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

across nations. Some have low tax rates, others

a steeply progressive ﬁscal system. Many coun-

tries have made the ﬁnancing of education and

health insurance the responsibility of the state.

Some, notably the United States, have left it in

large part to families, local communities, and

employers. The extent of implicit redistribution

through labor-market policies or the mix of

public goods also shows persistent differences.

Can these societal choices be explained without

appealing to exogenous differences in tastes,

technologies, or political systems?

Adding to the puzzle is the fact that redistri-

bution is often correlated with income inequal-

ity in just the opposite way than predicted by

standard politico-economic theory: among in-

dustrial democracies the more unequal ones

tend to redistribute less, not more. The arche-

typal case is that of the United States versus

Western Europe, but the observation holds

within the latter group as well; thus Scandina-

vian countries are both the most equal and the

most redistributive. In the developing world a

similar contrast is found in the incidence of

public education and health services, which is

much more egalitarian in East Asia than in Latin

America (e.g., South Korea versus Brazil).

Turning ﬁnally to time trends, it is rather strik-

ing that the welfare state is being cut back in

most industrial democracies at the same time

that an unprecedented rise in inequality is

occurring.

The aim of this paper is to develop a joint

theory of inequality and the social contract

which can contribute to resolving some of these

puzzles. In the process, it also seeks to reconcile

certain empirical ﬁndings of the recent literature

on political economy and growth. Several au-

thors, such as Alberto F. Alesina and Dani Ro-

drik (1994) or Torsten Persson and Guido

Tabellini (1994), have documented a negative

relationship between initial disparities of in-

come or wealth and subsequent aggregate

growth. The proposed explanation is that

greater inequality translates into a poorer me-

dian voter relative to the country’s mean in-

come, as in Alan H. Meltzer and Scott F.

Richard (1981). This leads to increased pressure

for redistributive policies, which in turn reduce

incentives for the accumulation of physical and

human capital. The cross-country data, how-

ever, do not seem very supportive of this expla-

nation. Roberto Perotti (1994, 1996), and most

* Woodrow Wilson School, Princeton University,

Princeton, NJ 08544, National Bureau of Economic Re-

search, and Centre for Economic Policy Research. This

paper was written while I was at New York University, and

visiting the Institut d’Economie Industrielle (IDEI-CERAS-

GREMAQ) in Toulouse, France. I am grateful for helpful

comments to Olivier Blanchard, Jason Cummins, John

Geanakoplos, Mark Gertler, Robert Hall, Ken Judd, Jean-

Charles Rochet, Julio Rotemberg, seminar participants at

the NBER Summer Institute, Stanford University, Princeton

University, and the Massachusetts Institute of Technology,

as well as to three referees. Financial support from the

National Science Foundation, the MacArthur Foundation,

and the C.V. Starr Center is gratefully acknowledged. Fre-

derico Ravenna provided excellent research assistance.

of the other studies reviewed in Be´nabou

(1996c), ﬁnd no relationship between inequality

and the share of transfers or government expen-

ditures in GDP. Among advanced countries the

effect is actually negative, as suggested by the

above examples (Francisco Rodriguez, 1998).

As to the effect of transfers on growth, most

studies yield estimates which are in fact signif-

icantly positive.

The point of departure for this paper is a

rather different view of both the role of the state

and the workings of the political process. When

capital and insurance markets are imperfect, a

variety of policies which redistribute wealth

from richer to poorer agents can have a positive

net effect on aggregate output, growth, or more

generally ex ante welfare. Examples considered

here will include social insurance through pro-

gressive taxes and transfers, state funding of

public education, and residential integration.

Net efﬁciency gains lead to very different po-

litical economy consequences from those of

standard models: popular support for such re-

distributive policies decreases with inequality,

at least over some range. Intuitively, efﬁcient

redistributions meet with a wide consensus in a

fairly homogeneous society but face strong op-

position in an unequal one. Conversely, if

agents engage in any type of investment, capital

market imperfections imply that lower redistri-

bution translates into more persistent inequality.

The combination of these two mechanisms cre-

ates the potential for multiple steady states:

mutually reinforcing high inequality and low

redistribution, or low inequality and high redis-

tribution. Temporary shocks to the distribution

of income or the political system can then have

permanent effects.

I formalize these ideas in a stochastic growth

model with incomplete asset markets and het-

erogeneous agents who vote over redistributive

policies, whether ﬁscal or educational. In the

short run, redistribution is shown to be

U-shaped with respect to inequality; in the long

run they are negatively correlated across steady

states. There are two important ingredients in

the analysis. The ﬁrst one is that redistribution

enhance ex ante welfare, at least up to a point.

I thus examine policies which reduce the vari-

ance and possibly increase the mean of family

income, by providing insurance against idiosyn-

cratic shocks and relaxing credit constraints.

The second one is a simple extension of the

standard voting model, reﬂecting the fact that

some groups have more inﬂuence in the politi-

cal process than others. I present extensive ev-

idence that the propensities to vote, give

political contributions, work on campaigns, and

participate in most forms of political activity

rise with income and education. In the model

the pivotal agent is richer than the median, but

need not be richer than the mean. It should be

emphasized that I do not appeal to variations in

political rights or participation to explain coun-

tries’ different societal choices: this parameter

is kept ﬁxed across steady states. In the com-

parative statics analysis I vary the efﬁciency

costs and beneﬁts of redistribution on the one

hand (via the elasticity of labor supply and the

degree of risk aversion), the political system on

the other, so as to identify their respective con-

tributions to the results. In particular, I show

that there exists a critical level for the gain in ex

ante welfare from a redistributive policy, such

that: (i) below this threshold, no allocation of

political inﬂuence can sustain more than a sin-

gle social contract; (ii) above this threshold,

multiple steady states arise provided the politi-

cal weight of the rich is neither too large nor too

small.

When two “unequal societies” arise from

common fundamentals, they cannot be Pareto

ranked. As to macroeconomic performance, the

trade-off between tax distortions and liquidity-

constraint effects allows for two interesting

scenarios. One, termed “growth-enhancing re-

distributions,” is consistent with the positive

coefﬁcients of transfers in growth regressions,

as well as the contributions of education and

land policies in East Asian and Latin American

countries to their respective developments (or

lack thereof). The other, termed “eurosclerosis,”

explains how European voters can choose to

sacriﬁce more employment and growth to social

insurance than their American counterparts,

even though both populations have the same

basic preferences. Another prediction of the

Using panel data for 20 OECD countries and control

ling for national income, population, and the age distribu-

tion, Rodriguez ﬁnds that pretax inequality has a

signiﬁcantly negative effect on every major category of

social transfers as a fraction of GDP, as well as on the

capital tax rate.

97VOL. 90 NO. 1 BE

NABOU: INCOME DISTRIBUTION AND THE SOCIAL CONTRACT

model is that, depending on their source, exog-

enous shocks to income inequality will bring

about sharply different evolutions of the social

contract. Thus, an increased variability of sec-

toral shocks will lead to an expansion of the

social safety net, while a surge in immigration

may prompt large-scale cutbacks.

Some methodological features of the model

may also be worth mentioning. The ﬁrst is its

analytical tractability. Individual transitions are

linear, reﬂecting the absence of nonconvexities;

yet there is multiplicity. The distribution of

wealth remains lognormal, and closed-form so-

lutions are obtained. The second is the progres-

sivity of the redistributive schemes over which

agents vote. The third is the intuitive formaliza-

tion of political inﬂuence. These modelling de-

vices could be useful in other settings.

The paper is related to three strands of litera-

ture. The ﬁrst one emphasizes the political econ-

omy of redistribution (Giuseppe Bertola, 1993;

Perotti, 1993; Gilles Saint-Paul and Thierry Ver-

dier, 1993; Alesina and Rodrik, 1994; Persson and

Tabellini, 1994, 1996). The second one is con-

cerned with the ﬁnancing and accumulation of

human capital (Glenn C. Loury, 1981; Gerhard

Glomm and B. Ravikumar, 1992; Oded Galor and

Joseph Zeira, 1993; Be´nabou, 1996b; Steven N.

Durlauf, 1996a; Mark Gradstein and Moshe Just-

man, 1997; Raquel Ferna´ndez and Richard Rog-

erson, 1998). The third one stresses the wealth and

incentive constraints which bear on entrepreneur-

ial investment (Abhijit Banerjee and Andrew

Newman, 1993; Philippe Aghion and Patrick

Bolton, 1997; Thomas Piketty, 1997). Most di-

rectly related are the models in Be´nabou

(1996b, c), upon which I build, and the paper by

Saint-Paul (1994), which identiﬁes another mech-

anism through which capital market imperfections

can lead to multiple politico-economic regimes.

A rather different explanation for international

differences in redistribution is provided in Piketty

(1995), where agents’ imperfect learning of the

social mobility process allows different views of

the equity-efﬁciency trade-off to persist in the

long run.

The paper is organized as follows. Section I

explains the main ideas using the simplest pos-

sible setup, which treats the aggregate impact of

redistribution as exogenous. Section II presents

the actual economic model, and Section III the

political mechanism. Section IV focuses on an

endowment economy to study whether social

insurance will be more or less extensive in a

more unequal country. Section V solves the full

model with endogenous wealth dynamics. The

range of economic and political “fundamentals”

which allow multiple steady states is character-

ized, and the growth rates under alternative re-

gimes are compared. Section VI recasts the

model so as to explain differences in countries’

systems of education ﬁnance. Section VII dis-

cusses other applications such as altruism, res-

idential integration, and the mix of public

goods. Section VIII concludes. All proofs are

gathered in the Appendix.

I. A Simpliﬁed Presentation of the Main Ideas

As a prelude to the actual model, I present in

this section a very stylized reduced form which

provides a shortcut to the main intuitions and

results.

A. The Standard View

Let there be a continuum of agents, i 僆 [0,

1], with lognormally distributed endowments:

ln y

⬃ N(m, ⌬

). The lognormal is a good

approximation of empirical income distribu-

tions, leads to tractable results, and allows for

an unambiguous deﬁnition of inequality, as in-

creases in ⌬

shift the Lorenz curve outward.

This variance also measures the distance be-

tween median and per capita income: m ⫽ ln

y ⫺⌬

/2, where y ⬅ E[ y

]. Suppose now that

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-

dian income, relative to the mean, so that the politically

decisive middle class will reduce its transfers to the poor. If

exit from poverty requires some investment, credit con-

straints may then lead to multiple steady states: a large

underclass which persists due to low redistribution, or one

which is kept small by signiﬁcant transfers.

Piketty’s mechanism is similar to a collective form of

the bandit problem. Because individual experimentation is

costly, agents never fully learn the extent to which income

is affected by effort rather than predetermined by social

origins. The citizens of otherwise identical countries may

then end up with different distributions of beliefs over the

disincentive effects of redistribution.

98 THE AMERICAN ECONOMIC REVIEW MARCH 2000

agents are faced with the choice between two

stylized policies:

(P) laissez-faire: each consumes his own en-

dowment, c

⫽ y

, for all i.

(

P) complete redistribution: resources are

pooled, and everyone consumes c

⫽ y.

In this benchmark case where redistribution has

no aggregate impact, how many people are in

favor of it? Clearly, all those with endowment

below the mean, i.e., a proportion

(1) p ⫽ ⌽

冉

⌬

冊

⫽ ⌽

冉

⌬

冊

where ⌽⵺ is the cumulative distribution func-

tion of a standard normal. Because the income

distribution is right-skewed, the median is be-

low the mean, p ⬎

⁄

. A strict majority rule

would thus predict that redistribution should

always take place. In reality the poor vote with

lower probability than the rich and, to some

extent, money buys political inﬂuence. There-

fore the relevant threshold for redistribution to

occur may not be p* ⫽ 50 percent ⫽⌽(0), but

p* ⫽⌽(

␭

⬎ 0. For instance if the

␲

poorest agents never vote, ⌽(

␭

) ⫽ (1 ⫹

␲

)/2.

What is important and robust in (1) is therefore

not the level effect, p ⬎

⁄

, but the comparative

statics, ⭸p/⭸⌬⬎0: in a more unequal society there

is greater political support for redistribution. For

any degree of bias

␭

in the voting system, positive

or negative, the likelihood that redistribution takes

place increases with inequality—or more speciﬁ-

cally, skewness.

This result is only reinforced under the stan-

dard assumption that redistribution entails some

deadweight loss (endogenized later on), reduc-

ing available resources from y to ye

⫺B

, B ⬎ 0.

Given the choice between laissez-faire and shar-

ing this reduced pie, the extent of political sup-

port for the inefﬁcient redistribution is:

(2) p ⫽ ⌽

冉

⫺B ⫹ ⌬

⌬

冊

⫽ ⌽

冉

⫺

⌬

⫹

⌬

冊

Note that now p ⭵

⁄

but ⭸p/⭸⌬ is even more

positive than before. As inequality increases, so

does the likelihood that a policy which reduces

aggregate income gets implemented. This is, in

essence, the mechanism by which inequality

reduces growth in models such as Alesina and

Rodrik (1994), Persson and Tabellini (1994), or

some cases of Bertola (1993) and Perotti

(1993).

The idea that distributional conﬂict

hampers economic performance appears to be

reasonably well supported by the evidence: a

number of studies have conﬁrmed Persson and

Tabellini’s and Alesina and Rodrik’s ﬁndings of

a negative effect of inequality on growth.

This

correlation, however, does not seem to arise

through increased redistribution. Perotti (1994,

1996), Philip Keefer and Stephen F. Knack

(1995), and Peter H. Lindert (1996) ﬁnd no

relationship between the income share of the

middle class (which corresponds to the median

voter) and any tax rate or share of government

transfers in GDP. George R. G. Clarke (1992)

ﬁnds no correlation between any measure of

inequality and government consumption. As ca-

sual empiricism suggests, more unequal coun-

tries do not redistribute more. Among advanced

nations, they typically redistribute less (Rodri-

guez, 1998). Furthermore, the coefﬁcients on

transfers in growth regressions are most often

signiﬁcantly positive: see, among others, Shan-

tarayan Devarajan et al. (1993), Lindert (1996),

Xavier Sala-i-Martin (1996), and especially

Perotti (1994, 1996), who controls for the en-

dogeneity of redistribution. While none of these

ﬁndings should be viewed as deﬁnitive evi-

dence, altogether they do suggest that some-

thing important may be missing from the

traditional story.

B. Efﬁciency Gains and Redistribution:

The Static Case

In a world of incomplete insurance and loan

markets some policies with redistributive

Section III will formalize in more detail—as well as

present extensive evidence on—the inﬂuence of income and

human wealth on most forms of political activity.

Naturally, this simple reduced form fails to capture the

richness of the original models.

See Be´nabou (1996c) for a survey. As with nearly all

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-

bution. Thus, Section V, subsection B, will identify param-

eter conﬁgurations such that inequality and growth are

positively, or negatively, correlated across steady states.

99VOL. 90 NO. 1 BE

NABOU: INCOME DISTRIBUTION AND THE SOCIAL CONTRACT

features can have a positive effect on total wel-

fare, and even output (as the evidence on trans-

fers and growth may suggest). Ex ante welfare

gains, in turn, imply a political support that

varies with inequality in a radically different

way from the traditional one. Indeed, suppose

that by redistributing resources agents achieve

increased efﬁciency, so that each gets to con-

sume ye

. For now I continue to take B ⬎ 0as

exogenous, but later on I shall derive it from a

variety of channels: insurance, altruism, or

credit constraints on the accumulation of human

and physical capital. The fraction of people who

support an efﬁcient redistribution is:

(3) p ⫽ ⌽

冉

B ⫹ ⌬

⌬

冊

⫽ ⌽

冉

⌬

⫹

⌬

冊

It is of course always higher than (1) and (2),

but the important point is the one illustrated on

Figure 1.

