Discussion Papers In Economics
And Business

Inequality, Mobility and Redistributive Politics
Ryo Arawatari

Tetsuo Ono

Discussion Paper 09-12-Rev.

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANInequality, Mobility and Redistributive Politics
Ryo Arawatari

Tetsuo Ono

Discussion Paper 09-12-Rev.

February 2010

この研究は「大学院経済学研究科・経済学部記念事業」
基金より援助を受けた、記して感謝する。

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANInequality, Mobility and Redistributive Politics
Ryo Arawatariy

Tetsuo Onoz

Shinshu University

Osaka University

February 3, 2010

Abstract
This paper develops a model where income inequality and intergenerational mobility are jointly determined via redistributive politics. The model includes two key
factors: accessibility of tertiary education for poor-born agents and multiple selfful…lling expectations of agents. Given these factors, the model provides predictionsof cross-country di¤erences in inequality and mobility consistent with empirical observations.
Keywords: Inequality; Intergenerational mobility; Redistribution; Markov perfect political equilibrium; Overlapping generations
JEL Classi…cation: D70; H55; I38.

Arawatari acknowledges …nancial support of the Japan Society for the Promotion of Science (JSPS)
through a Grant-in-Aid for Young Scientists (Start-up) (No. 21830046). Ono acknowledges the …nancial
support of the Japan Society for the Promotion of Science (JSPS) through a Grant-in-Aid for Young
Scientists (B) (No. 20730208).
yFaculty of Economics, Shinshu University, 3-1-1, Asahi, Matsumoto, Nagano, 390-8621 Japan. Email: ryo_a@shinshu-u.ac.jpzGraduate School of Economics, Osaka University, 1-7, Machikaneyama, Toyonaka, Osaka 560-0043,
Japan. E-mail: tono@econ.osaka-u.ac.jp1

1

Introduction

The extent to which economic status passes from one generation to the next is a measure of intergenerational mobility. Mobility aﬀects income inequality within a generation
through changes in the economic status of each agent. In turn, income inequality aﬀects
intergenerational mobility through changes in the incentive to invest in education. Therefore, there is a mutual link between inequality and mobility within and across generations(Hassler, Rodriguez Mora and Zeira, 2007).
Earlier studies have attempted to derive the correlation between inequality and mobility in the presence of ﬁnancial constraints (Banerjee and Newman, 1993; Owen and Weil,
1998; Maoz and Moav, 1999). In these models, the constraint creates a kind of pecuniary
increasing return that results in multiple equilibria, one characterized by low mobility
and high inequality, and the other by high mobility and low inequality. These models
also indicate a negative correlation between inequality and mobility, as shown by Davies,
Zhang and Zeng (2005), who examine the role of education systems in the determinationof inequality and mobility.
[Figure 1 about here.]
However, some empirical evidence shows that the negative correlation appears lessthan exact (Solon, 2002). As shown in Figure 1, a negative correlation is found in someOECD (Organization of Economic Cooperation and Development) countries with low
mobility and high inequality, including Italy, the United Kingdom (UK) and the UnitedStates (US), while Nordic countries are characterized by high mobility and low inequality.
However, despite considerable intergenerational mobility in Australia and Canada, thesecountries also score relatively highly on measures of cross-sectional inequality (Corak,
2006; D’Addio, 2007). This mixed evidence suggests that there is a need to developa theory that fully explains the correlation between inequality and mobility in OECD
countries.
To our knowledge, few papers discuss these cross-country diﬀerences. An exception
is Hassler, Rodriguez Mora and Zeira (2007), who demonstrate two opposing eﬀects: theﬁrst eﬀect is via the education sector, which produces the negative correlation; and thesecond eﬀect is via the production sector, which creates the positive correlation. We canuse these opposing eﬀects to demonstrate the properties of inequality and mobility in theabove-mentioned OECD countries.1
1

The eﬀects of changes in the production and education sectors in Hassler, Rodriguez Mora and Zeira(2007) are as follows. Changes in the production sector, such as skill-biased technological changes, aﬀect
the return to factors of production and thus impact on inequality between the skilled and the unskilled.
Increased inequality implies an incentive for the gains from education and thus strengthens the incentive2

Although Hassler, Rodriguez Mora and Zeira (2007) contribute to explaining crosscountry diﬀerences in inequality and mobility, the following two issues remain unresolved.
First, the link between inequality and mobility via redistributive politics, which is an
important policy issue in many OECD countries, is not included in their model. Second,
they measure inequality with the wage gap rather than the Gini coeﬃcient (which is affected by redistribution). Therefore, the question arises as to how inequality and mobility
are determined via redistributive politics within and across generations when inequality ismeasured by the Gini coeﬃcient in terms of after-tax-and-transfer income. The purposeof the current paper is to present a model that responds to this challenge.2
For the purpose of our analysis, we employ the political economy redistribution modelin Hassler, Storesletten and Zilibotti (2007). We extend this model by introducing intergenerational mobility whereby poor-born agents have a chance of becoming rich via
educational investment, and rich-born agents have a risk of becoming poor if they fail in
education. Furthermore, although poor-born agents have some opportunity for upward
mobility, they have disadvantages in terms of opportunities and the costs of education
relative to rich-born agents. When greater accessibility is ensured for poor-born agents,
the economy attains higher mobility and lower inequality. The parameter representing
accessibility of education is the ﬁrst key factor that accounts for cross-country diﬀerences
in inequality and mobility.
The second key factor is the multiple self-fulﬁlling expectations of agents (Hassler,
Storesletten and Zilibotti, 2007). In undertaking educational investment, young agents
have expectations of redistributive policies in their old age. When young agents hold the
expectation of greater redistribution, it provides a disincentive to engage in educational
investment, thereby resulting in a lower proportion of the rich. This implies a greaternumber of low-income young individuals, which in turn increases future demand for redistribution. The opposite mechanism applies when young agents have the expectationof lower redistribution in old age. Therefore, the economy may attain multiple equilibria,
including a rich-majority and a poor-majority equilibrium. The outcome depends on theto invest in education, which increases mobility. By contrast, changes in the education sector increase
access to education and thus increase mobility. This reduces the number of poor, unskilled workers,
which reduces inequality. Hassler, Rodriguez Mora and Zeira (2007) show that when the education sector
eﬀect is larger (smaller) than the production sector eﬀect, the economy displays a negative (positive)correlation between inequality and mobility.
2
In addition to Hassler, Rodriguez Mora and Zeira (2007), Bernasconi and Profeta (2007) and Ichino,
Karabarbounis and Moretti (2009) undertake analyses of intergenerational mobility and inequality. However, Bernasconi and Profeta (2007) focus on the dynamic motion of intergenerational mobility, and there
is no consideration of cross-country diﬀerences. By contrast, Ichino, Karabarbounis and Moretti (2009)
introduce political institutions into the Becker and Tomes (1979) framework of intergenerational mobility
and show that two societies with similar economic backgrounds may display diﬀerent levels of intergenerational mobility depending on their political institutions. However, the cross-country diﬀerences in income
inequality and mobility are abstracted from their analysis; their focus is on public education rather thanincome redistribution.

