Unequal Societies: Income Distribution and the Social Contract

By ROLAND BE´NABOU*

This paper develops a theory of inequality and the social contract aiming to explain how countries with similar economic and political “fundamentals” can sustain such different systems of social insurance, fiscal redistribution, and education finance as those of the United States and Western Europe. With imperfect credit and insurance markets some redistributive policies can improve ex ante welfare, and this implies that their political support tends to decrease with inequality. Conversely, with credit constraints, lower redistribution translates into more persistent inequality; hence the potential for multiple steady states, with mutually reinforcing high inequality and low redistribution, or vice versa. (JEL D31, E62, P16, O41, I22)

The social contract varies considerably Western Europe, but the observation holds across nations. Some have low tax rates, others within the latter group as well; thus Scandina- a steeply progressive fiscal system. Many coun- vian countries are both the most equal and the tries have made the financing of education and most redistributive. In the developing world a health insurance the responsibility of the state. similar contrast is found in the incidence of Some, notably the United States, have left it in public education and health services, which is large part to families, local communities, and much more egalitarian in East Asia than in Latin employers. The extent of implicit redistribution America (e.g., South Korea versus Brazil). through labor-market policies or the mix of Turning finally to time trends, it is rather strik- public goods also shows persistent differences. ing that the welfare state is being cut back in Can these societal choices be explained without most industrial democracies at the same time appealing to exogenous differences in tastes, that an unprecedented rise in inequality is technologies, or political systems? occurring. Adding to the puzzle is the fact that redistri- The aim of this paper is to develop a joint bution is often correlated with income inequal- theory of inequality and the social contract ity in just the opposite way than predicted by which can contribute to resolving some of these standard politico-economic theory: among in- puzzles. In the process, it also seeks to reconcile dustrial democracies the more unequal ones certain empirical findings of the recent literature tend to redistribute less, not more. The arche- on political economy and growth. Several au- typal case is that of the United States versus thors, such as Alberto F. Alesina and Dani Ro- drik (1994) or Torsten Persson and Guido Tabellini (1994), have documented a negative * Woodrow Wilson School, Princeton University, Princeton, NJ 08544, National Bureau of Economic Re- relationship between initial disparities of in- search, and Centre for Economic Policy Research. This come or wealth and subsequent aggregate paper was written while I was at New York University, and growth. The proposed explanation is that visiting the Institut d’Economie Industrielle (IDEI-CERAS- greater inequality translates into a poorer me- GREMAQ) in Toulouse, France. I am grateful for helpful dian voter relative to the country’s mean in- comments to Olivier Blanchard, Jason Cummins, John Geanakoplos, Mark Gertler, Robert Hall, Ken Judd, Jean- come, as in Alan H. Meltzer and Scott F. Charles Rochet, Julio Rotemberg, seminar participants at Richard (1981). This leads to increased pressure the NBER Summer Institute, Stanford University, Princeton for redistributive policies, which in turn reduce University, and the Massachusetts Institute of Technology, incentives for the accumulation of physical and as well as to three referees. Financial support from the National Science Foundation, the MacArthur Foundation, human capital. The cross-country data, how- and the C.V. Starr Center is gratefully acknowledged. Fre- ever, do not seem very supportive of this expla- derico Ravenna provided excellent research assistance. nation. Roberto Perotti (1994, 1996), and most 96 VOL. 90 NO. 1 BE´ NABOU: INCOME DISTRIBUTION AND THE SOCIAL CONTRACT 97 of the other studies reviewed in Be´nabou states. There are two important ingredients in (1996c), find no relationship between inequality the analysis. The first one is that redistribution and the share of transfers or government expen- enhance ex ante welfare, at least up to a point. ditures in GDP. Among advanced countries the I thus examine policies which reduce the vari- effect is actually negative, as suggested by the ance and possibly increase the mean of family above examples (Francisco Rodriguez, 1998).1 income, by providing insurance against idiosyn- As to the effect of transfers on growth, most cratic shocks and relaxing credit constraints. studies yield estimates which are in fact signif- The second one is a simple extension of the icantly positive. standard voting model, reflecting the fact that The point of departure for this paper is a some groups have more influence in the politi- rather different view of both the role of the state cal process than others. I present extensive ev- and the workings of the political process. When idence that the propensities to vote, give capital and insurance markets are imperfect, a political contributions, work on campaigns, and variety of policies which redistribute wealth participate in most forms of political activity from richer to poorer agents can have a positive rise with income and education. In the model net effect on aggregate output, growth, or more the pivotal agent is richer than the median, but generally ex ante welfare. Examples considered need not be richer than the mean. It should be here will include social insurance through pro- emphasized that I do not appeal to variations in gressive taxes and transfers, state funding of political rights or participation to explain coun- public education, and residential integration. tries’ different societal choices: this parameter Net efficiency gains lead to very different po- is kept fixed across steady states. In the com- litical economy consequences from those of parative statics analysis I vary the efficiency standard models: popular support for such re- costs and benefits of redistribution on the one distributive policies decreases with inequality, hand (via the elasticity of labor supply and the at least over some range. Intuitively, efficient degree of risk aversion), the political system on redistributions meet with a wide consensus in a the other, so as to identify their respective con- fairly homogeneous society but face strong op- tributions to the results. In particular, I show position in an unequal one. Conversely, if that there exists a critical level for the gain in ex agents engage in any type of investment, capital ante welfare from a redistributive policy, such market imperfections imply that lower redistri- that: (i) below this threshold, no allocation of bution translates into more persistent inequality. political influence can sustain more than a sin- The combination of these two mechanisms cre- gle social contract; (ii) above this threshold, ates the potential for multiple steady states: multiple steady states arise provided the politi- mutually reinforcing high inequality and low cal weight of the rich is neither too large nor too redistribution, or low inequality and high redis- small. tribution. Temporary shocks to the distribution When two “unequal societies” arise from of income or the political system can then have common fundamentals, they cannot be Pareto permanent effects. ranked. As to macroeconomic performance, the I formalize these ideas in a stochastic growth trade-off between tax distortions and liquidity- model with incomplete asset markets and het- constraint effects allows for two interesting erogeneous agents who vote over redistributive scenarios. One, termed “growth-enhancing re- policies, whether fiscal or educational. In the distributions,” is consistent with the positive short run, redistribution is shown to be coefficients of transfers in growth regressions, U-shaped with respect to inequality; in the long as well as the contributions of education and run they are negatively correlated across steady land policies in East Asian and Latin American countries to their respective developments (or lack thereof). The other, termed “eurosclerosis,” 1 Using panel data for 20 OECD countries and control- explains how European voters can choose to ling for national income, population, and the age distribu- sacrifice more employment and growth to social tion, Rodriguez finds that pretax inequality has a significantly negative effect on every major category of insurance than their American counterparts, social transfers as a fraction of GDP, as well as on the even though both populations have the same capital tax rate. basic preferences. Another prediction of the 98 THE AMERICAN ECONOMIC REVIEW MARCH 2000 model is that, depending on their source, exog- (1995), where agents’ imperfect learning of the enous shocks to income inequality will bring social mobility process allows different views of about sharply different evolutions of the social the equity-efficiency trade-off to persist in the contract. Thus, an increased variability of sec- long run.3 toral shocks will lead to an expansion of the The paper is organized as follows. Section I social safety net, while a surge in immigration explains the main ideas using the simplest pos- may prompt large-scale cutbacks. sible setup, which treats the aggregate impact of Some methodological features of the model redistribution as exogenous. Section II presents may also be worth mentioning. The first is its the actual economic model, and Section III the analytical tractability. Individual transitions are political mechanism. Section IV focuses on an linear, reflecting the absence of nonconvexities; endowment economy to study whether social yet there is multiplicity. The distribution of insurance will be more or less extensive in a wealth remains lognormal, and closed-form so- more unequal country. Section V solves the full lutions are obtained. The second is the progres- model with endogenous wealth dynamics. The sivity of the redistributive schemes over which range of economic and political “fundamentals” agents vote. The third is the intuitive formaliza- which allow multiple steady states is character- tion of political influence. These modelling de- ized, and the growth rates under alternative re- vices could be useful in other settings. gimes are compared. Section VI recasts the The paper is related to three strands of litera- model so as to explain differences in countries’ ture. The first one emphasizes the political econ- systems of education finance. Section VII dis- omy of redistribution (Giuseppe Bertola, 1993; cusses other applications such as altruism, res- Perotti, 1993; Gilles Saint-Paul and Thierry Ver- idential integration, and the mix of public dier, 1993; Alesina and Rodrik, 1994; Persson and goods. Section VIII concludes. All proofs are Tabellini, 1994, 1996). The second one is con- gathered in the Appendix. cerned with the financing and accumulation of human capital (Glenn C. Loury, 1981; Gerhard I. A Simplified Presentation of the Main Ideas Glomm and B. Ravikumar, 1992; Oded Galor and Joseph Zeira, 1993; Be´nabou, 1996b; Steven N. As a prelude to the actual model, I present in Durlauf, 1996a; Mark Gradstein and Moshe Just- this section a very stylized reduced form which man, 1997; Raquel Ferna´ndez and Richard Rog- provides a shortcut to the main intuitions and erson, 1998). The third one stresses the wealth and results. incentive constraints which bear on entrepreneur- ial investment (Abhijit Banerjee and Andrew A. The Standard View Newman, 1993; Philippe Aghion and Patrick Bolton, 1997; Thomas Piketty, 1997). Most di- Let there be a continuum of agents, i ʦ [0, rectly related are the models in Be´nabou 1], with lognormally distributed endowments: (1996b, c), upon which I build, and the paper by ln yi ϳ (m, ⌬2). The lognormal is a good Saint-Paul (1994), which identifies another mech- approximationN of empirical income distribu- anism through which capital market imperfections tions, leads to tractable results, and allows for can lead to multiple politico-economic regimes.2 an unambiguous definition of inequality, as in- A rather different explanation for international creases in ⌬2 shift the Lorenz curve outward. differences in redistribution is provided in Piketty This variance also measures the distance be- tween median and per capita income: m ϭ ln y Ϫ⌬2/ 2, where y ϵ E[ yi]. Suppose now that 2 Saint-Paul points out that increases in inequality whose adverse impact is concentrated in the lower tail of the income distribution may be accompanied by a rise in me- 3 Piketty’s mechanism is similar to a collective form of dian income, relative to the mean, so that the politically the bandit problem. Because individual experimentation is decisive middle class will reduce its transfers to the poor. If costly, agents never fully learn the extent to which income exit from poverty requires some investment, credit con- is affected by effort rather than predetermined by social straints may then lead to multiple steady states: a large origins. The citizens of otherwise identical countries may underclass which persists due to low redistribution, or one then end up with different distributions of beliefs over the which is kept small by significant transfers. disincentive effects of redistribution. VOL. 90 NO. 1 BE´ NABOU: INCOME DISTRIBUTION AND THE SOCIAL CONTRACT 99 agents are faced with the choice between two does the likelihood that a policy which reduces stylized policies: aggregate income gets implemented. This is, in ( ) laissez-faire: each consumes his own en- essence, the mechanism by which inequality dowment,P ci ϭ yi, for all i. reduces growth in models such as Alesina and ( ˆ ) complete redistribution: resources are Rodrik (1994), Persson and Tabellini (1994), or pooled,P and everyone consumes ci ϭ y. some cases of Bertola (1993) and Perotti In this benchmark case where redistribution has (1993).5 The idea that distributional conflict no aggregate impact, how many people are in hampers economic performance appears to be favor of it? Clearly, all those with endowment reasonably well supported by the evidence: a below the mean, i.e., a proportion number of studies have confirmed Persson and Tabellini’s and Alesina and Rodrik’s findings of ⌬2/2 ⌬ a negative effect of inequality on growth.6 This (1) p ϭ ⌽ ϭ ⌽ , ͩ ⌬ ͪ ͩ 2 ͪ correlation, however, does not seem to arise through increased redistribution. Perotti (1994, where ⌽٪ is the cumulative distribution func- 1996), Philip Keefer and Stephen F. Knack tion of a standard normal. Because the income (1995), and Peter H. Lindert (1996) find no distribution is right-skewed, the median is be- relationship between the income share of the low the mean, p Ͼ 1⁄2. A strict majority rule middle class (which corresponds to the median would thus predict that redistribution should voter) and any tax rate or share of government always take place. In reality the poor vote with transfers in GDP. George R. G. Clarke (1992) lower probability than the rich and, to some finds no correlation between any measure of extent, money buys political influence. There- inequality and government consumption. As ca- fore the relevant threshold for redistribution to sual empiricism suggests, more unequal coun- occur may not be p* ϭ 50 percent ϭ⌽(0), but tries do not redistribute more. Among advanced p* ϭ⌽(␭), ␭ Ͼ 0. For instance if the ␲ nations, they typically redistribute less (Rodri- poorest agents never vote, ⌽(␭) ϭ (1 ϩ ␲)/2.4 guez, 1998). Furthermore, the coefficients on What is important and robust in (1) is therefore transfers in growth regressions are most often not the level effect, p Ͼ 1⁄2, but the comparative significantly positive: see, among others, Shan- statics, Ѩp/Ѩ⌬Ͼ0: in a more unequal society there tarayan Devarajan et al. (1993), Lindert (1996), is greater political support for redistribution. For Xavier Sala-i-Martin (1996), and especially any degree of bias ␭ in the voting system, positive Perotti (1994, 1996), who controls for the en- or negative, the likelihood that redistribution takes dogeneity of redistribution. While none of these place increases with inequality—or more specifi- findings should be viewed as definitive evi- cally, skewness. dence, altogether they do suggest that some- This result is only reinforced under the stan- thing important may be missing from the dard assumption that redistribution entails some traditional story. deadweight loss (endogenized later on), reduc- ing available resources from y to yeϪB, B Ͼ 0. B. Efficiency Gains and Redistribution: Given the choice between laissez-faire and shar- The Static Case ing this reduced pie, the extent of political sup- port for the inefficient redistribution is: In a world of incomplete insurance and loan markets some policies with redistributive ϪB ϩ ⌬2/2 B ⌬ (2) p ϭ ⌽ ϭ ⌽ Ϫ ϩ . ͩ ⌬ ͪ ͩ ⌬ 2 ͪ 5 Naturally, this simple reduced form fails to capture the 1 richness of the original models. Note that now p ѥ ⁄2 but Ѩp/Ѩ⌬ is even more 6 See Be´nabou (1996c) for a survey. As with nearly all positive than before. As inequality increases, so growth regressions this finding is robust to some changes in specification but not to others. This caveat should be kept in mind, but in any case such a correlation is not essential to my results, which primarily involve inequality and redistri- 4 Section III will formalize in more detail—as well as bution. Thus, Section V, subsection B, will identify param- present extensive evidence on—the influence of income and eter configurations such that inequality and growth are human wealth on most forms of political activity. positively, or negatively, correlated across steady states. 100 THE AMERICAN ECONOMIC REVIEW MARCH 2000

FIGURE 1. INEQUALITY AND POLITICAL SUPPORT FOR REDISTRIBUTION Note: B ϭ 5 percent, 0, Ϫ5 percent. features can have a positive effect on total wel- stand to lose from the redistribution increases. fare, and even output (as the evidence on trans- At high enough levels of inequality, however, fers and growth may suggest). Ex ante welfare the standard skewness effect eventually domi- gains, in turn, imply a political support that nates: there are so many poor that they impose varies with inequality in a radically different redistribution no matter what its aggregate im- way from the traditional one. Indeed, suppose pact may be. There is thus no monotonic rela- that by redistributing resources agents achieve tionship between income inequality or the increased efficiency, so that each gets to con- relative position of the median agent (both mea- sume yeB. For now I continue to take B Ͼ 0as sured here by ⌬2) and the likelihood of exogenous, but later on I shall derive it from a redistribution.7 variety of channels: insurance, altruism, or To relate the extent of popular support for a credit constraints on the accumulation of human policy to actual outcomes one needs to specify and physical capital. The fraction of people who the mechanism through which preferences are support an efficient redistribution is: aggregated. I shall continue to assume that re- distribution occurs if p exceeds a threshold p* B ϩ ⌬2/2 B ⌬ ϵ⌽(␭), where ␭ reflects the degree to which (3) p ϭ ⌽ ϭ ⌽ ϩ . ͩ ⌬ ͪ ͩ ⌬ 2 ͪ financial or human wealth contributes to politi- cal influence. More generally, the probability of implementation could be some increasing func- It is of course always higher than (1) and (2), tion of p. Figure 1 shows that only if redistri- but the important point is the one illustrated on bution’s aggregate impact B is positive can the Figure 1.