PROPOSITION 1: When a redistributive pol-

icy generates gains in ex ante efﬁciency, polit-

ical support for it initially declines with

inequality. In the present framework, the rela-

tionship is U-shaped.

The intuition is simple: when dispersion is

relatively small compared to the average gain,

there is near-unanimous support for the policy.

As inequality rises, the proportion of those who

stand to lose from the redistribution increases.

At high enough levels of inequality, however,

the standard skewness effect eventually domi-

nates: there are so many poor that they impose

redistribution no matter what its aggregate im-

pact may be. There is thus no monotonic rela-

tionship between income inequality or the

relative position of the median agent (both mea-

sured here by ⌬

) and the likelihood of

redistribution.

To relate the extent of popular support for a

policy to actual outcomes one needs to specify

the mechanism through which preferences are

aggregated. I shall continue to assume that re-

distribution occurs if p exceeds a threshold p*

⬅⌽(

␭

), where

␭

reﬂects the degree to which

ﬁnancial or human wealth contributes to politi-

cal inﬂuence. More generally, the probability of

implementation could be some increasing func-

tion of p. Figure 1 shows that only if redistri-

bution’s aggregate impact B is positive can the

The fact that inequality affects political support for

redistribution in opposite ways, depending on the sign of its

aggregate impact, is essentially independent of distribu-

tional assumptions. For any symmetric distribution F( x)

with mean

␮

, a symmetric, mean-preserving spread leads to

a decline in popular support p ⫽ F(

␮

⫹ B) for all B ⬎ 0,

and an increase for all B ⬍ 0. With lognormally distributed

wealth, a rise in ⌬ combines this general effect of dispersion

with a fall in m ⫽

␮

⫺⌬

/2 due to increased skewness;

hence the U shape. These properties remain true when B is

a function of ⌬, as long as ⌬B⬘(⌬)/B(⌬) ⬍ 1. Such will be

the case when B is endogenized later on.

FIGURE 1. INEQUALITY AND POLITICAL SUPPORT FOR REDISTRIBUTION

Note: B ⫽ 5 percent, 0, ⫺5 percent.

100 THE AMERICAN ECONOMIC REVIEW MARCH 2000

policy be abandoned as the result of greater

inequality—no matter how biased the political

system might be. Conversely, a positive

␭

needed for the political outcome to reﬂect the

drop in popular support. The full model will

conﬁrm the joint importance of efﬁciency gains

and wealth bias in shaping a declining relation-

ship between inequality and redistribution. The

arguments seen here for a zero-one policy

choice will then apply to marginal changes in

the tax rate, i.e., to every electoral contest be-

tween

␶

and

␶

⫹ d

␶

Finally, consider the dynamic implications of

Proposition 1. A society which starts with

enough wealth disparity to ﬁnd itself below p*

on the U-shaped curve of Figure 1 will not

implement the redistributive policy; as a result,

high inequality will persist into the next period

or generation. Conversely, low inequality cre-

ates wide political support for efﬁcient policies

which prevent disparities from growing. The

dynamic feedback operates whenever some

form of investment is credit constrained, so that

current resources affect future earnings. Thus

the same type of market imperfections which

can give rise to a (partly) decreasing relation-

ship between inequality and transfers also pro-

vide the second ingredient required for multiple

steady states. These dynamics are represented

on Figure 2 (which will be formally derived

later on), for a continuous rate of redistribution

␶

僆 [0, 1]. The U-shaped curve

␶

⫽ T(⌬)is

similar to that of Figure 1, while the declining

⌬⫽D(

␶

) locus arises from the accumulation

mechanism.

The main part of the paper, to which I now

turn, will show that all the intuitions obtained in

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 僆 I ⬅ [0, 1]. In gen-

eration t, adult i combines his (human or

physical) capital endowment k

with effort l

produce output, subject to an independently

and identically distributed (i.i.d.) productivity

shock z

(4) y

⫽ z

共k

兲

␥

共l

兲

␦

Taxes and transfers, speciﬁed below, trans-

form this gross income y

into a disposable

income yˆ

which ﬁnances both the adult’s

consumption, c

, and his investment or edu-

cational bequest, e

(5) yˆ

⫽ c

⫹ e

(6) k

t ⫹ 1

⫽

␬␰

t ⫹ 1

共k

兲

␣

共e

兲

␤

Capital depreciates geometrically at the rate 1 ⫺

␣

ⱖ

␤␥

, and investment is subject to i.i.d.

productivity shocks

␰

. There is no loan market

for ﬁnancing individual investment or educa-

tional projects (e.g., children cannot be held

responsible for the debts of their parents) and no

insurance or securities market where the idio-

syncratic risks z

and

␰

t⫹1

could be diversiﬁed

away. These are extreme forms of market incom-

pleteness, but all that really matters is that there be

some imperfections.

Both shocks are assumed to

be lognormal with mean one, and initial endow-

ments lognormally distributed across families:

thus ln z

⬃ N(⫺v

/2, v

), ln

␰

⬃ N(⫺w

/2, w

)

and ln k

⬃ N(m

, ⌬

Perotti (1994) provides evidence that credit-market

frictions reduce aggregate investment, especially where the

income share of the bottom 40 percent is low. Additional

evidence on asset-market incompleteness as a constraint on

investment decisions in education and farming is discussed

in Be´nabou (1996c).

FIGURE 2. DYNAMICS AND STEADY STATES

101VOL. 90 NO. 1 BE

NABOU: INCOME DISTRIBUTION AND THE SOCIAL CONTRACT

Adults have preferences deﬁned over their

own consumption and effort, as well as their

child’s endowment of capital. Following David

M. Kreps and Evan L. Porteus (1979), Larry G.

Epstein and Stanley E. Zin (1989), and Philippe

Weil (1990), these preferences are deﬁned re-

cursively over an individual’s lifetime; see Fig-

ure 3. Thus, upon discovering his productivity,

, agent i chooses effort, consumption, and

savings so as to maximize:

(7) ln V

⬅ max

兵共1 ⫺

␳

兲关ln c

⫺ 共l

兲

␩

兴

⫹

␳

ln 共共E

关共k

t ⫹ 1

兲

r⬘

兩k

, z

兴兲

1/r⬘

兲其.

The disutility of effort is measured by

␩

⬎ 1,

which corresponds to an intertemporal elasticity

of labor supply equal to 1/(

␩

⫺ 1). The discount

factor

␳

deﬁnes the relative weights of the

adult’s own felicity and of his bequest motive,

while his (relative) risk aversion with respect to

the child’s endowment k

t⫹1

is 1 ⫺ r⬘. At the

beginning of period t, however, when evaluat-

ing and voting over redistributive policies, the

agent has not yet learned z

. In other words, he

knows his type (k

, z

) imperfectly.

The result-

ing uncertainty over his ex post utility level V

reﬂected in his ex ante preferences,

(8) U

⬅ ln 共E

关共V

兲

兩k

兴

1/r

兲,

with a risk-aversion coefﬁcient of 1 ⫺ r. When

r ⫽ 0 preferences are time separable and

1/(1 ⫺ r) coincides with the intertemporal

elasticity of substitution in consumption, which

by (7) is ﬁxed at one.

The more general spec-

iﬁcation allows 1 ⫺ r to parametrize the insur-

ance value of redistributive policies, just as the

labor-supply elasticity 1/(

␩

⫺ 1) parametrizes

the effort distortions. Varying these parameters

will make it possible to study how the set of

politico-economic equilibria is shaped by the

costs and beneﬁts of redistribution—the central

issue of this paper. Finally, it should be noted

that while I have assumed overlapping genera-

Therein lies the crucial difference between

␰

and z

rather than in the fact that one affects human capital and the

other production; the latter roles could be switched. Also,

because the policies which I shall consider provide no

insurance against

␰

t⫹ 1

(only against z

), the value of 1 ⫺ r⬘

will turn out not to play an important role in the analysis. It

could therefore be set (for instance) to zero, to one, or to

1 ⫺ r, which is the more essential risk-aversion parameter

deﬁned in (8) below.

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-

man (1993), Galor and Zeira (1993), Saint-Paul and Verdier

(1993), Aghion and Bolton (1997), or Gradstein and Just-

man (1997)].

FIGURE 3. PREFERENCES AND THE TIMING OF DECISIONS

102 THE AMERICAN ECONOMIC REVIEW MARCH 2000

tions with “imperfect” altruism, many of pa-

per’s results can also be derived with inﬁnitely

lived agents.

B. Fiscal Policy

Taxes and transfers map agent i’s market

income y

into a disposable a income yˆ

, accord

ing to the following scheme:

(9) yˆ

⬅ 共y

兲

1 ⫺

␶

共y˜

兲

␶

The break-even level y˜

is determined by the gov

ernment’s budget constraint: net transfers must

sum to zero or, denoting per capita income by y

(10)

冕

共y

兲

1 ⫺

␶

共y˜

兲

␶

di ⫽ y

The elasticity

␶

of posttax income measures the

degree of progressivity (or regressivity) of ﬁscal

policy.

An alternative interpretation is that of

wage compression through labor-market insti-

tutions favorable to workers with relatively low

skills. Conﬁscatory rates

␶

⬎ 1 must be ex

cluded as not incentive compatible, but nothing

in principle prevents a regressive tax, so I shall

allow it. Restricting

␶

to be nonnegative would

require dealing with corner solutions but would

not change the nature of any result.

PROPOSITION 2: Given a tax rate

␶

, agents

in generation t choose a common labor supply

and savings rate:

⫽

␹

共1 ⫺

␶

兲

␩

⫽ syˆ

where

␹

␩

⬅ (

␦

␩

)(1 ⫺

␳

⫹

␳␤

)/(1 ⫺

␳

) and s ⬅

␳␤

/(1 ⫺

␳

⫹

␳␤

Because taxes are progressive rather than

merely proportional they affect effort, l

⫽

␶

), in spite of the fact that utility is logarith

mic in consumption.

I will refer from here on

to 1/

␩

as “the” elasticity of labor supply.

C. Redistribution and Accumulation

Given Proposition 2, and substituting (9) into

(6), capital accumulation simpliﬁes to:

(11)

ln k

t ⫹ 1

⫽ ln

␰

t ⫹ 1

⫹

␤

共1 ⫺

␶

兲 ln z

⫹ ln

␬

⫹

␤

ln s ⫹ 共

␣

⫹

␤␥

共1 ⫺

␶

兲兲 ln k

⫹

␤␦

共1 ⫺

␶

兲 ln l

⫹

␤␶

ln y˜

Due to the symmetry of agents’ effort and sav-

ings decisions, wealth and income remain log-

normally distributed over time. If ln k

⬃ N(m

⌬

), the government’s budget constraint (10)

easily yields the break-even point of the redis-

tributive scheme (see the Appendix):

(12) ln y˜

⫽

␥

⫹

␦

ln l

⫹ 共2 ⫺

␶

兲

␥

⌬

⫹共1 ⫺

␶

兲v

/2.

See Be´nabou (1996a, 1999). The inﬁnite-horizon ver

sion of the preferences used here is obtained by replacing

t⫹ 1

with V

t⫹ 1

in (7), with r⬘⫽r and (8) unchanged.

When

␶

⬎ 0 the marginal rate rises with pretax

income, and agents with average income are made better

off: y˜

⬎ y

. Measuring progressivity by the (local) elastic

ity of after-tax to pretax income was ﬁrst proposed by

Richard A. Musgrave and Tun Thin (1948). Ulf Jakobsson

(1976) and Nanak C. Kakwani (1977) showed this to be the

“right” measure of equalization: the posttax distribution

induced by a ﬁscal scheme Lorenz-dominates the one in-

duced by another (for all pretax distributions), if and only if

the ﬁrst scheme’s elasticity is everywhere smaller. A “con-

stant residual progression” scheme similar to (9) turns out to

have been used in a couple of earlier but static models of

insurance or risk taking (Martin S. Feldstein, 1969; S. M.

Kanbur, 1979; Mats Persson, 1983).

They would also affect savings, were it not for adults’

simple bequest motive. Even with inﬁnitely lived agents,

however, the savings distortion can be fully offset by a

balanced-budget combination of consumption taxes and in-

vestment subsidies; moreover, this can be shown to be

Pareto optimal (Be´nabou, 1996a, 1999). In any case, one

distortion is enough to demonstrate how the trade-off be-

tween the costs and beneﬁts of redistribution shapes the

range of politico-economic equilibria.

It is indeed the uncompensated elasticity to the net-

of-tax rate 1 ⫺

␶

, and varies monotonically with the usual

intertemporal elasticity of substitution for proportional vari-

ations in the real wage, 1/(

␩

⫺ 1).

103VOL. 90 NO. 1 BE

NABOU: INCOME DISTRIBUTION AND THE SOCIAL CONTRACT

From (11), we then obtain two simple differ-

ence equations which govern the evolution of

the economy:

(13) m

t⫹ 1

⫽ 共

␣

⫹

␤␥

兲m

⫹

␤␦

ln l

⫹

␤␶

共2 ⫺

␶

兲共

␥

⌬

⫹ v

兲/2

⫹ ln 共

␬

␤

兲 ⫺ 共w

⫹

␤

兲/2

(14) ⌬

t ⫹ 1

⫽ 共

␣

⫹

␤␥

共1 ⫺

␶

兲兲

⌬

⫹

␤

共1 ⫺

␶

兲

⫹ w

The effect of redistribution on the dynamics of

inequality is clear: the progressivity rate

␶

termines the persistence of family wealth,

␣

⫹

␤␥

(1 ⫺

␶

). The impact on the dynamics of

aggregate income is more complex, as it in-

volves a trade-off between labor supply and

credit-constraint effects:

PROPOSITION 3: The distribution of pretax

income at time t is ln y

⬃ N(

␥

⫹

␦

ln l

⫺

/2,

␥

⌬

⫹ v

), where m

and ⌬

evolve

according to the linear difference equations

(13)–(14) and l

⫽

␹

(1 ⫺

␶

)

␩

. The growth

rate of per capita income is:

(15)

ln 共y

t ⫹ 1

兲 ⫽ ln

␬

˜ ⫺ 共1 ⫺

␣

⫺

␤␥

兲ln y

⫹

␦

共ln l

t ⫹ 1

⫺

␣

ln l

兲

⫺ L

(

␶

/2⫺L

⌬

(

␶

)

␥

⌬

/2,

where ln

␬

˜ ⬅

␥

(ln

␬

⫹

␤

ln s) ⫺

␥

(1 ⫺

␥

/2 is a constant and

(

␶

)⬅

␤␥

(1⫺

␤␥

)(1⫺

␶

)

⬎0,

⌬

(

␶

)⬅

␣

⫹

␤␥

(1⫺

␶

)

⫺(

␣

⫹

␤␥

(1⫺

␶

))

⬎0.

The term in ⫺ln y

reﬂects the standard

convergence effect; it disappears under con-

stant aggregate returns, namely when

␣

⫹

␤␥

⫽ 1orwhen

␬

is replaced by an appro-

priate spillover

␬

(see Section V, subsection

B). The next term, capturing the effects of

labor supply on accumulation, is also of a

“representative agent” nature. The terms in

(

␶

) and L

⌬

(

␶

), on the other hand, represent

growth losses speciﬁc to the heterogenous

agent economy with imperfect credit markets.

Suppose ﬁrst that adults in generation t are ex

ante identical (⌬

⫽ 0). Everyone then faces

the same accumulation technology, with con-

cavity (decreasing returns)

␤␥

(1 ⫺

␤␥

). The

shocks z

generate ex post income disparities

(partially offset by redistribution), which

credit constraints translate into inefﬁcient

variations in investment, reducing overall

growth by L

(

␶

/2. Consider now dispari

ties in initial endowments,

␥

⌬

. When

␣

⫽ 0

these have the same effect as income shocks:

⬅ L

⌬

. The marginal return to investment is

higher for the poor than for the rich, because

they are more severely liquidity constrained.

