social mobility and the demand for redistribution: …rbenabou/papers/d8zkmee3.pdf · social...

SOCIAL MOBILITY AND THE DEMAND FORREDISTRIBUTION 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 ldquoprospect of upward mobilityrdquo (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 probablydominated by the demand for social insurance

ldquoIn the future everyone will be world-famous for fteen minutesrdquo[Andy Warhol 1968]

INTRODUCTION

The following argument is among those commonly advancedto 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 averageit 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 policies1 For instance Okun [1975 p 49]relates that ldquoIn 1972 a storm of protest from blue-collar workersgreeted Senator McGovernrsquos proposal for conscatory estate taxesThey 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 UniversitatPompeu Fabra Princeton University and Universite de Toulouse for their helpfulcomments The rst author gratefully acknowledges nancial support from theNational Science Foundation (SBRndash9601319) and the MacArthur FoundationBoth authors are grateful for research support to the C V Starr Center at NewYork University

1 See for example Roemer [1998] or Putterman [1996] The prospect ofupward mobility hypothesis is also related to the famous ldquotunnel effectrdquo 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 othersThere 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(eg 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

copy 2001 by the President and Fellows of Harvard College and the Massachusetts Institute ofTechnologyThe Quarterly Journal of Economics May 2001

447

The silent majority did not want the yacht clubs closed forever totheir children and grandchildren while those who had alreadybecome members kept sailing alongrdquo

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 mobilitymdashaform of what Marxist writers refer to as ldquofalse consciousnessrdquo2

The ldquoprospect of upward mobilityrdquo (POUM) hypothesis hasto 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 averagemdashfor in-stance the median votermdashexpect to become richer than averageThis ldquooptimisticrdquo 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 tomorrowrsquos expected income lies some-where between todayrsquos 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 thusappearsmdashand economists have often concludedmdashthat 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

patible with everyone holding realistic views of their incomeprospects3

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) tomorrowrsquos expected income is anincreasing and concave function of todayrsquos income The moreconcave the transition function and the longer the length of timefor which taxes are preset the lower the demand for redistribu-tion Even the median votermdashin fact even an arbitrarily poorvotermdashmay 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-statedistribution 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 currentresources 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 ldquotechnologyrdquo 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 ldquovotingagainst wealth taxation to preserve the good fortune of onersquos 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 rangerdquo

449SOCIAL MOBILITY AND REDISTRIBUTION POUM

The key role played by concavity in the POUM mechanismmay be best understood by rst considering a very stripped-downexample Suppose that agents decide today between ldquolaissez-fairerdquo and complete sharing with respect to next periodrsquos incomeand that the latter is a deterministic function of current incomey 9 = f( y) for all y in some interval [0y ] Without loss of gener-ality normalize f so that someone with income equal to theaverage m maintains that same level tomorrow ( f( m ) = m ) Asshown 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 fmdashmore specicallyJensenrsquos inequalitymdashmeans that the losses of the rich sum tomore than the gains of the poor therefore tomorrowrsquos per capitaincome m 9 is below m An agent with mean initial income or evensomewhat poorer can thus rationally expect to be richer thanaverage in the next period and will therefore oppose futureredistributions

To provide an alternative interpretation let us now normal-ize the transition function so that tomorrowrsquos and todayrsquos mean

FIGURE IConcave Transition Function

Note that A lt B therefore m 9 lt f( m ) The gure also applies to the stochasticcase with f replaced by Ef everywhere


incomes coincide m 9 = m The concavity of f can then be inter-preted as saying that y 9 is obtained from y through a progressivebalanced 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 y 9 of a person with initial incomem is strictly greater than m hence greater than the average ofy 9 across agents This person and those with initial incomesnot too far below will therefore be hurt if future incomes areredistributed4

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 ldquomake it bigrdquo is some-what implicit in casual descriptions of the POUM hypothesismdashsuch as Okunrsquos 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 incomesThe 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 balancebetween 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 redistributivepolicies 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 (eg

4 The concavity of f implies that f(x)x is decreasing which corresponds to ldquotaxrdquoprogressivity and Lorenz equalization on any interval [yIy] such that yI f 9 (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 Jensenrsquos inequality given the normalization m 9 = 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


Alesina and Rodrik [1994] and Persson and Tabellini [1994]) Akey mechanism in this class of models is that of a poor medianvoter who chooses high tax rates or other forms of expropriationwhich 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 ratemdashindependently of any deadweight lossconsiderations

While sharing the same general motivation as Piketty[1995a 1995b] our approach is quite different Pikettyrsquos mainconcern 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 whoknow 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 invariant5 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]


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 [ [01] whose initial levels ofincome lie in some interval X ordm [0y ] 0 lt y ` 6 An incomedistribution is dened as a continuous function F X reg [01] suchthat F(0) = 0 F( y ) = 1 and m F ordm X y dF lt ` We shall denoteby F the class of all such distributions and by F+ the subset ofthose whose median mF ordm min [F 2 1(12)] is below their meanWe shall refer to such distributions as positively skewed andmore generally we shall measure ldquoskewnessrdquo 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 regR + which assigns to each pretax income and initial distribution alevel of disposable income r( y F) while preserving total incomeX r( y F) dF( y) = m F 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 commonframework

The class of redistributive schemes used in a vast majorityof political economy models is that of proportional schemeswhere all incomes are taxed at the rate t and the collectedrevenue is redistributed in a lump-sum manner7 We denote this

6 More generally the income support could be any interval [yI y ] yI $ 0We 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-dimensionalone thereby allowing the use of the median votertheorem By contrast when unrestricted nonlinear redistributive schemes are


class as P ordm r t u 0 t 1 where r t ( y F) ordm (1 2 t ) y + t m F forall y [ X and F [ F We shall mostly work with just the twoextreme members of P namely r0 and r1 Clearly r0 correspondsto the ldquolaissez-fairerdquo policy whereas r1 corresponds to ldquocompleteequalizationrdquo

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 r t and r t 9 with 0 t lt t 9 1 Second r0 and r1 arein a certain sense ldquofocalrdquo members of P since in the simplestframework where one abstracts from taxesrsquo 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 individualsrsquo votingbehavior also incorporates concerns about their future incomesFinally 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 inertiaand status quo bias that characterize the policy-making processespecially 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 policymust 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 redistribution8

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 asOkunrsquos citation) Moreover these dynamic voting games are notoriously intrac-table so the nearly universal practice in political economy models is to assume


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 u t+ 1i ) t 5 0 T 2 1

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

tion of a random shock Q t+ 1i 9 We require this stochastic process

to have the following properties(i) The random variables Q t

i (it) [ [0 1] 3 1 Thave a common probability distribution function P withsupport V

(ii) The function f X 3 V reg X is continuous with awell-dened expectation E Q [f( z Q )] on X

(iii) Future income increases with current income in thesense of rst-order stochastic dominance for any(yy 9 ) [ X 2 the conditional distribution M(y 9 u y) ordm prob( u [ V u f(y u ) y 9 ) is decreasing in y with strictmonotonicity 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 (eg 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 genderrace 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 canessentially 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 ( Q t

i = Q tj

for all i j in [0 1])

ldquomyopicrdquo 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 policiesBenabou [1999 2000] develops such a model using specic functional and distri-butional assumptions


purely idiosyncratic shocks (the Q ti rsquos are independent across

agents and sum to zero) and all cases in between10

The second condition is a minor technical requirement Thethird condition implies that expected income E Q [ 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 u t

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 DYNAMICS 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 incomestream of an individual with initial endowment y [ X is then yf( y) f 2( y) f t( y) and for any initial F [ F thecross-sectional distribution of incomes in period t is Ft ordm F + f 2 tA 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

IIA Two-Period Analysis

To distill our main argument into its most elementary formwe focus rst on a two-period (or overlapping generations) sce-nario where individuals vote ldquotodayrdquo (date 0) over alternativeredistribution schemes that will be enacted only ldquotomorrowrdquo (date1) For instance the predominant motive behind agentsrsquo 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 randomvariables and thus treat each Q t

i as jointly measurable for any t Consequentlythe law of large numbers and Fubinirsquos theorem are applied as usual


subject to the tax policy designed by the current generationAccordingly 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 m F1

Suppose now that f [ T that is it is concave but not afne Thenby Jensenrsquos inequality

(3) f( m F0) 5 f S E

X

y dF0 D EX

f dF0 5 m F1

so the agent with average income at date 0 will oppose date 1redistributions On the other hand it is clear that f (0) lt m F1

sothere must exist a unique yf in (0 m F0

) such that f ( yf ) ordm m F1 Of

course yf = f 2 1( m F 0+ f 2 1) also depends on F0 but for brevity we donot make this dependence explicit in the notation Since f isstrictly increasing it is clear that yf acts as a tipping point inagentsrsquo attitudes toward redistributions bearing on future in-come Moreover since Jensenrsquos inequalitymdashwith respect to alldistributions F0mdash characterizes concavity the latter is both nec-essary and sufcient for the prospect of upward mobility hypothe-sis to be valid under any linear redistribution scheme11

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

(a) f is concave (but not afne) ie f [ T(b) For any income distribution F0 [ F there exists a unique

yf lt m F0such that all agents in [0 yf ) vote for r1 over r0

while all those in (yf y ] vote for r0 over r1 Yet another way of stating the result is that f [ T if it is

skewness-reducing for any initial F0 next periodrsquos distributionF1 = F0 + f 2 1 is such that F1( m F1) lt F0( m F 0

) 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( m F 0

) 2 F1( m F 1) =F0( m F 0

) 2 F0( yf ) gt 0 The underlying intuition also suggests

11 For any r t and r t 9 in P such that 0 t lt t 9 1 agent y [ X votes for r t 9over r t iff (1 2 t ) f( y) + t m F 1

lt (1 2 t 9 ) f( y) + t 9 m F 1 which in turn holds if and

only if (2) holds Thus as noted earlier nothing is lost by focusing only on the twoextreme schemes in P namely r0 and r1


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 Acirc 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 Acirc gif and only if f + g 2 1 [ T

PROPOSITION 2 Let F0 [ F and f g [ T Then f Acirc g implies thatyf lt yg

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 howeverthat for any given pretax income distribution F0 there exists atransition function f which is ldquoconcave enoughrdquo that a majority ofvoters choose laissez-faire over redistribution12 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 thetransition function must be sufciently concave to make the date1 income distribution F1 negatively skewed Indeed if yf =f 2 1( m F1

) lt mF0 then m F1

lt f(mF 0) = mF 1

There are two reasonswhy 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 1mdashlike every other result in the paper concerning median incomemF 0

mdash holds in fact for any arbitrary income cutoff below m F 0(see the proof in the

Appendix)


a stochastic version of Theorem 1 where F1 can remain asskewed as one desires Second it may in fact not be necessarythat the cutoff yf 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 ldquoone person one voterdquo ideal needs to be for redistributivepolicies or parties advocating them to be defeated

IIB 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 morerealistic 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

d tf t( y) Ot= 0

T

d t m Ft

where we recall that f t denotes the tth iteration of f and Ft ordmF0 + f 2 t is the period t income distribution with mean m F t

We shall see that there again exists a unique tipping point

yf (T) such that all agents with initial income less than yf (T) votefor r1 while all those richer than yf (T) vote for r0 When thepolicy has no lasting effects this point coincides with the meanyf (0) = m F 0

When future incomes are factored in the coalition infavor of laissez-faire expands yf (T) lt m F0

for T $ 1 In fact themore farsighted voters are or the longer the duration of theproposed tax scheme the less support for redistribution there willbe yf (T) is strictly decreasing in T If agents care enough aboutfuture 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]


(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 2when forecasting incomes further into the future the one-steptransition f gets compounded into f 2 f T etc and each ofthese functions is more concave than its predecessor13

III INCOME DYNAMICS AND VOTING UNDER UNCERTAINTY

The assumption that individuals know their future incomeswith 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 processyt+ 1

i = f( yti u t+ 1

i ) satisfying the basic requirements (i)ndash(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 T P

be the set of transition functions such that prob[ u u f( z u ) [ T ]= 1 It is clear that for any f in T P

(iv) The expectation E Q [f( z Q )] is concave (but not afne)on X

For some of our purposes the requirement that f [ T P 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


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

IIIA 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) E Q i[ f(yi Q i)] E[ m F1]

where the subscript Q i on the left-hand side indicates that theexpectation is taken only with respect to Q i for given yi Whenshocks are purely idiosyncratic the future mean m F 1

is 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[m F1] 5 E F E

0

1

f(yj Q j) dj G5 E

0

1

E[ f(yj Q j)] dj

5 EX

E Q i[ f(y Q i)] dF0(y)

by Fubinirsquos theorem This is less than the expected income of anagent whose initial endowment is equal to the mean level m F 0

whenever f( y u )mdashor more generally E Q i[ f( y Q i)]mdashis concavein y

(6) EX

E Q i[ f(y Q i)] dF0(y) E Q i[ f( m F0 Q i)]

Consequently there must again exist a nonempty interval ofincomes [ yf m F 0

] in which agents will oppose redistribution withthe cutoff yf dened by

E Q i[ f(yf Q i)] 5 E[ m F1]

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


whether enough of it can still cause r0 to beat r1 under majorityvoting observe that the inequality in (6) involves only the ex-pected transition function E Q i[ f( y Q i)] rather than f itself Thisleads us to replace the ldquomore concave thanrdquo relation with a ldquomoreconcave in expectation thanrdquo relation Given any probability dis-tribution P we dene this ordering on the class T P as

f AcircP g if and only if E Q [ f( z Q )] Acirc E Q [g( z Q )]

where Q is any random variable with distribution P14 It is easilyshown that f AcircP g implies that yf lt yg In fact making f concaveenough in expectation will as before drive the cutoff yf below themedian mF 0

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 F1mdashin sharp contrast to whatoccurred in the deterministic case In particular a sufcientlyskewed distribution of shocks will ensure that F1 [ F+ withoutaffecting the cutoff yf This dichotomy between expectations andrealizations is the second key component of the POUM mecha-nism and allows us to establish a stochastic generalization ofTheorem 115

THEOREM 3 For any F0 [ F+ and any s [ (0 1) there exists amobility process ( f P) with f [ T P such that F1( m F1

) $ sand under pairwise majority voting r0 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 E Q [ f( z Q )] is necessary and suf-cient for the political support behind the laissez-faire policy toincrease when individualsrsquo voting behavior takes into accounttheir future income prospects If f is concave enough in expecta-

14 Interestingly Acirc and AcircP 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 AcircP 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 everyonersquos expected income coincides with themean) precisely because it is nowhere strictly concave


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

IIIB Steady-State Distributions

The presence of idiosyncratic uncertainty is not only realisticbut 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 stablepermanent ones as well

