Discussion Papers In Economics
And Business

Inequality, Mobility and Redistributive Taxation in a
Finance-constrained Economy
Ryo Arawatari and Tetsuo Ono
Discussion Paper 09-28

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANInequality, Mobility and Redistributive Taxation in a
Finance-constrained Economy
Ryo Arawatari and Tetsuo Ono
Discussion Paper 09-28

September 2009

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 Taxation in a
Finance-constrained Economy
Ryo Arawatari∗and Tetsuo Ono†
September 2009

Abstract
This paper presents a simple model that displays a joint determination of income
inequality and intergenerational mobility aﬀected by redistributive taxation. The
model shows that a larger redistribution improves equality and utility and enhances
mobility when the poor are ﬁnancially constrained, however it creates a trade-oﬀ
between the rich and the poor in terms of utility when the poor are ﬁnancially unconstrained. The model also shows that the size of the redistribution as well as wage
inequality play key roles in explaining the cross-country diﬀerences in inequality andmobility among OECD countries.
Keywords: inequality; Gini coeﬃcient; intergenerational mobility; redistributive
taxation; ﬁnancial constraints.
JEL Classification: D31, H23, O23.

∗

Faculty of Economics, Shinshu University, 3-1-1, Asahi, Matsumoto, Nagano, 390-8621 Japan. Email:ryo_a@shinshu-u.ac.jp†
Corresponding Author: Graduate School of Economics, Osaka University, 1-7, Machikaneyama, Toyonaka, Osaka 560-0043, Japan. Email: tono@econ.osaka-u.ac.jp

1

Introduction

Income inequality and intergenerational mobility are strongly related to each other (Piketty,
2000). On the one hand, greater inequality leads to underinvestment in education by thepoor, thereby resulting in lower upward mobility of poor-born individuals. On the other
hand, lower upward mobility implies a larger number of poor, thereby resulting in higher
inequality in the economy. This mutual link implies a negative correlation between inequality and mobility. This correlation is supported by empirical evidence among OECD
countries (OECD, 2006; D’Addio, 2007): Nordic countries are characterized by low inequality and high mobility while Italy, the United Kingdom and the United States arefeatured by high inequality and low mobility.
Redistributive institutions (such as income redistribution and public education) may
have the potential to play an important role in determining inequality and mobility (Checchi, Ichino and Rustichini, 1999; Iyigun, 1999; Hassler, Rodriguez Mora and Zeira, 2007;
Ichino, Karabarbounis and Moretti, 2008). For example, income redistribution from therich to the poor can expand the resources of the poor available for educational investment, which may in turn increase upward mobility and reduce inequality in the economy.
However, income redistribution has a negative income eﬀect on the rich, which may resultin a trade-oﬀ between the rich and the poor in terms of welfare. Therefore, the following
questions arise. How do redistributive institutions aﬀect inequality, mobility and welfare?
Under what conditions does redistribution improve the equality, mobility and welfare ofeach type of individual?
The aim of this study is to provide a simple theory of inequality, mobility and redistributive taxation that can be used to address the above-mentioned questions. For
the purpose of analysis, we develop a two-period-lived overlapping-generations modelthat introduces endogenous intergenerational mobility via educational investment. In
this framework, parents are assumed to invest in education for their children. Poor-born
individuals may have a chance of becoming rich via educational investment by parents.
However, poor parents would be faced with ﬁnancial constraints, thereby providing theirchildren with a lower level of educational investment compared with rich parents. To
resolve this ﬁnancial problem, we consider lump-sum transfers ﬁnanced by taxation of therich. This redistribution from the rich to the poor may relax the ﬁnancial constraints ofthe poor, thereby resulting in higher mobility and lower inequality.
Under the above-mentioned framework, we ﬁnd that the economy attains multiple, stable steady-state equilibria when the redistributive tax rate is set within a certain range.
One is a poverty trap where all individuals are poor; the other is the equilibrium wheresome are rich who are ﬁnancially unconstrained while the others are poor who are ﬁnancially constrained. The multiple stable steady-state equilibria arise because of a kind of1