When

␣

⬎ 0, however, the preexisting capital

stocks k

represent complementary inputs

which generate differential returns to invest-

ment, and thereby reduce the desirability of

equalizing resources. Thus L

⌬

(

␶

)ismini

mized for

␶

⫽ (1 ⫺

␣

⫺

␤␥

)/(1 ⫺

␤␥

), which

decreases with

␣

D. Individual Welfare

I now turn from the evolution of the economy

as a whole to individuals’ evaluations of alter-

native policies. Given the optimal labor supply

and savings responses to a tax rate

␶

, (7) gives

agent i’s ex post welfare V

for any productivity

realization z

. Since he must vote before learn

ing z

, his preferences over

␶

are then deﬁned

by the ex ante utility U

, according to (8).

PROPOSITION 4: Given a rate of ﬁscal pro-

gressivity

␶

, agent i’s intertemporal welfare is:

(16) U

⫽ u៮

⫹ A共

␶

兲共ln k

⫺ m

兲

⫹ C共

␶

兲 ⫺ 共1⫺

␳

⫹

␳␤

兲

⫻共1 ⫺

␶

兲

共

␥

⌬

⫹ Bv

兲/2,

104 THE AMERICAN ECONOMIC REVIEW MARCH 2000

where u៮

is independent of the policy

␶

and:

(17) A共

␶

兲 ⬅

␳␣

⫹ 共1 ⫺

␳

⫹

␳␤

兲

␥

共1 ⫺

␶

兲,

(18) C共

␶

兲 ⬅ 共1 ⫺

␳

兲共

␦

ln l共

␶

兲 ⫺ l共

␶

兲

␩

兲

⫹

␳␤␦

ln l共

␶

兲,

(19) B ⬅ 1 ⫺ r ⫹

␳

r共1 ⫺

␤

兲 ⱖ 0.

The ﬁrst term in U

depends only on the state

variables m

and ⌬

and on the (endogenous

but constant) investment rate s. The second

term, which disappears through aggregation,

makes clear the redistributive effects of tax

policy, including its impact on the persistence

of social positions,

␣

⫹

␤␥

(1 ⫺

␶

). The last

two terms represent the aggregate welfare

cost and aggregate welfare beneﬁt of a pro-

gressivity rate

␶

. Thus C(

␶

), which is max

imized for

␶

⫽ 0, reﬂects the distortions in

labor supply entailed by such a policy. Con-

versely, the term ⫺(1 ⫺

␶

)

(

␥

⌬

⫹ Bv

which is maximized for

␶

⫽ 1, embodies the

efﬁciency gains which arise from better in-

surance and the redistribution of resources

from low to high marginal-product invest-

ments. Note that B is now endogenous, and

monotonically related to risk aversion, 1 ⫺ r.

More generally, by varying 1 ⫺ r,

␤

, v

, and

␩

, the net ex ante efﬁciency gain, (1 ⫺

␳

⫹

␳␤

)(1 ⫺

␶

)

/2 ⫺ C(

␶

), can be made

arbitrarily large or small relative to preexist-

ing income inequality,

␥

⌬

To make the role of market incompleteness

more explicit I consider again each source of

heterogeneity in turn. By (16), idiosyncratic

uncertainty lowers everyone’s utility by (1 ⫺

␳

⫹

␳␤

) B(1 ⫺

␶

)

/2. When

␤

⫽ 1 this

simpliﬁes to (1 ⫺ r)(1 ⫺

␶

)

/2: with

constant returns, the ex ante value of redistri-

bution stems from the insurance it provides.

In general, it also contributes to efﬁciency

through the relaxation of credit constraints.

This is best seen with risk-neutral parents

who care only about their offspring: when

r ⫽

␳

⫽ 1 the utility loss is

␤

(1 ⫺

␤

)(1 ⫺

␶

)

/2, which is the shortfall in expected

(and aggregate) bequests resulting from idio-

syncratic resource shocks and a concave

investment technology. Turning now to pre-

existing inequality, the loss in aggregate wel-

fare is (1 ⫺

␳

⫹

␳␤

)(1 ⫺

␶

)

␥

⌬

/2; it

embodies two effects, only one of which is

due to market incompleteness. First, reallo-

cating investment resources towards the poor

again increases the growth rate of total

wealth, ln (k

t⫹ 1

). Second, there is the stan

dard effect of concave (logarithmic) utility

functions, whereby average welfare increases

when individual consumptions (of c

and

t⫹ 1

) are distributed more equally.

Equiv-

alently in this model, it captures the effect of

skewness, as in Section I: the median agent,

who is poorer than average, gains when con-

sumption and bequests are redistributed pro-

gressively.

III. The Political System

I now turn to the determination of the equi-

librium policy. Each generation chooses, before

the individual productivity shocks z

are real

ized, the rate of ﬁscal progressivity

␶

to which

it will be subject. Agent i’s ideal policy would

thus be to maximize U

These individual

preferences are aggregated through a political

process in which some groups have more inﬂu-

ence than others.

One can rewrite (1 ⫺

␳

⫹

␳␤

)

␥

(1 ⫺

␶

)

␳

[

␤␥

(1 ⫺

␶

)

⫺ (

␣

⫹

␤␥

(1 ⫺

␶

))

] ⫹ (1 ⫺

␳

)

␥

(1 ⫺

␶

)

⫹

␳

(

␣

⫹

␤␥

(1 ⫺

␶

))

⫽⫺

␳

ln (k

t⫹ 1

/ln k

) ⫹ (1 ⫺

␳

)var

j僆 I

[ln c

]

⫹

␳

var

j僆 I

[ln k

t⫹ 1

] ⫹

␮

, where var

j僆 I

[ 䡠 ] denotes a

cross-sectional variance,

␮

is independent of

␶

, and I have

set v ⫽ w ⫽

␦

⫽ 0 for notational simplicity.

Due to the overlapping-generations structure, no in

tertemporal strategic considerations are involved. Inﬁnite

horizons, by contrast, would generate a dynamic game

where voters try to inﬂuence future political outcomes

␶

t⫹ k

by altering the evolution of the wealth distribution ⌬

t⫹ k

through their choice of

␶

[see (14)]. This problem is noto

riously intractable, so the standard practice is to assume that

voters are “myopic,” either ignoring their inﬂuence on fu-

ture outcomes or—as here—not caring about it due to a

limited bequest motive. Notable exceptions are Saint-Paul

(1994) and Gene M. Grossman and Elhanan Helpman

(1996). Alternatively, Per Krusell et al. (1997) look for

numerical solutions. In Be´nabou (1996a) I considered a

different form of political myopia and obtained results sim-

ilar to those presented here. In short, agents with fully

dynastic preferences choose a constitution (namely a con-

stant sequence {

␶

t⫹ k

⫽

␶

}

k⫽ 0

⬁

), ignoring the fact that

future generations may revise it. In any steady state, how-

ever, [⌬

⫽⌬

t⫹ 1

in (14)], every generation, given the

same choice set as its predecessors, validates the existing

social contract.

105VOL. 90 NO. 1 BE

NABOU: INCOME DISTRIBUTION AND THE SOCIAL CONTRACT

A. Preferred Policies

I ﬁrst consider the simpler case where labor

supply is inelastic, 1/

␩

⫽ 0. The utility function

is then quadratic in

␶

, and maximized at:

(20)

1 ⫺

␶

⫽

␥

⌬

⫹ Bv

␥

max 兵ln k

⫺ m

,0其

Voters below the median desire the maximum

feasible tax rate,

␶

⫽ 1, which is also the ex

ante efﬁcient one in the absence of distortions.

Voters above the median desire a tax rate

␶

⬍

1 which decreases with their initial wealth and

increases with the variance of productivity

shocks, for both insurance and investment rea-

sons (concavity of preferences and concavity of

the technology).

With endogenous labor supply, complete re-

distribution is never chosen, as it would lead to

zero effort and output. Agent i’s desired policy

is given by the ﬁrst-order condition ⭸U

/⭸

␶

⫽

0, or:

(21) 共1 ⫺

␶

兲共

␥

⌬

⫹ Bv

兲 ⫺

␥

共ln k

⫺ m

兲

⫺

␦

␩

冉

␶

1 ⫺

␶

冊

⫽ 0.

This quadratic equation always has a unique

solution less than 1, which will be denoted

␶

PROPOSITION 5: Each agent’s utility U

strictly concave in the policy

␶

. His preferred

tax rate

␶

decreases with his wealth k

and

increases with Bv

. Finally, 兩

␶

兩 decreases with

the labor-supply elasticity 1/

␩

These results are intuitive. Lower personal

wealth or greater ex ante efﬁciency beneﬁts

from redistribution increase an agent’s de-

mand for such policies. A more elastic labor

supply increases the deadweight loss from

taxes and transfers, whether progressive or

regressive, which cause individuals to distort

their labor supply away from the ﬁrst-best

level. Finally, as a prelude to the analysis of

political equilibrium, note how (21) embodies

the same intuitions as the stylized model of

Section I. For any

␶

⬍ 1 the proportion of

agents who would like further redistribution

at the margin,

(22) p共

␶

, ⌬

兲

⬅ card

再

冏

⭸U

⭸

␶

⬎ 0

冎

⫽⌽

冉

共1 ⫺

␶

兲Bv

⫺

␦␶

/共

␩

共1 ⫺

␶

兲兲

␥

⌬

⫹共1 ⫺

␶

兲

␥

⌬

冊

is U-shaped in ⌬

, provided the resulting gain

in ex ante efﬁciency (1 ⫺

␶

) Bv

dominates

the distortion

␦␶

␩

(1 ⫺

␶

)). Otherwise, p(

␶

⌬

) is strictly increasing in ⌬

. Also as on

Figure 1, variations in popular support for

efﬁcient increases in

␶

all take place above the

50-percent level, so they will inﬂuence policy

outcomes only under some departure from the

pure “one person, one vote” democratic ideal.

B. Wealth and Political Inﬂuence

It is well known that poor and less educated

individuals have a relatively low propensity to

contributions. These are among the facts docu-

mented in Tables 1 and 2, which present data

from Steven J. Rosenstone and John M. Hansen

(1993); see also Raymond E. Wolﬁnger and

Rosenstone (1980), Thomas B. Edsall (1984),

or Margaret M. Conway (1991). For each form

of participation in electoral or governmental

politics, the representation ratio of a given so-

cioeconomic group is the ratio between its share

of the population engaged in this activity and its

share of the general population. Thus the poor-

est 16 percent account for only 0.76 ⫻ 16 ⫽

12.2 percent of the votes and 0.25 ⫻ 16 ⫽ 4.0

percent of the number of campaign contributors,

while the richest 5 percent account for 1.27 ⫻

More speciﬁcally, for ln k

ⱕ m

⫹

␥

⌬

⫹ Bv

␥

the

ideal

␶

is positive and decreases towards zero as 1/

␩

rises.

For ln k

⬎ m

⫹

␥

⌬

⫹ Bv

␥

␶

⬍ 0 and it increases

towards zero as 1/

␩

rises.

106 THE AMERICAN ECONOMIC REVIEW MARCH 2000

5 ⫽ 6.4 percent of the votes and 3.25 ⫻ 5 ⫽

16.3 percent of contributors.

The data in Tables 1 and 2 are striking in

several respects. The propensity to participate in

every reported form of political activity rises

with income and education. For voting itself the

tendency is relatively moderate, whereas for

contributing to political campaigns it is drastic.

In the latter case the actual bias is still under-

stated since the data reﬂects only the number of

contributions, and not their amounts. It is intu-

itive that the wealthy should be overrepresented

in money-intensive channels of political inﬂu-

ence: such lobbying is a form of collective

investment where liquidity constraints are even

more likely to bind than usual. One might have

expected poorer, less skilled agents to have a

countervailing advantage for attending meet-

ings, working on campaigns, writing Congress,

and other time-intensive activities for which

they have a lower opportunity cost. But, re-

markably, the pro-wealth (ﬁnancial and human)

bias is here again not only positive, but ex-

tremely strong.

I shall not seek to explain the source of these

biases, only to model them in a plausible and

convenient manner.

Let each agent’s opinion

be affected by a relative weight, or probability

of voting,

␻

/兰

␻

dj. If individual preferences

are single-peaked and the preferred policy is

monotonic in wealth, or more generally if pref-

Put differently, the representation ratios are the slopes

of the (piecewise linear) Lorenz curve which describes the

concentration, by income or education, of a given form of

political inﬂuence.

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

worth emphasizing again that I shall not appeal to differ-

ences in the allocation of political power or inﬂuence to

explain why redistribution varies across countries (although

the model can readily incorporate this “easier” explanation).

ABLE 1—POLITICAL PARTICIPATION BY INCOME

Political activity:

Electoral Politics,

1952–1988

Total fraction

taking part

(in percent)

Representation ratios by percentile family income

Pivot p*

(in percent)0–16 17–33 34–67 68–95 96–100

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

Political activity:

Total fraction

taking part

(in percent)

Representation ratios by years of education (with

corresponding percentage of population)

Pivot p*

(in percent)

0–8

(20.1)

9–11

(16.3)

(32.8)

13–15

(16.8)

16⫹

(14)

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*.

107VOL. 90 NO. 1 BE

NABOU: INCOME DISTRIBUTION AND THE SOCIAL CONTRACT

erences satisfy a single-crossing condition (as in

Joshua S. Gans and Michael Smart, 1996), a

median-voter-type result applies, but where the

median is computed on an appropriately renor-

malized population. With a lognormal distribu-

tion, the following schemes yield particularly

simple results.

PROPOSITION 6: Suppose that agents i 僆 [0,

1] have preferences U(k

␶

) over some policy

variable

␶

僆⺢, such that: for all k ⬍ k⬘ and

␶

⬍

␶

⬘,ifU(k⬘,

␶

⬘) ⬎ U(k⬘,

␶

) then U(k,

␶

⬘) ⬎

U(k,

␶

(1) If an agent’s political weight depends on

his rank in the wealth distribution,

␻

⫽

␻

), the pivotal voter is the one with rank

p* ⫽⌽(

␭

) and (log) wealth ln k* ⫽ m ⫹

␭

⌬, where

␭

⭴ 0 is deﬁned by (兰

⌽(

␭

)

␻

(p)

dp)/(兰

␻

(p) dp) ⫽

⁄

(2) If an agent’s political weight depends on

the absolute level of his wealth, with

␻

⫽

)

␭

for some

␭

⭴ 0, the pivotal voter has

rank p* ⫽⌽(

␭

⌬) and (log) wealth ln k*

⫽ m ⫹

␭

⌬

Ordinal schemes ensure that each person’s

weight and the identity of the pivotal voter

remain invariant when the distribution of

wealth shifts due to growth, or when it be-

comes more unequal.

Previous discussions

have often assumed that political rights or

inﬂuence depend on one’s absolute level of

wealth. This was motivated by historical ex-

amples such as voting franchises restricted to

citizens owning enough property (Saint-Paul

and Verdier, 1993; Persson and Tabellini,

1994) or costly membership in a ruling elite

(Alberto Ades and Verdier, 1996). I ﬁnd it

more plausible that even such cutoffs should

be relative ones, keeping up with aggregate

growth and the competitive nature of bids for

political inﬂuence. I shall therefore focus on

ordinal schemes, but the second part of Prop-

osition 6 shows that absolute income effects

are just as easy to capture; the case

␭

⫽ 1, for

instance, corresponds to a “one dollar, one

vote” rule. Moreover, this alternative formu-

lation would only reinforce the paper’s re-

sults, as it implies that the political system

becomes more biased towards the wealthy as

inequality rises.

In the last columns of Tables 1 and 2, I

interpolated the empirical

␻

(p) function to

compute the position of the pivotal agent p* for

each separate form of political participation,

i.e., as if it were the only one that mattered.

No data exist which would allow me to weigh

them by their relative importance in determin-

ing the political outcome. For voting, the wealth

bias is moderate, with the pivot at the 56th

percentile rather than the usually assumed me-

dian. For all other forms of inﬂuence it is much

stronger, with the pivotal agent always above

the 60th percentile, and quite often the 70th.

