Bénabou e Ok (2001)

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SOCIAL MOBILITY AND THE DEMAND FOR

REDISTRIBUTION: THE POUM HYPOTHESIS*

ROLAND BENABOU AND EFE A. OK

This paper examines the often stated idea that the poor do not support highlevels of redistribution because of the hope that they, or their offspring, may makeit up the income ladder. This prospect of upward mobility (POUM) hypothesis isshown to be fully compatible with rational expectations, and fundamentallylinked to concavity in the mobility process. A steady-state majority could even besimultaneously poorer than average in terms of current income, and richer thanaverage in terms of expected future incomes. A rst empirical assessment sug-gests, on the other hand, that in recent U. S. data the POUM effect is probably

dominated by the demand for social insurance.

In the future, everyone will be world-famous for fteen minutes

[Andy Warhol 1968].

INTRODUCTION

The following argument is among those commonly advanced

to explain why democracies, where a relatively poor majorityholds the political power, do not engage in large-scale expropria-tion and redistribution. Even people with income below average,it is said, will not support high tax rates because of the prospectof upward mobility: they take into account the fact that they, ortheir children, may move up in the income distribution and there-fore be hurt by such policies.1 For instance, Okun [1975, p. 49]

relates that: In 1972 a storm of protest from blue-collar workersgreeted Senator McGoverns proposal for conscatory estate taxes.

They apparently wanted some big prizes maintained in the game.

* We thank Abhijit Banerjee, Jess Benhabib, Edward Glaeser, Levent Kock-esen, Ignacio Ortuno-Ortin, three anonymous referees, seminar participants atthe Massachusetts Institute of Technology, and the Universities of Maryland andMichigan, New York University, the University of Pennsylvania, Universitat

Pompeu Fabra, Princeton University, and Universite de Toulouse for their helpfulcomments. The rst author gratefully acknowledges nancial support from theNational Science Foundation (SBR9601319), and the MacArthur Foundation.Both authors are grateful for research support to the C. V. Starr Center at New

York University.1. See, for example, Roemer [1998] or Putterman [1996]. The prospect of

upward mobility hypothesis is also related to the famous tunnel effect of Hirsch-man [1973], although the argument there is more about how people make infer-ences about their mobility prospects from observing the experience of others.There are of course several other explanations for the broader question of why thepoor do not expropriate the rich. These include the deadweight loss from taxation

(e.g., Meltzer and Richard [1981]), and the idea that the political system is biasedagainst the poor [Peltzman 1980; Benabou 2000]. Putterman, Roemer, and Syl-

vestre [1999] provide a review.

2001 by the President and Fellows of Harvard College and the Massachusetts Institute ofTechnology.

The Quarterly Journal of Economics, May 2001

447

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The silent majority did not want the yacht clubs closed forever to

their children and grandchildren while those who had already

become members kept sailing along.

The question we ask in this paper is simple: does this storymake sense with economic agents who hold rational expectationsover their income dynamics, or does it require that the poorsystematically overestimate their chances of upward mobilityaform of what Marxist writers refer to as false consciousness?2

The prospect of upward mobility (POUM) hypothesis has,to the best of our knowledge, never been formalized, which israther surprising for such a recurrent theme in the politicaleconomy of redistribution. There are three implicit premises be-hind this story. The rst is that policies chosen today will, to someextent, persist into future periods. Some degree of inertia orcommitment power in the setting of scal policy seems quitereasonable. The second assumption is that agents are not too riskaverse, for otherwise they must realize that redistribution pro-

vides valuable insurance against the fact that their income maygo down as well as up. The third and key premise is that individ-uals or families who are currently poorer than averagefor in-stance, the median voterexpect to become richer than average.This optimistic view clearly cannot be true for everyone belowthe mean, barring the implausible case of negative serial corre-

lation. Moreover, a standard mean-reverting income processwould seem to imply that tomorrows expected income lies some-where between todays income and the mean. This would leavethe poor of today still poor in relative terms tomorrow, andtherefore demanders of redistribution. And even if a positivefraction of agents below the mean today can somehow expect to beabove it tomorrow, the expected incomes of those who are cur-

rently richer than they must be even higher. Does this not thenrequire that the number of people above the mean be foreverrising over time, which cannot happen in steady state? It thusappearsand economists have often concludedthat the intui-tion behind the POUM hypothesis is awed, or at least incom-

2. If agents have a tendency toward overoptimistic expectations (as sug-gested by a certain strand of research in social psychology), this will of coursereinforce the mechanism analyzed in this paper, which operates even under fullrationality. It might be interesting, in future research, to extend our analysis ofmobility prospects to a setting where agents have (endogenously) self-servingassessments of their own abilities, as in Benabou and Tirole [2000].

448 QUARTERLY JOURNAL OF ECONOMICS

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patible with everyone holding realistic views of their incomeprospects.3

The contribution of this paper is to formally examine theprospect of upward mobility hypothesis, asking whether andwhen it can be valid. The answer turns out to be surprisinglysimple, yet a bit subtle. We show that there exists a range ofincomes below the mean where agents oppose lasting redistribu-tions if (and, in a sense, only if) tomorrows expected income is anincreasing and concave function of todays income. The moreconcave the transition function, and the longer the length of time

for which taxes are preset, the lower the demand for redistribu-tion. Even the median voterin fact, even an arbitrarily poor

votermay oppose redistribution if either of these factors is largeenough. We also explain how the concavity of the expected tran-sition function and the skewness of idiosyncratic income shocksinteract to shape the long-run distribution of income. We con-struct, for instance, a simple Markov process whose steady-state

distribution has three-quarters of the population below meanincome, so that they would support purely contemporary redis-tributions. Yet when voters look ahead to the next period, two-thirds of them have expected incomes above the mean, and thissuper-majority will therefore oppose (perhaps through constitu-tional design) any redistributive policy that bears primarily onfuture incomes.

Concavity of the expected transition function is a rathernatural property, being simply a form of decreasing returns: ascurrent income rises, the odds for future income improve, but ata decreasing rate. While this requirement is stronger than simplemean reversion or convergence of individual incomes, concavetransition functions are ubiquitous in economic models andeconometric specications. They arise, for instance, when current

resources affect investment due to credit constraints and theaccumulation technology has decreasing returns; or when someincome-generating individual characteristic, such as ability, ispassed on to children according to a similar technology. Inparticular, the specication of income dynamics most widely usedin theoretical and empirical work, namely the log linear ar(1)process, has this property.

3. For instance, Putterman, Roemer, and Sylvestre [1999] state that votingagainst wealth taxation to preserve the good fortune of ones family in the futurecannot be part of a rational expectations equilibrium, unless the deadweight lossfrom taxation is expected to be large or voters are risk loving over some range.

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The key role played by concavity in the POUM mechanism

may be best understood by rst considering a very stripped-downexample. Suppose that agents decide today between laissez-faire and complete sharing with respect to next periods income,and that the latter is a deterministic function of current income:

y9 = f(y), for all y in some interval [0,y#]. Without loss of gener-ality, normalize f so that someone with income equal to theaverage, m, maintains that same level tomorrow ( f(m) = m). As

shown in Figure I, everyone who is initially poorer will then seehis income rise, and conversely all those who are initially richerwill experience a decline. The concavity of fmore specically,Jensens inequalitymeans that the losses of the rich sum tomore than the gains of the poor; therefore, tomorrows per capitaincome m9 is below m. An agent with mean initial income, or evensomewhat poorer, can thus rationally expect to be richer than

average in the next period, and will therefore oppose futureredistributions.

To provide an alternative interpretation, let us now normal-ize the transition function so that tomorrows and todays mean

FIGURE IConcave Transition Function

Note that A < B; therefore m9 < f(m). The gure also applies to the stochasticcase, with f replaced by Ef everywhere.


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incomes coincide: m9 = m. The concavity of f can then be inter-preted as saying that y9 is obtained from y through a progressive,balanced budget, redistributive scheme, which shifts the Lorenzcurve upward and reduces the skewness of the income distribu-tion. As is well-known, such progressivity leaves the individualwith average endowment better off than under laissez-faire, be-cause income is taken disproportionately from the rich. Thismeans that the expected income y9 of a person with initial incomem is strictly greater than m, hence greater than the average of

y9 across agents. This person, and those with initial incomesnot too far below, will therefore be hurt if future incomes areredistributed.4

Extending the model to a more realistic stochastic settingbrings to light another important element of the story, namelythe skewness of idiosyncratic income shocks. The notion that liferesembles a lottery where a lucky few will make it big is some-what implicit in casual descriptions of the POUM hypothesis

such as Okuns. But, in contrast to concavity, skewness in itselfdoes nothing to reduce the demand for redistribution; in particu-lar, it clearly does not affect the distribution of expected incomes.The real role played by such idiosyncratic shocks, as we show, isto offset the skewness-reducing effect of concave expected transi-tions functions, so as to maintain a positively skewed distributionof income realizations (especially in steady state). The balance

between the two forces of concavity and skewness is what allowsus to rationalize the apparent risk-loving behavior, or overopti-mism, of poor voters who consistently vote for low tax rates due tothe slim prospects of upward mobility.

