revenue requirement function R : A ! R.
For each (w1 ; w2 ; :::; wk ) 2 A, R(w1 ; w2 ; :::; wk ) represents the total taxes duefrom a draw. For example, if the revenues from the income tax are used to…nance a good such as schooling, then R(w1 ; w2 ; :::; wk ) can be seen as: the percapita revenue requirement for providing schooling to the draw (w1 ; w2 ; :::; wk )multiplied by k.11
Although we shall begin by taking revenue requirements as a primitive, inthe end we will justify this postulate by deriving revenue requirements fromthe technology for producing a public good. But this is simply an important
example illustrating where aggregate revenue requirements come from.
It is important to be clear about the interpretation of R. One easy interpretation is that the taxing authority provides a schedule giving the taxes
owed by any draw. There are several reasons that revenue requirements mightdi¤er among draws, including di¤erences in taste for a public good that is implicitly provided, income or wealth di¤erences, a non-constant marginal costfor production of the public good, di¤erences in the cost of revenue collection,and so forth.
The government and the agents in the economy know the prior distribution
10

Note that f ( ) plays almost no role in the development to follow, in contrast with its
preeminent role in the standard optimal income tax model. It may be interpreted as a
subjective distribution describing the planner beliefs about the characteristics of the agentsin the economy, but that consideration is immaterial for the model presented here. We have
implicitly assumed that the abilities are drawn independently, but since we never use this,correlation would also be permissible provided that the support of the joint distribution is…xed at [w; w]k . In multistage voting in a representative democracy, the equilibria are likely
to be a function of f , as is often the case in signaling games. We expect to study thatproblem in the future.
11
Actually, regarding schooling, there is a separate literature on the political economy ofpublic supplements for such goods. The formal structure is slightly di¤erent from what we
consider in this paper; see Gouveia (1997).

10

f of types of agents in the economy12 as well as the mapping R. Before moving
on to consider the game - theoretic structure of the problem, it is necessary toobtain some facts about the set of tax systems that are feasible for any draw
in A. These are the only tax systems that can be proposed, for otherwise
the voters and social planner would know more about the draw than thatit consists of k people drawn from the distribution with density f . Voters
can use their private information (their endowment) when voting, but not in
constructing the feasible set. For otherwise either each voter will vote over adi¤erent feasible set, or information will be transmitted just in the constructionof the feasible set.
An individual revenue requirement 13 is a function g : [w; w] ! R that takes
w to tax liability. It is a lump sum tax function.
Clearly, there will generally be a range of individual revenue requirements
consistent with any map R. Our next job is to describe this set formally. Fix
k and R. Let
g : [w; w] ! R j g is measurable,

G
k
X
i=1

g(wi ))

R(w1 ; w2 ; :::; wk ) a.s. (w1 ; w2 ; :::; wk ) 2 A

G is the set of all individual revenue requirements that collect enoughP
revenue to satisfy R. G 6= ; if almost surely for (w1 ; w2 ; :::; wk ) 2 A, ki=1 wi
R(w1 ; w2 ; :::; wk ). The constraint that the aggregate revenue requirement besatis…ed for each draw restricts the feasible set G signi…cantly.

2.2

From Collective to Individual Revenue Requirements

In order to examine the set of feasible individual revenue requirements described above, more structure needs to be introduced. It is obvious that somefeasible g’s will raise strictly more taxes than necessary to meet R(w1 ; w2 ; :::; wk )for almost any (w1 ; w2 ; :::; wk ). For example, take any g 2 G. Then for > 0,P
g + will satisfy ki=1 [g(wi ) + ] R(w1 ; w2 ; :::; wk ) a.s. (w1 ; w2 ; :::; wk ) 2 A,
but clearly the consumers would all prefer g to g+ . The point is that some individual revenue requirements functions can be dominated unanimously, andthus can be eliminated from consideration. We now search for the minimal elements of the set G. We call the set of such elements G ; it is the12

Actually, all they need to know is the support of that distribution.
Even though this is simply a tax function on endowments, we will reserve the terminology“tax function” for an environment with incentives to simplify the exposition.
13

11

set of elements of G that are not dominated by any other element of G
pointwise. In other words, we search for individual revenue requirements
0g2G
G with the following property: there is no g such that almost surely
Pk 0
for (w1 ; w2 ; :::; wk ) 2 A, R(w1 ; w2 ; :::; wk )i=1 g (wi ); almost surely for0
w 2 [w; w], g (w) g(w); and there exists a set of positive Lebesgue measure0in [w; w] where g (w) < g(w).
0
To this end, de…ne a binary relation
over G by g
g if and only if0g(w) g (w) for almost all w 2 [w; w]. Let
GfB

Gj B is a maximal totally ordered subset of Gg:

By Hausdor¤’s Maximality Theorem (see Rudin (1974, p. 430)), G 6= ;. Finally, de…neG

g 0 (w) a.s.g:
fg : [w; w] ! Rj 9B 2 G such that g(w) = inf0
g 2B

G is nonempty.
If g 2 GnG is proposed as an alternative to g 2 G , 9g 0 2 G that is
unanimously weakly preferred to g.
Before turning to the model with labor disincentives, it is important to notethat G, G , and G all depend implicitly on the underlying revenue requirement
function R, and that the function R is common knowledge, so G, G , and G
are as well.

2.3

Notation for the Optimal Income Tax Model

Having dispensed with preliminaries, we now turn to the voting model withincentives based on Mirrlees (1971). The three goods in the model are acomposite consumption good, whose quantity is denoted by c; labor, whose
quantity is denoted by l; and a pure public good, whose quantity is denoted byx. Consumers have an endowment of 1 unit of labor/leisure, no consumption
good, and no public good.14 Let u : R+ [0; 1] R+ [w; w] ! R be theutility functions of the agents, writing u(c; l; x; w) as the utility function of typew, where u is twice continuously di¤erentiable. Subscripts represent partial
derivatives of u with respect to the appropriate arguments. The parameter w,
an agent’s type, is now to be interpreted as the wage rate or productivity ofan agent. Thus w is the value of an agent of type w’s endowment of labor.
14

It would be easy to add an endowment of consumption good for consumers, but thatwould complicate notation.

12

The gross income earned by an agent of type w is y = w l and it equals
consumption when there are no taxes.
A tax system is a function : R ! R that takes y to tax liability. A
net income function: R ! R corresponds to a given by the formula
(y) y
(y).
First we discuss the typical consumer’s problem under the premise that theconsumer does not lie about its type, and later turn to incentive problems. A
consumer of type w 2 [w; w] is confronted with the following maximization
problem in this model:

max u(c; l; x; w) subject to w l
c;l

(w l)

and subject to cc with ; x given,
0, l

0, l

1.

For …xed , we call arguments that solve this optimization problem c(w)and l(w) (omitting and x) as is common in the literature. De…ne y(w)w l(w).
The public good …nanced by the revenue raised through the income tax is
usually excluded from models of optimal income taxation due to the complexity
introduced, but here the cost of the public good will be used to derive therevenue requirements function. Let the cost function for the public good interms of consumption good be H(x), which is assumed to be C 2 .
The basic set of tax functions for the optimal income tax model is de…ned
as:T f : R+ ! Rj is measurableg
As is standard in the literature, for 2 T we shall generally write (y) todenote the tax liability of a worker earning income y.
These basic assumptions will be maintained throughout the remainder ofthis paper.
We will now use ideas inspired by Bergstrom and Cornes (1983) to obtaina unique Pareto optimal level of public good for each draw, so the revenue
requirement function is well-de…ned.
The major assumption that we make to obtain results, beyond requiringsu¢ cient revenue to …nance the public good for each draw, is that utility isquasi-linear and separable to a certain degree:15
15

In this case we are also using w as a taste parameter. That interpretation is quite
common in both the optimal tax literature and the literature on self-selection.

