Discussion Papers In Economics
And Business

A Second Chance at Success: A Political Economy
Perspective
Ryo Arawatari and Tetsuo Ono
Discussion Paper 08-04-Rev.

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANA Second Chance at Success: A Political Economy
Perspective
Ryo Arawatari and Tetsuo Ono
Discussion Paper 08-04-Rev.

July 2008

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A Second Chance at Success: A Political Economy
Perspective∗
Ryo Arawatari†and Tetsuo Ono‡
Graduate School of Economics, Osaka University
July 2008

Abstract
This paper characterizes a stationary Markov-perfect political equilibrium where
agents vote over income taxation that distorts educational investment. Agents become rich or poor through educational investment, and the poor have a second
chance at success. The results show the following concerning the cost of a second
chance. First, when the cost is low, the economy is characterized by high levels of
upward mobility and inequality, and a low tax burden supported by the poor withprospects for upward mobility. Second, when the cost is high, there are multiple
equilibria with various patterns of upward mobility, inequality and redistribution.
Numerical examples show that the shift from a high-cost economy to a low-costeconomy may reduce social welfare.
Key words: Second chance; Political economy; Inequality; Upward mobility;
Intragenerational mobility.
JEL Classification: D30; D72; H55.

∗

We would like to thank two anonymous referees, Karl Farmer, John Hassler, Kei-ichi Koda, Ronald
Wendner and seminar participants at Hokkaido University, Kyusyu Sangyo University, Nagoya University,
Osaka University, Osaka Prefecture University, University of Groningen, University of Graz, the Spring
Meeting of the Japanese Economic Association (May 2008, Sendai), the North American Summer Meetingof the Econometric Society (June 2008, Pittsburgh), and the Far Eastern and South Asian Meeting of
the Econometric Society (July 2008, Singapore) for their useful comments and suggestions. Arawatariacknowledges financial support from the Research Fellowships for Young Scientists of the Japan Society
for the Promotion of Science (JSPS). Ono acknowledges financial support from the JSPS through a
Grant-in-Aid for Young Scientists (B) (No. 20730208). All remaining errors are ours.
†
Corresponding author. Ryo Arawatari, Graduate School of Economics, Osaka University, 1-7,
Machikaneyama, Toyonaka, Osaka 560-0043, Japan. Email: ege001ar@mail2.econ.osaka-u.ac.jp‡
Tetsuo Ono, Graduate School of Economics, Osaka University, 1-7, Machikaneyama, Toyonaka, Osaka
560-0043, Japan. Email: tono@econ.osaka-u.ac.jp

1

Introduction

Why do Western countries have different welfare programs, even though they share asimilar background? Who supports redistribution in these countries? How is income
inequality related to redistributive politics? Hassler et al. (2003) and Hassler, Storesletten and Zilibotti (2007) developed a model that responds to these questions based on a
feedback mechanism between individual decisions and redistributive politics (Glomm andRavikumar, 1995; Saint-Paul and Verdier, 1997; Benabou, 2000). In this, the feedback
mechanism creates the joint determination of inequality and redistribution and resultsin multiple equilibria: a pro-welfare state and an anti-welfare state. In the pro-welfare(anti-welfare) state, expectations of higher (lower) redistribution lead to lower (higher)
educational investment, and thus a majority of the poor (rich) support higher (lower)
redistribution. The state realized then depends on the expectations of agents.
The model in Hassler et al. (2003) and Hassler, Storesletten and Zilibotti (2007)
presents cross-country differences in welfare programs among Western countries sharing
similar backgrounds. However, they abstract from intragenerational mobility in theirmodel and hence preclude the predictions of the POUM (prospect of upward mobility)
hypothesis supported by US data (Benabou and Ok, 2001; Alesina and La Ferrara, 2005):
namely, the poor do not support higher redistribution because of a hope for upward
mobility in the future. In order to explain the differences in welfare programs among
Western countries, there is a need to develop a model that includes an equilibrium that
supports the POUM hypothesis as well as multiple equilibria with different patterns ofredistribution, mobility, and inequality.
Independently of the above-mentioned studies, Quadrini (1999) succeeded in providing
a model, including the POUM hypothesis, based on an endogenous growth model that
produces multiple equilibria. This model encompasses a growing equilibrium with highmobility, high inequality and low redistribution (supporting the POUM hypothesis), anda stagnant economy with low mobility, low inequality and high redistribution (supportingthe case of a pro-welfare state). A key to this result is the assumption that the growth rateof the economy affects agents’ preferences over redistribution policies and thus changes
the ability of the agents to know their position in the future distribution of income.
The multiple equilibria in Quadrini (1999) predict a negative correlation between redistribution and inequality, and a positive correlation between mobility and inequality.
However, while the former is supported by empirical evidence (Kristov, Lindert and McClelland, 1992; Rodriguez, 2004), the latter is not necessarily supported by observable
data. Figure 1 illustrates the scatter plot of upward mobility (OECD, 1996) and the Gini
index (Föster and Mira d’Ercole, 2005), which appears to show a positive correlation be-2

tween mobility and inequality among sample countries.1 For example, the United States
features high mobility and inequality and Finland features low mobility and inequality.
The difference between these could be explained by the multiple equilibria in Quadrini(1999). However, other European countries display various patterns of upward mobility
and inequality. For example, for workers aged 35-49 years, inequality is greater in the
United States than in Denmark and Sweden, but the pattern of mobility is similar.2 We
cannot explain this property using the theory in Quadrini (1999).
[Figure 1 about here.]
Motivated by the above-mentioned discrepancy between theory and evidence, we develop a model that includes the equilibrium supporting the POUM hypothesis as well as
multiple equilibria describing different patterns of mobility and inequality among European countries. For this purpose, the paper utilizes a politicoeconomic framework thatincorporates the endogenous determination of income distribution by individuals’ educational investment (Hassler, Storesletten and Zilibotti, 2007) based on the concept of astationary Markov-perfect equilibrium.3 We extend their model by introducing a second
chance at success. Agents who live for two periods, youth and old age, can become richor poor by undertaking costly investment in their youth. Successful agents can retain
this status over their life cycle. Unsuccessful agents, however, have second chances andthus can become rich in old age through reinvestment in education. Within this extended
framework, we explore: (i) the relation between upward mobility and inequality and (ii)the welfare implications of upward mobility in society.
We obtain the following two results not shown by previous research. First, we consider
majority voting over redistribution policy and obtain the following result regarding theequilibrium properties. When the costs of a second chance are low, there is a unique, poormajority equilibrium characterized by a low tax burden on the decisive voter and highlevels of upward mobility and inequality: this supports the POUM hypothesis. Although
the majority are poor, they support a low tax burden for redistribution because of thehope for upward mobility in the future. In contrast, when the costs of a second chance are
high, there are multiple equilibria. The first is a poor-majority equilibrium featuring lower
1

We include limited, old data here because the available data on upward mobility are restricted to theeight countries provided by the OECD (1996) for the period 1986–1991.
2
The observation for Denmark, Sweden, and the United States is supported by empirical work by
Aaberge et al. (2002), who showed that inequality is greater in the United States than in Scandinavian
countries (Denmark, Norway and Sweden), even though the pattern of earnings mobility is remarkably
similar.
3
Examples of studies utilizing a stationary Markov-perfect equilibrium include Krusell, Quadrini andRı́os-Rull (1997), Grossman and Helpman (1998), Azariadis and Galasso (2002), Hassler et al. (2003),
Forni (2005), Hassler et al. (2005), Song (2005), Gonzalez-Eiras and Niepelt (2008), Hassler, Storesletten
and Zilibotti (2007), and Bassetto (2008).

3

levels of mobility and inequality; the second is a rich-majority equilibrium featuring higher
levels of mobility and inequality. In the rich-majority equilibrium, mobility depends onthe costs of a second chance, and the tax rate on the young is indeterminate in that it
depends on the expectations of agents. The multiplicity of equilibria and the differences in
the costs and expectations jointly provide an explanation for different patterns of mobility
and inequality in European countries.
Second, we show the welfare implications of policies enhancing upward mobility. Mobility may be a good thing because higher upward mobility means that the current statusof the poor is less persistent. However, mobility may also be viewed as a bad thing because
higher mobility means larger income fluctuations and thus economic insecurity (Atkinson, Bourguignon and Morrisson, 1992). Our framework captures the former aspect bymodeling upward mobility and characterizing an equilibrium that supports the POUM
hypothesis, but abstracts from the latter aspect by assuming linear utility functions andno downward mobility. Therefore, an intuitive prediction is that upward mobility is goodfor the economy. However, our analysis shows that a low-cost economy characterizedby high upward mobility may be inferior to a high-cost economy featuring low upward
mobility, at least in terms of social welfare.
The key to the second result is the education disincentive effect. The lower cost ofa second chance implies a higher probability of being successful in old age. This gives
agents a disincentive to invest in education in their youth and results in a lower number ofsuccessful young agents and thus a smaller tax base. To keep tax revenue from the young,the decisive voter imposes a higher tax rate on successful young. The smaller number ofsuccessful young agents combined with a higher tax rate on the young jointly reduces the
expected utility of the young and thus social welfare.
The organization of this paper is as follows. Section 2 develops the model. Section
3 characterizes the political equilibrium based on the concept of a stationary Markovperfect equilibrium. Sections 4 and 5 undertake the numerical analysis. Section 4 analyzes
the correlation between inequality and upward mobility, while Section 5 investigates thewelfare implications of policies that reduce the costs of a second chance. Section 6 shows
how the analysis and the results are changed when alternative assumptions are employed.
Section 7 provides some concluding remarks. All proofs are given in the Appendix.