From this evidence it seems safe to conclude

that the decisive political group is located above

the median in terms of income and human

wealth. Depending on the relative efﬁcacy of

the different channels of political inﬂuence it

may even be above the mean, which in the U.S.

income distribution falls around the 63rd

percentile.

C. Political Equilibrium, Inequality,

and Redistribution

We are now ready to establish the paper’s

ﬁrst main result. By virtue of Propositions 5 and

6, the policy outcome is obtained by simply

setting ln k

⫺ m

⫽

␭

⌬

in the ﬁrst-order

condition ⭸U

/⭸

␶

⫽ 0, or equivalently p(

␶

⌬

) ⫽ p* ⫽⌽(

␭

) in (22). As before, I consider

ﬁrst the case where labor supply is inelastic

(1/

␩

⫽ 0); for

␭

⬎ 0, this yields:

Also, for any ordinal scheme

␻

⵺ the associated

␭

is a

sufﬁcient statistic: it is as if the bottom 2⌽(

␭

) ⫺ 1 voters [or

the top 1 ⫺ 2⌽(

␭

) when

␭

⬍ 0] systematically abstained.

Note that even though

␭

⬎ 0 is the more empirically

relevant case, the model will be solved for all

␭

⭵ 0.

The weights

␻

are obtained by rescaling each group’s

representation ratios by the population’s average propensity

to participate in the activity under consideration (given in

column 1). Concerning the relationship between voters’

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

axis of rich/opposed to redistribution versus poor/favoring

redistribution.

We shall see below that the ﬁrst condition, but not the

second, is required for redistribution to decline with in-

equality in the model.

108 THE AMERICAN ECONOMIC REVIEW MARCH 2000

(23)

1 ⫺

␶

⫽

␥

⌬

⫹ Bv

␭␥

⌬

The equilibrium tax rate is clearly U-shaped in ⌬

and minimized where

␥

⌬

⫽ Bv

. Similarly, in

the general case it is the unique solution T(⌬

) ⬍

1 to the quadratic equation derived from (21):

(24) 共1 ⫺

␶

兲

冉

␥

⌬

⫹ Bv

␥

⌬

冊

⫺

␦

␩␥

⌬

冉

␶

1 ⫺

␶

冊

⫽

␭

PROPOSITION 7: The rate of ﬁscal progres-

sivity

␶

⫽ T(⌬

) chosen in generation t has the

following features:

(1)

␶

increases with the ex ante efﬁciency gain

from redistribution (gross of distortions)

, and decreases with the political inﬂu

ence of wealth,

␭

(2) 兩

␶

兩 decreases with the elasticity of labor

supply 1/

␩

(3) For

␭

⬎ 0,

␶

is U-shaped with respect to

inequality ⌬

. It starts at the ex ante efﬁ

cient rate T(0) ⫽ 1 ⫺ 2(1 ⫹

公

1 ⫹ 4

␩

␦

)

⫺1

, declines to a mini

mum at some ⌬⬎0, then rises back to-

wards T(⬁) ⫽ 1. The larger Bv

, the wider

the range [0, ⌬) where ⭸

␶

/⭸⌬

⬍ 0. For

␭

ⱕ 0,

␶

is increasing in ⌬

The ﬁrst two results show that the equilib-

rium tax rate depends on the costs and beneﬁts

of redistribution, as well as on the allocation of

political inﬂuence, in a very intuitive manner.

The third result provides an endogenously de-

rived analogue to Figure 1, with the continuous

policy

␶

now replacing the proportion of people

supporting redistribution in a zero-one decision.

It also conﬁrms several claims made earlier

about the result that redistribution may decline

with inequality. First, it is not predicated on the

pivotal agent being richer than the mean, or

becoming richer relative to the mean. Second, it

is more likely to occur the larger the ex ante

welfare gain from redistribution, e.g., the larger

Third, some bias

␭

⬎ 0 with respect to

pure majority rule is needed as well, because the

median agent always wants to push redistribu-

tion beyond its range of efﬁciency. As shown in

(22), it is only within that range that political

support for a tax increase, p(

␶

, ⌬

), can decline

with higher inequality.

It is worth noting that the distinctive non-

monotonic relationship predicted by the model

has recently been tested by Paolo Figini (1999),

who found in cross-country regressions a sig-

niﬁcant U-shaped effect of income inequality

on the shares of tax revenues and government

expenditures in GDP.

The above results apply equally in an endow-

ment economy (

␤

⫽ 0) and in the presence of

accumulation. In the ﬁrst case the efﬁciency

gains arise from insurance. In the second they

also reﬂect the reallocation of resources to

agents whose marginal product of investment is

higher, due to tighter liquidity constraints. In

studying which social contracts emerge in the

long run, I shall consider each case in turn.

IV. Inequality and Social Insurance

in an Endowment Economy

Should we expect a more generous welfare

state in countries with greater disparities of in-

come and wealth, as predicted by standard mod-

els, or a less generous one, as a comparison

between Sweden and the United States would

suggest? To study the political economy of pure

social insurance, let us focus on an endowment

economy (

␤

⫽ 0).

Each dynasty’s endowment

then simply follows a geometric AR(1) pro

More speciﬁcally, if Bv

⬎

␭

/4 then

␶

⬎ 0 and

⭸

␶

/⭸

␩

⬎ 0. If Bv

⬍

␭

/4, then ⭸

␶

/⭸

␩

has the sign of

␶

With respect to the ﬁrst claim, note that ln k* ⫺ ln

E[k] ⫽

␭

⌬⫺⌬

/2 is increasing only on [0,

␭

] and

positive only on [0, 2

␭

]. Neither interval coincides with, nor

contains, [0, ⌬]. The second claim follows from Proposition

7, but even when Bv

⫽ 0 one sees from (22) that for all

⌬ 僆 [0, ⌬] a marginal rise in

␶

above T(⌬) increases ex ante

efﬁciency. These gains now arise from lowering the effort

distortions due to regressionary taxes, as Bv

⫽ 0 implies

T(0) ⫽ 0, hence

␶

⬍ 0 on [0, ⌬].

The important assumption here is the absence of in

surance markets. The incompleteness of the loan market is

inessential, and indeed this section’s results remain un-

changed with

␳

⫽ 0. This one-shot model is close to that of

109VOL. 90 NO. 1 BE

NABOU: INCOME DISTRIBUTION AND THE SOCIAL CONTRACT

cess with serial correlation

␣

[see (6)], and the

equilibrium policy is T(⌬

). In the long run

inequality converges to ⌬

⬁

⬅ w

/(1 ⫺

␣

and the tax rate therefore to

␶

⬁

⬅ T(⌬

⬁

). When

␩

⫽ 0, for instance,

(25)

1 ⫺

␶

⬁

⫽

␭

冋

␥

冑

1 ⫺

␣

⫹ 共1 ⫺ 共1 ⫺

␳

兲r兲

冉

冊

冉

冑

1 ⫺

␣

␥

冊

册

More generally, the following results are imme-

diate (focussing on the empirically relevant case

␭

⬎ 0).

PROPOSITION 8: The steady-state rate of ﬁs-

cal progressivity

␶

⬁

increases with agents’ de

gree of risk aversion 1 ⫺ r and with income

uncertainty v

, but is U-shaped with respect to

the variability w

and the persistence

␣

of the

endowment process. It decreases with the polit-

ical inﬂuence of wealth

␭

, and declines in ab-

solute value with the labor-supply elasticity 1/

␩

Recalling that steady-state income dispersion

␥

/(1 ⫺

␣

) ⫹ v

, the above results

indicate that the relationship between inequality

and redistribution is not likely to be monotonic.

What matters is not just the amount of income

inequality, but also its source. To the extent that

high income disparities in some countries re-

ﬂect larger uninsurable shocks (or more imper-

fect insurance markets), higher taxes and

transfers should be observed. But if greater in-

equality is due to more persistent wealth dy-

namics or to greater ex ante heterogeneity at the

time of the policy decision—correlated for in-

stance with ethnic or regional differences—the

reverse correlation may be observed.

While

greater persistence makes income more vari-

able, it also increases the number of agents for

whom the value of insurance is more than offset

by their vested interest in the status quo.

V. History-Dependent Social Contracts

A. Dynamics and Multiple Steady States

I now turn to the paper’s second main idea,

sketched at the end of Section I: if more in-

equality leads to less redistribution, and if in-

vestment resources depend on past transfers,

multiple steady states can arise. To demonstrate

this point I solve the full model with capital

accumulation subject to wealth constraints (

␤

⬎

0). The critical difference with the endowment

economy is that the wealth distribution is now

endogenous, through the effect of ﬁscal policy

on persistence,

␣

⫹

␤

(1 ⫺

␶

). The joint

evolution of inequality and policy is thus de-

scribed by the recursive dynamical system:

(26)

再

␶

⫽ T共⌬

兲

⌬

t ⫹ 1

⫽ D共⌬

␶

兲

where T(⌬

) is the unique solution less than one

to (24), while D(⌬

␶

) is given by (14). Under

a time-invariant policy, in particular, long-run

inequality decreases with redistribution:

(27) ⌬

⬁

⫽

⫹

␤

共1 ⫺

␶

兲

1 ⫺ 共

␣

⫹

␤␥

共1 ⫺

␶

兲兲

⬅ D

共

␶

兲.

A steady-state equilibrium is an intersection of

this downward-sloping locus, ⌬⫽D(

␶

), with

the U-shaped curve

␶

⫽ T(⌬) described in

Proposition 7; see Figure 2. Substituting (27)

into (24), this corresponds to a rate of tax pro-

gressivity solving the equation:

(28) f共

␶

兲 ⬅ 共1 ⫺

␶

兲

冉

␥

D共

␶

兲

⫹ Bv

␥

D共

␶

兲

冊

⫺

␦

␩␥

D共

␶

兲

冉

␶

1 ⫺

␶

冊

⫽

␭

Persson (1983), except that he assumes no initial heteroge-

neity (⌬

⫽ 0).

A related result obtains in Persson and Tabellini

(1996), where two regions bargain over the degree of risk

sharing in the federal constitution. In both models the un-

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-

ponent. See also George Casamata et al. (1997) for a study

of how ex post political equilibrium constrains the ex ante

design of public health insurance systems.

110 THE AMERICAN ECONOMIC REVIEW MARCH 2000

This is a polynomial equation of degree eight,

and therefore quite complex. Yet, by exploiting

geometric intuitions on the shape of f, one can

establish a series of results which formalize the

paper’s main ideas.

As illustrated on Figure

4, two countries with the same economic and

political fundamentals can nonetheless evolve

into different societies, provided:

(a) the ex ante welfare beneﬁts of redistribution

are high enough, relative to the costs;

(b) the political power of the wealthy lies in

some intermediate range.

THEOREM 1: Let 1 ⫺

␣

⬍ 2

␤␥

. When the

normalized efﬁciency gain B ⬅ 1 ⫺ r (1 ⫺

␳

⫹

␳␤

) is below some critical value B (equivalent-

ly, when risk-aversion 1 ⫺ r is less than some

1 ⫺ r), there is a unique, stable, steady state.

When B ⬎ B, on the other hand, there exist

␭

and

␭

៮

with 0 ⬍

␭

⬍

␭

៮

, such that:

(1) For each

␭

in [

␭

៮

] there are (at least) two

stable steady states.

Inequality is lower,

and social mobility higher, under a more

redistributive social contract.

(2) For

␭

⬍

␭

⬎

␭

៮

the steady state is

unique.

Where multiple steady states occur, history

matters. Temporary shocks to the distribution of

wealth (immigration, educational discrimina-

tion, shifts in demand or technology) as well as

to the political system (slavery, voting-rights

restrictions) can permanently move society

from one equilibrium to the other, or more

generally have long-lasting effects on the econ-

omy and the social contract.

This history-dependence contrasts sharply

with traditional politico-economic models,

The intuition behind the proofs is the following. Con

sider f(

␶

) as a function of 1 ⫺

␶

僆 (0, (1 ⫺

␣

␤␥

). Since

D is monotonic, the ﬁrst fraction in (28) is U-shaped in 1 ⫺

␶

. After multiplication by 1 ⫺

␶

the product typically

becomes N-shaped, so that it will have three intersections

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(⌬) ⫺ D

⫺1

(⌬) on Figure

2. Conversely, the last term in (28), reﬂecting effort distor-

tions, tends to make f strictly increasing in 1 ⫺

␶

(at least

where

␶

⬎ 0), and therefore works towards uniqueness.

Speciﬁcally, there are 2 ⱕ n ⱕ 4 stable steady states.

If 1/

␩

is small enough then n ⱕ 3, and if

␣

is also small

enough then n ⫽ 2. See the theorem’s proof in the Appen-

dix.

FIGURE 4. MULTIPLICITY OR UNIQUENESS OF STABLE STEADY STATES

Note: The dotted line corresponds to 1/

␩

⬘⬎1/

␩

111VOL. 90 NO. 1 BE

NABOU: INCOME DISTRIBUTION AND THE SOCIAL CONTRACT

where countries can deviate at most temporarily

from a common steady-state level of inequality

and redistribution (given stable “fundamen-

tals”). In particular, ﬁscal policy operates there

as a stabilizing force on the distribution of

wealth: more inequality today means more re-

distribution, hence less inequality tomorrow.

Here, on the contrary, there emerges in the long

run a negative correlation between inequality

and redistribution, as indeed one observes be-

tween the United States and Europe, or among

advanced countries in general (Rodriguez,

1998).

Also of interest is the predicted negative cor-

relation between inequality and social mobility,

which is consistent with the results obtained by

Robert Erikson and John H. Goldthorpe (1992)

for a sample of 15 developed countries. But

what of the conventional wisdom of the United

States as an exceptionally mobile society? In

fact, most econometric studies of intergenera-

tional income mobility ﬁnd either no signiﬁcant

difference, or even somewhat greater mobility

in European “welfare states”: see Anders Bjo¨rk-

lund and Markus Ja¨ntti (1997a) for a survey.

Indeed, the most extreme forms of social im-

mobility at the lower end, such as urban ghettos

or the persistence of welfare dependency across

generations, do seem more exacerbated in

American society. Things could well be differ-

ent for the middle class, so a more satisfactory

comparison across countries would take into

account such nonlinearities in the mobility pro-

cess (e.g., Suzanne J. Cooper et al., 1994).

These remain beyond the scope of the present

model and of most existing comparative studies.

Having demonstrated how the sustainabil-

ity of different societal choices depends on the

importance of risk aversion and credit con-

straints (both summarized in B), as well as on

the allocation of power in the political system

(

␭

), I now consider the role of income and

endowment shocks (v

and w

) and that of tax

distortions (1/

␩

). Due to the complexity of the

problem, their effects on the multiplicity

threshold B are studied under additional as-

sumptions.

PROPOSITION 9: Let 1/

␩

⫽ 0. The efﬁciency

threshold for multiplicity B is a decreasing

function of v /w, with lim

v/w3 0

(B) ⫽⫹⬁and

lim

v/w3 ⫹⬁

(B) ⫽ 0.