3

expectations of agents.
We can explain the cross-country diﬀerences in inequality and mobility using these two
key factors. The ﬁrst key factor, accessibility of education, creates a negative correlation
between inequality and mobility. In the poor-majority equilibrium, greater accessibility ofeducation gives poor-born individuals an incentive to invest in education, thereby resulting
in higher mobility and lower inequality. This prediction ﬁts the empirical evidence of atleast some countries, where Italy, the UK and the US are low-accessibility economies, andNordic countries are high-accessibility economies (Schütz, Ursprung and Wößmann, 2008;
OECD, 2008a)
The rich-majority equilibrium emerges when accessibility lies beyond a critical level.
That is, given greater accessibility of education, the political economy achieves multiple political equilibria. The rich-majority equilibrium attains greater inequality than thepoor-majority equilibrium (representing Nordic countries) because the former features
lower redistribution as supported by the rich. However, both equilibria attain higher
mobility and lower inequality compared with the poor-majority equilibrium with low accessibility (as represented by Italy, the UK and the US). Therefore, in our model economy,
Australia and Canada can be interpreted as representing the rich-majority equilibrium.
Our analysis also contributes to the literature on multiple equilibria in inequality andmobility (Banerjee and Newman, 1993; Owen and Weil, 1998; Maoz and Moav, 1999;
Mookherjee and Napel, 2007). Extant models show that initial conditions matter for thedetermination of the long-run steady state. By contrast, our paper shows that the expectations of agents as well as the initial conditions matter. In particular, we demonstrate
that: (i) two economies sharing the same initial conditions could converge to diﬀerent
equilibria in the long run because the state of the economy depends on the expectationsof agents; and (ii) without any structural change, an economy at one of the equilibria may
move to the other because of changes in the expectations of agents. These results help
provide an explanation for why developed countries sharing similar backgrounds attaindiﬀerent inequality and mobility over time, and why a country experiences changes ininequality and mobility over time, even though its economic structure has changed little.
Apart from this body of work, the current analysis also relates to the literature on inequality and redistributive politics (Fernandez and Rogerson, 1995; Piketty, 1995; Hassler
et al., 2003; Hassler, Storesletten and Zilibotti, 2007; Ono and Arawatari, 2008; Arawatari
and Ono, 2008, 2009). In these studies, the following issues are abstracted from the analysis: a mutual link between inequality and mobility (Fernandez and Rogerson, 1995),
dynamic aspects of inequality and mobility (Piketty, 1995), and intergenerational mobility (Hassler et al., 2003; Hassler, Storesletten and Zilibotti, 2007; Ono and Arawatari,2008, 2009). Our previous work (Arawatari and Ono, 2008, 2009) demonstrates mobility
using the framework in Hassler, Storesletten and Zilibotti (2007). However, the focus
4there is on earning mobility over the life cycle. By contrast, the current paper presents
the dynamic aspects of the correlation between intergenerational mobility and inequalityaﬀected by redistributive politics.
The organization of the paper is as follows. Section 2 develops the model. Section 3
focuses on some period t and provides the characterization of the period−t political equilibrium. Section 4 characterizes the dynamic political equilibrium deﬁned as a sequence of
period−t political equilibriums. Using the characterization of political equilibrium in Sections 3 and 4, Section 5 focuses on the steady state and demonstrates how the accessibilityof education aﬀects inequality, mobility and redistribution. Section 6 provides empiricalimplications of the numerical result. Section 7 provides some concluding remarks.

2

The Model

The model is a two-period overlapping-generations model based on Hassler, Storesletten
and Zilibotti (2007). Time is discrete and denoted by t = 0, 1, 2, · · · . The economy consistsof a continuum of agents living for two periods, youth and old age. Each generation has
a unit mass. The main departure from the model in Hassler, Storesletten and Zilibotti(2007) is that agents are heterogeneous at birth. Some are born into poor families, whileothers are born into rich families. Poor-born agents have the chance of becoming rich via
educational investment, while rich-born agents face the risk of becoming poor if they failin education. Therefore, there is intergenerational mobility in this economy.
Let ut denote the size of the old poor in period t. Among the young born in period t,
ut are born into poor families and 1 − ut into rich families. Young agents can aﬀect theirprospects in life using educational investment. Those who are successful in education
become rich, and obtain a high wage, normalized to one, for both periods. Those who are
unsuccessful in education become poor and obtain a low wage, normalized to zero, forboth periods. Figure 2 illustrates the timing of events and the distribution of rich andpoor across the generations.
[Figure 2 about here.]
The opportunities for, and costs of, education depend on the status of the families
into which agents are born. Let ert and ept denote the probabilities of educational success(i.e., becoming rich) for rich-born and poor-born agents, respectively. Those who areborn into rich families have full access to education and can increase the probability ert ofremaining rich by undertaking costly investment with the cost function (ert )2 . However,those who are born into poor families have limited access to education. With probability
1 − μ ∈ [0, 1], they have no opportunity of education. With probability μ, they have an
5

opportunity, but the cost of education is given by γ · (ept ) where γ > 1. This assumption
implies that the educational cost for the poor-born young is higher than for the rich-born
young. Therefore, poor-born young agents have a disadvantage in terms of opportunities
and costs.
The assumption of γ and μ is motivated by the following observations. Hassler andRodriguez Mora (2000) and Roemer (2004) argue that the distribution of innate andsocial assets among individuals is not independent between generations and that parents
aﬀect their children’s chances for acquisition of income through innate and social heritage.
For example, rich-born children can achieve a high score with low costs thanks to theirhigh innate ability received from their parents. In addition, they can apply promisingstrategies for success thanks to precise information given by their educated parents. In
our framework, an intergenerational heritage of innate and social assets is captured by
the parameter γ, which represents the diﬀerence in costs of education for skill acquisition.
Empirical evidence suggests that the above-mentioned family background eﬀects diﬀer
across countries (OECD, 2008). The diﬀerence comes from the variation in equality of opportunities related to school systems like early tracking and preschool education (Schütz,
Ursprung and Wößmann, 2008). Based on the two international student achievement
tests, the Third International Mathematics and Science Study (TIMSS) and its replication
for a partly diﬀerent set of countries (TIMSS-repeat), Schütz, Ursprung and Wößmann
(2008) show that the family-background eﬀect is larger (i.e., equality of opportunity islower) the earlier a country tracks its students into diﬀerent school types of ability and theshorter is the preschool education. In our framework, equality of opportunity is capturedby the parameter μ, which demonstrates the degree of accessibility to education for thepoor-born children.3
The numbers of agents who experience upward and downward mobility are given by
Mtup = ut μept and Mtdown = (1 − ut )(1 − ert ), respectively. The poor-born agents of size ut
have the opportunity of education with probability μ, and they succeed in education andbecome rich with a probability of ept . The rich-born agents of size 1 − ut fail in education
and become poor with a probability of 1 − ert . Because the size of each generation isunity, Mtup and Mtdown also indicate the proportions of agents experiencing upward and3

It should be noted that the current paper abstracts from the pecuniary link between the budgetary
constraints of parents and the educational investments of children. Instead, the paper focuses on the
nonpecuniary link between parents and children in terms of education acquisition. The reason behindthe focus on the nonpecuniary link is twofold. First, the nonpecuniary link plays an important role indetermination of educational acquisition as shown by Schütz, Ursprung and Wößmann (2008) and OECD
(2008a). Second, the long-run state becomes dependent on initial conditions when the pecuniary link is
introduced into the model (see, for example, Banerjee and Newman, 1993; Owen and Weil, 1998; andMaoz and Moav, 1999). In other words, two economies sharing similar initial conditions converge to thesame equilibrium. This prediction fails to explain the variation in inequality and mobility among someOECD countries sharing similar economic backgrounds.

6

downward mobility, respectively.
There is no storage technology in this economy. Each agent uses his/her endowments
within the period. The government provides lump-sum transfers, s, ﬁnanced by taxes
levied on the rich. The tax rates are age dependent, τ o for the old and τ y for the young.
The tax rates are also determined before the young agents decide on their investments.
Therefore, the expected utility functions of agents alive at time t are given as follows:
Vto,r = (1 − τto ) + st ,(1)

Vto,p = st ,

(2)



o
Vty,r = ert · (1 − τty ) − (ert )2 + st + β · ert · (1 − τt+1
) + st+1 ,




o
Vty,p = μ · ept · (1 − τty ) − γ · (ept )2 + st + β · μ · ept · (1 − τt+1
) + st+1 ,

(3)
(4)

where Vto,r , Vto,p , Vty,r and Vty,p denote the utility of the old rich and the old poor, andthe expected utility of the young born into rich families and the young born into poorfamilies, respectively, and the parameter β ∈ (0, 1) is a discount factor. The utility levelsof Vto,r , Vto,p , Vty,r and Vty,p are computed prior to the success or failure of individuals.
Given these preferences, a young agent born into a rich family chooses ert to maximize
Vty,r , and a young agent born into a poor family chooses ept to maximize Vty,p . Therefore,
the optimal investment of the rich-born and poor-born young agents are respectively given
by:

1 
o
· (1 − τty ) + β · (1 − τt+1
) ,
2

1 
p∗ y
o
o
e (τt , τt+1 ) =· (1 − τty ) + β · (1 − τt+1) .
2γoer∗ (τty , τt+1)=oo
) < er∗ (τty , τt+1
) ≤ 1,
Given the assumption of γ > 1, it holds that 0 ≤ ep∗ (τty , τt+1o
∀ τty , τt+1
∈ [0, 1].
Agents with the same family background choose the same investment, implying thatthe proportion of the old poor in period t + 1, ut+1 , is given by:


 


o
o
ut+1 ≡ (1 − ut ) · 1 − er∗ (τty , τt+1
) + ut · μ · 1 − ep∗ (τty , τt+1
) + (1 − μ)




μ
1
o
) .
= 1 − · (1 − ut ) + · ut · (1 − τty ) + β · (1 − τt+1
2
γ

(5)

Thus, the proportion of old poor (i.e., the proportion of agents who were unsuccessful intheir youth), ut+1 , depends on the tax levied on the rich young agents in period t, τty , theo, and the proportion of old poor, ut .
tax levied on the old rich agents in period t + 1, τt+1
The tax revenues from rich agents are transferred to every agent in a lump-sum fashion.