Let P 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 yf lt m F 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 m F have expected incomesgreater than m F as do all those who start above this mean giventhat the number of people on either side of m F 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( yf F) strictly exceeds the number who actually endup with realized incomes above the mean 1 2 F( m F ) wheneverf is concave in expectation This result is apparent in Figure IIwhich provides additional intuition by plotting each agentrsquos ex-pected income path over future dates E[ yt

i u y0i ] In the long run

everyonersquos expected income converges to the population mean m F but this convergence is nonmonotonic for all initial endowments

16 If the inequality yf lt m F is required to hold only for the steady-statedistribution(s) F induced by f and P rather than for all initial distributionsconcavity of E Q [ f( z Q )] is still a sufcient condition but no longer a necessaryone Nonetheless some form of concavity ldquoon averagerdquo is still required so to speakif E Q [ f( z Q )] were linear or convex we would have yf $ m F for all distributionsincluding stationary ones


in some interval (yI F y F) around m F In particular for y0i

[(yI F m F) 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 a3with a1 lt a2 lt a3 The transition probabilities between thesestates are independent across agents and given by the Markovmatrix

(7) M 5 F 1 2 r r 0ps 1 2 s (1 2 p)s0 q 1 2 q

G

where ( pqrs) [ (0 1)4 The invariant distribution induced by

FIGURE IIExpected Future Income under a Concave Transition Function

(Semi-Logarithmic Scale)


M over a1 a2 a3 is found by solving p M = p It will be denotedby p = ( p 1 p 2p 3) with mean m ordm p 1a1 + p 2a2 + (1 2 p 1 2p 2)a3 We require that the mobility process and associated steadystate satisfy the following conditions

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

current income yti17

(b) The median income level is a2 p 1 lt 12 lt p 1 + p 2 (c) The median agent is poorer than the mean a2 lt m (d) The median agent has expected income above the mean

E[yt+ 1i u yt

i = a2] gt 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 ( pqrs a1 a2 a3) for (a)ndash(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 isnegatively 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 wellFirst let p = 55 q = 6 r = 5 and s = 7 leading to thetransition matrix

M 5 F 5 5 0385 3 315

0 6 4G

and the stationary distribution ( p 1 p 2 p 3) = (334423) 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) = (160003600091000) we obtain a

17 Put differently we posit that M = [mkl]3 3 3 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]


rather good t with the data especially in light of the modelrsquosextreme 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 constitutiondesigned to implement a zero tax rate for all future generationseven though

mdashno deadweight loss concern enters into votersrsquo calculationsmdashthree-quarters of the population is always poorer than

averagemdashthe pivotal middle class which accounts for most of the

laissez-faire coalition knows that its children have lessthan a one in three chance of ldquomaking itrdquo 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 ($) 35353 36000 35000Mean family income ($) 42652 41872 42260Standard deviation of family incomes ($) 29203 28138 27499Share of bottom 100 p p 1 = 33 of

population () 1162 1282 mdashShare of middle 100 p p 2 = 44 of

population () 3936 3746 mdashShare of top 100 p p 3 = 24 of

population () 4902 4972 mdashShare of bottom 100 p p 1 = 39 of

population () 1486 mdash 185Share of middle 100 p p 2 = 37 of

population () 3612 mdash 308Share of top 100 p p 3 = 24 of

population () 4902 mdash 508Intergenerational correlation of log-incomes 035 to 055 45 051

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 46 108 166 238and 443 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


same key features of the data and which we shall use later onwhen studying the effects of risk aversion With ( pqrs) =(45637) and (a1 a2 a3) = (200003500090000) the tran-sition matrix is now

M 5 F 7 3 0315 3 385

0 6 4G

and the invariant distribution is ( p 1 p 2 p 3) = (393724) 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

IIIC 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 presentvalue 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 Q I t

i ordm ( Q 1i Q t

i) the random sequence of shockswhich she receives up to date t and by uI t

i ordm ( u 1i u t

i) asample realization Given a one-step transition function f [ T Pher income in period t is

(8) yti 5 f( f( f( y0

i u 1i ) u 2

i ) u ti) ordm f t( y0

i uI ti)

t 5 1 T

where f t( y0i uI t

i) now denotes the t-step transition function Underlaissez-faire the expected present value of this income streamover the political horizon is

VT( y0i ) ordm E Q i E Q i F O

t= 0

T

d tyti U y0

i G 5 Ot= 0

T

d tE Q I ti f t(y Q I t

i)

5 Ot= 0

T

d tE Q I tf t(y Q I t)

where we suppressed the index i on the random variables Q I ti since

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


) k= 1t P( u k) on V t Under the policy r1 on the other hand agent

irsquos expected income at each t is the expected mean E Q I t[ m F t

] whichby Fubinirsquos theorem is also the mean expected income The re-sulting payoff is

Ot= 0

T

d tE[ m F t] 5 O

t= 0

T

d t S E0

1

EQ tj [ f t(y0

i Q I tj)] dj D

5 Ot= 0

T

d t EX

E Q I t[ f t(y Q I t)] dF0(y)

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

(9) V T( 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 [ T P everyfunction f t( y uI t

i) t $ 1 inherits this property Naturally so dothe weighted average S t= 0

T d tf t( y uI ti) and its expectation V T ( 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 mobilityprocess ( f P) with f [ T P there exists a unique yf (T) lt m F0

such that all agents in [0 yf (T)) vote for r1 over r0 while allthose in ( yf (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 ldquomore concave thanrdquorelation 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 Q I t+ 1)


AcircP t+ 1 f t( z Q I t) that is

E Q 1 E Q t+ 1[ f t+ 1( z Q 1 Q t+ 1)]

Acirc E Q 1 E Q t[ f t( z Q 1 Q t)]

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 taxscheme 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 ) = y a f ( u ) where a [ (01) and f can be an arbitrary function This is the familiar loglinear model of income mobility ln yt+ 1

i = a ln yti + ln e t+ 1

i whichis widely used in the empirical literature

IV EXTENDING THE BASIC FRAMEWORK

IVA Risk-Aversion

When agents are risk averse the fact that redistributivepolicies provide insurance against idiosyncratic shocks increasesthe breadth of their political support Consequently the cutoffseparating 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 voteat date 0 over the tax scheme for date 1 Denoting by U theirutility function the cutoff falls below the mean if

E Q [U( f(EF0[y] Q ))] U(E Q EF0

[ f(y Q )])

where EF0[ y] = m F 0

denotes the expectation with respect to theinitial distribution F0 Observe that f( z u ) [ T if and only if theleft-hand side is greater (for all U and P) than E Q [U(EF 0

[ f( yQ )])] But the concavity of U 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-


tions but the cutoff is not determined by any simple composite ofthe two18

One can on the other hand assess the outcome of this ldquobattleof the curvaturesrdquo 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 aversionis only 035 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 lowside19 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

IVB Nonlinear Taxation

Our analysis so far has mostly focused on an all-or-nothingpolicy 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 shalltherefore 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 between05 and 10


with the nonexistence of a voting equilibrium in a multidimen-sional policy space Even the simplest cases are subject to thiswell-known problem20

One can still however restrict voters to a binary policychoice (eg 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-Ortinrsquos [1995] result for the static casemay be reversed once mobility prospects are taken into account

Recall that a general redistribution scheme was dened as amapping r X 3 F reg R + which preserves total income In whatfollows we shall take the economyrsquos 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 reg R with collected revenue X T( y)dF rebated lump-sum to all agents and the normalization T(0) ordm0 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 withthe 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 (respectivelyconcave)21

To analyze how nonlinear taxation interacts with the POUMmechanism we maintain the simple two-period setup of subsec-tions IIA and IIIA 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 if T is convex (marginally progressive) then itis also progressive in the average sense ie T( x)x is increasing Similarly if Tis concave it is regressive both in the marginal and in the average sense


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 andyF0

the income level where agents are indifferent between imple-menting rprog or rreg today Given a mobility process f [ T P let yfdenote as before the income level where agents are indifferentbetween implementing rprog or rreg next period The basic POUMhypothesis is that yf lt yF0

so that expectations of mobilityreduce the political support for the more redistributive policy byF0( yF 0

) 2 F0( yf ) We shall establish two propositions that pro-vide sufcient conditions for an even stronger result namelyyF0

$ m F 0gt yf

22 In other words whereas the coalition favoringprogressivity 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 itraises 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 T 9prog(0) $ T 9reg(0) If

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

the political cutoffs for current and future redistributions aresuch that yF 0

$ m F 0gt yf

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 Treg 2 Tprog happens to be linear Theabove discussion implicitly assumes that voter preferences satisfy a single-cross-ing condition so that yF0

and yf exist and are indeed tipping points Such will bethe case in our formal analysis


the differential in an agentrsquos (or her childrsquos) future expected taxbill must be concave in her current income

Given the other assumptions the boundary conditionT 9prog(0) $ T 9reg(0) implies that the tax differential Tprog 2 Treg isalways positive and increases with pretax income This require-ment is always satised when Treg ordm 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 distributionLet Tprog and Treg be progressive and regressive tax schemesraising the same amount of steady-state revenue X TprogdF = X Treg dF If


with E Q [(Tprog 2 Treg)( f ( y Q ))] $ 0 then the politicalcutoffs for current and future redistributions are such thatyF $ m F gt yf

The concavity requirement and its economic interpretationare the same as above The boundary condition now simply statesthat agents close to the maximum income y face a higher expectedfuture 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 effective23 Note also that all previous results comparing com-plete redistribution and laissez-faire (Tprog( y) ordm y Treg( y) ordm 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 whetherthe POUM hypothesis is theoretically sound in spite of its ap-

23 Recall that Propositions 5 and 6 provide sufcient conditions for yf ltm F 0

yF 0 They are thus probably stronger than actually necessary especially for

the basic POUM result which is only that yf lt yF 0


parently paradoxical nature As we have seen the answer isafrmative The next question that naturally arises is whetherthe POUM effect is present in the actual data and if so whetherit is large enough to matter for redistributive politics

Our purpose here is not to carry out a large-scale empiricalstudy or detailed calibration but to show how the POUM effectcan be measured quite simply from income mobility and inequal-ity data We shall again start with the benchmark case of risk-neutral agents then introduce risk aversion The rst questionthat we ask is thus the following at any given horizon what is theproportion of agents who have expected future incomes strictlyabove the mean In particular does it increase with the length ofthe forecast horizon and does it eventually rise above 50percent24

Rather than impose a specic functional form on the incomeprocess we shall use here the more exible description providedby empirical mobility matrices These are often estimated fortransitions between income quintiles which is too coarse a gridfor our purposes especially around the median We shall there-fore use the more disaggregated data compiled by Hungerford[1993] from the Panel Study on Income Dynamics (PSID)namely

(a) interdecile mobility matrices for the periods 1969ndash1976and 1979ndash1986 denoted M69

76 and M7986 respectively Each

of those is in fact computed in two different ways usingannual family incomes and using ve-year averages cen-tered on the rst and last years of the transition periodso as to reduce measurement error

(b) mean income for each decile in 1969 and 1979 We shalltreat each decile as homogeneous and denote the vectorsof relative incomes as a69 and a79 A ldquoprimerdquo will denotetransposition

Let us start by examining these two income distributions

a 969 5 (211 410 566 696 822 947

1104 1302 1549 2393)

24 Recall that there is no reason a priori (ie absent some concavity in thetransition function) why either effect should be observed in the data since theseare not general features of stationary processes Rather than just ldquomean-revert-ingrdquo the expected income dynamics must be ldquomean-crossing from belowrdquo oversome range as in Figure II


a 979 5 (179 358 523 669 801 933

1084 1289 1588 2576)

In both years the median group earned approximately 80 percentof mean income while those with the average level of resourceswere located somewhere between the sixtieth and seventiethpercentiles More precisely by linear interpolation we can esti-mate the size of the redistributive coalition to be 634 percent in1969 and 644 percent in 1979

Next we apply the appropriate empirical transition matrix tocompute the vector of conditionally expected relative incomes t 37 years ahead namely (M69

76)t z a69 or (M7986)t z a79 for t = 1

3 The estimated rank of the cutoff yf t where expected futureincome equals the population mean is then obtained by linearinterpolation of these decile values By iterating a seven-yeartransition matrix to compute mobility over 14 and 21 years weare once again treating the transition process as stationarySimilarly by using the initial income distribution vectors we areabstracting from changes in the decilesrsquo relative incomes duringthe transition period These are obviously simplifying approxima-tions imposed by the limitations of the data25

The results presented in Table IIa are consistent across allspecications the POUM effect appears to be a real feature of theprocess of socioeconomic mobility in the United Statesmdash even atrelatively short horizons but especially over longer ones26 Itaffects approximately 35 percent of the population over sevenyears and 10 percent over fourteen years This is far from neg-ligible especially since the differential rates of political partici-pation according to socioeconomic class observed in the UnitedStates imply that the pivotal agent is almost surely located above

25 As a basic robustness check we used the composite matrix M6976

z M7986 to

recompute the fourteen-year transitions We also applied the transition matricesM69

76 M6976

z M7 98 6 and relevant iterations to the (slightly more unequal) income

distribution a7 9 instead of a69 The results (not reported here) did not change26 Tracing the effect back to its source one can also examine to what extent

expected future income is concave in current income The expected transition is infact concave over most but not all of its domain of the nine slopes dened by theten decile values only three are larger than their predecessor when we use (M6 9

7 6 a69) and only two when we use (M79

86 a79) Recall that while concavity at everypoint is always a sufcient condition for the POUM effect it is a necessary oneonly if one requires yf lt m F 0

to hold for any initial F0 For a given initialdistribution such as the one observed in the data there must be simply ldquoenoughrdquoconcavity on average so that Jensenrsquos inequality is satised This is the situationencountered here


the fty-fth percentile and probably above the sixtieth27 Over ahorizon of approximately twenty years mobility prospects offsetthe entire 13ndash15 percent point interval between mean and me-dian incomes so that if agents are risk-neutral a strict majoritywill oppose such long-run redistributions Thus in both 1969 and1979 64 percent of the population was poorer than average interms of current income and yet 51 percent could rationally seethemselves as richer than average in terms of expected incometwo decades down the road

Upward mobility prospects for the poor however representonly one of the forces that determine the equilibrium rate ofredistributionmdashalongside with deadweight losses the politicalsystem and especially risk aversion which was seen to work inthe opposite direction The second question we consider is there-fore how the POUM effect compares in magnitude with the de-mand for social insurance To that end let us now assume thatagents have constant-relative risk aversion b gt 0 Using the sameprocedure and horizons of t 3 7 years as before we now compareeach decilersquos 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 resultspresented 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 campaignsetc) 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 mostvalues 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 634 618 542 488Five-year averages 634 608 564 529




about b = 025 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 overa horizon of ten to twenty years