This result appears on Figure 4. The intu-

ition is that income uncertainty interacts with

market incompleteness in generating efﬁ-

ciency gains from redistribution, as reﬂected

by the term Bv

in (16). For a given B,

multiplicity therefore occurs when v

is large

enough compared to the variance w

of the

shocks which agents learn prior to choosing

policy. The concrete implications are impor-

tant: depending on their source, changes in

the economic environment which have similar

short-run effects on income inequality can

bring about radically different evolutions of

the social contract. Thus, an increase in the

variability of sectoral shocks (similar to v

)

may lead to an expansion of the welfare state

and public education. Conversely, a surge in

immigration which results in a greater hetero-

geneity of the population (similar to a rise in

) may lead to cutbacks or even a large-

scale dismantling. In the ﬁrst case the policy

response will mitigate the shock’s impact on

In Persson and Tabellini (1994) income inequality,

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

allows any distribution of initial endowments to persist

indeﬁnitely. Incorporating idiosyncratic shocks would nor-

mally lead to a unique steady-state distribution. Uniqueness

also obtains in Perotti (1993) and Saint-Paul and Verdier

(1993). In Saint-Paul and Verdier (1992) greater inequality

again results in higher taxes and spending on public educa-

tion (which is problematic in view of the empirical evi-

dence), but this stabilizing effect is more than offset by a

divergence in the incentives of rich and poor to invest

privately in additional human capital. Hence two possible

steady states: high (low) inequality and public education

expenditures, with private accumulation by the rich only (by

both classes).

Multiple steady states due to a negative impact of

inequality on redistribution is the distinguishing feature

which the present model shares with Saint-Paul (1994).

Consistent with the general argument that this entails redis-

tributions which increase the size of the pie, Saint-Paul’s

model has the property that transfers raise aggregate income

in the long run.

For instance, Kenneth A. Couch and Thomas A. Dunn

(1997) ﬁnd greater mobility—especially in terms of educa-

tion—in Germany than in the United States, and Bjo¨rklund

and Ja¨ntti (1997b) ﬁnd similar results for Sweden. Aldo

Rustichini et al. (1999), on the other hand, ﬁnd lower

mobility in Italy than in the United States.

112 THE AMERICAN ECONOMIC REVIEW MARCH 2000

long-run inequality, while in the second it

will aggravate it.

Just as greater beneﬁts of redistribution in-

crease the likelihood of multiplicity, greater

distortions reduce it. This is also illustrated

on Figure 4, where the efﬁciency threshold B

shifts up as 1/

␩

rises. The formal proposition

is somewhat more complicated, as it is only

for positive values of

␶

that labor-supply dis-

tortions rise with 1/

␩

I shall not present it

here due to space constraints, and because a

somewhat similar result appears in Theorem 2

below. Basically, when regressive ﬁscal pol-

icy is ruled out, or more generally for econ-

omies that operate in a region where

␶

⬎ 0,

distortions do rise monotonically with 1/

␩

and the scope for multiple regimes corre-

spondingly declines.

B. Growth Implications of Different

Social Contracts

The steady states corresponding to two dif-

ferent social contracts are clearly not Pareto

rankable. How do they compare in terms of

aggregate performance? Recall from (15) that

the effect of

␶

on short-run growth reﬂects the

trade-off between tax distortions and credit-

constraint effects; taking limits shows that this

remains true for long-run income. Moreover,

any comparison of long-run levels is easily

transposed to long-term growth rates, through

knowledge spillovers or public goods comple-

menting private investment. For instance, let

␬

in (6) be replaced by (

␬

)

␾

, where the human or

physical capital aggregate

(29)

␬

⬅

冉

冕

共k

兲

␥

冊

␥

captures external effects of the economic envi-

ronment on accumulation, other than those of

policy. As

␬

does not enter into the determina

tion of the politico-economic equilibrium (⌬

␶

), all previous results remain unchanged, with

␬

simply replaced by

␬

everywhere. The pres

ence of the spillover only affects the growth rate

along each equilibrium trajectory, transforming

for instance ﬁnite steady states (when 0 ⱕ

␾

⬍

1 ⫺

␣

⫺

␤␥

) into endogenous growth paths

(when

␾

⫽ 1 ⫺

␣

⫺

␤␥

The following

results therefore apply equally to short- and to

long-run economic growth.

PROPOSITION 10: A more redistributive so-

cial contract

(1) has higher income growth when 1/

␩

⫽

␣

⫽

0 and

␤␥

⬍ 1;

(2) has lower income growth when 1/

␩

⬎ 0,

␣

⫽ 0 and

␤␥

⫽ 1.

Since both conditions are compatible with

Theorem 1’s requirement that 1 ⫺

␣

⬍ 2

␤␥

they allow the comparison of steady states cor-

responding to different, self-sustaining values

␶

. Two interesting empirical scenarios can

thus be accounted for by the model.

Case 1: Growth-enhancing redistributions.—

The fact that all equilibria have (endogenously)

the same savings rate makes clear that the faster

growth under the more redistributive social

contract arises from a more efﬁcient allocation

of investment expenditures.

Tax distortions,

meanwhile, remain relatively small. This sce-

nario is particularly relevant for human capital

investment (which is considered in more detail

in the next section) and public health expendi-

tures, where the contrasted paths followed by

East Asia and Latin America come to mind.

Where effort is distorted by regressionary taxes and

transfers, on the contrary, a rise in

␶

represents an efﬁciency

gain which increases with 1/

␩

: recall that C(

␶

) is maxi-

mized for

␶

⫽ 0.

Because it aggregates individual contributions with

the same elasticity of substitution as total output,

␬

heterogeneity neutral, in the sense that it does not introduce

any additional effects of income distribution on growth. It

just makes more permanent those due to imperfect credit

markets, by reducing or even eliminating the “convergence”

term ⫺(1 ⫺

␣

⫺

␤␥

)lny

from (15). Alternative constant

elasticity of substitution (CES) aggregates with elasticities

other than 1/(1 ⫺

␥

) could easily be dealt with, as in

Be´nabou (1996b).

The equality of investment rates is true in the inﬁnite-

horizon version of the model as well. A higher

␶

implies a

lower private savings rate, but this is exactly offset by a

higher equilibrium rate of consumption taxation and invest-

ment subsidization.

113VOL. 90 NO. 1 BE

NABOU: INCOME DISTRIBUTION AND THE SOCIAL CONTRACT

More generally, it offers a potential explana-

tion, in a context of endogenous policy choice,

for the fact that regression estimates of the

effects of social and educational transfers on

growth are often signiﬁcantly positive.

Case 2: Eurosclerosis and the welfare

state.—In this converse case, the credit-constraint

effect is weak compared to the tax distortions.

European countries, it is often argued, have cho-

sen a higher degree of social insurance and com-

pression of inequalities than the United States, at

the cost of higher unemployment and slower

growth.

Whether this is viewed as enlightened

policy or dismal “Eurosclerosis,” it begs the ques-

tion of why voters on both sides of the Atlantic

would choose such different points on the equity-

efﬁciency, or insurance-growth, trade-off. In my

model, Europeans choose more redistribution than

Americans not because they are intrinsically more

risk averse, but because in more homogeneous

societies there is less erosion of the consensus

over social insurance mechanisms which, ex ante,

would be valued enough to compensate for lesser

growth prospects.

Consider ﬁnally average welfare, which here

is also that of the median voter; see (16). Since

multiplicity requires some political bias (

␭

⬎ 0)

it is clear from (21) that, in each steady state, a

marginal increase in redistribution would raise

兰

di. This corresponds to the requirement,

in the stylized model of Section I, of an aggre-

gate gain from the policy

P relative to P.

Comparing steady states, on the other hand,

involves discrete variations in

␶

. When tax

distortions are small enough (1/

␩

is low), a

more redistributive steady state does have

higher total welfare; but in general this need not

be the case.

VI. Explaining International Differences

in Education Finance

A. Alternative Systems of School Funding

Education ﬁnance provides perhaps the most

compelling case of a redistributive policy with

positive efﬁciency implications. Loan market

imperfections are more likely to affect invest-

ment in human than in physical capital, which

can serve as collateral. The same is true for

decreasing returns. The ﬁnancing of elementary

and secondary education also constitutes a strik-

ing example of international differences in re-

distributive policy. Japan and most European

countries have state-funded public schools,

which essentially equalize expenditures across

pupils. The United States, in contrast, relies in

large part on local ﬁnancing; because commu-

nities are heavily income segregated, expendi-

tures reﬂect parental resources to a large extent,

making education a quasi-private good. In Be´n-

abou (1996b) I show how a move from local to

state funding of schools can raise the economy’s

long-run income, or even its long-term growth

rate. Calibrating a model with local funding to

U.S. data, Ferna´ndez and Rogerson (1998) ﬁnd

that a move to state ﬁnance could raise steady-

state GDP by about 3 percent. Whether or not

one subscribes to this view, differences in na-

tional education systems which persist for over

a century represent a puzzle.

To examine this issue, let k

now speciﬁcally

represent human wealth. The term (k

)

␣

in (6)

captures the transmission of human capital or

ability within the family, while the shocks

␰

t⫹1

represent the unpredictable component of innate

talent. Finally, instead of progressive taxes and

transfers I now consider progressive subsidies

to educational investment. Thus (9) is replaced

by yˆ

⫽ y

, while in (6) parental savings e

are

replaced by the net (after-tax) resources in-

vested in their child’s education, namely:

(30) eˆ

⫽ e

共y˜

兲

␶

with y˜

still determined by (10). Because agents

It is only in recent years that European growth has

fallen short of U.S. growth, but unemployment has been

higher in Europe for nearly two decades.

In Glomm and Ravikumar (1992) private ﬁnance of

education leads to higher long-run growth than public fund-

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-

man (1997) study similar issues in a model with endogenous

labor supply, then examine voters’ choice among different

funding regimes. They obtain a unique equilibrium.

114 THE AMERICAN ECONOMIC REVIEW MARCH 2000

will still choose a common savings rate, the

government’s budget constraint, which is now

(31)

冕

共eˆ

⫺ e

兲 di ⫽ 0,

will again be satisﬁed. The progressivity rate

␶

is the elasticity of the tax price of education

with respect to wealth. Given agents’ savings

behavior, e

⫽ sy

, it also measures the extent

to which education is publicly and equally

provided: thus

␶

⫽ 0 corresponds to private

ﬁnance, while

␶

⫽ 1 is equivalent to a Euro-

pean-style system where universal public ed-

ucation is funded by a proportional income

tax.

PROPOSITION 11: Given a rate of education

ﬁnance progressivity

␶

, agents in generation t

choose a common labor supply and savings

rate: e

/yˆ

⫽

␳␤

/(1 ⫺

␳

⫹

␳␤

) ⬅ s, as before,

while:

⫽

冉

␦

␩

冊

␩

冉

1 ⫺

␳

⫹

␳␤

共1 ⫺

␶

兲

1 ⫺

␳

冊

␩

The effort distortion is smaller than with ﬁscal

redistribution, because the education-based pol-

icy leaves untouched the part of their income

which adult agents consume. Up to that differ-

ence in l

⫽ l(

␶

), individual and aggregate

wealth dynamics are exactly identical to (11)–

(15), and the resulting ex ante utility function

resembles closely the one which arose under

ﬁscal redistribution.

PROPOSITION 12: Given a rate of education

ﬁnance progressivity

␶

, agent i’s intertemporal

welfare is:

(32) U

⫽ u៮

⫹ A共

␶

兲共ln k

⫺ m

兲 ⫹ C共

␶

兲

⫺

␳␤

共1 ⫺

␶

兲

␥

⌬

⫹ B共

␶

兲v

/2,

where u៮

is independent of the policy

␶

and

(33) A共

␶

兲 ⬅

␳␣

⫹ 共1 ⫺

␳

⫹

␳␤

共1 ⫺

␶

兲兲

␥

(34) C共

␶

兲 ⬅ 共1 ⫺

␳

兲共

␦

ln l共

␶

兲 ⫺ l共

␶

兲

␩

兲

⫹

␳␤␦

ln l共

␶

兲

(35) B共

␶

兲 ⬅ ⫺共1 ⫺

␳

⫹

␳␤

共1 ⫺

␶

兲

⫹ r共1 ⫺

␳

⫹

␳␤

共1 ⫺

␶

兲兲

Two differences with Proposition 4 are worth

mentioning. On the cost side, C(

␶

) is the

same function of effort l(

␶

) as before, but l(

␶

)

is now different. In particular, distortions re-

main bounded, so U

may be maximized at

␶

⫽ 1 for a poor enough agent. The other

difference occurs in the beneﬁts term. For

␳

⫽ 1, B(

␶

) ⫽⫺

␤

(1 ⫺ r

␤

)(1 ⫺

␶

)

as in

(16), and with the same interpretation in

terms of insurance and reallocation of liquid-

ity-constrained investments. But in general

␶

) is no longer proportional to ⫺(1 ⫺

␶

)

and when r ⬎ 0 it is not even monotonic in

␶

While the education model is sufﬁciently

close to the tax model to ensure that the

political equilibrium remains qualitatively

similar, the formal analysis is made more

difﬁcult by these differences. Moreover, de-

riving an exact analogue to Theorem 1 would

be repetitious. I shall therefore focus instead

on a simpler case, which yields new and more

explicit comparative statics results.

B. Sustainability of Centralized and

Decentralized Education Systems

From here on, policy is restricted to two

options. Under laissez-faire or decentralized

funding,

␶

⫽ 0, education expenditures are

determined by family or community resourc-

es; the two are essentially equivalent when

communities are stratiﬁed by socioeconomic

status. Public funding of education corre-

sponds to

␶

⫽ 1 or more generally to

␶

⫽

␶

៮,

where 0 ⬍

␶

៮ ⱕ 1. Given an initial distribution

of human capital ⌬

, this system is adopted if

(

␶

៮ ) ⬎ U

(0) for at least a fraction p* ⬅

115VOL. 90 NO. 1 BE

NABOU: INCOME DISTRIBUTION AND THE SOCIAL CONTRACT

⌽(

␭

) of the population. Setting ln k

⫺ m

⫽

␭

⌬

in (32), this means:

(36)

␭

⬍

冉

共B共

␶

៮ 兲 ⫺ B共0兲兲

冉

冊

⫺ 共C共0兲

⫺ C共

␶

៮ 兲兲

冊

␳␤␶

៮

␥

⌬

⫹

冉

2 ⫺

␶

៮

冊

␥

⌬

Intuitively, the political inﬂuence of wealth must

not be too large compared to the aggregate wel-

fare beneﬁt of redistributive education ﬁnance

(relative to laissez-faire). Preexisting inequality

raises the hurdle which public policy must over-

come, as reﬂected by the term 1/

␥

⌬

multiplying

the net welfare beneﬁt. This effect tends to make

adoption of state ﬁnance more difﬁcult where it

has not previously been in place (e.g., the United

States), because of the greater human capital dis-

parities which result over time from a decentral-

ized system. Conversely, the term in

␥

⌬

incorporates the combined effects of skewness

and credit constraints, which both intensify the

demand for redistribution. As a result of these

offsetting forces the right-hand side of (36) has the

usual U shape in ⌬

, 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 ⌬

with ⌬

⬁

⫽

␶

៮), given by (27). Public funding of education

(partial or complete) is thus a steady state when:

(37)

␭

⬍

冉

共B共

␶

៮ 兲 ⫺ B共0兲兲

冉

冊

⫺ 共C共0兲 ⫺ C共

␶

៮ 兲兲

冊

⫻

冉

␳␤␶

៮

␥

D共

␶

៮ 兲

冊

⫹

冉

2 ⫺

␶

៮

冊

␥

D共

␶

៮ 兲 ⬅

␭

៮

Conversely, private or local ﬁnancing is a

steady state, with inequality ⌬

⬁

⫽ D(0), when:

(38)

␭

⬎

冉

共B共

␶

៮ 兲 ⫺ B共0兲兲

冉

冊

⫺ 共C共0兲 ⫺ C共

␶

៮ 兲兲

冊

⫻

冉

␳␤␶

៮

␥

D共0兲

冊

⫹

冉

2 ⫺

␶

៮

冊

␥

D共0兲 ⬅

␭

The two regimes can coexist if and only if

␭

⬍

␭

៮

, which occurs when the differential gain

(39) B共

␶

៮ 兲 ⫺ B共0兲 ⫽

␳␤␶

៮ 关2 ⫺

␶

៮

⫺ r共2 ⫺ 2

␳

⫹

␳␤

共2 ⫺

␶

៮ 兲兲兴

exceeds the differential cost

(40) C共0兲 ⫺ C共

␶

៮ 兲 ⫽

␦

␩

冋

共1 ⫺

␳

⫹

␳␤

兲 ln

冉

1 ⫺

␳

⫹

␳␤

1 ⫺

␳

⫹

␳␤

共1 ⫺

␶

៮ 兲

冊

⫺

␳␤␶

៮

册

by a sufﬁcient amount, speciﬁed below. It is

easily seen that the distortion C(0) ⫺ C(

␶

៮)is

positive and increasing 1/

␩

. I will assume that

␶

៮) ⬎ B(0), so that there is an actual gain to

compare to this cost. Such is the case provided

agents care enough about insurance (1 ⫺ r ⱖ

1) or about investment in their children’s edu-

cation (

␳

(2 ⫹

␤

) ⱖ 1), or alternatively provided

␶

៮ is not too large.