pecuniary increasing returns stemming from ﬁnancial constraints (Ljungqvist, 1993; Galor
and Zeira, 1993; Banerjee and Newman, 1993; Freeman, 1996; Ghatak and Jiang, 2002;
Mookherjee and Ray, 2003). The economy converges to one of the steady-state equilibria
depending on the initial distribution. Policymakers can prevent the economy falling into
the poverty trap by controlling the redistributive tax rate.
Given the characterization of equilibria, we consider the eﬀects of redistributive taxation on inequality, mobility and utility of each type of individual. Raising the redistributive tax rate relaxes the constraint of the poor, thereby resulting in higher mobility andlower inequality. In addition, the tax increase is beneﬁcial from the viewpoint of utility
because the poor purely beneﬁt from a higher tax rate, whereas the rich are neutral tothis increase: an increase in the tax burden of the rich is completely oﬀset by an increasein lump-sum transfers. A further increase in tax shifts the economy to the equilibrium
where both the rich and the poor are ﬁnancially unconstrained. In this new equilibrium,
an increase in tax results in a trade-oﬀ between the rich and the poor in terms of utility.
Therefore, a tax increase is beneﬁcial from the viewpoint of equality, mobility and utility
as long as the economy stays in the equilibrium where the poor are ﬁnancially constrained.
The analysis and results of this paper provide possible explanations for the following
two empirical ﬁndings. First, as mentioned above, Nordic countries are characterized byhigh redistributive tax rates, low inequality and high mobility, while Italy, the United
Kingdom and the United States are characterized by low redistributive tax rates, highinequality and low mobility (OECD, 2006; D’Addio, 2007). This evidence can be explained by the model prediction that a country with a higher redistributive tax rate ischaracterized by lower inequality and higher mobility when poor individuals are ﬁnancially constrained. Therefore, income redistribution from the rich to the poor is a key toexplaining the negative correlation between inequality and mobility among a sample ofOECD countries.
Second, when the focus is on a group of countries with low mobility and high inequality, the negative correlation between inequality and mobility does not necessarily hold:
Italy, compared with the United States, displays less income inequality and less intergenerational mobility (Checchi, Ichino and Rustichini, 1999). This puzzling evidence can be
explained when the diﬀerence in wage inequality is considered together with the diﬀerencein the size of the redistribution. In the current framework, a country with lower wage
inequality and a higher redistributive tax rate, such as Italy, attains lower inequality andlower mobility.
The current paper relates to two strands of literature. First, some authors explaincross-country diﬀerence in inequality and mobility by showing multiple equilibria createdby a kind of pecuniary increasing returns stemming from ﬁnancial constraints (Owen and2

Weil, 1998; Mookherjee and Napel, 2007; Napel and Schneider, 2008). However, redistributive institutions are omitted from these studies. In contrast, this paper introduces
redistributive taxation into a model with pecuniary increasing returns, and shows how
redistribution aﬀects the pattern of equilibria, inequality, mobility and welfare.
Second, many studies on inequality and mobility focus on public education as redistributive institutions (for example, Owen and Weil, 1998; Iyigun, 1999; Checchi, Ichino
and Rustichini, 1999; Davis, Zhang and Zeng, 2005; Hassler, Rodriguez Mora and Zeira,2007; Ichino, Karabarbounis and Moretti, 2008). In contrast, this paper focuses on income
redistribution as a device for relaxing the ﬁnancial constraints of the poor for educational
investment. Maoz and Moav (1999) also introduce redistributive taxation into their model,
but their analysis is limited to a brief illustration of ﬁgure and provides no welfare implications of redistributive taxation. In contrast, this paper provides analytically a welfare
implication of redistributive taxation via inequality and mobility.
The organization of this paper is as follows. Section 2 develops the model. Section 3
provides the characterization of equilibrium. Section 4 examines the eﬀects of redistributive taxation on inequality, mobility and welfare. Section 5 provides concluding remarks.