With the important exceptions of Hirschman [1973] andPiketty [1995a, 1995b], the economic literature on the implica-tions of social mobility for political equilibrium and redistributive

policies is very sparse. For instance, mobility concerns are com-pletely absent from the many papers devoted to the links betweenincome inequality, redistributive politics, and growth (e.g.,

4. The concavity offimplies that f(x)/x is decreasing, which corresponds to taxprogressivity and Lorenz equalization, on any interval [yI,y#] such that yIf9(yI ) # f(yI ).This clearly applies in the present case, where yI = 0 and f(0) $ 0 since incomeis nonnegative. When the boundary condition does not hold, concavity is consis-tent with (local or global) regressivity. At the most general level, a concave schemeis thus one that redistributes from the extremes toward the mean. This is theeconomic meaning of Jensens inequality, given the normalization m9 = m. Inpractice, however, most empirical mobility processes are clearly progressive (inexpectation). The progressive case discussed above and illustrated in Figure I isthus really the relevant one for conveying the key intuition.


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Alesina and Rodrik [1994] and Persson and Tabellini [1994]). Akey mechanism in this class of models is that of a poor median

voter who chooses high tax rates or other forms of expropriation,which in turn discourage accumulation and growth. We show thatwhen agents vote not just on the current scal policy but on onethat will remain in effect for some time, even a poor median votermay choose a low tax rateindependently of any deadweight lossconsiderations.

While sharing the same general motivation as Piketty[1995a, 1995b], our approach is quite different. Pikettys main

concern is to explain persistent differences in attitudes towardredistribution. He therefore studies the inference problem ofagents who care about a common social welfare function, butlearn about the determinants of economic success only throughpersonal or dynastic experimentation. Because this involvescostly effort, they may end up with different long-run beliefs overthe incentive costs of taxation. We focus instead on agents who

know the true (stochastic) mobility process and whose main con-cern is to maximize the present value of their aftertax incomes, orthat of their progeny. The key determinant of their vote is there-fore how they assess their prospects for upward and downwardmobility, relative to the rest of the population.

The paper will formalize the intuitions presented above link-ing these relative income prospects to the concavity of the mobil-

ity process and then examine their robustness to aggregate un-certainty, longer horizons, discounting, risk aversion, andnonlinear taxation. It will also present an analytical example thatdemonstrates how a large majority of the population can besimultaneously below average in terms of current income andabove average in terms of expected future income, even thoughthe income distribution remains invariant.5 Interestingly, a simu-

lated version of this simple model ts some of the main featuresof the U. S. income distribution and intergenerational persistencerather well. It also suggests, on the other hand, that the POUMeffect can have a signicant impact on the political equilibriumonly if agents have relatively low degrees of risk aversion.

Finally, the paper also makes a rst pass at the empiricalassessment of the POUM hypothesis. Using interdecile mobility

5. Another analytical example is the log linear, lognormal ar(1) processcommonly used in econometric studies. Complete closed-form solutions to themodel under this specication are provided in Benabou and Ok [1998].


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matrices from the PSID, we compute over different horizons theproportion of agents who have expected future incomes above themean. Consistent with the theory, we nd that this laissez-fairecoalition grows with the length of the forecast period, to reach amajority for a horizon of about twenty years. We also nd, how-ever, that these expected income gains of the middle class arelikely to be dominated, under standard values of risk aversion, bythe desire for social insurance against the risks of downwardmobility or stagnation.

I. PRELIMINARIES

We consider an endowment economy populated by a contin-uum of individuals indexed by i [ [0,1], whose initial levels ofincome lie in some interval X [0,y#], 0 < y# # .6 An incomedistribution is dened as a continuous function F: X [0,1] suchthat F(0 ) = 0, F(y#) = 1 and mF *X y dF < . We shall denote

by F the class of all such distributions, and by F+ the subset ofthose whose median, mF min [F

21(12)], is below their mean.We shall refer to such distributions as positively skewed, andmore generally we shall measure skewness in a random vari-able as the proportion of realizations below the mean (minus ahalf), rather than by the usual normalized third moment.

A redistribution scheme is dened as a function r : X 3 F

R + which assigns to each pretax income and initial distribution alevel of disposable income r(y; F), while preserving total income:*X r(y; F) dF(y) = mF. We thus abstract from any deadweightlosses that such a scheme might realistically entail, in order tobetter highlight the different mechanism which is our focus. Bothrepresent complementary forces reducing the demand for redis-tribution, and could potentially be combined into a common

framework.The class of redistributive schemes used in a vast majority

of political economy models is that of proportional schemes,where all incomes are taxed at the rate t and the collectedrevenue is redistributed in a lump-sum manner.7 We denote this

6. More generally, the income support could be any interval [yI ,y#], yI $ 0.We choose yI = 0 for notational simplicity.

7. See, for instance, Meltzer and Richard [1981], Persson and Tabellini[1996], or Alesina and Rodrik [1994]. Proportional schemes reduce the votingproblem to a single-dimensional one, thereby allowing the use of the median votertheorem. By contrast, when unrestricted nonlinear redistributive schemes are


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class as P {rt u0 # t # 1}, where rt(y; F) (1 2 t)y + tmF forall y [ X and F [ F. We shall mostly work with just the twoextreme members ofP, namely, r0 and r1. Clearly, r0 correspondsto the laissez-faire policy, whereas r1 corresponds to completeequalization.

Our focus on these two polar cases is not nearly so restrictiveas might initially appear. First, the analysis immediately extendsto the comparison between any two proportional redistributionschemes, say rt and rt 9, with 0 # t < t9 # 1. Second, r0 and r1 arein a certain sense focal members of P since, in the simplest

framework where one abstracts from taxes distortionary effectsas well as their insurance value, these are the only candidates inthis class that can be Condorcet winners. In particular, for anydistribution with median income below the mean, r1 beats everyother linear scheme under majority voting if agents care onlyabout their current disposable income. We shall see that thisconclusion may be dramatically altered when individuals voting

behavior also incorporates concerns about their future incomes.Finally, in Section IV we shall extend the analysis to nonlinear(progressive or regressive) schemes, and show that our mainresults remain valid.

As pointed out earlier, mobility considerations can enter intovoter preferences only if current policy has lasting effects. Suchpersistence is quite plausible given the many sources of inertia

and status quo bias that characterize the policy-making process,especially in an uncertain environment. These include constitu-tional limits on the frequency of tax changes, the costs of formingnew coalitions and passing new legislation, the potential for pro-longed gridlock, and the advantage of incumbent candidates andparties in electoral competitions. We shall therefore take suchpersistence as given, and formalize it by assuming that tax policy

must be set one period in advance, or more generally preset for Tperiods. We will then study how the length of this commitmentperiod affects the demand for redistribution.8

allowed, there is generally no voting equilibrium (in pure strategies): the core ofthe associated voting game is empty.

8. Another possible channel through which current tax decisions might in-corporate concerns about future redistributions is if voters try to inuence futurepolitical outcomes by affecting the evolution of the income distribution, throughthe current tax rate. This strategic voting idea has little to do with the POUMhypothesis as discussed in the literature (see references in footnote 1, as well asOkuns citation). Moreover, these dynamic voting games are notoriously intrac-table, so the nearly universal practice in political economy models is to assume


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The third and key feature of the economy is the mobilityprocess. We shall study economies where individual incomes orendowments yt

i evolve according to a law of motion of the form,

(1) yt+ 1i 5 f(yt

i, ut+ 1i ), t 5 0, . . . , T2 1,

where f is a stochastic transition function and ut+ 1i is the realiza-

tion of a random shock Qt+ 1i .9 We require this stochastic process

to have the following properties:(i) The random variables Qt

i , (i,t) [ [0, 1] 3 {1, . . . , T},have a common probability distribution function P, with

support V.(ii) The function f : X 3 V X is continuous, with a

well-dened expectation EQ[f( z ; Q)] on X.(iii) Future income increases with current income, in the

sense of rst-order stochastic dominance: for any(y,y9) [ X2 , the conditional distribution M(y9uy) prob({u [ V uf(y; u) # y9}) is decreasing in y, with strict

monotonicity on some nonempty interval in X.The rst condition means that everyone faces the same un-

certain environment, which is stationary across periods. Put dif-ferently, current income is the only individual-level state variablethat helps predict future income. While this focus on unidimen-sional processes follows a long tradition in the study of socioeco-nomic mobility (e.g., Atkinson [1983], Shorrocks [1978], Conlisk

[1990], and Dardanoni [1993]), one should be aware that it isfairly restrictive, especially in an intragenerational context. Itmeans, for instance, that one abstracts from life-cycle earningsproles and other sources of lasting heterogeneity such as gender,race, or occupation, which would introduce additional state vari-ables into the income dynamics. This becomes less of a concernwhen dealing with intergenerational mobility, where one can

essentially think of the two-period case, T = 1, as representingoverlapping generations. Note, nally, that condition (i) puts norestriction on the correlation of shocks across individuals; it al-lows for purely aggregate shocks (Qt

i = Qtj

for all i, j in [0, 1]),

myopic voters. In our model the issue does not arise since we focus on endow-ment economies, where income dynamics are exogenous.