7

The government budget is balanced in each period so that it can be expressed as:o, ut ),
2st = W (τto , ut ) + Z(τty , τt+1

where:W (τto , ut ) ≡ (1 − ut ) · τto and


o
o
o
Z(τty , τt+1
, ut ) ≡ (1 − ut ) · er∗ (τty , τt+1
) + ut · μ · ep∗ (τty , τt+1
) · τty
are the tax revenues ﬁnanced by the old rich and the young rich, respectively. The function
Z, representing tax revenue from the young rich, is rewritten as:
1o, ut ) = ·Z(τty , τt+1
2

where:

μ(1 − ut ) + ut ·γ ty , τ o ),
· Z(τ
t+1

(6)



o
o
 ty , τt+1
) ≡ (1 − τty ) + β · (1 − τt+1
) · τty .
Z(τEquation (6) implies that, given the state variable ut , maximizing the revenue from theo ty , τ o ).
, ut ) is equivalent to maximizing Z(τyoung Z(τty , τt+1
t+1

3

Period-t Political Equilibrium

This section considers the determination of redistribution policy and the distributionof rich and poor in some period t. Section 3.1 provides the deﬁnition of a period-t
political equilibrium based on the concept of a stationary Markov-perfect equilibriumwith majority voting. Sections 3.2 and 3.3 characterize the period-t political equilibriumby focusing on the pattern of taxation and the distribution of rich and poor. The dynamic
equilibrium sequence of redistribution and the distribution of rich and poor are analyzed
in the following section.

3.1

Deﬁnition of Period-t Political Equilibrium

Following Hassler, Storesletten and Zilibotti (2007), we assume that elections are held at
the end of each period and the elected politician sets tax rates for the following period.
The old have no interest in the following period’s tax rates and thus abstain from voting.
At the end of each period, the young know their wage realization in the following period.
Therefore, this assumption of voting is observationally equivalent to assuming that agents
vote over current taxes at the beginning of each period but that only the old vote (Hassler
et al., 2003; Hassler, Storesletten and Zilibotti, 2007). We employ the latter interpretation8

in the following analysis.
ooWith the optimal investments er∗ (τty , τt+1
) and ep∗ (τty , τt+1
) and the government budget

constraint, the indirect utility functions of the old rich and the old poor are respectively
given by:μ
 ty , τ o ) ,(1 − ut ) + ut ·
· Z(τ
t+1
2
2
γ


1
1
μ
o,p
o
o
 ty , τt+1
Vt = · W (τt , ut ) + · (1 − ut ) + ut ·
) ,
· Z(τ
2
2
γ
1
Vto,r = (1 − τto ) + ·1
W (τto , ut ) + ·where the term in the ﬁrst line, (1 − τto ), is the after-tax income of the old rich, and theterm (1/2) · W + (1/2) · {(1 − ut ) + ut · (μ/γ)} · Z is the lump-sum transfer.
The present paper focuses on stationary Markov-perfect equilibria with majority voting. The proportion of old poor (ut ) summarizes the state of the economy, and the identityof a decisive voter depends on this proportion. An oﬃce-seeking politician elected by voters sets policies to maximize the utility of the larger group. Given these features, we
provide the deﬁnition of the period-t political equilibrium as follows.

Deﬁnition 1: A period-t (stationary) Markov-perfect political equilibrium is a tripletof functions {T o , T y , U }, where T o : [0, 1] → [0, 1] and T y are two public policy
rules, τto = T o (ut ) and τty = T y , and U : [0, 1] → [0, 1] is a private decision rule,
ut+1 = U (τty ), such that given ut , the following functional equations hold.
1. T o (ut ) = arg maxτto ∈[0,1] W dec (τto , ut ) (dec = o, r; o, p), where:Wdec(τto , ut ) =W o,r ≡ (1 − τto ) + 12 · W (τto , ut ) if ut ≤ 1/2,
if ut > 1/2.
W o,p ≡ 12 · W (τto , ut )



 


o
o
) + ut · μ · 1 − ep∗ (τty , τt+1
) + (1 − μ) ,2. U (τty , ut ) = (1 − ut ) · 1 − er∗ (τty , τt+1o
= T o (U (τty )).
with τt+1oo ty , τt+1
3. T y = arg maxτto ∈[0,1] Z(τ) subject to τt+1
= T o (U (τty , ut )).

The ﬁrst equilibrium condition requires the decisive voter to choose τto to maximize
the utility of the old rich if ut < 1/2 and the utility of the old poor if ut > 1/2. In thecase of an equal number, the old rich are assumed to be decisive. The second equilibrium
condition implies that all young individuals optimally choose their investments givenoτty and τt+1
, under rational expectations about future taxes and distributions of types.
The third equilibrium condition requires the decisive old voter to choose τty to maximize

9

revenue from the young. Rational voters also understand that their choice over current
redistribution aﬀects future redistribution via the private decision rule and public policy.

3.2

The Determination of T o and U

We solve the equilibrium conditions recursively. Condition 1 deﬁnes a one-to-one mappingfrom the state variable to the equilibrium choice of taxation of the old: τto = T o (ut ).
Suppose that the majority are the old rich: ut ≤ 1/2. The objective function of the
majority is given by W o,r = (1 − τto ) + W (τto , ut )/2, which is strictly decreasing in τto :∂W o,r (τto , ut )/∂τto < 0. This implies that the old rich pay more than they receive because
poor agents pay no tax, but the revenue is distributed equally between the rich and thepoor. Therefore, the old rich prefer τto = 0.
Alternatively, suppose that the majority are the old poor: ut > 1/2. The objectivefunction of the majority is given by W o,p = W (τto , ut )/2, which is strictly increasing in τto :∂W o,p (τto , ut )/∂τto > 0. Therefore, the old poor prefer τto = 1. The mapping that satisﬁes
equilibrium condition 1 is summarized as follows:
T o (ut ) =0 if ut ≤ 1/2,
1 if ut > 1/2.

(7)

Next, we rewrite equilibrium condition 2 by substituting it into the optimal investooments er∗ (τty , τt+1
) and ep∗ (τty , τt+1). This yields the following functional equation:μ(1 − ut ) + · ut · {(1 − τty ) + β · (1 − T o (U (τty , ut )))} ,2γ

1
U (τty , ut ) = 1 − ·(8)where T o (·) ∈ {0, 1} is given by (7). Because T o (·) ∈ {0, 1}, any solution of (8) must be
a combination of the following two linear functions:⎧⎨U p (τty , ut ) ≡ 1 − 12 · (1 − ut ) + μγ · ut · (1 − τty ),⎩ U r (τty , ut ) ≡ 1 − 1 · (1 − ut ) + μ · ut · (1 − τty + β),2γowhere U p (τty , ut ) is the proportion of the old poor when τt+1
= 1 is expected, and U r (τty , ut )ois the proportion of the old poor when τt+1
= 0 is expected.

Under the assumption of rational expectations, any solution to the functional equation(8) must be such that:
U (τty , ut ) ={U p (τty , ut ), U r (τty , ut )} for τty ≤ τ̂ y (ut ),U p (τty , ut )10

for τty > τ̂ y (ut ),(9)where τ̂ y (ut ) is given by:τ̂ y (ut ) ≡(1 − ut ) + μγ · ut

· (1 + β) − 1(1 − ut ) + μγ · ut.

The process of deriving the solution (9) is as follows. Suppose that young agents ino= 0. With this expectation, educational investments by the richperiod t expect τt+1

born young and the poor-born young are er∗ (τty , 0) = {1 − τty + β} /2 and ep∗ (τty , 0) ={1 − τty + β} /2γ, respectively, and the size of the old poor in period t + 1 is ut+1 =oU r (τty , ut ). By (7), the expectation of τt+1
= 0 is rational if ut+1 = U r (τty , ut ) ≤ 1/2, thatois, if τty ≤ τ̂ y (ut ). Suppose, instead, that young agents born in period t expect τt+1
= 1.