PROPOSITION 1: When a redistributive pol- 7 The fact that inequality affects political support for icy generates gains in ex ante efficiency, polit- redistribution in opposite ways, depending on the sign of its ical support for it initially declines with aggregate impact, is essentially independent of distribu- tional assumptions. For any symmetric distribution F( x) inequality. In the present framework, the rela- with mean ␮, a symmetric, mean-preserving spread leads to tionship is U-shaped. a decline in popular support p ϭ F(␮ ϩ B) for all B Ͼ 0, and an increase for all B Ͻ 0. With lognormally distributed The intuition is simple: when dispersion is wealth, a rise in ⌬ combines this general effect of dispersion with a fall in m ϭ ␮ Ϫ⌬2/ 2 due to increased skewness; relatively small compared to the average gain, hence the U shape. These properties remain true when B is there is near-unanimous support for the policy. a function of ⌬, as long as ⌬BЈ(⌬)/B(⌬) Ͻ 1. Such will be As inequality rises, the proportion of those who the case when B is endogenized later on. VOL. 90 NO. 1 BE´ NABOU: INCOME DISTRIBUTION AND THE SOCIAL CONTRACT 101

this section carry over to a fully specified inter- temporal model of individual behavior and col- lective choice, where the welfare gains and losses from redistribution are endogenous.

II. Incomplete Asset Markets and Progressive Taxation

A. Technology, Preferences, and Decisions

The economy is populated by overlapping- generations families, i ʦ I ϵ [0, 1]. In gen- FIGURE 2. DYNAMICS AND STEADY STATES eration t, adult i combines his (human or i i physical) capital endowment kt with effort lt to produce output, subject to an independently policy be abandoned as the result of greater and identically distributed (i.i.d.) productivity i inequality—no matter how biased the political shock zt: system might be. Conversely, a positive ␭ is i i i ␥ i ␦ needed for the political outcome to reflect the (4) yt ϭ zt͑kt͒ ͑lt͒ . drop in popular support. The full model will confirm the joint importance of efficiency gains Taxes and transfers, specified below, trans- i and wealth bias in shaping a declining relation- form this gross income yt into a disposable i ship between inequality and redistribution. The income yˆ t which finances both the adult’s i arguments seen here for a zero-one policy consumption, ct, and his investment or edu- i choice will then apply to marginal changes in cational bequest, et: the tax rate, i.e., to every electoral contest be- i i i tween ␶ and ␶ ϩ d␶. (5) yˆ t ϭ ct ϩ et Finally, consider the dynamic implications of i i i ␣ i ␤ Proposition 1. A society which starts with (6) kt ϩ 1 ϭ ␬␰t ϩ 1͑kt͒ ͑et͒ . enough wealth disparity to find itself below p* on the U-shaped curve of Figure 1 will not Capital depreciates geometrically at the rate 1 Ϫ implement the redistributive policy; as a result, ␣ Ն ␤␥, and investment is subject to i.i.d. i high inequality will persist into the next period productivity shocks ␰t. There is no loan market or generation. Conversely, low inequality cre- for financing individual investment or educa- ates wide political support for efficient policies tional projects (e.g., children cannot be held which prevent disparities from growing. The responsible for the debts of their parents) and no dynamic feedback operates whenever some insurance or securities market where the idio- i i form of investment is credit constrained, so that syncratic risks zt and ␰t ϩ 1 could be diversified current resources affect future earnings. Thus away. These are extreme forms of market incom- the same type of market imperfections which pleteness, but all that really matters is that there be can give rise to a (partly) decreasing relation- some imperfections.8 Both shocks are assumed to ship between inequality and transfers also pro- be lognormal with mean one, and initial endow- vide the second ingredient required for multiple ments lognormally distributed across families: i 2 2 i 2 2 steady states. These dynamics are represented thus ln zt ϳ (Ϫv /2, v ), ln ␰t ϳ (Ϫw /2, w ) i N 2 N on Figure 2 (which will be formally derived and ln k0 ϳ (m0, ⌬0). later on), for a continuous rate of redistribution N ␶ ʦ [0, 1]. The U-shaped curve ␶ ϭ T(⌬)is similar to that of Figure 1, while the declining 8 Perotti (1994) provides evidence that credit-market ⌬ϭD(␶) locus arises from the accumulation frictions reduce aggregate investment, especially where the income share of the bottom 40 percent is low. Additional mechanism. evidence on asset-market incompleteness as a constraint on The main part of the paper, to which I now investment decisions in education and farming is discussed turn, will show that all the intuitions obtained in in Be´nabou (1996c). 102 THE AMERICAN ECONOMIC REVIEW MARCH 2000

FIGURE 3. PREFERENCES AND THE TIMING OF DECISIONS

i Adults have preferences defined over their ing uncertainty over his ex post utility level Vt is own consumption and effort, as well as their reflected in his ex ante preferences, child’s endowment of capital. Following David M. Kreps and Evan L. Porteus (1979), Larry G. i i r i 1/r Epstein and Stanley E. Zin (1989), and Philippe (8) Ut ϵ ln ͑Et ͓͑Vt͒ ͉kt͔ ͒, Weil (1990), these preferences are defined re- cursively over an individual’s lifetime; see Fig- ure 3. Thus, upon discovering his productivity, with a risk-aversion coefficient of 1 Ϫ r. When i zt, agent i chooses effort, consumption, and r ϭ 0 preferences are time separable and savings so as to maximize: 1/(1 Ϫ r) coincides with the intertemporal elasticity of substitution in consumption, which i i i ␩ 10 (7) ln Vt ϵ max ͕͑1 Ϫ ␳͓͒ln ct Ϫ ͑lt͒ ͔ by (7) is fixed at one. The more general spec- i i lt,ct ification allows 1 Ϫ r to parametrize the insur- ance value of redistributive policies, just as the i rЈ i i 1/rЈ ϩ ␳ ln ͑͑Et ͓͑kt ϩ 1͒ ͉kt, zt͔͒ ͖͒. labor-supply elasticity 1/(␩ Ϫ 1) parametrizes the effort distortions. Varying these parameters will make it possible to study how the set of The disutility of effort is measured by ␩ Ͼ 1, politico-economic equilibria is shaped by the which corresponds to an intertemporal elasticity costs and benefits of redistribution—the central of labor supply equal to 1/(␩ Ϫ 1). The discount issue of this paper. Finally, it should be noted factor ␳ defines the relative weights of the that while I have assumed overlapping genera- adult’s own felicity and of his bequest motive, while his (relative) risk aversion with respect to i the child’s endowment kt ϩ 1 is 1 Ϫ rЈ. At the because the policies which I shall consider provide no i i beginning of period t, however, when evaluat- insurance against ␰t ϩ 1 (only against zt), the value of 1 Ϫ rЈ ing and voting over redistributive policies, the will turn out not to play an important role in the analysis. It i could therefore be set (for instance) to zero, to one, or to agent has not yet learned zt. In other words, he 1 r, which is the more essential risk-aversion parameter i i 9 Ϫ knows his type (kt, zt) imperfectly. The result- defined in (8) below. 10 This last assumption is made for analytical tractability, as in much of the literature on income distribution dynamics [e.g., Glomm and Ravikumar (1992), Banerjee and New- 9 i i Therein lies the crucial difference between ␰t and zt, man (1993), Galor and Zeira (1993), Saint-Paul and Verdier rather than in the fact that one affects human capital and the (1993), Aghion and Bolton (1997), or Gradstein and Just- other production; the latter roles could be switched. Also, man (1997)]. VOL. 90 NO. 1 BE´ NABOU: INCOME DISTRIBUTION AND THE SOCIAL CONTRACT 103 tions with “imperfect” altruism, many of pa- and savings rate: per’s results can also be derived with infinitely 11 1/␩ lived agents. lt ϭ ␹͑1 Ϫ ␶t ͒

i i B. Fiscal Policy et ϭ syˆ t

Taxes and transfers map agent i’s market i i ␩ income yt into a disposable a income yˆ t, accord- where ␹ ϵ (␦/␩)(1 Ϫ ␳ ϩ ␳␤)/(1 Ϫ ␳) and s ϵ ing to the following scheme: ␳␤/(1 Ϫ ␳ ϩ ␳␤).

i i 1 Ϫ ␶t ␶t (9) yˆ t ϵ ͑yt͒ ͑y˜ t ͒ . Because taxes are progressive rather than merely proportional they affect effort, lt ϭ l(␶t), in spite of the fact that utility is logarith- 13 The break-even level y˜t is determined by the gov- mic in consumption. I will refer from here on ernment’s budget constraint: net transfers must to 1/␩ as “the” elasticity of labor supply.14 sum to zero or, denoting per capita income by yt: C. Redistribution and Accumulation 1 (10) yi 1 Ϫ ␶t y˜ ␶t di ϭ y . Given Proposition 2, and substituting (9) into ͵ ͑ t͒ ͑ t ͒ t 0 (6), capital accumulation simplifies to: (11) The elasticity ␶ of posttax income measures the t i i i degree of progressivity (or regressivity) of fiscal ln kt ϩ 1 ϭ ln ␰t ϩ 1 ϩ ␤͑1 Ϫ ␶t ͒ ln zt ϩ ln ␬ policy.12 An alternative interpretation is that of i wage compression through labor-market insti- ϩ ␤ ln s ϩ ͑␣ ϩ ␤␥͑1 Ϫ ␶t ͒͒ ln kt tutions favorable to workers with relatively low skills. Confiscatory rates ␶t Ͼ 1 must be ex- ϩ ␤␦͑1 Ϫ ␶t ͒ ln lt ϩ ␤␶t ln y˜ t . cluded as not incentive compatible, but nothing in principle prevents a regressive tax, so I shall Due to the symmetry of agents’ effort and sav- allow it. Restricting ␶t to be nonnegative would ings decisions, wealth and income remain log- i require dealing with corner solutions but would normally distributed over time. If ln kt ϳ (mt, 2 N not change the nature of any result. ⌬t ), the government’s budget constraint (10) easily yields the break-even point of the redis- PROPOSITION 2: Given a tax rate ␶t, agents tributive scheme (see the Appendix): in generation t choose a common labor supply 2 2 (12) ln y˜t ϭ ␥mt ϩ ␦ ln lt ϩ ͑2 Ϫ ␶t͒␥ ⌬t /2

11 2 See Be´nabou (1996a, 1999). The infinite-horizon ver- ϩ͑1 Ϫ ␶t͒v /2. sion of the preferences used here is obtained by replacing i i kt ϩ 1 with Vt ϩ 1 in (7), with rЈϭr and (8) unchanged. 12 When ␶t Ͼ 0 the marginal rate rises with pretax income, and agents with average income are made better 13 They would also affect savings, were it not for adults’ off: y˜ t Ͼ yt. Measuring progressivity by the (local) elastic- simple bequest motive. Even with infinitely lived agents, ity of after-tax to pretax income was first proposed by however, the savings distortion can be fully offset by a Richard A. Musgrave and Tun Thin (1948). Ulf Jakobsson balanced-budget combination of consumption taxes and in- (1976) and Nanak C. Kakwani (1977) showed this to be the vestment subsidies; moreover, this can be shown to be “right” measure of equalization: the posttax distribution Pareto optimal (Be´nabou, 1996a, 1999). In any case, one induced by a fiscal scheme Lorenz-dominates the one in- distortion is enough to demonstrate how the trade-off be- duced by another (for all pretax distributions), if and only if tween the costs and benefits of redistribution shapes the the first scheme’s elasticity is everywhere smaller. A “con- range of politico-economic equilibria. stant residual progression” scheme similar to (9) turns out to 14 It is indeed the uncompensated elasticity to the net- have been used in a couple of earlier but static models of of-tax rate 1 Ϫ ␶t, and varies monotonically with the usual insurance or risk taking (Martin S. Feldstein, 1969; S. M. intertemporal elasticity of substitution for proportional vari- Kanbur, 1979; Mats Persson, 1983). ations in the real wage, 1/(␩ Ϫ 1). 104 THE AMERICAN ECONOMIC REVIEW MARCH 2000

From (11), we then obtain two simple differ- priate spillover ␬t (see Section V, subsection ence equations which govern the evolution of B). The next term, capturing the effects of the economy: labor supply on accumulation, is also of a “representative agent” nature. The terms in (13) mtϩ1 ϭ ͑␣ ϩ ␤␥͒mt ϩ ␤␦ ln lt v(␶) and ⌬(␶), on the other hand, represent growthL lossesL specific to the heterogenous 2 2 2 ϩ ␤␶t͑2 Ϫ ␶t͒͑␥ ⌬t ϩ v ͒/2 agent economy with imperfect credit markets. Suppose first that adults in generation t are ex ␤ 2 2 2 ϩ ln ͑␬s ͒ Ϫ ͑w ϩ ␤v ͒/2 ante identical (⌬t ϭ 0). Everyone then faces the same accumulation technology, with con- 2 2 2 (14) ⌬t 1 ϭ ͑␣ ϩ ␤␥͑1 Ϫ ␶t ͒͒ ⌬t cavity (decreasing returns) ␤␥(1 Ϫ ␤␥). The ϩ i shocks zt generate ex post income disparities 2 2 2 2 ϩ ␤ ͑1 Ϫ ␶t ͒ v ϩ w . (partially offset by redistribution), which credit constraints translate into inefficient variations in investment, reducing overall 2 The effect of redistribution on the dynamics of growth by v(␶t)v /2. Consider now dispari- L 2 2 inequality is clear: the progressivity rate ␶t de- ties in initial endowments, ␥ ⌬t . When ␣ ϭ 0 termines the persistence of family wealth, ␣ ϩ these have the same effect as income shocks: ␤␥(1 Ϫ ␶t). The impact on the dynamics of v ϵ ⌬. The marginal return to investment is aggregate income is more complex, as it in- higherL L for the poor than for the rich, because volves a trade-off between labor supply and they are more severely liquidity constrained. credit-constraint effects: When ␣ Ͼ 0, however, the preexisting capital i stocks kt represent complementary inputs which generate differential returns to invest- PROPOSITION 3: The distribution of pretax ment, and thereby reduce the desirability of i income at time t is ln yt ϳ (␥mt ϩ ␦ ln lt Ϫ equalizing resources. Thus ⌬(␶)ismini- 2 2 2 2 N 2 L v /2, ␥ ⌬t ϩ v ), where mt and ⌬t evolve mized for ␶ ϭ (1 Ϫ ␣ Ϫ ␤␥)/(1 Ϫ ␤␥), which according to the linear difference equations decreases with ␣. 1/␩ (13)–(14) and lt ϭ ␹(1 Ϫ ␶t) . The growth rate of per capita income is: D. Individual Welfare

(15) I now turn from the evolution of the economy as a whole to individuals’ evaluations of alter- ln ͑yt ϩ 1 /yt ͒ ϭ ln ␬˜ Ϫ ͑1 Ϫ ␣ Ϫ ␤␥͒ln yt native policies. Given the optimal labor supply and savings responses to a tax rate ␶t, (7) gives ␦ ln l ␣ ln l i ϩ ͑ t ϩ 1 Ϫ t ͒ agent i’s ex post welfare Vt for any productivity i realization zt. Since he must vote before learn- ␶ 2 ␶ ␥2 2 i Ϫ v ( t )v /2Ϫ ⌬ ( t ) ⌬t /2, ing zt, his preferences over ␶t are then defined L L i by the ex ante utility Ut, according to (8). where ln ␬˜ ϵ ␥(ln ␬ ϩ ␤ ln s) Ϫ ␥(1 Ϫ ␥)w2/2 is a constant and PROPOSITION 4: Given a rate of fiscal pro- (␶)ϵ␤␥(1Ϫ␤␥)(1Ϫ␶)2Ͼ0, gressivity ␶ , agent i’s intertemporal welfare is: Lv t (␶)ϵ␣ϩ␤␥(1Ϫ␶)2Ϫ(␣ϩ␤␥(1Ϫ␶))2Ͼ0. L⌬ i i (16) Ut ϭ u៮ t ϩ A͑␶t ͒͑ln kt Ϫ mt ͒