THEOREM 2: If the gain in ex ante welfare

from progressive public ﬁnancing of education,

relative to private or decentralized ﬁnancing, is

large enough, namely

共

␶

៮ 兲 ⫺ B共0兲 ⱖ 共C共0兲 ⫺ C共

␶

៮ 兲兲共2/v

兲

⫹ G共

␶

៮ , v

兲 ⬅ B,

where G(

␶

៮, v

) ⬎ 0 is given in the Appen

dix, then there exist 0 ⬍

␭

⬍

␭

៮

such that:

(1) For each

␭

in [

␭

៮

] both public and de-

centralized school funding are stable steady

states. The ﬁrst regime has lower inequality

and greater social mobility than second.

(2) For

␭

⬍

␭

public funding is the only steady

state, while for

␭

⬎

␭

៮

it is decentralized

funding.

The efﬁciency threshold for multiplicity B

decreases with income uncertainty v

and in

creases with endowment variability w

, with

116 THE AMERICAN ECONOMIC REVIEW MARCH 2000

lim

v/w3 0

(B) ⫽⫹⬁. It rises with the labor-

supply elasticity 1/

␩

These results demonstrate how such different

systems of school ﬁnance as those of the United

States and Western European countries can be

self-perpetuating, once arisen from historical

circumstances.

As to which one leads to faster

growth, this depends once again on the trade-off

between the positive impact of redistributive

education ﬁnance on wealth constraints and its

adverse effect on incentives: Proposition 10 ap-

plies unchanged. Because the distortions are

less severe than with ﬁscal policy, however, the

likelihood that state funding enhances growth is

now greater.

Theorem 2 also makes clear show how the

efﬁciency threshold for multiplicity B (or equiv-

alently, the minimal risk aversion 1 ⫺ r) varies

with the costs of redistributive school ﬁnance,

as well as with different sources of income

inequality. These results can again be repre-

sented on Figure 4, with B ⬅ B(

␶

៮) ⫺ B(0) now

on the vertical axis, and the same interpretations

as for Proposition 9. The only difference is that

for B ⬍ B there might be no steady state, as

␭

៮

is then less than

␭

. The economy can instead be

shown to cycle between the two regimes, as in

Gradstein and Justman (1997).

Actual in-

stances of such cycling are hard to come by, and

indeed Theorem 1 indicates that nonexistence is

largely an artifact of restricting the policy

␶

discrete values.

Generalizing Theorem 2 to any policy pair {

␶

៮} with

0 ⱕ

␶

⬍

␶

៮ ⱕ 1 is straightforward.

Equations (37)–(38) show that for

␭

ⰻ [

␭

៮

␭

] there is

a unique (and intuitive) steady state, but for

␭

僆 [

␭

៮

␭

] there

is none. The model of Gradstein and Justman (1997) cor-

responds to the case where 1 ⫺ r ⫽ 1 (time-separable,

logarithmic utility), the disutility of effort ⫺l

␩

is replaced

by ln (1 ⫺ l ),

␶

僆 {0, 1} (pure public or private system),

␭

⫽ 0 (pure democracy), and v

⫽ 0 (no uncertainty at the

time of voting). From Theorem 2 we see that the restrictions

␭

⫽ 0 and v

⫽ 0 preclude multiple equilibria.

With a continuous

␶

the analogue of equation (28) for

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-

tributive schemes used in the paper, namely (9) and (30), are

not driving the results: for

␶

⫽ 0 and

␶

⫽

␶

៮ ⬅ 1 the

FIGURE 5. THE NUMBER OF STEADY STATES

Note: The dotted lines correspond to 1/

␩

⬘⬎1/

␩

117VOL. 90 NO. 1 BE

NABOU: INCOME DISTRIBUTION AND THE SOCIAL CONTRACT

I now return to the central issue, namely the

coexistence of multiple regimes. The range of

political systems which allow this indetermi-

nacy is illustrated on Figure 5.

PROPOSITION 13: The scope for the political

system to generate multiple equilibria increases

with the efﬁciency beneﬁts of redistribution and

decreases with their efﬁciency costs: as B ⬅

␶

៮) ⫺ B(0) increases, due to greater risk aver-

sion or more decreasing returns, both

␭

and

␭

៮

rise

but the interval [

␭

៮

] widens. A higher labor-

supply elasticity 1/

␩

has the opposite effects.

VII. Other Applications

A. Concern for Equity

Apart from social insurance and capital market

imperfections, one of the main reasons for income

redistribution is simply that most people dislike

living in a society which is too unequal. This may

be due to pure altruism or to the fact that inequal-

ity generates social tensions, crime, and similar

problems which have direct costs. To capture

these ideas one can simply augment (7) as fol-

lows:

(7⬘)ln

⫽ ln V

⫺共A/2兲

⫻关共1 ⫺

␳

兲var

j 僆 I

关ln c

⫺共l

兲

␩

兴

⫹

␳

var

j 僆 I

关ln k

t ⫹ 1

兴兴.

The coefﬁcient A represents everyone’s aversion

to disparities in felicity, measured by the cross-

sectional variances of consumption (including lei-

sure) and bequests. It is easily seen that the

economy’s laws of motion remain unchanged and

the political equilibrium quite similar. In an en-

dowment economy (

␤

⫽ 0), for instance, the only

difference is that the ubiquitous

␥

⌬

⫹ Bv

replaced by (1 ⫹ A)

␥

⌬

⫹ (B ⫹ A)v

: inequal

ity aversion is equivalent to a simultaneous in-

crease in risk aversion and in the concavity of

aggregate welfare (previously logarithmic). Thus,

with 1/

␩

⫽ 0 the steady-state tax rate becomes

(25⬘)

1 ⫺

␶

⬁

⫽

␭

冋

共1 ⫹ A兲

冉

␥

冑

1⫺

␣

冊

⫹ (1⫹A⫺共1⫺

␳

兲r兲

冉

冊

冉

冑

1⫺

␣

␥

冊

册

If one observed two countries, the ﬁrst with low

pretax inequality yet extensive redistribution,

the other with the reverse situation, one would

indeed be tempted to conclude that the citizens

of the ﬁrst country were more altruistic, or their

poor better organized politically. In fact it could

be that preferences are identical and political

institutions equivalent (same A and

␭

), but that

the second country’s more unequal distribution

reﬂects a more persistent income process. This

could be due to exogenous factors, as with

␣

here, or be endogenous, as in the case of mul-

tiple steady states.

B. The Mix of Public Goods

Some public services such as the legal sys-

tem, the protection of property, prisons, etc.,

beneﬁt citizens largely in proportion to their

levels of wealth or investment. Others have

more uniformly or even regressively distributed

beneﬁts. Klaus Deininger and Lyn Squire

(1995) ﬁnd in cross-country regressions that

public investment affects income growth

equally for all quintiles, while public schooling

beneﬁts the bottom 40 percent most, the middle

class to a lesser extent, and the rich not at all.

Consider therefore a government choosing (at

the margin) a single public good or service from

a menu of options: if g

is spent on a good with

characteristics (

␬

␣

␤

␥

), the private sector

faces the accumulation technology k

t⫹1

⫽

␬␰

t⫹1

)

␣

)

␤

( g

)

␥

. A public good or institu

tion with high

␣

⫹

␤

makes wealth more per-

sistent, so relatively well-off agents may prefer

it to an alternative which has higher overall

productivity (larger

␬

␣

⫹

␤

⫹

␥

). The

problem is thus analogous to the earlier ones,

implying that countries can sustain different

choices without any underlying differences in

tastes. In reality, many public goods are pro-

vided simultaneously and the debate is over the

appropriate mix, but the same intuitions should

remain applicable.

geometric speciﬁcation and the standard linear one coincide

exactly.

118 THE AMERICAN ECONOMIC REVIEW MARCH 2000

C. The Socioeconomic Structure of Cities

The presence in human capital accumulation of

peer effects, role models, and other neighborhood

interactions implies that residential stratiﬁcation

increases the persistence of income disparities

across families (e.g., Be´nabou, 1993; Durlauf,

1996a). Urban ghettos are but the most extreme

example of this phenomenon, which is the subject

of a large empirical literature.

Moreover, I show

in Be´nabou (1993, 1996b) that equilibrium segre-

gation generally tends to be inefﬁciently high. The

two conditions identiﬁed in this paper for multiple

politico-economic regimes are thus again satisﬁed,

leading to the following predictions: (a) more

highly segregated cities are also those where so-

cioeconomic disparities are greater; (b) in such

cities public policy (on housing, schooling, trans-

port, or infrastructure) will tend to accommodate

and even facilitate segregation, while in better

integrated (and more equal) cities more public

support and resources will be mobilized to prevent

further polarization.

VIII. Conclusion

In this paper I have asked how countries

with similar preferences and technologies, as

well as equally democratic political systems,

can nonetheless make very different choices

with respect to ﬁscal progressivity, social in-

surance, and education ﬁnance. The proposed

answer is a simple theory of inequality and

the social contract, based on two mechanisms

which arise naturally in the absence of com-

plete insurance and credit markets. First, re-

distributions which would increase ex ante

welfare command less political support in an

unequal society than in a more homogeneous

one. A lower rate of redistribution, in turn,

increases inequality of future incomes due to

wealth constraints on investment in human or

physical capital. This leads to two stable

steady states, the archetypes for which could

be the United States and Western Europe: one

with high inequality yet low redistribution,

the other with the reverse conﬁguration.

These two societies are not Pareto rankable,

and which one has faster income growth

depends on the balance between tax distor-

tions to effort and the greater productivity of

investment resources (particularly in educa-

tion) reallocated to more severely credit-

constrained agents.

These ideas were formalized in a stochastic

growth model with missing markets, progres-

sive ﬁscal or education ﬁnance policy, and a

more realistic political system than the stan-

dard median voter setup. The resulting distri-

butional dynamics remain simple enough to

allow a number of extensions. In Be´nabou

(1999) I develop and calibrate an inﬁnite-

horizon version of the incomplete markets

model, then use it to quantify the effects of

ﬁscal and educational redistribution on

growth, risk, and welfare. Another interesting

problem is to endogenize the kind of wealth-

biased political mechanism used here, where

those with more resources command more

inﬂuence; Franc¸ois Bourguignon and Verdier

(1997) and Rodriguez (1998) are recent ex-

amples of such models. Finally, the original

question of why the social contract differs

across countries, and whether these choices

are sustainable in the long run, remains an

important topic for further research.

Recent references include Cooper et al. (1994), George J.

Borjas (1995), and Giorgio Topa (1997). Christopher Jencks

and Susan E. Mayer (1990) provide an extensive survey of

earlier empirical studies, and Charles F. Manski (1993) a

critical discussion of methodology. Indeed the identiﬁcation of

these social spillovers remains the subject of some contro-

versy; see for instance William N. Evans et al. (1992).

At the heart of this class of models (e.g., Be´nabou,

1996b) lies a general law of motion for human capital of the

form: h

t⫹1

⫽ F(

␰

t⫹1

, h

, L

, H

), where L

and H

are human

capital averages capturing respectively local externalities (e.g.,

peer effects) and economy-wide interactions (production

complementarities, knowledge spillovers, etc.). Where families

have sorted into socioeconomically homogeneous communi-

ties, L

⫽ h

for all i,soh

t⫹1

⫽ F(

␰

t⫹1

, h

, H

). Conversely,

under perfect integration L

⫽ L

, hence h

t⫹1

⫽ F(

␰

t⫹1

, L

, H

). The similarity with the cases

␶

⫽ 0 and

␶

⫽ 1inthe

present model is readily apparent. An alternative source of

multiplicity is explored in Durlauf (1996b), where it is only

when socioeconomic disparities are not too large that rich

families are willing to share with poorer ones the ﬁxed costs of

running a community and its schools.

119VOL. 90 NO. 1 BE

NABOU: INCOME DISTRIBUTION AND THE SOCIAL CONTRACT

APPENDIX

PROOF OF PROPOSITION 2:

Once agent i knows his productivity z

, hence also his pre- and posttax incomes y

⫽ z

)

␥

)

␦

and yˆ

⫽ (y

)

1⫺

␶

( y˜

), his decision problem takes the form:

(A1) ln V

⫽ max

␯

兵共1 ⫺

␳

兲关ln 共共1 ⫺

␯

兲yˆ

兲⫺ l

␩

兴⫹共

␳

/r⬘兲ln E

关共k

t⫹ 1

兲

r⬘

兴兩 k

t⫹ 1

⫽

␬␰

t⫹ 1

共k

兲

␣

共

␯

yˆ

兲

␤

其

⫽ max

␯

兵共1 ⫺

␳

兲 ln 共1 ⫺

␯

兲⫹

␳␤

␯

其 ⫹ max

兵⫺共1 ⫺

␳

兲l

␩

⫹共1 ⫺

␳

⫹

␳␤

兲共1 ⫺

␶

兲

␦

ln l其

⫹

␳

共ln

␬

⫺ 共1 ⫺ r⬘兲w

/2兲⫹ 关

␳␣

⫹ 共1 ⫺

␳

⫹

␳␤

兲

␥

共1 ⫺

␶

兲兴ln k

⫹共1 ⫺

␳

⫹

␳␤

兲关共1⫺

␶

兲 ln z

⫹

␶

ln y˜

兴,

where

␯

⬅ e

/yˆ

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:

Let us start by computing the redistributive scheme’s cutoff level y˜

.Iflnk

⬃ N(m

, ⌬

), then

(4) implies that aggregate income is:

(A2) ln y

⫽ ln E关z

兴 ⫹ ln E关共k

兲

␥

兴 ⫹

␦

ln l

⫽

␥

⫹

␦

ln l

⫹

␥

⌬

/2.