By (7), this expectation is rational if ut+1 = U p (τty , ut ) > 1/2. The expectation is rational
for any τty ∈ [0, 1] and ut ∈ [0, 1] because it always holds that ut+1 = U p (τty , ut ) >1/2. Consequently, there are multiple solutions for the range of τty ≤ τ̂ y (ut ) as long asτ̂ y (ut ) ≥ 0 holds. τ̂ y (ut ) ≥ 0 holds if μ/γ ≥ 1/(1 + β), or, if μ/γ < 1/(1 + β) andut ∈ [0, β/(1 + β)(1 − μ/γ)].4[Figure 3 about here.]
As depicted in Figure 3, there are multiple, self-fulﬁlling expectations of U for the rangeof τty ∈ [0, τ̂ y (ut )]. Which particular U arises in equilibrium depends on the expectationsof agents. To illustrate U in equilibrium, we follow Hassler, Storesletten and Zilibotti(2007) and introduce the critical rate of τty : θ ∈ [0, τ̂ty (ut )]. The rate θ, which depends onthe expectations of agents, is the highest tax rate that yields a majority of the old rich.
For τty > θ, the majority is the old poor. However, for τty ∈ (0, θ], the majority is either
the rich or the poor.
With the expectation parameter θ, the functional solution is reduced as follows. If
μ/γ < 1/(1 + β) and ut ∈ (β/(1 + β)(1 − μ/γ), 1], then τ̂ y (ut ) < 0 holds. The solution
to (8) is uniquely given by U (τty , ut ) = U p (τty , ut ) (Fig. 3(a)). If μ/γ ≥ 1/(1 + β), or if
μ/γ < 1/(1 + β) and ut ∈ [0, β/(1 + β)(1 − μ/γ)], then τ̂ y (ut ) ≥ 0 holds (Fig. 3(b)). The
solution is given by:
U (τty , ut ) ={U p (τty , ut ), U r (τty , ut )} if τty ∈ (0, θ]
U p (τty , ut )4 y

if τty ∈ (θ, 1].

τ̂ (ut ) ≥ 0 is rewritten as ut ≤ β/(1 − μ/γ)(1 + β). This condition of ut holds ∀ut ∈ [0, 1] ifβ/(1 − μ/γ)(1 + β) ≥ 1, that is, if μ/γ ≥ 1/(1 + β). If μ/γ < 1/(1 + β), then τ̂ y (ut ) ≥ 0 holds forut ∈ [0, β/(1 − μ/γ)(1 + β)].

11The functional solution is therefore summarized as follows:⎧1⎪
U p (τty , ut )if μγ < 1+βand ut ∈⎪⎨U (τty , ut ) =⎪⎪⎩{U p (τty , ut ), U r (τty , ut )} for τty ∈ (0, θ]
U p (τty , ut )for τty ∈ (θ, 1]β,1(1+β)(1−μ/γ)otherwise.
(10)

3.3

The Determination of T y and the Characterization of the
Period−t Political Equilibria

Given the characterization of T o and U satisfying equilibrium conditions 1 and 2, respectively, we consider the political determination of τty that satisﬁes equilibrium condition
3. Because there are two possible cases of a majority, we introduce the corresponding deﬁnition of the political equilibria: a poor-majority equilibrium and a rich-majority
equilibrium. When ut > 1/2, there is a poor-majority equilibrium where agents expectoτt+1= 1 and choose τty to induce a majority of the poor in period t + 1. When ut ≤ 1/2,othere is a rich-majority equilibrium where agents expect τt+1
= 0 and choose τty to inducea majority of the rich in period t + 1.

The objective of the decisive voter is to choose the τty that maximizes revenue fromthe young Z. Given the state variable ut , maximizing Z is equivalent to maximizing Z̃ aspresented in (6). Therefore, the objective function of the decisive voter is given by Z̃(τty , 1)ooif agents expect τt+1
= 1, and is given by Z̃(τty , 0) if agents expect τt+1
= 0. These two ty , 1) is
objective functions demonstrate Laﬀer curves with the following properties: (i) Z(τ ty , 0) is a hump-shaped
a hump-shaped function of τty , with a maximum at τty = 1/2; (ii) Z(τ ty , 0) > Z(τ ty , 1) ∀τty ∈ (0, 1]function of τty , with a maximum at τty = (1 + β)/2; (iii) Z(τ(see Figure 4).
[Figure 4 about here.]
The properties of the objective functions imply that the decisive voter may maximize
 ty , 0); i.e., by setting
revenue from the young by taking the top of the Laﬀer curve Z(τoτty = (1 + β)/2 under the expectation of τt+1
= 0. However, this choice is inconsistent
with the second equilibrium condition because the choice of τty is limited to the range of(0, θ] where θ ≤ τ̂ y (ut ) < (1 + β)/2. By contrast, the decisive voter can take the top ofo ty , 1) by setting τty = 1/2 under the expectation that τt+1
the other Laﬀer curve Z(τ= 1 isconsistent with the second equilibrium condition. Therefore, the revenue from the youngois maximized by setting τty = θ under the expectation of τt+1
= 0, or by setting τty = 1/2o= 1. The former choice produces a higher level of revenue
under the expectation of τt+112

 where: 0) ≥ Z(1/2,than the latter if and only if Z(θ,1); that is, if and only if θ ≥ θ,

θ(β)
=

(1 + β) −β(β + 2)> 0.
2

Given ut and θ(β),
we obtain the following characterization of the period-t politicalequilibria.
Proposition 1(i) For any ut ∈ [0, 1], there exists a set of poor-majority equilibria such that ∀t, T o isgiven by (7), U (τty , ut ) is given by (10), and T y = 1/2. The equilibrium outcome iso= 1, and ut+1 = 1−((1 − ut ) + (μ/γ) · ut ) /4.
unique and such that ∀t, τty = 1/2, τt+1(ii) Suppose that β ≥ 1/4 and ut ∈ [0, û] hold where:β(β + 2) − (1 − β)

.
û ≡

(1 − μ/γ) (1 + β) + β(β + 2)


There exists a set of rich-majority equilibria such that ∀t, T o is given by (7),U (τty , ut ) is given by (10), and T y = θ ∈ [θ(β),τ̂ y (ut )]. The equilibrium outcomeo
is indeterminate such that ∀t, τty = θ ∈ [θ(β),τ̂ y (ut )], τt+1
= 0, and ut+1 = 1 −{(1 − ut ) + (μ/γ) · ut } (1 − θ + β)/2.

Proof. See the appendix.
[Figure 5 about here.]
Figure 5 indicates the set of (β, ut ) satisfying the condition given in Proposition 1.
When the size of poor-born agents is large such that ut > û (see the area P.1(i)), a
unique poor-majority equilibrium exists. Although the poor-born agents have the chanceof becoming rich via educational investment, the rich can never become a majority inperiod t because the number of poor-born agents is large. Therefore, when ut > û holds,
voting that induces a future majority of the poor is the only option. The economy thenfeatures a unique equilibrium with a poor majority who prefer 100% taxation on the old.
The young are taxed at the top of the Laﬀer curve, conditional on the expectation of
100% taxation when old.
When the size of poor-born agents is small, such that ut ≤ û (see the area P.1(i)&(ii)),
there is still an option that induces a future majority of the poor. However, there is analternative option that induces a future majority of the rich by setting τty = θ. The
equilibrium realized depends on the expectation of agents. The rich-majority equilibrium13

is sustained as an equilibrium if Z(θ, 0) ≥ Z(1//2, 1); that is, if the expectation parameterθ is above the critical level θ(β).
Given that the upper bound of θ is τ̂ y (ut ), there exists arich-majority equilibrium if θ is set within the range [θ(β),τ̂ y (ut )]. This set is nonempty
if and only if ut ≤ û, where û is nonnegative if and only if β ≥ 1/4.
As illustrated in Figure 5, the condition given in Proposition 1(ii) requires a high β
and a low ut . A high β implies that agents attach a high weight to their utility in theirold age, thereby having a strong incentive to invest in education. This results in a large
number of the old who have been successful in their youth. A low ut indicates a large
number of rich-born young agents, thereby implying a large number of future rich agents.
Given these two factors, the economy can attain a rich-majority equilibrium when agents
expect no taxation on the old.