The term in Ϫln yt reflects the standard convergence effect; it disappears under con- ϩ C͑␶t ͒ Ϫ ͑1Ϫ ␳ ϩ ␳␤͒ stant aggregate returns, namely when ␣ ϩ 2 2 2 2 ␤␥ ϭ 1orwhen␬ is replaced by an appro- ϫ͑1 Ϫ ␶t ͒ ͑␥ ⌬t ϩ Bv ͒/2, VOL. 90 NO. 1 BE´ NABOU: INCOME DISTRIBUTION AND THE SOCIAL CONTRACT 105 where u៮ t is independent of the policy ␶t and: existing inequality, the loss in aggregate wel- 2 2 2 fare is (1 Ϫ ␳ ϩ ␳␤)(1 Ϫ ␶) ␥ ⌬t /2; it (17) A͑␶͒ ϵ ␳␣ ϩ ͑1 Ϫ ␳ ϩ ␳␤͒␥͑1 Ϫ ␶͒, embodies two effects, only one of which is due to market incompleteness. First, reallo- (18) C͑␶͒ ϵ ͑1 Ϫ ␳͒͑␦ ln l͑␶͒ Ϫ l͑␶͒␩͒ cating investment resources towards the poor again increases the growth rate of total ϩ ␳␤␦ ln l͑␶͒, wealth, ln (kt ϩ 1/kt). Second, there is the stan- dard effect of concave (logarithmic) utility (19) B ϵ 1 Ϫ r ϩ ␳r͑1 Ϫ ␤͒ Ն 0. functions, whereby average welfare increases i when individual consumptions (of ct and i 15 kt ϩ 1) are distributed more equally. Equiv- i The first term in Ut depends only on the state alently in this model, it captures the effect of 2 variables mt and ⌬t and on the (endogenous skewness, as in Section I: the median agent, but constant) investment rate s. The second who is poorer than average, gains when con- term, which disappears through aggregation, sumption and bequests are redistributed pro- makes clear the redistributive effects of tax gressively. policy, including its impact on the persistence of social positions, ␣ ϩ ␤␥(1 Ϫ ␶). The last III. The Political System two terms represent the aggregate welfare cost and aggregate welfare benefit of a pro- I now turn to the determination of the equi- gressivity rate ␶t. Thus C(␶t), which is max- librium policy. Each generation chooses, before i imized for ␶t ϭ 0, reflects the distortions in the individual productivity shocks zt are real- labor supply entailed by such a policy. Con- ized, the rate of fiscal progressivity ␶t to which 2 2 2 2 versely, the term Ϫ(1 Ϫ ␶t) (␥ ⌬t ϩ Bv ), it will be subject. Agent i’s ideal policy would i 16 which is maximized for ␶t ϭ 1, embodies the thus be to maximize Ut. These individual efficiency gains which arise from better in- preferences are aggregated through a political surance and the redistribution of resources process in which some groups have more influ- from low to high marginal-product invest- ence than others. ments. Note that B is now endogenous, and monotonically related to risk aversion, 1 Ϫ r. 15 2 2 2 More generally, by varying 1 Ϫ r, ␤, v2, and One can rewrite (1 Ϫ ␳ ϩ ␳␤)␥ (1 Ϫ ␶) as ␳[␤␥ (1 Ϫ ␶)2 Ϫ (␣ ϩ ␤␥(1 Ϫ ␶))2] ϩ (1 Ϫ ␳)␥2(1 Ϫ ␶)2 ϩ ␳(␣ ϩ 1/␩, the net ex ante efficiency gain, (1 Ϫ ␳ ϩ 2 j 2 2 ␤␥(1 Ϫ ␶)) ϭϪ␳ ln (kt ϩ 1/ln kt) ϩ (1 Ϫ ␳)varj ʦ I[ln ct] ␳␤)(1 Ϫ ␶) B /2 Ϫ C(␶), can be made j v ϩ ␳varj ʦ I[ln kt ϩ 1] ϩ ␮t, where varj ʦ I[ ⅐ ] denotes a arbitrarily large or small relative to preexist- cross-sectional variance, ␮t is independent of ␶t, and I have ing income inequality, ␥2 2. set v ϭ w ϭ ␦ ϭ 0 for notational simplicity. ⌬t 16 To make the role of market incompleteness Due to the overlapping-generations structure, no in- tertemporal strategic considerations are involved. Infinite more explicit I consider again each source of horizons, by contrast, would generate a dynamic game heterogeneity in turn. By (16), idiosyncratic where voters try to influence future political outcomes ␶t ϩ k 2 uncertainty lowers everyone’s utility by (1 Ϫ by altering the evolution of the wealth distribution ⌬t ϩ k 2 2 through their choice of ␶ [see (14)]. This problem is noto- ␳ ϩ ␳␤) B(1 Ϫ ␶) v / 2. When ␤ ϭ 1 this t 2 2 riously intractable, so the standard practice is to assume that simplifies to (1 Ϫ r)(1 Ϫ ␶) v / 2: with voters are “myopic,” either ignoring their influence on fu- constant returns, the ex ante value of redistri- ture outcomes or—as here—not caring about it due to a bution stems from the insurance it provides. limited bequest motive. Notable exceptions are Saint-Paul In general, it also contributes to efficiency (1994) and Gene M. Grossman and Elhanan Helpman through the relaxation of credit constraints. (1996). Alternatively, Per Krusell et al. (1997) look for numerical solutions. In Be´nabou (1996a) I considered a This is best seen with risk-neutral parents different form of political myopia and obtained results sim- who care only about their offspring: when ilar to those presented here. In short, agents with fully r ϭ ␳ ϭ 1 the utility loss is ␤(1 Ϫ ␤)(1 Ϫ dynastic preferences choose a constitution (namely a con- 2 2 stant sequence {␶ ␶ }ϱ ), ignoring the fact that ␶) v /2, which is the shortfall in expected t ϩ k ϭ t k ϭ 0 future generations may revise it. In any steady state, how- (and aggregate) bequests resulting from idio- 2 2 ever, [⌬t ϭ⌬t ϩ 1 in (14)], every generation, given the syncratic resource shocks and a concave same choice set as its predecessors, validates the existing investment technology. Turning now to pre- social contract. 106 THE AMERICAN ECONOMIC REVIEW MARCH 2000

A. Preferred Policies regressive, which cause individuals to distort their labor supply away from the first-best I first consider the simpler case where labor level. Finally, as a prelude to the analysis of supply is inelastic, 1/␩ ϭ 0. The utility function political equilibrium, note how (21) embodies i Ut is then quadratic in ␶t, and maximized at: the same intuitions as the stylized model of Section I. For any ␶ Ͻ 1 the proportion of 2 2 2 1 ␥ ⌬t ϩ Bv agents who would like further redistribution (20) i ϭ i . at the margin, 1 Ϫ ␶t ␥ max ͕ln kt Ϫ mt ,0͖

(22) p͑␶, ⌬t ͒ Voters below the median desire the maximum i i feasible tax rate, ␶t ϭ 1, which is also the ex ѨUt ϵ card i Ͼ 0 ante efficient one in the absence of distortions. ͭ ͯ Ѩ␶ ͮ i Voters above the median desire a tax rate ␶t Ͻ 1 which decreases with their initial wealth and ͑1 Ϫ ␶͒Bv2 Ϫ ␦␶/͑␩͑1 Ϫ ␶͒͒ increases with the variance of productivity ϭ⌽ shocks, for both insurance and investment rea- ͩ ␥⌬t sons (concavity of preferences and concavity of the technology). ϩ͑1 Ϫ ␶͒␥⌬ , With endogenous labor supply, complete re- tͪ distribution is never chosen, as it would lead to zero effort and output. Agent i’s desired policy is U-shaped in ⌬t, provided the resulting gain i 2 is given by the first-order condition ѨUt/Ѩ␶ ϭ in ex ante efficiency (1 Ϫ ␶) Bv dominates 0, or: the distortion ␦␶/(␩(1 Ϫ ␶)). Otherwise, p(␶, ⌬t) is strictly increasing in ⌬t. Also as on 2 2 2 i (21) ͑1 Ϫ ␶͒͑␥ ⌬t ϩ Bv ͒ Ϫ ␥ ͑ln kt Ϫ mt ͒ Figure 1, variations in popular support for efficient increases in ␶ all take place above the ␦ ␶ 50-percent level, so they will influence policy Ϫ ϭ 0. ␩ ͩ 1 Ϫ ␶ͪ outcomes only under some departure from the pure “one person, one vote” democratic ideal.

This quadratic equation always has a unique i solution less than 1, which will be denoted ␶t. B. Wealth and Political Influence

It is well known that poor and less educated i PROPOSITION 5: Each agent’s utility Ut is individuals have a relatively low propensity to strictly concave in the policy ␶t. His preferred register, turn out to vote, and give political i i tax rate ␶t decreases with his wealth kt and contributions. These are among the facts docu- 2 i increases with Bv . Finally, ͉␶t͉ decreases with mented in Tables 1 and 2, which present data the labor-supply elasticity 1/␩.17 from Steven J. Rosenstone and John M. Hansen (1993); see also Raymond E. Wolfinger and These results are intuitive. Lower personal Rosenstone (1980), Thomas B. Edsall (1984), wealth or greater ex ante efficiency benefits or Margaret M. Conway (1991). For each form from redistribution increase an agent’s de- of participation in electoral or governmental mand for such policies. A more elastic labor politics, the representation ratio of a given so- supply increases the deadweight loss from cioeconomic group is the ratio between its share taxes and transfers, whether progressive or of the population engaged in this activity and its share of the general population. Thus the poor-

17 i 2 2 est 16 percent account for only 0.76 ϫ 16 ϭ More specifically, for ln kt Յ mt ϩ ␥⌬t ϩ Bv /␥ the i 12.2 percent of the votes and 0.25 ϫ 16 ϭ 4.0 ideal ␶t is positive and decreases towards zero as 1/␩ rises. i 2 2 i percent of the number of campaign contributors, For ln kt Ͼ mt ϩ ␥⌬t ϩ Bv /␥, ␶t Ͻ 0 and it increases towards zero as 1/␩ rises. while the richest 5 percent account for 1.27 ϫ VOL. 90 NO. 1 BE´ NABOU: INCOME DISTRIBUTION AND THE SOCIAL CONTRACT 107

TABLE 1—POLITICAL PARTICIPATION BY INCOME

Political activity: Total fraction Representation ratios by percentile family income Electoral Politics, taking part Pivot p* 1952–1988 (in percent) 0–16 17–33 34–67 68–95 96–100 (in percent) Vote 66.1 0.76 0.90 1.00 1.16 1.27 55.5 Try to influence others 26.7 0.63 0.79 0.98 1.25 1.54 60.6 Contribute money 8.9 0.25 0.51 0.98 1.54 3.25 73.6 Attend meetings 7.8 0.49 0.73 0.93 1.31 2.27 65.7 Work on campaign 4.6 0.48 0.74 0.85 1.37 2.42 67.6

Source: Rosenstone and Hansen (1993), Table 8-2, plus my computation of p*.

TABLE 2—POLITICAL PARTICIPATION BY EDUCATION

Representation ratios by years of education (with Total fraction corresponding percentage of population) taking part 0–8 9–11 12 13–15 16ϩ Pivot p* Political activity: (in percent) (20.1) (16.3) (32.8) (16.8) (14) (in percent) Electoral Politics, 1952–1988 Vote 66.1 0.85 0.83 1.00 1.12 1.26 55.8 Try to influence others 26.7 0.61 0.75 0.94 1.33 1.61 63.5 Contribute money 8.9 0.33 0.51 0.87 1.37 2.41 73.7 Attend meetings 7.8 0.48 0.50 0.85 1.43 2.14 72.2 Work on campaign 4.6 0.48 0.50 0.87 1.33 2.25 72.0 Governmental Politics, 1976–1988 Sign petition 34.8 0.34 [4 0.87 3] 1.44 69.5 Attend local meeting 18.0 0.31 [4 0.78 3] 1.46 73.3 Write Congress 14.6 0.38 [4 0.72 3] 1.56 75.9

Source: Rosenstone and Hansen (1993), Tables 8-1 and 8-2, plus my computation of p*.

5 ϭ 6.4 percent of the votes and 3.25 ϫ 5 ϭ countervailing advantage for attending meet- 16.3 percent of contributors.18 ings, working on campaigns, writing Congress, The data in Tables 1 and 2 are striking in and other time-intensive activities for which several respects. The propensity to participate in they have a lower opportunity cost. But, re- every reported form of political activity rises markably, the pro-wealth (financial and human) with income and education. For voting itself the bias is here again not only positive, but ex- tendency is relatively moderate, whereas for tremely strong. contributing to political campaigns it is drastic. I shall not seek to explain the source of these In the latter case the actual bias is still under- biases, only to model them in a plausible and stated since the data reflects only the number of convenient manner.19 Let each agent’s opinion contributions, and not their amounts. It is intu- be affected by a relative weight, or probability i 1 j itive that the wealthy should be overrepresented of voting, ␻ /͐0 ␻ dj. If individual preferences in money-intensive channels of political influ- are single-peaked and the preferred policy is ence: such lobbying is a form of collective monotonic in wealth, or more generally if pref- investment where liquidity constraints are even more likely to bind than usual. One might have expected poorer, less skilled agents to have a 19 John E. Roemer (1998) shows how the presence of a second dimension in the political game (morals, religion) can result in a similar kind of bias, by splitting the coalition which would naturally arise in favor of redistribution. It is 18 Put differently, the representation ratios are the slopes worth emphasizing again that I shall not appeal to differ- of the (piecewise linear) Lorenz curve which describes the ences in the allocation of political power or influence to concentration, by income or education, of a given form of explain why redistribution varies across countries (although political influence. the model can readily incorporate this “easier” explanation). 108 THE AMERICAN ECONOMIC REVIEW MARCH 2000 erences satisfy a single-crossing condition (as in are just as easy to capture; the case ␭ ϭ 1, for Joshua S. Gans and Michael Smart, 1996), a instance, corresponds to a “one dollar, one median-voter-type result applies, but where the vote” rule. Moreover, this alternative formu- median is computed on an appropriately renor- lation would only reinforce the paper’s re- malized population. With a lognormal distribu- sults, as it implies that the political system tion, the following schemes yield particularly becomes more biased towards the wealthy as simple results. inequality rises. In the last columns of Tables 1 and 2, I interpolated the empirical ␻( p) function to PROPOSITION 6: Suppose that agents i ʦ [0, compute the position of the pivotal agent p* for 1] have preferences U(ki , ␶) over some policy each separate form of political participation, variable ␶ ʦ ޒ, such that: for all k Ͻ kЈ and i.e., as if it were the only one that mattered.21 ␶ Ͻ ␶Ј,ifU(kЈ, ␶Ј) Ͼ U(kЈ, ␶) then U(k, ␶Ј) Ͼ No data exist which would allow me to weigh U(k, ␶). them by their relative importance in determin- ing the political outcome. For voting, the wealth (1) If an agent’s political weight depends on bias is moderate, with the pivot at the 56th his rank in the wealth distribution, ␻i ϭ percentile rather than the usually assumed me- ␻( pi), the pivotal voter is the one with rank dian. For all other forms of influence it is much p* ϭ⌽(␭) and (log) wealth ln k* ϭ m ϩ stronger, with the pivotal agent always above ⌽(␭) ␭⌬, where ␭ Ѥ 0 is defined by (͐0 ␻( p) the 60th percentile, and quite often the 70th. 1 1 dp)/(͐0 ␻( p) dp) ϭ ⁄2. From this evidence it seems safe to conclude (2) If an agent’s political weight depends on that the decisive political group is located above the absolute level of his wealth, with ␻i ϭ the median in terms of income and human (ki)␭ for some ␭ Ѥ 0, the pivotal voter has wealth. Depending on the relative efficacy of rank p* ϭ⌽(␭⌬) and (log) wealth ln k* the different channels of political influence it ϭ m ϩ ␭⌬2. may even be above the mean, which in the U.S. income distribution falls around the 63rd Ordinal schemes ensure that each person’s percentile.22 weight and the identity of the pivotal voter remain invariant when the distribution of C. Political Equilibrium, Inequality, wealth shifts due to growth, or when it be- and Redistribution comes more unequal.20 Previous discussions have often assumed that political rights or We are now ready to establish the paper’s influence depend on one’s absolute level of first main result. By virtue of Propositions 5 and wealth. This was motivated by historical ex- 6, the policy outcome is obtained by simply i amples such as voting franchises restricted to setting ln kt Ϫ mt ϭ ␭⌬t in the first-order i citizens owning enough property (Saint-Paul condition ѨUt/Ѩ␶ ϭ 0, or equivalently p(␶, and Verdier, 1993; Persson and Tabellini, ⌬t) ϭ p* ϭ⌽(␭) in (22). As before, I consider 1994) or costly membership in a ruling elite first the case where labor supply is inelastic (Alberto Ades and Verdier, 1996). I find it (1/␩ ϭ 0); for ␭ Ͼ 0, this yields: more plausible that even such cutoffs should be relative ones, keeping up with aggregate growth and the competitive nature of bids for 21 The weights ␻i are obtained by rescaling each group’s political influence. I shall therefore focus on representation ratios by the population’s average propensity ordinal schemes, but the second part of Prop- to participate in the activity under consideration (given in column 1). Concerning the relationship between voters’ osition 6 shows that absolute income effects income and their political preferences, Nolan M. McCarthy et al. (1997) provide substantial evidence that U.S. politics are highly—and increasingly—unidimensional, along the Also, for any ordinal scheme ␻٪ the associated ␭ is a axis of rich/opposed to redistribution versus poor/favoring 20 sufficient statistic: it is as if the bottom 2⌽(␭) Ϫ 1 voters [or redistribution. the top 1 Ϫ 2⌽(␭) when ␭ Ͻ 0] systematically abstained. 22 We shall see below that the first condition, but not the Note that even though ␭ Ͼ 0 is the more empirically second, is required for redistribution to decline with in- relevant case, the model will be solved for all ␭ ѥ 0. equality in the model. VOL. 90 NO. 1 BE´ NABOU: INCOME DISTRIBUTION AND THE SOCIAL CONTRACT 109