The level of transfers y˜

which satisﬁes the government budget constraint (10) is then given by:

␶

ln y˜

⫽ ln y

⫺

␦

共1 ⫺

␶

兲ln l

⫺ ln E关共z

兲

1⫺

␶

兴 ⫺ ln E关共k

兲

␥

共1 ⫺

␶

兲

兴

⫽ ln y

⫺

␦

共1 ⫺

␶

兲ln l

⫹ 共共1 ⫺

␶

兲⫺共1 ⫺

␶

兲

兲v

/2 ⫺ 共共1 ⫺

␶

兲

␥

⫹共1 ⫺

␶

兲

␥

⌬

/2兲,

since the z

’s and k

’s are independent. Thus:

(A3)

␶

ln y˜

⫽

␥␶

⫹

␦␶

ln l

⫹

␶

共2 ⫺

␶

兲

␥

⌬

/2⫹

␶

共1 ⫺

␶

兲v

/2,

as claimed in (12). Equation (14) follows from taking variances in (11), while (13) follows from taking

averages with

␶

ln y˜

replaced by (A3). Finally, combining both laws of motions with (A2) yields:

ln y

t⫹ 1

⫽

␦

ln l

t⫹ 1

⫹

␥

关共

␣

⫹

␤␥

兲m

⫹

␤␦

ln l

⫹

␤␶

共2 ⫺

␶

兲共

␥

⌬

⫹ v

兲/2 ⫹ ln 共

␬

␤

兲⫺ 共w

⫹

␤

兲/2兴

⫹

␥

关共

␣

⫹

␤␥

共1⫺

␶

兲兲

⌬

⫹

␤

共1 ⫺

␶

兲

⫹ w

兴/2

⫽

␥

共ln

␬

⫹

␤

ln s ⫺ 共1 ⫺

␥

兲w

/2兲⫹

␦

共ln l

t⫹ 1

⫺

␣

ln l

兲⫹ 共

␣

⫹

␤␥

兲关

␥

⫹

␦

⫹

␥

⌬

/2兴

⫺

␤␥

关1 ⫺

␶

共2 ⫺

␶

兲⫺

␤␥

共1 ⫺

␶

兲

兴v

⫺关

␣

⫹

␤␥

⫺共

␣

⫹

␤␥

共1 ⫺

␶

兲兲

⫺

␤␥␶

共2 ⫺

␶

兲兴

␥

⌬

⫽ ln

␬

˜ ⫹

␦

共ln l

t ⫹ 1

⫺

␣

ln l

兲 ⫹ 共

␣

⫹

␤␥

兲ln y

⫺

␤␥

共1 ⫺

␤␥

兲共1⫺

␶

兲

/2 ⫺ L

⌬

(

␶

)

␥

⌬

/2,

hence the result, given the deﬁnitions of ln

␬

˜, L

(

␶

) and L

⌬

(

␶

PROOF OF PROPOSITION 4:

Substituting the optimal l

⫽ l

and

␯

⫽ s into (A1), and denoting ln

␬

⬘⬅ln

␬

⫺ (1 ⫺ r⬘)w

/2, yields:

120 THE AMERICAN ECONOMIC REVIEW MARCH 2000

(A4) ln V

⫽

␳

␬

⬘ ⫹ 共1 ⫺

␳

兲 ln 共1 ⫺ s兲⫹

␳␤

ln s ⫺ 共1 ⫺

␳

兲l

␩

⫹ 共1 ⫺

␳

⫹

␳␤

兲共1 ⫺

␶

兲

␦

ln l

⫹关

␳␣

⫹ 共1⫺

␳

⫹

␳␤

兲

␥

共1 ⫺

␶

兲兴ln k

⫹ 共1 ⫺

␳

⫹

␳␤

兲关共1 ⫺

␶

兲 ln z

⫹

␶

ln y˜

兴.

Thus conditional on k

,lnV

is normally distributed, with variance (1 ⫺

␳

⫹

␳␤

)

(1 ⫺

␶

)

. This

implies:

(A5) U

⬅

ln E关共V

兲

兩k

兴 ⫽ E关ln V

兩k

兴 ⫹ r共1⫺

␳

⫹

␳␤

兲

共1 ⫺

␶

兲

/2.

Therefore, to obtain U

one simply needs to replace in (A4) the term in ln z

by ⫺(1 ⫺

␳

⫹

␳␤

)(1 ⫺

␶

)[1 ⫺ r(1 ⫺

␳

⫹

␳␤

)(1 ⫺

␶

)]v

/2. Finally, substituting in the value of

␶

ln y˜

from (A3) yields

the claimed result, with

(A6) u៮

⬅ ln关共1 ⫺ s兲

1 ⫺

␳

␳␤

兴 ⫹

␳

共ln

␬

⫺ 共1⫺ r⬘兲w

/2兲

⫹共

␳␣

⫹ 共1 ⫺

␳

⫹

␳␤

兲

␥

兲m

⫹ 共1 ⫺

␳

⫹

␳␤

兲

␥

⌬

/2.

PROOF OF PROPOSITION 5:

We can rewrite (21) as a second-degree polynomial in x ⬅ 1 ⫺

␶

(A7) P共x兲 ⬅ x

共

␥

⌬

⫹ Bv

兲⫺ 共

␥

共ln k

⫺ m

兲 ⫺

␦

␩

兲x⫺

␦

␩

⫽ 0,

which always has two real roots of opposite sign. Since

␶

ⱕ 1 necessarily, the relevant root is x

⫽ 1 ⫺

␶

⬎ 0, and at that point P⬘(x

) ⬎ 0. It is easy to compute

␶

explicitly, but for comparative

statics one can simply use the implicit function theorem. Since ⭸P/⭸(Bv

) ⬎ 0 ⬎ ⭸P/⭸ln k

and

P⬘(x

) ⬎ 0, the theorem implies that ⭸ x

/⭸ln k

⬎ 0 ⬎ ⭸ x

/⭸(Bv

). Similarly, ⭸ x

/⭸⌬

⬍ 0.

Finally, ⫺⭸P/⭸(1/

␩

) ⫽

␦

(1 ⫺ x), so ⭸ x

/⭸(1/

␩

) has the sign of 1 ⫺ x

⫽

␶

, or equivalently

⭸

␶

/⭸(1/

␩

) has the sign of ⫺

␶

. Finally,

␶

⬎ 0 if and only if x

⬍ 1, which means P(1) ⬎ 0, or

␥

(ln k

⫺ m

) ⬍

␥

⌬

⫹ Bv

PROOF OF PROPOSITION 6:

Let us index agents by their log-wealth,

␪

⬅ ln k

, and denote its cumulative distribution function

by F(

␪

). For any weighting scheme

␻

⫽ g(

␪

), the proportion of votes cast by agents with

␪

ⱕ

␪

(more generally, their total political weight) is G(

␪

)/G(⬁), where G(

␪

) ⬅兰

⫺⬁

␪

g( z) dF(z). For

an ordinal scheme, g(z) ⫽

␻

(F(z)), so G(

␪

) ⫽兰

␪

)

␻

(p) dp. Given the single-crossing condition

satisﬁed by preferences, the agent with log-wealth

␪

* and rank p* ⫽ F(

␪

*) deﬁned by G(

␪

*)/

G(⬁) ⫽

⁄

is clearly pivotal (see Gans and Smart, 1996). In the lognormal case F(

␪

) ⫽⌽((

␪

⫺

m)/⌬), so if we deﬁne

␭

⬅⌽

⫺1

(p*) then

␪

* ⫽ m ⫹

␭

⌬. This wealth level is the same as if the

whole distribution of

␪

⫽ ln k were shifted up by

␭

⌬. Let us now turn to the cardinal scheme g(

␪

) ⫽

␭␪

. Simple derivations show that

(A8) G共

␪

兲 ⬅

冕

⫺⬁

␪

␭

dF共z兲 ⫽ e

␭

共m ⫹

␭

⌬

/2兲

䡠 F共

␪

⫺

␭

⌬

兲,

hence G(

␪

)/G(⬁) ⫽ F(

␪

⫺

␭

⌬

). The whole distribution is thus shifted by

␭

⌬

, and so is the

solution to G(

␪

*)/G(⬁) ⫽

⁄

PROOF OF PROPOSITION 7:

The equilibrium tax rate is the one preferred by the agent with ln k

⫺ m

⫽

␭

⌬

, so claims (1)

and (2) follow directly from the properties of

␶

established in Proposition 5. To establish the third

121VOL. 90 NO. 1 BE

NABOU: INCOME DISTRIBUTION AND THE SOCIAL CONTRACT

claim let us rewrite that agent’s ﬁrst-order condition, ⭸U

/⭸

␶

⫽ 0, in terms of x

⬅ 1 ⫺

␶

. By (A7),

is the unique positive root of the polynomial:

(A9) Q共x兲 ⬅ x

共

␥

⌬

⫹ Bv

兲 ⫺ 共

␥␭

⌬

⫺

␦

␩

兲x ⫺

␦

␩

⫽ 0.

Now, Q⬘(x

) ⬎ 0 and ⭸Q/⭸⌬

⫽ 2x

␥

⌬

⫺

␥␭

x,so⭸ x

/⭸⌬

has the sign of

␭

⫺ 2

␥

⌬

Therefore ⭸

␶

/⭸⌬

⬎ 0 if and only if x

⬎

␭

␥

⌬

. For

␭

ⱕ 0 this is always true, hence

␶

is strictly

increasing in ⌬

. For

␭

⬎ 0, on the other hand, the condition is equivalent to:

⌬

Q共

␭

␥

⌬

兲 ⫽ 共

␭

␥

兲

共

␥

⌬

⫹ Bv

兲 ⫺ 共

␥␭

⌬

⫺

␦

␩

兲共

␭

⌬

␥

兲 ⫺ 共

␦

␩

兲⌬

⬍ 0 N

R共⌬

兲 ⬅ ⫺共

␭

⫹ 4

␦

␩␭

兲⌬

⫹ 2共

␦

␩␥

兲⌬

⫹

␭

␥

⬍ 0.

This second-degree polynomial in ⌬

has two real roots of opposite sign. Denoting ⌬ the positive one

[with, clearly, ⭸⌬/⭸(Bv

) ⬎ 0], we conclude that ⭸

␶

/⭸⌬

⬎ 0 if and only if ⌬

⬎⌬. Thus

␶

indeed U-shaped in ⌬

, and its limiting values at ⌬

⫽ 0 and as ⌬

3 ⬁ are readily obtained from

(A9). Finally, recall that

␶

⬎ 0 if and only

␥

(ln k

⫺ m

) ⬍

␥

⌬

⫹ Bv

; therefore

(A10)

␶

⬎ 0 N

␥

⌬

⫺

␥␭

⌬

⫹ Bv

⬎ 0.

When Bv

⬎

␭

/4 the condition always holds, so

␶

⬎ 0. When Bv

⬍

␭

/4 there is a range [⌬⬘,

⌬⬙] 傺 (0, 2

␭

) such that

␶

⬍ 0 if and only if ⌬

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 ⌬⫽D(

␶

) cuts the curve

␶

⫽ 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⫹1

⫽ D(⌬

, T(⌬

)). A

ﬁxed-point ⌬* ⫽ D(⌬*, T(⌬*)) is stable if and only if (dD(⌬

, T(⌬

))/d⌬

)

⌬⫽⌬*

⬍ 1, or:

(A11) D

共⌬*,

␶

*兲⫹T⬘共⌬*兲D

(⌬*,

␶

*)⬍1,

where

␶

* ⬅ T(⌬*) and a j subscript denotes a jth partial derivative. Now, the function

␶

⫽ T(⌬)

is implicitly given by the ﬁrst-order condition (24), or:

(A12)

␺

共

␶

, ⌬兲 ⬅ 共1 ⫺

␶

兲

冉

␥

⌬

⫹ Bv

␥

冊

⫺

␦

␩␥

冉

␶

1 ⫺

␶

冊

⫺

␭

⌬ ⫽ 0,

therefore T⬘(⌬) ⫽⫺(

␺

)(T(⌬), ⌬) and the stability condition becomes:

(A13) D

(⌬*,

␶

*)⫺

冉

␺

(

␶

*, ⌬*)

␺

(

␶

*, ⌬*)

冊

(⌬*,

␶

*)⬍1.

Next, recall from (28) that f is deﬁned by: f(

␶

) ⫺

␭

⫽

␺

(

␶

, D(

␶

))/D(

␶

), so that f⬘⬍0 if and only

␺

⫹ (

␺

⫺

␺

/D) D⬘⬍0. Finally, D(

␶

) is deﬁned by (27) as the (unique) ﬁxed-point solution

to D(

␶

) ⫽ D(D(

␶

); therefore: D⬘⫽D

/(1 ⫺ D

). Substituting D⬘ and using the fact that

␺

(

␶

⌬*) ⫽ 0 at a steady state, this becomes:

f⬘共

␶

*兲 ⬍ 0 N

␺

共

␶

*, ⌬*兲共1 ⫺ D

(⌬*,

␶

*))⫹

␺

(

␶

*, ⌬*)D

(⌬*,

␶

*)⬍0,

122 THE AMERICAN ECONOMIC REVIEW MARCH 2000

which is the same as (A13), hence the result in terms of the slope of f. Its translation into the

condition that T⬘(⌬*) ⬎ (D

⫺1

)⬘(⌬*) is immediate.

We now come to actually solving for steady states. It will be more convenient here to work with

the variable x ⬅

␤␥

(1 ⫺

␶

) 僆 [0, ⬁). Accordingly, let us deﬁne:

(A14) ⌬共x兲 ⬅

冑

⫹ x

␥

1 ⫺ 共

␣

⫹ x兲

⫽ D共

␶

兲,

and rewrite the equation f(

␶

) ⫽

␭

as:

(A15)

␸

共x兲 ⬅ x

冉

⌬共x兲 ⫹

␥

⌬共x兲

冊

⫺

冉

␤␦

␩␥

⌬共x兲

冊冉

␤␥

⫺ x

冊

⫽

␭␤

A stable steady state is now an intersection of the function

␸

(x) with the horizontal

␭␤

, from below. Since

␸

(0) ⱕ 0 (it equals ⫺⬁ for 1/

␩

⬎ 0, or 0 for 1/

␩

⫽ 0) while

␸

(1 ⫺

␣

) ⫽⫹⬁, for any

␭

⬎ 0 there is

always at least one stable equilibrium x 僆 (0, 1 ⫺

␣

), with 0 ⬍⌬(x) ⬍⫹⬁. Moreover, the total number

of intersections must always be odd, with n intersections from below (stable equilibria), alternating with

n ⫺ 1 intersections from above (unstable equilibria). Multiple intersections (n ⬎ 0) will actually occur,

for some nonempty interval of values of

␭

, if and only if

␸

⵺ is nonmonotonic. Indeed, since

␸

⬘(0) ⬎ 0

[this is easily veriﬁed from (A15)] and

␸

(1 ⫺

␣

) ⫽⫹⬁, nonmonotonicity is equivalent to the property

of having at least one strict local maximum, followed by one strict local minimum, in (0, 1 ⫺

␣

). Given

the boundary values of

␸

, multiple equilibria then occur if and only if

␭

belongs to the range [

␭

៮

], where:

(A16)

再

␭

⬅

␤

⫺1

min 兵

␸

共x兲兩x is a strict local minimum of

␸

共x兲其 ⬎ 0

␭

៮

⬅

␤

⫺1

max 兵

␸

共x兲兩x is a strict local minimum of

␸

共x兲其 ⬎ 0.

That

␭

៮

and

␭

are both always positive follows from the fact that for 1/

␩

⫽ 0,

␸

(x) ⬎ 0 for all x ⬎

0 [see (A15)], while for 1/

␩

⬎ 0, if

␸

(x) ⬍ 0 then

␸

⬘( x) ⬎ 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) ⬍ 0 and

␸

is not monotonic. This, in

turn, would imply that there exists values of

␭

⬍ 0 for which (A15) has at least two solutions. But

such solutions are also intersections of the curves ⌬⫽D(

␶

) and

␶

⫽ T(⌬); the former is always

decreasing, and we saw earlier that, for all

␭

⬍ 0, the latter is always increasing. Multiple

intersections are therefore impossible.

Theorem 1 will now be proved by characterizing the set B ⬅ {B ⱖ 0兩

␸

is nonmonotonic on (0,

1 ⫺

␣

)}, then studying its variations with the parameters v, w, and

␩

LEMMA 2: Let 1 ⫺

␣

⬍ 2

␤␥

. The set B ⬅ {B ⱖ 0兩? x 僆 (0, 1 ⫺

␣

␸

⬘( x) ⬍ 0} is a nonempty

interval of the form B ⫽ (B, ⫹⬁) with B ⬎ 0, or B ⫽ [0, ⫹⬁).