2

The Model

The model is based on Hassler, Storesletten and Zilibotti (2007). Time is discrete anddenoted by t = 1, 2, · · · . The economy consists of a continuum of agents living for two
periods, youth and old age. Each generation has a unit mass. Agents are identical at4birth. Young agents can affect their prospects in life with educational investment. In
particular, they either become rich or poor, and by undertaking costly investment, canincrease the probability ey of becoming rich in youth. The cost of investment, which ismeasured in terms of disutility, is given by (ey )2 .
At the beginning of each period, there are two types of old agents: those who weresuccessful in their youth and those who were not. Successful agents can retain this statusthroughout their lifetime; unsuccessful agents have a second chance at success. A proportion e0 ∈ (0, 1) of those who were unsuccessful in youth become rich in old age by
undertaking costly investment e0 . The cost of investment is given by ηo · (eo )2 , ηo > 0.
The parameter ηo represents the cost of getting a second chance. A lower ηo leads to alower cost of reinvestment and thus a higher probability of being successful in old age.
Rich agents earn a high wage, normalized to unity, whereas poor agents earn a low wage,
normalized to zero. Figure 2 illustrates the timing of events and the distribution of therich and poor.
[Figure 2 about here.]
Following conventional terminology, we refer to the first period of life as youth andthe second period as old age. Educational investment in youth, ey , is interpreted as study
at tertiary education institutions such as universities and college; educational investment
in old age, e0 , is interpreted as study at professional schools for career development orefforts at skill accumulation in on-the-job training. An individual may start in a lowpaying job, but with continued opportunity for training, he/she will be able to advanceinto a higher-paying position or move to a different, higher-paying job.
There is no storage technology in this economy. Each individual uses his/her endowments within the period. The government provides lump-sum transfers, s, financed bytaxes 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 determined before the young agents decide on their investment.
Therefore, the expected utility functions of agents alive at time t are given as follows:
Vtos = (1 − τto ) + st ,(1)

Vtou = eot · (1 − τto ) − ηo · (eot )2 + st ,(2)

Vty = eyt · (1 − τty ) − (eyt )2 + st
¤
£
©
ª
o
o
+ β eyt · (1 − τt+1
) + (1 − eyt ) eot+1 · (1 − τt+1
) − ηo · (eot+1 )2 + st+1 ,(3)

where Vtos , Vtou and Vty denote the utility of the old who were successful in youth, theutility of the old who were unsuccessful in youth, and the utility of the young. The utility
levels of Vtou and Vty are computed prior to individual success or failure. The parameterβ ∈ (0, 1) is a discount factor.
5Given these preferences, an old agent who was unsuccessful in youth has a second
chance at success, and thus chooses eot to maximize Vtou ; and a young agent in period tchooses eyt to maximize Vty by taking account of the optimal investment in his/her oldage. Therefore, optimal investments by the old and the young are given by, respectively:eo∗(1 − τto )o(τt ) =
,
2ηo

e

y∗

1
o
(τty , τt+1
)= ·
2

·
(1 − τty ) + β

½
¾¸
1
o
o
2
(1 − τt+1 ) −.
· (1 − τt+1 )4ηo

We make the following assumption.
Assumption 1: ηo > 1.
This assumption means that the cost of education in old age is higher than in youth:
this reflects the fact that it is harder for agents to get educated as they grow older. Theoassumption is sufficient to ensure that eo∗ (τto ) and ey∗ (τty , τt+1
) are set within the range
4(0, 1).
Because young agents are ex ante identical, agents of the same cohort choose the sameinvestment, implying that the proportion of the old who were unsuccessful in youth isgiven by:y∗

ut+1 ≡ 1 − e1o(τty , τt+1)=1− ·
2·½(1 − τty ) + βo(1 − τt+1)−

1o· (1 − τt+1
)2
4ηo

¾¸
.

Thus, the proportion of the old who were unsuccessful in youth, ut+1 , depends on the tax
levied on the successful young agents in period t, τty , and the tax levied on the successfuloold agents in period t + 1, τt+1.
The tax revenues from successful agents are transferred to every agent in a lump-sum
fashion. The government budget is balanced in each period so that it can be expressedas:o) · τty
2st = [(1 − ut ) + ut · eo∗ (τto )] · τto + ey∗ (τty , τt+1o),
= W (τto , ut ) + Z(τty , τt+1

where W (τto , ut ) ≡ [(1 − ut ) + ut · eo∗ (τto )]·τto is the revenue financed by the rich old agents,oo) · τty is the tax revenue financed by rich young agents.5) ≡ ey∗ (τty , τt+1
and Z(τty , τt+1
4

Expanding the range of ηo from (1, ∞) to (1/2, ∞) does not qualitatively affect the result shown
below. However, we obtain a new result when ηo is set below 1/2: the decisive voter will always prefer notaxation on the old: τto = 0. All of the old agents can become rich after all because eo∗ (0) = 1/2ηo ≥ 1
if ηo ≤ 1/2. Because this result is trivial, we limit the range of ηo to (1, ∞).
5
An alternative method of transfer is a targeted transfer to old or poor agents. In addition, we can63Political Equilibrium

This section characterizes a political equilibrium where agents vote on taxation period
by period. Section 3.1 provides the definition of a political equilibrium based on theconcept of a stationary Markov-perfect equilibrium with majority voting. Sections 3.2
and 3.3 provide the characterization of the political equilibrium classified according to
the pattern of taxation on the old and the distribution of income.

3.1

Definition of Political Equilibrium

Following Hassler, Storesletten and Zilibotti (2007), we assume that agents vote on current
taxes at the beginning of each period but that only the old vote. At the time of voting,young agents are identical, while old agents are heterogeneous; there are two types, those
who were successful in their youth and those who were unsuccessful. Therefore, ourassumption focuses on the conflict between the two types of old agents. For more discussion
about the role of this assumption, see Section IIB in Hassler et al. (2003) and Footnote
6 in Hassler, Storesletten and Zilibotti (2007).
oWith the optimal investments eo∗ (τto ) and ey∗ (τty , τt+1
) and the government budget
constraint, the indirect utility of the old who were successful and unsuccessful in youth
are given by, respectively:¢
1 ¡o· W (τto , ut ) + Z(τty , τt+1
) ,
2
o 2
¢
(1
−
τ
1 ¡
)
t
o
Vtou =+ · W (τto , ut ) + Z(τty , τt+1
) ,
4ηo
2
Vtos = (1 − τto ) +where the term in the first line, (1 − τto ), is the after-tax income of the old who weresuccessful in youth, the term in the second line, (1 − τto )2 /4ηo , is the expected net benefit
from the second challenge for the old who were unsuccessful in youth, and the term
(W + Z)/2 is the lump-sum transfer.
The current paper focuses on stationary Markov-perfect equilibria with majority voting. The proportion of the old who were unsuccessful in youth (ut ) summarizes the state of
the economy; the identity of 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 now provide the definition of the political equilibrium as follows.
Definition: A (stationary Markov-perfect) political equilibrium is defined as a triplet
consider alternative means of government spending such as the provision of public goods. However, weabstract from these alternatives and adopt a lump-sum transfer to all agents in order to keep the analysis
as simple as possible and to focus our attention on the political determination of taxes affected by thepresence of a second chance.

7

of 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 u0 , the following functional equations hold.
1. T o (ut ) = arg maxτto ∈[0,1] W dec (τto , ut ) (dec = os, ou), where:
(
W

dec

(τto , ut ) =W os ≡ (1 − τto ) + 12 · W (τto , ut ) if ut ≤ 1/2,
W ou ≡(1−τto )2+ 12 · W (τto , ut )4ηoif ut > 1/2.

oo
2. U (τty ) = 1 − ey∗ (τty , τt+1
= T o (U (τty )).
), with τt+1oo
3. T y = arg maxτto ∈[0,1] Z(τty , τt+1
) subject to τt+1
= T o (U (τty )).

The first equilibrium condition requires the decisive voter to choose τto to maximize theutility of the old who were successful (if ut < 1/2) or unsuccessful (if ut > 1/2) in youth. In
the case of an equal number of successful and unsuccessful agents (i.e., ut = 1/2), the old
who were successful in youth are assumed to be decisive. The second equilibrium conditionoimplies that all young individuals choose their investment optimally, given τty 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 revenue
from the young. Rational voters 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 now solve the equilibrium conditions recursively. Condition 1 defines a one-to-one
mapping from the state variable to the equilibrium choice of taxation of the old: τto =T o (ut ). Suppose that the majority are the old who were successful in youth: ut ≤ 1/2.
The objective function of the majority is given by W os (τto , ut ) = (1 − τto ) + W (τto , ut )/2
that is strictly decreasing in τto : ∂W dec (τto , ut )/∂τto < 0.6 . This implies that the old who
were successful in youth pay more than they receive because the unsuccessful agents pay
no tax, but the revenue is distributed equally between the successful and unsuccessful
agents. Therefore, the old who were successful in youth prefer τto = 0.
Alternatively, suppose that the majority are the old who were unsuccessful in youth:
ut > 1/2. The objective function of the majority is given by W ou (τto , ut ) = (1 − τto )2 /4ηo +W (τto , ut )/2 that is strictly convex in τto : ∂ 2 W dec (τto , ut )/∂τto2 = (1 − ut )/2ηo > 0. This
6

When ut ≤ 1/2, the differentiation of W dec with respect to τto is:∂W dec /∂τto = (−1) · [1 − {(1 − ut ) + ut (1 − τto )/2ηo } /2] − ut τto /4ηo ,where (1 − ut ) + ut (1 − τto )/2ηo < 1 under Assumption 1. Therefore, we have ∂W dec /∂τto < 0 if ut ≤ 1/2.

8

implies that T o (ut ) = 0 if W ou (0, ut ) ≥ W ou (1, ut ), i.e., if ut ≥ 1 − 1/2ηo ; and T o (ut ) = 1
if W ou (0, ut ) ≤ W ou (1, ut ), i.e., if ut ≤ 1 − 1/2ηo . In order to understand the result, let
us focus on the condition that derives T o (ut ) = 0. This condition means that the size ofthe tax base (1 − ut ) is small and the cost of education aimed at a second chance (ηo ) islow. The unsuccessful agents expect a low return from taxation on the old and a highprobability of being taxed in old age. Therefore, they prefer τto = 0 if ut ≥ 1 − 1/2ηo .
The mapping that satisfies equilibrium condition 1 is summarized as follows:(
T o (ut ) =0
1

if ut ≤ 1/2 or 1 − 1/2ηo ≤ ut ≤ 1,
if 1/2 < ut ≤ 1 − 1/2ηo .

(4)[Figure 3 about here.]
Figure 3 illustrates (4) that satisfies equilibrium condition 1. Suppose that the decisive
voter is an old agent who was successful in youth: ut ≤ 1/2. He/she sets τto = 0 because
his/her marginal cost of taxation is greater than his/her marginal benefit of taxation,∂(W/2)/∂τto . Alternatively, suppose that the decisive voter is an old agent who was
unsuccessful in youth: ut > 1/2. He/she sets either τto = 0 or 1 depending on themagnitude of the correlation between the marginal benefit of taxation, ∂(W/2)/∂τto , andthe marginal cost of taxation, (1 − τto )/2ηo . The marginal benefit is smaller than themarginal cost for ut ∈ [1 − 1/2ηo , 1]; and the marginal benefit is larger than the marginal
cost for ut ∈ (1/2, 1 − 1/2ηo ].7
Next, we rewrite equilibrium condition 2 by substituting in the optimal investmentoey∗ (τty , τt+1). This yields the following functional equation:

1
U (τty ) = 1 − ·2·½(1 − τty ) + β ·(1 − To(U (τty ))) −1· (1 − T o (U (τty )))2
4ηo

¾¸
,

(5)

where T o (·) ∈ {0, 1} is given by (4). Because T o (·) ∈ {0, 1}, any solution of (5) must be
a combination of the two linear functions: U (τty ) = 1 − (1 − τty + β · (1 − 1/4ηo )) /2 andU (τty ) = 1 − (1 − τty )/2.
We derive the solution to the functional equation (5) by assuming rational expecta-

7

Our model approximates Hassler, Storesletten and Zilibotti (2007) by assuming the prohibitively
high cost of education, ηo → ∞. Under this approximation, the critical level of ut illustrated in Figure
3, 1 − 1/2ηo , approaches one; the unsuccessful agents no longer prefer τto = 0 if ηo → ∞. Therefore,
Hassler, Storesletten and Zilibotti (2007) abstracted from the case where unsuccessful agents prefer notaxation on the old. In contrast, we produce this case by introducing a second chance into their model.
It is shown below that the case supports the POUM hypothesis proposed by Benabou and Ok (2001).