9. We thus consider only endowment economies, but the POUM mechanismremains operative when agents make effort and investment decisions, and thetransition function varies endogenously with the chosen redistributive policies.Benabou [1999, 2000] develops such a model, using specic functional and distri-butional assumptions.


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purely idiosyncratic shocks (the Qti s are independent across

agents and sum to zero), and all cases in between.10

The second condition is a minor technical requirement. Thethird condition implies that expected income EQ[ f( z ; Q)] riseswith current income, which is what we shall actually use in theresults. We impose the stronger distributional monotonicity forrealism, as all empirical studies of mobility (intra- or intergen-erational) nd income to be positively serially correlated andtransition matrices to be monotone. Thus, given the admittedlyrestrictive assumption (i), (iii) is a natural requirement to impose.

We shall initially focus the analysis on deterministic incomedynamics (where ut

i is just a constant, and therefore dropped fromthe notation), and then incorporate random shocks. While thestochastic case is obviously of primary interest, the deterministicone makes the key intuitions more transparent, and providesuseful intermediate results. This two-step approach will also helphighlight the fundamental dichotomy between the roles of con-

cavity in expectations and skewness in realizations.

II. INCOME D YNAMICS AND VOTING UNDER CERTAINTY

It is thus assumed for now that individual pretax incomes orendowments evolve according to a deterministic transition func-tion f, which is continuous and strictly increasing. The income

stream of an individual with initial endowment y [ X is then y,f(y), f2(y), . . . , ft(y), . . . , and for any initial F [ F thecross-sectional distribution of incomes in period t is Ft F + f

2t. A particularly interesting class of transition functions for thepurposes of this paper is the set of all concave (but not afne)transition functions; we denote this set by T .

II.A. Two-Period AnalysisTo distill our main argument into its most elementary form,

we focus rst on a two-period (or overlapping generations) sce-nario, where individuals vote today (date 0) over alternativeredistribution schemes that will be enacted only tomorrow (date1). For instance, the predominant motive behind agents votingbehavior could be the well-being of their offspring, who will be

10. Throughout the paper we shall follow the common practice of ignoring thesubtle mathematical problems involved with continua of independent random

variables, and thus treat each Qti as jointly measurable, for any t. Consequently,

the law of large numbers and Fubinis theorem are applied as usual.


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subject to the tax policy designed by the current generation. Accordingly, agent y [ X votes for r1 over r0 if she expects herperiod 1 earnings to be below the per capita average:

(2) f(y) , EX

f dF0 5 mF1.

Suppose now that f [ T ; that is, it is concave but not afne. Then,by Jensens inequality,

(3) f(mF0) 5 f S EXy dF0D . EX f dF0 5 mF1,so the agent with average income at date 0 will oppose date 1redistributions. On the other hand, it is clear that f(0) < mF

1, so

there must exist a unique y*f in (0, mF0) such that f(y*f) mF

1. Of

course y*f = f21(mF

0+f21) also depends on F0 but, for brevity, we do

not make this dependence explicit in the notation. Since f isstrictly increasing, it is clear that y*f acts as a tipping point inagents attitudes toward redistributions bearing on future in-come. Moreover, since Jensens inequalitywith respect to alldistributions F0characterizes concavity, the latter is both nec-essary and sufcient for the prospect of upward mobility hypothe-sis to be valid, under any linear redistribution scheme.11

PROPOSITION 1. The following two properties of a transition func-tion f are equivalent:

(a) f is concave (but not afne); i.e., f [ T.(b) For any income distribution F0 [ F there exists a unique

y*f < mF0

such that all agents in [0, y*f) vote for r1 over r0,while all those in (y*f, y#] vote for r0 over r1 .

Yet another way of stating the result is thatf

[T

if it isskewness-reducing: for any initial F0, next periods distributionF1 = F0 + f

21 is such that F1(mF1) < F0(mF0). Compared with thestandard case where individuals base their votes solely on howtaxation affects their current disposable income, popular supportfor redistribution thus falls by a measure F0(mF0) 2 F1(mF1) =

F0(mF0) 2 F0(y*f) > 0. The underlying intuition also suggests

11. For any rt and rt9 in P such that 0 # t < t9 # 1, agent y [ X votes for rt9over rt iff (1 2 t) f(y) + tmF1 < (1 2 t9) f(y) + t9mF 1, which in turn holds if andonly if (2) holds. Thus, as noted earlier, nothing is lost by focusing only on the twoextreme schemes in P, namely r0 and r1 .


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that the more concave is the transition function, the fewer peopleshould vote for redistribution. This simple result will turn out tobe very useful in establishing some of our main propositions onthe outcome of majority voting and on the effect of longer politicalhorizons.

We shall say that f [ T is more concave than g [ T , andwrite f g, if and only if f is obtained from g through anincreasing and concave (not afne) transformation; that is, ifthere exists an h [ T such that f = h + g. Put differently, f gif and only if f + g21 [ T .

PROPOSITION 2. Let F0 [ F and f, g [ T . Then f g implies thaty*f < y*g.

The underlying intuition is, again, straightforward: the de-mand for future scal redistribution is lower under the transitionprocess which reduces skewness by more. Can prospects of up-ward mobility be favorable enough for r0 to beat r1 under major-

ity voting? Clearly, the outcome of the election depends on theparticular characteristics of f and F0 . One can show, however,that for any given pretax income distribution F0 there exists atransition function f which is concave enough that a majority of

voters choose laissez-faire over redistribution.12 When combinedwith Proposition 2, it allows us to show the following, moregeneral result.

THEOREM 1. For any F0 [ F+ , there exists an f [ T such that r0beats r1 under pairwise majority voting for all transitionfunctions that are more concave than f, and r1 beats r0 for alltransition functions that are less concave than f.

This result is subject to an obvious caveat, however: for amajority of individuals to vote for laissez-faire at date 0, the

transition function must be sufciently concave to make the date1 income distribution F1 negatively skewed. Indeed, if y*f =

f21(mF1) < mF

0, then mF

1

< f(mF0) = mF

1. There are two reasons

why this is far less problematic than might initially appear. Firstand foremost, it simply reects the fact that we are momentarilyabstracting from idiosyncratic shocks, which typically contributeto reestablishing positive skewness. Section III will thus present

12. In this case, r0 is the unique Condorcet winner in P. Note also thatTheorem 1like every other result in the paper concerning median incomemF0holds in fact for any arbitrary income cutoff below mF0 (see the proof in the

Appendix).


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a stochastic version of Theorem 1, where F1 can remain asskewed as one desires. Second, it may in fact not be necessarythat the cutoff y*f fall all the way below the median for redistri-bution to be defeated. Even in the most developed democracies itis empirically well documented that poor individuals have lowerpropensities to vote, contribute to political campaigns, and oth-erwise participate in the political process, than rich ones. Thegeneral message of our results can then be stated as follows: themore concave the transition function, the smaller the departurefrom the one person, one vote ideal needs to be for redistributive

policies, or parties advocating them, to be defeated.

II.B. Multiperiod Redistributions

In this subsection we examine how the length of the horizonover which taxes are set and mobility prospects evaluated affectsthe political support for redistribution. We thus make the more

realistic assumption that the tax scheme chosen at date 0 willremain in effect during periods t = 0, . . . , T, and that agentscare about the present value of their disposable income streamover this entire horizon. Given a transition function f and adiscount factor d [ (0, 1], agent y [ X votes for laissez-faire overcomplete equalization if

(4) Ot= 0

T

dtft(y) . Ot= 0

T

dtmFt,

where we recall that ft denotes the tth iteration of f and Ft F0 + f

2t is the period t income distribution, with mean mFt.We shall see that there again exists a unique tipping point

y*f (T) such that all agents with initial income less than y*f(T) vote

for r1, while all those richer than y*f(T) vote for r0. When thepolicy has no lasting effects, this point coincides with the mean:

y*f (0) = mF0. When future incomes are factored in, the coalition infavor of laissez-faire expands: y*f(T) < mF0for T $ 1. In fact, themore farsighted voters are, or the longer the duration of theproposed tax scheme, the less support for redistribution there willbe: y*f(T) is strictly decreasing in T. If agents care enough about

future incomes, the increase in the vote for r0 can be enough toensure its victory over r1.