2 2 2 1 ␥ ⌬t ϩ Bv becoming richer relative to the mean. Second, it (23) ϭ . 1 Ϫ ␶ ␭␥⌬ is more likely to occur the larger the ex ante t t welfare gain from redistribution, e.g., the larger Bv2.24 Third, some bias ␭ Ͼ 0 with respect to The equilibrium tax rate is clearly U-shaped in ⌬t, pure majority rule is needed as well, because the and minimized where ␥2⌬2 ϭ Bv2. Similarly, in median agent always wants to push redistribu- the general case it is the unique solution T(⌬t) Ͻ tion beyond its range of efficiency. As shown in 1 to the quadratic equation derived from (21): (22), it is only within that range that political support for a tax increase, p(␶, ⌬t), can decline 2 2 2 ␥ ⌬t ϩ Bv with higher inequality. (24) ͑1 Ϫ ␶t ͒ It is worth noting that the distinctive non- ͩ ␥⌬t ͪ monotonic relationship predicted by the model has recently been tested by Paolo Figini (1999), ␦ ␶t Ϫ ϭ ␭. who found in cross-country regressions a sig- ␩␥⌬t ͩ 1 Ϫ ␶tͪ nificant U-shaped effect of income inequality on the shares of tax revenues and government expenditures in GDP. PROPOSITION 7: The rate of fiscal progres- The above results apply equally in an endow- sivity ␶t ϭ T(⌬t) chosen in generation t has the ment economy (␤ ϭ 0) and in the presence of following features: accumulation. In the first case the efficiency gains arise from insurance. In the second they (1) ␶t increases with the ex ante efficiency gain also reflect the reallocation of resources to from redistribution (gross of distortions) agents whose marginal product of investment is Bv2, and decreases with the political influ- higher, due to tighter liquidity constraints. In ence of wealth, ␭. studying which social contracts emerge in the (2) ͉␶t͉ decreases with the elasticity of labor long run, I shall consider each case in turn. supply 1/␩.23 (3) For ␭ Ͼ 0, ␶t is U-shaped with respect to IV. Inequality and Social Insurance inequality ⌬t. It starts at the ex ante effi- in an Endowment Economy cient rate T(0) ϭ 1 Ϫ 2(1 ϩ ͌1 ϩ 4␩Bv2/␦)Ϫ1, declines to a mini- Should we expect a more generous welfare mum at some ⌬Ͼ0, then rises back to- state in countries with greater disparities of in- wards T(ϱ) ϭ 1. The larger Bv2, the wider come and wealth, as predicted by standard mod- the range [0, ⌬) where Ѩ␶t /Ѩ⌬t Ͻ 0. For els, or a less generous one, as a comparison ␭ Յ 0, ␶t is increasing in ⌬t. between Sweden and the United States would suggest? To study the political economy of pure The first two results show that the equilib- social insurance, let us focus on an endowment rium tax rate depends on the costs and benefits economy (␤ ϭ 0).25 Each dynasty’s endowment i of redistribution, as well as on the allocation of kt then simply follows a geometric AR(1) pro- political influence, in a very intuitive manner. The third result provides an endogenously de- rived analogue to Figure 1, with the continuous 24 With respect to the first claim, note that ln k* Ϫ ln policy ␶ now replacing the proportion of people E[k] ϭ ␭⌬Ϫ⌬2/ 2 is increasing only on [0, ␭] and supporting redistribution in a zero-one decision. positive only on [0, 2␭]. Neither interval coincides with, nor It also confirms several claims made earlier contains, [0, ⌬]. The second claim follows from Proposition 7, but even when Bv2 ϭ 0 one sees from (22) that for all about the result that redistribution may decline ⌬ ʦ [0, ⌬] a marginal rise in ␶ above T(⌬) increases ex ante with inequality. First, it is not predicated on the efficiency. These gains now arise from lowering the effort pivotal agent being richer than the mean, or distortions due to regressionary taxes, as Bv2 ϭ 0 implies T(0) ϭ 0, hence ␶t Ͻ 0 on [0, ⌬]. 25 The important assumption here is the absence of in- surance markets. The incompleteness of the loan market is 23 2 2 More specifically, if Bv Ͼ ␭ /4 then ␶t Ͼ 0 and inessential, and indeed this section’s results remain un- 2 2 Ѩ␶t /Ѩ␩ Ͼ 0. If Bv Ͻ ␭ /4, then Ѩ␶t /Ѩ␩ has the sign of ␶t. changed with ␳ ϭ 0. This one-shot model is close to that of 110 THE AMERICAN ECONOMIC REVIEW MARCH 2000 cess with serial correlation ␣ [see (6)], and the greater persistence makes income more vari- equilibrium policy is T(⌬t). In the long run able, it also increases the number of agents for 2 2 2 inequality converges to ⌬ϱ ϵ w /(1 Ϫ ␣ ), whom the value of insurance is more than offset and the tax rate therefore to ␶ϱ ϵ T(⌬ϱ). When by their vested interest in the status quo. 1/␩ ϭ 0, for instance, V. History-Dependent Social Contracts

1 w ␥ A. Dynamics and Multiple Steady States (25) ϭ 2 1 Ϫ ␶ϱ ␭ 1 Ϫ ␣ ͫ ͱ I now turn to the paper’s second main idea, sketched at the end of Section I: if more in- v 2 ͱ1 Ϫ ␣2 ϩ ͑1 Ϫ ͑1 Ϫ ␳͒r͒ . equality leads to less redistribution, and if in- ͩ wͪ ͩ ␥ ͪͬ vestment resources depend on past transfers, multiple steady states can arise. To demonstrate More generally, the following results are imme- this point I solve the full model with capital diate (focussing on the empirically relevant case accumulation subject to wealth constraints (␤ Ͼ of ␭ Ͼ 0). 0). The critical difference with the endowment economy is that the wealth distribution is now PROPOSITION 8: The steady-state rate of fis- endogenous, through the effect of fiscal policy cal progressivity ␶ϱ increases with agents’ de- on persistence, ␣ ϩ ␤(1 Ϫ ␶t). The joint gree of risk aversion 1 Ϫ r and with income evolution of inequality and policy is thus de- uncertainty v2, but is U-shaped with respect to scribed by the recursive dynamical system: the variability w2 and the persistence ␣ of the endowment process. It decreases with the polit- ␶ ϭ T͑⌬ ͒ (26) t t ical influence of wealth ␭, and declines in ab- ͭ ⌬t ϩ 1 ϭ ͑⌬t , ␶t ͒ solute value with the labor-supply elasticity 1/␩. D

Recalling that steady-state income dispersion where T(⌬t) is the unique solution less than one 2 2 2 2 is ␥ w /(1 Ϫ ␣ ) ϩ v , the above results to (24), while (⌬t, ␶t) is given by (14). Under indicate that the relationship between inequality a time-invariantD policy, in particular, long-run and redistribution is not likely to be monotonic. inequality decreases with redistribution: What matters is not just the amount of income inequality, but also its source. To the extent that w2 ϩ ␤2͑1 Ϫ ␶͒2v2 (27) ⌬2 ϭ ϵ D2͑␶͒. high income disparities in some countries re- ϱ 1 Ϫ ͑␣ ϩ ␤␥͑1 Ϫ ␶͒͒2 flect larger uninsurable shocks (or more imper- fect insurance markets), higher taxes and transfers should be observed. But if greater in- A steady-state equilibrium is an intersection of equality is due to more persistent wealth dy- this downward-sloping locus, ⌬ϭD(␶), with namics or to greater ex ante heterogeneity at the the U-shaped curve ␶ ϭ T(⌬) described in time of the policy decision—correlated for in- Proposition 7; see Figure 2. Substituting (27) stance with ethnic or regional differences—the into (24), this corresponds to a rate of tax pro- reverse correlation may be observed.26 While gressivity solving the equation:

␥2D͑␶͒2 ϩ Bv2 (28) f͑␶͒ ϵ ͑1 Ϫ ␶͒ Persson (1983), except that he assumes no initial heteroge- ͩ ␥D͑␶͒ ͪ neity (⌬0 ϭ 0). 26 A related result obtains in Persson and Tabellini ␦ ␶ (1996), where two regions bargain over the degree of risk Ϫ ϭ ␭. sharing in the federal constitution. In both models the un- ␩␥D͑␶͒ ͩ 1 Ϫ ␶ͪ derlying assumption is the inability to make transfers con- tingent only on unpredictable innovations to individual or regional income, as distinguished from its permanent com- of how ex post political equilibrium constrains the ex ante ponent. See also George Casamata et al. (1997) for a study design of public health insurance systems. VOL. 90 NO. 1 BE´ NABOU: INCOME DISTRIBUTION AND THE SOCIAL CONTRACT 111

FIGURE 4. MULTIPLICITY OR UNIQUENESS OF STABLE STEADY STATES Note: The dotted line corresponds to 1/␩ЈϾ1/␩.

This is a polynomial equation of degree eight, ly, when risk-aversion 1 Ϫ r is less than some and therefore quite complex. Yet, by exploiting 1 Ϫ r), there is a unique, stable, steady state. geometric intuitions on the shape of f, one can When B Ͼ B, on the other hand, there exist ␭ establish a series of results which formalize the and ␭៮ with 0 Ͻ ␭ Ͻ ␭៮, such that: paper’s main ideas.27 As illustrated on Figure 4, two countries with the same economic and (1) For each ␭ in [␭, ␭៮] there are (at least) two political fundamentals can nonetheless evolve stable steady states.28 Inequality is lower, into different societies, provided: and social mobility higher, under a more redistributive social contract. (a) the ex ante welfare benefits of redistribution (2) For ␭ Ͻ ␭ or ␭ Ͼ ␭៮ the steady state is are high enough, relative to the costs; unique. (b) the political power of the wealthy lies in some intermediate range. Where multiple steady states occur, history matters. Temporary shocks to the distribution of THEOREM 1: Let 1 Ϫ ␣ Ͻ 2␤␥. When the wealth (immigration, educational discrimina- normalized efficiency gain B ϵ 1 Ϫ r (1 Ϫ ␳ ϩ tion, shifts in demand or technology) as well as ␳␤) is below some critical value B (equivalent- to the political system (slavery, voting-rights restrictions) can permanently move society from one equilibrium to the other, or more 27 The intuition behind the proofs is the following. Con- generally have long-lasting effects on the econ- sider f(␶) as a function of 1 Ϫ ␶ ʦ (0, (1 Ϫ ␣)/␤␥). Since omy and the social contract. D is monotonic, the first fraction in (28) is U-shaped in 1 Ϫ ␶. After multiplication by 1 Ϫ ␶ the product typically This history-dependence contrasts sharply becomes N-shaped, so that it will have three intersections with traditional politico-economic models, with the horizontal ␭, for some range of values of ␭. The larger B, the more pronounced this N shape, which is also that of the vertical difference T(⌬) Ϫ DϪ1(⌬) on Figure 28 Specifically, there are 2 Յ n Յ 4 stable steady states. 2. Conversely, the last term in (28), reflecting effort distor- If 1/␩ is small enough then n Յ 3, and if ␣ is also small tions, tends to make f strictly increasing in 1 Ϫ ␶ (at least enough then n ϭ 2. See the theorem’s proof in the Appen- where ␶ Ͼ 0), and therefore works towards uniqueness. dix. 112 THE AMERICAN ECONOMIC REVIEW MARCH 2000 where countries can deviate at most temporarily Indeed, the most extreme forms of social im- from a common steady-state level of inequality mobility at the lower end, such as urban ghettos and redistribution (given stable “fundamen- or the persistence of welfare dependency across tals”). In particular, fiscal policy operates there generations, do seem more exacerbated in as a stabilizing force on the distribution of American society. Things could well be differ- wealth: more inequality today means more re- ent for the middle class, so a more satisfactory distribution, hence less inequality tomorrow.29 comparison across countries would take into Here, on the contrary, there emerges in the long account such nonlinearities in the mobility pro- run a negative correlation between inequality cess (e.g., Suzanne J. Cooper et al., 1994). and redistribution, as indeed one observes be- These remain beyond the scope of the present tween the United States and Europe, or among model and of most existing comparative studies. advanced countries in general (Rodriguez, Having demonstrated how the sustainabil- 1998).30 ity of different societal choices depends on the Also of interest is the predicted negative cor- importance of risk aversion and credit con- relation between inequality and social mobility, straints (both summarized in B), as well as on which is consistent with the results obtained by the allocation of power in the political system Robert Erikson and John H. Goldthorpe (1992) (␭), I now consider the role of income and for a sample of 15 developed countries. But endowment shocks (v2 and w2) and that of tax what of the conventional wisdom of the United distortions (1/␩). Due to the complexity of the States as an exceptionally mobile society? In problem, their effects on the multiplicity fact, most econometric studies of intergenera- threshold B are studied under additional as- tional income mobility find either no significant sumptions. difference, or even somewhat greater mobility in European “welfare states”: see Anders Bjo¨rk- lund and Markus Ja¨ntti (1997a) for a survey.31 PROPOSITION 9: Let 1/␩ ϭ 0. The efficiency threshold for multiplicity B is a decreasing function of v/w, with limv/w 3 0(B) ϭϩϱand 29 In Persson and Tabellini (1994) income inequality, limv/w 3 ϩϱ(B) ϭ 0. hence the equilibrium policy, depends only on the exoge- nous distribution of talent. In Bertola (1993) and Alesina and Rodrik (1994) the deterministic nature of the models This result appears on Figure 4. The intu- allows any distribution of initial endowments to persist indefinitely. Incorporating idiosyncratic shocks would nor- ition is that income uncertainty interacts with mally lead to a unique steady-state distribution. Uniqueness market incompleteness in generating effi- also obtains in Perotti (1993) and Saint-Paul and Verdier ciency gains from redistribution, as reflected (1993). In Saint-Paul and Verdier (1992) greater inequality by the term Bv2 in (16). For a given B, again results in higher taxes and spending on public educa- multiplicity therefore occurs when v2 is large tion (which is problematic in view of the empirical evi- 2 dence), but this stabilizing effect is more than offset by a enough compared to the variance w of the divergence in the incentives of rich and poor to invest shocks which agents learn prior to choosing privately in additional human capital. Hence two possible policy. The concrete implications are impor- steady states: high (low) inequality and public education tant: depending on their source, changes in expenditures, with private accumulation by the rich only (by both classes). the economic environment which have similar 30 Multiple steady states due to a negative impact of short-run effects on income inequality can inequality on redistribution is the distinguishing feature bring about radically different evolutions of which the present model shares with Saint-Paul (1994). the social contract. Thus, an increase in the Consistent with the general argument that this entails redis- variability of sectoral shocks (similar to 2) tributions which increase the size of the pie, Saint-Paul’s v model has the property that transfers raise aggregate income may lead to an expansion of the welfare state in the long run. and public education. Conversely, a surge in 31 For instance, Kenneth A. Couch and Thomas A. Dunn immigration which results in a greater hetero- (1997) find greater mobility—especially in terms of educa- geneity of the population (similar to a rise in tion—in Germany than in the United States, and Bjo¨rklund 2 and Ja¨ntti (1997b) find similar results for Sweden. Aldo w ) may lead to cutbacks or even a large- Rustichini et al. (1999), on the other hand, find lower scale dismantling. In the first case the policy mobility in Italy than in the United States. response will mitigate the shock’s impact on VOL. 90 NO. 1 BE´ NABOU: INCOME DISTRIBUTION AND THE SOCIAL CONTRACT 113

long-run inequality, while in the second it ␶t), all previous results remain unchanged, with will aggravate it. ␬ simply replaced by ␬t everywhere. The pres- Just as greater benefits of redistribution in- ence of the spillover only affects the growth rate crease the likelihood of multiplicity, greater along each equilibrium trajectory, transforming distortions reduce it. This is also illustrated for instance finite steady states (when 0 Յ ␾ Ͻ on Figure 4, where the efficiency threshold B 1 Ϫ ␣ Ϫ ␤␥) into endogenous growth paths shifts up as 1/␩ rises. The formal proposition (when ␾ ϭ 1 Ϫ ␣ Ϫ ␤␥).33 The following is somewhat more complicated, as it is only results therefore apply equally to short- and to for positive values of ␶ that labor-supply dis- long-run economic growth. tortions rise with 1/␩.32 I shall not present it here due to space constraints, and because a PROPOSITION 10: A more redistributive so- somewhat similar result appears in Theorem 2 cial contract below. Basically, when regressive fiscal pol- icy is ruled out, or more generally for econ- (1) has higher income growth when 1/␩ ϭ ␣ ϭ omies that operate in a region where ␶ Ͼ 0, 0 and ␤␥ Ͻ 1; distortions do rise monotonically with 1/␩, (2) has lower income growth when 1/␩ Ͼ 0, and the scope for multiple regimes corre- ␣ ϭ 0 and ␤␥ ϭ 1. spondingly declines.