13

Assumptions:u(c; l; x; w) = c + b(l; w) + r(x; w)
We assume throughout that @b=@l

0 a.s., @ 2 b=@l2 < 0, @b(l;w)
@l

w,
l=1
2

b(1; w) b(0; w)
is weakly increasing in w, @r=@x > 0, @ r=@x2 <w, @b(l;w)@l0; dH(x)=dx > 0 and d2 H(x)=dx2 0.
From this, it follows that utility is strictly monotonic in consumption commodity (a good) and labor (a bad). The assumption @b(l;w)
w is a@l
l=1(weak) boundary condition on utility. There are constraints on labor supply (0 l 1) that could bind. The assumption implies that the constraint
w is the lower boundaryl 1 does not bind. In contrast, b(1; w) b(0; w)
condition in the context of quasi-linear utility. The assumption that @b(l;w)
is@lweakly increasing in w is the single crossing property. There are several more
remarks to be made. First, if we had more than one e¢ cient level of public good possible for given parameters, as is standard in public goods models
without the Bergstrom-Cornes type of assumptions, then we would have another dimension to vote over, namely the level of the public good. Generally
speaking, this would cause Condorcet cycles and thus no Condorcet winner.
Second, if we made utility more general, for example allowing the subutility
function r(x; w) to depend on consumption good c or labor l or both, then the
public good level and hence the aggregate revenue requirement function would
depend on the tax function, and that tax function would depend on the public
good level and hence the aggregate revenue requirement function. Thus, theaggregate revenue requirement would not be exogenous and likely not uniquelyde…ned. Third, when production of the public good is not constant returns toscale, there is a potential issue of pro…t distribution. However, when utility
is quasi-linear, pro…ts can be distributed to consumers without a¤ecting the…rst order conditions for optimization.
The bottom line is that something has to be done to shut down the feedback
between tax liabilities and the optimal level of the public good. The Bergstrom
and Cornes (1983) speci…cation is a natural starting point and actually is moregeneral than some of the separability assumptions used in the optimal nonlinear
income tax literature.

14

3

Voting Over Income Taxes

3.1

Preliminaries

The next step in our analysis is to …nd the Pareto e¢ cient level of public
goods provision for each draw using the Lindahl-Samuelson condition for our
specialized economy, a technique pioneered by Bergstrom and Cornes (1983).
Let (w1 ; w2 ; :::; wk ) 2 A, and let ci and li denote the consumption andlabor supply of the ith member of the draw respectively. Then production
possibilities for this given draw are:kX
i=1

Fix (w1 ; w2 ; :::; wk ) 2 A.

wi li

k
X

ci

(1)

H(x):

i=1

We de…ne an allocation to be interior if thekXassociated level of public good x satis…es x > 0 and H(x) <wi . Giveni=1

our assumptions, a necessary and su¢ cient condition for an interior ParetokX
optimum is:@r(x; wi )=@x jx=0 > dH(x)=dx jx=0 and there is x such that
i=1

k
X
i=1

@r(x; wi )=@x jx=x < dH(x)=dx jx=x and H(x) <kXwi . More usefully, we

i=1

shall assume the following su¢ cient boundary condition:
For all w 2 [w; w], @r(x; w)=@x jx=0 > dH(x)=dx jx=0 , and k @r(x; w)=@x jx=H 1 (kw) <dH(x)=dx jx=H 1 (kw) .
The …rst part of the condition says that for each type, at zero public goodlevel, the marginal willingness to pay exceeds the marginal cost. The second
part of the condition says that for each type, at public good level that is the
maximum possible for a draw of only the lowest type, the sum of marginal
willingnesses to pay is less than the marginal cost.
See Bergstrom and Cornes (1983, p. 1757) for a detailed explanation of
why we need to restrict the analysis to interior allocations. Brie‡y, the issue
is boundary optima where some consumer has no private good. In this case,there may be multiple e¢ cient levels of public good production. It is ruled out
by the second part of our su¢ cient condition. The entire su¢ cient condition
allows the use of the Lindahl-Samuelson condition with equality. Without
the …rst part, a unique e¢ cient level of public good (possibly 0) could still beobtained, but its characterization is more di¢ cult. Boundary Pareto optima
15

could occur, for example, if the sum of the marginal tax rate at zero income
and the marginal disutility of labor at zero labor supply is greater than 1 forsome type.
Lemma 1: Under the basic assumptions listed above, for any given draw
(w1 ; w2 ; :::wk ), there exists an interior Pareto optimal allocation; moreover, for
all interior Pareto optimal allocations, the public good level x is the same.
Proof: The Pareto optimal allocations are solutions to: max u(c1 ; l1 ; x; w1 )ui for i = 2; 3; :::k and subject to (1) where thesubject to u(ci ; li ; x; wi )maximum is taken over ci , li , (i = 1; :::; k) and x.16 The Lindahl-Samuelson
condition for this problem is:kX@r(x; wi )=@x = dH(x)=dx.

(2)

i=1

Given our assumptions on r and H, there exists a unique interior levelof public good x that solves (2). Since this equation is independent of ci
and li for all i, the interior Pareto optimal level of public good provision is
independent of the distribution of income and consumption for the given draw.
For the class of utility functions de…ned above we can thus solve for x as
an (implicit) function of (w1 ; w2 ; :::; wk ), and obtain the revenue requirement
functionR(w1 ; w2 ; :::; wk ) H(x (w1 ; w2 ; :::; wk )):Let F

T be the feasible set de…ned by:
F
k
X
i=1

2 T j a.s. for (w1 ; w2 ; :::; wk ) 2 A,)(y(wi ))

R(w1 ; w2 ; :::; wk ) .

Notice that the feasible set does not depend on the level of public good x, sincerevenue requirements for Pareto e¢ cient x must be satis…ed for all draws, andhence for the draw that is actually realized.
16

Bergstrom and Cornes (1983, Theorem 3) show that in their context, all interior Pareto
optima can be found by solving the utilitarian maximization problem. The same holds inour context, but the direct analysis of interior Pareto optima here is short and self-contained.

16With this in hand, a straightforward de…nition of majority rule equilibrium
follows: a majority rule equilibrium is a mapping from pro…les of preferences
into outcomes such that there is no pro…le in which n > k=2 agents can agree
on an outcome that is strictly better for all n of them.
It is important to make a couple of remarks about this de…nition. First,
in the case where aggregate revenue requirements R are taken as primitiveand exogenous, we can simply take the public good level x to be …xed as anargument in consumers’ utility functions. Then there is no voting over the
public good component of the bundle. Second, why do we focus on the …rst
best level of public good provision?
Since our income tax is distortive, it is reasonable to inquire why we should
impose the Pareto e¢ cient level of public goods provision for each draw. To
be speci…c, a second-best, lower level of public good provision could be used,
where the amount of revenue that must be collected, and thus the distortion
imposed by the income tax, could be reduced. For example, a draw of allhigh types will require a high level of public good provision and thus high per
capita taxes. To ensure that high types do not try to mimic low types, the percapita tax on low types will generally have to be higher than it would be if thedraw were known to consist of all low types and the …rst best level of public
good provision for that draw is used. Thus, excess revenue will be generated
from some draws, speci…cally of all low types, and it might be more e¢ cient
to reduce the income tax for all draws and types and provide a second best,
lower level of public good and consequent lower tax for each draw.17 The
problem with this argument is that, ex post, after labor is supplied and taxes
are collected, the workers would like to voluntarily contribute lump sum toraise the level of public good back to …rst best, say through a constant percapita tax. In other words, they would like to renegotiate. So we stick to…rst best public good levels to avoid this problem. Naturally, this involves
some cooperative behavior ex post.
This example also illustrates how the incentive constraints a¤ect the feasible set of tax instruments since the tax on low types is higher than at …rst
best, where types are known. What we have presented here is a benchmark,in that the …rst best public good level is unique for each draw, thus cutting o¤feedback between public good level and income tax selection. Such feedback
would make our analysis much more di¢ cult.
So to summarize, …rst, the types are drawn and each person learns their
17

These interesting comments belong to Paolo Piacquadio.

17

type. The government constructs the set of feasible tax systems, namely thosethat will fund every Pareto optimal level of public goods without knowledgeof types in the draw, and commits to not using information it receives fromvoting over this set in later stages of the game. The players in the draw voteover income taxes, choosing a majority rule winner. Then the income tax is
implemented, workers choose their labor supplies, and the public good level isselected as the one that is Pareto optimal or renegotiation proof for the draw.
For example, the government could run a VCG mechanism that determines
rebates to the consumers.
Of course, in cases where R is a primitive and there is no public good in
the model, this entire argument is super‡uous.

3.2

The Main Result

To simplify notation, we shall abbreviate derivatives of functions of only one
variable using primes, e.g. H 0 (x) dH(x)=dx. Beyond uniqueness of the e¢ cient public goods level for each draw, further properties of aggregate revenue
requirements R, and the implied minimal individual revenue requirements g,
are needed for our main result. To be speci…c, we make assumptions implyingthat R is a strictly concave function of the sum of the types in a draw. Since
R is derived from e¢ cient levels of public good production, assumptions on
both the consumer and producer side of the model are required:
De…nition: A utility function - production function pair (u; H) is called
manageable if the following conditions hold:u(c; l; x; w) = c + b(l; w) + w s(br(x)), H(x) = m rb(x), where rb0 (x) > 0,
rb00 (x) 0, s0 (r) > 0, s00 (r) < 0, 2s00 (r)2 > s000 (r) s0 (r), m > 0.
Theorem 1: Let k 2 and let (u; H) be manageable. Then there exists
a majority rule equilibrium that is the favorite tax system of a median of thedraw.
Proof: See the Appendix.
Examples covered by this theorem include the following:
A. u(c; l; x; w) = c + b(l; w) + w s(x), H(x) = m x, where rb(x) = x,s0 (x) > 0, s00 (x) < 0, s000 (x) 0, and m > 0.
B. u(c; l; x; w) = c + b(l; w) + 1 w x1 , H(x) = m x , with > 1,1,
1

> 0. In this case, rb(x) = 1 x , s(r) =

18

1

r

1

.