4

Dynamic Political Equilibrium

This section investigates the motion of the distribution of rich and poor over time. The
deﬁnition of the dynamic political equilibrium is as follows.
Deﬁnition 2: A dynamic political equilibrium is a sequence of ut with the initial cono
) constitutes the period-t political equilibrium; anddition u0 such that (i) (τty , τt+1o(ii) the sequence of ut is determined by ut+1 = (1 − ut ) · 1 − er∗ (τty , τt+1
) + ut · oμ · 1 − ep∗ (τty , τt+1
) + (1 − μ) .
Based on the characterization of the period-t political equilibrium in Proposition 1, asequence of ut is characterized by:
ut+1 =
where:⎧⎨{U p (1/2, ut ) , U r (θ, ut )} if β ≥ 1/4 and ut ≤ û,U p (1/2, ut )otherwise,

(11)




μ
1
(1
−
u
,
=
1
−
,
u
)
+
·
u
t
t
t
2
 4
γ
⎩ U r (θ, ut ) = 1 − 1 (1 − ut ) + μ · ut (1 − θ + β).
2
γ
Up

1

Suppose that β ≥ 1/4 and ut ≤ û hold. There are multiple period-t political equilibriaowith (τty , τt+1
) = (1/2, 1) and (θ, 0). The economy attains a poor-majority equilibriumowith τty = 1/2 and ut+1 = U p (1/2, ut ) > 1/2 when agents expect τt+1
= 1, and arich-majority equilibrium with τty = θ and ut+1 = U r (θ, ut ) ≤ 1/2 when agents expectoτt+1= 0. Suppose, instead, that β ≥ 1/4 or ut ≤ û fails to hold. There is a uniqueo) = (1/2, 1), and ut+1 is uniquely given by U p (1/2, ut ).
period-t equilibrium with (τty , τt+1

For precise consideration of the dynamic motion of ut , we illustrate (11) in a ut −
ut+1 space. For this purpose, we present the properties of the functions U p (1/2, ut ) and14

U r (θ, ut ) as follows:13
μ= , Up,1 = 1 − ,424γ
1U r (τ̂ y (ut ), ut ) = ∀ut ∈ [0, 1],2 0) = 1 − 1 (1 + β) + β(β + 2) ,U r (θ,
4
Up

1
,0
2

(12)
 û) = 1 .
U r (θ,2

Based on these properties, we can illustrate the dynamic path of ut given by (11) as in
Figure 6.
[Figure 6 about here.]
In what follows, we focus on a steady-state equilibrium where the sequence of ut isstationary over time, and we show that there are multiple steady-state equilibria undera certain condition. The steady-state equilibria are illustrated by the crossing points of
the 45-degree line and ut+1 = {U p (1/2, ut ) , U r (θ, ut )}. We denote by ūp the steady-statelevel of the old poor in a poor-majority equilibrium, and we denote by ūr the steadystate level of the old poor in a rich-majority equilibrium. By deﬁnition, ūp and ūr satisfyūp = U p (1/2, ūp ) and ūr = U r (θ, ūr ); that is:3,
3 + μ/γ
1 − 12 · (1 − θ + β)rr,low 1∈ ū,ū (θ) =,2
1 − 12 · 1 − μγ · (1 − θ + β)ūp =where:

(13)
(14)




4 − (1 + β) + β(β + 2).
ūr,low ≡
4 − (1 − μ/γ) (1 + β) + β(β + 2)

Within the range of θ ∈ [θ(β),τ̂ y (ut )], ūr (θ) attains a minimum at θ = θ(β)
and amaximum at θ = τ̂ y (ut ) because ūr (θ) is increasing in θ. Direct calculation leads toūr (θ(β))= ūr,low and ūr (τ̂ y (ut )) = 1/2, implying that ūr (θ) ∈ ūr,low , 1/2 .
The following proposition establishes the condition for the existence and stability ofthe steady-state equilibria.
Proposition 2(i) Suppose that β < 1/4 holds, or that β ≥ 1/4 and û < 1/2 hold. There exists a
unique, globally stable poor-majority steady-state equilibrium.

15

(ii) Suppose that β ≥ 1/4 and û ≥ 1/2 hold. There exist multiple steady-state equilibria.
(a) The equilibrium path is determinate and converges to the poor-majority steadystate equilibrium if û ∈ [1/2, ūp ) and u0 ∈ (û, 1]; it is indeterminate otherwise.
Proof. The proof is immediate from Figure 6.
The properties of the steady-state equilibria depend on the parameter values andthe expectations of agents as illustrated in Figure 6. Panels (a) and (b) demonstratea unique, stable poor-majority steady-state equilibrium. The majority is always poor
along the equilibrium path in panel (a). By contrast, in panel (b), the majority could
be rich along the transition path but ﬁnally converge to the poor-majority steady-state
equilibrium.
Panels (c) and (d) demonstrate the multiple steady-state equilibria. Panel (c) indicates that given u0 > û, the equilibrium path deﬁnitely converges to the poor-majority
steady-state equilibrium. However, when u0 ≤ û, the equilibrium is indeterminate: theeconomy might attain the rich-majority steady-state equilibrium depending on the expectations of agents. By contrast, panel (d) illustrates a situation where the equilibrium isindeterminate for all initial condition of u. An economy that stays in the poor-majority
steady state might move to the rich-majority steady state because of changes in the expectations of agents. Moreover, the changes in the expectations of agents could produce
electoral cycles that move back and forth between the two equilibria as illustrated inpanel (d) of Figure 6. These expectations play a key role in the dynamic motion of theequilibrium path.
Figure 7 illustrates the set of parameters (μ/γ, β) classiﬁed according to the characterization of the steady states. As illustrated in Figure 7, given (μ/γ), a high β is required
for the existence of the rich-majority steady state. This is because a high β implies thatagents place large weight on their future income, thereby giving young agents an incentive to invest in education, and the successful (rich) agents then form a majority. On theother hand, given the discount factor, a high (μ/γ) is required for the existence of therich-majority steady state. A high (μ/γ) implies that the poor-born young agents have
many opportunities for education and that education costs are low. Therefore, young
agents have a strong incentive to invest in education, and this results in a majority of richagents.
[Figure 7 about here.]

5Comparative Statics Analysis

We have characterized the political equilibria and qualitatively assessed the impacts of γ
and μ on the determination of tax rates and the size of the poor. To facilitate understand16

ing, this section undertakes numerical analysis by focusing on the relative magnitude of
the two parameters μ and γ, (μ/γ), that represent accessibility of education for poor-born
agents. We examine how accessibility (μ/γ) aﬀects intergenerational mobility, income inequality, and the size of redistribution in the steady states.
For the purposes of this analysis, we assume a generation to be 20 years. This is
shorter than the usual assumption of, say, 30 years. We adopt a shorter generation length
because the current model assumes that agents work in both periods of life. The ﬁrst
and second periods correspond to ages 25–44 and 45–64 years, respectively. We assume asingle-period discount rate given by 0.96. Because agents under the current assumption
plan over generations that span 20 years, we discount the future by (0.96)20 .
We ﬁrst examine how accessibility of education, (μ/γ), aﬀects the pattern of politicalequilibria. In the current environment, there is a threshold level of (μ/γ) given by 0.6123,
as illustrated in Figure 8. For a low value of (μ/γ) such that (μ/γ) ∈ [0, 0.6123], there isa unique poor-majority steady state with 50% taxation on the young and 100% taxationon the old. A low value of (μ/γ) means a low μ and/or a high γ, evidencing the diﬃcultythat poor-born agents have in obtaining education. Because of this, the majority is always
the poor, who support 100% taxation on the old. By contrast, for a high value of (μ/γ)
such that (μ/γ) ∈ (0.6123, 1], there are multiple steady-state equilibria, comprising thepoor-majority equilibrium above and a rich-majority equilibrium with no taxation on theold. In the rich-majority equilibrium, the young and the old both have a lower tax burden
than in the poor-majority equilibrium.
[Figure 8 about here.]
In what follows, we demonstrate some numerical results for intergenerational mobility,
income inequality, and the size of redistribution and consider how these variables areaﬀected by the size of (μ/γ).