(b) on the other hand these expected income gains are smallenough compared with the risks of downward mobility or

TABLE IIbINCOME PERCENTILE OF THE POLITICAL CUTOFF RISK AVERSION = 010

Horizon (years) 0 7 14 21





TABLE IIcINCOME PERCENTILE OF THE POLITICAL CUTOFF RISK AVERSION = 025






TABLE IIdINCOME PERCENTILE OF THE POLITICAL CUTOFF RISK AVERSION = 050







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 underlyingHungerfordrsquos [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 resultssuggest that this is probably not a major problem A secondcaveat is that the data obviously pertain to a particular countrynamely the United States and a particular period One may noteon the other hand that the two sample periods 1969 ndash1976 and1979 ndash1986 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 Stillthings might well be different elsewhere such as in a developingeconomy 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 familiesrsquo income trajectories

The above exercise should thus be taken as representing onlya rst pass at testing for the POUM effect in actual mobility dataMore empirical work will hopefully follow using different orbetter data to further investigate the issue of how peoplersquos 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 LatinAmerican countries respectively Both studies nd a signicantcorrelation 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 ofthe political economy of redistribution is perfectly compatiblewith rational expectations and fundamentally linked to an intui-tive feature of the income mobility process namely concavityVoters poorer than average may nonetheless opt for a low tax rate


if the policy choice bears sufciently on future income and if thelatterrsquos 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 votersThe 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 votersrsquoproblem 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 bythe 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 Jensenrsquos inequality we observe that f Acirc g impliesthat

f( yf ) 5 EX

f dF0 = EX

h(g) dF0 hS EX

g dF0 D5 h( g( yg)) 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 m F 0

More generally we shall be interested in any income cutoff h lt


m F0 Therefore dene for any a [ [01] and any h [ (0 m F0

) thefunction

(A1) g h a ( y) ordm min y h 1 a (y 2 h )

which clearly is an element of T It is clear that

EX

g h 0 dF0 h m F0 5 EX

g h 1 dF0

so by continuity there exists a unique a ( h ) [ (01) which solves

(A2) EX

g h a ( h ) dF0 5 h

Finally let f = g h a ( h ) so that m F1= h by (A2) Adding the

constant m F 02 h to the function f so as to normalize m F 1

= m F0

would of course not alter any of what follows It is clear from (A1)that everyone with y lt h prefers r0 to r1 while the reverse is truefor everyone with y gt h so yf = h By Proposition 2 thereforethe 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

lt m F 0yields the claimed results for majority voting QED

Proof of Theorem 2

For each t = 1 T f t [ T so by Proposition 1 there isa unique yf t [ (0 m F ) such that f t( yf t) = m F t

Moreover since f T

Acirc f T 2 1 Acirc Acirc f Proposition 2 implies that yf T lt yf T2 1 lt ltyf lt yf 0 ordm m F0

The concavity of f also implies f( m F t) gt m F t+ 1

forall t from which it follows by a simple induction that

(A3) f t( m F0) $ m Ft

5 f t( yf t)

with strict inequality for t gt 1 and t lt T respectively Let usnow dene the operators V T T reg T and WT T reg R as follows

(A4) V T( f ) ordm Ot= 0

T

d tf t and

W T( f ) ordm EX

V T( f ) dF0 = Ot= 0

T

d t m Ft

Agent y achieves utility V T ( f )( y) under laissez-faire and utility


WT ( f ) under the redistributive policy Moreover (A3) impliesthat

V T( f )( m F0) 5 O

t= 0

T

d tf t( m F0) O

t= 0

T

d t m Ft Ot= 0

T

d tf t( yf t)

5 V T( f )( yf T)

for any T $ 1 Since V T ( f )[ is clearly continuous and increas-ing there must therefore exist a unique yf (T) [ ( yf T m F0

) suchthat

(A5) V T( f )( yf (T)) 5 Ot= 0

T

d t m Ft5 W T( f )

But since yf T+ 1 lt yfT we have yf T+ 1 lt yf (T) This implies thatm FT+ 1

= f T + 1( yf T+ 1) lt f T + 1( yf (T)) and hence

V T+ 1( f )( yf (T)) 5 Ot= 0

T+ 1

d tf t( yf (T)) gt Ot= 0

T+ 1

d t m Ft

5 V T+ 1( f )(yf (T 1 1))

Therefore yf (T + 1) lt yf (T) must hold By induction weconclude that yf (T 9 ) lt yf (T) whenever T 9 gt T part (a) of thetheorem is proved

To prove part (b) we shall use again the family of piecewiselinear functions g h a dened in (A1) Let us rst observe that thetth iteration of g h a is simply

(A6) ( g h a )t( y) ordm min y h 1 a t(y 2 h ) 5 gh a t(y)

In particular both g h 1 y deg y and g h 0 y deg min y h areidempotent Therefore

V T( g h 1)( h ) 5 Ot= 0

T

d t h Ot= 0

T

d t m F0 5 W T( g h 1)

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


h 1 Ot= 1

T

d t h 5 V T( g h 0)( h ) gt W T(gh 0)

5 m F0 1 Ot= 1

T

d t EX

min y h dF0

or equivalently

(A7)m F0 2 h

h 2 X min y h dF0 O

t= 1

T

d t 5d (1 2 d T )

1 2 d

This last inequality is clearly satised for ( d 1T) close enough to(1 0) In that case we have WT ( g h 1) gt S t= 0

T d t h gt WT ( g h 0)Next it is clear from (A4) and (A1) that WT ( g h a ) is continuousand strictly increasing in a Therefore there exists a uniquea ( h ) [ (0 1) such that WT ( g h a ( h )) = S t= 0

T d t h This means thatunder the transition function f ordm g h a ( h ) we have

W T( f ) 5 Ot= 0

T

d tf t( h ) 5 V T( f )( h )

so that an agent with initial income h is just indifferent betweenreceiving her laissez-faire income stream equal to h in everyperiod and the stream of mean incomes m F t

Moreover underlaissez-faire an agent with initial y lt h receives y in every periodwhile an agent with y gt h receives h + a t( y 2 h ) gt h Thereforeh is the cutoff yf (T) separating those who support r0 from thosewho support r1 given f ordm g h a ( h ) This proves the rst statementin part (b) of the theorem

Finally by part (a) of the theorem increasing (decreasing)the horizon T will reduce (raise) the cutoff yf (T) below (above) h Applying these results to the particular choice of a cutoff equal tomedian income h ordm mF0

completes the proof

QED

Proof of Theorem 3

As in the proof Theorem 1 let F0 [ F+ and consider anyincome cutoff h lt m F 0

Recall the function g h a ( h )( y) which wasdened by (A1) and (A2) so as to ensure that m F1

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


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 [ (01) and let Q be a randomvariable taking values 0 and 1 with probabilities p and 1 2 p Forany e [ (0 h ) we dene f X 3 V reg R + as follows

c if 0 y h 2 e f( y u ) ordm y for all uc if h 2 e lt y h

f (y u ) ordm H h 2 e 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 ) ordm H h 2 e if u 5 0 (probability p)

h 1a (y 2 h ) 1 p e


By construction E Q [ f( y Q )] = g( y) for all y [ X thereforeE Q [ 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( x u y) ordm P( u [ V u f( 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 X

h( x) dM( x u y) must be (strictly) increasing in y for any (strictly)increasing function h X reg R this latter form of the property iseasily veried from the above denition of f( y u )

Because E Q [ f( z Q )] = g so that m F1= h by (A2) the cutoff

between the agents who prefer r0 and those who prefer r1 is yf =h This tipping point can be set to any value below m F 0

(ie 1 2F0( h ) can be made arbitrarily small) while simultaneously en-suring that F1( m F1

) gt s Indeed

F1( m F1) 5 F1( h ) = p E

X

1 f( x0) h dF0(x)

1 (1 2 p) EX

1 f( x1) h dF0(x)

5 p 1 (1 2 p) F0( h 2 p e )

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


transition function f [ T is more concave than f in expectationif and only if E Q [ f( z Q )] Acirc E Q [ f( z Q )] = g Proposition 2 thenimplies that the fraction of agents who support redistributionunder f is greater than F0( h ) The reverse inequalities holdwhenever f AcircP f As before choosing the particular cutoff h =mF0

lt m F 0yields the claimed results pertaining to majority

voting for any distribution F0 [ F+ QED


Dene ht ordm E Q 1 E Q t

f t( z Q 1 Q t) t = 1 andobserve that ht [ T and h t+ 1 Acirc h t 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 yF 0 By the mean value theorem there exists

a point yF 0[ X such that

(A8) (Tprog 2 Treg)(yF0) 5 E

X

(Tprog 2 Treg) dF0

which by (10) means that rprog( yF0) = rreg( yF0

) Now since Tprog2 Treg is convex Jensenrsquos inequality allows us to write

(A9) (Tprog 2 Treg)( m F0)

EX

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

With (Tprog 2 Treg) 9 (0) $ 0 the convexity of Tprog 2 Treg impliesthat it is a nondecreasing function on [0y ] In the trivial casewhere it is identically zero indifference obtains at every point sochoosing yF = m F0

immediately yields the results Otherwise twocases 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 (A9) is strict Under eitherscenario (A8) implies that yF 0

is the unique cutoff point betweenthe supporters of the two (static) policies and (A9) requires thatyF0

$ m F 0


2) Properties of yf By the mean value theorem there existsa point yf [ X such that

(A10) E Q [(Tprog 2 Treg)( f(yf Q ))]

5 EX

E Q [(Tprog 2 Treg)( f(y Q ))] dF0(y)

or equivalently E Q [rprog( f( yf Q ))] = E Q [rreg( f( yf Q ))]Moreover

(A11) EX


E Q [(Tprog 2 Treg)( f( m F0 Q ))]

by Jensenrsquos inequality for the concave (not afne) functionE Q [(Tprog 2 Treg) + f( z Q )] This function is also nondecreasingsince Tprog 2 Treg has this property and f [ T P is increasing iny with probability one Therefore yf is indeed the cutoff pointsuch that E Q [rprog( f( y Q ))] E Q [rreg( f( y Q )] as y yf andmoreover (A11) requires that m F 0

gt yfQED


1) Properties of yF The proofs of (A8) and (A9) proceed asabove except that the right-hand side of (A8) is now equal to zerodue to the equal-revenue constraint Since Tprog 2 Treg is convexequal 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 (0yF ) then positive on ( yF y ) Therefore yF

is indeed the threshold such that rprog( y) rreg( y) as y yF andmoreover by (A8) it must be that m F yF

2) Properties of yf Since the distribution F is by denitioninvariant under the mobility process ( fP) so is the revenueraised by either tax scheme This implies that the functionE Q [(Tprog 2 Treg)( f( yf Q ))] like Tprog 2 Treg sums to zero over[0y ]

(A12) EX

E Q [(Tprog 2 Treg)( f(y Q ))] dF(y)


5 EX

EV

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

5 EX

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

Therefore by the mean value theorem there is at least one pointyf where

E Q [(Tprog 2 Treg)( f(yfQ ))] 5 0

Furthermore since this same function is concave (not afne)Jensenrsquos inequality implies that

(A13) E Q [(Tprog 2 Treg)( f( m FQ ))]

EX

E Q [(Tprog 2 Treg)( f(yQ ))] dF(y) 5 0

Finally observe that since the function E Q [(rprog 2 rreg)( f( z Q ))] is convex (not afne) on [0y ] equal to zero at yf andnonnegative at the right boundary y it must rst be negative on(0 yf ) then positive on ( yf y ) Therefore yf is indeed the uniquetipping point meaning that E Q [rprog( f( y Q ))] E Q [rreg( f( yQ ))] as y yf moreover by (A13) we must have m F gt yf

QED

PRINCETON UNIVERSITY THE NATIONAL BUREAU OF ECONOMIC RESEARCH AND THE

CENTRE FOR ECONOMIC POLICY RESEARCH

NEW YORK UNIVERSITY

REFERENCES

Alesina A and D Rodrik ldquoDistributive Politics and Economic Growthrdquo Quar-terly Journal of Economics CIX (1994) 465ndash490

Atkinson A B ldquoThe Measurement of Economic Mobilityrdquo in Social Justice andPublic Policy (Cambridge MIT Press 1983)

Benabou R ldquoTax and Education Policy in a Heterogeneous Agent EconomyWhat Levels of Redistribution Maximize Growth and Efciencyrdquo NBERWorking Paper No 7132 Econometrica forthcoming

mdashmdash ldquoUnequal Societies Income Distribution and the Social Contractrdquo AmericanEconomic Review XC (2000) 96ndash129

Benabou R and E A Ok ldquoSocial Mobility and the Demand for RedistributionThe POUM Hypothesisrdquo NBER Working Paper No 6795 1998

Benabou R and J Tirole ldquoSelf-Condence Intrapersonal Strategiesrdquo CEPRDiscussion Paper No 2580 October 2000

Conlisk J ldquoMonotone Mobility Matricesrdquo Journal of Mathematical Sociology XV(1990) 173ndash191


Cooper S S Durlauf and P Johnson ldquoOn the Evolution of Economic Statusacross Generationsrdquo American Statistical Association (Business and Econom-ics Section) Papers and Proceedings (1998) 50ndash58

Dardanoni V ldquoMeasuring Social Mobilityrdquo Journal of Economic Theory LXI(1993) 372ndash394

Gourinchas P O and J Parker ldquoConsumption over the Life Cyclerdquo NBERWorking Paper No 7271 1999 Econometrica forthcoming

Graham C and S Pettinato ldquoAssessing Hardship and Happiness Trends inMobility and Expectations in the New Market Economiesrdquo mimeo BrookingsInstitution 1999

Hirschman A O ldquoThe Changing Tolerance for Income Inequality in the Courseof Economic Development (with a Mathematical Appendix by Michael Roth-schild)rdquo Quarterly Journal of Economics LXXXVII (1973) 544ndash566

Hungerford T L ldquoU S Income Mobility in the Seventies and Eightiesrdquo Reviewof Income and Wealth XXXI (1993) 403ndash417

Keilson J and A Ketser ldquoMonotone Matrices and Monotone Markov ChainsrdquoStochastic Processes and their Applications V (1977) 231ndash241

Marhuenda F and I Ortuno-Ortin ldquoPopular Support for Progressive TaxationrdquoEconomics Letters XLVIII (1995) 319ndash324

Meltzer A H and S F Richard ldquoA Rational Theory of the Size of GovernmentrdquoJournal of Political Economy LXXXIX (1981) 914ndash927

Mulligan C ldquoSome Evidence on the Role of Imperfect Markets for the Transmis-sion of Inequalityrdquo Parental Priorities and Economic Inequality (Chicago ILUniversity of Chicago Press 1997)