Example 1 below is covered by B, with

= 3,

= 2.

A complicating factor in proving such theorems is that aggregate revenue
requirements derived here depend not only on the …rst derivative of the costfunction, but on its level as well.
An important assumption used in the theorem is that utility from thepublic good is multiplicative in type w. Although this is a strong assumption,
it is often used in empirical work; see for example Bishop and Timmins (2019,equation (5)).

3.3

A Reiteration of the Key Assumptions
Functional form and regularity assumptions are imposed on utility andproduction. These are common in both the optimal income tax andpolitical economy literatures.
Robustness of the tax system to the draw.
The government doesn’t remember revelations from previous stages of
the game.
Assume that the Pareto e¢ cient level of public goods provision is implemented. This can be produced using a renegotiation proof outcome or
implementing a VCG mechanism for rebates at the end of the game.

3.4

Outline of the Proof of the Main Result

Before proceeding into details of the proof, we outline its structure:
All feasible g 2 G take a very speci…c form, and any two elements of G
cross at most once (Lemma 2).
The implementations of any given g 2 G are Pareto ordered, and thebest of these, called 2 T , is well-de…ned (Lemma 3).
Any two elements of T satisfy a single crossing property inherited from
G (Lemma 4).
For any k-tuple (w1 ; :::; wk ), de…neto be the optimal tax schedulefrom the point of view of the median type(s).

19Now suppose there is another strictly preferred by a majority. By
Lemma 4, andsatisfy single-crossing.
The net income functions induced by and also satisfy single crossing,whereas the indi¤erence curves induced over gross and after tax income
satisfy single crossing in type, so is not majority defeated by (Lemma5); see Gans and Smart (1996, Figure 1).

3.5

Single Crossing Individual Revenue Requirements

With a view toward future extensions of Theorem 1, we state some natural
assumptions on R that will be satis…ed by the revenue requirements derived in
the course of proving Theorem 1. For instance, we might wish simply to takerevenue requirements R as a primitive rather than derived from a model of apure public good. That generalization is covered in this section.
The …rst of these assumptions means that position in the draw (…rst, second, etc.) does not matter. All that matters in determining the revenue to be
extracted from a draw is which types are drawn from the distribution.
De…nition: A revenue requirement function R is said to be symmetricif for each k and for each (w1 ; w2 ; :::; wk ) 2 A, for any permutationoff1; 2; :::; kg, R(w1 ; w2 ; :::; wk ) = R(w (1) ; w (2) :::; w (k) ).
We will use the assumption that R is C 2 . This is not a strong assumption,
because the assumption that R is C 2 is generic in the appropriate topology;
that is, C 2 R’s will uniformly approximate any continuous R (Hirsch (1976,
Theorem 2.2)).18 We will also assume that R is smoothly monotonic:
De…nition: A revenue requirement function R is said to be smoothly
monotonic if for any (w1 ; w2 ; :::; wk ) 2 A, @R(w1 ; w2 ; :::; wk )=@wi > 0 for
i = 1; 2; :::; k.
This assumption requires that increasing the ability or wage of any individual in a draw increases the total tax liability of the draw. One could
successfully use weaker assumptions with this framework, but at a cost of
greatly complicating the proofs.19
18

This idea is also used to justify di¤erentiability in the smooth economies literature.
One particular case ruled out is the one of constant per capita revenues. In our model
this situation implies constant individual revenue requirements, i.e. a head tax, clearly anuninteresting situation even though it is …rst-best. It also includes the particular situation
19

20

A major step in our analysis that we have relegated to other papers that arecited in the bibliography, Berliant and Gouveia (2001) and Berliant and Page
(1996), is to implement the individual revenue requirement g using an income
tax, an indirect mechanism. A su¢ cient (and virtually necessary) conditionis that g be increasing in type, w.20 If g is anywhere decreasing in type, thenet income function can cut the indi¤erence curve of an agent, creating a gap
in the assignment of types to tax liability and ruining the implementation of gby an income tax. To use the …rst order approach to incentive compatibility,
for example, we must make further assumptions, namely the second orderconditions.21 These second order conditions are equivalent to the property
that g is increasing.
Turning next to aggregate revenue requirements R, we relate the property
of increasing R (@R(w1 ; w2 ; :::; wk )=@wi > 0 for i = 1; 2; :::; k) to increasing g 2
G . Suppose that there are w; w0 2 [w; w] with w0 > w. Then, by de…nition
Pof G , there is a draw (w1 ; w2 ; :::; wk ) with w = wi for some i and kj=1 g(wj ) =R(w1 ; w2 ; :::; wk ).
Now replace w with w0 , namely set wi = w0 , leaving
all other elements of the draw the same. Then R(w1 ; w2 ; :::; w0 ; :::; wk ) >R(w1 ; w2 ; :::; wk ). Since g is feasible, g(w1 ) + g(w2 ) + + g(w0 ) + + g(wk )P
R(w1 ; w2 ; :::; w0 ; :::; wk ) > R(w1 ; w2 ; :::; wk ) = kj=1 g(wj ), so g(w0 ) > g(w).
The next step is to introduce a set of assumptions where the elements ofthe set of feasible and minimal individual revenue requirements G are single
crossing,22 i.e. each pair of g’s will cross only once.23 They will be implied by
the postulates of Theorem 1. Thus, future generalizations of our main results
will likely use the lemmas below. The assumptions have collective revenue
requirements decreasing as a draw becomes more polarized.
De…nition: A revenue requirement function R(w1 ; :::wk ) is argumentwhere the government wants to raise zero …scal revenue. Constant per capita revenues canbe handled as a limit of the cases considered here.
20
The case g 0 (w) = 0 for some types w could be handled, but it creates some technicalproblems because g is not necessarily invertible.
21
We note that much of the recent literature on optimal taxation veri…es the second orderconditions ex post, not ex ante; see Kapiµcka (2013) for example.
22
In fact, under stronger assumptions, it is possible to show that the set of feasible andminimal individual revenue requirements is a singleton, rendering voting trivial. In thatanalysis, it’s useful to have the size of the draw, k, unknown to the planner as well. We
omit this analysis for the sake of brevity.
23
A G with single crossing g’s generates a trade-o¤ where raising more taxes from one
type of voter allows less revenue to be raised from another type, as in the conventional
income tax model.

21

additive if R(w1 ; w2 ; :::; wk )kXQ(wi ). Let Q0 denote XdQk

.

d

i=1

wi

i=1

The goal of the next lemma is to characterize minimal individual revenue
requirements, in other words elements of G , and show that they are single
crossing.
Lemma 2: Let k 2 and let the revenue requirement function R(w1 ; w2 ; :::; wk )be argument-additive with Q0 > 0 and Q00 < 0.
(a) There exists g : [w; w]2 ! R such that for all g 2 G , there exists
e for all w 2 [w; w] where g is de…nedw
e 2 [w; w] such that g(w) = g(w; w)
pointwise as follows:–For w
e

(w
e

(w
e

(w + w)=2:

A) g(w; w)
e = Q(k w)=ke
+ Q0 (k w)e (wB) g(w; w)
e = Q((k 1)w + w) ((kw) if w > w
e + (k 1) (w
e w).

w).

1)=k) Q(k w)
e + (k1) (we1) Q0 (k w)ew)
e if w

1) (w

w)
e

–For we < (w + w)=2:

e = Q(k w)=ke
+ Q0 (k w)e (wC) g(w; w)

D) g(w; w)
e = Q((k 1)w + w)w) if w < we (k 1) (w w)
e

((k

w)
e if ww
e + (kwe(k1)=k) Q(k w)
e + (k1) Q0 (k w)e(b) @g(w; w)=@w
e> 0 for each …xed w
e 2 [w; w] except at a …nite number ofpoints w 2 [w; w].
(c) 8w 2 [w; w], g(w; w)
e is single caved24 in w
e and attains a minimum atw
e = w.