THEOREM 2. Let F [ F+ and d [ (0, 1].


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(a) For all f [ T , the longer is the horizon T, the larger is theshare of votes that go to r0.

(b) For all d and T large enough, there exists an f [ T suchthat r0 ties with r1 under pairwise majority voting. More-over, r0 beats r1 if the duration of the redistributionscheme is extended beyond T, and is beaten by r1 if thisduration is reduced below T.

Simply put, longer horizons magnify the prospect of upwardmobility effect, whereas discounting works in the opposite direc-tion. The intuition is very simple, and related to Proposition 2:

when forecasting incomes further into the future, the one-steptransition f gets compounded into f2 , . . . , fT, etc., and each ofthese functions is more concave than its predecessor.13

III. INCOME D YNAMICS AND VOTING UNDER UNCERTAINTY

The assumption that individuals know their future incomes

with certainty is obviously unrealistic. Moreover, in the absenceof idiosyncratic shocks the cross-sectional distribution becomesmore equal over time, and eventually converges to a single mass-point. In this section we therefore extend the analysis to thestochastic case, while maintaining risk neutrality. The role ofinsurance will be considered later on.

Income dynamics are now governed by a stochastic process

yt+ 1i = f(yti ; ut+ 1i ) satisfying the basic requirements (i)(iii) dis-cussed in Section I, namely stationarity, continuity, and monoto-nicity. In the deterministic case the validity of the POUM conjec-ture was seen to hinge upon the concavity of the transitionfunction. The strictest extension of this property to the stochasticcase is that it should hold with probability one. Therefore, let TPbe the set of transition functions such that prob[{uuf( z ; u ) [ T }]

= 1. It is clear that, for any f in TP,(iv) The expectation EQ[f( z ; Q)] is concave (but not afne)

on X.For some of our purposes the requirement that f [ TP will be toostrong, so we shall develop our analysis for the larger set ofmobility processes that simply satisfy concavity in expectation.

13. The reason why d and T must be large enough in part (b) of Theorem 2 isthat redistribution is now assumed to be implemented right away, starting inperiod 0. If it takes effect only in period 1, as in the previous section, the resultsapply for all d and T $ 1.


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We shall therefore denote as T *P the set of transition functionsthat satisfy conditions (i) to (iv).

III.A. Two-Period Analysis

We rst return to the basic case where risk-neutral agentsvote in period 0 over redistributing period 1 incomes. Agent yi [X then prefers r0 to r1 if and only if

(5) EQi[f(yi; Qi)] . E[mF1],

where the subscript Qi on the left-hand side indicates that theexpectation is taken only with respect to Qi, for given yi. Whenshocks are purely idiosyncratic, the future mean mF1is determin-istic due to the law of large numbers; with aggregate shocks itremains random. In any case, the expected mean income at date 1is the mean expected income across individuals:

E[mF1] 5 E

F E01

f(yj

; Qj

) dj

G5 E

0

1

E[f(yj; Qj)] dj

5

EXEQi[f(y; Q

i

)] dF0(y),

by Fubinis theorem. This is less than the expected income of anagent whose initial endowment is equal to the mean level mF0,whenever f(y; u ) or, more generally, EQ i[ f(y; Q

i)]is concavein y:

(6) EX

EQi[f(y; Qi)] dF0(y) , EQi[f(mF0; Q

i)].

Consequently, there must again exist a nonempty interval ofincomes [y*f, mF0] in which agents will oppose redistribution, withthe cutoff y*f dened by

EQi[ f(y*

f; Q

i)] 5E

[mF1

].

The basic POUM result thus holds for risk-neutral agents whoseincomes evolve stochastically. To examine whether an appropri-ate form of concavity still affects the cutoff monotonically, and


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whether enough of it can still cause r0 to beat r1 under majority voting, observe that the inequality in (6) involves only the ex-pected transition function EQi[ f(y; Q

i)], rather than f itself. Thisleads us to replace the more concave than relation with a moreconcave in expectation than relation. Given any probability dis-tribution P, we dene this ordering on the class T *P as

f P g if and only if EQ[f( z ; Q)] EQ[g( z ; Q)],

where Q is any random variable with distribution P.14 It is easilyshown that f P g implies that y*f < y*g. In fact, making f concave

enough in expectation will, as before, drive the cutoffy*f below themedian mF0, or even below any chosen income level. Most impor-tantly, since this condition bears only on the mean of the randomfunction f( z ; Q), it puts essentially no restriction on the skewnessof the period 1 income distribution F1in sharp contrast to whatoccurred in the deterministic case. In particular, a sufcientlyskewed distribution of shocks will ensure that F1 [ F+ without

affecting the cutoff y*f. This dichotomy between expectations andrealizations is the second key component of the POUM mecha-nism, and allows us to establish a stochastic generalization ofTheorem 1.15

THEOREM 3. For any F0 [ F+ and any s [ (0, 1), there exists amobility process ( f, P) with f [ T *P such that F1(mF1) $ sand, under pairwise majority voting, r

0

beats r1

for all tran-sition functions in T *P that are more concave than f in expec-tation, while r1 beats r0 for all those that are less concavethan f in expectation.

Thus, once random shocks are incorporated, we reach essen-tially the same conclusions as in Section II, but with muchgreater realism. Concavity of EQ[ f( z ; Q)] is necessary and suf-

cient for the political support behind the laissez-faire policy toincrease when individuals voting behavior takes into accounttheir future income prospects. If f is concave enough in expecta-

14. Interestingly, and P are logically independent orderings. Even ifthere exists some h [ T such that f( z , u ) = h(g( z , u )) for all u, it need not bethat f P g.

15. The simplest case where the distribution of expectations and the distri-bution of realizations differ is that where future income is the outcome of awinner-take-all lottery. The rst distribution reduces to a single mass-point(everyone has the same expected payoff), whereas the second is extremely unequal(there is only one winner). Note, however, that this income process does not havethe POUM property (instead, everyones expected income coincides with themean), precisely because it is nowhere strictly concave.


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tion, then r0 can even be the preferred policy of a majority ofvoters.

III.B. Steady-State Distributions

The presence of idiosyncratic uncertainty is not only realistic,but is also required to ensure a nondegenerate long-run incomedistribution. This, in turn, is essential to show that our previousndings describe not just transitory, short-run effects, but stable,permanent ones as well.

LetP be a probability distribution of idiosyncratic shocks andf a transition function in T *P. An invariant or steady-state distri-bution of this stochastic process is an F [ F (not necessarilypositively skewed) such that

F(y) 5 EV

EX

1{ f(x, u )#y} dF(x) dP(u ) for all y [X,

where 1{ z } denotes the indicator function. Since the basic resultthat the coalition opposed to lasting redistributions includesagents poorer than the mean holds for all distributions in F, itapplies to invariant ones in particular: thus, y*f < mF.

16

This brings us back to the puzzle mentioned in the introduc-tion. How can there be a stationary distribution F where a posi-tive fraction of agents below the mean mF have expected incomes

greater than mF, as do all those who start above this mean, giventhat the number of people on either side of mF must remaininvariant over time?

The answer is that even though everyone makes unbiasedforecasts, the number of agents with expected income above themean, 1 2 F(y*f, F), strictly exceeds the number who actually endup with realized incomes above the mean, 1 2 F(mF), whenever

f is concave in expectation. This result is apparent in Figure II,which provides additional intuition by plotting each agents ex-pected income path over future dates, E[yt

i uy0i

]. In the long runeveryones expected income converges to the population mean mF,but this convergence is nonmonotonic for all initial endowments

16. If the inequality y*f

< mF

is required to hold only for the steady-statedistribution(s) F induced by f and P, rather than for all initial distributions,concavity of EQ[ f( z ; Q)] is still a sufcient condition, but no longer a necessaryone. Nonetheless, some form of concavity on average is still required, so to speak:if EQ [ f( z ; Q)] were linear or convex, we would have y*f $ mF for all distributions,including stationary ones.


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in some interval (yI F,y#F) around mF. In particular, for y0i

[(yI F,mF) expected income rst crosses the mean from below, thenconverges back to it from above. While such nonmonotonicity mayseem surprising at rst, it follows from our results that all con-cave (expected) transition functions must have this feature.