B. Growth Implications of Different Since both conditions are compatible with Social Contracts Theorem 1’s requirement that 1 Ϫ ␣ Ͻ 2␤␥, they allow the comparison of steady states cor- The steady states corresponding to two dif- responding to different, self-sustaining values ferent social contracts are clearly not Pareto of ␶. Two interesting empirical scenarios can rankable. How do they compare in terms of thus be accounted for by the model. aggregate performance? Recall from (15) that the effect of ␶t on short-run growth reflects the Case 1: Growth-enhancing redistributions.— trade-off between tax distortions and credit- The fact that all equilibria have (endogenously) constraint effects; taking limits shows that this the same savings rate makes clear that the faster remains true for long-run income. Moreover, growth under the more redistributive social any comparison of long-run levels is easily contract arises from a more efficient allocation transposed to long-term growth rates, through of investment expenditures.34 Tax distortions, knowledge spillovers or public goods comple- meanwhile, remain relatively small. This sce- menting private investment. For instance, let ␬ nario is particularly relevant for human capital ␾ in (6) be replaced by (␬t) , where the human or investment (which is considered in more detail physical capital aggregate in the next section) and public health expendi- tures, where the contrasted paths followed by East Asia and Latin America come to mind. 1/␥ 1 (29) ␬ ϵ ͑ki͒␥di t ͵ t 33 Because it aggregates individual contributions with ͩ 0 ͪ the same elasticity of substitution as total output, ␬ is t heterogeneity neutral, in the sense that it does not introduce any additional effects of income distribution on growth. It captures external effects of the economic envi- just makes more permanent those due to imperfect credit ronment on accumulation, other than those of markets, by reducing or even eliminating the “convergence” term Ϫ(1 Ϫ ␣ Ϫ ␤␥)lnyt from (15). Alternative constant policy. As ␬t does not enter into the determina- elasticity of substitution (CES) aggregates with elasticities tion of the politico-economic equilibrium (⌬t, other than 1/(1 Ϫ ␥) could easily be dealt with, as in Be´nabou (1996b). 34 The equality of investment rates is true in the infinite- 32 Where effort is distorted by regressionary taxes and horizon version of the model as well. A higher ␶ implies a transfers, on the contrary, a rise in ␶ represents an efficiency lower private savings rate, but this is exactly offset by a gain which increases with 1/␩: recall that C(␶) is maxi- higher equilibrium rate of consumption taxation and invest- mized for ␶ ϭ 0. ment subsidization. 114 THE AMERICAN ECONOMIC REVIEW MARCH 2000

More generally, it offers a potential explana- positive efficiency implications. Loan market tion, in a context of endogenous policy choice, imperfections are more likely to affect invest- for the fact that regression estimates of the ment in human than in physical capital, which effects of social and educational transfers on can serve as collateral. The same is true for growth are often significantly positive. decreasing returns. The financing of elementary and secondary education also constitutes a strik- Case 2: Eurosclerosis and the welfare ing example of international differences in re- state.—In this converse case, the credit-constraint distributive policy. Japan and most European effect is weak compared to the tax distortions. countries have state-funded public schools, European countries, it is often argued, have cho- which essentially equalize expenditures across sen a higher degree of social insurance and com- pupils. The United States, in contrast, relies in pression of inequalities than the United States, at large part on local financing; because commu- the cost of higher unemployment and slower nities are heavily income segregated, expendi- growth.35 Whether this is viewed as enlightened tures reflect parental resources to a large extent, policy or dismal “Eurosclerosis,” it begs the ques- making education a quasi-private good. In Be´n- tion of why voters on both sides of the Atlantic abou (1996b) I show how a move from local to would choose such different points on the equity- state funding of schools can raise the economy’s efficiency, or insurance-growth, trade-off. In my long-run income, or even its long-term growth model, Europeans choose more redistribution than rate. Calibrating a model with local funding to Americans not because they are intrinsically more U.S. data, Ferna´ndez and Rogerson (1998) find risk averse, but because in more homogeneous that a move to state finance could raise steady- societies there is less erosion of the consensus state GDP by about 3 percent. Whether or not over social insurance mechanisms which, ex ante, one subscribes to this view, differences in na- would be valued enough to compensate for lesser tional education systems which persist for over growth prospects. a century represent a puzzle.36 i Consider finally average welfare, which here To examine this issue, let kt now specifically i ␣ is also that of the median voter; see (16). Since represent human wealth. The term (kt) in (6) multiplicity requires some political bias (␭ Ͼ 0) captures the transmission of human capital or i it is clear from (21) that, in each steady state, a ability within the family, while the shocks ␰t ϩ 1 marginal increase in redistribution would raise represent the unpredictable component of innate 1 i ͐0 Ut di. This corresponds to the requirement, talent. Finally, instead of progressive taxes and in the stylized model of Section I, of an aggre- transfers I now consider progressive subsidies gate gain from the policy ˆ relative to . to educational investment. Thus (9) is replaced P P i i i Comparing steady states, on the other hand, by yˆ t ϭ yt, while in (6) parental savings et are involves discrete variations in ␶. When tax replaced by the net (after-tax) resources in- distortions are small enough (1/␩ is low), a vested in their child’s education, namely: more redistributive steady state does have higher total welfare; but in general this need not i i i ␶t be the case. (30) eˆ t ϭ et͑y˜ t /yt͒ ,

VI. Explaining International Differences in Education Finance with y˜ t still determined by (10). Because agents

A. Alternative Systems of School Funding 36 In Glomm and Ravikumar (1992) private finance of Education finance provides perhaps the most education leads to higher long-run growth than public fund- compelling case of a redistributive policy with ing, as it generates better incentives to accumulate human wealth. Be´nabou (1996b) shows that allowing for idiosyn- cratic shocks (e.g., children’s ability) tends to reverse this ranking, as do economy-wide spillovers. Gradstein and Just- 35 It is only in recent years that European growth has man (1997) study similar issues in a model with endogenous fallen short of U.S. growth, but unemployment has been labor supply, then examine voters’ choice among different higher in Europe for nearly two decades. funding regimes. They obtain a unique equilibrium. VOL. 90 NO. 1 BE´ NABOU: INCOME DISTRIBUTION AND THE SOCIAL CONTRACT 115 will still choose a common savings rate, the (33) A͑␶͒ ϵ ␳␣ ϩ ͑1 Ϫ ␳ ϩ ␳␤͑1 Ϫ ␶͒͒␥ government’s budget constraint, which is now

1 (34) C͑␶͒ ϵ ͑1 Ϫ ␳͒͑␦ ln l͑␶͒ Ϫ l͑␶͒␩͒ (31) eˆ i Ϫ ei di ϭ 0, ͵ ͑ t t͒ 0 ϩ ␳␤␦ ln l͑␶͒ will again be satisfied. The progressivity rate (35) B͑␶͒ ϵ Ϫ͑1 Ϫ ␳ ϩ ␳␤͑1 Ϫ ␶͒2͒ ␶t is the elasticity of the tax price of education with respect to wealth. Given agents’ savings r 1 Ϫ ␳ ϩ ␳␤ 1 Ϫ ␶ 2. i i ϩ ͑ ͑ ͒͒ behavior, et ϭ syt, it also measures the extent to which education is publicly and equally provided: thus ␶ ϭ 0 corresponds to private Two differences with Proposition 4 are worth finance, while ␶ ϭ 1 is equivalent to a Euro- mentioning. On the cost side, C(␶) is the pean-style system where universal public ed- same function of effort l(␶) as before, but l(␶) ucation is funded by a proportional income is now different. In particular, distortions re- i tax. main bounded, so Ut may be maximized at ␶ ϭ 1 for a poor enough agent. The other difference occurs in the benefits term. For PROPOSITION 11: Given a rate of education ␳ ϭ 1, B(␶) ϭϪ␤(1 Ϫ r␤)(1 Ϫ ␶)2 as in finance progressivity ␶t, agents in generation t (16), and with the same interpretation in choose a common labor supply and savings terms of insurance and reallocation of liquid- i i rate: et/yˆ t ϭ ␳␤/(1 Ϫ ␳ ϩ ␳␤) ϵ s, as before, ity-constrained investments. But in general while: B(␶) is no longer proportional to Ϫ(1 Ϫ ␶)2, and when r Ͼ 0 it is not even monotonic in ␶. 1/␩ 1/␩ ␦ 1 Ϫ ␳ ϩ ␳␤͑1 Ϫ ␶t ͒ While the education model is sufficiently l ϭ . t ͩ ␩ͪ ͩ 1 Ϫ ␳ ͪ close to the tax model to ensure that the political equilibrium remains qualitatively similar, the formal analysis is made more The effort distortion is smaller than with fiscal difficult by these differences. Moreover, de- redistribution, because the education-based pol- riving an exact analogue to Theorem 1 would icy leaves untouched the part of their income be repetitious. I shall therefore focus instead which adult agents consume. Up to that differ- on a simpler case, which yields new and more ence in lt ϭ l(␶t), individual and aggregate explicit comparative statics results. wealth dynamics are exactly identical to (11)– (15), and the resulting ex ante utility function resembles closely the one which arose under B. Sustainability of Centralized and fiscal redistribution. Decentralized Education Systems

PROPOSITION 12: Given a rate of education From here on, policy is restricted to two finance progressivity ␶t, agent i’s intertemporal options. Under laissez-faire or decentralized welfare is: funding, ␶ ϭ 0, education expenditures are determined by family or community resourc- es; the two are essentially equivalent when i i (32) Ut ϭ u៮t ϩ A͑␶t͒͑ln kt Ϫ mt͒ ϩ C͑␶t͒ communities are stratified by socioeconomic status. Public funding of education corre- 2 2 2 2 Ϫ ␳␤͑1 Ϫ ␶t ͒ ␥ ⌬t ϩ B͑␶t ͒v /2, sponds to ␶ ϭ 1 or more generally to ␶ ϭ ␶៮, where 0 Ͻ ␶៮ Յ 1. Given an initial distribution of human capital ⌬t, this system is adopted if i i where u៮ t is independent of the policy ␶t and Ut(␶៮) Ͼ Ut(0) for at least a fraction p* ϵ 116 THE AMERICAN ECONOMIC REVIEW MARCH 2000

i ⌽(␭) of the population. Setting ln kt Ϫ mt ϭ The two regimes can coexist if and only if ␭ Ͻ ␭⌬t in (32), this means: ␭៮, which occurs when the differential gain

v2 (39) B͑␶៮͒ Ϫ B͑0͒ ϭ ␳␤␶៮͓2 Ϫ ␶៮ (36) ␭ Ͻ ͑B͑␶៮͒ Ϫ B͑0͒͒ Ϫ ͑C͑0͒ ͩ ͩ 2 ͪ Ϫ r͑2 Ϫ 2␳ ϩ ␳␤͑2 Ϫ ␶៮͔͒͒ 1/␳␤␶៮ 2 Ϫ ␶៮ Ϫ C͑␶៮͒͒ ϩ ␥⌬t . ͪ ␥⌬t ͩ 2 ͪ exceeds the differential cost

Intuitively, the political influence of wealth must ␦ (40) C͑0͒ Ϫ C͑␶៮͒ ϭ ͑1 Ϫ ␳ not be too large compared to the aggregate wel- ␩ ͫ fare benefit of redistributive education finance (relative to laissez-faire). Preexisting inequality 1 Ϫ ␳ ϩ ␳␤ raises the hurdle which public policy must over- ϩ ␳␤͒ ln Ϫ ␳␤␶៮ ͩ 1 Ϫ ␳ ϩ ␳␤͑1 Ϫ ␶៮͒ͪ ͬ come, as reflected by the term 1/␥⌬t multiplying the net welfare benefit. This effect tends to make adoption of state finance more difficult where it by a sufficient amount, specified below. It is has not previously been in place (e.g., the United easily seen that the distortion C(0) Ϫ C(␶៮)is States), because of the greater human capital dis- positive and increasing 1/␩. I will assume that parities which result over time from a decentral- B(␶៮) Ͼ B(0), so that there is an actual gain to ized system. Conversely, the term in ␥⌬t compare to this cost. Such is the case provided incorporates the combined effects of skewness agents care enough about insurance (1 Ϫ r Ն and credit constraints, which both intensify the 1) or about investment in their children’s edu- demand for redistribution. As a result of these cation (␳(2 ϩ ␤) Ն 1), or alternatively provided offsetting forces the right-hand side of (36) has the ␶៮ is not too large. usual U shape in ⌬t, and is in fact very similar to (3) in the stylized model of Section I. To focus now on the long run, let us replace ⌬t with ⌬ϱ ϭ THEOREM 2: If the gain in ex ante welfare D(␶៮), given by (27). Public funding of education from progressive public financing of education, (partial or complete) is thus a steady state when: relative to private or decentralized financing, is large enough, namely (37) ͑␶៮͒ Ϫ B͑0͒ Ն ͑C͑0͒ Ϫ C͑␶៮͒͒͑2/v2͒ v2 ␭ Ͻ ͑B͑␶៮͒ Ϫ B͑0͒͒ Ϫ ͑C͑0͒ Ϫ C͑␶៮͒͒ 2 2 ͩ ͩ 2 ͪ ͪ ϩ G͑␶៮, v /w ͒ ϵ B,

1/␳␤␶៮ 2 Ϫ ␶៮ ϫ ϩ ␥D͑␶៮͒ ϵ ␭៮ . where G(␶, 2/w2) 0 is given in the Appen- ͩ ␥D͑␶៮͒ͪ ͩ 2 ͪ ៮ v Ͼ dix, then there exist 0 Ͻ ␭ Ͻ ␭៮ such that:

Conversely, private or local financing is a (1) For each ␭ in [␭, ␭៮] both public and de- steady state, with inequality ⌬ϱ ϭ D(0), when: centralized school funding are stable steady states. The first regime has lower inequality and greater social mobility than second. (38) (2) For ␭ Ͻ ␭ public funding is the only steady state, while for ␭ Ͼ ␭៮ it is decentralized v2 funding. ␭ Ͼ ͑B͑␶៮͒ Ϫ B͑0͒͒ Ϫ ͑C͑0͒ Ϫ C͑␶៮͒͒ ͩ ͩ 2 ͪ ͪ The efficiency threshold for multiplicity B 2 1/␳␤␶៮ 2 Ϫ ␶៮ decreases with income uncertainty v and in- ϫ ϩ ␥D͑0͒ ϵ ␭. creases with endowment variability w2, with ͩ ␥D͑0͒ͪ ͩ 2 ͪ VOL. 90 NO. 1 BE´ NABOU: INCOME DISTRIBUTION AND THE SOCIAL CONTRACT 117