(d) Any pair of functions in G will cross once: for any w;
e w
e0 2 [w; w], there
exist w;b w
b0 2 [w; w] such that g(w; w)
e > g(w; we0 ) implies g(w; w)
e > g(w; we0 ) forall w 2 [w; w),
b g(w; w)e = g(w; we0 ) for all w 2 [w;b w
b0 ] and g(w; w)
e < g(w; we0 )for all w 2 (wb0 ; w].
Proof: See the Appendix.

The implication of our feasibility approach in this case is that feasible tax
functions turn out to be parameterized by w.
e The intuition for this result is24

A function g is single-caved if

g is single peaked.

22quite simple. Consider (for the moment) the case where the distribution ofendowments is not bounded above or below. Since the revenue requirement Q
is concave, so is the per capita revenue requirement Q=k. But then, only thetangents to Q=k can be tax functions, since any linear combination of taxes has
to be greater than or equal to the per capita requirement. The w’e s correspondto the arguments of the per capita revenue functions at the tangency points.
The statement of the theorem is slightly more complex because this intuition
may not work near the bounds w and w.
Note that the marginal rates in branch B are lower than the rates inbranches A and C (the tangent branches), that in turn are lower than those inbranch D. In the argument-additivity case, concavity implies that per-capita
revenue requirements decrease with the polarization of the draw.
Notice that the shape of the distribution of endowments f does not havein itself any relevant information to predict the shape of the income tax schedules chosen by majority rule, since we have not used it anywhere. Revenue
requirements function R is all that is needed.25

4Single Crossing Optimal Tax Functions

Next, some results from the literature on optimal income taxation and implementation theory are used to construct the best income tax function that
implements a given individual revenue requirement. The discussion will be
informal, but made formal in the lemmas and their proofs.
The problem confronting a worker/consumer of type w given net income
schedule is maxl u( (w l); l; x; w). Using the particular form of utility that
we have speci…ed, the …rst order condition from this problem is ddy w +@b=@l =0. Rearranging,d@b(l; w) 1
=
.
(3)
dy
@l
w
For this tax schedule, we want the consumer of type w to pay exactly thetaxes due, which are g(w) for some g 2 G . If g is strictly increasing, g isinvertible. If we assume (for the moment) that g(w) is continuously di¤erentiable, then g 1 , which maps tax liability to ability (or wage), is well-de…ned
and continuously di¤erentiable. Substituting into the last expression,
25

With these preliminary results in hand, it would be possible to prove that a majority
rule equilibrium exists for the endowment economy where there is no choice of labor supply;
voting occurs over feasible taxes, then types are truthfully revealed to the planner for tax
purposes. Since this not our main aim, for the sake of brevity it is omitted.

23

d
=
dy

@b( g 1 (yy); g 1 (y

))

@l

1
g

1 (y

)

( ; y).

(4)

As in Berliant (1992), a standard result from the theory of di¤erential
equations yields a family of solutions to this di¤erential equation.26 Berliant
and Gouveia (2001) show that (4) has global solutions if g 0 > 0, g(w) 0.27
Of course, as L’Ollivier and Rochet (1983) point out, the second orderconditions must be checked to ensure that solutions to (4) do not involve
bunching, which means that consumers do optimize in (4) at the tax liability
given by g.28 This was done in Berliant and Gouveia (2001), where the Revelation Principle29 was used to construct strictly increasing post tax income
26

The method used above originates with the signaling model in Spence (1974), further
developed by Riley (1979) and Mailath (1987). Equation (4) is best seen as de…ning an indirect mechanism where gross income is the signal sent by each agent to the planner, much asin Spence’s model education is the signal sent to the …rm. However, …nding the equilibria of
this game is only part of the problem. The remaining part of the problem relates to implementation. By this we mean that the social planner’s problem is to de…ne reward/penalty
functions that induce each type of agent to choose, in equilibrium, the behavior the planner
desires of that type of agent. A reference closer to our work is Guesnerie and La¤ont (1984).
However, there is a di¤erence between our results and the other literature on implementation using the di¤erentiable approach to the revelation principle. The di¤erence is that in
the other literature the principal cares only about implementing the action pro…les of theagents (labor supply schedules in our model). In contrast, we consider the implementationof explicit maps from types to tax liability. That is, the principal cares about agents’types,
which are hidden knowledge. These maps from types to tax liability are not action pro…les,and are motivated by the ability to pay approach in classical public …nance. They play thesame role here as reduced form auctions play in the auction literature.
27
The relationship between individual and aggregate revenue constraints, if R is the …xed
aggregate revenue requirement, is given in the continuum of agents model by:
Z w

g(w)f (w)dw

Rw

The point of Berliant and Gouveia (2001) is exactly to study implementation of individual
revenue requirements in terms of an income tax, show that they are Pareto ranked, and to
examine the properties of individual revenue requirements functions or their implementations
that correspond to optimal income taxes when there is an aggregate revenue constraint.
28
That is, we have a separating equilibrium.
29
In (4) the planner …rst chooses a net income function (y), the agents then take thechosen net income function as given and maximize utility by selecting a gross income level y(or the corresponding level of labor supply). This is the implementation approach describedin La¤ont (1988). The Revelation Principle allows us to write an equivalent mechanism
where agents are simply asked to report their type w. It is easier to check second orderconditions of the problem for this direct mechanism. They essentially say that both pre and24

functions (w) = y(w) g(w) that implement g(w), where g 0 (w) > 0. Since
we then have that y(w) is invertible, we immediately obtain (y) = (w(y))
and (y) = g(w(y)).
It is almost immediate from this development that the set of solutions to(4) for a given g is Pareto ranked. We focus on the best of these for each given
g. De…neT

f 2 Tjandis a solution to (4) for some g 2 G , (y) = y(y),Pareto dominates all other solutions to (4) for the given gg .

Any element of T has the property that the marginal tax rate for the top
ability w consumer type is zero.
From a practical viewpoint, for instance in solving examples such as thosepresented here, the use of these techniques and in particular equation (4)makes sense. However, for the general theory, in our application we do nothave the conditions required by Berliant and Gouveia (2001); for example, thestandard boundary condition is not satis…ed due to the quasi-linear form ofutility. Thus, we use Berliant and Page (1996), which is more general than
Berliant and Gouveia (2001).
Lemma 3: If each minimal, feasible individual revenue requirement function is non-decreasing, then any such function is implementable by an income
@b(

y(w)

;w)

w
1. Moretax. Without loss of generality, for all w 2 [w; w], w1@lover, the implementations are Pareto ranked, and there is a best one under
the Pareto ranking with the property that the marginal tax rate for the top
ability consumer type, if it exists, is zero.

Proof: See the Appendix.
Remarks: The theorem says that any non-decreasing and feasible revenue
requirement function can be implemented by a continuum of tax schedules.
These tax schedules are Pareto ranked and furthermore a maximal tax schedule
under the Pareto ranking exists. The result on the top marginal tax rate isextended to non-di¤erentiable functions in Berliant and Page (1996), but is alittle complicated and, in fact, irrelevant to our purpose here.
The next step is to characterize a class of individual revenue requirements
for which we will be able to obtain results. This class contains the cases
discussed in Theorem 1 and may possibly include other sets of assumptions.
post tax incomes should be increasing functions of w. In our case they are strictly increasing
functions and there is no bunching.

25

De…nition: A collection E of functions mapping [w; w] into R is calledstrongly single crossing if each g 2 E is:
1. Continuous.
2. Twice continuously di¤erentiable except possibly at a …nite number ofpoints.
3. dg=dw > 0 except possibly at a …nite number of points.
4. Individual revenue requirements cross each other only once, i.e. for any
b<w
b0 such that g(w) > g 0 (w)pair g; g 0 2 E, there exist w;
b w
b0 2 [w; w], w
implies g(w) > g 0 (w) for all w 2 [w; w),
b g(w) = g 0 (w) for all w 2 [w;b w
b0 ]and g(w) < g 0 (w) for all w 2 (wb0 ; w].

De…nition: A collection T 0 of functions mapping non-negative incomes totax liabilities is called single crossing if for all ; 0 2 T , letting y( ); y 0 ( ) be
the gross income functions associated with and 0 , respectively, for incomesy1 ; y2 ; y3 2 y([w; w])\y 0 ([w; w]), y1 < y2 < y3 , (y3 ) < 0 (y3 ) and (y2 ) > 0 (y2 )0implies (y1 )(y1 ).