This still leaves us with one of the most interesting ques-tions: can one nd income processes whose stationary distribu-

tion is positively skewed, but where a strict majority of thepopulation nonetheless opposes redistribution? The answer isafrmative, as we shall demonstrate through a simple Markovianexample. Let income take one of three values: X = {a1,a2 ,a3},with a1 < a2 < a3 . The transition probabilities between thesestates are independent across agents, and given by the Markovmatrix:

(7) M5

F1 2 r r 0

ps 1 2 s (1 2 p)s0 q 1 2 q G

,

where (p,q,r,s) [ (0, 1)4 . The invariant distribution induced by

FIGURE IIExpected Future Income under a Concave Transition Function

(Semi-Logarithmic Scale)


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M over {a1,a2,a3} is found by solving pM = p. It will be denotedby p = (p1,p2,p3), with mean m p1a1 + p2a2 + (1 2 p1 2p2)a3. We require that the mobility process and associated steadystate satisfy the following conditions:

(a) Next periods income yt+ 1i is stochastically increasing in

current income yti;17

(b) The median income level is a2 : p1 < 12 < p1 + p2;(c) The median agent is poorer than the mean: a2 < m;(d) The median agent has expected income above the mean:

E[yt+ 1i uyt

i = a2] > m.Conditions (b) and (c) together ensure that a strict majority of thepopulation would vote for current redistribution, while (b) and (d)together imply that a strict majority will vote against futureredistribution. In Benabou and Ok [1998] we provide sufcientconditions on (p,q,r,s; a1,a2,a3) for (a)(d) to be satised, andshow them to hold for a wide set of parameters. In the steadystate of such an economy the distribution of expected incomes is

negatively skewed, even though the distribution of actual incomesremains positively skewed and every one has rationalexpectations.

Granted that such income processes exist, one might stillask: are they at all empirically plausible? We shall present twospecications that match the broad facts of the U. S. incomedistribution and intergenerational persistence reasonably well.

First, let p = .55, q = .6, r = .5, and s = .7, leading to thetransition matrix,

M 5 F .5 .5 0.385 .3 .3150 .6 .4

Gand the stationary distribution (p1,p2,p3) = (.33,.44,.23). Thus, 77percent of the population is always poorer than average, yet 67percent always have expected income above average. In eachperiod, however, only 23 percent actually end up with realizedincomes above the mean, thus replicating the invariant distribu-tion. Choosing (a1,a2,a3) = (16000,36000,91000), we obtain a

17. Put differently, we posit that M = [mkl]333 is a monotone transitionmatrix, requiring row k + 1 to stochastically dominate row k : m1 1 $ m21 $ m31and m11 + m12 $ m21 + m22 $ m31 + m32. Monotone Markov chains wereintroduced by Keilson and Ketser [1977], and applied to the analysis of incomemobility by Conlisk [1990] and Dardanoni [1993].


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rather good t with the data, especially in light of the modelsextreme simplicity; see Table I, columns 1 and 2.

This income process also has more persistence for the lowerand upper income groups than for the middle class, which isconsistent with the ndings of Cooper, Durlauf, and Johnson[1994]. But most striking is its main political implication: a two-thirds majority of voters will support a policy or constitution

designed to implement a zero tax rate for all future generations,even though:

no deadweight loss concern enters into voters calculations;three-quarters of the population is always poorer than

average;the pivotal middle class, which accounts for most of the

laissez-faire coalition, knows that its children have less

than a one in three chance of making it into the upperclass.

The last column of Table I presents the results for a slightlydifferent specication, which also does a good job of matching the

TABLE IDISTRIBUTION AND PERSISTENCE OF INCOME IN THE UNITED STATES

Data (1990) Model 1 Model 2

Median family income ($) 35,353 36,000 35,000Mean family income ($) 42,652 41,872 42,260Standard deviation of family incomes ($) 29,203 28,138 27,499Share of bottom 100 p p1 = 33% of

population (%) 11.62 12.82 Share of middle 100 p p2 = 44% of

population (%) 39.36 37.46

Share of top 100 p p3 = 24% ofpopulation (%) 49.02 49.72

Share of bottom 100 p p1% = 39% ofpopulation (%) 14.86 18.5

Share of middle 100 p p2% = 37% ofpopulation (%) 36.12 30.8

Share of top 100 p p3% = 24% ofpopulation (%) 49.02 50.8

Intergenerational correlation of log-incomes 0.35 to 0.55 .45 0.51

Sources: median and mean income are from the 1990 U. S. Census (Table F-5). The shares presented hereare obtained by linear interpolation from the shares of the ve quintiles (respectively, 4.6, 10.8, 16.6, 23.8,and 44.3 percent) given for 1990 by the U. S. Census Bureau (Income Inequality Table 1). The variance iscomputed from the average income levels of each quintile in 1990 (Table F-3). Estimates of the intergenera-tional correlation from PSID or NLS data are provided by Solon [1992], Zimmerman [1992], and Mulligan[1995], among others.


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same key features of the data, and which we shall use later onwhen studying the effects of risk aversion. With (p,q,r,s) =(.45,.6,.3,.7), and (a1,a2,a3) = (20000,35000,90000), the tran-sition matrix is now

M5 F .7 .3 0.315 .3 .3850 .6 .4

G ,and the invariant distribution is (p1,p2,p3) = (.39,.37,.24). Themajority opposing future redistributions is now a bit lower, butstill 61 percent. Middle-class children now have about a 40 per-cent chance of upward mobility, and this will make a differencewhen we introduce risk aversion later on. Note that the source ofthese greater expected income gains is increased concavity in thetransition process, relative to the rst specication.

III.C. Multiperiod Redistributions

We now extend the general stochastic analysis to multiperiodredistributions, maintaining the assumption of risk neutrality (orcomplete markets). Agents thus care about the expected present

value of their net income over the T + 1 periods during which thechosen tax scheme is to remain in place. For any individual i, wedenote by QI t

i (Q1

i , . . . , Qti) the random sequence of shocks

which she receives up to date t, and by uIt

i (u

1

i , . . . , ut

i) asample realization. Given a one-step transition function f [ TP,her income in period t is

(8) yti 5 f(. . . , f( f(y0

i ; u1i ); u2

i ) . . . ; uti) ft(y0

i ; uI ti),

t 5 1, . . . , T,

whereft(y0i ; uI t

i) now denotes the t-step transition function. Under

laissez-faire, the expected present value of this income streamover the political horizon is

VT(y0i ) EQi . . . EQiF O

t= 0

T

dtytiU y0i G 5 O

t= 0

T

dtEQI tift(y; QI t

i)

5

Ot= 0T

dt

EQI tft

(y; QI t),

where we suppressed the index i on the random variables QI ti since

they all have the same probability distribution Pt(uI t)


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) k= 1t

P(uk) on Vt. Under the policy r1, on the other hand, agent

is expected income at each t is the expected mean EQIt[mF

t], which

by Fubinis theorem is also the mean expected income. The re-sulting payoff is

Ot= 0

T

dtE[mFt] 5 Ot= 0

T

dtS E0

1

EQ tj [ ft(y0

i ; QI tj)] djD

5

Ot= 0T

d

t

EXEQI t

[f

t(y; Q

It)]

dF0(y

),

so that agent i votes for r1 over r0 if and only if

(9) VT(y0i ) . E

X

VT(y) dF0(y).

It is easily veried that for transition functions which are concave(but not afne) in y with probability one, that is, for f [ TP, everyfunction ft(y; uI t

i), t $ 1, inherits this property. Naturally, so dothe weighted average St= 0

T dtft(y; uI ti) and its expectation VT(y),

for T $ 1. Hence, in this quite general setup, the now familiarresult:

PROPOSITION 3. Let F0 [ F, d [ (0, 1], T $ 1. For any mobility

process ( f, P) with f [ TP, there exists a unique y*f(T) < mF0such that all agents in [0, y*f(T)) vote for r1 over r0, while allthose in (y*f(T), 1] vote for r0 over r1.

Note that Proposition 3 does not cover the larger class oftransition processes T *P dened earlier, since V

T(y) need not beconcave if f is only concave in expectation. Can one obtain astronger result, similar to that of the deterministic case, namelythat the tipping point decreases as the horizon lengthens? Whilethis seems quite intuitive, and will indeed occur in the dataanalyzed in Section V, it may in fact not hold without relativelystrong additional assumptions. The reason is that the expectationoperator does not, in general, preserve the more concave thanrelation. A sufcient condition which insures this result is thatthe t + 1-step transition function be more concave in expectationthan the t-step transition function.

PROPOSITION 4. Let F0 [ F, d [ (0, 1], T $ 1, and let ( f, P) bea mobility process with f [ T *P. If, for all t, f

t+ 1( z ; QI t+ 1)


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Pt+ 1ft( z ; QI t), that is,

EQ1 . . . EQt+ 1[ft+ 1( z ; Q1, . . . , Qt+ 1)]

EQ1 . . . EQt[ft( z ; Q1, . . . , Qt)],

then the larger the political horizon T, the larger the share ofthe votes that go to r0 .

The interpretation is the same as that of Theorem 2(a): themore forward-looking the voters, or the more long-lived the tax

scheme, the lower is the political support for redistribution. Animmediate corollary is that this monotonicity holds when thetransition function is of the form f(y, u ) = yaf(u ), where a [ (0,1) and f can be an arbitrary function. This is the familiar loglinear model of income mobility, ln yt+ 1

i = aln yti + ln et+ 1

i , whichis widely used in the empirical literature.