9

tions:U (τty ) =

³
³
´´
y
1
1

1 − 2 · (1 − τt ) + β · 1 − 4ηo








1 − 12 · (1 − τty )

h
³
´i
1
0, β 1 − 4ηo , or
h
n ³
´ ³
´ o i
τty ∈ max β · 1 − 4η1o + 1 − η1o , 1 , 1 ,hiif τty ∈ 0, 1 − η1o .
if τty ∈Interpretation of the solution is as follows. Suppose that young agents in periodot expect τt+1
= 0. Under this expectation, young agents choose their investment asy∗ ye (τt , 0) = (1 − τty + β · (1 − 1/4ηo )) /2. By (4), this expectation is rational if 1 −ey∗ (τty , 0) ≤ 1/2 or 1 − 1/2ηo ≤ 1 − ey∗ (τty , 0) ≤ 1, that is, if τty ≤ β · (1 − 1/4ηo ) or
β · (1 − 1/4ηo ) + (1 − 1/ηo ) ≤ τty . Suppose, instead, that young agents in period t expectτt+1 = 1. Under this expectation, young agents choose their investment as ey∗ (τty , 1) =(1 − τty )/2. By (4), their expectation is rational if 1/2 < 1 − ey∗ (τty , 1) ≤ 1 − 1/2ηo , thatis, if τty ≤ 1 − 1/ηo .
The solution depends on the expectations of agents. Figure 4 illustrates four possible
cases of U that satisfy the second equilibrium condition. These cases are classified according to the magnitude of the correlation between β · (1 − 1/4ηo ) + (1 − 1/ηo ) and 1,
and between β · (1 − 1/4ηo ) and (1 − 1/ηo ).
[Figure 4 about here.]
As depicted in Figure 4, there are multiple, self-fulfilling expectations of U for acertain set of τty . Which U arises in equilibrium depends on the expectations of agents.
To illustrate U in equilibrium, we introduce the critical rate of τty : θ ≤ min{1 − 1/η0 , β ·(1 − 1/4ηo )}. The rate θ, which depends on the expectations of agents, is the highest tax
rate that yields a majority of the old who were successful in youth for the range of τty thatyields multiple expectations. For τty > θ, the majority is the old who were unsuccessful
in youth. However, for τty ∈ (0, θ), the majority are either successful or unsuccessful
depending on agents’ expectations. The function is thus reduced as follows:
U (τ y ) =
 nt
³
³
´´
o
y
y
1
1
1

·
(1
−
τ
)
+
β
·
1
−
,
1
−
·
(1
−
τ
)
1
−

t
t
2
4ηo
2




y
1

1 − 2 · (1 − τt )
³
³
´´
y
1
1

1 − 2 · (1 − τt ) + β · 1 − 4ηo







for τty ∈ (0, θ],³
ifor τty ∈ θ, 1 − η1o ,
³
³
´´
y
1
1
for τt ∈ 1 − ηo , β · 1 − 4ηo
, or
h ³
´ ³
´ i
τty ∈ β · 1 − 4η1o + 1 − η1o , 1 .
(6)

10

The third solution, U (τty ) = 1−((1 − τty ) + β · (1 − 1/4ηo )) /2, can be given provided thatthe set (1 − 1/ηo , β · (1 − 1/4ηo )) or [β · (1 − 1/4ηo ) + (1 − 1/ηo ) , 1] is nonempty.
Figure 4 reveals the following two features. First, if τty ≤ θ, there can be multiple,
self-fulfilling expectations. If agents expect not to be taxed when old, they choose highinvestment, and this leads to a majority of agents who were successful in youth; theyowill set τt+1
= 0. If, instead, they expect high future taxes, they choose low investment,
and this leads to a majority of the agents who were unsuccessful in youth; they will set
o
τt+1
= 1.
Second, suppose that the decisive voter chooses a sufficiently large τty such that τty ≥β · (1 − 1/4ηo ) + (1 − 1/ηo ) (see Panels (a) and (c)). In this case, the majority in thenext period will be unsuccessful agents, but they will choose no taxation on the old:o
τt+1
= 0. The intuition of this result is as follows. Suppose that the proportion of agents
who were unsuccessful in youth is high such that ut ∈ [1 − 1/2η0 , 1]. By (4), the decisive
voter chooses no taxation on the old: τto = 0. Given τto = 0, if the decisive old voter
chooses a high tax rate on the young, this choice gives young individuals a disincentive
to invest in education, thereby resulting in a higher proportion of unsuccessful agents:
ut+1 ∈ [1 − 1/2η0 , 1]. By (4), the tax rate on the old in the next period is given byoτt+1= 0: this satisfies the stationary property of the function T o .

3.3

The Determination of T y and the Characterization of the
Political Equilibria

Given the characterization of T o and U satisfying equilibrium conditions 1 and 2, respectively, we now consider the political determination of τty that satisfies equilibrium
condition 3. Because there are two possible cases of a majority, we introduce corresponding definitions of the political equilibria: a poor-majority equilibrium and a rich-majority
equilibrium. When the majority are poor (i.e., the old who were unsuccessful in youth),othere are two sorts of equilibria: a poor-majority equilibrium where agents expect τt+1=0yand choose τt to induce a majority of poor at time t + 1, and a poor-majority equilibriumo= 1 and choose τty to induce a majority of poor at time t + 1.
where agents expect τt+1
When the majority are rich (i.e., the old who were successful in their youth), there is ao
= 0 and choose τty to induce a majority
rich-majority equilibrium where agents expect τt+1
of rich at time t + 1.
We first characterize a poor-majority equilibrium with no taxation on the old.
Proposition 1(i) Suppose that β ≤ (2/ηo − 1)/(1 − 1/4ηo ) holds. There exists a set of poor-majority
equilibria with no taxation on the old such that ∀t, T o is given by (4), U (τty ) is given11

by (6), and T y = (1 + β · (1 − 1/4ηo )) /2. The equilibrium outcome is unique, suchthat ∀t, τty = (1 + β · (1 − 1/4ηo )) /2, τto = 0, ut = 1 − (1 + β · (1 − 1/4ηo )) /4, and2st = {1 + β · (1 − 1/4ηo )}2 /8.
(ii) Suppose that β > (2/ηo − 1)/(1 − 1/4ηo ), β ≥ {ηo /4 − (1 − 1/ηo )}/(1 − 1/4ηo ),and β ≤ (1/ηo )/(1 − 1/4ηo ) hold. There exists a set of poor-majority equilibria
with no taxation on the old such that ∀t, T o is given by (4), U (τty ) is given by(6), and T y = β · (1 − 1/4ηo ) + (1 − 1/ηo ). The equilibrium outcome is unique,such that ∀t, τty = β · (1 − 1/4ηo ) + (1 − 1/ηo ), τto = 0, ut = 1 − 1/2ηo , and2st = {β · (1 − 1/4ηo ) + (1 − 1/ηo )}/2ηo .
Proof. See the Appendix.
[Figure 5 about here.]
The area P.1(i) in Figure 5 indicates the set of parameters (β, ηo ) satisfying theequilibrium condition in Proposition 1(i). The assumption of Proposition 1(i), β ≤(2/ηo − 1)/(1 − 1/4ηo ), implies that it is rational for the decisive voter to choose aninterior solution τty = arg maxτty ∈[0,1] Z(τty , 0) = (1 + β · (1 − 1/4ηo ))/2 under the expectaotion of τt+1
= 0. The young are taxed at the top of the Laffer curve, conditional on their
expectation of no taxation when old (see Panel (a) of Figure 6).
[Figure 6 about here.]
The area P.1(ii) in Figure 5 illustrates the set of parameters (β, ηo ) satisfying theequilibrium condition in Proposition 1(ii). The first assumption of Proposition 1(ii),β > (2/ηo − 1)/(1 − 1/4ηo ), implies that the interior solution is irrational: the decisive
voter is unable to set the interior solution but is able to set a corner solution τty =β · (1 − 1/4ηo ) + (1 − 1/ηo ). Under this situation, there are (at most) two alternative
options that may dominate the corner solution: voting that induces a future majority ofpoor (unsuccessful) by setting τty = arg max Z(τty , 1) = 1/2; and voting that induces afuture majority of rich (successful) by setting τty = β·(1−1/4ηo ) or τty = θ(≤ β·(1−1/4ηo )).
These alternatives cannot dominate the corner solution if they provide less tax revenue
from the young; that is, the second and third assumptions in Proposition 1(ii) hold.
As illustrated in Figure 5, given the discount factor β, the poor-majority equilibriumwith no taxation on the old requires a low ηo , i.e., a low cost of a second chance. A
low ηo implies a high probability of being successful in old age, which gives individuals adisincentive to invest in education in youth. The unsuccessful agents form a majority, butthey do not support a high tax rate (i.e., 100% taxation) on the old because of the hope for
upward mobility through a second chance. The unsuccessful agents take into account the12

fact that with a high probability they can move up the income distribution and therefore
be harmed by the high tax rate. Hence, given the low cost of a second chance, there is aunique political equilibrium outcome that supports the POUM hypothesis.
Next, we provide a characterization of a poor-majority equilibrium with 100% taxationon the old. For this characterization, we introduce a critical level of θ, θ̃, such that
Z(1/2, 1) ≷ Z(θ, 0) if and only if θ ≶ θ̃, where:³e ηo ) ≡
θ(β,

1+β·

1 − 4η1o

r

´
−

³
´ n ³
´
o
β · 1 − 4η1o · β · 1 − 4η1o + 2
2.

Proposition 2
Suppose that (i) (1/ηo )/(1 − 1/4ηo ) < β ≤ (1 − 1/ηo )/(1 − 1/4ηo ) or (ii) β ≤(1/ηo )/(1 − 1/4ηo ) and β < {ηo /4 − (1 − 1/ηo )}/(1 − 1/4ηo ). Given the expectationof θ ∈ (0, θ̃), there exists a set of poor-majority equilibria with 100% taxation on
the old such that ∀t, T o is given by (4), U (τty ) is given by (6), and T y = 1/2. The
equilibrium outcome is unique and such that ∀t, τty = 1/2, τto = 1, ut = 3/4, and2st = 3/8.
Proof. See the Appendix.
The area P.2 in Figure 5 indicates the set of parameters that attains a poor-majority
equilibrium with 100% taxation on the old. From this figure, we can see that β ≤ (1 −1/ηo )/(1 − 1/4ηo ) is a necessary condition for the existence of a poor-majority equilibriumwith 100% taxation on the old. In order to understand the role of this necessary condition,
suppose alternatively that β > (1 − 1/ηo )/(1 − 1/4ηo ) holds. In this case, it is rational forothe decisive voter to set τty = β · (1 − 1/4ηo ) under the expectation of τt+1
= 0. Because
the condition β > (1 − 1/ηo )/(1 − 1/4ηo ) is rewritten as Z (β · (1 − 1/4ηo ), 0) > Z(1/2, 1),osetting τty = 1/2 under the expectation of τt+1
= 0 is not sustained as an equilibrium.
Based on this argument, we suppose that β ≤ (1−1/ηo )/(1−1/4ηo ) holds, and we seek
additional conditions that ensure the existence of a poor-majority equilibrium with 100%
taxation on the old. Under the current situation, there are at most two alternative options
that may dominate the choice of τty = 1/2: setting τty = θ(≤ β · (1 − 1/4ηo )) under theo= 0 and setting τty = β · (1 − 1/4ηo ) + (1 − 1/ηo ) under the expectation
expectation of τt+1o
= 0 (see Panel (b) of Figure 6). The condition θ ∈ (0, θ̃) implies that the formerof τt+1
alternative provides less tax revenue, condition (i) in Proposition 2 implies that the latter
alternative is irrational, and condition (ii) implies that the latter alternative is rational
but provides less tax revenue. Therefore, given θ ∈ (0, θ̃), the choice of τty = 1/2 undero= 1 is sustained as an equilibrium when the one of conditions (i)the expectation of τt+1
and (ii) hold.