Lemma 4 proves that when individual revenue requirements are strongly
single crossing, the income tax systems in T cross at most once.
Lemma 4: Suppose that minimal individual revenue requirements, G ,are strongly single crossing. Then their best implementations T are single
crossing.
Proof: See the Appendix.
Remarks: The notion of strongly single crossing is the analog of condition(SC) of Gans and Smart (1996) in this speci…c context. Outside of y([w; w]),
can be extended in an arbitrary fashion subject to incentive compatibility,
for example in a linear way.
Lemmas 3 and 4 are used to prove Lemma 5:
Lemma 5: Suppose that R implies strongly single crossing minimal individual revenue requirements, G . Then a majority rule equilibrium exists.
Proof: See the Appendix.
26

Strongly single crossing is used intensively to prove this. It has the implication that induced preferences over tax systems have properties shared bysingle peaked preferences over a one dimensional domain. The winners will be
the tax systems most preferred by the median voter (in the draw) out of tax
systems in T .
The proof consists of two parts. The …rst part shows that there is a tax
schedule that is weakly preferred to all others by the median voter. The second
part shows that this tax schedule is a majority rule winner. This second part
could be replaced by Gans and Smart (1996, Theorem 1). But it would takeas much space to verify the assumptions of that Theorem as it does to prove
our more specialized result directly.
Can majority rule equilibrium tax functions and optimal tax functions be
the same? There are two answers, depending on whether the interpretation of
optimal is Pareto optimal or utilitarian optimal.
Consider …rst the interpretation as Pareto optimal. In the terminology of
the introduction, income taxes are third best, so our restricted set of feasible
income taxes is actually fourth best. Evidently any majority rule equilibrium
will be fourth best e¢ cient, since an objection of the grand coalition is necessarily an objection of a majority. The hard part here is proving existence ofa majority rule winner, not showing that the majority rule winner is Pareto
optimal.30
Consider next the interpretation of e¢ ciency as (possibly weighted) utilitarian optima, as is common in the optimal tax literature. The answer isrelated to Bowen’s (1943) theorem and the extension by Bergstrom (1979).
We focus on the quasi-linear and additively separable (in public goods andlabor supply) case. First consider the expenditure or public good side of the
problem, which is closer to this literature. Without getting into the technicalities, the main intuition for the primary result, namely Bergstrom (1979,
Theorem 3), is that if the marginal willingness to pay for the public good of themedian voter is equal to the mean marginal willingness to pay for the draw,then an e¢ cient allocation can be achieved by a majority rule equilibrium.
As Bergstrom remarks, this would be rare, and in our view is likely in thecomplement of a generic set of utility functions. But clearly it is possible.
Inspired by this literature, we can use a similar argument here, in the30

It is common in the optimal income tax literature to use a Pareto weighted utilitarian optimization problem to …nd Pareto optima. We do not use this technique because Guesnerie(1995) showed that the utility possibilities set might be nonconvex in such settings.

27

context of labor supply. We must modify it because we address utility levels
rather than …rst order conditions for optima, and because labor supply is a
private good. Assume that31b(l; w) = bb(l w) = bb(y)Then the (indirect) utility function of every agent (neglecting public good level,that is irrelevant at this stage of the game) is:y(y) + bb(y)The …rst order condition for incentive compatibility is:1d (y) dbb(y)
+
=0
dy
dy

The interpretation is that for each income tax , each type of agent earns thesame income and enjoys the same level of utility, though they work di¤erent
hours.32 That is because the optimized gross income y = w l is the samefor each type w, but since w varies, so does l. Hence, what the median voter
selects as a best tax out of a feasible set will also be unanimously best andthus (weighted) utilitarian best. Again, this seems to be a possible but rareoccurrence.

5

ExamplesNext we provide a pair of simple examples that can be solved.
Example 1: Takel2u(c; l; x; w) = c
H(x) =

w

x 2
2

x2
2

The marginal cost of the public good is x. The marginal willingness to payof type w for the public good is w x 3 , so the total marginal willingness tokX3
pay for the draw (w1 ; w2 ; :::; wk ) is x
wi . Setting this equal to marginal
i=1

31
32

Notice that bb implicitly is a function of w.
We neglect second order conditions here for brevity and simplicity.

28

cost to solve for the Pareto e¢ cient level of public good provision (that will
be unique), we obtain:! 14
kX
x (w1 ; w2 ; :::; wk ) =wi
i=1

A reason why the isoelastic case might be interesting comes from the fact that
it is a suitable case for the purpose of carrying out empirical tests of the model,
given that the correct way to aggregate abilities (or tastes) in this particularcase is simply to sum them.
The aggregate revenue requirement function is:
v
u k
X
1u
wi
R(w1 ; w2 ; :::; wk ) = H (x (w1 ; w2 ; :::; wk )) = t
2
i=1

Next, take w = 1, w = 2, and let w
e be the median type of a draw. Then
as in Lemma 2, if k
2 and 1:5we
2, the minimal individual revenue
requirements are indexed by w
e and given by33
p 1
1
1
1
g(w; w)e =(k w)
e 2 =k + (k w)
e 2 (w w)
e
2
4
#
r
p "r
w
e
1
=
+
w
4
k
kw
e

In an endowment economy, this is the tax on endowments most preferred bytype we among those satisfying the aggregate revenue constraints. In thisparticular case, it is a linear tax. The next step is to implement it in an
optimal income tax economy.
Applying the …rst order approach to incentive compatibility34 given in thedi¤erential equation (4) above, ddy = 1 ddy , and g 1 (y) = w, the income
tax function is given by the solution to:35
d
=1
dy

2y
w2

Inverting g( ; w)e and solving for w in terms of ,
s
kw
e
w=4
w
e
33

34

0.

To keep calculations simple, we focus on draws where the median is at least 1:5.
w)
e
The second order condition for incentive compatibility will be satsi…ed because @g(w;
>
@w

35

Although we know that a solution exists and through any point it is unique, actuallysolving the ODE explicitly is another matter entirely.

29

so

d
=1
dy

2y
h q
4 kwe

w
e

i2

This ordinary di¤erential equation has a solution through every point. To
choose the best of these, take the one that has the marginal tax rate zero forthe top type w = 2. For the top type, it is the solution that goes throughpe ; 2 . This will be the majority rule equilibrium for any( ; y) = 4pkwe [2 + w]
draw with median we
1:5. Several comparative statics can be derived inthis example: The total tax paid by the top type is decreasing in the sizeof the economy (k), but increasing in the marginal utility of public good ( ).
The income of the top type is independent of both the size of the economy (k)and the marginal utility of the public good ( ). This is a consequence of theseparability assumption.
Example 2: One point of this example is that although we will restrict toquasi-linear utility functions for the general theory, that might not be necessary. Takeu(c; l; x; w) = min (c; w [1

l])

w

x 2
2

x2
2

H(x) =

e is unchanged
The aggregate revenue requirements function, and thus g( ; w),from Example 1. Setting c = w [1 l],
y

=w

y

Therefore,(y) = 2y= 2yand thus
(y) =

w
s
4

kw
e

2y + weq
1 + 4 kwe(y) + weComparative statics are apparent.
Remarks: The single crossing of individual revenue requirements resultsfrom the combination of: the assumptions on utility and cost of the public good,
and the idea that the aggregate revenue requirements must be satis…ed for any
draw. We prove in Lemma 4 that when we implement the individual revenue
30

requirements and impose second best e¢ ciency, the single crossing property isinherited by the income tax implementations. One common feature of our individual revenue requirement functions is that there is a switch point, indexed
by w
e in our examples here, that represents the individual revenue requirementthat minimizes that type’s tax liability among all individual revenue requirements satisfying the aggregate revenue requirements for all draws. This is not
actually necessary for our general results, and is not used in the proofs once
we obtain single crossing of individual revenue requirements. However, as
seen from Example 1, provided that g is strictly increasing, the optimization
point for type w under the (optimal) income tax framework will corresponde
Therefore, using the standard diagrams from optito tax liability g(w; w).
36
mal tax theory, the majority rule equilibrium will correspond to the best
implementation (solution to the ordinary di¤erential equation) of the revenue
requirement function that minimizes the tax liability of the median type ofe Thus, the switch point is inherited by the optimal income
the draw, g(w; w).
tax implementation of the individual revenue requirements. The fact that we
do not use the switch point once we have single crossing of individual revenue
requirements allows room for expansion of our results.

6

Conclusions

Two di¤erent but related issues deserve some discussion at the outset. The …rst
is whether information on the likelihood of each draw can be used. The second
is how to deal with possible excess revenues. As for the opposite situation ofinsu¢ cient revenues, the reader should note that imposing a penalty for not
meeting the requirement simply results in a new revenue requirement function.
We …rst discuss the information issue. One obvious possibility would beto de…ne as feasible all individual revenue requirement functions that generate
an expected revenue equal to or larger than the expected collective revenue
requirement. In contrast with what we use, this would be a single constraint
rather than a constraint for each draw. The problem with this notion is thatsingle crossing conditions for individual revenue requirements would likely failto be satis…ed for most cases, including the ones studied in this paper. But
one could consider weakening our feasibility restriction and still have enough“bite”to generate single crossing g’s. Here is a suggestion:
One option is to use a class of probability measures over draws and constrain
36

See Seade (1977).