5.1Intergenerational Mobility

From the deﬁnition in Section 2, the levels of upward and downward mobility are givenby:μo· {(1 − τty ) + β · (1 − τt+1
)},
2γ



1
y
down
r∗ y
o
o
(1 − τt ) + β · (1 − τt+1 ) ,= (1 − ut ) · (1 − e (τt , τt+1 )) = (1 − ut ) · 1 −Mt2o
Mtup = ut · μ · ep∗ (τty , τt+1
) = ut ·respectively. In the steady state, the distribution of rich and poor becomes stationary
over time. The upward and downward mobility levels are equal in the steady state:17

M up = M down . We denote by M̄ p and M̄ r the levels of intergenerational mobility in thepoor-majority steady state and the rich-majority steady state, respectively. M̄ p and M̄ r
are given by:M̄ p = up · μ · ep∗ (1/2, 1) = (1 − up ) · (1 − er∗ (1/2, 1)),M̄ r (θ) = ur · μ · ep∗ (θ, 0) = (1 − ur ) · (1 − er∗ (θ, 0)).
Figure 9 illustrates the numerical result.
[Figure 9 about here.]
To consider the eﬀect of (μ/γ) on M̄ j (j = p, r), we rewrite M̄ p and M̄ r (θ) as follows:M̄ p =
M̄ r (θ) =

μ/γ
3
·
,
4 3 + μ/γ
1 − 12 · (1 − θ + β)
1− 21 ·(1−θ+β)+1
μ·(1−θ+β)2γ, θ ∈ [θ̃(β), τ̂ y (ūr )].

These equations indicate that M̄ p and M̄ r (θ) are increasing in μ and decreasing in γ. For
poor-born agents, a higher μ yields greater opportunities for going on to education, whilea lower γ provides lower costs of receiving education. In turn, these eﬀects increase thenumber of poor-born agents who can become rich via education. Therefore, an increasein (μ/γ) enhances intergenerational mobility, as depicted in Figure 9.
Figure 9 demonstrates that for the case of (μ/γ) ∈ (0.6123, 1] featuring multiple
equilibria, mobility in the rich-majority steady state is higher than in the poor-majority
steady state:M̄ r (θ) = ūr (θ)μep∗ (θ, 0) > M̄ p = ūp μep∗ (1/2, 1) ∀θ ∈ [0, τ̂ty (ut )] .
To understand this relation, we focus on the two factors that determine mobility: thesize of the poor and the probability of success for poor-born agents. The size of the poor
is less than half in the rich-majority equilibrium, while it is more than half in the poormajority equilibrium: ūr (θ) ≤ 1/2 < ūp . The probability of success for poor-born agentsis also higher in the rich-majority equilibrium than in the poor-majority equilibrium:ep∗ (θ, 0) > ep∗ (1/2, 1). Therefore, two competing eﬀects determine the relative size ofM̄ r (θ) and M̄ p . Figure 9 shows that the eﬀect via the probability of success is greaterthan the eﬀect via the size of the poor, thereby resulting in M̄ r (θ) > M̄ p .

18

5.2

Income Inequality

We use the Gini coeﬃcient as a measure of inequality in our framework. In particular,
we focus on the income inequality among young agents rather than among young and oldagents for the following reasons. First, income inequality among young agents provides
a qualitatively similar result to that among young and old agents because the status ofeach agent persists over the life cycle. Second, focusing on the inequality among young
agents enables us intuitively and graphically to illustrate the numerical result.
We calculate the Gini coeﬃcient among young agents in terms of after-tax-and-transfer
income. In each period, there are two types of young agents: young rich agents with per
capita income given by (1 − τty ) + st , and young poor agents with per capita income given
by st . Panel (a) of Figure 10 summarizes information about per capita income, the sizeof the population and the total income for each type of young agent. Panel (b) of Figure
10 illustrates a Lorenz curve. The Gini coeﬃcient among young agents is calculated by
A/(A + B) in panel (b) where A is the shaded area and B is the unshaded area.
[Figure 10 about here.]
Panel (b) shows that two factors determine the Gini coeﬃcient: (i) the ratio of therich to the poor denoted by the population ratio (PR), and (ii) the ratio of the per capita
income of the rich to the per capita income of the poor, denoted by the income ratio(IR). The role of the ﬁrst factor is intuitive: given the IR factor eﬀect, a lower PR (i.e., asmaller number of the rich) results in the larger size of the area A and thus a higher Ginicoeﬃcient. The second factor arises from the ratio of the slopes (A) and (B). The ratio
is rewritten as:
Slope (B)
Slope (A)=(Total income of the young rich)/(Total income of the young)1 − ut+1(Total income of the young poor)/(Total income of the young)
ut+1
Per capita income of the young rich=
Per capita income of the young poor
(1 − τty ) + st
=
.
st

−1

×

Given the PR factor eﬀect, a higher IR implies a higher ratio of (Slope (B)/Slope (A)),
a larger size of the area A, and thus a higher Gini coeﬃcient.
[Figure 11 about here.]
These factors can be used to explain the following predictions of the numerical result
depicted in Figure 11. First, in the poor-majority equilibrium, a higher (μ/γ) results in
19

a lower level of inequality. An increase in (μ/γ) leads to a larger proportion of the rich,thereby creating a positive PR eﬀect on equality. In addition, a larger proportion of therich yields a larger share of taxpayers, thereby resulting in a greater redistribution thatcreates a positive IR eﬀect on equality. Because of these positive eﬀects on equality, ahigher (μ/γ) leads to a lower Gini coeﬃcient in the poor-majority equilibrium.
Second, when we compare the rich- and poor-majority equilibria, the rich-majority
equilibrium attains a higher or a lower level of inequality depending on the relative sizeof the PR and IR eﬀects. The rich-majority equilibrium attains a larger proportion of
the rich, which produces a positive PR eﬀect on equality. However, the rich-majority
equilibrium realizes lower tax rates on the young and old, both of which result in lesser
redistribution and thus a negative IR eﬀect on equality.
For the case of (μ/γ) ∈ (0.6123, 1] featured by multiple equilibria, the positive PR
eﬀect is outweighed by the negative IR eﬀect, as the rich-majority equilibrium attains
a higher Gini coeﬃcient than the poor-majority equilibrium. However, when the richmajority equilibrium with a high (μ/γ) (for example, μ/γ = 0.9) is compared with thepoor-majority equilibrium with a low (μ/γ) (for example, μ/γ = 0.1), the positive PR
eﬀect overrides the negative IR eﬀect, as the rich-majority equilibrium attains a lower Ginicoeﬃcient than the poor-majority equilibrium. Therefore, the accessibility of education,
represented by (μ/γ), plays a key role in determining the relative degree of inequality
between the rich-majority and poor-majority equilibria.

5.3

The Size of Redistribution

We use the size of transfer, s, rather than the tax rates, as a measure of redistribution
for the following reason. In the political economy literature, the skewness of income
distribution aﬀects the political determination of tax rates, and tax rates and the numberof the rich (i.e., the tax base) determine the size of transfer. In the current framework, theeﬀect of the skewness of tax rates is abstracted except at the threshold level of u = 1/2.
That is, the political determination of the tax rate on the old is featured by a binary choiceof τ o ∈ {0, 1} because of simple majority voting by the rich and the poor in the absenceof tax distortion. Given the binary choice and the resulting lack of a skewness eﬀect,
it is not appropriate to focus on tax rates when we compare cross-country diﬀerenceof redistribution. Therefore, we instead focus on the size of transfers as a measure of
redistribution.
We denote by s̄p and s̄r the size of transfers in the poor-majority steady state and in
the rich-majority steady state, respectively. From the deﬁnition in Section 2, s̄p and s̄r20are calculated as:


1
1
p
p
W (1, ū ) + Z, 1, ū
s̄ =
2
2


1
1
p
, 1 + ūp μep∗=
(1 − ū ) + (1 − ūp )er∗221s̄r (θ) = [W (0, ūr ) + Z (θ, 0, ūr )]2
1= {(1 − ūr )er∗ (θ, 0) + ūr μep∗ (θ, 0)} · θ.
2
p

1
,1
2




1
·
,
2

These equations indicate that two factors determine the size of transfer: the tax rates andthe number of the rich (i.e., the tax base). Figure 12 illustrates the numerical result.
[Figure 12 about here.]
We ﬁnd the following three observations from Figure 12. First, in the poor-majority
equilibrium, a higher (μ/γ) results in a larger size of transfer. An increase in (μ/γ) leads
to a larger proportion of successful, rich agents, thereby creating a positive tax base eﬀect
on the size of transfer. Second, for the case of (μ/γ) ∈ (0.6123, 1] featured by multiple
equilibria, the poor-majority equilibrium attains a larger size of transfer than the richmajority equilibrium because the former is characterized by higher tax rates.
Third, when the rich-majority equilibrium with a high (μ/γ) (for example, μ/γ = 0.9)
is compared with the poor-majority equilibrium with a low (μ/γ) (for example, μ/γ =0.1), the rich-majority equilibrium attains a larger size of transfer than the poor-majority
equilibrium because of a larger tax base eﬀect. Therefore, the accessibility of educationaﬀects the relative size of transfer between the rich-majority and poor-majority equilibria.

