Optimal nonlinear income taxation with a finite population

Journal of Economic Theory 132 (2007) 548–556www.elsevier.com/locate/jet

Notes, Comments, and Letters to the Editor

Optimal nonlinear income taxation with a finitepopulation

Jonathan Hamilton∗, Steven SlutskyDepartment of Economics, Warrington College of Business, University of Florida, Gainesville, FL 32611-7140,

USA

Received 8 January 2004; final version received 26 July 2005Available online 13 September 2005

Abstract

In the standard optimal income taxation problem, tax payments depend only on each consumer’sown actions. Piketty [J. Econ. Theory 61 (1993) 23–41] shows that, if one individual’s tax scheduledepends on others’ actions and the government knows the exact ability distribution, it can implementany undistorted allocation as the unique revelation game outcome. If some individuals misrevealtheir types, Piketty’s mechanism may assign infeasible allocations. We require that tax schedulesmust balance the government budget for every possible vector of revelations. When individuals revealtheir type by simple announcements, all undistorted allocations can be still implemented, even withoff-equilibrium feasibility constraints.© 2005 Elsevier Inc. All rights reserved.

JEL classification: D82; H21

Keywords: Optimal income taxation; Mechanism design; Balanced-budget mechanisms

1. Introduction

Since Mirrlees [8], the optimal income taxation literature has incorporated his essentialinsight that the government cannot perfectly identify individuals’ types and assess personal-ized lump-sum taxes. As a consequence, tax schedules must satisfy incentive compatibility(or self-selection) constraints. Such tax schedules are formulated so that an individual’s

∗ Corresponding author. Fax: +1 352 392 7860.E-mail address: [email protected] (J. Hamilton).

0022-0531/$ - see front matter © 2005 Elsevier Inc. All rights reserved.doi:10.1016/j.jet.2005.07.005

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J. Hamilton, S. Slutsky / Journal of Economic Theory 132 (2007) 548–556 549

tax payments do not depend on the behavior of others. In a continuum model, Hammond[7] proves that a social planner cannot improve on this solution by incorporating interde-pendence across taxpayers, even having knowledge of the precise distribution of types inthe economy. Stiglitz [10] and others alternatively model the economy as consisting of afinite number of different types, but continue to assume there is a large (infinite) number ofindividuals of each type, and they retain the Mirrlees approach in which the consumptionbundles assigned to individuals depend only on their own choices.

When there is a finite number of consumers (not just of types), researchers have longknown, at least informally, that it is possible to improve on the optimal allocations foundin the Mirrlees–Stiglitz approach. If the government knew precisely the number of peopleof each type, it could announce that, were the set of reports received not to match thetrue distribution, then all taxpayers would be executed. Truthful reporting of types by allindividuals would be an equilibrium even if the resulting consumption bundles violate self-selection constraints. One objection to this approach is that it supports multiple equilibria.Due to the extreme nature of the punishment, any set of announcements which matchesthe true distribution in aggregate numbers is a Nash equilibrium, even if some individualsmisreport their types. Starting from any such pattern with some misrevelation, no one whois falsely revealing her type would be willing to deviate by telling the truth since this wouldtrigger the harsh penalty. Hence, while truthful revelation is a Nash equilibrium, a situationin which an equal number of more able and less able individuals misreveal their types isalso a Nash equilibrium.

In an important and provocative paper, Piketty [9] shows that, with a finite population, asocial planner can design a mechanism whose unique Nash equilibrium implements undis-torted allocations, even those which violate the Stiglitz self-selection constraints. 1 Pikettythus shows that the problem of a multiplicity of Nash equilibria need not be severe. If thepunishments are not too large, then the unique Nash equilibrium is for everyone to revealtruthfully. Piketty establishes uniqueness by constructing a mechanism whose equilibriumcan be found by sequential elimination of strictly dominated strategies. In his analysis,Piketty assumes a form of revelation which differs from the standard mechanism design ap-proach in which individuals announce their types. Piketty instead assumes that individualscommit to an action such as hours worked before they learn their tax which will depend onthe actions of everyone. In effect, individuals reveal their type by taking an action whichcannot be altered even if some individuals misreveal and move the game off the equilibriumpath.