13

As illustrated in Figure 5, the condition given in Proposition 2 requires a low value ofβ and a high value of ηo . First, a low value of β implies that agents attach a low weight tothe utility in their old age, thereby having a weak incentive to invest in education in their
youth. This results in a larger number of the old who were unsuccessful in youth, thereby
leading to a majority of the poor. Second, a high value of ηo implies a low probability of
being successful in old age for the old who were unsuccessful in their youth. This provides
them with a low prospect of upward mobility through a second chance and gives them an
incentive to prefer 100% taxation on the rich old. Therefore, given a set of a low β anda high ηo satisfying the conditions in Proposition 2, there exists a unique poor-majority
equilibrium featuring 100% taxation on the old.
Finally, we provide the condition for the existence of a rich-majority equilibrium.
Proposition 3(i) Suppose that β > max{(1/ηo )/(1 − 1/4ηo ), (1 − 1/ηo )/(1 − 1/4ηo )} holds. There exists
a set of rich-majority equilibria such that ∀t, T o is given by (4), U (τty ) is given by(6), and T y = β · (1 − 1/4ηo ). The equilibrium outcome is unique and such that ∀t,τty = β · (1 − 1/4ηo ), τto = 0, ut = 1/2, and 2st = β · (1 − 1/4ηo )/2.
(ii) Suppose that β > (1/ηo )/(1 − 1/4ηo ), β ≥ (1/4)/(1 − 1/4ηo ) and β ≤ (1 − 1/ηo )/(1 −1/4ηo ) holds. There exists a set of rich-majority equilibria such that ∀t, T o is givenby (4), U (τty ) is given by (6), and T y = θ ∈ [θ̃, β · (1 − 1/4ηo )]. The equilibriumoutcome is indeterminate, such that ∀t, τty = θ, τto = 0, ut = 1 − (1 − θ + β · (1 −1/4ηo ))/2, and 2st = (1 − θ + β · (1 − 1/4ηo ))θ/2.
Proof. See the Appendix.
The area P.3(i) in Figure 5 indicates the set of parameters (β, ηo ) satisfying the equilibrium condition in Proposition 3(i). The assumption β > (1 − 1/ηo )/(1 − 1/4ηo ) impliesothat it is rational for voters to set τty = β · (1 − 1/4ηo ) under the expectation of τt+1
= 0.
When this choice is available, there are at most two alternatives that may dominateo= 1 and setting
the current choice: setting τty = 1/2 under the expectation of τt+1yy
τt = (1 + β · (1 − 1/4ηo )) /2 or τt = β · (1 − 1/4ηo ) + (1 − 1/ηo ) under the expectationo= 0. The first alternative always provides less tax revenue; the second alternative
τt+1
is irrational under the assumption of β > (1/ηo )/(1 − 1/4ηo ). Therefore, the choice ofo= 0 is sustained as an equilibrium whenτty = β · (1 − 1/4ηo ) under the expectation of τt+1
the condition in Proposition 3(i) holds.
The area P.3(ii) in Figure 5 indicates the set of parameters (β, ηo ) satisfying theequilibrium condition in Proposition 3(ii). The third assumption β ≤ (1−1/ηo )/(1−1/4ηo )implies that it is irrational for voters to set τty = β · (1 − 1/4ηo ) under the expectation14

oof τt+1= 0; the first assumption β > (1/ηo )/(1 − 1/4ηo ) implies that it is irrational for
voters to set τty = (1 + β · (1 − 1/4ηo )) /2 or τty = β · (1 − 1/4ηo ) + (1 − 1/ηo ) under theoexpectation τt+1
= 0. The remaining alternative is to set τty = 1/2 under the expectationoof τt+1= 1. This alternative provides less tax revenue than the current choice if Z(θ, 0) ≥Z(1/2, 1), i.e., if θ ∈ [θ̃, β · (1 − 1/4ηo )]. This set is nonempty if the second assumption,β ≥ (1/4)/(1 − 1/4ηo ), holds. Therefore, given θ ∈ [θ̃, β · (1 − 1/4ηo )], the choice of τty = θounder the expectation of τt+1
= 0 is sustained as an equilibrium when the condition in

Proposition 3(ii) holds.
The next proposition summarizes the results established thus far. The proof of this
proposition is immediately obvious from Figure 5.
Proposition 4(i) If β ∈ [{ηo /4 − (1 − 1/ηo )}/(1 − 1/4ηo ), (1/ηo )/(1 − 1/4ηo )], there exists a uniquepoor-majority equilibrium with no taxation on the old as in Proposition 1.
(ii) If β > max{1/ηo /(1 − 1/4ηo ), (1 − 1/ηo )/(1 − 1/4ηo )}, there exists a unique richmajority equilibrium with no taxation on the old as in Proposition 3(i).
(iii) If β > (1/ηo )/(1 − 1/4ηo ), β ≥ (1/4)/(1 − 1/4ηo ) and β < (1 − 1/ηo )/(1 − 1/4ηo ),the equilibrium is indeterminate. There exists both a poor-majority equilibrium with
100% taxation on the old as in Proposition 2 and a set of rich-majority equilibria
as in Proposition 3(ii).
(iv) If β < {ηo /4 − (1 − 1/ηo )}/(1 − 1/4ηo ) and β ≤ (1/4)/(1 − 1/4ηo ), there exists a
unique poor-majority equilibrium with 100% taxation on the old as in Proposition 2.
By focusing on the parameter ηo , we now provide an interpretation of the result established in Proposition 4. To facilitate understanding, we assume the following economic
environment. We take a generation to be 20 years in length because the current model
assumes that agents work for both periods of their life. The first and the second periods
correspond to, for example, ages 21–40 and 41–60 years, respectively. Our selection of
β is based on the single period discount rate provided by Kydland and Prescott (1982)
of 0.96. Because the agents in our model plan over generations that span 20 years, we
discount the future by (0.96)20 .
Figure 5 illustrates the equilibrium properties under the current economic environment. There is a threshold level of ηo , 2.5124, which determines the equilibrium properties. When ηo is below the threshold level, the unsuccessful agents can succeed in asecond chance with a high probability. This implies that unsuccessful agents have a highprospect of upward mobility in the future, which gives the young a weak incentive to15

invest in education. Therefore, a low ηo leads to a large number of agents who were unsuccessful in youth. They prefer no taxation on the old because given a high probabilityof being successful in old age, they wish to avoid being taxed. The POUM hypothesis is
supported in this case.
When ηo is above the threshold level, there is no equilibrium that supports the POUM
hypothesis. This is because given a high cost of a second chance, there is no possibility ofchanging status from poor to rich by attempting a second chance. The agents who were
unsuccessful in youth are expected to remain poor in old age. Therefore, for ηo > 2.5124,
there is a poor-majority equilibrium with 100% taxation on rich old agents. However,there is also a rich-majority equilibrium because of the multiple self-fulfilling expectations. Which particular equilibrium is realized depends on the expectations of agents.
Expectations of 100% (0%) taxation on the old lead to low (high) educational investment
and thus a majority of the poor (rich); the majority of the poor (rich) then support 100%(0%) taxation and thus high (low) redistribution.

4

Upward Mobility and Inequality

This section examines the correlation between upward mobility and inequality. Section 4.1
undertakes a numerical analysis of upward mobility and inequality. Section 4.2 explores
the empirical implications of the numerical results.

4.1

Numerical Analysis

We define the Gini coefficient as a measure of inequality in the current framework andthen undertake a numerical analysis under the economic environment of β = (0.96)20 .
We calculate the Gini coefficient in terms of after tax-and-transfer income (see the Appendix for the method of calculation). Figure 7 illustrates the numerical results of the
upward mobility rate (i.e., the probability of being successful in old age, eo ) and the Gini
coefficient.
The result illustrated in Figure 7 provides the following prediction for upward mobility
and inequality. When the cost is low, such that ηo < 2.5124, the economy attains a uniquepoor-majority equilibrium featuring high levels of upward mobility and inequality. The
low cost of a second chance then enhances the upward mobility of unsuccessful agents.
However, it gives young individuals a disincentive to invest in education, which results ina greater number of unsuccessful young agents and thus a larger level of inequality among
them. Therefore, the lower cost of a second chance results in higher levels of upward
mobility and inequality.
[Figure 7 about here.]16

When the cost of a second chance is high, such that ηo ≥ 2.5124, the economy attains multiple equilibria depending on the expectations of agents: one is a poor-majority
equilibrium featuring lower levels of upward mobility and inequality; the other is a richmajority equilibrium featuring higher levels of upward mobility and inequality. In thepoor-majority equilibrium, agents expect higher redistribution in the future, and this expectation is realized as 100% taxation on the old. The upward mobility rate becomes zero,and thus old agents are equal in the sense that all of them obtain st units of after taxand-transfer income. In contrast, in the rich-majority equilibrium, agents expect lower
redistribution in the future, and this expectation is realized as no taxation on the old.
The upward mobility rate is given by eot = 1/2ηo , and thus there is inequality among old
agents. Therefore, the rich-majority equilibrium attains higher levels of upward mobility
and inequality compared with the poor-majority equilibrium.
A noteworthy feature of the result in Figure 7 is that a decrease in ηo around thethreshold level of ηo = 2.5124 leads to a sharp increase in inequality. The mechanism
behind this is as follows. The objective of the decisive voter is to maximize tax revenue
from the young. This objective is achieved by: (i) the larger size of the tax base associatedwith a lower tax rate that enhances educational investment by the young, or (ii) the smaller
size of the tax base associated with a higher tax rate that gives the young a disincentive
to invest in education. When ηo is above the threshold level, the young have an incentive
to invest in education because they expect a lower probability of being successful in oldage. The decisive voter can expect a large number of successful young agents and thus
attempts to maximize the tax revenue from the young based on the first option. However,when ηo is below the threshold level, the young have a disincentive to invest in education
because they expect upward mobility in the future with a high probability. The decisive
voter then maximizes revenue from the young based on the second option. Therefore, at
ηo = 2.5124, a reduction in ηo leads to a discontinuous increase in the tax rate on theyoung and a discontinuous decrease in the number of successful young agents (see Figure8), both of which cause a sharp increase in inequality.
[Figure 8 about here.]
Another noteworthy feature is that countries included in the rich-majority equilibria
could present different patterns of mobility and inequality. To understand this, let us
consider the effects of θ and ηo on mobility and inequality in the rich-majority equilibria.
The parameter θ, representing the expectations of agents, has no effect on mobility but
has a critical effect on inequality via taxation on the young. The parameter ηo , representing the cost of a second chance, has a primary effect on mobility but has a minor effecton inequality. These effects imply that depending on θ and ηo , the rich-majority equilibrium demonstrates a positive or a negative relation between mobility and inequality. For
17example, a country with a low ηo features a high level of mobility but either a high or alow level of inequality depending on the expectations of agents.