6

Empirical Implications of Mobility and Inequality

The numerical analysis in Section 5 shows that intergenerational mobility and inequality
depend on the parameter (μ/γ) representing the accessibility of education for poor-born
agents. In this subsection, we focus on (μ/γ) and investigate the empirical implications of
the model by comparing the model predictions in Figures 9, 11 and 12 with the empiricalevidence (Subsection 6.1). We then provide some empirical evidence on the cross-country
diﬀerences of educational accessibility that supports our interpretation of the numericalresult (Subsection 6.2).

6.1

Interpretation of the Numerical Result

The numerical results in Figures 9 and 11 show that where the poor-majority equilibrium is concerned, the political economy tends to generate a negative correlation between
21inequality and mobility, as shown in previous work (Maoz and Moav, 1999; Owen andWeil, 1998; Hassler, Rodriguez Mora and Zeira, 2007; Mookherjee and Napel, 2007). Thisprediction also ﬁts some empirical data in Figure 1. Here, Italy, the United Kingdom
and the United States feature high inequality and low mobility, while the Nordic countries (Denmark, Sweden, Finland and Norway) feature low inequality and high mobility.
Therefore, the substantial diﬀerence in inequality and mobility among those countries areproduced by diﬀerences in the accessibility of education.
The negative correlation between inequality and mobility, however, appears less thanexact (Solon, 2002). Despite considerable intergenerational mobility in Australia (D’Addio,2007) and Canada (Corak, 2006), these countries also score relatively highly on measuresof cross-sectional inequality (see Figure 1). One possible explanation is that they are
represented by a rich-majority equilibrium with a certain level of μ/γ(≥ 0.6123). To
understand this argument, let us compare the poor-majority equilibrium with μ/γ = 0.9(representing the Nordic countries) and the rich-majority equilibrium with μ/γ = 0.7 (representing Australia and Canada). Both these equilibria obtain similar levels of mobility:
the diﬀerence in mobility rates is less than 2% (see Figure 9). However, the rich-majority
equilibrium with μ/γ = 0.9 attains a much higher Gini coeﬃcient than the poor-majority
equilibrium with μ/γ = 0.7, the diﬀerence being more than 10% (see Figure 11). These
properties, the similarity in mobility rates and the substantial diﬀerence in inequality
between two groups of countries, are produced by multiple, self-fulﬁlling expectations ofagents.
The above-mentioned classiﬁcation of countries also ﬁts the empirical evidence on
the size of redistribution. When the poor majority is considered, a country with higher
accessibility of education (such as the Nordic countries) shows a larger size of redistribution compared with one with lower accessibility of education (such as Italy, the United
Kingdom and the United States) because of the tax base eﬀect. When the focus is on
the rich-majority equilibrium representing Australia and Canada, the equilibrium showsa smaller size of redistribution compared with the poor-majority equilibrium with highaccessibility (such as the Nordic countries) because of lower tax rates, but shows a larger
size of redistribution compared with the poor-majority equilibrium with low accessibility(such as Italy, the United Kingdom and the United States) because of the tax base eﬀect.
This model prediction of the size of redistribution ﬁts in well with the empirical evidence
on the size of redistribution reported by OECD (2008b).

22

6.2

Evidence on the Cross-country Diﬀerence of Educational
Accessibility

Our interpretation of the numerical result depends on the assumption that Australia,
Canada and Nordic countries have higher accessibility of education (i.e., a higher μ/γ)
than Italy, the United Kingdom and the United States. We here present some empiricalevidence that supports this assumption.
Based on the two international student achievement tests, the Third International
Mathematics and Science Study (TIMSS) and its replication for a partly diﬀerent set ofcountries (TIMSS-repeat), Schütz, Ursprung and Wößmann (2008) report that England,
Scotland and the United States are included in the top 10 countries with the least equalityof educational opportunity among the sample of 29 OECD countries, whereas Australia,
Canada, Denmark, Finland, Italy, Norway, and Sweden are ranked lower than eleventh.
This empirical result indicates that the United Kingdom and the United States provide
less equality of educational opportunity than other countries. OECD (2008a) supports thisevidence using the data from the OECD program for International Student Assessment(PISA) 2000 survey. Finland provides more equitable access to higher education thanthe United Kingdom among 10 countries including Austria, Finland, France, Germany,
Ireland, Italy, the Netherlands, Portugal, Spain and the United Kingdom.
We should note that equality and mobility are low in Italy, despite fairly equitable
access to education, a fact less easy to explain with our model. However, some empiricalstudies suggest a strong regional variation in educational accessibility in Italy, whichmight result in low equality and mobility. For example, Checchi and his collaboratorsshow that the southern area in Italy presents a stronger eﬀect of the family backgroundon the test scores of high school students (Checchi and Peragine, 2005; Checchi, Fiorio,
Leonardi, 2007) and a higher inequality of educational opportunity (Checchi and Peragine,2009) compared with the northern area. Therefore, this regional variation in educational
accessibility, which is not captured by Schütz, Ursprung and Wößmann (2008) and OECD
(2008a), could be a possible explanation for low equality and mobility in Italy.

7

Conclusion

This paper presents a politico-economic model that analyzes the mutual link between income inequality and intergenerational mobility within and across generations. The focusof this paper is the correlation between inequality and mobility. Empirical studies haveshown that this correlation across countries is negative for most of the OECD, but it
does not ﬁt Australia and Canada, as these counties feature both high mobility and highinequality. The current analysis demonstrates two key factors that explain the empiri23

cal evidence: the accessibility of education that provides a negative correlation between
inequality and mobility, and the multiple, self-fulﬁlling expectations that produce an equilibrium characterized by both high inequality and high mobility. Our model can also be
used to demonstrate the dynamic motion of inequality and mobility within each country. Under the condition that the economy attains multiple political equilibria, we showthat two economies sharing the same initial conditions can converge to diﬀerent equilibria
depending on the expectations of agents, and that without any structural change, theeconomy at one of the equilibria may move to the other with a change in the expectationsof agents. The expectations of agents then play a key role in the determination of thelong-run state of the economy.
To obtain these results, we simplify the analysis by adopting a simple linear, riskneutral utility function. We do not consider an alternative utility function; for example
a concave, risk-averse utility function. In addition, we focus on income inequality among
young agents, rather than income inequality among young and old agents. Furthermore,
the parameters γ and μ representing educational accessibility for poor-born individuals
are exogenously given and thus reﬂect no political decisions. However, we believe that thepresent analysis provides a suitable framework for understanding cross-country diﬀerences
in inequality and mobility as well as changes in inequality and mobility over time asaﬀected by redistributive politics and the self-fulﬁlling expectations of agents.

24Appendix
Proof of Proposition 1o(i) Suppose that at time t, agents know that τty = 1/2 and expect that τt+1
= 1. Weobtain:1,1+ ut · μ · 1 − ep∗
ut+1 = (1 − ut ) · 1 − e
2


1
μ
= 1 − · (1 − ut ) + · ut
4
γ
1
> ,
2
r∗

1
,1
2


+ (1 − μ)where the inequality in the third line comes from μ ∈ [0, 1] and γ > 1. By (7), ut+1 > 1/2
oimplies that τt+1
= 1, fulﬁlling initial expectations. Therefore, a poor-majority equilib-

rium exists if the decisive voter ﬁnds it optimal to set τty = 1/2.
Suppose that γ/μ > 1 + β and ut ∈ (β/(1 + β)(1 − μ/γ), 1] hold. The condition (10)oimplies that the rational expectation is uniquely given by τt+1
= 1 for τty ∈ [0, 1]. The
revenue from the young Z(τty , 1) is maximized by setting τty = 1/2. Suppose, instead,that γ/μ ≤ 1 + β holds or that γ/μ > 1 + β and ut ∈ [0, β/(1 + β)(1 − μ/γ)] hold. The
condition (10) implies that there are multiple, self-fulﬁlling expectations for τty ∈ [0, θ].
Setting τty = 1/2 is optimal if Z(1/2, 1) > Z(θ, 0); that is, if θ ∈ [0, θ̃).
o(ii) Suppose that agents know that τty = θ ≤ τ̂ y (ut ) and expect τt+1
= 0. Then:

ut+1 = (1 − ut ) · (1 − er∗ (θ, 0)) + ut · {μ · (1 − ep∗ (θ, 0)) + (1 − μ)}1μ= 1 − · (1 − ut ) + · ut · (1 − θ + β)
2
γ