2

The model

Consider a discrete time economy where time is denoted by t = 0, 1, 2, · · · . The economy
is made up of overlapping generations of individuals, each of whom has a single parent
and a single child. Each generation has a unit mass. Each individual lives and works forexactly two periods, youth and old age. There is no storage technology in this economy;each individual uses his/her income for consumption and/or educational investment forchildren.
Individuals are heterogeneous at birth. Among the young born in period t, 1 − ut
are born into rich families and ut into poor families. Parents in rich families are skilled
and receive high wages, indexed by w > 0, whereas parents in poor families are unskilled
and receive low wages, normalized to zero. The proportion τ ∈ (0, 1) of the wage w of
the rich is taxed by the government and the tax revenue is distributed within and acrossgenerations in a lump-sum fashion. Therefore, the after-tax-and-transfer income for theskilled (rich) is w(1 − τ ) + st , whereas that for the unskilled (poor) is st , where τ is thetime-invariant labor income tax rate and st is the per capita lump-sum transfer in periodt.
Individuals have preferences for consumption in youth and old age and the education of their children. Let erich(epoor) denote the level of educational investment by richtt(poor) parents. Parents cannot borrow money for educational investment because they3are deceased in the next period. The investment is ﬁnanced by their own income. As
demonstrated below, the level of educational investment is equivalent to the probability ofsuccess in education. The rich-born individuals succeed in education and become skilled(rich) with probability erich, whereas they fail in education and become unskilled (poor)twith probability 1 − erich. Similarly, the poor-born individuals succeed in education andt, whereas they fail in education and become
become skilled (rich) with probability epoortunskilled (poor) with probability 1 − epoor. Therefore, there is intergenerational mobilitytin this economy. Figure 1 illustrates the timing of events and the distribution of rich andpoor within and across generations.
[Figure 1 about here.]
Individuals gain utility from consumption in both periods of life and from educational
investment in their children. In youth, individuals use up all their income for consumption because of the assumption of no storage technology. This implies that there is nointertemporal interaction with regard to individuals’ decisions. Therefore, we can focus
on the decision making on consumption and education of the old individuals.
The preferences of the rich old and the poor old over consumption and education are
speciﬁed by the following quasilinear utility functions:+ γ log erich
Vto,rich = w(1 − τ ) + st − erich
,
t
t


1
Vto,poor = st − epoor.
+ γ log epoort
μ t
For the rich, a part of the after-tax-and-transfer income is devoted to educational invest, and the rest w(1−τ )+st −erichis consumed. For the poor, the marginal cost ofment erichtteducation is 1/μ(> 1), which is assumed to be larger than that for the rich. The transfer
minus the cost of education, st − epoor/μ, is consumed. The term γ log eit (i = rich, poor)t
indicates the utility of educational investment where the parameter γ represents the degree of educational concern. We impose the following assumption with regard to theparameters γ and μ.
Assumption 1. γ ∈ (0, 1) and μ ∈ (0, 1).
The assumption of γ ∈ (0, 1) ensures that the probability of success eit (i = rich, poor)
is set within the range [0, 1]. The assumption of μ ∈ (0, 1) implies that the marginal costof educational investment is higher for the poor old than the rich old. This assumption
implies that poor-born individuals must pay more costs than the rich-born individuals
to obtain a certain level of education. In other words, rich-born individuals have an
advantage over poor-born individuals in being educated (Becker and Tomes, 1979; Solon,
42004; Blumkin and Sadka, 2005; Hassler, Rodriguez Mora and Zeira, 2007). For example,
rich agents may have a home environment more suitable for education, along with betterknowledge of things naturally provided to their children.
The rich old maximize their utility Vto,rich subject to the ﬁnancial constraint w(1−τ )+st ≥ erich; the poor old maximize their utility Vto,poor subject to the ﬁnancial constraintt/μ; . Solving these problems leads to the probabilities of success for the rich-bornst ≥ epoort
and the poor-born individuals, given by, respectively:

erich
=
t

=
epoor
t

if γ ≤ w(1 − τ ) + stγw(1 − τ ) + st if γ > w(1 − τ ) + stγμif γ ≤ st

μst

if γ > st

,

(1)
(2)