Okun A Equality and Efciency The Big Tradeoff (Washington DC The Brook-ings Institution 1975)

Peltzman S ldquoThe Growth of Governmentrdquo Journal of Law and Economics XXIII(1980) 209ndash287

Persson T and G Tabellini ldquoIs Inequality Harmful for Growth Theory andEvidencerdquo American Economic Review LXXXIV (1994) 600ndash621

Piketty T ldquoSocial Mobility and Redistributive Politicsrdquo Quarterly Journal ofEconomics CX (1995a) 551ndash583

mdashmdash ldquoRedistributive Responses to Distributive Trendsrdquo mimeo MassachusettsInstitute of Technology 1995b

Putterman L ldquoWhy Have the Rabble Not Redistributed the Wealth On theStability of Democracy and Unequal Propertyrdquo in Property Relations Incen-tives and Welfare J E Roemer ed (London Macmillan 1996)

Putterman L J Roemer and J Sylvestre ldquoDoes Egalitarianism Have a FuturerdquoJournal of Economic Literature XXXVI (1996) 861ndash902

Ravallion M and M Lokshin ldquoWho Wants to Redistribute Russiarsquos TunnelEffect in the 1990rsquosrdquo Journal of Public Economics LXXVI (2000) 87ndash104

Roemer J ldquoWhy the Poor Do Not Expropriate the Rich in Democracies An OldArgument in New Garbrdquo Journal of Public Economics LXX (1998) 399ndash426

Shorrocks A ldquoThe Measurement of Mobilityrdquo Econometrica XLV (1978)1013ndash1024

Solon G ldquoIntergenerational Income Mobility in the United Statesrdquo AmericanEconomic Review LXXXII (1992) 393ndash408

United States Bureau of the Census ldquoCurrent Population Reportsrdquo and dataavailable at httpwwwcensusgovftppubhheswwwincomehtml U S De-partment of Commerce Income Statistics Branch HHES Division Washing-ton DC 20233-8500

Zimmerman D ldquoRegression Toward Mediocrity in Economic Staturerdquo AmericanEconomic Review LXXXII (1992) 409ndash429


The silent majority did not want the yacht clubs closed forever totheir children and grandchildren while those who had alreadybecome members kept sailing alongrdquo

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 mobilitymdashaform of what Marxist writers refer to as ldquofalse consciousnessrdquo2

The ldquoprospect of upward mobilityrdquo (POUM) hypothesis hasto 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 averagemdashfor in-stance the median votermdashexpect to become richer than averageThis ldquooptimisticrdquo 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 tomorrowrsquos expected income lies some-where between todayrsquos 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 thusappearsmdashand economists have often concludedmdashthat 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]
























I PRELIMINARIES















(1) yt+ 1i 5 f( yt

i u t+ 1i ) t 5 0 T 2 1








i = Q tj


















(2) f( y) EX

f dF0 5 m F1


(3) f( m F0) 5 f S E

X

y dF0 D EX

f dF0 5 m F1











) 2 F1( m F 1) =F0( m F 0












) lt mF0 then m F1

lt f(mF 0) = mF 1




Appendix)





(4) Ot= 0

T

d tf t( y) Ot= 0

T

d t m Ft














i = f( yti u t+ 1










(5) E Q i[ f(yi Q i)] E[ m F1]



E[m F1] 5 E F E

0

1

f(yj Q j) dj G5 E

0

1

E[ f(yj Q j)] dj

5 EX




(6) EX




E Q i[ f(yf Q i)] 5 E[ m F1]

















F( y) 5 EV

EX












(7) M 5 F 1 2 r r 0ps 1 2 s (1 2 p)s0 q 1 2 q

G









E[yt+ 1i u yt



M 5 F 5 5 0385 3 315

0 6 4G






















M 5 F 7 3 0315 3 385

0 6 4G




i ordm ( Q 1i Q t


i ordm ( u 1i u t


(8) yti 5 f( f( f( y0

i u 1i ) u 2


i uI ti)

t 5 1 T

where f t( y0i uI t



t= 0

T

d tyti U y0

i G 5 Ot= 0

T


i)

5 Ot= 0

T








Ot= 0

T

d tE[ m F t] 5 O

t= 0

T

d t S E0

1

EQ tj [ f t(y0

i Q I tj)] dj D

5 Ot= 0

T

d t EX



(9) V T( y0i ) E

X

VT( y) dF0( y)











E Q 1 E Q t+ 1[ f t+ 1( z Q 1 Q t+ 1)]







IVA Risk-Aversion



[ f(y Q )])
















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












$ m F 0gt yf






$ m F 0gt yf


























a 969 5 (211 410 566 696 822 947

1104 1302 1549 2393)



a 979 5 (179 358 523 669 801 933

1084 1289 1588 2576)



76)t z a69 or (M7986)t z a79 for t = 1




z M7986 to


76 M6976












































VI CONCLUSION





APPENDIX



f( yf ) 5 EX

f dF0 = EX

h(g) dF0 hS EX

g dF0 D5 h( g( yg)) 5 f( yg)


QED

Proof of Theorem 1





) thefunction



EX


g h 1 dF0


(A2) EX

g h a ( h ) dF0 5 h



= m F0



Proof of Theorem 2


Moreover since f T




(A3) f t( m F0) $ m Ft

5 f t( yf t)



T

d tf t and

W T( f ) ordm EX

V T( f ) dF0 = Ot= 0

T

d t m Ft




V T( f )( m F0) 5 O

t= 0

T

d tf t( m F0) O

t= 0

T

d t m Ft Ot= 0

T

d tf t( yf t)

5 V T( f )( yf T)


) suchthat

(A5) V T( f )( yf (T)) 5 Ot= 0

T

d t m Ft5 W T( f )



V T+ 1( f )( yf (T)) 5 Ot= 0

T+ 1


T+ 1

d t m Ft

5 V T+ 1( f )(yf (T 1 1))





V T( g h 1)( h ) 5 Ot= 0

T

d t h Ot= 0

T

d t m F0 5 W T( g h 1)



h 1 Ot= 1

T

d t h 5 V T( g h 0)( h ) gt W T(gh 0)

5 m F0 1 Ot= 1

T

d t EX

min y h dF0

or equivalently

(A7)m F0 2 h

h 2 X min y h dF0 O

t= 1

T

d t 5d (1 2 d T )

1 2 d




W T( f ) 5 Ot= 0

T

d tf t( h ) 5 V T( f )( h )




completes the proof

QED

Proof of Theorem 3









c if h y y


h 1a (y 2 h ) 1 p e







) gt s Indeed

F1( m F1) 5 F1( h ) = p E

X

1 f( x0) h dF0(x)

1 (1 2 p) EX

1 f( x1) h dF0(x)

5 p 1 (1 2 p) F0( h 2 p e )














X

(Tprog 2 Treg) dF0




EX





$ m F 0




5 EX



(A11) EX




gt yfQED





(A12) EX



5 EX

EV


5 EX






EX



QED



NEW YORK UNIVERSITY

REFERENCES























































I PRELIMINARIES















(1) yt+ 1i 5 f( yt

i u t+ 1i ) t 5 0 T 2 1








i = Q tj


















(2) f( y) EX

f dF0 5 m F1


(3) f( m F0) 5 f S E

X

y dF0 D EX

f dF0 5 m F1











) 2 F1( m F 1) =F0( m F 0












) lt mF0 then m F1

lt f(mF 0) = mF 1




Appendix)





(4) Ot= 0

T

d tf t( y) Ot= 0

T

d t m Ft














i = f( yti u t+ 1










(5) E Q i[ f(yi Q i)] E[ m F1]



E[m F1] 5 E F E

0

1

f(yj Q j) dj G5 E

0

1

E[ f(yj Q j)] dj

5 EX




(6) EX




E Q i[ f(yf Q i)] 5 E[ m F1]

















F( y) 5 EV

EX












(7) M 5 F 1 2 r r 0ps 1 2 s (1 2 p)s0 q 1 2 q

G









E[yt+ 1i u yt



M 5 F 5 5 0385 3 315

0 6 4G






















M 5 F 7 3 0315 3 385

0 6 4G




i ordm ( Q 1i Q t


i ordm ( u 1i u t


(8) yti 5 f( f( f( y0

i u 1i ) u 2


i uI ti)

t 5 1 T

where f t( y0i uI t



t= 0

T

d tyti U y0

i G 5 Ot= 0

T


i)

5 Ot= 0

T








Ot= 0

T

d tE[ m F t] 5 O

t= 0

T

d t S E0

1

EQ tj [ f t(y0

i Q I tj)] dj D

5 Ot= 0

T

d t EX



(9) V T( y0i ) E

X

VT( y) dF0( y)











E Q 1 E Q t+ 1[ f t+ 1( z Q 1 Q t+ 1)]







IVA Risk-Aversion



[ f(y Q )])
















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












$ m F 0gt yf






$ m F 0gt yf


























a 969 5 (211 410 566 696 822 947

1104 1302 1549 2393)



a 979 5 (179 358 523 669 801 933

1084 1289 1588 2576)



76)t z a69 or (M7986)t z a79 for t = 1




z M7986 to


76 M6976












































VI CONCLUSION





APPENDIX



f( yf ) 5 EX

f dF0 = EX

h(g) dF0 hS EX

g dF0 D5 h( g( yg)) 5 f( yg)


QED

Proof of Theorem 1





) thefunction



EX


g h 1 dF0


(A2) EX

g h a ( h ) dF0 5 h



= m F0



Proof of Theorem 2


Moreover since f T




(A3) f t( m F0) $ m Ft

5 f t( yf t)



T

d tf t and

W T( f ) ordm EX

V T( f ) dF0 = Ot= 0

T

d t m Ft




V T( f )( m F0) 5 O

t= 0

T

d tf t( m F0) O

t= 0

T

d t m Ft Ot= 0

T

d tf t( yf t)

5 V T( f )( yf T)


) suchthat

(A5) V T( f )( yf (T)) 5 Ot= 0

T

d t m Ft5 W T( f )



V T+ 1( f )( yf (T)) 5 Ot= 0

T+ 1


T+ 1

d t m Ft

5 V T+ 1( f )(yf (T 1 1))





V T( g h 1)( h ) 5 Ot= 0

T

d t h Ot= 0

T

d t m F0 5 W T( g h 1)



h 1 Ot= 1

T

d t h 5 V T( g h 0)( h ) gt W T(gh 0)

5 m F0 1 Ot= 1

T

d t EX

min y h dF0

or equivalently

(A7)m F0 2 h

h 2 X min y h dF0 O

t= 1

T

d t 5d (1 2 d T )

1 2 d




W T( f ) 5 Ot= 0

T

d tf t( h ) 5 V T( f )( h )




completes the proof

QED

Proof of Theorem 3









c if h y y


h 1a (y 2 h ) 1 p e







) gt s Indeed

F1( m F1) 5 F1( h ) = p E

X

1 f( x0) h dF0(x)

1 (1 2 p) EX

1 f( x1) h dF0(x)

5 p 1 (1 2 p) F0( h 2 p e )














X

(Tprog 2 Treg) dF0




EX





$ m F 0




5 EX



(A11) EX




gt yfQED





(A12) EX



5 EX

EV


5 EX






EX



QED



NEW YORK UNIVERSITY

REFERENCES









































(1) yt+ 1i 5 f( yt

i u t+ 1i ) t 5 0 T 2 1








i = Q tj


















(2) f( y) EX

f dF0 5 m F1


(3) f( m F0) 5 f S E

X

y dF0 D EX

f dF0 5 m F1











) 2 F1( m F 1) =F0( m F 0












) lt mF0 then m F1

lt f(mF 0) = mF 1




Appendix)





(4) Ot= 0

T

d tf t( y) Ot= 0

T

d t m Ft














i = f( yti u t+ 1










(5) E Q i[ f(yi Q i)] E[ m F1]



E[m F1] 5 E F E

0

1

f(yj Q j) dj G5 E

0

1

E[ f(yj Q j)] dj

5 EX




(6) EX




E Q i[ f(yf Q i)] 5 E[ m F1]

















F( y) 5 EV

EX












(7) M 5 F 1 2 r r 0ps 1 2 s (1 2 p)s0 q 1 2 q

G









E[yt+ 1i u yt



M 5 F 5 5 0385 3 315

0 6 4G






















M 5 F 7 3 0315 3 385

0 6 4G




i ordm ( Q 1i Q t


i ordm ( u 1i u t


(8) yti 5 f( f( f( y0

i u 1i ) u 2


i uI ti)

t 5 1 T

where f t( y0i uI t



t= 0

T

d tyti U y0

i G 5 Ot= 0

T


i)

5 Ot= 0

T








Ot= 0

T

d tE[ m F t] 5 O

t= 0

T

d t S E0

1

EQ tj [ f t(y0

i Q I tj)] dj D

5 Ot= 0

T

d t EX



(9) V T( y0i ) E

X

VT( y) dF0( y)











E Q 1 E Q t+ 1[ f t+ 1( z Q 1 Q t+ 1)]







IVA Risk-Aversion



[ f(y Q )])
















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












$ m F 0gt yf






$ m F 0gt yf


























a 969 5 (211 410 566 696 822 947

1104 1302 1549 2393)



a 979 5 (179 358 523 669 801 933

1084 1289 1588 2576)



76)t z a69 or (M7986)t z a79 for t = 1




z M7986 to


76 M6976












































VI CONCLUSION





APPENDIX



f( yf ) 5 EX

f dF0 = EX

h(g) dF0 hS EX

g dF0 D5 h( g( yg)) 5 f( yg)


QED

Proof of Theorem 1





) thefunction



EX


g h 1 dF0


(A2) EX

g h a ( h ) dF0 5 h



= m F0



Proof of Theorem 2


Moreover since f T




(A3) f t( m F0) $ m Ft

5 f t( yf t)



T

d tf t and

W T( f ) ordm EX

V T( f ) dF0 = Ot= 0

T

d t m Ft




V T( f )( m F0) 5 O

t= 0

T

d tf t( m F0) O

t= 0

T

d t m Ft Ot= 0

T

d tf t( yf t)

5 V T( f )( yf T)


) suchthat

(A5) V T( f )( yf (T)) 5 Ot= 0

T

d t m Ft5 W T( f )



V T+ 1( f )( yf (T)) 5 Ot= 0

T+ 1


T+ 1

d t m Ft

5 V T+ 1( f )(yf (T 1 1))





V T( g h 1)( h ) 5 Ot= 0

T

d t h Ot= 0

T

d t m F0 5 W T( g h 1)



h 1 Ot= 1

T

d t h 5 V T( g h 0)( h ) gt W T(gh 0)

5 m F0 1 Ot= 1

T

d t EX

min y h dF0

or equivalently

(A7)m F0 2 h

h 2 X min y h dF0 O

t= 1

T

d t 5d (1 2 d T )