FIGURE 5. THE NUMBER OF STEADY STATES Note: The dotted lines correspond to 1/␩ЈϾ1/␩. limv/w 3 0(B) ϭϩϱ. It rises with the labor- sented on Figure 4, with B ϵ B(␶៮) Ϫ B(0) now supply elasticity 1/␩. on the vertical axis, and the same interpretations as for Proposition 9. The only difference is that These results demonstrate how such different for B Ͻ B there might be no steady state, as ␭៮ systems of school finance as those of the United is then less than ␭. The economy can instead be States and Western European countries can be shown to cycle between the two regimes, as in self-perpetuating, once arisen from historical Gradstein and Justman (1997).38 Actual in- circumstances.37 As to which one leads to faster stances of such cycling are hard to come by, and growth, this depends once again on the trade-off indeed Theorem 1 indicates that nonexistence is between the positive impact of redistributive largely an artifact of restricting the policy ␶ to education finance on wealth constraints and its discrete values.39 adverse effect on incentives: Proposition 10 ap- plies unchanged. Because the distortions are less severe than with fiscal policy, however, the 38 Equations (37)–(38) show that for ␭ ԫ [␭៮, ␭] there is a unique (and intuitive) steady state, but for ␭ ʦ [␭៮, ␭] there likelihood that state funding enhances growth is is none. The model of Gradstein and Justman (1997) cor- now greater. responds to the case where 1 Ϫ r ϭ 1 (time-separable, Theorem 2 also makes clear show how the logarithmic utility), the disutility of effort Ϫl␩ is replaced efficiency threshold for multiplicity B (or equiv- by ln (1 Ϫ l ), ␶ ʦ {0, 1} (pure public or private system), 2 alently, the minimal risk aversion 1 Ϫ r) varies ␭ ϭ 0 (pure democracy), and v ϭ 0 (no uncertainty at the time of voting). From Theorem 2 we see that the restrictions with the costs of redistributive school finance, ␭ ϭ 0 and v2 ϭ 0 preclude multiple equilibria. as well as with different sources of income 39 With a continuous ␶ the analogue of equation (28) for inequality. These results can again be repre- education funding is easily shown to always have at least one stable steady state. Conversely, an analogue of Theorem 2 can be derived for fiscal policy, with the progressivity rate ␶ restricted to {0, ␶៮}. This shows that the nonlinear redis- 37 Generalizing Theorem 2 to any policy pair {␶, ␶៮} with tributive schemes used in the paper, namely (9) and (30), are 0 Յ ␶ Ͻ ␶៮ Յ 1 is straightforward. not driving the results: for ␶ ϭ 0 and ␶ ϭ ␶៮ ϵ 1 the 118 THE AMERICAN ECONOMIC REVIEW MARCH 2000

I now return to the central issue, namely the 1 w ␥ coexistence of multiple regimes. The range of (25Ј) ϭ ͑1 ϩ ͒ 1 Ϫ ␶ ␭ 2 political systems which allow this indetermi- ϱ ͫ A ͩ ͱ1Ϫ␣ ͪ nacy is illustrated on Figure 5. v 2 ͱ1Ϫ␣2 ϩ (1ϩ Ϫ͑1Ϫ␳͒r͒ . PROPOSITION 13: The scope for the political A ͩ wͪ ͩ ␥ ͪͬ system to generate multiple equilibria increases with the efficiency benefits of redistribution and decreases with their efficiency costs: as B ϵ If one observed two countries, the first with low B(␶៮) Ϫ B(0) increases, due to greater risk aver- pretax inequality yet extensive redistribution, sion or more decreasing returns, both ␭ and ␭៮ rise the other with the reverse situation, one would but the interval [␭, ␭៮] widens. A higher labor- indeed be tempted to conclude that the citizens supply elasticity 1/␩ has the opposite effects. of the first country were more altruistic, or their poor better organized politically. In fact it could VII. Other Applications be that preferences are identical and political institutions equivalent (same and ␭), but that A. Concern for Equity the second country’s more unequalA distribution reflects a more persistent income process. This Apart from social insurance and capital market could be due to exogenous factors, as with ␣ imperfections, one of the main reasons for income here, or be endogenous, as in the case of mul- redistribution is simply that most people dislike tiple steady states. living in a society which is too unequal. This may be due to pure altruism or to the fact that inequal- B. The Mix of Public Goods ity generates social tensions, crime, and similar problems which have direct costs. To capture Some public services such as the legal sys- these ideas one can simply augment (7) as fol- tem, the protection of property, prisons, etc., lows: benefit citizens largely in proportion to their levels of wealth or investment. Others have ˆ i i (7Ј)lnV t ϭ ln V tϪ͑ /2͒ more uniformly or even regressively distributed A benefits. Klaus Deininger and Lyn Squire j j ␩ (1995) find in cross-country regressions that ϫ͓͑1 Ϫ ␳͒varj ʦ I ͓ln c tϪ͑lt͒ ͔ public investment affects income growth j equally for all quintiles, while public schooling ϩ ␳ varj ʦ I ͓ln kt ϩ 1͔͔. benefits the bottom 40 percent most, the middle class to a lesser extent, and the rich not at all. The coefficient represents everyone’s aversion Consider therefore a government choosing (at to disparities inA felicity, measured by the cross- the margin) a single public good or service from sectional variances of consumption (including lei- a menu of options: if gt is spent on a good with sure) and bequests. It is easily seen that the characteristics (␬, ␣, ␤, ␥), the private sector i economy’s laws of motion remain unchanged and faces the accumulation technology kt ϩ 1 ϭ i i ␣ i ␤ ␥ the political equilibrium quite similar. In an en- ␬␰t ϩ 1(kt) (et) ( gt) . A public good or institu- dowment economy (␤ ϭ 0), for instance, the only tion with high ␣ ϩ ␤ makes wealth more per- 2 2 2 difference is that the ubiquitous ␥ ⌬t ϩ Bv is sistent, so relatively well-off agents may prefer 2 2 2 replaced by (1 ϩ )␥ ⌬t ϩ (B ϩ )v : inequal- it to an alternative which has higher overall ity aversion is equivalentA to a simultaneousA in- productivity (larger ␬ or ␣ ϩ ␤ ϩ ␥). The crease in risk aversion and in the concavity of problem is thus analogous to the earlier ones, aggregate welfare (previously logarithmic). Thus, implying that countries can sustain different with 1/␩ ϭ 0 the steady-state tax rate becomes choices without any underlying differences in tastes. In reality, many public goods are pro- vided simultaneously and the debate is over the geometric specification and the standard linear one coincide appropriate mix, but the same intuitions should exactly. remain applicable. VOL. 90 NO. 1 BE´ NABOU: INCOME DISTRIBUTION AND THE SOCIAL CONTRACT 119

C. The Socioeconomic Structure of Cities well as equally democratic political systems, can nonetheless make very different choices The presence in human capital accumulation of with respect to fiscal progressivity, social in- peer effects, role models, and other neighborhood surance, and education finance. The proposed interactions implies that residential stratification answer is a simple theory of inequality and increases the persistence of income disparities the social contract, based on two mechanisms across families (e.g., Be´nabou, 1993; Durlauf, which arise naturally in the absence of com- 1996a). Urban ghettos are but the most extreme plete insurance and credit markets. First, re- example of this phenomenon, which is the subject distributions which would increase ex ante of a large empirical literature.40 Moreover, I show welfare command less political support in an in Be´nabou (1993, 1996b) that equilibrium segre- unequal society than in a more homogeneous gation generally tends to be inefficiently high. The one. A lower rate of redistribution, in turn, two conditions identified in this paper for multiple increases inequality of future incomes due to politico-economic regimes are thus again satisfied, wealth constraints on investment in human or leading to the following predictions: (a) more physical capital. This leads to two stable highly segregated cities are also those where so- steady states, the archetypes for which could cioeconomic disparities are greater; (b) in such be the United States and Western Europe: one cities public policy (on housing, schooling, trans- with high inequality yet low redistribution, port, or infrastructure) will tend to accommodate the other with the reverse configuration. and even facilitate segregation, while in better These two societies are not Pareto rankable, integrated (and more equal) cities more public and which one has faster income growth support and resources will be mobilized to prevent depends on the balance between tax distor- further polarization.41 tions to effort and the greater productivity of investment resources (particularly in educa- VIII. Conclusion tion) reallocated to more severely credit- constrained agents. In this paper I have asked how countries These ideas were formalized in a stochastic with similar preferences and technologies, as growth model with missing markets, progres- sive fiscal or education finance policy, and a more realistic political system than the stan- 40 Recent references include Cooper et al. (1994), George J. dard median voter setup. The resulting distri- Borjas (1995), and Giorgio Topa (1997). Christopher Jencks butional dynamics remain simple enough to and Susan E. Mayer (1990) provide an extensive survey of earlier empirical studies, and Charles F. Manski (1993) a allow a number of extensions. In Be´nabou critical discussion of methodology. Indeed the identification of (1999) I develop and calibrate an infinite- these social spillovers remains the subject of some contro- horizon version of the incomplete markets versy; see for instance William N. Evans et al. (1992). model, then use it to quantify the effects of 41 At the heart of this class of models (e.g., Be´nabou, 1996b) lies a general law of motion for human capital of the fiscal and educational redistribution on i i i i i growth, risk, and welfare. Another interesting form: htϩ1 ϭ F(␰tϩ1, ht, Lt, Ht), where Lt and Ht are human capital averages capturing respectively local externalities (e.g., problem is to endogenize the kind of wealth- peer effects) and economy-wide interactions (production biased political mechanism used here, where complementarities, knowledge spillovers, etc.). Where families those with more resources command more have sorted into socioeconomically homogeneous communi- i i i i i i influence; Franc¸ois Bourguignon and Verdier ties, Lt ϭ ht for all i,sohtϩ1 ϭ F(␰tϩ1, ht, ht, Ht). Conversely, i j i i under perfect integration Lt ϭ Lt ϭ Lt, hence htϩ1 ϭ F(␰tϩ1, (1997) and Rodriguez (1998) are recent ex- i ht, Lt, Ht). The similarity with the cases ␶ ϭ 0 and ␶ ϭ 1inthe amples of such models. Finally, the original present model is readily apparent. An alternative source of question of why the social contract differs multiplicity is explored in Durlauf (1996b), where it is only when socioeconomic disparities are not too large that rich across countries, and whether these choices families are willing to share with poorer ones the fixed costs of are sustainable in the long run, remains an running a community and its schools. important topic for further research. 120 THE AMERICAN ECONOMIC REVIEW MARCH 2000

APPENDIX

PROOF OF PROPOSITION 2: i i i i ␥ i ␦ Once agent i knows his productivity zt, hence also his pre- and posttax incomes yt ϭ zt(kt) (lt) i i 1 Ϫ ␶t and yˆ t ϭ ( yt) ( y˜ t), his decision problem takes the form:

i i ␩ i rЈ i i i ␣ i ␤ (A1) ln V tϭ max͕͑1 Ϫ ␳͓͒ln ͑͑1 Ϫ ␯͒yˆt͒Ϫ l ͔ϩ͑␳/rЈ͒ln Et͓͑ktϩ1͒ ͔ ͉ ktϩ1ϭ ␬␰ tϩ1͑kt͒ ͑␯yˆ t͒ ͖ l,␯ ␩ ϭ max͕͑1 Ϫ ␳͒ ln ͑1 Ϫ ␯͒ϩ ␳␤ ln ␯͖ ϩ max͕Ϫ͑1 Ϫ ␳͒l ϩ͑1 Ϫ ␳ ϩ ␳␤͒͑1 Ϫ ␶t͒␦ ln l͖ ␯ l 2 i ϩ ␳͑ln ␬ Ϫ ͑1 Ϫ rЈ͒w /2͒ϩ ͓␳␣ ϩ ͑1 Ϫ ␳ϩ ␳␤͒␥͑1 Ϫ ␶t ͔͒ln kt i ϩ͑1 Ϫ ␳ ϩ ␳␤͓͒͑1Ϫ ␶t ͒ ln zt ϩ ␶t ln y˜ t ͔,

i i i where ␯t ϵ et/yˆ t is the savings rate. Strict concavity in ␯ and l is easily verified, and the first-order conditions directly yield the stated results. PROOF OF PROPOSITION 3: i 2 Let us start by computing the redistributive scheme’s cutoff level y˜ t.Iflnkt ϳ (mt, ⌬t ), then (4) implies that aggregate income is: N

i i ␥ 2 2 (A2) ln yt ϭ ln E͓zt͔ ϩ ln E͓͑kt͒ ͔ ϩ ␦ ln lt ϭ ␥mt ϩ ␦ ln lt ϩ ␥ ⌬t /2.

The level of transfers y˜ t which satisfies the government budget constraint (10) is then given by:

i 1Ϫ␶t i ␥͑1 Ϫ ␶t ͒ ␶t ln y˜ t ϭ ln yt Ϫ ␦͑1 Ϫ ␶t ͒ln ltϪ ln E͓͑zt͒ ͔ Ϫ ln E͓͑kt͒ ͔

2 2 2 2 2 ϭ ln yt Ϫ ␦͑1 Ϫ ␶t͒ln lt ϩ ͑͑1 Ϫ ␶t͒Ϫ͑1 Ϫ ␶t͒ ͒v /2 Ϫ ͑͑1 Ϫ ␶t͒␥mtϩ͑1 Ϫ ␶t͒ ␥ ⌬t /2͒,

i i since the zt’s and kt’s are independent. Thus:

2 2 2 (A3) ␶t ln y˜ t ϭ ␥␶t mt ϩ ␦␶t ln ltϩ ␶t ͑2 Ϫ ␶t ͒␥ ⌬t /2ϩ ␶t ͑1 Ϫ ␶t ͒v /2,

as claimed in (12). Equation (14) follows from taking variances in (11), while (13) follows from taking averages with ␶t ln y˜t replaced by (A3). Finally, combining both laws of motions with (A2) yields:

2 2 2 ␤ 2 2 ln ytϩ1 ϭ ␦ ln ltϩ1 ϩ ␥͓͑␣ ϩ ␤␥͒mtϩ ␤␦ ln lt ϩ ␤␶t͑2 Ϫ ␶t͒͑␥ ⌬t ϩ v ͒/2 ϩ ln ͑␬s ͒Ϫ ͑w ϩ ␤v ͒/2͔

2 2 2 2 2 2 2 ϩ ␥ ͓͑␣ ϩ ␤␥͑1Ϫ ␶t ͒͒ ⌬t ϩ ␤ ͑1 Ϫ ␶t ͒ v ϩ w ͔/2

2 2 2 ϭ ␥͑ln ␬ ϩ ␤ ln s Ϫ ͑1 Ϫ ␥͒w /2͒ϩ ␦͑ln ltϩ1 Ϫ ␣ ln lt͒ϩ ͑␣ ϩ ␤␥͓͒␥mt ϩ ␦lt ϩ ␥ ⌬t /2͔

2 2 Ϫ ␤␥͓1 Ϫ ␶t ͑2 Ϫ ␶t ͒Ϫ ␤␥͑1 Ϫ ␶t ͒ ͔v /2

2 2 2 Ϫ͓␣ ϩ ␤␥Ϫ͑␣ ϩ ␤␥͑1 Ϫ ␶t ͒͒ Ϫ ␤␥␶t ͑2 Ϫ ␶t ͔͒␥ ⌬t /2

ϭ ln ␬˜ ϩ ␦͑ln lt ϩ 1 Ϫ ␣ ln lt ͒ ϩ ͑␣ϩ ␤␥͒ln yt

Ϫ ␤␥͑1 Ϫ ␤␥͒͑1Ϫ ␶ ͒2v2/2 Ϫ (␶ )␥2⌬2/2, t L⌬ t t hence the result, given the definitions of ln ␬˜, (␶) and (␶). Lv L⌬ PROOF OF PROPOSITION 4: i i 2 Substituting the optimal lt ϭ lt and ␯t ϭ s into (A1), and denoting ln ␬Јϵln ␬ Ϫ (1 Ϫ rЈ)w /2, yields: VOL. 90 NO. 1 BE´ NABOU: INCOME DISTRIBUTION AND THE SOCIAL CONTRACT 121

i ␩ (A4) ln V t ϭ ␳ ln ␬Ј ϩ ͑1 Ϫ ␳͒ ln ͑1 Ϫ s͒ϩ ␳␤ ln s Ϫ ͑1 Ϫ ␳͒lt ϩ ͑1 Ϫ ␳ϩ ␳␤͒͑1 Ϫ ␶t͒␦ ln lt

i i ϩ͓␳␣ ϩ ͑1Ϫ ␳ ϩ ␳␤͒␥͑1 Ϫ ␶t͔͒ln k t ϩ ͑1 Ϫ ␳ ϩ ␳␤͓͒͑1 Ϫ ␶t͒ ln z t ϩ ␶t ln y˜t͔.