31

the expectation of revenues for each probability measure. Expected revenue
according to f would be one particular member of this class. The class could
be chosen to generate a continuum of constraints, binding enough for the single
crossing result to survive, and we would be back to our initial setup althoughwith di¤erent feasibility conditions. This is similar to a model of government
behavior using ambiguity aversion or Knightian uncertainty. Perhaps this
could be justi…ed as a way to aggregate risk averse voter preferences overbudget de…cits.
We now address the issue of excess revenue. Consider …rst the case of utility
quasi-linear in consumption good that we have used throughout this paper. It
is possible to return the ex post excess revenue in a lump-sum fashion, as there
are no income e¤ects. Or the government could use them for another purpose,such as production of yet another public good.
When we consider general preferences and technologies the problem becomes more di¢ cult. Clearly, the excess revenue cannot be returned to taxpayers in a lump sum fashion, as it will a¤ect their behavior in optimizing
against the income tax. However, once we deviate from quasi-linear utility,
other issues would arise before we get to this point, most importantly thepresence of multiple Pareto optimal levels of public good provision. From thepoint of view of applications, analysis of these more general models will bemuch more di¢ cult.
The bottom line is whether the alternative models have more to o¤er. Is
it better to restrict ourselves to …xed revenue and voting over a parameter ofa prespeci…ed functional form for taxes (as in the previous literature), which
are also generally Pareto dominated, or is the model proposed here a usefulcomplement? Di¤erences of opinion are clearly possible.
We note here that unlike much of the earlier literature on voting over linear
taxes, the majority equilibria are not likely to be linear taxes without strong
assumptions on utility functions and on the structure of incentives. The reason
is simple: in the optimal income tax model, Pareto optimality requires that the
top ability individuals face a marginal tax rate of zero.37 Therefore, poll taxes
are the only linear taxes that could possibly be equilibria. In our model, such
37

We know of only one case where an optimal tax is linear: Snyder and Kramer (1988).
But this and other results derived in that paper are due to the use of a peculiar model thatdeparts signi…cantly from the other models used in the study of income taxation. There
are no income nor substitution e¤ects on e¤ort induced by taxation up to the point where
workers switch to the underground sector, and from that point on the same holds since, by
de…nition, income realized in the underground sector is not taxed.

32

taxes are not generally majority rule equilibria, since consumers at the lower
ability end of the spectrum will object. All majority rule equilibria derived
in this paper are second best Pareto optimal (for a given individual revenue
requirement), and hence satisfy the property that the top marginal tax rate iszero.
In that sense, the results obtained here are a step forward relative to Romer(1975) and Roberts (1977). In another sense, they also improve on Snyder andKramer (1988) by using a standard optimal income tax model as the framework
to obtain the results.
Although majority rule equilibrium taxes in our model have the property
that the top marginal tax rate is zero, a property in common with second best
Pareto e¢ cient taxes, in our model the majority rule equilibrium taxes satisfy
a revenue requirement for each draw (or equivalently, an individual revenue
requirement), stronger than a single aggregate revenue requirement. Thus, ourmajority rule equilibrium taxes are not necessarily second best Pareto e¢ cient.
Notice also that our majority rule equilibrium di¤ers by draw, whereas thesecond best Pareto e¢ cient tax systems for the aggregate distribution fromwhich the population is drawn does not. However, if we restrict to tax systems
and public good levels that are feasible in our sense, majority rule equilibria
will be Pareto e¢ cient for the draw among the feasible tax systems in our
restricted sense; the latter are third best.
There are a few strategies that may be productive in pursuing research on
voting over taxes.38 One strategy is to use probabilistic voting models suchas in Ledyard (1984). Another is to take advantage of the structure built inthis paper and, with our results in hand, look at multi-stage games in which
players’actions at the earlier stages might transmit information about types.
Of course, it might be necessary to look at re…nements of the Nash equilibrium
concept to narrow down the set of equilibria to those that are reasonable (atleast imposing subgame perfection as a criterion).
A two-stage game of interest is one in which k is …xed and each player ina draw proposes a tax system in T (simultaneously). The second stage of thegame proceeds as in the single stage game above, with voting restricted to only
those tax systems in T that were proposed in the …rst stage.
A three stage game of interest is one in which k is again …xed and theplayers in a draw elect representatives and who then propose tax systems and38

For a discussion of dynamics in both the general literature and a model closely related
to this one, see Berliant and Boyer (2021).

33

proceed as in the two stage game (see Baron and Ferejohn (1989)).
Work remains to be done in obtaining comparative statics results, as in
the examples. Finally, the predictive power of the model will be the subjectof empirical research. That will certainly be the focus of future work.

7

Appendix

7.1

Proof of Lemma 2

For part (b), it is straightforward to prove by direct calculation that 8w
e 2
e is continuous everywhere, continuously di¤erentiable in w (ex[w; w], g(w; w)
cept where the transition is made between branches), and has positive derivative in w. Below, we will also use the fact that for 8w 2 [w; w], g(w; w)
e iscontinuously di¤erentiable in w
e (except where the transition is made between
branches).
To provide some intuition for the next part of the proof, it is important
to inquire: Where does the relation w Q w
e + (k 1) (we w) (for branches
A and B) come from? It simply describes when it is possible or impossible
to construct a draw containing w so that the average ability is equal to w.
e
Beyond that, the function Q(kw)=k will be crucial in the proof below.
Now we proceed to prove part (a). First we shall show that g( ; w)e isfeasible and minimal for each w,
e in other words it is in G , and then showthat there are no other functions that are feasible and minimal. Fix a draw
(w1 ; w2 ; :::; wk ). To consider branches A and B separately to begin, de…ne:IL
IH

fi j wiw
e + (kfi j wi > w
e + (k

1) (w
e

w)g

1) (w
e

w)g

Let kL be the number of elements of IL . Further, let
1 X
wL
wi
kL i2I
L

Since Q is concave, if wiw
e + (k 1) (weg(w; w)e is the tangent line to Q(kw)=k at w = w:eg(wi ; w)
e = Q(k w)=ke
+ Q0 (k w)(we iw), branch A yields thatw)e

Therefore, since g(w; w)
e is linear in w on branch A,X
g(wi ; w)e = kL g(wL ; w)
e
i2IL

34

Q(kwi )=k.

So if for all i = 1; :::; k, i 2 IL , thenkX
i=1

g(wi ; w)e = k g(wL ; w)ekX
Q(kwL ) = Q(wi ) = R(w1 ; :::; wk )i=1

In particular, if we set the draw to be wi0
min fwi ; w
e + (kthenkXg(wi0 ; w)eR(w10 ; :::; wk0 )

1) (w
e

w)g,
(5)

i=1

For branch B, let w
bj 2 [we + (k 1) (we w); w] for j 2 IH and w
bj = wj for
j 2 IL . Using concavity of Q and setting i 2 IH ,@ X
g(wj ; w)e jwj =wbj = Q0 ((k 1)w+w
bj )@wi j2I
H

k
X
@
Q(
w
bj ) =Q(wj ) jwj =wbj 8j@wi j=1
j=1
0kX

Integrating via a line integral and applying Stokes’theorem,Xeg(we + (k 1) (we w); w)]e[g(wj ; w)
j2IH=

X Z wjj2IH@
g(wbj ; w)d
e w
bj
bj
w+(k
e
1) (w
e w) @ w

X Z wj

j2IH

0

Q(

w+(k
e
1) (w
e w)

k
X
= Q(
wj )
j=1

= R(w1 ; :::; wk )

Q(

k
X

k
X
j=1

wj0 )

w
bj )dwbj

j=1

R(w10 ; :::; wk0 )Adding this inequality to (5) yields feasibility of g( ; w).
eFeasibility of
branches C and D is proved similarly.
We now prove that the g( ; w)
e are minimal among those feasible. Consider
branch A. Clearly, if a draw consists of k individuals of type w,
e g(w;e w)
e isminimal by de…nition. To show that g(w; w)
e is minimal, suppose the opposite.
Take h(w) with h(w)g(w; w)e and with strict inequality for some w1 2
[w; w+(ke1) (we w)] (the case w1 2 (w+(ke1)(we w); w] will be consideredin the next paragraph). It is possible to construct a draw (w1 ; w2 ; :::wk ) withe1) (we w)] for i = 2; :::k. Then, R(w1 ; w2 ; :::wk ) =mean we and wi 2 [w; w+(k
k
k
k
X
X
X
Q(k w)e =g(wi ; w).
e Buth(wi ) <g(wi ; w),e so h(w) is not feasible.
i=1

i=1

i=1

Similar reasoning holds for branch C.