1 2 d




W T( f ) 5 Ot= 0

T

d tf t( h ) 5 V T( f )( h )




completes the proof

QED

Proof of Theorem 3









c if h y y


h 1a (y 2 h ) 1 p e







) gt s Indeed

F1( m F1) 5 F1( h ) = p E

X

1 f( x0) h dF0(x)

1 (1 2 p) EX

1 f( x1) h dF0(x)

5 p 1 (1 2 p) F0( h 2 p e )














X

(Tprog 2 Treg) dF0




EX





$ m F 0




5 EX



(A11) EX




gt yfQED





(A12) EX



5 EX

EV


5 EX






EX



QED



NEW YORK UNIVERSITY

REFERENCES














































(2) f( y) EX

f dF0 5 m F1


(3) f( m F0) 5 f S E

X

y dF0 D EX

f dF0 5 m F1











) 2 F1( m F 1) =F0( m F 0












) lt mF0 then m F1

lt f(mF 0) = mF 1




Appendix)





(4) Ot= 0

T

d tf t( y) Ot= 0

T

d t m Ft














i = f( yti u t+ 1










(5) E Q i[ f(yi Q i)] E[ m F1]



E[m F1] 5 E F E

0

1

f(yj Q j) dj G5 E

0

1

E[ f(yj Q j)] dj

5 EX




(6) EX




E Q i[ f(yf Q i)] 5 E[ m F1]

















F( y) 5 EV

EX












(7) M 5 F 1 2 r r 0ps 1 2 s (1 2 p)s0 q 1 2 q

G









E[yt+ 1i u yt



M 5 F 5 5 0385 3 315

0 6 4G






















M 5 F 7 3 0315 3 385

0 6 4G




i ordm ( Q 1i Q t


i ordm ( u 1i u t


(8) yti 5 f( f( f( y0

i u 1i ) u 2


i uI ti)

t 5 1 T

where f t( y0i uI t



t= 0

T

d tyti U y0

i G 5 Ot= 0

T


i)

5 Ot= 0

T








Ot= 0

T

d tE[ m F t] 5 O

t= 0

T

d t S E0

1

EQ tj [ f t(y0

i Q I tj)] dj D

5 Ot= 0

T

d t EX



(9) V T( y0i ) E

X

VT( y) dF0( y)











E Q 1 E Q t+ 1[ f t+ 1( z Q 1 Q t+ 1)]







IVA Risk-Aversion



[ f(y Q )])
















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












$ m F 0gt yf






$ m F 0gt yf


























a 969 5 (211 410 566 696 822 947

1104 1302 1549 2393)



a 979 5 (179 358 523 669 801 933

1084 1289 1588 2576)



76)t z a69 or (M7986)t z a79 for t = 1




z M7986 to


76 M6976












































VI CONCLUSION





APPENDIX



f( yf ) 5 EX

f dF0 = EX

h(g) dF0 hS EX

g dF0 D5 h( g( yg)) 5 f( yg)


QED

Proof of Theorem 1





) thefunction



EX


g h 1 dF0


(A2) EX

g h a ( h ) dF0 5 h



= m F0



Proof of Theorem 2


Moreover since f T




(A3) f t( m F0) $ m Ft

5 f t( yf t)



T

d tf t and

W T( f ) ordm EX

V T( f ) dF0 = Ot= 0

T

d t m Ft




V T( f )( m F0) 5 O

t= 0

T

d tf t( m F0) O

t= 0

T

d t m Ft Ot= 0

T

d tf t( yf t)

5 V T( f )( yf T)


) suchthat

(A5) V T( f )( yf (T)) 5 Ot= 0

T

d t m Ft5 W T( f )



V T+ 1( f )( yf (T)) 5 Ot= 0

T+ 1


T+ 1

d t m Ft

5 V T+ 1( f )(yf (T 1 1))





V T( g h 1)( h ) 5 Ot= 0

T

d t h Ot= 0

T

d t m F0 5 W T( g h 1)



h 1 Ot= 1

T

d t h 5 V T( g h 0)( h ) gt W T(gh 0)

5 m F0 1 Ot= 1

T

d t EX

min y h dF0

or equivalently

(A7)m F0 2 h

h 2 X min y h dF0 O

t= 1

T

d t 5d (1 2 d T )

1 2 d




W T( f ) 5 Ot= 0

T

d tf t( h ) 5 V T( f )( h )




completes the proof

QED

Proof of Theorem 3









c if h y y


h 1a (y 2 h ) 1 p e







) gt s Indeed

F1( m F1) 5 F1( h ) = p E

X

1 f( x0) h dF0(x)

1 (1 2 p) EX

1 f( x1) h dF0(x)

5 p 1 (1 2 p) F0( h 2 p e )














X

(Tprog 2 Treg) dF0




EX





$ m F 0




5 EX



(A11) EX




gt yfQED





(A12) EX



5 EX

EV


5 EX






EX



QED



NEW YORK UNIVERSITY

REFERENCES







































) lt mF0 then m F1

lt f(mF 0) = mF 1




Appendix)





(4) Ot= 0

T

d tf t( y) Ot= 0

T

d t m Ft














i = f( yti u t+ 1










(5) E Q i[ f(yi Q i)] E[ m F1]



E[m F1] 5 E F E

0

1

f(yj Q j) dj G5 E

0

1

E[ f(yj Q j)] dj

5 EX




(6) EX




E Q i[ f(yf Q i)] 5 E[ m F1]

















F( y) 5 EV

EX












(7) M 5 F 1 2 r r 0ps 1 2 s (1 2 p)s0 q 1 2 q

G









E[yt+ 1i u yt



M 5 F 5 5 0385 3 315

0 6 4G






















M 5 F 7 3 0315 3 385

0 6 4G




i ordm ( Q 1i Q t


i ordm ( u 1i u t


(8) yti 5 f( f( f( y0

i u 1i ) u 2


i uI ti)

t 5 1 T

where f t( y0i uI t



t= 0

T

d tyti U y0

i G 5 Ot= 0

T


i)

5 Ot= 0

T








Ot= 0

T

d tE[ m F t] 5 O

t= 0

T

d t S E0

1

EQ tj [ f t(y0

i Q I tj)] dj D

5 Ot= 0

T

d t EX



(9) V T( y0i ) E

X

VT( y) dF0( y)











E Q 1 E Q t+ 1[ f t+ 1( z Q 1 Q t+ 1)]







IVA Risk-Aversion



[ f(y Q )])
















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












$ m F 0gt yf






$ m F 0gt yf


























a 969 5 (211 410 566 696 822 947

1104 1302 1549 2393)



a 979 5 (179 358 523 669 801 933

1084 1289 1588 2576)



76)t z a69 or (M7986)t z a79 for t = 1




z M7986 to


76 M6976












































VI CONCLUSION





APPENDIX



f( yf ) 5 EX

f dF0 = EX

h(g) dF0 hS EX

g dF0 D5 h( g( yg)) 5 f( yg)


QED

Proof of Theorem 1





) thefunction



EX


g h 1 dF0


(A2) EX

g h a ( h ) dF0 5 h



= m F0



Proof of Theorem 2


Moreover since f T




(A3) f t( m F0) $ m Ft

5 f t( yf t)



T

d tf t and

W T( f ) ordm EX

V T( f ) dF0 = Ot= 0

T

d t m Ft




V T( f )( m F0) 5 O

t= 0

T

d tf t( m F0) O

t= 0

T

d t m Ft Ot= 0

T

d tf t( yf t)

5 V T( f )( yf T)


) suchthat

(A5) V T( f )( yf (T)) 5 Ot= 0

T

d t m Ft5 W T( f )



V T+ 1( f )( yf (T)) 5 Ot= 0

T+ 1


T+ 1

d t m Ft

5 V T+ 1( f )(yf (T 1 1))





V T( g h 1)( h ) 5 Ot= 0

T

d t h Ot= 0

T

d t m F0 5 W T( g h 1)



h 1 Ot= 1

T

d t h 5 V T( g h 0)( h ) gt W T(gh 0)

5 m F0 1 Ot= 1

T

d t EX

min y h dF0

or equivalently

(A7)m F0 2 h

h 2 X min y h dF0 O

t= 1

T

d t 5d (1 2 d T )

1 2 d




W T( f ) 5 Ot= 0

T

d tf t( h ) 5 V T( f )( h )




completes the proof

QED

Proof of Theorem 3









c if h y y


h 1a (y 2 h ) 1 p e







) gt s Indeed

F1( m F1) 5 F1( h ) = p E

X

1 f( x0) h dF0(x)

1 (1 2 p) EX

1 f( x1) h dF0(x)

5 p 1 (1 2 p) F0( h 2 p e )














X

(Tprog 2 Treg) dF0




EX





$ m F 0




5 EX



(A11) EX




gt yfQED





(A12) EX



5 EX

EV


5 EX






EX



QED



NEW YORK UNIVERSITY

REFERENCES




































(4) Ot= 0

T

d tf t( y) Ot= 0

T

d t m Ft














i = f( yti u t+ 1










(5) E Q i[ f(yi Q i)] E[ m F1]



E[m F1] 5 E F E

0

1

f(yj Q j) dj G5 E

0

1

E[ f(yj Q j)] dj

5 EX




(6) EX




E Q i[ f(yf Q i)] 5 E[ m F1]

















F( y) 5 EV

EX












(7) M 5 F 1 2 r r 0ps 1 2 s (1 2 p)s0 q 1 2 q

G









E[yt+ 1i u yt



M 5 F 5 5 0385 3 315

0 6 4G






















M 5 F 7 3 0315 3 385

0 6 4G




i ordm ( Q 1i Q t


i ordm ( u 1i u t


(8) yti 5 f( f( f( y0

i u 1i ) u 2


i uI ti)

t 5 1 T

where f t( y0i uI t



t= 0

T

d tyti U y0

i G 5 Ot= 0

T


i)

5 Ot= 0

T








Ot= 0

T

d tE[ m F t] 5 O

t= 0

T

d t S E0

1

EQ tj [ f t(y0

i Q I tj)] dj D

5 Ot= 0

T

d t EX



(9) V T( y0i ) E

X

VT( y) dF0( y)











E Q 1 E Q t+ 1[ f t+ 1( z Q 1 Q t+ 1)]







IVA Risk-Aversion



[ f(y Q )])
















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












$ m F 0gt yf






$ m F 0gt yf


























a 969 5 (211 410 566 696 822 947

1104 1302 1549 2393)



a 979 5 (179 358 523 669 801 933

1084 1289 1588 2576)



76)t z a69 or (M7986)t z a79 for t = 1




z M7986 to


76 M6976












































VI CONCLUSION





APPENDIX



f( yf ) 5 EX

f dF0 = EX

h(g) dF0 hS EX

g dF0 D5 h( g( yg)) 5 f( yg)


QED

Proof of Theorem 1





) thefunction



EX


g h 1 dF0


(A2) EX

g h a ( h ) dF0 5 h



= m F0



Proof of Theorem 2


Moreover since f T




(A3) f t( m F0) $ m Ft

5 f t( yf t)



T

d tf t and

W T( f ) ordm EX

V T( f ) dF0 = Ot= 0

T

d t m Ft




V T( f )( m F0) 5 O

t= 0

T

d tf t( m F0) O

t= 0

T

d t m Ft Ot= 0

T

d tf t( yf t)

5 V T( f )( yf T)


) suchthat

(A5) V T( f )( yf (T)) 5 Ot= 0

T

d t m Ft5 W T( f )



V T+ 1( f )( yf (T)) 5 Ot= 0

T+ 1


T+ 1

d t m Ft

5 V T+ 1( f )(yf (T 1 1))





V T( g h 1)( h ) 5 Ot= 0

T

d t h Ot= 0

T

d t m F0 5 W T( g h 1)



h 1 Ot= 1

T

d t h 5 V T( g h 0)( h ) gt W T(gh 0)

5 m F0 1 Ot= 1

T

d t EX

min y h dF0

or equivalently

(A7)m F0 2 h

h 2 X min y h dF0 O

t= 1

T

d t 5d (1 2 d T )

1 2 d




W T( f ) 5 Ot= 0

T

d tf t( h ) 5 V T( f )( h )




completes the proof

QED

Proof of Theorem 3









c if h y y


h 1a (y 2 h ) 1 p e







) gt s Indeed

F1( m F1) 5 F1( h ) = p E

X

1 f( x0) h dF0(x)

1 (1 2 p) EX

1 f( x1) h dF0(x)

5 p 1 (1 2 p) F0( h 2 p e )














X

(Tprog 2 Treg) dF0




EX





$ m F 0




5 EX



(A11) EX




gt yfQED





(A12) EX



5 EX

EV


5 EX






EX



QED



NEW YORK UNIVERSITY

REFERENCES







































i = f( yti u t+ 1










(5) E Q i[ f(yi Q i)] E[ m F1]



E[m F1] 5 E F E

0

1

f(yj Q j) dj G5 E

0

1

E[ f(yj Q j)] dj

5 EX




(6) EX




E Q i[ f(yf Q i)] 5 E[ m F1]

















F( y) 5 EV

EX












(7) M 5 F 1 2 r r 0ps 1 2 s (1 2 p)s0 q 1 2 q

G









E[yt+ 1i u yt



M 5 F 5 5 0385 3 315

0 6 4G






















M 5 F 7 3 0315 3 385

0 6 4G




i ordm ( Q 1i Q t


i ordm ( u 1i u t


(8) yti 5 f( f( f( y0

i u 1i ) u 2


i uI ti)

t 5 1 T

where f t( y0i uI t



t= 0

T

d tyti U y0

i G 5 Ot= 0

T


i)

5 Ot= 0

T








Ot= 0

T

d tE[ m F t] 5 O

t= 0

T

d t S E0

1

EQ tj [ f t(y0

i Q I tj)] dj D

5 Ot= 0

T

d t EX



(9) V T( y0i ) E

X

VT( y) dF0( y)











E Q 1 E Q t+ 1[ f t+ 1( z Q 1 Q t+ 1)]







IVA Risk-Aversion



[ f(y Q )])
















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












$ m F 0gt yf






$ m F 0gt yf


























a 969 5 (211 410 566 696 822 947

1104 1302 1549 2393)



a 979 5 (179 358 523 669 801 933

1084 1289 1588 2576)



76)t z a69 or (M7986)t z a79 for t = 1




z M7986 to


76 M6976












































VI CONCLUSION





APPENDIX



f( yf ) 5 EX

f dF0 = EX

h(g) dF0 hS EX

g dF0 D5 h( g( yg)) 5 f( yg)


QED

Proof of Theorem 1





) thefunction



EX


g h 1 dF0


(A2) EX

g h a ( h ) dF0 5 h



= m F0



Proof of Theorem 2


Moreover since f T




(A3) f t( m F0) $ m Ft

5 f t( yf t)