PROOF:

Let us differentiate (A15):

␸

⬘共x兲 ⬅

␸

共x兲

⫹ x⌬⬘共x兲

冉

1 ⫺

␥

⌬

共x兲

冊

⫺

␤␦

␩␥

冋

冉

⫺

␤␥

冊

⌬共x兲

⫺

冉

␤␥

⫺ 1

冊

⌬⬘共x兲

⌬

共x兲

册

⫽ ⌬共x兲 ⫹

␥

⌬共x兲

⫺

冉

␤␦

␩␥

⌬共x兲

冊冉

␤␥

⫺ x

冊

⫹ x⌬⬘共x兲

冉

1 ⫺

␥

⌬

共x兲

冊

⫺

␤␦

␩␥

冋

冉

⫺

␤␥

冊

⌬共x兲

⫺

冉

␤␥

⫺ 1

冊

⌬⬘共x兲

⌬

共x兲

册

123VOL. 90 NO. 1 BE

NABOU: INCOME DISTRIBUTION AND THE SOCIAL CONTRACT

Grouping terms,

␸

⬘( x) ⬍ 0 if and only if:

(A17)

␥

冉

x⌬⬘共x兲

⌬共x兲

⫺ 1

冊

⬎

冉

x⌬⬘共x兲

⌬共x兲

⫹ 1

冊

⌬

共x兲 ⫹

冉

␤␦

␩␥

冊冉

␤␥

⫹

冉

␤␥

⫺ 1

冊

⌬⬘共x兲

⌬共x兲

冊

We now establish the lemma through two intermediate claims.

Claim 1: If (A17) is satisﬁed for some B ⱖ 0 at some x 僆 (0, 1 ⫺

␣

), then x⌬⬘(x)/⌬(x) ⬎ 1. As a

consequence, (A17) is satisﬁed at x for all B⬘⬎B.

PROOF:

If ⌬⬘(x)/⌬(x) ⫺ 1/x ⱕ 0 the left-hand-side of (A17) is nonpositive, so on the right-hand side

it must be that

␤␥

/x ⫺ 1 ⬍ 0. But then:

␤␥

⫹

冉

␤␥

⫺ 1

冊

⌬⬘共x兲

⌬共x兲

⬎

␤␥

⫹

冉

␤␥

⫺ 1

冊

⫽

␤␥

⫺ x

Since x ⬍ 1 ⫺

␣

⬍ 2

␤␥

, this implies that the right-hand side of (A17) is positive, a contradiction.

Claim 2: There exists an xˆ 僆 (0, 1 ⫺

␣

) such that x⌬⬘(x)/⌬(x) ⬍ 1 on (0, xˆ) and x⌬⬘(x)/⌬(x) ⬎

1on(xˆ, ⫹⬁). As a consequence, for any x ⬎ xˆ, (A17) holds for B large enough.

PROOF:

Let us denote from here on

␻

⬅ v /

␥

w. Since ⌬

(x) ⫽ w

(1 ⫹

␻

)/(1 ⫺ (

␣

⫹ x)

(A18)

⌬⬘共x兲

⌬共x兲

⫽

␻

1 ⫹

␻

⫹

␣

⫹ x

1 ⫺ 共

␣

⫹ x兲

Therefore x⌬⬘(x)/⌬(x) ⬎ 1 if and only if

␻

共1 ⫺ 共

␣

⫹ x兲

兲 ⫹ x共

␣

⫹ x兲共1 ⫹

␻

兲 ⬎ 共1 ⫹

␻

兲共1 ⫺共

␣

⫹ x兲

兲 N

␻

共

␣

⫹ x兲 ⫺ 共1 ⫺ 共

␣

⫹ x兲

兲 ⫹ x共

␣

⫹ x兲 ⬎ 0 N

␻

共

␣

⫹ x兲⫹共

␣

⫹ x兲共

␣

⫹ 2x兲⫺ 1 ⬎ 0.

This last expression is clearly increasing in x on (0, 1 ⫺

␣

), from

␣

⫺ 1 ⬍ 0atx ⫽ 0to

␻

(1 ⫺

␣

)

⫹ 1 ⫺

␣

⬎ 0atx ⫽ 1 ⫺

␣

. This proves Claim 2 which, together with Claim 1, establishes that

the set B is a nonempty interval of the form (B, ⫹⬁)or[B, ⫹⬁). Moreover, note that its

complement, ⺢

⫹

⶿B ⬅ {B ⱖ 0兩@ x 僆 (0, 1 ⫺

␣

␸

⬘( x) ⱖ 0}, is a closed set because

␸

⬘ is

continuous in B at every point. This implies that either B ⫽ (B, ⫹⬁) with B ⬎ 0, or else B ⫽ [0,

⫹⬁), and thereby ﬁnishes to establish Lemma 2. From (A17)–(A18), moreover, it is clear that B

depends on v, w, and

␩

only through v/w and

␩

; let it therefore be denoted as B(v/w;

␩

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 ⬍ B(v /w;

␩

), there is a unique, stable, steady state. For B ⬎ B(v/w;

␩

)

the same is true if

␭

ⰻ [

␭

៮

], while for

␭

僆 [

␭

៮

␭

] there are n 僆 {2, 3, 4} stable steady states.

Moreover, if 1/

␩

is small enough then n ⱕ 3, and if

␣

is also small enough then n ⫽ 2.

PROOF:

By deﬁnition, for B ⬍ B(v/w;

␩

) the function

␸

is strictly increasing everywhere, so the steady

124 THE AMERICAN ECONOMIC REVIEW MARCH 2000

state is unique. The same is true for B ⫽ B(v /w;

␩

) ⬎ 0, since then B ⫽ (B, ⫹⬁). The other

measure-zero case, B ⫽ B(v /w;

␩

) ⫽ 0, is too special to be of interest. Now, for B ⬎ B(v /w;

␩

), we saw earlier that there must be n stable equilibria and n ⫺ 1 unstable ones in the interval

(0, 1 ⫺

␣

). To examine what values n can take, rewrite

␸

(x) ⫽

␭␤

as:

(A19) S共x兲 ⬅

冋

冉

⌬

共x兲 ⫹

␥

冊

⫺

冉

␤␦

␩␥

冊

共

␤␥

⫺ x兲

册

⫺关

␭␤

x⌬共x兲兴

⫽ 0.

By (A14), ⌬

(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 ⌬

(x), we therefore obtain

a polynomial S*(x) of degree 2 ⫻ (2 ⫹ 2) ⫽ 8, which can have at most 8 real roots. But we saw

in the discussion following (A15) that

␸

(x) ⫽

␭␤

must have an odd number of solutions on (0, 1 ⫺

␣

), with n intersections from above and n ⫺ 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/

␩

⫽ 0 we can simplify (A19) by x

, leaving for S*(x) only a polynomial of degree 6; this rules

out n ⫽ 4. When, in addition,

␣

⫽ 0, note from (A14) that ⌬(x) depends on x only through x

.Asa

consequence, the sixth-degree polynomial S*(x) is also a polynomial of degree 3 in x

, so it has at most

three real roots. This rules out n ⫽ 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 ⫺

␣

). The preceding results therefore also apply for 1/

␩

and

␣

small enough.

This concludes the proof of Theorem 1.

PROOF OF PROPOSITION 9:

When 1/

␩

⫽ 0 the condition for

␸

⬘( x) ⬍ 0, namely (A17), becomes:

(A20)

␥

⬎

冉

x⌬⬘共x兲/⌬共x兲 ⫹ 1

x⌬⬘共x兲/⌬共x兲 ⫺ 1

冊

⌬

共x兲,

with the requirement that the denominator must be positive. Using (A18), this can be rewritten as

B ⬎

冉

⫹

␻

⫺2

共

␣

⫹ x兲 ⫺

␻

⫺2

共1 ⫺ 共

␣

⫹ x兲共

␣

⫹ 2x兲兲

冊

⫻

冉

共2 ⫺ 共

␣

⫹ x兲共

␣

⫹ 2x兲兲 ⫹

␻

⫺2

共1 ⫺

␣

共

␣

⫹ x兲兲

1 ⫺ 共

␣

⫹ x兲

冊

⬅ ⌫共x,

␻

兲

where

␻

⬅ v /

␥

w. It is easily veriﬁed that the each of the bracketed functions is increasing in

␻

⫺2

therefore:

(A21) B共

␻

兲 ⬅ inf兵⌫共x,

␻

兲兩x 僆共0, 1 ⫺

␣

兲 and x

共

␣

⫹ x兲 ⬎

␻

⫺2

共1 ⫺ 共

␣

⫹ x兲共

␣

⫹ 2x兲兲其

is strictly positive, and decreasing in

␻

. Observe next that, as

␻

tends to inﬁnity, ⌫( x,

␻

) approaches

⌫( x, ⫹⬁) ⫽ x(2 ⫺ (

␣

⫹ x)(

␣

⫹ 2x))/(1 ⫺ (

␣

⫹ x)

), whose inﬁmum value on (0, 1 ⫺

␣

)is

zero; therefore, lim

␻

3 ⫹⬁

␻

) ⫽ 0. Finally, for any x with x

(

␣

⫹ x) ⬎

␻

⫺2

(1 ⫺ (

␣

⫹ x)(

␣

⫹

2x)), note that ⌫(x,

␻

) ⬎

␻

⫺2

/(1 ⫺

␣

), which tends to inﬁnity as

␻

tends to 0. Therefore

lim

␻

3 0

␻

) ⫽⫹⬁.

PROOF OF PROPOSITION 10:

Given lognormality, (29) becomes ln

␬

⫽ m

⫹

␥

⌬

/2 ⫽ (ln y

⫺

␦

ln l

␥

, so substituting

␬

˜ into the growth equation (15) yields:

(A22) ln y

t⫹ 1

⫺ 共

␣

⫹

␤␥

⫹

␾

兲ln y

⫽ ln

␬

៮⫹

␦

共ln l

t⫹ 1

⫺ 共

␣

⫹

␾

兲ln l

兲⫺ L

共

␶

兲v

/2⫺L

⌬

(

␶

)

␥

⌬

/2,

125VOL. 90 NO. 1 BE

NABOU: INCOME DISTRIBUTION AND THE SOCIAL CONTRACT

with ln

␬

៮ ⬅

␤␥

ln s ⫺

␥

(1 ⫺

␥

/2. In a steady state, if

␣

⫹

␤␥

⫹

␾

⬍ 1 the left-hand side equals

(1 ⫺

␣

⫺

␤␥

⫺

␾

) times the output level ln y

⬁

; when

␣

⫹

␤␥

⫹

␾

⫽ 1 it becomes equal to the

asymptotic growth rate, lim

t3 ⫹⬁

ln( y

t⫹ 1

). As to the right-hand side, in a steady state with

␶

⫽

␶

it becomes:

(A23) g

⬁

共

␶

兲 ⬅ ln

␬

៮ ⫹

␤␥␦

ln l共

␶

兲 ⫺ L

(

␶

/2⫺L

⌬

(

␶

)

␥

␶

)

/2.

For

␣

⫽ 0 we saw earlier that L

⌬

(

␶

) ⫽ L

(

␶

) ⫽

␤␥

(1 ⫺

␤␥

)(1 ⫺

␶

)

; therefore L

(

␶

/2 ⫹

⌬

(

␶

)

␥

␶

)

/2 is strictly decreasing in

␶

. Now, with 1/

␩

⫽ 0 the labor-supply term is constant, therefore

⬁

(

␶

) is strictly increasing in

␶

; this proves the proposition’s ﬁrst claim. Conversely, when

␤␥

⫽ 1, then

(

␶

) ⫽ L

⌬

⫽ 0; with 1/

␩

⬎ 0, g

⬁

(

␶

), like ln l(

␶

), is then decreasing in

␶

; hence the second claim.

PROOF OF PROPOSITION 11:

Once agent i knows his productivity z

, hence also his income y

⫽ z

)

␥

)

␦

and his investment

subsidy rate eˆ

⫽ (y˜

)

␶

, his decision problem takes the form:

(A24) ln V

⫽ max

␯

兵共1 ⫺

␳

兲关ln 共共1 ⫺

␯

兲y

兲 ⫺ l

␩

兴⫹ 共

␳

/r⬘兲ln关E

共k

t ⫹ 1

兲

r⬘

兴兩 k

t ⫹ 1

⫽

␬␰

t ⫹ 1

共k

兲

␣

共eˆ

兲

␤

其

⫽ max

␯

兵共1 ⫺

␳

兲 ln 共1 ⫺

␯

兲 ⫹

␳␤

␯

其⫹ max

兵⫺共1 ⫺

␳

兲l

␩

⫹ 共1 ⫺

␳

⫹

␳␤

共1 ⫺

␶

兲兲

␦

ln l其

⫹

␳

共ln

␬

⫺ 共1 ⫺ r⬘兲w

/2兲⫹ 关

␳␣

⫹ 共1 ⫺

␳

⫹

␳␤

共1⫺

␶

兲兲

␥

兴ln k

⫹共1 ⫺

␳

⫹

␳␤

共1 ⫺

␶

兲兲ln z

⫹

␳␤␶

ln y˜

where

␯

⬅ e

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:

Substituting the optimal l

⫽ l

and

␯

⫽ s into (A24) and denoting ln

␬

⬘⬅ln

␬

⫺ (1 ⫺ r⬘)w

yields:

(A25) ln V

⫽

␳

␬

⬘ ⫹ 共1 ⫺

␳

兲ln 共1 ⫺ s兲⫹

␳␤

ln s

⫺共1 ⫺

␳

兲l

␩

⫹ 共1 ⫺

␳

⫹

␳␤

共1 ⫺

␶

兲兲

␦

ln l

⫹关

␳␣

⫹ 共1⫺

␳

⫹

␳␤

共1 ⫺

␶

兲兲

␥

兴ln k

⫹共1 ⫺

␳

⫹

␳␤

共1 ⫺

␶

兲兲ln z

⫹

␳␤␶

ln y˜

Thus conditional on k

,lnV

is normally distributed, with variance (1 ⫺

␳

⫹

␳␤

(1 ⫺

␶

))

. This implies:

(26) U

⬅

ln E关共V

兲

兩 k

兴 ⫽ E关ln V

兩 k

兴⫹ r共1 ⫺

␳

⫹

␳␤

共1 ⫺

␶

兲兲

/2.

Therefore, to obtain U

one simply needs to replace in (A25) the term in ln z

by ⫺(1 ⫺

␳

⫹

␳␤

(1 ⫺

␶

))[1 ⫺ r(1 ⫺

␳

⫹

␳␤

(1 ⫺

␶

))]v

/2. Finally, substituting in the value of

␶

ln y˜

from (A3) yields

the claimed result, with:

(A27) u៮

⬅ ln 关共1 ⫺ s兲

1 ⫺

␳

␳␤

兴 ⫹

␳

共ln

␬

⫺共1 ⫺ r⬘兲w

/2兲

⫹共

␳␣

⫹ 共1 ⫺

␳

⫹

␳␤

兲

␥

兲m

⫹

␳␤␥

⌬

/2.

126 THE AMERICAN ECONOMIC REVIEW MARCH 2000

PROOFS OF THEOREM 2 AND PROPOSITION 13:

By (37) and (38),

␭

៮

⬎

␭

if and only if:

(A28) B共

␶

៮ 兲 ⫺ B共0兲 ⫺ 共C共0兲 ⫺ C共

␶

៮ 兲兲

冉

冊

⬎

␳␤␶

៮

冉

2 ⫺

␶

៮

冊

␥

D共0兲D共

␶

៮ 兲

which yields Theorem 2, with:

(A29) G共

␶

៮ , v

兲 ⬅

␳␤␶

៮ 共2 ⫺

␶

៮ 兲

␥

冉

冑

⫹

␤

1 ⫺ 共

␣

⫹

␤␥

兲

冊冉

冑

⫹

␤

共1 ⫺

␶

៮ 兲

1 ⫺ 共

␣

⫹

␤␥

共1 ⫺

␶

៮ 兲兲

冊

Note that since G(

␶

៮, v

) ⬎ 0, if B(

␶

៮) ⫺ B(0) ⬎ B then it must be that

␭

⬎ 0. Finally, both

␭

and

␭

៮

are clearly increasing in B ⬅ B(

␶

៮) ⫺ B(0) and in ⫺(C(0) ⫺ 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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