35

Now consider branch B. Take h(w) with h(w) g(w; w)
e and with strict
inequality for some w1 2 (we + (k 1)(we w); w]. The logic used for branches
A and C does not hold in this case: it is not possible to …nd k 1 ability
levels in this interval to construct a draw with mean w.
e Consider a draw with
wj = w for j = 2; 3; :::k. For this draw,kX
i=1

g(wi ; w)e= Q((k 1) w + w1 ) ((k 1)=k) Q(k w)
e + (kkX+
Q(k w)=ke+ Q0 (k w)e (w w)e1) Q0 (k w)
e (w
e

w)

1) Q0 (k w)
e (w
e

w)

i=2

= Q((k 1) w + w1 ) ((k 1)=k) Q(k w)
e + (kk 1+
Q(k w)e(k 1)Q0 (k w)(
e w
e w)
k
= Q((k 1) w + w1 )Hence, for the draw (w1 ; w; :::; w),Pk

i=1 h(wi ) <

k
X
i=1

g(wi ; w)
e = Q((k1)

w + w1 ) = R(w1 ; w; :::; w), so h is not feasible. Similar reasoning holds for
Branch D.
Next, suppose there is h 2 G that is not of the form given in the statementof the Lemma. Then for each w
e 2 [(w + w)=2; w] there is some w 2 [w; w]
with h(w) < g(w; w).
e Fix any such w.
e Repeating the arguments just above
showing that g(w; w)
e is minimal in the cases of Branch A (w w
e + (k 1)(we w)) and Branch B (w > w
e + (k 1) (we w)), it follows that h is not
feasible, a contradiction. Similar reasoning hold for Branches C and D.
Next we prove part (c). To prove single cavedness in w,
e one need onlydi¤erentiate g(w; w)
e with respect to the parameter w,
e …xing w. For branches
A and C we obtain:@g(w; w)e
= Q00 (k w)e k (w@we

w).
e

The derivative above is positive if w < we and negative for w > w.
e
For branch B we have:@g(w; w)e= k (k@we1) Q00 (k w)e (wew) < 0.

@g(w; w)e= k (k@we1) Q00 (k w)e (wew) > 0.

which applies only for w > w
e + (kFinally, for branch D we get:

1) (w
e

36

w). Otherwise Branch A applies.

which applies only for w < w
e + (k 1) (we w). Otherwise Branch C applies.
The results above imply that arg minwe g(w; w)
e = w.
Finally, consider part (d). We claim that these g’s are single crossing. To
see this, …rst note that from the de…nition of g(w; w)
e in the statement of thee@g(w;w)
Lemma, direct calculation yields that @w is weakly decreasing in w
e for each0000
e and g(w; we ) cross twice, there exist w; w ; w 2 [w; w],
w. Therefore, if g(w; w)
000
w < w < w such that g(w; w)
e = g(w; we0 ), g(w0 ; w)e 6= g(w0 ; we0 ), g(w00 ; w)
e =
00
0
0
0
0
g(w ; we ). But this cannot happen in each case: w
e = w,e w
e < w,e w
e > w.
e
The …rst case is obvious, and the other cases use the fundamental theorem of
w)
e
w
e0 )
and @g(w;.
calculus applied to integrals of @g(w;
@w
@w

7.2

Proof of Lemma 3

We employ Berliant and Page (1996) Theorems 1 and 2 to prove Lemma 3.
Notice …rst that since H(x) 0, if we focus on draws of identical individuals,R
0 and g0. So we restrict to taxes that are non-negative. The key
assumptions to be veri…ed are the boundary conditions and a single crossing
property. We will be precise. For the quasi-linear utility we use, we mustverify the following boundary conditions on the utility function, numbered (3)and (4) in that paper:
For all w 2 [w; w], for all 0t0y, there exists y 0 , t0

t

y0

y
y
t0 + b( ; w) y t + b( ; w)w
w
For all w 2 [w; w], for all 0 y w and 0 t y, for all 0
y0

there exists 0

t0

y with (6)00yt0 + b( ; w)w

y 0 with y 0yy 0 < y,(7)

yt + b( ; w)
w

Fix w 2 [w; w]. For condition (6), we must show that there is y 0y0such that y 0 t0 + b( yw ; w)
y t + b( wy ; w). We will show, in addition,that without loss of generality, w1y(w)
1 @b( w ;w)0@b( yw ;w)@limplementation, w
1.
@ly
1 @b( w ;w)
Next we show that if w @l<yt + b( wy ; w) = yb

1. This will imply that for the
1, there exists yb, tyb@b( w ;w)t + b( wyb ; w) and w1 @l

yb < y with

1.
First, we know from the boundary condition that b(1; w) b(0; w)
w.
R 1 @b(l;w)
Using the fundamental theorem of calculus, 0 @l dl = b(1; w) b(0; w).
Since @b(l;w)< 0 is weakly increasing in w, b(1; w) b(0; w) is weakly increasing@lw.
in w, so b(1; w) b(0; w)w

37

Rearranging, w + b(1; w)b(0; w).

@b( yw ;w)
=
@l
y 00
@b( w ;w)
Note that w1 @l

Let y solve

w if thesolution is positive, set y = 0 otherwise.
< 1 for
y 00
00
00
00
y > y , so y + b( w ; w) is decreasing in y . The worst utility level is given by
y 00 = w, or w + b(1; w) b(0; w) using the start of this paragraph. So by theintermediate value theorem, there is a yb with y + b( wy ; w) = yb + b( wyb ; w) and@b( yb ;w)

w
y , so w1 @l
1, and without loss of generality we can take y = yb.
Next we prove (6). For t0 = t, this is trivial (take y 0 = y), so take0 t0 < t y.
First we show that (6) holds for t0 in a neighborhood of t, and then expand
to any t0 .
2
1 @b(l;w)< 0 and w1 @b(l;w)
1, w1 @b(l;w)
1, so for
Using @ b(l;w)
@l2
@l
w
@l
@l
l=1
0
ye < ye,Z ye
Z ye
1 @b( wyb ; w)
ye0
ye
1db
y = ye0 ye
(8)
db
y>
b( ; w) b( ; w) =
w
w
@l
ye0 w
ye0

yb

Now take y 0 with 0t0y0 < yw. Using (8),yy00y t + b( ; w) > y t + b( ; w)w
w
0
So for a size neighborhood of t, for all t with t
t0 < t in that neighbor2
hood, the conclusion of (6) holds. In fact, given the assumption @ b(l;w)< 0,@l2
the size of the neighborhood expands as y decreases, so that we can use thesame size as applies to y and t. Therefore, given any t0 < t and y, we can= t2 with t0k = t0 so that
construct a …nite chain y1 = y, y2 = y 0 = t, t01 = t
the conclusion of (6) holds for each element of the chain and by transitivity
holds for the original data, namely y, t, and t0 .
The proof of (7) proceeds in the same manner. As shown above, for y 0 < y,yy0t + b( ; w) > y 0 t + b( ; w)
w
w
0 0
0
0
0
Thus, for all (t ; y ) where tt and tyy in a neighborhood of (t; y),the conclusion of (7) holds. Then construct a …nite chain as before.
The single crossing condition in Berliant and Page (1996) has two parts:yw0 > w, y > y 0 and y 0

t0 + b(y0 0;w ) > yw0t + b(y; w0 )w0y0
yt0 + b( ; w) > y t + b( ; w)
w
w
0
y
y
w0 > w, y 0 > y and y 0 t0 + b( ; w) > y t + b( ; w)
w
w
0
y
y
implies y 0 t0 + b( 0 ; w0 ) > y t + b( 0 ; w0 )
w
w

(9)

implies y 0

38

(10)

A su¢ cient condition for (9) to hold is:y0b( ; w)w

y0 0b( 0 ; w )wyb( ; w)wb(y; w0 )
0
w

or

y0
y
y0 0
y
0
b( 0 ; w ) b( 0 ; w ) b( ; w) b( ; w)
w
w
w
w
We now proceed to prove that.
Z y=w0Z y=w0@b(l; w)y@b(l; w0 )
y
y0 0
y0
0
b( 0 ; w ) b( 0 ; w ) =dl
dl = b( 0 ; w) b( 0 ; w)
w
w
@l
@l
w
w
y 0 =w0y 0 =w0ZZ yy@b( wyb0 ; w) 1@b( wyb ; w) 1
y
y0
db
y
=
b(
;
w)
b(
; w)
db
y
=
@l
w0
@l
w
w
w
y0
y0
A su¢ cient condition for (10) to hold is:
b(

y0 0
;w )
w0

y0
b( ; w)wb(y; w0 )w0yb( ; w)
w

or

y0 0
y
y0
y
0
;
w
)
b(
;
w
)
b(
; w) b( ; w)
0
0
w
w
w
w
This follows from the argument just above, replacing y with y 0 and y 0 with y.
b(