We tighten Piketty’s analysis by imposing budget balance on the off-equilibrium pathallocations implemented when some individuals misreveal their types. Restrictions on suchoff-equilibrium outcomes can affect whether the government is able to sustain an undistortedallocation in which all individuals report truthfully. The announced tax plan must balancethe budget in all possible circumstances. This prevents the government from running asurplus or a deficit on or off the equilibrium path. A deficit is infeasible, hence would not bebelieved by individuals naming their actions.A surplus also would seriously affect revelation

1 Guesnerie [5] discusses the relationship between mechanisms and tax systems. He establishes that tax systemscan be viewed as mechanisms which satisfy anonymity restrictions. Such mechanisms cannot exploit correlationsamong individuals’ types as Piketty and we do.

550 J. Hamilton, S. Slutsky / Journal of Economic Theory 132 (2007) 548–556

incentives. Simply throwing away the excess would generate questions of credibility quiteeasily. 2

Section 2 introduces the basic model of the economy in which the planner wants toundertake redistribution. We describe the problems that can exist with respect to off-the-equilibrium-path allocations in the Piketty approach. In Section 3, we analyze the impactof imposing budget balance restriction off the equilibrium path. Section 4 contains ourconclusions.

2. The model

Stiglitz [10] formulated the standard optimal nonlinear income tax model for discretetypes. Brito et al. [1] present a general version of this model.Assume that there are two typesof individuals with the number of each type equal to ni . Let n denote the total population,where n = n1 +n2. Each type i, i = 1, 2, has a utility function Ui(Xi) which is continuousand quasiconcave. The vector Xi is a net trade bundle with two components, the firstbeing labor income and the second consumption. There is a linear production technologyspecifying feasible pairs of net trades, p · (n1X1 + n2X2)�0. Without loss of generality,we specify the units of the commodities so that p = (−1, 1). A net trade pair for these typesinvolves no redistribution if p · Xi = 0.

The utility functions can differ across types because of taste, productivity, or endowmentdifferences. We assume single crossing with the slope of the indifference curves for type1’s through any bundle X steeper than that for type 2’s. Because effort Xi

1 is a bad, the

indifference curves have positive slopes with MRSi = − �Ui/�Xi1

�Ui/�Xi2. For each type, we assume

that the marginal utility of consumption goes to infinity as consumption goes to zero. Thus,for any bundle X with p · X < 0, there is a bundle (0, X̂2) which is indifferent to it forsome X̂2 > 0. Every indifference curve through a bundle X∗ below the line p · X = 0must cross that line at some level of X1 less than X∗

1 . Individuals also are assumed to havea maximum possible labor supply L̄ and thus each type has a maximum possible pre-taxincome denoted X̄1

1 and X̄21 where X̄1

1 < X̄21. As Xi

1 approaches X̄i1, the marginal disutility

of labor goes to negative infinity. Hence, at any bundle X∗ with p ·X∗ < 0, the indifferencecurve for type i through X∗ must cross the line p · X = 0 at some Xi

1 between X∗1 and X̄i

1.The social planner assigns net trade bundles. If the social planner had full information

about individuals’ types, the full information outcomes would be the solution to:

(I) MaxX1,X2

�n1U1(X1) + (1 − �)n2U2(X2) s.t. p · (n1X1 + n2X2)�0.