4.2

Empirical Implications

The result in Figure 7 shows that the political economy tends to generate a positive
correlation between mobility and inequality as in Quadrini (1999) but may present various
patterns of mobility and inequality in the rich-majority equilibria. In this section, we
consider the scatter plot of upward mobility and inequality illustrated in Figure 1, andshow that the correlation in Figure 1 is in line with the model predictions.8
As shown in Figure 1, among the eight countries presented, the United States displays
the highest upward mobility for workers aged 25–34 years, the second-highest upward
mobility for workers aged 35–49 years, and the highest inequality. Finland shows the lowest upward mobility for workers aged 25–34 years, the second-lowest upward mobility forworkers aged 35–49 years, and the lowest inequality. The remaining six countries generally show lower levels of upward mobility and inequality when compared with the UnitedStates, and higher levels of upward mobility and inequality when compared with Finland.
These observations suggest a positive correlation between mobility and inequality.
However, there are some exceptions to this view: for instance, high mobility and low
inequality are observed in Sweden for workers aged 25–34 years, and in Denmark and Sweden for workers aged 35–49 years. Empirical work by Aaberge et al. (2002) supports thisobservation by showing that inequality is greater in the United States than in Scandinavia,
even though the pattern of mobility is remarkably similar. One possible interpretation ofthis finding is that Scandinavia is represented by the rich-majority equilibrium featuringhigh mobility and low inequality caused by low levels of θ and ηo . Therefore, the various
patterns of mobility and inequality in the rich-majority equilibria, which were not shownin Quadrini (1999), are key factors in explaining the relation between the United Statesand Scandinavia.
From Figures 1 and 7, we can draw some further inferences about the relative ηoand θ across countries included in the rich-majority equilibria. For example, Sweden
and the United Kingdom feature low levels of ηo because they show high levels of upward
mobility. However, they might have different expectations of future taxation because they8

An alternative way of connecting the model predictions to empirical evidence is to focus on tax burden.
However, this is not necessarily appropriate because the current model has the following limitations. First,
the binary choice of the tax rate on the old makes the model analytically tractable but fails to demonstraterealistic tax rates. Second, the preferences of the young are not reflected in the political determinationof the tax rate on the young because the model assumes that the young do not participate in voting.
These limitations imply that the tax rates analytically derived from the model do not fit the empirical
data well. Therefore, we have decided to observe the inequality in terms of after-tax income instead of
directly focusing on tax rates.

18

show substantial differences in inequality. As another example, Germany and Italy may
feature high levels of ηo because they show low levels of upward mobility. However, theymight also have different expectations of θ as the data in Figure 1 shows that inequality
is lower in Germany than in Italy.
Our analysis of inequality and intragenerational mobility supplements the analysis ofinequality and intergenerational mobility in Hassler, Rodriguez Mora and Zeira (2007).
They developed a model including intergenerational mobility from parents to children
through altruistic investment in education, and showed that a reduction in the cost of
upward mobility has the primary effect of enhancing upward mobility. This is in common
with Hassler, Rodriguez Mora and Zeira (2007) and the current analysis. However, thesecondary effect and its implications differ completely from those in the current paper. In
Hassler, Rodriguez Mora and Zeira (2007), the secondary effect is a decrease in the gainfrom education. This effect decreases upward mobility. They showed that the primaryeffect generally overrides the secondary effect so that a reduction in the cost of education
results in improvements in both upward mobility and equality (see Proposition 3 in Hassler, Rodriguez Mora and Zeira, 2007). In contrast, in our framework, the secondary effectprovides young individuals with a disincentive to invest in education, and this overrides
the primary effect. Therefore, the correlation between inequality and upward mobility
depends on whether the focus is on intragenerational or intergenerational mobility.

5

Welfare Analysis

Based on the analysis given in Section 3, we consider the welfare implications of a second
chance by focusing on the parameter representing the cost of a second chance, ηo . While
policies to decrease ηo benefit old agents who were unsuccessful in youth, they will not
be welfare improving if the agents who were successful in their youth or young agentsare made worse off. To examine the total welfare effect of these policies, we introduce a
Benthamite social welfare function Γ defined by:Γ(τ y , τ o ) = V y (τ y , τ o ) + (1 − u(τ y , τ o )) · V os (τ y , τ o ) + u(τ y , τ o ) · V ou (τ y , τ o ),where V os , V ou and V y are given by (1), (2) and (3), respectively. Using this function, we
undertake comparative statics analysis of ηo via numerical examples under the assumptionof β = (0.96)20 . We examine the social welfare implications of polices that reduce ηo toenhance upward mobility, and consider whether a low-cost economy is superior to a highcost economy in terms of social welfare.
[Figure 9 about here.]19

We calculate the educational investment by the young, the tax rates on the young
and old, and the utility of each type of agent and social welfare by taking the value ofηo from one to infinity. We have already presented educational investment by the old in
Figure 7, and the tax rate on the young and the educational investment by the young in
Figure 8. Figure 9 illustrates the expected utility of young agents (Panel (a)), and social
welfare (Panel (b)). We provide an interpretation of the simulation results by focusing
on the parameter ηo . In particular, we consider a larger set of ηo to a smaller set of ηoin turn and provide the welfare implications of policies that reduce ηo to enhance upward
mobility.
First, consider the case of ηo ∈ [2.5124, ∞): there are multiple equilibria. If the
expectation θ is larger than the critical level θ̃, there is an equilibrium where the majorityoof voters are the successful agents who prefer τty = θ and τt+1
= 0. However, if the
expectation θ is lower than the critical level θ̃, there is an equilibrium where the majorityoof voters are the unsuccessful agents who prefer τty = 1/2 and τt+1
= 1. The former
equilibrium is superior to the latter in terms of social welfare because in the former, thesuccessful agents benefit from no taxation in their old age.
Second, consider the case of ηo ∈ (1, 2.5124). There is a unique equilibrium where themajority of voters are the unsuccessful agents who prefer τty = β · (1 − 1/4ηo ) + (1 − 1/ηo )oor (1 + β · (1 − 1/4ηo )) /2 and τt+1
= 0. A noteworthy feature is that the current casemay show a lower level of social welfare compared with the case of ηo ∈ [2.5124, ∞). In
particular, a decrease in ηo around the threshold level of ηo = 2.5124 leads to a sharp
decrease in social welfare. The mechanism behind this result is as follows. At ηo = 2.5124,

a decrease in ηo produces a discontinuous increase in the tax rate on the young and a
discontinuous decrease in the number of successful young agents. Both of these effects
were explained in Section 4.1. These effects result in a sharp decrease in the expectedutility of the young and thus in social welfare. This result implies that the United States(as represented by the low-cost economy) may be inferior to some European countries (asrepresented by the high-cost economy), at least in terms of social welfare.
The empirical validity of numerical analysis so far can be checked by comparing themodel prediction of the number of successful young agents with the empirical data. Wetake the graduation rate in tertiary education (especially tertiary type A programs including universities and college) as a proxy variable for the proportion of successful young
agents because we regard education in youth as study at tertiary education institutions.
The graduation rates in 2005 of the countries concerned are as follows: 45.5% for Denmark 47.3% for Finland, 19.9% for Germany, 41.0% for Italy, 37.7% for Sweden, 39.4% for
the UK, and 34.2% for the US (OECD, 2007).9 With the second-lowest graduation rate,
9

The data for France are not available.

20the US has a lower graduation rate than European countries such as Iceland, Ireland,
the Netherlands and Norway (OECD, 2007). These results imply that it is reasonable toconsider that the US is represented by the low-cost economy featuring a lower graduation
rate whereas some European countries are represented by the high-cost economy featuring
higher graduation rates.

6

Extensions and Further Analysis

The analysis so far has assumed that: (i) the cost function in youth is specified by (ey )2 ,and (ii) the same wages are paid to individuals who were successful in youth and individuals who were unsuccessful in youth but became successful in old age. This section briefly
considers how the analysis and the results are changed when each of these assumptions isrelaxed.

6.1

The Generalization of the Investment Cost Function in YouthThe analysis in the previous sections assumed that the investment cost functions in youth
and old age are given by (ey )2 and ηo (eo )2 , respectively. Given these specifications, we
focused on the parameter ηo representing the cost in old age, and analyzed the role of ηo forthe determination of political equilibria. In this subsection, we generalize the investment
cost function in youth by assuming ηy (ey )2 , where ηy (> 0) is a parameter representing thecost in youth, and we consider the role of ηy for the determination of political equilibria.10
Given the cost functions ηy (ey )2 and ηo (eo )2 , educational investments in youth and old
age are given by, respectively:
¾¸
·
½
1
1
y
o
o
2
=
(1 − τt ) + β (1 − τt+1 ) −(1 − τt+1 )
,
2ηy
4ηo
1
o
e0∗
(1 − τto ).
t (τt ) =
2ηo

y
o
ey∗
t (τt , τt+1 )A lower value of ηy implies a lower cost in youth, thereby yielding a higher probabilityof being successful in youth. However, the value of ηy does not affect the probability of
being successful in old age because the educational cost in youth has no relevance to theinvestment in old age.
The equations presented imply that the impact of the cost in youth is limited to the
incentive of educational investment in youth and thus is limited to the determination of
the majority. For cases of high values of ηy , with ηy = 1 as a special case, young indi10

The authors would like to thank an anonymous referee for suggesting the important role of education
costs in youth.

21

viduals have a weak incentive to invest in education; this results in a similar equilibrium
characterization to that presented in Section 3. However, for cases of low values of ηy ,young individuals have a strong incentive to invest in education even if they expect hightax rates. The economy attains a unique rich-majority equilibrium.11 Therefore, thegeneralization of the cost function in youth has no impact on the mobility but has aneffect on the characterization of political equilibria in that a rich-majority equilibrium ismore likely to hold when the cost in youth is lower. Because the focus of this paper is on
upward mobility and its impact on the welfare, the previous sections assumed ηy = 1 andconcentrated on the parameter ηo that affects mobility.

6.2

Experience Premiums

One of the main results in this paper is that an equilibrium exists that supports the POUM
hypothesis: the majority are poor but they support low redistribution because of the hopeof upward mobility in the future. A key to this result is the education disincentive effectproduced by low costs of a second chance in old age. Low costs of a second chance provide
young individuals with a disincentive for educational investment in youth because they
expect that they will become successful in old age with a high probability, even if they
fail in education in their youth.
The education disincentive effect is mitigated when experience premiums are considered. The analysis so far has assumed that the same wages are paid to individuals who
were successful in youth and individuals who were unsuccessful in youth but became successful in old age. However, in the real world, the former would be more experienced thanthe latter and thus would receive experience premiums in old age. The experience premiums give young individuals an incentive to invest in education in youth, which mitigates
the education disincentive effect created by low costs of a second chance.12 This raises thefollowing question: does the model still predict an equilibrium that supports the POUM
hypothesis when experience premiums are considered? In other words, how robust is theequilibrium that supports the POUM hypothesis in the presence of experience premiums?
In order to answer these questions, we assume here that the old who were successful inyouth receive w(> 1) units of wages in old age: w − 1 units of wages represent experience
premiums. Under this setting, the educational investment in youth is given by:
1
y
o
ey∗
t (τt , τt+1 ) =2·½(1 − τty ) + βow(1 − τt+1)−1o
(1 − τt+1
)2
4ηo

¾¸
.