7.3

Proof of Lemma 439

Let g 0 and g 00 be the elements of G , and let ( 0 ; y 0 ; 0 ) and ( 00 ; y 00 ; 00 ) be
the tax, gross income, and net income functions associated with g 0 and g 00 ,respectively. We ignore x and r(x; w) in this proof, due to additive separability
and the structure of the game. The proof is by contradiction. Suppose that
there exist incomes y1 < y2 < y3 with 0 (y1 ) < 00 (y1 ), 0 (y2 ) > 00 (y2 ) and0(y3 ) < 00 (y3 ). Then by the intermediate value theorem applied to utility
0 0di¤erences as a function of w, there exists wa such that y 0 (wa )(y (wa )) +00 00b(y 0 (wa )=wa ; wa ) = y 00 (wa )(y (wa )) + b(y 00 (wa )=wa ; wa ), y 00 (wa ) > y 0 (wa ),and 0 (y 0 (wa )) < 00 (y 0 (wa )). Moreover, g 0 (wa ) = 0 (y 0 (wa )) < 00 (y 0 (wa )).
@b(y 00 (w)

;w)

00

00

w
1 and ddy = 1 ddy
0 and since
Now by (3) and Lemma 3, w1@l00 0
y 00 (wa ) > y 0 (wa ), g 00 (wa ) = 00 (y 00 (wa ))(y (wa )) > 0 (y 0 (wa )) = g 0 (wa ).
There also exists wb > wa with y 0 (wb ) 0 (y 0 (wb ))+b(y 0 (wb )=wb ; wb ) = y 00 (wb )00 00(y (wb )) + b(y 00 (wb )=wb ; wb ), y 0 (wb ) > y 00 (wb ), 0 (y 00 (wb )) > 00 (y 00 (wb )), and
00 0(y (wb )) > 0 (y 0 (wb )). Hence 0 (y 00 (wb )) > 00 (y 00 (wb )) = g 00 (wb ) and since
0 00
y 0 (wb ) > y 00 (wb ), g 0 (wb ) = 0 (y 0 (wb ))(y (wb )) > 00 (y 00 (wb )) = g 00 (wb ).

39

To see how this critical proof works, it is useful to draw the graphs from optimal taxation,
net income as a function of gross income, that are standard in the literature; see Seade (1977).

39

Using strongly single crossing, g 0 (w) > g 00 (w).
By construction of T , 0 (y 0 (w)) > 00 (y 00 (w)). Note that since the marginal tax rate at y 0 (w) and y 00 (w) is zero, what we have are essentially lumpsum taxes at the top ability level. Hence, y 00 (w) 00 (y 00 (w))+b(y 00 (w)=w; w) >0 0
y 0 (w)(y (w)) + b(y 0 (w)=w; w). Zero income e¤ects implies y 0 (w) = y 00 (w).
Moreover, 0 (y 00 (w)) > 00 (y 0 (w)). Since 00 (y 0 (wb )) > 0 (y 0 (wb )), there exists y , y 00 (w) > y > y 0 (wb ) with 0 (y ) = 00 (y ), so there exists wc withy 0 (wc ) 0 (y 0 (wc ))+b(y 0 (wc )=wc ; wc ) = y 00 (wc ) 00 (y 00 (wc ))+b(y 00 (wc )=wc ; wc ),y 00 (wc ) > y 0 (wc ), and 0 (y 0 (wc )) < 00 (y 0 (wc )). As above, g 0 (wc ) = 0 (y 0 (wc )) <00 0(y (wc )) and since y 00 (wc ) > y 0 (wc ), g 00 (wc ) > g 0 (wc ).
This contradicts strongly single crossing. So the hypothesis is false, andthe lemma is established.

7.4

Proof of Lemma 5

De…nition: Let C 1 be the space of continuously di¤erentiable functions (withdomain [w; w] and range R) endowed with the uniform topology. We consider
T to be a subset of this space by extending any 2 T to the whole domain,
if necessary, in a C 1 and linear fashion.
Fix 2 T . First we claim that 0 ddy 1. From @b=@l 0 a.s., (3), andLemma 3, we obtain that 0 ddy = w1 @b
1. Since ddy = 1 ddy , the claim is@lproved. So every 2 T is Lipschitz in income with constant 1, and thus T isequicontinuous. Since k g(w) R(w; w; :::; w) 0, T is also norm bounded
by w. Using Ascoli’s theorem (see Munkres (1975, p. 290)), T (the closure of
T in C 1 ) is compact.40
Fix k and let (w1 ; w2 ; :::; wk ) 2 A. For any 2 T , let v( ; w) = maxy u(y(y); y=w), the utility induced by the tax system for type w. It is easy toverify that for each w, v( ; w) is continuous in its …rst argument.
Letbe a maximal element of T using v( ; wM ) as the objective, where
wM is the median ability level in (w1 ; w2 ; :::; wk ) if k is odd, and wM 2
[wk=2 ; wk=2+1 ] (where the wage rates are ordered in an increasing fashion) if
k is even. Using Lemmas 2 and 3,
2T .
Now suppose there exists 2 T such that there is a subset D of fw1 ; w2 ; :::; wk gwith v( ; w) > v( ; w) for all w 2 D and where the cardinality of D is greaterthan k=2. Then using Lemma 3, we can take to be in T without loss of40

An alternative proof, pointed out by a referee, would show that the best implementationsof g(w; w)
e are continuous in w.
e40

generality. Using Lemma 4, and are single crossing, or alternatively, theirafter tax income functions y(y) and y(y) are single crossing. Noticealso that, due to non-inferiority of consumption good, indi¤erence curves (asa function of gross and after tax income) are single crossing in w; see Seade(1977, footnote 8). Thus, there exist intervals W; W 0[w; w] such that W
0c be the smallest interval (in theand W partition [w; w] and D W . Let W
c and its complement W
c 0 are both intervals,
sense of set inclusion) such that Wc.
c and Wc 0 partition [w; w], and D W
W
c . Hence D cannot contain a majority
Then by de…nition of , wM 2
= W
of the draw, a contradiction. Hence the hypothesis is false andcannot be
defeated by any other feasible tax system.

7.5

Proof of Theorem 1

For a draw (w1 ; w2 ; :::; wk ) 2 A, the Lindahl-Samuelson condition for thismodel is:kX
wi s0 (br(x)) rb0 (x) = m rb0 (x)i=1Hence,

2

13

0

C7
6
B
6
B m C7C7
6
B
x = rb 1 6s0 1 B k
C76
B X C7
4
@
w A5
i

i=1

and thus

0

1

B
C
B m C
B
C
R(w1 ; w2 ; :::; wk ) = m s0 1 B k
C
BX C
@
wA
i

i=1

Hence, R is argument additive. Computing the …rst derivative,

d

dR
k
X
i=1

=
wi

m2

0
B
B
@

0

B
m
BX
k
@

B
B
s00 s0 1

41

i=1

1
11
wi

CC
CC
CC
AA

>0
k
X
i=1

wi

!2

Computing the second derivative,
d2 R
k
X

d

i=1

wi

!2

0

11

0

B
B
B
B m
2s00 Bs0 1 B X
@
@ k

wi

i=1

CC
CC
CC
AA

2

= m

where

2

k
X

wi

i=1

!

0

i=1

0

+ s000 (r)

i=1

11

0

i=1

wi

0

B
B
B
B m
+ s000 Bs0 1 B X
@
@ k

6 B
B
6 00 B 0 1 B m
6s Bs B X
4 @
@ k

k
X

4s00 (r)

wi

i=1

2

2s00 (r)
= m2

k
X

!

wi

CC
CC
CC
AA

k
X
i=1

wi

11
wi

CC
CC
CC
AA

!2

32

0

B
B
B
B
00
s Bs 0
B
B
@

0m

7
7
7
5

m
s00 (r)

!2 32
5

0

1

B
C
B m C
B
C
r = s0 1 B k
C
BX C
@
wA
i

i=1

Thus,

d2 R
d
2s00 (r)

k
X

wi

i=1
k
X

i=1
00

!2 < 0 if and only ifwi!

< s000 (r)mors00 (r)2s (r)2 > s000 (r) s0 (r)The last expression holds by assumption. Therefore, R is argument additive
with negative second derivative. The result then follows from Lemmas 2 and
5.
42

11

CC
B
CC
B
CC
B
CC
m
1B
CC
B k
B X CC
B
CC
@
w AA
i

i=1

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47