i i 2 2 2 Thus conditional on kt,lnVt is normally distributed, with variance (1 Ϫ ␳ ϩ ␳␤) (1 Ϫ ␶t) v . This implies: 1 (A5) Ui ϵ ln E͓͑Vi͒r͉ki͔ ϭ E͓ln Vi͉ki͔ ϩ r͑1Ϫ ␳ ϩ ␳␤͒2͑1 Ϫ ␶ ͒2v2/2. t r t t t t t

i i Therefore, to obtain Ut one simply needs to replace in (A4) the term in ln zt by Ϫ(1 Ϫ ␳ ϩ ␳␤)(1 Ϫ 2 ␶t)[1 Ϫ r(1 Ϫ ␳ ϩ ␳␤)(1 Ϫ ␶t)]v / 2. Finally, substituting in the value of ␶t ln y˜ t from (A3) yields the claimed result, with

1 Ϫ ␳ ␳␤ 2 (A6) u៮ t ϵ ln͓͑1 Ϫ s͒ s ͔ ϩ ␳͑ln ␬ Ϫ ͑1Ϫ rЈ͒w /2͒

2 2 ϩ͑␳␣ ϩ ͑1 Ϫ ␳ϩ ␳␤͒␥͒mt ϩ ͑1 Ϫ ␳ ϩ ␳␤͒␥ ⌬t /2. PROOF OF PROPOSITION 5: We can rewrite (21) as a second-degree polynomial in x ϵ 1 Ϫ ␶,

2 2 2 2 i (A7) P͑x͒ ϵ x ͑␥ ⌬t ϩ Bv ͒Ϫ ͑␥͑ln kt Ϫ mt ͒ Ϫ ␦/␩͒xϪ ␦/␩ ϭ 0,

i i which always has two real roots of opposite sign. Since ␶t Յ 1 necessarily, the relevant root is xt i i i ϭ 1 Ϫ ␶t Ͼ 0, and at that point PЈ( xt) Ͼ 0. It is easy to compute ␶t explicitly, but for comparative 2 i statics one can simply use the implicit function theorem. Since ѨP/Ѩ(Bv ) Ͼ 0 Ͼ ѨP/Ѩln kt and i i i i 2 i PЈ( xt) Ͼ 0, the theorem implies that Ѩ xt/Ѩln kt Ͼ 0 Ͼ Ѩ xt/Ѩ(Bv ). Similarly, Ѩ xt/Ѩ⌬t Ͻ 0. i i i Finally, ϪѨP/Ѩ(1/␩) ϭ ␦(1 Ϫ x), so Ѩ xt/Ѩ(1/␩) has the sign of 1 Ϫ xt ϭ ␶t, or equivalently i i i i Ѩ␶t/Ѩ(1/␩) has the sign of Ϫ␶t. Finally, ␶t Ͼ 0 if and only if xt Ͻ 1, which means P(1) Ͼ 0, or i 2 2 2 ␥(ln kt Ϫ mt) Ͻ ␥ ⌬t ϩ Bv .

PROOF OF PROPOSITION 6: Let us index agents by their log-wealth, ␪i ϵ ln ki , and denote its cumulative distribution function by F(␪ ). For any weighting scheme ␻i ϭ g(␪i), the proportion of votes cast by agents with ␪i Յ ␪ ␪ (more generally, their total political weight) is G(␪ )/G(ϱ), where G(␪ ) ϵ͐Ϫϱ g( z) dF( z). For F(␪ ) an ordinal scheme, g( z) ϭ ␻(F( z)), so G(␪ ) ϭ͐0 ␻( p) dp. Given the single-crossing condition satisfied by preferences, the agent with log-wealth ␪* and rank p* ϭ F(␪*) defined by G(␪*)/ G(ϱ) ϭ 1⁄2 is clearly pivotal (see Gans and Smart, 1996). In the lognormal case F(␪ ) ϭ⌽((␪ Ϫ m)/⌬), so if we define ␭ ϵ⌽Ϫ1( p*) then ␪* ϭ m ϩ ␭⌬. This wealth level is the same as if the whole distribution of ␪ ϭ ln k were shifted up by ␭⌬. Let us now turn to the cardinal scheme g(␪ ) ϭ e␭␪. Simple derivations show that

␪ (A8) G͑␪͒ ϵ ͵ e␭z dF͑z͒ ϭ e␭͑m ϩ ␭⌬2/2͒ ⅐ F͑␪ Ϫ ␭⌬2͒, Ϫϱ

hence G(␪ )/G(ϱ) ϭ F(␪ Ϫ ␭⌬2). The whole distribution is thus shifted by ␭⌬2, and so is the solution to G(␪*)/G(ϱ) ϭ 1⁄2.

PROOF OF PROPOSITION 7: i The equilibrium tax rate is the one preferred by the agent with ln kt Ϫ mt ϭ ␭⌬t, so claims (1) i and (2) follow directly from the properties of ␶t established in Proposition 5. To establish the third 122 THE AMERICAN ECONOMIC REVIEW MARCH 2000

i claim let us rewrite that agent’s first-order condition, ѨUt/Ѩ␶ ϭ 0, in terms of xt ϵ 1 Ϫ ␶t. By (A7), xt is the unique positive root of the polynomial:

2 2 2 2 (A9) Q͑x͒ ϵ x ͑␥ ⌬t ϩ Bv ͒ Ϫ ͑␥␭⌬t Ϫ ␦/␩͒x Ϫ ␦/␩ ϭ 0.

2 2 Now, QЈ( xt) Ͼ 0 and ѨQ/Ѩ⌬t ϭ 2x ␥ ⌬t Ϫ ␥␭x,soѨ xt /Ѩ⌬t has the sign of ␭ Ϫ 2␥xt⌬t. Therefore Ѩ␶t /Ѩ⌬t Ͼ 0 if and only if xt Ͼ ␭/2␥⌬t. For ␭ Յ 0 this is always true, hence ␶t is strictly increasing in ⌬t. For ␭ Ͼ 0, on the other hand, the condition is equivalent to:

2 2 2 2 2 2 ⌬t Q͑␭/2␥⌬t ͒ ϭ ͑␭/2␥͒ ͑␥ ⌬t ϩ Bv ͒ Ϫ ͑␥␭⌬t Ϫ ␦/␩͒͑␭⌬t /2␥͒ Ϫ ͑␦/␩͒⌬t Ͻ 0 N

2 2 2 R͑⌬t ͒ ϵ Ϫ͑␭ ϩ 4␦/␩␭͒⌬t ϩ 2͑␦/␩␥͒⌬t ϩ ␭Bv /␥ Ͻ 0.

This second-degree polynomial in ⌬t has two real roots of opposite sign. Denoting ⌬ the positive one 2 [with, clearly, Ѩ⌬/Ѩ(Bv ) Ͼ 0], we conclude that Ѩ␶t/Ѩ⌬t Ͼ 0 if and only if ⌬t Ͼ⌬. Thus ␶t is indeed U-shaped in ⌬t, and its limiting values at ⌬t ϭ 0 and as ⌬t 3 ϱ are readily obtained from i i 2 2 2 (A9). Finally, recall that ␶t Ͼ 0 if and only ␥(ln kt Ϫ mt) Ͻ ␥ ⌬t ϩ Bv ; therefore

2 2 2 (A10) ␶t Ͼ 0 N ␥ ⌬t Ϫ ␥␭⌬t ϩ Bv Ͼ 0.

2 2 2 2 When Bv Ͼ ␭ /4 the condition always holds, so ␶t Ͼ 0. When Bv Ͻ ␭ /4 there is a range [⌬Ј, ⌬Љ] ʚ (0, 2␭) such that ␶t Ͻ 0 if and only if ⌬t is in that interval. PROOF OF THEOREM 1: Let us start with a lemma characterizing stable and unstable steady states.

LEMMA 1: A stable steady state is a point ␶* where the function f(␶) cuts the horizontal ␭ from above, or equivalently a point (⌬*, ␶*) where the curve ⌬ϭD(␶) cuts the curve ␶ ϭ T(⌬) from above. An unstable steady state corresponds in each case to an intersection from below. PROOF: The dynamical system (26) reduces to a one-dimensional recursion: ⌬t ϩ 1 ϭ (⌬t, T(⌬t)). A fixed-point ⌬* ϭ (⌬*, T(⌬*)) is stable if and only if (d (⌬ , T(⌬ ))/d⌬ ) D Ͻ 1, or: D D t t t ⌬ϭ⌬* (A11) ͑⌬*, ␶*͒ϩTЈ͑⌬*͒ (⌬*, ␶*)Ͻ1, D1 D2 where ␶* ϵ T(⌬*) and a j subscript denotes a jth partial derivative. Now, the function ␶ ϭ T(⌬) is implicitly given by the first-order condition (24), or:

2 2 2 ␥ ⌬ ϩ Bv ␦ ␶t (A12) ␺͑␶, ⌬͒ ϵ ͑1 Ϫ ␶͒ Ϫ Ϫ ␭⌬ ϭ 0, ͩ ␥ ͪ ␩␥ ͩ 1 Ϫ ␶tͪ therefore TЈ(⌬) ϭϪ(␺2/␺1)(T(⌬), ⌬) and the stability condition becomes:

␺2 (␶*, ⌬*) (A13) 1 (⌬*, ␶*)Ϫ 2 (⌬*, ␶*)Ͻ1. D ͩ␺1 (␶*, ⌬*)ͪD

Next, recall from (28) that f is defined by: f(␶) Ϫ ␭ ϭ ␺(␶, D(␶))/D(␶), so that fЈϽ0 if and only if ␺1 ϩ (␺2 Ϫ ␺/D) DЈϽ0. Finally, D(␶) is defined by (27) as the (unique) fixed-point solution to D(␶) ϭ (D(␶), ␶); therefore: DЈϭ 2/(1 Ϫ 1). Substituting DЈ and using the fact that ␺(␶*, ⌬*) ϭ 0 atD a steady state, this becomes:D D

fЈ͑␶*͒ Ͻ 0 N ␺ ͑␶*, ⌬*͒͑1 Ϫ (⌬*, ␶*))ϩ␺ (␶*, ⌬*) (⌬*, ␶*)Ͻ0, 1 D1 2 D2 VOL. 90 NO. 1 BE´ NABOU: INCOME DISTRIBUTION AND THE SOCIAL CONTRACT 123 which is the same as (A13), hence the result in terms of the slope of f. Its translation into the condition that TЈ(⌬*) Ͼ (DϪ1)Ј(⌬*) is immediate. We now come to actually solving for steady states. It will be more convenient here to work with the variable x ϵ ␤␥(1 Ϫ ␶) ʦ [0, ϱ). Accordingly, let us define:

w2 ϩ x2v2/␥2 (A14) ⌬͑x͒ ϵ ϭ D͑␶͒, ͱ1 Ϫ ͑␣ ϩ x͒2 and rewrite the equation f(␶) ϭ ␭ as:

Bv2/␥2 ␤␦ ␤␥ Ϫ x (A15) ␸͑x͒ ϵ x ⌬͑x͒ ϩ Ϫ ϭ ␭␤. ͩ ⌬͑x͒ ͪ ͩ ␩␥⌬͑x͒ͪͩ x ͪ

A stable steady state is now an intersection of the function ␸(x) with the horizontal ␭␤, from below. Since ␸(0) Յ 0 (it equals Ϫϱ for 1/␩ Ͼ 0, or 0 for 1/␩ ϭ 0) while ␸(1 Ϫ ␣) ϭϩϱ, for any ␭ Ͼ 0 there is always at least one stable equilibrium x ʦ (0, 1 Ϫ ␣), with 0 Ͻ⌬(x) Ͻϩϱ. Moreover, the total number of intersections must always be odd, with n intersections from below (stable equilibria), alternating with n Ϫ 1 intersections from above (unstable equilibria). Multiple intersections (n Ͼ 0) will actually occur, for some nonempty interval of values of ␭, if and only if ␸٪ is nonmonotonic. Indeed, since ␸Ј(0) Ͼ 0 [this is easily verified from (A15)] and ␸(1 Ϫ ␣) ϭϩϱ, nonmonotonicity is equivalent to the property of having at least one strict local maximum, followed by one strict local minimum, in (0, 1 Ϫ ␣). Given the boundary values of ␸, multiple equilibria then occur if and only if ␭ belongs to the range [␭, ␭៮], where:

␭ ϵ ␤Ϫ1 min ͕␸͑x͉͒x is a strict local minimum of ␸͑x͖͒ Ͼ 0 (A16) ͭ ␭៮ ϵ ␤Ϫ1 max ͕␸͑x͉͒x is a strict local minimum of ␸͑x͖͒ Ͼ 0.

That ␭៮ and ␭ are both always positive follows from the fact that for 1/␩ ϭ 0, ␸( x) Ͼ 0 for all x Ͼ 0 [see (A15)], while for 1/␩ Ͼ 0, if ␸( x) Ͻ 0 then ␸Ј( x) Ͼ 0 necessarily. This last property can be verified directly from (A15) and (A17) below, or more intuitively by observing that if it were not true, there would be a subinterval of values of x where ␸( x) Ͻ 0 and ␸ is not monotonic. This, in turn, would imply that there exists values of ␭ Ͻ 0 for which (A15) has at least two solutions. But such solutions are also intersections of the curves ⌬ϭD(␶) and ␶ ϭ T(⌬); the former is always decreasing, and we saw earlier that, for all ␭ Ͻ 0, the latter is always increasing. Multiple intersections are therefore impossible. Theorem 1 will now be proved by characterizing the set ϵ {B Ն 0͉␸ is nonmonotonic on (0, 1 Ϫ ␣)}, then studying its variations with the parameters vB, w, and ␩.

LEMMA 2: Let 1 Ϫ ␣ Ͻ 2␤␥. The set ϵ {B Ն 0͉? x ʦ (0, 1 Ϫ ␣), ␸Ј( x) Ͻ 0} is a nonempty interval of the form ϭ (B, ϩϱ) withB B Ͼ 0, or ϭ [0, ϩϱ). B B PROOF: Let us differentiate (A15):

␸͑x͒ Bv2/␥2 ␤␦ 1 2␤␥ 1 ␤␥ ⌬Ј͑x͒ ␸Ј͑x͒ ϵ ϩ x⌬Ј͑x͒ 1 Ϫ Ϫ Ϫ Ϫ Ϫ 1 x ͩ ⌬2͑x͒ ͪ ␩␥ ͫͩ x x2 ͪ ⌬͑x͒ ͩ x ͪ ⌬2͑x͒ͬ

Bv2/␥2 ␤␦ ␤␥ Ϫ x Bv2/␥2 ϭ ⌬͑x͒ ϩ Ϫ ϩ x⌬Ј͑x͒ 1 Ϫ ⌬͑x͒ ͩ ␩␥⌬͑x͒ͪͩ x2 ͪ ͩ ⌬2͑x͒ ͪ

␤␦ 1 2␤␥ 1 ␤␥ ⌬Ј͑x͒ Ϫ Ϫ Ϫ Ϫ 1 . ␩␥ ͫͩ x x2 ͪ ⌬͑x͒ ͩ x ͪ ⌬2͑x͒ͬ 124 THE AMERICAN ECONOMIC REVIEW MARCH 2000

Grouping terms, ␸Ј( x) Ͻ 0 if and only if:

Bv2 x⌬Ј͑x͒ x⌬Ј͑x͒ ␤␦ ␤␥ ␤␥ ⌬Ј͑x͒ (A17) Ϫ 1 Ͼ ϩ 1 ⌬2͑x͒ ϩ ϩ Ϫ 1 . ␥2 ͩ ⌬͑x͒ ͪ ͩ ⌬͑x͒ ͪ ͩ ␩␥ͪͩ x2 ͩ x ͪ ⌬͑x͒ ͪ

We now establish the lemma through two intermediate claims.