⎞
⎛
μ


)
+
·
u
(1
−
u
t
t (1 + β) − 1γ
μ1⎠≤ 1 − · (1 − ut ) + · ut · ⎝1 + β −2γ(1 − ut ) + μγ · ut1
= ,2
where the inequality in the third line comes from θ ≤ τ̂ y (ut ). By (7), this implies thato
τt+1
= 0, fulﬁlling initial expectations. Therefore, a rich-majority equilibrium exists if thedecisive voter ﬁnds it optimal to set τty = θ ∈ [θ(β),τ̂ y (ut )].
o
The condition (10) implies that the expectation of τt+1
= 0 can be rational if (i)γ/μ ≤ 1 + β, or (ii) γ/μ > 1 + β and ut ∈ [0, β/(1 + β)(1 − μ/γ)]. For each case, settingτty = θ is optimal if and only if Z(θ, 0) ≥ Z(1/2, 1); that is, θ ∈ [θ(β),τ̂ y (ut )]. Theset [θ(β),τ̂ y (ut )] is nonempty if and only if ut ∈ [0, û], where û < β/(1 + β)(1 − μ/γ)

25
always holds and û ≥ 0 ⇔ β ≥ 1/4. Therefore, given θ ∈ [θ(β),τ̂ y (ut )], there exists arich-majority equilibrium if β ≥ 1/4 and ut ∈ [0, û].
26

References[1] Arawatari, R., and Ono, T., 2008. The Political Economy of Occupational Mobility
and Redistribution Policies. Discussion Papers in Economics and Business, Osaka
University, No. 08–18.
[2] Arawatari, R., and Ono, T., 2009. A Second Chance at Success:

A Po-

litical Economy Perspective. Journal of Economic Theory 144, 1249–1277.
doi:10.1016/j.jet.2008.11.002[3] Banerjee, A.V., and Newman, A.F., 1993. Occupational Choice and the Process ofDevelopment. Journal of Political Economy 101, 274–298.
[4] Becker, G., and Tomes, N., 1979. An Equilibrium Theory of the Distribution ofIncome and Intergenerational Mobility. Journal of Political Economy 87, 1153–1189.
[5] Bernasconi, M., and Profeta, P., 2007. Redistribution or Education? The Political
Economy of the Social Race. CESifo Working Paper No. 1934.
[6] Checchi, D., Fiorio, C.V., and Leonardi, M., 2007. Intergenerational Persistence in
Educational Attainment in Italy, University of Milan.
[7] Checchi, D., and Peragine, V., 2005. Regional Disparities and Inequality of Opportunity: The Case of Italy. IZA working paper 1874.
[8] Checchi, D., and Peragine,V., 2009, Inequality of Opportunity in Italy, Journal of
Economic Inequality. doi: 10.1007/s10888-009-9118-3[9] Corak, M., 2006. Do Poor Children Become Poor Adults? IZA Discussion Paper,
No. 1993.
[10] D’Addio, A.C., 2007. Mobility or Immobility across Generations? A Review of the
Evidence for OECD Countries. OECD Social, Employment and Migration Working
Paper, No. 52.
[11] Davies, J.B., Zhang, J., and Zeng, J., 2005. Intergenerational Mobility under Private
vs. Public Education. Scandinavian Journal of Economics 107, 399–417.
[12] Fernandez, R., and Rogerson, R., 1995. On the Political Economy of Education
Subsidies. Review of Economic Studies 62, 249–262.
[13] Hassler, J., and Rodriguez Mora, J.V., 2000. Intelligence, Social Mobility, and
Growth. American Economic Review 90, 888–908.
27

[14] Hassler, J., Rodriguez Mora, J.V., and Zeira, J., 2007. Inequality and Mobility.
Journal of Economic Growth 12, 235–259.
[15] Hassler, J., Storesletten, K., Rodriguez Mora, J.V., and Zilibotti, F., 2003. The
Survival of the Welfare State. American Economic Review 93, 87–112.
[16] Hassler, J., Storesletten, K., and Zilibotti, F., 2007. Democratic Public Good Provision. Journal of Economic Theory 133, 127–151.
[17] Ichino, A., Karabarbounis, L., and Moretti, E., 2009. The Political Economy ofIntergenerational Income Mobility.
http://www.people.fas.harvard.edu/˜karabarb/mao.pdf[18] Lefranc, A., Ojima, F., and Yoshida, T., 2008. The Intergenerational Transmissionof Income and Education: A Comparison of Japan and France. EUI Working Paper
RSCAS 2008/25.
[19] Maoz, Y., and Moav, O., 1999. Intergenerational Mobility and the Process of Development. Economic Journal 109, 677–697.
[20] Mookherjee, D., and Napel, S., 2007. Intergenerational Mobility and Macroeconomic
History Dependence. Journal of Economic Theory 137, 49–78.
[21] OECD, 2005. Society at a Glance 2005. OECD, Paris.
[22] OECD, 2006. Society at a Glance 2006. OECD, Paris.
[23] OECD, 2008a. Education at a Glance 2008. OECD, Paris.
[24] OECD, 2008b. Society at a Glance 2008. OECD, Paris.
[25] Ono, T., and Arawatari, R., 2008. Dynamic Political Economy of Redistribution
Policy: The Role of Education Costs. Economics Bulletin 4–13, 1–13.
[26] Owen, A., and Weil, D., 1998. Intergenerational Earnings Mobility, Inequality and
Growth. Journal of Monetary Economics 41, 71–104.
[27] Piketty, T., 1995. Social Mobility and Redistributive Politics. Quarterly Journal of
Economics 110, 551–584.
[28] Roemer, J., 2004. Equal Opportunity and Intergenerational Mobility: Going Beyond
Intergenerational Income Transition Matrices, in Miles Corak (ed.), Generational
Income Mobility in North America and Europe, Cambridge: Cambridge University
Press.
28[29] Schütz, G., Ursprung, H.W., and Wößmann, L., 2008. Education Policy and Equalityof Opportunity. Kyklos 61, 279–308.
[30] Solon, G., 2002. Cross-country Diﬀerences in Intergenerational Earnings Mobility.
Journal of Economic Perspectives 16, 59–66.

29

Figure 1: This ﬁgure is a scatter plot of the Gini coeﬃcients and intergenerational earnings elasticity for 13 OECD countries (Australia, Canada, Germany, Denmark, Spain,
Finland, France, Italy, Japan, Norway, Sweden, the United Kingdom and the UnitedStates). All data except for Japan are from OECD (2006). The Gini coeﬃcient and theintergenerational earnings elasticity of Japan are from OECD (2005) and Lefranc, Ojima
and Yoshida (2008).

30

Figure 2: Timing of events and the distribution of rich and poor.

31

¡

¡

¡

¡

Figure 3: The equilibrium decision rule ut+1 = U (τty ). The solid lines are graphs of
U satisfying equilibrium condition 2. Panel (a) illustrates the case of γ/μ > 1 + β
and ut ∈ (β/(1 + β)(1 − μ/γ), 1]; panel (b) illustrates the case of γ/μ > 1 + β andut ∈ [0, β/(1 + β)(1 − μ/γ)], or the case of γ/μ ≤ 1 + β.

32

Figure 4: The graph of Z under the set of parameters provided in Proposition 1. Panel(a) illustrates the case where ut > û; panel (b) illustrates the case where ut ≤ û.

33

¡ ¡
1¡

¡

Figure 5: The set of (β, ut ) classiﬁed according to the characterization of period-t politicalequilibria.

34

Figure 6: The characterization of dynamic political equilibria. Panel (a) illustrates thecase of β < 1/4; panel (b) illustrates the case of β ≥ 1/4 and û < 1/2; panel (c)illustrates the case of β ≥ 1/4 and û ∈ [1/2, ūp ); panel (d) illustrates the case of β ≥ 1/4
and û ∈ [ūp , 1].

35

Figure 7: The set of parameters (μ/γ, β) classiﬁed according to the characterization of
the steady states.

36

Figure 8: The tax rate on the young (panel (a)) and the tax rate on the old (panel (b)).

37

Figure 9: Intergenerational mobility in the steady state.

38

Figure 10: Information about per capita income, the size of the population and total
income for each type of young agent (panel (a)) and the Lorenz curve (panel (b)).

39

Figure 11: Gini coeﬃcients among young agents.

40

Figure 12: Size of transfer in the steady state.

41