T

d tf t and

W T( f ) ordm EX

V T( f ) dF0 = Ot= 0

T

d t m Ft




V T( f )( m F0) 5 O

t= 0

T

d tf t( m F0) O

t= 0

T

d t m Ft Ot= 0

T

d tf t( yf t)

5 V T( f )( yf T)


) suchthat

(A5) V T( f )( yf (T)) 5 Ot= 0

T

d t m Ft5 W T( f )



V T+ 1( f )( yf (T)) 5 Ot= 0

T+ 1


T+ 1

d t m Ft

5 V T+ 1( f )(yf (T 1 1))





V T( g h 1)( h ) 5 Ot= 0

T

d t h Ot= 0

T

d t m F0 5 W T( g h 1)



h 1 Ot= 1

T

d t h 5 V T( g h 0)( h ) gt W T(gh 0)

5 m F0 1 Ot= 1

T

d t EX

min y h dF0

or equivalently

(A7)m F0 2 h

h 2 X min y h dF0 O

t= 1

T

d t 5d (1 2 d T )

1 2 d




W T( f ) 5 Ot= 0

T

d tf t( h ) 5 V T( f )( h )




completes the proof

QED

Proof of Theorem 3









c if h y y


h 1a (y 2 h ) 1 p e







) gt s Indeed

F1( m F1) 5 F1( h ) = p E

X

1 f( x0) h dF0(x)

1 (1 2 p) EX

1 f( x1) h dF0(x)

5 p 1 (1 2 p) F0( h 2 p e )














X

(Tprog 2 Treg) dF0




EX





$ m F 0




5 EX



(A11) EX




gt yfQED





(A12) EX



5 EX

EV


5 EX






EX



QED



NEW YORK UNIVERSITY

REFERENCES




































(5) E Q i[ f(yi Q i)] E[ m F1]



E[m F1] 5 E F E

0

1

f(yj Q j) dj G5 E

0

1

E[ f(yj Q j)] dj

5 EX




(6) EX




E Q i[ f(yf Q i)] 5 E[ m F1]

















F( y) 5 EV

EX












(7) M 5 F 1 2 r r 0ps 1 2 s (1 2 p)s0 q 1 2 q

G









E[yt+ 1i u yt



M 5 F 5 5 0385 3 315

0 6 4G






















M 5 F 7 3 0315 3 385

0 6 4G




i ordm ( Q 1i Q t


i ordm ( u 1i u t


(8) yti 5 f( f( f( y0

i u 1i ) u 2


i uI ti)

t 5 1 T

where f t( y0i uI t



t= 0

T

d tyti U y0

i G 5 Ot= 0

T


i)

5 Ot= 0

T








Ot= 0

T

d tE[ m F t] 5 O

t= 0

T

d t S E0

1

EQ tj [ f t(y0

i Q I tj)] dj D

5 Ot= 0

T

d t EX



(9) V T( y0i ) E

X

VT( y) dF0( y)











E Q 1 E Q t+ 1[ f t+ 1( z Q 1 Q t+ 1)]







IVA Risk-Aversion



[ f(y Q )])
















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












$ m F 0gt yf






$ m F 0gt yf


























a 969 5 (211 410 566 696 822 947

1104 1302 1549 2393)



a 979 5 (179 358 523 669 801 933

1084 1289 1588 2576)



76)t z a69 or (M7986)t z a79 for t = 1




z M7986 to


76 M6976












































VI CONCLUSION





APPENDIX



f( yf ) 5 EX

f dF0 = EX

h(g) dF0 hS EX

g dF0 D5 h( g( yg)) 5 f( yg)


QED

Proof of Theorem 1





) thefunction



EX


g h 1 dF0


(A2) EX

g h a ( h ) dF0 5 h



= m F0



Proof of Theorem 2


Moreover since f T




(A3) f t( m F0) $ m Ft

5 f t( yf t)



T

d tf t and

W T( f ) ordm EX

V T( f ) dF0 = Ot= 0

T

d t m Ft




V T( f )( m F0) 5 O

t= 0

T

d tf t( m F0) O

t= 0

T

d t m Ft Ot= 0

T

d tf t( yf t)

5 V T( f )( yf T)


) suchthat

(A5) V T( f )( yf (T)) 5 Ot= 0

T

d t m Ft5 W T( f )



V T+ 1( f )( yf (T)) 5 Ot= 0

T+ 1


T+ 1

d t m Ft

5 V T+ 1( f )(yf (T 1 1))





V T( g h 1)( h ) 5 Ot= 0

T

d t h Ot= 0

T

d t m F0 5 W T( g h 1)



h 1 Ot= 1

T

d t h 5 V T( g h 0)( h ) gt W T(gh 0)

5 m F0 1 Ot= 1

T

d t EX

min y h dF0

or equivalently

(A7)m F0 2 h

h 2 X min y h dF0 O

t= 1

T

d t 5d (1 2 d T )

1 2 d




W T( f ) 5 Ot= 0

T

d tf t( h ) 5 V T( f )( h )




completes the proof

QED

Proof of Theorem 3









c if h y y


h 1a (y 2 h ) 1 p e







) gt s Indeed

F1( m F1) 5 F1( h ) = p E

X

1 f( x0) h dF0(x)

1 (1 2 p) EX

1 f( x1) h dF0(x)

5 p 1 (1 2 p) F0( h 2 p e )














X

(Tprog 2 Treg) dF0




EX





$ m F 0




5 EX



(A11) EX




gt yfQED





(A12) EX



5 EX

EV


5 EX






EX



QED



NEW YORK UNIVERSITY

REFERENCES















































F( y) 5 EV

EX












(7) M 5 F 1 2 r r 0ps 1 2 s (1 2 p)s0 q 1 2 q

G









E[yt+ 1i u yt



M 5 F 5 5 0385 3 315

0 6 4G






















M 5 F 7 3 0315 3 385

0 6 4G




i ordm ( Q 1i Q t


i ordm ( u 1i u t


(8) yti 5 f( f( f( y0

i u 1i ) u 2


i uI ti)

t 5 1 T

where f t( y0i uI t



t= 0

T

d tyti U y0

i G 5 Ot= 0

T


i)

5 Ot= 0

T








Ot= 0

T

d tE[ m F t] 5 O

t= 0

T

d t S E0

1

EQ tj [ f t(y0

i Q I tj)] dj D

5 Ot= 0

T

d t EX



(9) V T( y0i ) E

X

VT( y) dF0( y)











E Q 1 E Q t+ 1[ f t+ 1( z Q 1 Q t+ 1)]







IVA Risk-Aversion



[ f(y Q )])
















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












$ m F 0gt yf






$ m F 0gt yf


























a 969 5 (211 410 566 696 822 947

1104 1302 1549 2393)



a 979 5 (179 358 523 669 801 933

1084 1289 1588 2576)



76)t z a69 or (M7986)t z a79 for t = 1




z M7986 to


76 M6976












































VI CONCLUSION





APPENDIX



f( yf ) 5 EX

f dF0 = EX

h(g) dF0 hS EX

g dF0 D5 h( g( yg)) 5 f( yg)


QED

Proof of Theorem 1





) thefunction



EX


g h 1 dF0


(A2) EX

g h a ( h ) dF0 5 h



= m F0



Proof of Theorem 2


Moreover since f T




(A3) f t( m F0) $ m Ft

5 f t( yf t)



T

d tf t and

W T( f ) ordm EX

V T( f ) dF0 = Ot= 0

T

d t m Ft




V T( f )( m F0) 5 O

t= 0

T

d tf t( m F0) O

t= 0

T

d t m Ft Ot= 0

T

d tf t( yf t)

5 V T( f )( yf T)


) suchthat

(A5) V T( f )( yf (T)) 5 Ot= 0

T

d t m Ft5 W T( f )



V T+ 1( f )( yf (T)) 5 Ot= 0

T+ 1


T+ 1

d t m Ft

5 V T+ 1( f )(yf (T 1 1))





V T( g h 1)( h ) 5 Ot= 0

T

d t h Ot= 0

T

d t m F0 5 W T( g h 1)



h 1 Ot= 1

T

d t h 5 V T( g h 0)( h ) gt W T(gh 0)

5 m F0 1 Ot= 1

T

d t EX

min y h dF0

or equivalently

(A7)m F0 2 h

h 2 X min y h dF0 O

t= 1

T

d t 5d (1 2 d T )

1 2 d




W T( f ) 5 Ot= 0

T

d tf t( h ) 5 V T( f )( h )




completes the proof

QED

Proof of Theorem 3









c if h y y


h 1a (y 2 h ) 1 p e







) gt s Indeed

F1( m F1) 5 F1( h ) = p E

X

1 f( x0) h dF0(x)

1 (1 2 p) EX

1 f( x1) h dF0(x)

5 p 1 (1 2 p) F0( h 2 p e )














X

(Tprog 2 Treg) dF0




EX





$ m F 0




5 EX



(A11) EX




gt yfQED





(A12) EX



5 EX

EV


5 EX






EX



QED



NEW YORK UNIVERSITY

REFERENCES




































(7) M 5 F 1 2 r r 0ps 1 2 s (1 2 p)s0 q 1 2 q

G









E[yt+ 1i u yt



M 5 F 5 5 0385 3 315

0 6 4G






















M 5 F 7 3 0315 3 385

0 6 4G




i ordm ( Q 1i Q t


i ordm ( u 1i u t


(8) yti 5 f( f( f( y0

i u 1i ) u 2


i uI ti)

t 5 1 T

where f t( y0i uI t



t= 0

T

d tyti U y0

i G 5 Ot= 0

T


i)

5 Ot= 0

T








Ot= 0

T

d tE[ m F t] 5 O

t= 0

T

d t S E0

1

EQ tj [ f t(y0

i Q I tj)] dj D

5 Ot= 0

T

d t EX



(9) V T( y0i ) E

X

VT( y) dF0( y)











E Q 1 E Q t+ 1[ f t+ 1( z Q 1 Q t+ 1)]







IVA Risk-Aversion



[ f(y Q )])
















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












$ m F 0gt yf






$ m F 0gt yf


























a 969 5 (211 410 566 696 822 947

1104 1302 1549 2393)



a 979 5 (179 358 523 669 801 933

1084 1289 1588 2576)



76)t z a69 or (M7986)t z a79 for t = 1




z M7986 to


76 M6976












































VI CONCLUSION





APPENDIX



f( yf ) 5 EX

f dF0 = EX

h(g) dF0 hS EX

g dF0 D5 h( g( yg)) 5 f( yg)


QED

Proof of Theorem 1





) thefunction



EX


g h 1 dF0


(A2) EX

g h a ( h ) dF0 5 h



= m F0



Proof of Theorem 2


Moreover since f T




(A3) f t( m F0) $ m Ft

5 f t( yf t)



T

d tf t and

W T( f ) ordm EX

V T( f ) dF0 = Ot= 0

T

d t m Ft




V T( f )( m F0) 5 O

t= 0

T

d tf t( m F0) O

t= 0

T

d t m Ft Ot= 0

T

d tf t( yf t)

5 V T( f )( yf T)


) suchthat

(A5) V T( f )( yf (T)) 5 Ot= 0

T

d t m Ft5 W T( f )



V T+ 1( f )( yf (T)) 5 Ot= 0

T+ 1


T+ 1

d t m Ft

5 V T+ 1( f )(yf (T 1 1))





V T( g h 1)( h ) 5 Ot= 0

T

d t h Ot= 0

T

d t m F0 5 W T( g h 1)



h 1 Ot= 1

T

d t h 5 V T( g h 0)( h ) gt W T(gh 0)

5 m F0 1 Ot= 1

T

d t EX

min y h dF0

or equivalently

(A7)m F0 2 h

h 2 X min y h dF0 O

t= 1

T

d t 5d (1 2 d T )

1 2 d




W T( f ) 5 Ot= 0

T

d tf t( h ) 5 V T( f )( h )




completes the proof

QED

Proof of Theorem 3









c if h y y


h 1a (y 2 h ) 1 p e







) gt s Indeed

F1( m F1) 5 F1( h ) = p E

X

1 f( x0) h dF0(x)

1 (1 2 p) EX

1 f( x1) h dF0(x)

5 p 1 (1 2 p) F0( h 2 p e )














X

(Tprog 2 Treg) dF0




EX





$ m F 0




5 EX



(A11) EX




gt yfQED





(A12) EX



5 EX

EV


5 EX






EX



QED



NEW YORK UNIVERSITY

REFERENCES





































E[yt+ 1i u yt



M 5 F 5 5 0385 3 315

0 6 4G






















M 5 F 7 3 0315 3 385

0 6 4G




i ordm ( Q 1i Q t


i ordm ( u 1i u t


(8) yti 5 f( f( f( y0

i u 1i ) u 2


i uI ti)

t 5 1 T

where f t( y0i uI t



t= 0

T

d tyti U y0

i G 5 Ot= 0

T


i)

5 Ot= 0

T








Ot= 0

T

d tE[ m F t] 5 O

t= 0

T

d t S E0

1

EQ tj [ f t(y0

i Q I tj)] dj D

5 Ot= 0

T

d t EX



(9) V T( y0i ) E

X

VT( y) dF0( y)











E Q 1 E Q t+ 1[ f t+ 1( z Q 1 Q t+ 1)]







IVA Risk-Aversion



[ f(y Q )])
















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












$ m F 0gt yf






$ m F 0gt yf


























a 969 5 (211 410 566 696 822 947

1104 1302 1549 2393)



a 979 5 (179 358 523 669 801 933

1084 1289 1588 2576)



76)t z a69 or (M7986)t z a79 for t = 1




z M7986 to


76 M6976












































VI CONCLUSION





APPENDIX



f( yf ) 5 EX

f dF0 = EX

h(g) dF0 hS EX

g dF0 D5 h( g( yg)) 5 f( yg)


QED

Proof of Theorem 1





) thefunction



EX


g h 1 dF0


(A2) EX

g h a ( h ) dF0 5 h



= m F0



Proof of Theorem 2


Moreover since f T




(A3) f t( m F0) $ m Ft

5 f t( yf t)



T

d tf t and

W T( f ) ordm EX

V T( f ) dF0 = Ot= 0

T

d t m Ft




V T( f )( m F0) 5 O

t= 0

T

d tf t( m F0) O

t= 0

T

d t m Ft Ot= 0

T

d tf t( yf t)

5 V T( f )( yf T)


) suchthat

(A5) V T( f )( yf (T)) 5 Ot= 0

T

d t m Ft5 W T( f )



V T+ 1( f )( yf (T)) 5 Ot= 0

T+ 1


T+ 1

d t m Ft

5 V T+ 1( f )(yf (T 1 1))





V T( g h 1)( h ) 5 Ot= 0

T

d t h Ot= 0

T

d t m F0 5 W T( g h 1)



h 1 Ot= 1

T

d t h 5 V T( g h 0)( h ) gt W T(gh 0)