Choice of the parameter � allows this problem to describe any full-information efficientallocation. Let Xi(n1, �) denote the solution to (I) for any n1 and �, where the total

2 This issue of whether a budget surplus can simply be thrown away is an issue in Groves [3]–Clarke [2]mechanisms which can be set up to run a surplus, but cannot be set up to balance the budget without losingthe dominant strategy property. The Groves–Ledyard [4] mechanism satisfies budget balance, but at the cost thattruthful revelation is only a Nash equilibrium. In both of these analyses, the main focus is equilibrium budgetbalance and not off-equilibrium properties, although Groves–Ledyard balances the budget in both circumstances.


population n is fixed. There exists an �0 such that the solution to (I) entails no redistri-bution (p · X1(n1, �0) = p · X2(n1, �0) = 0). For � > �0, there will be transfers from type2’s to type 1’s (n1p · X1(n1, �) = −n2p · X2(n1, �) > 0) and the reverse for � < �0. 3

Without full information, the planner knows the utility functions, Ui(Xi), and the num-bers ni of each type, but not individuals’ types. To induce individuals to report truthfully, theplanner offers each individual a choice from the same pair of net trade bundles, where eachbundle is intended for a particular consumer type and where the bundles must satisfy self-selection or incentive compatibility constraints: U1(X1)�U1(X2) and U2(X2)�U2(X1).We call this problem (II).

The allocations X1 and X2 solving (II) are the bundles assigned to individuals in equi-librium. In this specification, nothing explicit is said about allocations if the number ofpeople who reveal themselves as each type differs from the number known to be that typeby the planner. With a continuum of individuals, this is not an issue because a solution to(II) is a Nash equilibrium since any individual has a mass of zero. Hence, if one individualmisreveals, the planner will not observe this, either directly or through any resulting bud-get imbalance. With a finite number of individuals as well as types, this will not be true.Even if only one individual misreveals her type, the planner at some stage of the gamewill detect that the revealed pattern of types does not accord with what it knows for certainto be the actual pattern. If the net trade bundles of the two types use different resources(p · X1 �= p · X2), misrevelation causes either a surplus or a deficit. Since the planner willdetect misrevelation by even one individual, it can condition the bundles offered to eachtype on what others reveal (this conditioning must be anonymous in the sense that it onlydepends on the aggregate announcements of others). In fact, if a deficit results, the plannernot only can but must alter taxes to maintain feasibility.

Piketty formalized this for a finite population. Let N1 be the number who reveal them-selves as type 1 and for any 0�N1 �n1+n2, let (X1(N1), X2(N1)) be the bundles assignedto the two types. He imposed that truthful revelation is a dominance-solvable equilibrium,which guarantees that the Nash equilibrium is unique. The conditions for dominance solv-ability are:

U1(X1(N1 + 1))�U1(X2(N1)), 0�N1 �n1 + n2 − 1, (1)

U2(X2(N1))�U2(X1(N1 + 1)), n1 �N1 �n1 + n2 − 1. (2)

Condition (1) imposes that telling the truth is a dominant strategy for type 1s. Condition(2) then makes it a dominant strategy for type 2s to reveal truthfully, given that type 1s aretelling the truth. 4 In addition, Piketty assumed that individuals do not simply announcetheir types but must perform an action that reveals the type they claim to be. Once the action

3 In the action revelation game, the planner cannot sustain any allocation where X11(n1, �) = X2

1(n1, �), thatis where both types earn the same amount in the first-best allocation with truthful revelation. As long as leisure isstrictly increasing in lump-sum income when X1 > 0, this occurs only for a single value of �. Since the plannercould sustain the first-best allocation for a value of � arbitrarily close to this value, we ignore this case in ourformal analysis. Note that this issue only arises for transfers toward type 2, given normality of leisure.

4 Conditions (1) and (2) are appropriate when � is such that redistribution is from type 2’s to type 1’s. If thereverse holds, then a set of analogous conditions would be needed where truth telling is a dominant strategy for2’s and then 1’s tell the truth if 2’s do.


is performed, it cannot be changed, so the planner is constrained to have the same level ofthat action in all bundles for a particular type for every N1 value. For example, an individualmay reveal his type by earning the income designated for that type. The planner then canadjust the remaining components of an individual’s consumption bundle depending uponwhat others reveal, but cannot alter the income level. Formally, assume that Xi

1, i = 1, 2,is the labor income component of the net trade vector that is observed by the planner todetermine an individual’s type. This component must be the same in the bundles assignedin all revelation outcomes:

Xi1(N

1) = Xi1(n

1) and X21(N1) = X2

1(n1), 0�N1 �n1 + n2. (3)

In Piketty, the planner’s optimization problem, denoted as (III), then is to choose thevectors X1(N1) and X2(N1) to maximize �n1U1(X1(n1))+ (1−�)n2U2(X2(n1)) subjectto p ·(n1X1(n1)+n2X2(n1))�0 and conditions (1–3). Note that, while the planner choosesallocations for every N1, which enter the incentive constraints, only the truthful revelationbundles X1(n1) and X2(n1) enter the objective function and resource constraint. Pikettydemonstrates that the solution to (III) is the same as to (I).

To see the intuition behind Piketty’s result, consider the simplest case of an economycontaining one individual of each type. 5 Let X1(1, �) and X2(1, �) denote the solutionto problem (I) for an arbitrary �. Let Xa and Xb denote the bundles chosen by the twoindividuals by name and not type. After individuals reveal their type by choosing an action,there are three possible information sets assuming that the individuals have chosen actionsconsistent with the first best allocations Xi(1, �): both individuals have chosen the action oftype 1, Xa

1 = Xb1 = X1

1(1, �), both have chosen the action of type 2, Xa1 = Xb

1 = X21(1, �),

or exactly one individual has chosen the action of each type, Xa1 = X1

1(1, �) and Xb1 =

X21(1, �) or Xa

1 = X21(1, �) and Xb

1 = X11(1, �). In the first two cases, the planner knows

one person has lied but does not know who it is. The planner must treat them identicallysince there is no basis upon which to distinguish them. In the third case, both may have liedor both may have told the truth. The planner may have beliefs about whether both have liedbut cannot observe this. Hence, the planner must assign the same bundle to the individualchoosing X1

1(1, �) whether that individual is named a or b.Given these information sets, the planner chooses consumption levels X2(0), (X1

2(1),

X22(1)), and X2(2). If the full-information solution is to be the Nash equilibrium, then the

information set on the equilibrium path must be that with one individual choosing eachincome level and X1

2(1) = X12(1, �) and X2

2(1) = X22(1, �). For this to be an equilibrium,

the policies (X2(0), (X12(1), X2

2(1)), X2(2)) must induce the individuals to choose the rightaction.

If p · X1(1, �) > 0, the off-equilibrium bundles must satisfy the dominance solvabilityconditions (1–2) of problem (III) which in this special case reduce to:

U1(X11(1, �), X1

2(1))�U1(X21(1, �), X2(0)), (4)

U1(X11(1, �), X2(2))�U1(X2

1(1, �), X22(1)), (5)

U2(X21(1, �), X2

2(1))�U2(X11(1, �), X2(2)). (6)

5 Piketty actually shows this for more than two types.


(Con

sum

ptio

n)

(Labor Income)

X1(1,α)

)2(X1

U1

X1

X2

X2(1,α)

U1

U2

Fig. 1.

A solution to (I) will also solve (III) if it is feasible in (III). Feasibility in (III) follows if(4–6) are satisfied. To satisfy (4), X2(0) can equal any amount below the type 1 indifferencecurve through X1(1, �) on the vertical line through X2

1(1, �). Because of single crossing, onthe vertical line through X1

1(1, �), the indifference curve of the type 1’s through the X2(1, �)

bundle must be below the type 2 indifference curve through X2(1, �). Setting X2(2) equalto any amount above the type 1 indifference curve and below the type 2 indifference curvesatisfies (5) and (6). See Fig. 1.