Eqs. (1) and (2) show that the probabilities of success depend on whether the old are
ﬁnancially constrained or not. When the after-tax-and-transfer income of the rich is above
the critical level such that γ ≤ w(1 − τ ) + st , the rich old can realize their preferred levelof investment erich= γ. The probability of success is independent of the redistributivettax. In contrast, when the after-tax-and-transfer income is below the critical level γ,
the rich old are ﬁnancially constrained, consume nothing and invest all their income into
education. In this case, the probability of success is dependent on the redistributive tax.
A similar argument holds for the poor old, but the after-tax-and-transfer income is given
by st which is lower than that of the rich old. In other words, the poor old are more
likely to be ﬁnancially constrained compared with the rich old. Figure 2 illustrates therelationship between probabilities of success and after-tax-and-transfer income for the richand the poor old.
[Figure 2 about here.]In each period, all skilled workers are equally taxed and total tax revenue is equallydivided within and across generations. Thus, the government budget constraint is:
2st = [(1 − ut ) min{γ, w(1 − τ ) + st } + ut min{γμ, μst }] wτ + (1 − ut )wτ.

(3)

The left-hand side shows the total amount of redistribution. The right-hand side shows
the tax revenue from the rich. The ﬁrst term on the right-hand side is the tax revenue fromthe rich (skilled) young, and the second term on the right-hand side is the tax revenue
from the rich (skilled) old.
Individuals with the same family background choose the same educational investment,
implying that the proportion of the poor old in the beginning of period t + 1, ut+1 is given
5

by:
ut+1 = (1 − ut )(1 − erich) + ut (1 − epoor
).
t
t

(4)

The ﬁrst term on the right-hand side is the number of unsuccessful individuals who areborn into rich families, and the second term is the number of unsuccessful individuals whoare born into poor families. This equation plays a key role for the analysis of dynamic
motion of inequality and mobility.

3

Equilibrium analysis

This section investigates the dynamic motion of the distribution of the poor. For thispurpose, we ﬁrst deﬁne a competitive equilibrium, as follows.
Definition 1. A competitive equilibrium is a sequence {ut }∞t=0 with the initial conditionu0 ∈ [0, 1] and the time-invariant tax rate τ ∈ [0, 1] such that: (a) the educational
investments by the rich old and the poor old are given by (1) and (2), respectively;(b) the government budget constraint is balanced every period such that (3) issatisﬁed; and (c) given u0 ∈ [0, 1], the distribution of the poor ut follows (4).
As demonstrated in Figure 2, there are three cases to consider: (i) the case where
both rich and poor individuals are ﬁnancially unconstrained; (ii) the case where only thepoor are ﬁnancially constrained; and (iii) the case where both rich and poor individuals
are ﬁnancially constrained. For each case, we derive the condition for the existence of asteady state where a competitive-equilibrium sequence of ut is stationary over time. Then,
we present a dynamic characterization of the equilibrium path and derive the stability
condition of each steady state.

3.1

The case where both the rich and the poor are financiallyunconstrained

The rich and the poor are ﬁnancially unconstrained if γ ≤ st (see Figure 2). From (1)and (2), the probabilities of success for the rich-born and poor-born individuals are given
by, respectively:= γ; epoor
= γμ.
erich
t
t
The probability of success for the poor-born individuals is lower than that for the richborn individuals because the cost of educational investment is higher for the poor-born
individuals.

6

The government budget constraint (3) is now given by:st =1[(1 + γ) − {1 + γ(1 − μ)}ut ] wτ.
2

Given this size of the redistribution, the condition γ ≤ st is rewritten as:γ ≤ st ⇔ ut ≤ uL (τ ) ≡(1 + γ)wτ − 2γ.
{1 + γ(1 − μ)}wτ

(5)

When the number of poor ut satisﬁes (5), the motion of ut is given by:
ut+1 = φ1 (ut ) ≡ (1 − γ) + γ(1 − μ)ut if ut ≤ uL (τ ),(6)with the following properties:
φ1 (0) = 1 − γ, φ1 (1) = 1 − γμ, φ1 (·) = γ(1 − μ), φ1 (·) = 0.
Panel (a) of Figure 3 illustrates the graph of ut+1 = φ1 (ut ).
[Figure 3 about here.]
We denote the steady-state level of u satisfying ut+1 = φ1 (ut ) by ū1 and the corresponding size of the transfer by s̄1 . The following lemma establishes the condition for the
existence of the steady state with u = ū1 .
Lemma 1. There exists a stable steady-state equilibrium where both the rich and thepoor are ﬁnancially unconstrained if:
1 − γ(1 − μ)
≤ τ.
μw