This equation implies that a higher level of experience premiums results in a larger amount11
12

Ono and Arawatari (2008) show this result in a model without second chances.
The authors would like to thank an anonymous referee for pointing this out.

22

of educational investment in youth. Given this equation, we obtain the following resultthat provides the conditions for the existence of an equilibrium that supports the POUM
hypothesis.
Proposition 5(i) If β ≤ (2/ηo w − 1)/(w − 1/4ηo ), there exists a poor-majority equilibrium with theooutcome of τty = [1+β(w−1/4ηo )]/2, τt+1
= 0 and ut+1 = 1−[1+β(w−1/4ηo )]/4(>
1/2).
(ii) If β > (2/ηo w − 1)/(w − 1/4ηo ), β ≥ {ηo w/4 − (1 − 1/ηo w)}/(w − 1/4ηo ) andβ ≤ (1/ηo w)/(w − 1/4ηo ), there exists a poor-majority equilibrium with the outcomeoof τty = β(w − 1/4ηo ) + (1 − 1/ηo w), τt+1
= 0 and ut+1 = 1 − 1/2ηo w(> 1/2).
Proof. See the supplementary explanation in Section 9.
The first and second statements of Proposition 5 correspond to the first and second
statements of Proposition 1, respectively. Proposition 5 implies that when β is given by(0.96)20 as in the previous sections, there exists an equilibrium that supports the POUM
hypothesis if ηo satisfies:13
β = (0.96)

20

1/ηo w≤, i.e., ηo ∈w − 1/4ηoµ¸
1 w + (0.96)20 /4,.
w(0.96)20 w

The set is nonempty if and only if w < 4/3 · (0.96)20 ≈ 1.589. Therefore, the existence ofthe equilibrium that supports the POUM hypothesis is ensured as long as the experience
premiums are below 58% of the wage in youth.

7

Conclusion

This paper provides a framework for the analysis of the dynamic political economy ofredistribution when agents who failed in education early in life have a second chance for
upward mobility later in life. In this framework, the analysis considers repeated voting
over redistribution policy and provides an analytical characterization of the stationary
Markov-perfect political equilibrium by focusing on the parameter representing the cost ofa second chance. In addition, we undertake a comparative static analysis of this parametervia numerical example and present how the parameter affects (i) the correlation between
inequality and upward mobility, and (ii) the determination of social welfare.
The results established in this paper are as follows. When the cost of a second chance
is low, there is a unique, poor-majority equilibrium characterized by high levels of upward mobility and inequality with a low tax burden on the decisive voter. Although the13

We assume ηo > 1/w. This assumption corresponds to Assumption 1 with the specification of w = 1.

23

majority are the poor who benefit from redistribution, they do not support a high tax
rate because of a hope for upward mobility in the future. This prediction is consistent
with empirical observations on the US economy by Benabou and Ok (2001) and Alesina
and La Ferrara (2005). However, when the cost of a second chance is high, the economy
displays multiple equilibria representing many European countries: one is a rich-majority
equilibrium characterized by higher levels of upward mobility and inequality; the anotheris a poor-majority equilibrium characterized by lower levels of upward mobility and inequality. Furthermore, mobility and inequality are lower in the high-cost economy thanin the low-cost economy representing the United States. Based on this characterization ofpolitical equilibrium, we investigate the welfare implications of upward mobility and showthat in terms of social welfare, the low-cost economy featuring high upward mobility may
be inferior to the high-cost economy featuring low upward mobility. This implies thatenhancing upward mobility is not necessarily good for the economy.
In order to obtain these results, we simplified the analysis by adopting a simple lumpsum transfer scheme. We did not consider alternative policy methods, for example, transfers targeted at the elderly or the poor. In addition, we abstracted from downward mobility in the analysis. Furthermore, some results are based on numerical analysis because of
the difficulty in solving the model analytically. However, the current paper does provide
a framework for understanding: (i) cross-country differences in inequality and upward
mobility from the viewpoint of political economy, and (ii) the role of upward mobility
from the viewpoint of social welfare.

24

8

Appendix

8.1

Proof of Proposition 1(i) Suppose that under the assumption of β ≤ (2/ηo − 1)/(1 − 1/4ηo ), agents know thato
τty = (1 + β · (1 − 1/4ηo ))/2 and expect τt+1
= 0. Then:

µ
1−e

y∗

¶
¶¶
µ
µ
1
1 + β · (1 − 1/4ηo )1.
, 0 = ut+1 = 1 −1+β· 1−2
4
4ηoBecause the assumption β ≤ (2/ηo −1)/(1−1/4ηo ) is rewritten as 1−(1+β·(1−1/4ηo ))/4 ≥o
1 − 1/2ηo , ut+1 ≥ 1 − 1/2ηo holds for all t. By (4), this implies that τt+1
= 0, fulfilling
initial expectations. Therefore, under the assumption of β ≤ (2/ηo − 1)/(1 − 1/4ηo ), there
exists a poor-majority equilibrium with no taxation on the old if the decisive voter finds
it optimal to set τty = (1 + β · (1 − 1/4ηo ))/2.
To establish that setting τty = (1 + β · (1 − 1/4ηo ))/2 is optimal for the decisive voter,
we note the following properties of the function Z : (i) Z(τty , 0) is concave in τty andis maximized at τty = arg maxτty ∈[0,1] Z(τty , 0) = (1 + β · (1 − 1/4ηo ))/2; (ii) Z(τty , 0) >Z(τty , 1)∀τty ∈ [0, 1]. These properties imply that setting τty = (1 + β · (1 − 1/4ηo ))/2ois optimal if this setting is rational under the expectation of τt+1
= 0, i.e., if β · (1 −1/4ηo ) + (1 − 1/ηo ) ≤ (1 + β · (1 − 1/4ηo ))/2 ≤ 1. The first inequality is rewritten asβ ≤ (2/ηo − 1)/(1 − 1/4ηo ), which is equivalent to the current assumption; and the second
inequality always holds under the assumption of β ∈ [0, 1] and ηo > 1 (Assumption 1).
(ii) Suppose that at time t, agents know that τty = β · (1 − 1/4ηo ) + (1 − 1/ηo ) and expectoτt+1= 0. Then:

¶ µ
¶ ¶
µ µ
1
1
1
.
+ 1−
, 0 = ut+1 = 1 −
1−e
β· 1−
4ηo
ηo
2ηo
y∗

o
By (4), this implies that τt+1
= 0, fulfilling initial expectations. Therefore, a poor-majority
equilibrium exists if the decisive voter finds it optimal to set τty = β·(1−1/4ηo )+(1−1/ηo ).
To establish that setting τty = β · (1 − 1/4ηo ) + (1 − 1/ηo ) is optimal for the decisive
voter, we first note that setting τty = (1 + β · (1 − 1/4ηo ))/2 is not rational under the first

assumption:β > (2/ηo − 1)/(1 − 1/4ηo ).

(7)

Given this condition, we consider the following two cases: β > (1 − 1/ηo )/(1 − 1/4ηo ) andβ ≤ (1 − 1/ηo )/(1 − 1/4ηo ).
First, suppose that a set of parameters satisfies β · (1 − 1/4ηo ) > (1 − 1/ηo ), i.e.:β > (1 − 1/ηo )/(1 − 1/4ηo ),25(8)

as illustrated in Panel (a) of Figure 4. Under this situation, the relevant payoff functionis Z(τty , 0) or Z(τty , 1) for τty ∈ (0, θ], Z(τty , 1) for τty ∈ (θ, 1 − 1/ηo ], and Z(τty , 0) forτty ∈ (1 − 1/ηo , β · (1 − 1/4ηo )]. Given the properties of the payoff function demonstrated
in the proof of Proposition 1(i), an alternative option is to choose τty = β · (1 − 1/4ηo )ounder the expectation of τt+1
= 0. Thus, under the assumptions of (7) and (8), settingyτt = β · (1 − 1/4ηo ) + (1 − 1/ηo ) is sustained as an equilibrium if Z(β · (1 − 1/4ηo ) + (1 −1/ηo ), 0) ≥ Z(β · (1 − 1/4ηo ), 0), i.e., if:
β≤

1/ηo
.
1 − 1/4ηo(9)Second, suppose that a set of parameters satisfies β · (1 − 1/4ηo ) ≤ (1 − 1/ηo ), i.e.:β ≤ (1 − 1/ηo )/(1 − 1/4ηo ),(10)

as illustrated in Panel (c) of Figure 4. Under this situation, the relevant payoff function isZ(τty , 0) or Z(τty , 1) for τty ∈ (0, θ] and Z(τty , 1) for τty ∈ (θ, 1 − 1/ηo ]. Given the propertiesof the payoff function, there are two alternatives: setting τty = θ(≤ β · (1 − 1/4ηo ))ounder the expectation of τt+1
= 0 and setting τty = arg max Z(τty , 1) = 1/2 under theoexpectation of τt+1
= 1. Thus, under the assumptions of (7) and (10), setting τty =β · (1 − 1/4ηo ) + (1 − 1/ηo ) is sustained as an equilibrium if:

Z(β · (1 − 1/4ηo ) + (1 − 1/ηo ), 0) ≥ Z(β · (1 − 1/4ηo ), 0),and:Z(β · (1 − 1/4ηo ) + (1 − 1/ηo ), 0) ≥ Z(1/2, 1),that is, if:β≤ηo /4 − (1 − 1/ηo )1/ηoand β ≥.
1 − 1/4ηo1 − 1/4ηo(11)

In sum, there exists a poor-majority equilibrium featured by τty = β · (1 − 1/4ηo ) + (1 −o
= 0 if (7), (8) and (9) hold or if (7), (10) and (11) hold. These conditions
1/ηo ) and τt+1
are reduced to (7) and (11).
¥8.2

Proof of Proposition 2

Under the set of parameters given in Proposition 2, ηo > 2, i.e., 1/2 < 1 − 1/ηo holds (seeo= 1 and θ < 1/2, agents canFigure 5). This implies that given the expectations of τt+1
yychoose τt to attain the top of the Laffer curve Z(τt , 1) : arg max Z(τty , 1) = 1/2.
o= 1. Then,
Suppose that at time t, agents know that τty = 1/2 and expect that τt+126