This analysis placed no restrictions on the consumption component of the off-equilibriumbundles (Xi(Ni), Ni �= ni) other than what is needed to sustain the desired first-bestallocation as a unique Nash equilibrium. Instead consider that the only allowed bundles arethose exactly balancing the budget in all states. This extends the resource constraints to

p · (N1X1(N1) + (n − N1)X2(N1)) = 0, 0�N1 �n1 + n2. (7)

If we add constraint (7) to problem (III) to get an optimization problem denoted as(IV), Piketty’s result no longer holds. Some full information optimal allocations cannot besustained. To see this, consider the case of one individual of each type. Since the type 1indifference curve through X1(1, �) lies above the line p · X = 0, and condition (4) issatisfied by any bundle (X2

1(1, �), X2(0)) below that indifference curve, then X2(0) canalways be chosen to balance the budget. However, as shown in Fig. 1, no X2(2) may existwhich satisfies (5) and (6) and which balances the budget. In Fig. 1, every such bundle liesabove the p · X = 0 line, and hence runs a deficit. In that example, when both individualsreport they are type 1, no feasible allocation exists to sustain that first best allocation.Alternatively, the interval might lie below the p · X = 0 line and the full informationallocation cannot be sustained unless a surplus is run when the type 2 individual mimics theother type’s action. From single crossing, at X1(1, �) the type 2 indifference curves throughX2(1, �) must lie above the type 1 indifference curve through X2(1, �). Hence, a deficit isnecessary if that type 1 indifference curve lies above the p · X = 0 line at X1

1(1, �) and


a surplus is necessary if that type 2 indifference curves lies below the p · X = 0 line atX1

1(1, �). Otherwise budget balance is possible.

3. Announcement revelation and off-equilibrium budget balance

In contrast to action revelation, with announcement revelation, the planner has suffi-cient flexibility in specifying off-equilibrium bundles that imposing off-equilibrium budgetbalance is not a binding restriction. 6

Theorem. Under announcement revelation, for any full information optimal allocation,there exist X1(N1) and X2(N1) for all N1 which satisfy budget balance and which satisfythe dominance solvability constraints (1–2).

Proof. First, consider � such that no redistribution is done or redistribution is from type 2sto type 1s. Then, the bundle X2(n1, �) lies on or below the line p · X = 0. Consider thefollowing allocations to each type as a function of N1:

for N1 < n1 + 1, X1(N1) = X1(n1, �)

X2(N1) solves MaxX2

U2(X2) s.t. (n − N1)p · X2 = −N1p · X1(n1, �)

for N1 = n1 + 1, X1(n1 + 1) satisfies p · X1(n1 + 1) = 0,

U1(X1(n1 + 1)) > U1(X2(n1, �))

and U2(X1(n1 + 1)) < U2(X2(n1, �)),

X2(n1 + 1) solves MaxX2

U2(X2) s.t. p · X2 = 0

for N1 > n1 + 1, X1(N1) solves MaxX1

U1(X1) s.t. p · X1 = 0,

X2(N1) solves MaxX2

U2(X2) s.t. p · X2 = 0.

By construction, budget balance is satisfied for all N1 and, for N1 = n1, the allocationsare the first best bundles X1(n1, �) and X2(n1, �). That a bundle X1(n1 + 1) exists whichsatisfies the requirements given for it follows from the assumptions on preferences. Thebundle X2(n1, �) satisfies p ·X2(n1, �) < 0. The indifference curves of both types throughX2(n1, �) must intersect the line p · X = 0. From the single crossing assumption, thetype 1 indifference curve intersects p · X = 0 closer to the origin than does the type 2indifference curve. Any bundle on p · X = 0 between the two indifference curves that gothrough X2(n1, �) satisfies the requirements for X1(n1 + 1).

For dominance solvability, consider a single type 1 individual and let N̂1 be the numberof other individuals who have revealed themselves as type 1s. For any N̂1 �= n1, the type

6 The announcement revelation problem with budget balance is the same as (IV), except that condition (3) isdeleted.


1, by revealing truthfully, would get a bundle on some budget line p · X = K �0, which isbest on that line for type 1s. By misrevealing, the type 1 individual would get a bundle onsome budget line p · X = C�0, which is best on that line for type 2s. Clearly, the bundleunder truthful revelation is strictly better than under misrevelation. If N̂1 = n1, then byconstruction, the bundle under truthful revelation (X1(n1 + 1)) is superior to that undermisrevelation (X2(n1, �)). Hence, (1) is satisfied and revealing truthfully is a dominantstrategy for any type 1.