(7)

The number of poor and size of the transfer are given by, respectively:
ū1 ≡

1−γ
γμ
∈ (0, 1); s̄1 ≡wτ.
1 − γ(1 − μ)
1 − γ(1 − μ)Proof. There exists a steady-state level of u satisfying ut+1 = φ1 (ut ) if ū1 ≤ uL (τ ).
This condition is rewritten as (7). Because φ1 (·) = γ(1 − μ) ∈ (0, 1) for all ut ∈ [0, 1],
the steady-state equilibrium with u = ū1 is stable. The level of ū1 is calculated by setting
ut = ut+1 = ū1 in (6). The level of s̄1 is calculated by substituting u = ū1 into thegovernment budget constraint.

The condition (7) requires that the redistributive tax rate should be higher than orequal to the critical rate {1 − γ(1 − μ)}/μw. This requirement ensures that the old receive7

a large enough transfer to ﬁnance educational investment. The condition (7) also requiresthat the wage rate of the skilled, w, should be higher than (1 − γ)/μ + γ (see Figure 4).
Otherwise, the critical tax rate {1 − γ(1 − μ)}/μw becomes greater than one, implyingthat there is no tax rate τ ∈ (0, 1) satisfying condition (7).
[Figure 4 about here.]

3.2

The case where only the poor are financially constrained

We next consider the case where the rich are ﬁnancially unconstrained whereas the poor
are constrained. From Figure 2, this case holds if st < γ ≤ w(1 − τ ) + st . From (1) and(2), the probabilities of success for the rich-born and poor-born individuals are given by,respectively:= γ; epoor= μst .
erichtt
The probability of success for the rich-born individuals is independent of redistribution
and is given by γ. In contrast, the probability of success for the poor-born individuals
depends on the size of the redistribution because the poor are ﬁnancially constrained
under the condition st < γ ≤ w(1 − τ ) + st .
The government budget constraint (3) is now given by 2st = [(1 − ut )γ + ut μst ]wτ +(1 − ut )wτ, that is:st =(1 − ut )(1 + γ)wτ.
2 − ut μwτ

We impose the following assumption to ensure st > 0.
Assumption 2. w ∈ (0, 2].
Given the above-mentioned size of the redistribution, the conditions st < γ and γ ≤w(1 − τ ) + st are rewritten as follows:(1 + γ)wτ − 2γ,
{1 + γ(1 − μ)}wτ
2(w − γ) − (1 − γ)wτ.
γ ≤ w(1 − τ ) + st ⇔ ut ≤ uH (τ ) ≡[1 + γ(1 − μ) + μw(1 − τ )]wτst < γ ⇔ ut > uL (τ ) ≡

(8)
(9)

When the number of poor ut satisﬁes (8) and (9), the motion of ut is given by:
ut+1 = φ2 (ut ) if uL (τ ) < ut ≤ uH (τ ),
where



(1 − ut )(1 + γ)wτφ (ut ) ≡ (1 − ut )(1 − γ) + ut 1 − μ,
2 − ut μwτ
2

8

(10)

with the following properties:
φ2 (ut ) = −1 +2(1 + γ)(2 − μwτ ) 2
, φ (ut ) > 0,
(2 − ut μwτ )2

φ2 (0) = 1 − γ, φ2 (1) = 1, φ2 (uL (τ )) = φ1 (uL (τ )).
Panel (b) of Figure 3 illustrates the graph of ut+1 = φ2 (ut ).
We denote the steady-state level of u satisfying ut+1 = φ2 (ut ) by ū2 and the corresponding size of the transfer by s̄2 . The following lemma establishes the condition for the
existence of the steady state with u = ū2 .
Lemma 2. There exists a stable steady-state equilibrium where only the poor are ﬁnancially constrained if:
1−γ+ γ ≤ w and τ ∈μ

1 − γ 1 − γ(1 − μ)
,
μw
μw


.