1 − ey∗ (1/2, 1) = ut+1 = 3/4. Because the condition in Proposition 2 ensures that ηo >o2, ut+1 = 3/4 ∈ (1/2, 1 − 1/2ηo ] holds for all t. By (4), this implies that τt+1
= 1,
fulfilling initial expectations. Therefore, there exists a poor-majority equilibrium with
100% taxation on the old if the decisive voter finds it optimal to set τty = 1/2.
To establish that setting τty = 1/2 is optimal for the decisive voter, we consider
the following two cases: β > (1 − 1/ηo )/(1 − 1/4ηo ) and β ≤ (1 − 1/ηo )/(1 − 1/4ηo ).
First, suppose that a set of parameters satisfies β · (1 − 1/4ηo ) > (1 − 1/ηo ), i.e., β >(1−1/ηo )/(1−1/4ηo ) as illustrated in Panels (a) and (b) of Figure 4. Under this situation,osetting τty = β ·(1−1/4ηo ) is rational under the expectation of τt+1
= 0 (see Panels (a) and(b) of Figure 4). It holds that Z(β·(1−1/4ηo ), 0) ≥ Z(1/2, 1) ⇔ β ≥ (1−1/ηo )/(1−1/4ηo ),implying that setting τty = 1/2 is not sustained as an equilibrium.
Alternatively, suppose that a set of parameters satisfies β · (1 − 1/4ηo ) ≤ (1 − 1/ηo ),i.e.:β ≤ (1 − 1/ηo )/(1 − 1/4ηo ),(12)
as illustrated in Panels (c) and (d) of Figure 4. Under this situation, there are twooalternative options: (i) setting τty = θ(≤ β · (1 − 1/4ηo )) under the expectation of τt+1
=0
y(see Panels (c) and (d) of Figure 4); and (ii) setting τt = β·(1−1/4ηo )+(1−1/ηo ) under theoexpectation of τt+1
= 0 provided that (1+β ·(1−1/4ηo ))/2 < β ·(1−1/4ηo )+(1−1/ηo ) ≤ 1
holds (Panel (c) of Figure 4).14 Therefore, under the condition of (12), setting τty = 1/2
is sustained as an equilibrium if:µ
Z(θ, 0) < Z
or:

¶
µ
¶ µ
¶
1
1
1
, 1 and 1 < β · 1 −
+ 1−
,
2
4ηo
ηo

³
´ ³
´
¢ 1+β·(1−1/4ηo )
1
1
,
1
,
<
β
·
1
−
+
1
−
≤ 1,
2
2
4ηo
η
¡1 ¢ oand Z (β · (1 − 1/4ηo ) + (1 − 1/ηo ), 0) < Z 2 , 1 .

Z(θ, 0) < Z

(13)

¡1

(14)

The conditions in (13) are reduced to:θ < θ̃ and β > (1/ηo )/(1 − 1/4ηo ),(15)

and the condition in (14) is reduced to:θ < θ̃, β ≤ (1/ηo )/(1 − 1/4ηo ) and β < {ηo /4 − (1 − 1/ηo )}/(1 − 1/4ηo ),

(16)

q

1
1
1
e ηo ) ≡ 1+β·(1− 4ηo )− β·(1− 4ηo )·{β·(1− 4ηo )+2} . From Figure 5, we
where θ̃ is defined by:θ(β,2can see that the second and third conditions in (16) imply (12). Thus, given θ < θ̃, there

If β · (1 − 1/4ηo ) + (1 − 1/ηo ) ≤ (1 + β · (1 − 1/4ηo ))/2 holds, setting τty = (1 + β · (1 − 1/4ηo ))/2
ounder the expectation of τ+1= 0 dominates all the other options.
14

27

o
exists an equilibrium featured by τty = 1/2 and τt+1
= 1 if (12) and (15) hold; or if (16)holds.

8.3¥

Proof of Proposition 3(i) Suppose that at time t, agents know that τty = β · (1 − 1/4ηo ) and expect thato
τt+1
= 0. Then, 1 − ey∗ (β · (1 − 1/4ηo ), 0) = ut+1 = 1/2. By (4), this implies thato
τt+1
= 0, fulfilling initial expectations. Therefore, there exists a rich-majority equilibriumowith τty = β · (1 − 1/4ηo ) and τt+1
= 0 if the decisive voter finds it optimal to setyτt = β · (1 − 1/4ηo ).
To establish that setting τty = β·(1−1/4ηo ) is optimal for the decisive voter, we consider
the following two cases: β · (1 − 1/4ηo ) > (1 − 1/ηo ) and β · (1 − 1/4ηo ) ≤ (1 − 1/ηo ).

First, suppose that a set of parameters satisfies β · (1 − 1/4ηo ) > (1 − 1/ηo ), i.e.:β > (1 − 1/ηo )/(1 − 1/4ηo ),(17)

as illustrated in Panels (a) and (b) of Figure 4. If β · (1 − 1/4ηo ) + (1 − 1/ηo ) ≤ 1, i.e., ifβ ≤ (1/ηo )/(1 − 1/4ηo ), there is an alternative option τty = β · (1 − 1/4ηo ) + (1 − 1/ηo )ounder the expectation of τt+1
= 0 (see Panel (a) of Figure 4). This dominates the initialychoice τt = β · (1 − 1/4ηo ) because β ≤ (1/ηo )/(1 − 1/4ηo ) is equivalent to Z(β · (1 −1/4ηo ) + (1 − 1/ηo ), 0) ≥ Z(β · (1 − 1/4ηo ), 0). However, if β · (1 − 1/4ηo ) + (1 − 1/ηo ) > 1,
i.e., if:β > (1/ηo )/(1 − 1/4ηo ),(18)

it is irrational for voters to set τty = β · (1 − 1/4ηo ) + (1 − 1/ηo ) under the expectation ofoτt+1= 0. An alternative option is to set τty = arg max Z(τty , 1) = 1/2 or τty = 1 − 1/ηoounder the expectation of τt+1
= 1. Given the properties of the payoff function, thisalternative is dominated by the initial choice. Therefore, there exists a rich-majorityoequilibrium featured by τty = β · (1 − 1/4ηo ) and τt+1
= 0 if (17) and (18) hold.

Second, suppose that a set of parameters satisfies:β ≤ (1 − 1/ηo )/(1 − 1/4ηo ),as illustrated in Panels (c) and (d) of Figure 4. In this case, the relevant payoff functionis Z(τty , 0) or Z(τty , 1) for τty ∈ (0, θ] and Z(τty , 1) for τty ∈ (θ, 1 − 1/ηo ]. Therefore, there
is no choice of τty = β · (1 − 1/4ηo ) in this case.
(ii) First, suppose that a set of parameters satisfies β · (1 − 1/4ηo ) > (1 − 1/ηo ), i.e.,β > (1 − 1/ηo )/(1 − 1/4ηo ) as illustrated in Panels (a) and (b) of Figure 4. It is possible too= 0. However, it is also possible toset τty = θ(< 1 − 1/ηo ) under the expectation of τt+128

oset τty = β · (1 − 1/4ηo ) under the expectation of τt+1
= 0, and this alternative dominates

the initial one because Z(β · (1 − 1/4ηo ), 0) > Z(θ, 0) holds for all t. Therefore, there is noorich-majority equilibrium featured by τty = θ and τt+1
= 0 if β > (1 − 1/ηo )/(1 − 1/4ηo ).
Next, suppose that a set of parameters satisfies β · (1 − 1/4ηo ) ≤ (1 − 1/ηo ), i.e.:β ≤ (1 − 1/ηo )/(1 − 1/4ηo ),(19)

as illustrated in Panels (c) and (d) of Figure 4. It is possible to set τty = θ(< 1−1/ηo ) underothe expectation of τt+1
= 0. If β · (1 − 1/4ηo ) + (1 − 1/ηo ) ≤ 1 holds (see Panel (c)), there
is an alternative option of setting τty = β · (1 − 1/4ηo ) + (1 − 1/ηo ) under the expectationoof τt+1= 0: this dominates the initial option. However, if: β · (1 − 1/4ηo ) + (1 − 1/ηo ) > 1,
i.e., if:β > (1/ηo )/(1 − 1/4ηo ),(20)

as illustrated in Panel (d) of Figure 3, an alternative option is to set τty = arg max Z(τty , 1) =o
1/2 under the expectation of τt+1
= 1. Setting τty = θ is sustained as an equilibrium if
Z(θ, 0) ≥ Z(1/2, 1), i.e., if θ ≥ θ̃. Therefore, under the condition of (19) and (20), thereoexists a rich-majority equilibrium featured by τty = θ and τt+1
= 0 if θ is set within therange [θ̃, β · (1 − 1/4ηo )]. This set is nonempty if and only if θ̃ ≤ β · (1 − 1/4ηo ), i.e., if
and only if β ≥ (1/4)/(1 − 1/4ηo ). ¥8.4

The Gini Index

This section provides an explanation for the calculation of the Gini Index in Section
4. We calculate the Gini index in terms of the after-tax-and-transfer income. In eachperiod, there are three types of agents: young rich agents with per capita income given by(1 − τty ) + st , old rich agents with per capita income given by (1 − τto ) + st , and young andold poor agents with per capita income given by st . Panel (a) of Figure 10 summarizes
information about per capita income, the size of the population and the total income foreach type of agent. For an equilibrium with τto = 0, per capita income levels are rankedby (1 − τto ) + st > (1 − τty ) + st > st . For an equilibrium with τto = 1, per capita income
levels are ranked by (1 − τty ) + st > (1 − τto ) + st > st . Based on this rank, we canillustrate the Lorenz curve and calculate the corresponding Gini coefficient. Panel (b)of Figure 10 illustrates an example of the Lorenz curve for the equilibrium with τto = 0.
The Gini coefficient is calculated by A/(A + B), where A is the shaded area and B is thenon-shaded area in Panel (b). The Gini index is equal to the Gini coefficient multipliedby 100.
[Figure 10 about here.]29

9Supplementary Explanation for Section 6.2

We modify the model presented in Section 2 by assuming that the old who were successful
in youth earn experience premiums in old age. Specifically, they earn a wage w, which isgreater than one: w > 1. Under this setting, the expected utility functions of agents aliveat time t are given as follows:
Vtos = w(1 − τto ) + st ,Vtou = eot · (1 − τto ) − ηo · (eot )2 + st ,Vty = eyt · (1 − τty ) − (eyt )2 + st
£
©
ª
¤
o
o
+ β eyt · w(1 − τt+1
) + (1 − eyt ) eot+1 · (1 − τt+1
) − ηo · (eot+1 )2 + st+1 .
The optimal investments by the old and the young are given by, respectively:(1 − τto )
,
2ηo
·
½
¾¸
1
1
y
y∗ y
o
o
o
2
e (τt , τt+1 ) = · (1 − τt ) + β w(1 − τt+1 ) −· (1 − τt+1 ).
24ηoeo∗ (τto ) =We make the following assumption.
Assumption A: (i) ηo w > 1; (ii) βw < 1.
The first assumption includes Assumption 1 as a special case of w = 1. The secondoassumption ensures that ey∗ (τty , τt+1
) is set within the range (0, 1).
o
The government budget constraint is given by 2st = W (τto , ut ) + Z(τty , τt+1), where
W (τto , ut ) ≡ [(1 − ut )w + ut · eo∗ (τto )] · τto is the revenue financed by the rich old agents,ooand Z(τty , τt+1
) ≡ ey∗ (τty , τt+1
) · τty is the tax revenue financed by the rich young agents.
The first equilibrium condition is T o (ut ) = arg maxτto ∈[0,1] W dec (τto , ut ) (dec = os, ou),

where:(
Wdec(τto , ut ) =W os ≡ w(1 − τto ) + 21 · W (τto , ut ) if ut ≤ 1/2,
W ou ≡(1−τto )2+ 12 · W (τto , ut )4ηoif ut > 1/2.