5 m F0 1 Ot= 1

T

d t EX

min y h dF0

or equivalently

(A7)m F0 2 h

h 2 X min y h dF0 O

t= 1

T

d t 5d (1 2 d T )

1 2 d




W T( f ) 5 Ot= 0

T

d tf t( h ) 5 V T( f )( h )




completes the proof

QED

Proof of Theorem 3









c if h y y


h 1a (y 2 h ) 1 p e







) gt s Indeed

F1( m F1) 5 F1( h ) = p E

X

1 f( x0) h dF0(x)

1 (1 2 p) EX

1 f( x1) h dF0(x)

5 p 1 (1 2 p) F0( h 2 p e )














X

(Tprog 2 Treg) dF0




EX





$ m F 0




5 EX



(A11) EX




gt yfQED





(A12) EX



5 EX

EV


5 EX






EX



QED



NEW YORK UNIVERSITY

REFERENCES



















































M 5 F 7 3 0315 3 385

0 6 4G




i ordm ( Q 1i Q t


i ordm ( u 1i u t


(8) yti 5 f( f( f( y0

i u 1i ) u 2


i uI ti)

t 5 1 T

where f t( y0i uI t



t= 0

T

d tyti U y0

i G 5 Ot= 0

T


i)

5 Ot= 0

T








Ot= 0

T

d tE[ m F t] 5 O

t= 0

T

d t S E0

1

EQ tj [ f t(y0

i Q I tj)] dj D

5 Ot= 0

T

d t EX



(9) V T( y0i ) E

X

VT( y) dF0( y)











E Q 1 E Q t+ 1[ f t+ 1( z Q 1 Q t+ 1)]







IVA Risk-Aversion



[ f(y Q )])
















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












$ m F 0gt yf






$ m F 0gt yf


























a 969 5 (211 410 566 696 822 947

1104 1302 1549 2393)



a 979 5 (179 358 523 669 801 933

1084 1289 1588 2576)



76)t z a69 or (M7986)t z a79 for t = 1




z M7986 to


76 M6976












































VI CONCLUSION





APPENDIX



f( yf ) 5 EX

f dF0 = EX

h(g) dF0 hS EX

g dF0 D5 h( g( yg)) 5 f( yg)


QED

Proof of Theorem 1





) thefunction



EX


g h 1 dF0


(A2) EX

g h a ( h ) dF0 5 h



= m F0



Proof of Theorem 2


Moreover since f T




(A3) f t( m F0) $ m Ft

5 f t( yf t)



T

d tf t and

W T( f ) ordm EX

V T( f ) dF0 = Ot= 0

T

d t m Ft




V T( f )( m F0) 5 O

t= 0

T

d tf t( m F0) O

t= 0

T

d t m Ft Ot= 0

T

d tf t( yf t)

5 V T( f )( yf T)


) suchthat

(A5) V T( f )( yf (T)) 5 Ot= 0

T

d t m Ft5 W T( f )



V T+ 1( f )( yf (T)) 5 Ot= 0

T+ 1


T+ 1

d t m Ft

5 V T+ 1( f )(yf (T 1 1))





V T( g h 1)( h ) 5 Ot= 0

T

d t h Ot= 0

T

d t m F0 5 W T( g h 1)



h 1 Ot= 1

T

d t h 5 V T( g h 0)( h ) gt W T(gh 0)

5 m F0 1 Ot= 1

T

d t EX

min y h dF0

or equivalently

(A7)m F0 2 h

h 2 X min y h dF0 O

t= 1

T

d t 5d (1 2 d T )

1 2 d




W T( f ) 5 Ot= 0

T

d tf t( h ) 5 V T( f )( h )




completes the proof

QED

Proof of Theorem 3









c if h y y


h 1a (y 2 h ) 1 p e







) gt s Indeed

F1( m F1) 5 F1( h ) = p E

X

1 f( x0) h dF0(x)

1 (1 2 p) EX

1 f( x1) h dF0(x)

5 p 1 (1 2 p) F0( h 2 p e )














X

(Tprog 2 Treg) dF0




EX





$ m F 0




5 EX



(A11) EX




gt yfQED





(A12) EX



5 EX

EV


5 EX






EX



QED



NEW YORK UNIVERSITY

REFERENCES




































Ot= 0

T

d tE[ m F t] 5 O

t= 0

T

d t S E0

1

EQ tj [ f t(y0

i Q I tj)] dj D

5 Ot= 0

T

d t EX



(9) V T( y0i ) E

X

VT( y) dF0( y)











E Q 1 E Q t+ 1[ f t+ 1( z Q 1 Q t+ 1)]







IVA Risk-Aversion



[ f(y Q )])
















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












$ m F 0gt yf






$ m F 0gt yf


























a 969 5 (211 410 566 696 822 947

1104 1302 1549 2393)



a 979 5 (179 358 523 669 801 933

1084 1289 1588 2576)



76)t z a69 or (M7986)t z a79 for t = 1




z M7986 to


76 M6976












































VI CONCLUSION





APPENDIX



f( yf ) 5 EX

f dF0 = EX

h(g) dF0 hS EX

g dF0 D5 h( g( yg)) 5 f( yg)


QED

Proof of Theorem 1





) thefunction



EX


g h 1 dF0


(A2) EX

g h a ( h ) dF0 5 h



= m F0



Proof of Theorem 2


Moreover since f T




(A3) f t( m F0) $ m Ft

5 f t( yf t)



T

d tf t and

W T( f ) ordm EX

V T( f ) dF0 = Ot= 0

T

d t m Ft




V T( f )( m F0) 5 O

t= 0

T

d tf t( m F0) O

t= 0

T

d t m Ft Ot= 0

T

d tf t( yf t)

5 V T( f )( yf T)


) suchthat

(A5) V T( f )( yf (T)) 5 Ot= 0

T

d t m Ft5 W T( f )



V T+ 1( f )( yf (T)) 5 Ot= 0

T+ 1


T+ 1

d t m Ft

5 V T+ 1( f )(yf (T 1 1))





V T( g h 1)( h ) 5 Ot= 0

T

d t h Ot= 0

T

d t m F0 5 W T( g h 1)



h 1 Ot= 1

T

d t h 5 V T( g h 0)( h ) gt W T(gh 0)

5 m F0 1 Ot= 1

T

d t EX

min y h dF0

or equivalently

(A7)m F0 2 h

h 2 X min y h dF0 O

t= 1

T

d t 5d (1 2 d T )

1 2 d




W T( f ) 5 Ot= 0

T

d tf t( h ) 5 V T( f )( h )




completes the proof

QED

Proof of Theorem 3









c if h y y


h 1a (y 2 h ) 1 p e







) gt s Indeed

F1( m F1) 5 F1( h ) = p E

X

1 f( x0) h dF0(x)

1 (1 2 p) EX

1 f( x1) h dF0(x)

5 p 1 (1 2 p) F0( h 2 p e )














X

(Tprog 2 Treg) dF0




EX





$ m F 0




5 EX



(A11) EX




gt yfQED





(A12) EX



5 EX

EV


5 EX






EX



QED



NEW YORK UNIVERSITY

REFERENCES


































E Q 1 E Q t+ 1[ f t+ 1( z Q 1 Q t+ 1)]







IVA Risk-Aversion



[ f(y Q )])
















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












$ m F 0gt yf






$ m F 0gt yf


























a 969 5 (211 410 566 696 822 947

1104 1302 1549 2393)



a 979 5 (179 358 523 669 801 933

1084 1289 1588 2576)



76)t z a69 or (M7986)t z a79 for t = 1




z M7986 to


76 M6976












































VI CONCLUSION





APPENDIX



f( yf ) 5 EX

f dF0 = EX

h(g) dF0 hS EX

g dF0 D5 h( g( yg)) 5 f( yg)


QED

Proof of Theorem 1





) thefunction



EX


g h 1 dF0


(A2) EX

g h a ( h ) dF0 5 h



= m F0



Proof of Theorem 2


Moreover since f T




(A3) f t( m F0) $ m Ft

5 f t( yf t)



T

d tf t and

W T( f ) ordm EX

V T( f ) dF0 = Ot= 0

T

d t m Ft




V T( f )( m F0) 5 O

t= 0

T

d tf t( m F0) O

t= 0

T

d t m Ft Ot= 0

T

d tf t( yf t)

5 V T( f )( yf T)


) suchthat

(A5) V T( f )( yf (T)) 5 Ot= 0

T

d t m Ft5 W T( f )



V T+ 1( f )( yf (T)) 5 Ot= 0

T+ 1


T+ 1

d t m Ft

5 V T+ 1( f )(yf (T 1 1))





V T( g h 1)( h ) 5 Ot= 0

T

d t h Ot= 0

T

d t m F0 5 W T( g h 1)



h 1 Ot= 1

T

d t h 5 V T( g h 0)( h ) gt W T(gh 0)

5 m F0 1 Ot= 1

T

d t EX

min y h dF0

or equivalently

(A7)m F0 2 h

h 2 X min y h dF0 O

t= 1

T

d t 5d (1 2 d T )

1 2 d




W T( f ) 5 Ot= 0

T

d tf t( h ) 5 V T( f )( h )




completes the proof

QED

Proof of Theorem 3









c if h y y


h 1a (y 2 h ) 1 p e







) gt s Indeed

F1( m F1) 5 F1( h ) = p E

X

1 f( x0) h dF0(x)

1 (1 2 p) EX

1 f( x1) h dF0(x)

5 p 1 (1 2 p) F0( h 2 p e )














X

(Tprog 2 Treg) dF0




EX





$ m F 0




5 EX



(A11) EX




gt yfQED





(A12) EX



5 EX

EV


5 EX






EX



QED



NEW YORK UNIVERSITY

REFERENCES












































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












$ m F 0gt yf






$ m F 0gt yf


























a 969 5 (211 410 566 696 822 947

1104 1302 1549 2393)



a 979 5 (179 358 523 669 801 933

1084 1289 1588 2576)



76)t z a69 or (M7986)t z a79 for t = 1




z M7986 to


76 M6976












































VI CONCLUSION





APPENDIX



f( yf ) 5 EX

f dF0 = EX

h(g) dF0 hS EX

g dF0 D5 h( g( yg)) 5 f( yg)


QED

Proof of Theorem 1





) thefunction



EX


g h 1 dF0


(A2) EX

g h a ( h ) dF0 5 h



= m F0



Proof of Theorem 2


Moreover since f T




(A3) f t( m F0) $ m Ft

5 f t( yf t)



T

d tf t and

W T( f ) ordm EX

V T( f ) dF0 = Ot= 0

T

d t m Ft




V T( f )( m F0) 5 O

t= 0

T

d tf t( m F0) O

t= 0

T

d t m Ft Ot= 0

T

d tf t( yf t)

5 V T( f )( yf T)


) suchthat

(A5) V T( f )( yf (T)) 5 Ot= 0

T

d t m Ft5 W T( f )



V T+ 1( f )( yf (T)) 5 Ot= 0

T+ 1


T+ 1

d t m Ft

5 V T+ 1( f )(yf (T 1 1))





V T( g h 1)( h ) 5 Ot= 0

T

d t h Ot= 0

T

d t m F0 5 W T( g h 1)



h 1 Ot= 1

T

d t h 5 V T( g h 0)( h ) gt W T(gh 0)

5 m F0 1 Ot= 1

T

d t EX

min y h dF0

or equivalently

(A7)m F0 2 h

h 2 X min y h dF0 O

t= 1

T

d t 5d (1 2 d T )

1 2 d




W T( f ) 5 Ot= 0

T

d tf t( h ) 5 V T( f )( h )




completes the proof

QED

Proof of Theorem 3









c if h y y


h 1a (y 2 h ) 1 p e







) gt s Indeed

F1( m F1) 5 F1( h ) = p E

X

1 f( x0) h dF0(x)

1 (1 2 p) EX

1 f( x1) h dF0(x)

5 p 1 (1 2 p) F0( h 2 p e )














X

(Tprog 2 Treg) dF0




EX





$ m F 0




5 EX



(A11) EX




gt yfQED





(A12) EX



5 EX

EV


5 EX






EX



QED



NEW YORK UNIVERSITY

REFERENCES






































$ m F 0gt yf






$ m F 0gt yf


























a 969 5 (211 410 566 696 822 947

1104 1302 1549 2393)



a 979 5 (179 358 523 669 801 933

1084 1289 1588 2576)



76)t z a69 or (M7986)t z a79 for t = 1




z M7986 to


76 M6976












































VI CONCLUSION





APPENDIX



f( yf ) 5 EX

f dF0 = EX

h(g) dF0 hS EX

g dF0 D5 h( g( yg)) 5 f( yg)


QED

Proof of Theorem 1





) thefunction



EX


g h 1 dF0


(A2) EX

g h a ( h ) dF0 5 h



= m F0



Proof of Theorem 2


Moreover since f T




(A3) f t( m F0) $ m Ft

5 f t( yf t)



T

d tf t and

W T( f ) ordm EX

V T( f ) dF0 = Ot= 0

T

d t m Ft




V T( f )( m F0) 5 O

t= 0

T

d tf t( m F0) O

t= 0

T

d t m Ft Ot= 0

T

d tf t( yf t)

5 V T( f )( yf T)


) suchthat

(A5) V T( f )( yf (T)) 5 Ot= 0

T

d t m Ft5 W T( f )



V T+ 1( f )( yf (T)) 5 Ot= 0

T+ 1


T+ 1

d t m Ft

5 V T+ 1( f )(yf (T 1 1))





V T( g h 1)( h ) 5 Ot= 0

T

d t h Ot= 0

T

d t m F0 5 W T( g h 1)



h 1 Ot= 1

T

d t h 5 V T( g h 0)( h ) gt W T(gh 0)

5 m F0 1 Ot= 1

T

d t EX

min y h dF0

or equivalently

(A7)m F0 2 h

h 2 X min y h dF0 O

t= 1

T

d t 5d (1 2 d T )

1 2 d




W T( f ) 5 Ot= 0

T

d tf t( h ) 5 V T( f )( h )




completes the proof

QED

Proof of Theorem 3









c if h y y


h 1a (y 2 h ) 1 p e







) gt s Indeed

F1( m F1) 5 F1( h ) = p E

X

1 f( x0) h dF0(x)

1 (1 2 p) EX

1 f( x1) h dF0(x)

5 p 1 (1 2 p) F0( h 2 p e )














X

(Tprog 2 Treg) dF0




EX





$ m F 0




5 EX



(A11) EX




gt yfQED





(A12) EX



5 EX

EV


5 EX






EX



QED



NEW YORK UNIVERSITY

REFERENCES






















































a 969 5 (211 410 566 696 822 947

1104 1302 1549 2393)



a 979 5 (179 358 523 669 801 933

1084 1289 1588 2576)