IV. E XTENDING THE BASIC FRAMEWORK

IV.A. Risk-Aversion

When agents are risk averse, the fact that redistributivepolicies provide insurance against idiosyncratic shocks increasesthe breadth of their political support. Consequently, the cutoff

separating those who vote for r0 from those who prefer r1 may beabove or below the mean, depending on the relative strength ofthe prospect of mobility and the risk-aversion effects. While thetension between these two forces is very intuitive, no generalcharacterization of the cutoff in terms of the relative concavity ofthe transition and utility functions can be provided. To under-stand why, consider again the simplest setup where agents vote

at date 0 over the tax scheme for date 1. Denoting by U theirutility function, the cutoff falls below the mean if

EQ[U(f(EF0[y]; Q))] . U(EQEF0[f(y; Q)]),

where EF0[y] = mF

0denotes the expectation with respect to the

initial distribution F0. Observe that f( z , u ) [ T if and only if theleft-hand side is greater (for all U and P) than EQ[U(EF

0[ f(y;

Q)])]. But the concavity ofU, namely risk aversion, is equivalentto the fact that this latter expression is also smaller than theright-hand side of the above inequality. The curvatures of thetransition and utility functions clearly work in opposite direc-


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tions, but the cutoff is not determined by any simple composite ofthe two.18

One can, on the other hand, assess the outcome of this battleof the curvatures quantitatively. To that effect, let us return tothe simulations of the Markovian model reported in Table I, andask the following question: how risk averse can the agents withmedian income a2 be, and still vote against redistribution offuture incomes based on the prospects of upward mobility? As-suming CRRA preferences and comparing expected utilities un-der r0 and r1 , we nd that the maximum degree of risk aversion

is only 0.35 under the specication of column 2, but rises to 1under that of column 3. The second number is well within therange of plausible estimates, albeit still somewhat on the lowside.19 While the results from such a simple model need to beinterpreted with caution, these numbers do suggest that, withempirically plausible income processes, the POUM mechanismwill sustain only moderate degrees of risk aversion. The under-

lying intuition is simple: to offset risk aversion, the expectedincome gain from the POUM mechanism has to be larger, whichmeans that the expected transition function must be more con-cave. In order to maintain a realistic invariant distribution, theskewed idiosyncratic shocks must then be more important, whichis of course disliked by risk-averse voters.

IV.B. Nonlinear TaxationOur analysis so far has mostly focused on an all-or-nothing

policy decision, but we explained earlier that it immediatelyextends to the comparison of any two linear redistributionschemes. We also provided results on voting equilibrium withinthis class of linear policies. In practice, however, both taxes andtransfers often depart from linearity. In this subsection we shall

therefore extend the model to the comparison of arbitrary pro-gressive and regressive schemes, and demonstrate that our mainconclusions remain essentially unchanged.

When departing from linear taxation (and before mobilityconsiderations are even introduced), one is inevitably confronted

18. One case where complete closed-form results can be obtained is for the loglinear specication with lognormal shocks; see Benabou and Ok [1998].

19. Conventional macroeconomic estimates and values used in calibratedmodels range from .5 to 4, but most cluster between .5 and 2. In a recent detailedstudy of the income and consumption proles of different education and occupa-tion groups, Gourinchas and Parker [1999] estimate risk aversion to lie between0.5 and 1.0.


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with the nonexistence of a voting equilibrium in a multidimen-sional policy space. Even the simplest cases are subject to thiswell-known problem.20

One can still, however, restrict voters to a binary policychoice (e.g., a new policy proposal versus the status quo), andderive results on pairwise contests between alternative nonlinearschemes. This is consistent with our earlier focus on pairwisecontests between linear schemes, and will most clearly demon-strate the main new insights. In particular, we will show howMarhuenda and Ortuno-Ortins [1995] result for the static case

may be reversed once mobility prospects are taken into account.Recall that a general redistribution scheme was dened as a

mapping r : X 3 F R + which preserves total income. In whatfollows, we shall take the economys initial income distribution asgiven, and denote disposable income r(y; F) as simply r(y). Aredistribution scheme r can then equivalently be specied bymeans of a tax function T : X R , with collected revenue *X T(y)

dF rebated lump-sum to all agents, and the normalization T(0) 0 imposed without loss of generality. Thus,

(10) r(y) 5 y 2 T(y) 1 EX

T(y) dF, for all y [X.

We shall conne our attention to redistribution schemes with

the following standard properties: T and r are increasing andcontinuous on X, with *X T dF $ 0 and r $ 0. We shall say thatsuch a redistribution scheme is progressive (respectively, re-

gressive) if its associated tax function is convex (respectively,concave).21

To analyze how nonlinear taxation interacts with the POUMmechanism, we maintain the simple two-period setup of subsec-

tions II.A and III.A, and consider voters faced with the choicebetween a progressive redistribution scheme and a regressive one(which could be laissez-faire). The key question is then whether

20. For instance, if the policy space is that of piecewise linear, balancedschemes with just two tax brackets (which has dimension three), there is never aCondorcet winner. Even if one restricts the dimensionality further by imposing azero tax rate for the rst bracket, which then corresponds to an exemption, theproblem remains: there exist many (positively skewed) distributions for whichthere is no equilibrium.

21. Recall that with T(0) = 0, ifT is convex (marginally progressive), then itis also progressive in the average sense; i.e., T(x)/x is increasing. Similarly, if Tis concave, it is regressive both in the marginal and in the average sense.


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political support for the progressive scheme is lower when theproposed policies are to be enacted next period, rather than in thecurrent one. We shall show that the answer is positive, providedthat the transition function f is concave enough relative to thecurvature of the proposed (net) redistributive policy.

To be more precise, let rprog and rreg be any progressive andregressive redistribution schemes, with associated tax functionsTprog and Treg. Let F0 denote the initial income distribution, and

yF0

the income level where agents are indifferent between imple-menting rprog or rreg today. Given a mobility process f [ TP, let y*fdenote, as before, the income level where agents are indifferentbetween implementing rprog or rreg next period. The basic POUMhypothesis is that y*f < yF

0, so that expectations of mobility

reduce the political support for the more redistributive policy byF0(yF0) 2 F0(y*f). We shall establish two propositions that provide sufcient conditions for an even stronger result, namelyyF

0$ mF

0

> y*f.22 In other words, whereas the coalition favoring

progressivity in current scal policy extends to agents even richerthan the mean [Marhuenda and Ortuno-Ortin 1995], the coalitionopposing progressivity in future scal policy extends to agentseven poorer than the mean.

The rst proposition places no restrictions on F0 but focuseson the case where the progressive scheme Tprog involves (weakly)higher tax rates than Treg at every income level. In particular, it

raises more total revenue.

PROPOSITION 5. Let F0 [ F, let ( f, P) be a mobility process withf [ T *P, and let Tprog, Treg be progressive and regressive taxschemes with T9prog(0) $ T9reg(0). If

EQ[(Tprog 2 Treg)(f(y, Q))] is concave (not affine),

the political cutoffs for current and future redistributions aresuch that yF

0$ mF

0

> y*f.

Note that Tprog 2 Treg is a convex function, so the keyrequirement is that f be concave enough, on average, to dominatethis curvature. The economic interpretation is straightforward:

22. The rst inequality is strict unless Tre g 2 Tprog happens to be linear. Theabove discussion implicitly assumes that voter preferences satisfy a single-cross-ing condition, so that yF0 and y*f exist and are indeed tipping points. Such will bethe case in our formal analysis.


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the differential in an agents (or her childs) future expected taxbill must be concave in her current income.

Given the other assumptions, the boundary conditionT9prog(0) $ T9reg(0) implies that the tax differential Tprog 2 Treg isalways positive and increases with pretax income. This require-ment is always satised when Treg 0, which corresponds to thelaissez-faire policy. On the other hand, it may be too strong if theregressive policy taxes the poor more heavily. It is dropped in thenext proposition, which focuses on economies in steady-state andtax schemes that raise equal revenues.

PROPOSITION 6. Let ( f, P) be a mobility process with f [ T *P, anddenote by F the resulting steady-state income distribution.Let Tprog and Treg be progressive and regressive tax schemesraising the same amount of steady-state revenue: *X TprogdF = *X Treg dF. If

EQ[(Tprog 2 Treg)(f(y, Q))] is concave (not affine)

with EQ [(Tprog 2 Treg)( f(y#, Q))] $ 0, then the politicalcutoffs for current and future redistributions are such that

yF $ mF > y*f.

The concavity requirement and its economic interpretationare the same as above. The boundary condition now simply statesthat agents close to the maximum incomey# face a higher expected

future tax bill under Tprog than under Treg. This requirement isboth quite weak and very intuitive.