After some calculation, we obtain the mapping that satisfies equilibrium condition 1 asfollows:(0
if ut ≤ 1/2 or 1 − 1/2ηo w ≤ ut ≤ 1,
T o (ut ) =(21)
1
if 1/2 < ut ≤ 1 − 1/2ηo w.
The second equilibrium condition is given by:1
U (τty ) = 1 − ·
2·(1 − τty ) + β ·

½
¾¸
1
2
y
y
o
o
w (1 − T (U (τt ))) −· (1 − T (U (τt )))
,
4ηo
30

where T o (·) ∈ {0, 1} is given by (21). After some calculation, we obtain the following
solution:U (τty ) =
 n
³
³
´´
o
y
y
1
1
1

·
(1
−
τ
)
+
β
·
w
−
,
1
−
·
(1
−
τ
)
1
−

t
t
2
4ηo
2




y
1

1 − 2 · (1 − τt )
³
³
´´
y
1
1

1 − 2 · (1 − τt ) + β · w − 4ηo







for τty ∈ (0, θ],³
ifor τty ∈ θ, 1 − ηo1w ,
³
³
´´
y
1
1
for τt ∈ 1 − ηo w , β · w − 4ηo
, or
h ³
´ ³
´ i
τty ∈ β · w − 4η1o + 1 − ηo1w , 1 .
(22)Given the first and the second equilibrium conditions presented above, we now consider
the political determination of τty that satisfies equilibrium condition 3, and characterizea poor-majority equilibrium with no taxation on the old, that is, an equilibrium that
supports the POUM hypothesis.
Proposition A(i) Suppose that β ≤ (2/ηo w − 1)/(w − 1/4ηo ) holds. There exists a set of poor-majority
equilibrium with no taxation on the old such that ∀t, T o is given by (21), U (τty )is given by (22), and T y = (1 + β · (w − 1/4ηo )) /2. The equilibrium outcome isunique and such that ∀t, τty = (1 + β · (w − 1/4ηo )) /2, τto = 0, and ut = 1 −(1 + β · (w − 1/4ηo )) /4.
(ii) Suppose that β > (2/ηo w − 1)/(w − 1/4ηo ), β ≥ {ηo w/4 − (1 − 1/ηo w)}/(w − 1/4ηo ),and β ≤ (1/ηo w)/(w − 1/4ηo ) hold. There exists a set of poor-majority equilibriumwith no taxation on the old such that ∀t, T o is given by (21), U (τty ) is given by (22),and T y = β · (w − 1/4ηo ) + (1 − 1/ηo w). The equilibrium outcome is unique and suchthat ∀t, τty = β · (w − 1/4ηo ) + (1 − 1/ηo w), τto = 0, and ut = 1 − 1/2ηo w(> 1/2).
Proof.
(i) Suppose that under the assumption of β ≤ (2/ηo w − 1)/(w − 1/4ηo ), agents know thato
= 0. Then:τty = (1 + β · (w − 1/4ηo ))/2 and expect τt+1

µ
y∗

1−e

¶
¶¶
µ
µ
1 + β · (w − 1/4ηo )1
1
1, 0 = ut+1 = 1 −≥1−.
1+β· w−
2
4
4ηo
2ηo w

o
= 0, fulfilling initial expectations.
By (21), this implies that τt+1
y
Setting τt = (1 + β · (w − 1/4ηo ))/2 is optimal for the decisive voter if this settingo= 0, i.e., if β · (w − 1/4ηo ) + (1 − 1/ηo w) ≤is feasible under the expectation of τt+131

(1+β ·(w−1/4ηo ))/2 ≤ 1. The first inequality is rewritten as β ≤ (2/ηo w−1)/(w−1/4ηo ),which is equivalent to the current assumption; and the second inequality always holds
under the assumption of βw < 1.
(ii) Suppose that at time t, agents know that τty = β · (w − 1/4ηo ) + (1 − 1/ηo w) ando
expect τt+1
= 0. Then:
µ
y∗

1−e

µ

1
β· w−
4ηo

¶

µ

1
+ 1−
ηo w

¶

¶
, 0 = ut+1 = 1 −

1
.
2ηo w

o
By (21), this implies that τt+1
= 0, fulfilling initial expectations. A poor-majority equilibrium exists if the decisive voter finds it optimal to set τty = β · (w − 1/4ηo ) + (1 − 1/ηo w).
To establish that setting τty = β · (w − 1/4ηo ) + (1 − 1/ηo w) is optimal for the decisive

voter, we first note that setting τty = (1 + β · (w − 1/4ηo ))/2 is not feasible under the first
assumption:β > (2/ηo w − 1)/(w − 1/4ηo ).
(23)
Given this condition, we consider the following two cases: β > (1 − 1/ηo w)/(w − 1/4ηo )and β ≤ (1 − 1/ηo w)/(w − 1/4ηo ).
First, suppose that a set of parameters satisfies β · (w − 1/4ηo ) > (1 − 1/ηo w), i.e.:β > (1 − 1/ηo w)/(w − 1/4ηo ).

(24)

Following the logic in the proof of Proposition 1(i), setting τty = β·(w−1/4ηo )+(1−1/ηo w)
is sustained as an equilibrium if Z(β · (w − 1/4ηo ) + (1 − 1/ηo w), 0) ≥ Z(β · (w − 1/4ηo ), 0),i.e., if:1/ηo w.
(25)β≤
w − 1/4ηoSecond, suppose that a set of parameters satisfies β · (w − 1/4ηo ) ≤ (1 − 1/ηo w), i.e.:β ≤ (1 − 1/ηo w)/(w − 1/4ηo ).

(26)

Following the logic in the proof of Proposition 1(ii), setting τty = β·(w−1/4ηo )+(1−1/ηo w)
is sustained as an equilibrium if:
Z(β · (w − 1/4ηo ) + (1 − 1/ηo w), 0) ≥ Z(β · (w − 1/4ηo ), 0),and:Z(β · (w − 1/4ηo ) + (1 − 1/ηo w), 0) ≥ Z(1/2, 1),32

that is, if:β≤1/ηo w
ηo w/4 − (1 − 1/ηo w)
and β ≥.
w − 1/4ηow − 1/4ηo(27)

In sum, there exists a poor-majority equilibrium featured by τty = β · (w − 1/4ηo ) +o(1 − 1/ηo w) and τt+1
= 0 if (23), (24) and (25) hold or if (23), (26) and (27) hold. These
conditions are reduced to (23) and (27).
¥33

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34

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35

30

35

45
40

USA
SWE

SWE

25

GBR

USA

35

DNK
GBR

20

DEU

DNK

FRA

ITA

30

ITA

FRA

FIN
DEU

10

25

15

FIN

15

20

25

30

35

40

15

20

25

30

35

40

Figure 1: The figure illustrates the scatter plot of the Gini indexes for eight OECD
countries (Denmark, Finland, Frace, Germany, Italy, Sweden, the United Kingdom, andthe United States). The data of the Gini indexes are from Figure 10 in Förster and Mirad’Ercole (2005). The data of upward mobility are from Table 3.7a in OECD (1996). Panel(a) depicts the mobility rate of aged 25-34 workers; Panel (b) depicts the mobility rate ofaged 35-49 workers.

36

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

37

Figure 3: The determination of τto .

38

³
´
¯ ¢ 1 ¡ 4´1o · 1 ¡ ´1o³´¯ ¢ 1 ¡ 4´1o > 1 ¡ ´1o´ih³U(¿ty ) = 1 ¡ 12 ¢ 1 ¡ ¿ty + ¯ ¢ 1 ¡ 4´1o´ih³U(¿ty ) = 1 ¡ 12 ¢ 1 ¡ ¿ty + ¯ ¢ 1 ¡ 41´oU(¿ty ) = 1 ¡ 12 ¢ (1 ¡ ¿ty )³´ ³´¯ ¢ 1 ¡ 41´o + 1 ¡ ´1o·U(¿ty ) = 1 ¡ 21 ¢ (1 ¡ ¿ty )

1
2

1
2

1
4´o

1
4´o
1
´o

1
4´o

1
4´o

1
´o

U(¿ty ) = 1 ¡ 21 ¢ (1 ¡ ¿ty )

>

1
´o

´i
h
³
U(¿ty ) = 1 ¡ 12 ¢ 1 ¡ ¿ty + ¯ ¢ 1 ¡ 4´1o´ih³U(¿ty ) = 1 ¡ 12 ¢ 1 ¡ ¿ty + ¯ ¢ 1 ¡ 4´1o³
´ ³´
¯ ¢ 1 ¡ 41´o + 1 ¡ ´1o1´oU(¿ty ) = 1 ¡ 21 ¢ (1 ¡ ¿ty )

1
2

1
2

1
´o

1
4´o

1
4´o
1
4´o

1
´o

1
´o
1
4´o

1
´o

Figure 4: The equilibrium decision rule ut+1 = U (τty ). The solod lines show the graphsof U satisfying equilibrium condition 2.

39

¯

¯=

¯=

¯=
¯=

1¡ ´1o11¡ 4´o³´´o1
4 ¡ 1¡ ´o
1
1¡4´o

1
4

1
1¡ 4´o1´o1¡ 4´1o2 ¡1¯ = ´o 11¡ 4´oFigure 5: The set of parameters (β, ηo ) classified according to the characterization ofpolitical equilibria.

40

³
´
1
¯ ¢ 1 ¡ 4´1
³
´1 ³
´
2
o
1 + 1¡ 1
1 ¡ ´1
¯
¢
1
¡
4´o
´o
o
³
³
´´
1 ¢ 1+¯ ¢ 1¡ 1
4´
2

³
´
1
¯ ¢ 1 ¡ 4´1
1
2
³
³
´´
o
1 ¢ 1+¯ ¢ 1¡ 1
1 ¡ ´1
4´o
2
o
³
´ ³
´
¯ ¢ 1 ¡ 4´1 o + 1 ¡ ´1ooFigure 6: The tax revenue from the young, Z, under the set of parameters provided in
Proposition 1. Panel (a) illustrates the case of Proposition 1(i); panel (b) illustrates thecase of Proposition 2(ii).

41

Figure 7: Numerial results for the case of β = (0.96)20 . The horizontal axis indicatesthe parameter ηo . Panel (a) depicts the upward mobility rate eo ; panel (b) depicts theGini index.

42

Figure 8: Numerical results for the case of β = (0.96)20 . Panel (a) depicts the tax rateon the young; Panel (b) depicts educational investment by the young.

43Figure 9: Numerical results for the case of β = (0.96)20 . Panel (a) depicts the expectedutility of the young agents; Panel (b) depicts the social welfare.

44

y

1¢e
2 t
o
n
y
y
1
o
2 ¢ (1 ¡ et¡1 ) ¢ (1 ¡ et ) + (1 ¡ et )

1¢
2

n

o
(1 ¡ eyt¡1 ) ¢ eot + eyt¡1Figure 10: Panel (a) summarizes the information about per capita income, the size of
population and total income for each type of agents. Panel (b) illustrates the Lorenz
curve for the case in which the tax rate on the old is zero.

45