76)t z a69 or (M7986)t z a79 for t = 1




z M7986 to


76 M6976












































VI CONCLUSION





APPENDIX



f( yf ) 5 EX

f dF0 = EX

h(g) dF0 hS EX

g dF0 D5 h( g( yg)) 5 f( yg)


QED

Proof of Theorem 1





) thefunction



EX


g h 1 dF0


(A2) EX

g h a ( h ) dF0 5 h



= m F0



Proof of Theorem 2


Moreover since f T




(A3) f t( m F0) $ m Ft

5 f t( yf t)



T

d tf t and

W T( f ) ordm EX

V T( f ) dF0 = Ot= 0

T

d t m Ft




V T( f )( m F0) 5 O

t= 0

T

d tf t( m F0) O

t= 0

T

d t m Ft Ot= 0

T

d tf t( yf t)

5 V T( f )( yf T)


) suchthat

(A5) V T( f )( yf (T)) 5 Ot= 0

T

d t m Ft5 W T( f )



V T+ 1( f )( yf (T)) 5 Ot= 0

T+ 1


T+ 1

d t m Ft

5 V T+ 1( f )(yf (T 1 1))





V T( g h 1)( h ) 5 Ot= 0

T

d t h Ot= 0

T

d t m F0 5 W T( g h 1)



h 1 Ot= 1

T

d t h 5 V T( g h 0)( h ) gt W T(gh 0)

5 m F0 1 Ot= 1

T

d t EX

min y h dF0

or equivalently

(A7)m F0 2 h

h 2 X min y h dF0 O

t= 1

T

d t 5d (1 2 d T )

1 2 d




W T( f ) 5 Ot= 0

T

d tf t( h ) 5 V T( f )( h )




completes the proof

QED

Proof of Theorem 3









c if h y y


h 1a (y 2 h ) 1 p e







) gt s Indeed

F1( m F1) 5 F1( h ) = p E

X

1 f( x0) h dF0(x)

1 (1 2 p) EX

1 f( x1) h dF0(x)

5 p 1 (1 2 p) F0( h 2 p e )














X

(Tprog 2 Treg) dF0




EX





$ m F 0




5 EX



(A11) EX




gt yfQED





(A12) EX



5 EX

EV


5 EX






EX



QED



NEW YORK UNIVERSITY

REFERENCES

































a 979 5 (179 358 523 669 801 933

1084 1289 1588 2576)



76)t z a69 or (M7986)t z a79 for t = 1




z M7986 to


76 M6976












































VI CONCLUSION





APPENDIX



f( yf ) 5 EX

f dF0 = EX

h(g) dF0 hS EX

g dF0 D5 h( g( yg)) 5 f( yg)


QED

Proof of Theorem 1





) thefunction



EX


g h 1 dF0


(A2) EX

g h a ( h ) dF0 5 h



= m F0



Proof of Theorem 2


Moreover since f T




(A3) f t( m F0) $ m Ft

5 f t( yf t)



T

d tf t and

W T( f ) ordm EX

V T( f ) dF0 = Ot= 0

T

d t m Ft




V T( f )( m F0) 5 O

t= 0

T

d tf t( m F0) O

t= 0

T

d t m Ft Ot= 0

T

d tf t( yf t)

5 V T( f )( yf T)


) suchthat

(A5) V T( f )( yf (T)) 5 Ot= 0

T

d t m Ft5 W T( f )



V T+ 1( f )( yf (T)) 5 Ot= 0

T+ 1


T+ 1

d t m Ft

5 V T+ 1( f )(yf (T 1 1))





V T( g h 1)( h ) 5 Ot= 0

T

d t h Ot= 0

T

d t m F0 5 W T( g h 1)



h 1 Ot= 1

T

d t h 5 V T( g h 0)( h ) gt W T(gh 0)

5 m F0 1 Ot= 1

T

d t EX

min y h dF0

or equivalently

(A7)m F0 2 h

h 2 X min y h dF0 O

t= 1

T

d t 5d (1 2 d T )

1 2 d




W T( f ) 5 Ot= 0

T

d tf t( h ) 5 V T( f )( h )




completes the proof

QED

Proof of Theorem 3









c if h y y


h 1a (y 2 h ) 1 p e







) gt s Indeed

F1( m F1) 5 F1( h ) = p E

X

1 f( x0) h dF0(x)

1 (1 2 p) EX

1 f( x1) h dF0(x)

5 p 1 (1 2 p) F0( h 2 p e )














X

(Tprog 2 Treg) dF0




EX





$ m F 0




5 EX



(A11) EX




gt yfQED





(A12) EX



5 EX

EV


5 EX






EX



QED



NEW YORK UNIVERSITY

REFERENCES





































































VI CONCLUSION





APPENDIX



f( yf ) 5 EX

f dF0 = EX

h(g) dF0 hS EX

g dF0 D5 h( g( yg)) 5 f( yg)


QED

Proof of Theorem 1





) thefunction



EX


g h 1 dF0


(A2) EX

g h a ( h ) dF0 5 h



= m F0



Proof of Theorem 2


Moreover since f T




(A3) f t( m F0) $ m Ft

5 f t( yf t)



T

d tf t and

W T( f ) ordm EX

V T( f ) dF0 = Ot= 0

T

d t m Ft




V T( f )( m F0) 5 O

t= 0

T

d tf t( m F0) O

t= 0

T

d t m Ft Ot= 0

T

d tf t( yf t)

5 V T( f )( yf T)


) suchthat

(A5) V T( f )( yf (T)) 5 Ot= 0

T

d t m Ft5 W T( f )



V T+ 1( f )( yf (T)) 5 Ot= 0

T+ 1


T+ 1

d t m Ft

5 V T+ 1( f )(yf (T 1 1))





V T( g h 1)( h ) 5 Ot= 0

T

d t h Ot= 0

T

d t m F0 5 W T( g h 1)



h 1 Ot= 1

T

d t h 5 V T( g h 0)( h ) gt W T(gh 0)

5 m F0 1 Ot= 1

T

d t EX

min y h dF0

or equivalently

(A7)m F0 2 h

h 2 X min y h dF0 O

t= 1

T

d t 5d (1 2 d T )

1 2 d




W T( f ) 5 Ot= 0

T

d tf t( h ) 5 V T( f )( h )




completes the proof

QED

Proof of Theorem 3









c if h y y


h 1a (y 2 h ) 1 p e







) gt s Indeed

F1( m F1) 5 F1( h ) = p E

X

1 f( x0) h dF0(x)

1 (1 2 p) EX

1 f( x1) h dF0(x)

5 p 1 (1 2 p) F0( h 2 p e )














X

(Tprog 2 Treg) dF0




EX





$ m F 0




5 EX



(A11) EX




gt yfQED





(A12) EX



5 EX

EV


5 EX






EX



QED



NEW YORK UNIVERSITY

REFERENCES



































APPENDIX



f( yf ) 5 EX

f dF0 = EX

h(g) dF0 hS EX

g dF0 D5 h( g( yg)) 5 f( yg)


QED

Proof of Theorem 1





) thefunction



EX


g h 1 dF0


(A2) EX

g h a ( h ) dF0 5 h



= m F0



Proof of Theorem 2


Moreover since f T




(A3) f t( m F0) $ m Ft

5 f t( yf t)



T

d tf t and

W T( f ) ordm EX

V T( f ) dF0 = Ot= 0

T

d t m Ft




V T( f )( m F0) 5 O

t= 0

T

d tf t( m F0) O

t= 0

T

d t m Ft Ot= 0

T

d tf t( yf t)

5 V T( f )( yf T)


) suchthat

(A5) V T( f )( yf (T)) 5 Ot= 0

T

d t m Ft5 W T( f )



V T+ 1( f )( yf (T)) 5 Ot= 0

T+ 1


T+ 1

d t m Ft

5 V T+ 1( f )(yf (T 1 1))





V T( g h 1)( h ) 5 Ot= 0

T

d t h Ot= 0

T

d t m F0 5 W T( g h 1)



h 1 Ot= 1

T

d t h 5 V T( g h 0)( h ) gt W T(gh 0)

5 m F0 1 Ot= 1

T

d t EX

min y h dF0

or equivalently

(A7)m F0 2 h

h 2 X min y h dF0 O

t= 1

T

d t 5d (1 2 d T )

1 2 d




W T( f ) 5 Ot= 0

T

d tf t( h ) 5 V T( f )( h )




completes the proof

QED

Proof of Theorem 3









c if h y y


h 1a (y 2 h ) 1 p e







) gt s Indeed

F1( m F1) 5 F1( h ) = p E

X

1 f( x0) h dF0(x)

1 (1 2 p) EX

1 f( x1) h dF0(x)

5 p 1 (1 2 p) F0( h 2 p e )














X

(Tprog 2 Treg) dF0




EX





$ m F 0




5 EX



(A11) EX




gt yfQED





(A12) EX



5 EX

EV


5 EX






EX



QED



NEW YORK UNIVERSITY

REFERENCES


































) thefunction



EX


g h 1 dF0


(A2) EX

g h a ( h ) dF0 5 h



= m F0



Proof of Theorem 2


Moreover since f T




(A3) f t( m F0) $ m Ft

5 f t( yf t)



T

d tf t and

W T( f ) ordm EX

V T( f ) dF0 = Ot= 0

T

d t m Ft




V T( f )( m F0) 5 O

t= 0

T

d tf t( m F0) O

t= 0

T

d t m Ft Ot= 0

T

d tf t( yf t)

5 V T( f )( yf T)


) suchthat

(A5) V T( f )( yf (T)) 5 Ot= 0

T

d t m Ft5 W T( f )



V T+ 1( f )( yf (T)) 5 Ot= 0

T+ 1


T+ 1

d t m Ft

5 V T+ 1( f )(yf (T 1 1))





V T( g h 1)( h ) 5 Ot= 0

T

d t h Ot= 0

T

d t m F0 5 W T( g h 1)



h 1 Ot= 1

T

d t h 5 V T( g h 0)( h ) gt W T(gh 0)

5 m F0 1 Ot= 1

T

d t EX

min y h dF0

or equivalently

(A7)m F0 2 h

h 2 X min y h dF0 O

t= 1

T

d t 5d (1 2 d T )

1 2 d




W T( f ) 5 Ot= 0

T

d tf t( h ) 5 V T( f )( h )




completes the proof

QED

Proof of Theorem 3









c if h y y


h 1a (y 2 h ) 1 p e







) gt s Indeed

F1( m F1) 5 F1( h ) = p E

X

1 f( x0) h dF0(x)

1 (1 2 p) EX

1 f( x1) h dF0(x)

5 p 1 (1 2 p) F0( h 2 p e )














X

(Tprog 2 Treg) dF0




EX





$ m F 0




5 EX



(A11) EX




gt yfQED





(A12) EX



5 EX

EV


5 EX






EX



QED



NEW YORK UNIVERSITY

REFERENCES


































V T( f )( m F0) 5 O

t= 0

T

d tf t( m F0) O

t= 0

T

d t m Ft Ot= 0

T

d tf t( yf t)

5 V T( f )( yf T)


) suchthat

(A5) V T( f )( yf (T)) 5 Ot= 0

T

d t m Ft5 W T( f )



V T+ 1( f )( yf (T)) 5 Ot= 0

T+ 1


T+ 1

d t m Ft

5 V T+ 1( f )(yf (T 1 1))





V T( g h 1)( h ) 5 Ot= 0

T

d t h Ot= 0

T

d t m F0 5 W T( g h 1)



h 1 Ot= 1

T

d t h 5 V T( g h 0)( h ) gt W T(gh 0)

5 m F0 1 Ot= 1

T

d t EX

min y h dF0

or equivalently

(A7)m F0 2 h

h 2 X min y h dF0 O

t= 1

T

d t 5d (1 2 d T )

1 2 d




W T( f ) 5 Ot= 0

T

d tf t( h ) 5 V T( f )( h )




completes the proof

QED

Proof of Theorem 3









c if h y y


h 1a (y 2 h ) 1 p e







) gt s Indeed

F1( m F1) 5 F1( h ) = p E

X

1 f( x0) h dF0(x)

1 (1 2 p) EX

1 f( x1) h dF0(x)

5 p 1 (1 2 p) F0( h 2 p e )














X

(Tprog 2 Treg) dF0




EX





$ m F 0




5 EX



(A11) EX




gt yfQED





(A12) EX



5 EX

EV


5 EX






EX



QED



NEW YORK UNIVERSITY

REFERENCES

































h 1 Ot= 1

T

d t h 5 V T( g h 0)( h ) gt W T(gh 0)

5 m F0 1 Ot= 1

T

d t EX

min y h dF0

or equivalently

(A7)m F0 2 h

h 2 X min y h dF0 O

t= 1

T

d t 5d (1 2 d T )

1 2 d




W T( f ) 5 Ot= 0

T

d tf t( h ) 5 V T( f )( h )




completes the proof

QED

Proof of Theorem 3









c if h y y


h 1a (y 2 h ) 1 p e







) gt s Indeed

F1( m F1) 5 F1( h ) = p E

X

1 f( x0) h dF0(x)

1 (1 2 p) EX

1 f( x1) h dF0(x)

5 p 1 (1 2 p) F0( h 2 p e )














X

(Tprog 2 Treg) dF0




EX





$ m F 0




5 EX



(A11) EX




gt yfQED





(A12) EX



5 EX

EV


5 EX






EX



QED



NEW YORK UNIVERSITY

REFERENCES





































c if h y y


h 1a (y 2 h ) 1 p e







) gt s Indeed

F1( m F1) 5 F1( h ) = p E

X

1 f( x0) h dF0(x)

1 (1 2 p) EX

1 f( x1) h dF0(x)

5 p 1 (1 2 p) F0( h 2 p e )














X

(Tprog 2 Treg) dF0




EX





$ m F 0




5 EX



(A11) EX




gt yfQED





(A12) EX



5 EX

EV


5 EX






EX



QED



NEW YORK UNIVERSITY

REFERENCES











































X

(Tprog 2 Treg) dF0




EX





$ m F 0




5 EX



(A11) EX




gt yfQED





(A12) EX



5 EX

EV


5 EX






EX



QED



NEW YORK UNIVERSITY

REFERENCES



































5 EX



(A11) EX




gt yfQED





(A12) EX



5 EX

EV


5 EX






EX



QED



NEW YORK UNIVERSITY

REFERENCES

































5 EX

EV


5 EX






EX



QED



NEW YORK UNIVERSITY

REFERENCES

































social mobility and the demand for redistribution: …rbenabou/papers/d8zkmee3.pdf · social...

Documents