Claim 1: If (A17) is satisfied for some B Ն 0 at some x ʦ (0, 1 Ϫ ␣), then x⌬Ј(x)/⌬(x) Ͼ 1. As a consequence, (A17) is satisfied at x for all BЈϾB. PROOF: If ⌬Ј( x)/⌬( x) Ϫ 1/x Յ 0 the left-hand-side of (A17) is nonpositive, so on the right-hand side it must be that ␤␥/x Ϫ 1 Ͻ 0. But then:

␤␥ ␤␥ ⌬Ј͑x͒ ␤␥ ␤␥ 1 2␤␥ Ϫ x ϩ Ϫ 1 Ͼ ϩ Ϫ 1 ϭ . x2 ͩ x ͪ ⌬͑x͒ x2 ͩ x ͪ x x2

Since x Ͻ 1 Ϫ ␣ Ͻ 2␤␥, this implies that the right-hand side of (A17) is positive, a contradiction.

Claim 2: There exists an xˆ ʦ (0, 1 Ϫ ␣) such that x⌬Ј( x)/⌬( x) Ͻ 1 on (0, xˆ) and x⌬Ј( x)/⌬( x) Ͼ 1on(xˆ, ϩϱ). As a consequence, for any x Ͼ xˆ, (A17) holds for B large enough. PROOF: Let us denote from here on ␻ ϵ v/␥w. Since ⌬2( x) ϭ w2(1 ϩ ␻2x2)/(1 Ϫ (␣ ϩ x)2),

⌬Ј͑x͒ ␻2x ␣ ϩ x (A18) ϭ ϩ . ⌬͑x͒ 1 ϩ ␻2x2 1 Ϫ ͑␣ ϩ x͒2

Therefore x⌬Ј( x)/⌬( x) Ͼ 1 if and only if

␻2x2͑1 Ϫ ͑␣ ϩ x͒2͒ ϩ x͑␣ ϩ x͒͑1 ϩ ␻2x2͒ Ͼ ͑1 ϩ ␻2x2͒͑1 Ϫ͑␣ ϩ x͒2͒ N

␻2x3͑␣ ϩ x͒ Ϫ ͑1 Ϫ ͑␣ ϩ x͒2͒ ϩ x͑␣ ϩ x͒ Ͼ 0 N

␻2x3͑␣ ϩ x͒ϩ͑␣ ϩ x͒͑␣ ϩ 2x͒Ϫ 1 Ͼ 0.

This last expression is clearly increasing in x on (0, 1 Ϫ ␣), from ␣2 Ϫ 1 Ͻ 0atx ϭ 0to␻2(1 Ϫ ␣)3 ϩ 1 Ϫ ␣ Ͼ 0atx ϭ 1 Ϫ ␣. This proves Claim 2 which, together with Claim 1, establishes that the set is a nonempty interval of the form (B, ϩϱ)or[B, ϩϱ). Moreover, note that its B ϵ {B Ն 0͉@ x ʦ (0, 1 Ϫ ␣), ␸Ј( x) Ն 0}, is a closed set because ␸Ј is گϩޒ ,complement continuous in B atB every point. This implies that either ϭ (B, ϩϱ) with B Ͼ 0, or else ϭ [0, ϩϱ), and thereby finishes to establish Lemma 2. FromB (A17)–(A18), moreover, it is clearB that B depends on v, w, and ␩ only through v/w and ␩v2; let it therefore be denoted as B(v/w; ␩v2). To conclude the proof of Theorem 1 as well as the additional claims in footnote 28 concerning the exact number of steady states, we shall make use of a last lemma.

LEMMA 3: For B Ͻ B(v/w; ␩v2), there is a unique, stable, steady state. For B Ͼ B(v/w; ␩v2) the same is true if ␭ ԫ [␭, ␭៮], while for ␭ ʦ [␭, ␭៮] there are n ʦ {2, 3, 4} stable steady states. Moreover, if 1/␩ is small enough then n Յ 3, and if ␣ is also small enough then n ϭ 2. PROOF: By definition, for B Ͻ B(v/w; ␩v2) the function ␸ is strictly increasing everywhere, so the steady VOL. 90 NO. 1 BE´ NABOU: INCOME DISTRIBUTION AND THE SOCIAL CONTRACT 125 state is unique. The same is true for B ϭ B(v/w; ␩v2) Ͼ 0, since then ϭ (B, ϩϱ). The other measure-zero case, B ϭ B(v/w; ␩v2) ϭ 0, is too special to be of interest.B Now, for B Ͼ B(v/w; ␩v2), we saw earlier that there must be n stable equilibria and n Ϫ 1 unstable ones in the interval (0, 1 Ϫ ␣). To examine what values n can take, rewrite ␸( x) ϭ ␭␤ as:

Bv2 ␤␦ 2 (A19) S͑x͒ ϵ x2 ⌬2͑x͒ ϩ Ϫ ͑␤␥ Ϫ x͒ Ϫ͓␭␤x⌬͑x͔͒2 ϭ 0. ͫ ͩ ␥2 ͪ ͩ ␩␥ͪ ͬ

By (A14), ⌬2( x) is a polynomial fraction in x whose numerator and denominator are both of degree 2. Multiplying the whole equation (A19) by the squared denominator of ⌬2( x), we therefore obtain a polynomial S*(x) of degree 2 ϫ (2 ϩ 2) ϭ 8, which can have at most 8 real roots. But we saw in the discussion following (A15) that ␸( x) ϭ ␭␤ must have an odd number of solutions on (0, 1 Ϫ ␣), with n intersections from above and n Ϫ 1 from below. The numbers of stable and unstable equilibria can therefore only be (1, 0), (2, 1), (3, 2), or (4, 3). When 1/␩ ϭ 0 we can simplify (A19) by x2, leaving for S*(x) only a polynomial of degree 6; this rules out n ϭ 4. When, in addition, ␣ ϭ 0, note from (A14) that ⌬(x) depends on x only through x2.Asa consequence, the sixth-degree polynomial S*(x) is also a polynomial of degree 3 in x2, so it has at most three real roots. This rules out n ϭ 3. Finally, since the polynomial S*(x), like (A19), is continuous with respect to 1/␩ and ␣, so is (generically with respect to the other parameters) its number of real zeroes in the interval (0, 1 Ϫ ␣). The preceding results therefore also apply for 1/␩ and ␣ small enough. This concludes the proof of Theorem 1. PROOF OF PROPOSITION 9: When 1/␩ ϭ 0 the condition for ␸Ј( x) Ͻ 0, namely (A17), becomes:

Bv2 x⌬Ј͑x͒/⌬͑x͒ ϩ 1 (A20) Ͼ ⌬2͑x͒, ␥2 ͩ x⌬Ј͑x͒/⌬͑x͒ Ϫ 1ͪ with the requirement that the denominator must be positive. Using (A18), this can be rewritten as

x2 ϩ ␻Ϫ2 B Ͼ ͩ x3͑␣ ϩ x͒ Ϫ ␻Ϫ2͑1 Ϫ ͑␣ ϩ x͒͑␣ ϩ 2x͒͒ͪ

x2͑2 Ϫ ͑␣ ϩ x͒͑␣ ϩ 2x͒͒ ϩ ␻Ϫ2͑1 Ϫ ␣͑␣ ϩ x͒͒ ϫ ϵ ⌫͑x, ␻͒ ͩ 1 Ϫ ͑␣ ϩ x͒2 ͪ where ␻ ϵ v/␥w. It is easily verified that the each of the bracketed functions is increasing in ␻Ϫ2, therefore:

(A21) B͑␻͒ ϵ inf͕⌫͑x, ␻͉͒x ʦ ͑0, 1 Ϫ ␣͒ and x3͑␣ ϩ x͒ Ͼ ␻Ϫ2͑1 Ϫ ͑␣ ϩ x͒͑␣ ϩ 2x͖͒͒ is strictly positive, and decreasing in ␻. Observe next that, as ␻ tends to infinity, ⌫( x, ␻) approaches ⌫( x, ϩϱ) ϭ x(2 Ϫ (␣ ϩ x)(␣ ϩ 2x))/(1 Ϫ (␣ ϩ x)2), whose infimum value on (0, 1 Ϫ ␣)is 3 Ϫ2 zero; therefore, lim␻ 3 ϩϱ B(␻) ϭ 0. Finally, for any x with x (␣ ϩ x) Ͼ ␻ (1 Ϫ (␣ ϩ x)(␣ ϩ 2x)), note that ⌫( x, ␻) Ͼ ␻Ϫ2/(1 Ϫ ␣2), which tends to infinity as ␻ tends to 0. Therefore lim␻ 3 0 B(␻) ϭϩϱ. PROOF OF PROPOSITION 10: 2 Given lognormality, (29) becomes ln ␬t ϭ mt ϩ ␥⌬t /2 ϭ (ln yt Ϫ ␦ ln lt)/␥, so substituting ln ␬˜ into the growth equation (15) yields:

(A22) ln y Ϫ ͑␣ ϩ ␤␥ ϩ ␾͒ln y ϭ ln ␬៮ϩ ␦͑ln l Ϫ ͑␣ ϩ ␾͒ln l ͒Ϫ ͑␶ ͒v2/2Ϫ (␶ )␥2⌬2/2, tϩ1 t tϩ1 t Lv t L⌬ t t 126 THE AMERICAN ECONOMIC REVIEW MARCH 2000

with ln ␬៮ ϵ ␤␥ ln s Ϫ ␥(1 Ϫ ␥)w2/ 2. In a steady state, if ␣ ϩ ␤␥ ϩ ␾ Ͻ 1 the left-hand side equals (1 Ϫ ␣ Ϫ ␤␥ Ϫ ␾) times the output level ln yϱ; when ␣ ϩ ␤␥ ϩ ␾ ϭ 1 it becomes equal to the asymptotic growth rate, limt 3 ϩϱ ln( yt ϩ 1/yt). As to the right-hand side, in a steady state with ␶t ϭ ␶ it becomes:

(A23) g ͑␶͒ ϵ ln ␬៮ ϩ ␤␥␦ ln l͑␶͒ Ϫ (␶)v2/2Ϫ (␶)␥2D(␶)2/2. ϱ Lv L⌬ 2 2 For ␣ ϭ 0 we saw earlier that ⌬(␶) ϭ v(␶) ϭ ␤␥(1 Ϫ ␤␥)(1 Ϫ ␶) ; therefore v(␶)v /2 ϩ (␶)␥2D(␶)2/2 is strictly decreasingL in ␶. Now,L with 1/␩ ϭ 0 the labor-supply term is constant,L therefore L⌬ gϱ(␶) is strictly increasing in ␶; this proves the proposition’s first claim. Conversely, when ␤␥ ϭ 1, then (␶) ϭ ϭ 0; with 1/␩ Ͼ 0, g (␶), like ln l(␶), is then decreasing in ␶; hence the second claim. Lv L⌬ ϱ PROOF OF PROPOSITION 11: i i i i ␥ i ␦ Once agent i knows his productivity zt, hence also his income yt ϭ zt(kt) (lt) and his investment i i i ␶t subsidy rate eˆ t/et ϭ ( y˜ t/yt) , his decision problem takes the form:

i i ␩ i rЈ i i i ␣ i ␤ (A24) ln Vt ϭ max ͕͑1 Ϫ ␳͓͒ln ͑͑1 Ϫ ␯͒yt͒ Ϫ l ͔ϩ ͑␳/rЈ͒ln͓Et ͑kt ϩ 1͒ ͔ ͉ kt ϩ 1 ϭ ␬␰t ϩ 1͑kt͒ ͑eˆ t͒ ͖ l,␯

␩ ϭ max ͕͑1 Ϫ ␳͒ ln ͑1 Ϫ ␯͒ ϩ ␳␤ ln ␯͖ϩ max͕Ϫ͑1 Ϫ ␳͒l ϩ ͑1 Ϫ ␳ϩ ␳␤͑1 Ϫ ␶t͒͒␦ ln l͖ ␯ l

2 i ϩ ␳͑ln ␬ Ϫ ͑1 Ϫ rЈ͒w /2͒ϩ ͓␳␣ ϩ ͑1 Ϫ ␳ ϩ ␳␤͑1Ϫ ␶t ͒͒␥͔ln kt

i ϩ͑1 Ϫ ␳ϩ ␳␤͑1 Ϫ ␶t ͒͒ln ztϩ ␳␤␶t ln y˜ t ,

i i i where ␯t ϵ et/yt is the savings rate. Strict concavity in ␯ and l is easily verified, and the first-order conditions directly yield the stated results.

PROOF OF PROPOSITION 12: i i 2 Substituting the optimal lt ϭ lt and ␯t ϭ s into (A24) and denoting ln ␬Јϵln ␬ Ϫ (1 Ϫ rЈ)w /2 yields:

i (A25) ln Vt ϭ ␳ ln ␬Ј ϩ ͑1 Ϫ ␳͒ln ͑1 Ϫ s͒ϩ ␳␤ ln s

␩ Ϫ͑1 Ϫ ␳͒lt ϩ ͑1 Ϫ ␳ϩ ␳␤͑1 Ϫ ␶t ͒͒␦ ln lt

i ϩ͓␳␣ ϩ ͑1Ϫ ␳ ϩ ␳␤͑1 Ϫ ␶t ͒͒␥͔ln kt

i ϩ͑1 Ϫ ␳ ϩ ␳␤͑1 Ϫ ␶t ͒͒ln ztϩ ␳␤␶t ln y˜ t .

i i 2 2 Thus conditional on kt,lnVt is normally distributed, with variance (1 Ϫ ␳ ϩ ␳␤(1 Ϫ ␶t)) v . This implies:

1 (26) Ui ϵ ln E͓͑Vi͒r ͉ ki͔ ϭ E͓ln Vi ͉ ki͔ϩ r͑1 Ϫ ␳ ϩ ␳␤͑1 Ϫ ␶ ͒͒2v2/2. t r t t t t t

i i Therefore, to obtain Ut one simply needs to replace in (A25) the term in ln zt by Ϫ(1 Ϫ ␳ ϩ ␳␤(1 Ϫ 2 ␶t))[1 Ϫ r(1 Ϫ ␳ ϩ ␳␤(1 Ϫ ␶t))]v / 2. Finally, substituting in the value of ␶t ln y˜ t from (A3) yields the claimed result, with:

1 Ϫ ␳ ␳␤ 2 (A27) u៮ t ϵ ln ͓͑1 Ϫ s͒ s ͔ ϩ ␳͑ln ␬Ϫ͑1 Ϫ rЈ͒w /2͒

2 2 ϩ͑␳␣ ϩ ͑1 Ϫ ␳ ϩ ␳␤͒␥͒mtϩ ␳␤␥ ⌬t /2. VOL. 90 NO. 1 BE´ NABOU: INCOME DISTRIBUTION AND THE SOCIAL CONTRACT 127

PROOFS OF THEOREM 2 AND PROPOSITION 13: By (37) and (38), ␭៮ Ͼ ␭ if and only if:

2 2 Ϫ ␶៮ (A28) B͑␶៮͒ Ϫ B͑0͒ Ϫ ͑C͑0͒ Ϫ C͑␶៮͒͒ Ͼ ␳␤␶៮ ␥2D͑0͒D͑␶៮͒ ͩ v2ͪ ͩ v2 ͪ which yields Theorem 2, with:

w2/v2 ϩ ␤2 w2/v2 ϩ ␤2͑1 Ϫ ␶៮͒2 (A29) G͑␶៮, v2/w2͒ ϵ ␳␤␶៮ ͑2 Ϫ ␶៮͒␥2 . ͩ ͱ1 Ϫ ͑␣ ϩ ␤␥͒2ͪͩͱ1 Ϫ ͑␣ ϩ ␤␥͑1 Ϫ ␶៮͒͒2ͪ

Note that since G(␶៮, v2/w2) Ͼ 0, if B(␶៮) Ϫ B(0) Ͼ B then it must be that ␭ Ͼ 0. Finally, both ␭ and ␭៮ are clearly increasing in B ϵ B(␶៮) Ϫ B(0) and in Ϫ(C(0) Ϫ C(␶៮)) (hence in ␩). Since the common coefficient of these two terms is 1/(␶៮D(␶៮)) in ␭៮ and 1/␶៮D(0) in ␭, Proposition 13 follows immediately.

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