Propositions 5 and 6 demonstrate that the essence of ourprevious analysis remains intact when we allow for plausiblenonlinear taxation policies. At most, progressivity may increasethe extent of concavity in f required for the POUM mechanism tobe effective.23 Note also that all previous results comparing com-

plete redistribution and laissez-faire (Tprog(y) y, Treg(y) 0),or more generally two linear schemes with different marginalrates, immediately obtain as special cases.

V. MEASURING POUM IN THE DATA

The main objective of this paper was to determine whether

the POUM hypothesis is theoretically sound, in spite of its ap-

23. Recall that Propositions 5 and 6 provide sufcient conditions for y*f 0. Using the sameprocedure and horizons oft 3 7 years as before, we now compareeach deciles expected utility under laissez-faire with the utility ofreceiving the mean income for sure. The threshold where theycoincide is again computed by linear interpolation. The results,presented in Tables IIb to IId, indicate that even small amountsof risk aversion will dominate upward mobility prospects in theexpected utility calculations of the middle class. Beyond values of

27. Using data on how the main forms of political participation (voting, tryingto inuence others, contributing money, participating in meetings and campaigns,etc.) vary with income and education, Benabou [2000] computes the resulting biaswith respect to the median. It is found to vary between 6 percent for votingpropensities and 24 percent for propensities to contribute money, with most

values being above 10 percent.

TABLE IIaINCOME PERCENTILE OF THE POLITICAL CUTOFF: RISK-NEUTRAL AGENTS

Forecast horizon(years) 0 7 14 21

1. Mobility matrix: M6976

Annual incomes 63.4 61.8 54.2 48.8Five-year averages 63.4 60.8 56.4 52.9




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about b = 0.25 the threshold ceases to decrease with T, becominginstead either U-shaped or even monotonically increasing.

Our ndings can thus be summarized as follows:(a) a sizable fraction of the middle class can rationally look

forward to expected incomes that rise above average over

a horizon of ten to twenty years;(b) on the other hand, these expected income gains are small

enough, compared with the risks of downward mobility or

TABLE IIbINCOME PERCENTILE OF THE POLITICAL CUTOFF: RISK AVERSION = 0.10

Horizon (years) 0 7 14 21





TABLE IIcINCOME PERCENTILE OF THE POLITICAL CUTOFF: RISK AVERSION = 0.25






TABLE IIdINCOME PERCENTILE OF THE POLITICAL CUTOFF: RISK AVERSION = 0.50



Annual incomes 63.4 65.1 67.3 81.3

Five-year averages 63.4 64.3 65.6 69.1 2. Mobility matrix: M79

86



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stagnation, that they are not likely to have a signicantimpact on the political outcome, unless voters have verylow risk aversion and discount rates.

Being drawn from such a limited empirical exercise, theseconclusions should of course be taken with caution. A rst con-cern might be measurement error in the PSID data underlyingHungerfords [1979] mobility tables. The use of family incomerather than individual earnings, and the fact that replacingyearly incomes with ve-year averages does not affect our results,suggest that this is probably not a major problem. A second

caveat is that the data obviously pertain to a particular country,namely the United States, and a particular period. One may note,on the other hand, that the two sample periods, 1969 1976 and19791986, witnessed very different evolutions in the distribu-tion of incomes (relatively stable inequality in the rst, risinginequality in the second), yet lead to very similar results. Still,things might well be different elsewhere, such as in a developing

economy on its transition path. Finally, there is our compoundingthe seven-year mobility matrices to obtain forecasts at longerhorizons. This was dictated by the lack of sufciently detaileddata on longer transitions, but may well presume too much sta-tionarity in families income trajectories.

The above exercise should thus be taken as representing onlya rst pass at testing for the POUM effect in actual mobility data.

More empirical work will hopefully follow, using different orbetter data to further investigate the issue of how peoples sub-

jective and, especially, objective income prospects relate to theirattitudes vis-a-vis redistribution. Ravallion and Lokshin [2000]and Graham and Pettinato [1999] are recent examples based onsurvey data from post-transition Russia and post-reform Latin

American countries, respectively. Both studies nd a signicant

correlation between (self-assessed) mobility prospects and atti-tudes toward laissez-faire versus redistribution.

VI. CONCLUSION

In spite of its apparently overoptimistic avor, the prospectof upward mobility hypothesis often encountered in discussions of

the political economy of redistribution is perfectly compatiblewith rational expectations, and fundamentally linked to an intui-tive feature of the income mobility process, namely concavity.

Voters poorer than average may nonetheless opt for a low tax rate


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if the policy choice bears sufciently on future income, and if thelatters expectation is a concave function of current income. Thepolitical coalition in favor of redistribution is smaller, the moreconcave the expected transition function, the longer the durationof the proposed tax scheme, and the more farsighted the voters.The POUM mechanism, however, is subject to several importantlimitations. First, there must be sufcient inertia or commitmentpower in the choice of scal policy, governing parties or institu-tions. Second, the other potential sources of curvature in votersproblem, namely risk aversion and nonlinearities in the tax sys-

tem, must not be too large compared with the concavity of thetransition function.

With the theoretical puzzle resolved, the issue now becomesan empirical one, namely whether the POUM effect is largeenough to signicantly affect the political equilibrium. We madea rst pass at this question and found that this effect is indeedpresent in the U. S. mobility data, but likely to be dominated by

the value of redistribution as social insurance, unless voters havevery low degrees of risk aversion. Due to the limitations inherentin this simple exercise, however, only more detailed empiricalwork on mobility prospects (in the United States and other coun-tries) will provide a denite answer.

APPENDIX

Proof of Proposition 2

By using Jensens inequality, we observe that f g impliesthat

f(y

*f

)5

EX f dF0

=

EXh

(g

)dF0 , h

S EXg dF0

D5 h(g(y*g)) 5 f(yg)for some h [ T . The proposition follows from the fact that f isstrictly increasing.

QED

Proof of Theorem 1

Let F0 [ F+ , so that the median mF0is below the mean mF0.More generally, we shall be interested in any income cutoff h h; so y*f = h. By Proposition 2, therefore,the fraction of agents who support redistribution is greater (re-spectively, smaller) than F(h) for all f [ T which are more(respectively, less) concave than f. In particular, choosing h =mF0< mF0yields the claimed results for majority voting. QED

Proof of Theorem 2

For each t = 1, . . . , T, ft [ T , so by Proposition 1 there isa unique y*ft[ (0, mF) such that f

t(y*ft) = mFt. Moreover, since fT

fT21 . . . f, Proposition 2 implies that y*fT< y*fT21< . . . mFt+ 1forall t, from which it follows by a simple induction that

(A.3) ft(mF0) $ mFt 5 ft(y*f t)

with strict inequality for t > 1 and t < T, respectively. Let usnow dene the operators VT : T T and WT : T R as follows:

(A.4) VT( f) Ot= 0

T

dtft and

WT(f) EXVT(f) dF0 = Ot= 0

T

dtmFt.

Agent y achieves utility VT( f)(y) under laissez-faire, and utility


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WT( f) under the redistributive policy. Moreover, (A.3) impliesthat

VT( f )(mF0) 5 Ot= 0

T

dtft(mF0) . Ot= 0

T

dtmFt . Ot= 0

T

dtft(y*ft)

5 VT( f)(y*fT)

for any T $ 1. Since VT( f)[ is clearly continuous and increas-

ing, there must therefore exist a uniquey*f(

T)

[(yf

T

,mF0) suchthat

(A.5) VT( f)(y*f(T)) 5 Ot= 0

T

dtmFt 5 WT( f).

But since y*fT+ 1< y*fT, we have y*fT+ 1< y*f(T). This implies that

mFT+ 1= fT+ 1(y*fT+ 1) < fT+ 1(y*f(T)), and hence

VT+ 1( f )(y*f(T)) 5 Ot= 0

T+ 1

dtft(y*f(T)) > Ot= 0

T+ 1

dtmFt

5 VT+ 1( f)(y*f(T1 1)).

Therefore, y*f(T + 1) < y*f(T) must hold. By induction, weconclude that y*f (T9) < y*f(T) whenever T9 > T; part (a) of thetheorem is proved.

To prove part (b), we shall use again the family of piecewiselinear functions gh,adened in (A.1). Let us rst observe that thetth iteration of gh,a is simply

(A.6) (gh,a)t

(y) min {y, h 1 at

(y 2 h)} 5 gh,at(y).

In particular, both gh,1 : y y and gh,0 : y min {y, h} areidempotent. Therefore,

VT(gh,1)(h) 5 Ot= 0

T

dth , Ot= 0

T

dtmF0 5 WT(gh,1).