Next, consider any type 2. Given that all 1s tell the truth, a type 2 will only face N̂1 �n1.For N̂1 > n1, by telling the truth, the type 2 gets the best type 2 bundle on the line p ·X = 0while by misrevealing the type 2 individual gets the best type 1 bundle on that line. Clearly,the type 2 is better off being truthful. For N̂1 = n1, the type 2 if truthful gets X2(n1, �) whileif misrevealing, gets X1(n1 + 1) which is worse by construction. Hence, (2) is satisfied anddominance solvability is shown. �

4. Conclusions

Three important implications follow from our analysis which are likely to apply to incen-tive problems with a finite number of agents other than optimal taxation. First, our analysisreinforces Piketty’s [9] result that it is possible for a principal to attain better outcomes thanare possible under standard self-selection constraints, even achieving undistorted outcomes.Second, to determine when this occurs, it is important to consider feasibility restrictions onallocations off the equilibrium path. Third, the extent to which such restrictions matter de-pends upon the specifics of how agents provide information. If agents make simple ex anteannouncements, as in the standard mechanism design literature based upon the revelationprinciple, then our main theorem shows that off-equilibrium restrictions are not binding.However, if individuals commit to some specific action that reveals their types, as in [9], thenthe off-equilibrium constraints significantly restrict what outcomes can be implemented. Asshown in [6], the set of implementable outcomes then differs significantly from that whicharises under standard self-selection constraints.

The consideration of off-equilibrium feasibility opens questions of what additional re-strictions should be imposed off (as well as on) the equilibrium path. One view of thesefeasibility requirements is that the principal makes announcements of what punishments itwill impose if misrevelation occurs. Announcements of infeasible actions are not credible,so the principal will be forced to take actions which differ from the announcements. If thiscan occur, then the principal might deviate from announcements for other reasons beyondinfeasibility. If the principal can deviate from announcements as long as truthful agentsare not harmed, Hamilton and Slutsky [6] show that the set of implementable undistortedoutcomes is significantly reduced.

Acknowledgments

A preliminary version of this paper was presented at the Southern Economic Associ-ation meetings and the Association for Public Economic Theory meetings and seminars


at Academia Sinica, Sungkyunkwan University, the University of Copenhagen, and theUniversity of British Columbia. We thank Michael Blake, Richard Romano, David Sap-pington, Art Snow, two anonymous referees, and the Associate Editor for comments. Wethank the Warrington College of Business for financial support for this research. All errorsare solely our responsibility.

References

[1] D. Brito, J. Hamilton, S. Slutsky, J. Stiglitz, Pareto efficient tax schedules, Oxford Econ. Pap. 42 (1990)61–77.

[2] E. Clarke, Multipart pricing of public goods, Public Choice 11 (1971) 17–33.[3] T. Groves, Incentives in teams, Econometrica 41 (1973) 617–663.[4] T. Groves, J. Ledyard, Optimal allocation of public goods: a solution to the “free rider” problem, Econometrica

45 (1977) 783–809.[5] R. Guesnerie, A Contribution to the Pure Theory of Taxation, Cambridge University Press, Cambridge, UK,

1995.[6] J. Hamilton, S. Slutsky, Partial commitment in optimal nonlinear income taxation, http://bear.cba.ufl.edu/

hamilton/working.html, 2005.[7] P. Hammond, Straightforward incentive compatibility in large economies, Rev. Econ. Stud. 46 (1979)

263–282.[8] J. Mirrlees, An exploration in the theory of optimal income taxation, Rev. Econ. Stud. 38 (1971) 175–208.[9] T. Piketty, Implementation of first-best allocations via generalized tax schedules, J. Econ. Theory 61 (1993)

23–41.[10] J. Stiglitz, Self-selection and Pareto efficient taxation, J. Public Econ. 17 (1982) 213–240.

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Optimal nonlinear income taxation with a finite population