On the other hand, when the transition function is gh,0, the voterwith initial income h prefers r0 (under which she receives h ineach period) to r1 , if and only if


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h 1 Ot= 1

T

dth 5 VT(gh,0)(h) > WT(gh,0)

5 mF0 1 Ot= 1

T

dt EX

min {y, h} dF0,

or, equivalently,

(A.7)

mF0 2 h

h 2 *X min {y, h} dF0 , Ot= 1T

dt

5

d(1 2 dT)

1 2 d .

This last inequality is clearly satised for (d, 1/T) close enough to(1, 0). In that case, we have WT(gh,1) > St= 0

T dth > WT(gh,0).Next, it is clear from (A.4) and (A.1) that WT(gh,a) is continuousand strictly increasing in a. Therefore, there exists a uniquea(h) [ (0, 1) such that WT(gh,a(h)) = St= 0

T dth. This means that,

under the transition function f gh,a(h), we have

WT( f) 5 Ot= 0

T

dtft(h) 5 VT( f)( h),

so that an agent with initial income h is just indifferent betweenreceiving her laissez-faire income stream, equal to h in every

period, and the stream of mean incomes mFt. Moreover, underlaissez-faire an agent with initial y < h receivesy in every period,while an agent with y > h receives h + at(y 2 h) > h. Therefore,h is the cutoff y*f(T) separating those who support r0 from thosewho support r1, given f gh,a(h). This proves the rst statementin part (b) of the theorem.

Finally, by part (a) of the theorem, increasing (decreasing)

the horizonT

will reduce (raise) the cutoffy*f (

T) below (above) h.Applying these results to the particular choice of a cutoff equal to

median income, h mF0, completes the proof.

QED

Proof of Theorem 3

As in the proof Theorem 1, let F0 [ F+ , and consider any

income cutoff h < mF0. Recall the function gh,a(h)(y) which wasdened by (A.1) and (A.2) so as to ensure that mF

1

= h. (Onceagain, adding any positive constant to f would not change any-thing.) For brevity, we shall now denote a(h) and gh,a(h) as just a


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and g. Let us now construct a stochastic transition functionwhose expectation is g and which, together with F0, results in apositively skewed F1. Let p [ (0,1), and let Q be a random

variable taking values 0 and 1 with probabilities p and 1 2 p. Forany e [ (0, h), we dene f : X 3 V R + as follows:

c if 0 # y # h 2 e, f(y; u ) y for all uc if h 2 e < y # h,

f(y; u)

Hh2e if u 5 0 (probability p)

y 2 p(h 2 e)

1 2 pif u 5 1 (probability 1 2 p)

c if h # y # y#,

f(y; u) H h 2 e if u 5 0 (probability p)h 1 a(y 2 h) 1 pe1 2 p

if u 5 1 (probability 1 2 p).

By construction, EQ[ f(y; Q)] = g(y) for all y [ X; therefore,EQ [ f( z ; Q)] = g [ T . It remains to be checked that f(y; Q) isstrictly stochastically increasing in y. In other words, for any x [

X, the conditional distribution M(xuy) P({u [ V uf(y; u ) # x})must be decreasing in y on X, and strictly decreasing on a non-empty subinterval of X. But this is equivalent to saying that *Xh(x) dM(xuy) must be (strictly) increasing in y, for any (strictly)increasing function h : X R ; this latter form of the property is

easily veried from the above denition of f(y; u ).Because EQ[ f( z ; Q)] = g, so that mF

1

= h by (A.2), the cutoffbetween the agents who prefer r0 and those who prefer r1 is y*f =h. This tipping point can be set to any value below mF

0(i.e., 1 2

F0(h) can be made arbitrarily small), while simultaneously en-suring that F1(mF1) > s. Indeed,

F1(mF1) 5 F1(h) = p EX

1{ f( x,0)#h} dF0(x)

1 (1 2 p) EX

1{ f( x,1)#h} dF0(x)

5 p 1(1 2 p

)F0

(h 2 pe

),

so by choosing p close to 1 this expression can be made arbitrarilyclose to 1, for any given h.

To conclude the proof, it only remains to observe that a


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transition function f*

[ T is more concave than f in expectationif and only if EQ [ f*(

z ; Q)] EQ [ f( z ; Q)] = g. Proposition 2 thenimplies that the fraction of agents who support redistributionunder f

*is greater than F0(h). The reverse inequalities hold

whenever f P f*. As before, choosing the particular cutoffh =

mF0

< mF0

yields the claimed results pertaining to majorityvoting, for any distribution F0 [ F+ .

QED

Proof of Proposition 4Dene ht EQ1. . . EQtf

t( z ; Q1, . . . , Qt), t = 1, . . . , andobserve that ht [ T and ht+ 1 ht for all t under the hypothesesof the proposition. The proof is thus identical to part (a) of The-orem 2, with ht playing the role of f

t.i


1) Properties of yF0. By the mean value theorem, there existsa point yF0[ X such that

(A.8) (Tprog 2 Treg)(yF0) 5 EX

(Tprog 2 Treg) dF0,

which by (10) means that rprog(yF0) = rreg(yF0). Now, since Tprog2 Treg is convex, Jensens inequality allows us to write

(A.9) (Tprog 2 Treg)(mF0)

# EX

(Tprog 2 Treg) dF0 5 (Tprog 2 Treg)(yF0).

With (Tprog 2 Treg)9(0) $ 0, the convexity ofTprog 2 Treg impliesthat it is a nondecreasing function on [0,y#]. In the trivial casewhere it is identically zero, indifference obtains at every point, sochoosing yF = mF

0immediately yields the results. Otherwise, two

cases are possible. Either Tprog 2 Treg is a strictly increasinglinear function; or else it is convex (not afne) on some subinter-

val, implying that the inequality in (A.9) is strict. Under eitherscenario (A.8) implies that yF

0is the unique cutoff point between

the supporters of the two (static) policies, and (A.9) requires thatyF

0$ mF

0.


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2) Properties of y*f. By the mean value theorem, there existsa point y*f [ X such that

(A.10) EQ[(Tprog 2 Treg)(f(y*f, Q))]

5 EX

EQ[(Tprog 2 Treg)( f(y, Q))] dF0(y),

or equivalently EQ[rprog( f(y*f,Q))] = EQ[rreg( f(y*f,Q))].Moreover,

(A.11) EX

EQ[(Tprog 2 Treg)( f(y, Q))] dF0(y)

, EQ[(Tprog 2 Treg)(f(mF0, Q))],

by Jensens inequality for the concave (not afne) functionEQ [(Tprog 2 Treg) + f( z , Q)]. This function is also nondecreasing,since Tprog 2 Treg has this property and f [ T P is increasing in

y with probability one. Therefore, y*f is indeed the cutoff pointsuch that EQ[rprog( f(y, Q))] : EQ [rreg( f(y, Q)] as y " y*f, andmoreover (A.11) requires that mF

0

> y*f.

QED


1) Properties of yF. The proofs of (A.8) and (A.9) proceed asabove, except that the right-hand side of (A.8) is now equal to zerodue to the equal-revenue constraint. Since Tprog 2 Treg is convex,equal to zero at y = 0, and has a zero integral, it must again bethe case barring the trivial case where it is zero everywhere) thatit is rst negative on (0,yF), then positive on (yF,y#). Therefore,yF

is indeed the threshold such that rprog(y) : rreg(y) as y " yF, andmoreover, by (A.8) it must be that mF # yF.

2) Properties of y*f. Since the distribution F is, by denition,invariant under the mobility process ( f,P), so is the revenueraised by either tax scheme. This implies that the functionEQ [(Tprog 2 Treg)( f(y*f, Q))], like Tprog 2 Treg, sums to zero over[0,y#]:

(A.12) EX

EQ[(Tprog 2 Treg)(f(y, Q))] dF(y)


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5 EX

EV

(Tprog 2 Treg)(f(y,u )) dF(y) dP(u )

5 EX

(Tprog 2 Treg)(x) dF(x) = 0.

Therefore, by the mean value theorem there is at least one pointy*f where

EQ[(Tprog 2 Treg)(f(y*f,Q))] 5 0.Furthermore, since this same function is concave (not afne),Jensens inequality implies that

(A.13) EQ[(Tprog 2 Treg)(f(mF,Q))]

. EX

EQ[(Tprog 2 Treg)(f(y,Q))] dF(y) 5 0.

Finally, observe that since the function EQ [(rprog 2 rreg)( f( z ,Q))] is convex (not afne) on [0,y#], equal to zero at y*f andnonnegative at the right boundary y#, it must rst be negative on(0, y*f), then positive on (y*f, y#). Therefore,y*f is indeed the uniquetipping point, meaning that EQ[rprog( f(y, Q))] : EQ[rreg( f(y,Q))] as y " y*f; moreover, by (A.13) we must have mF > y*f.

QED

PRINCETON UNIVERSITY, THE NATIONAL BUREAU OF ECONOMIC RESEARCH, AND THE

CENTRE FOR ECONOMIC POLICY RESEARCH

NEW YORK UNIVERSITY

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