(11)

The number of poor and size of the transfer are given by, respectively:
ū2 ≡

1−γ 2
1−γ
; s̄ ≡ wτ −.
μwτμProof. There exists a steady-state level of u satisfying ut+1 = φ2 (ut ) if uL (τ ) < ū2
and ū2 ≤ uH (τ ). These inequality conditions are rewritten as:τ<

1 − γ(1 − μ)1−γ
and+ γ ≤ w.
μw
μWith the condition ū2 < 1 ⇔ (1 − γ)/μw < τ, we obtain (11).
As illustrated in Panel (b) of Figure 3, given the properties of φ2 (ut ) > 0 and φ2 (1) =1, the graph of ut+1 = φ2 (ut ) crosses the ut+1 = ut line from above at ut = ū2 , implyingthat the steady-state equilibrium with ut = ū2 is stable. The level of ū2 is calculated by
setting ut = ut+1 = ū2 in (10). The level of s̄2 is calculated by substituting u = ū2 into
the government budget constraint.
The area marked by (L2) in Figure 4 indicates the set of parameters (w, τ ) satisfying
the condition (11). The condition requires that the wage of the skilled should be higher
than (1 − γ)/μ + γ. Similar to the argument in Subsection 3.1, this requirement ensuresthat there is a tax rate τ ∈ (0, 1) that satisﬁes the condition (11). In addition, thecondition (11) requires that, given the wage rate, the redistributive tax rate should behigher than (1 − γ)/μw but lower than {1 − γ(1 − μ)}/μw. When the distributive tax
rate is below the critical rate (1 − γ)/μw, the size of the redistribution is small; the skilledindividuals cannot leave a part of their endowments for their consumption. In contrast,9when the distributive tax rate is above the critical rate {1 − γ(1 − μ)}/μw, the size ofthe redistribution is large; the poor can ﬁnance educational investment and leave a partof their endowments for their consumption. Therefore, the condition in (11) requires thatthe distributive tax rate should be set within the range ((1 − γ)/μw, {1 − γ(1 − μ)}/μw)
for the existence of the steady state where only the poor are ﬁnancially constrained.

3.3

The case where both the rich and the poor are financiallyconstrained

Finally, we consider the case where the rich and the poor are ﬁnancially constrained. Thiscase holds if w(1 − τ ) + st < γ (see Figure 2). From (1) and (2), the probabilities ofsuccess for the rich-born and poor-born individuals are given by, respectively:erich= w(1 − τ ) + st ; epoor= st .
tt
The probabilities of success for the rich-born and the poor-born individuals depend on thesize of the redistribution because both types of the old are ﬁnancially constrained under
the condition w(1 − τ ) + st < γ.
The government budget constraint is now given by 2st = [(1 − ut ){w(1 − τ ) + st } +ut μst ]wτ + (1 − ut )wτ, which is rewritten as:st =(1 − ut ){w(1 − τ ) + 1}wτ.
2 − {(1 − ut ) + ut μ}wτGiven this size of the redistribution, we can rewrite the condition w(1 − τ ) + st < γ asfollows:
w(1 − τ ) + st < γ ⇔ uH (τ ) < ut .

(12)

When the number of poor ut satisﬁes (12), the motion of ut is given by:
ut+1 = φ3 (ut ) if uH (τ ) < ut ,

(13)

where
φ3 (ut ) ≡ (1 − ut ) (1 − w(1 − τ )) + ut − [(1 − ut ) + μut ](1 − ut ){w(1 − τ ) + 1}wτ,
2 − {(1 − ut ) + ut μ}wτwith the following properties:
φ3 (·) > 0, φ3 (·) < 0, φ3 (0) =102(1 − w) 3
, φ (1) = 1.
2 − wτ

Panel (c) of Figure 3 illustrates the graph of ut+1 = φ3 (ut ).
We denote the steady-state level of u satisfying ut+1 = φ3 (ut ) by ū3 and the corresponding size of the transfer by s̄3 . The following lemma establishes the condition for the
existence of the steady state with u = ū3 .
Lemma 3. There exists an unstable steady-state equilibrium where both the rich and thepoor are ﬁnancially constrained if:
w−11−γ≤ τ and+ γ < w.
(1 − μ)wμ(14)

The number of poor and the size of the transfer are given by, respectively:
ū3 ≡

w−1
(1 − μ)wτ − (w − 1)
; s̄3 ≡.
(1 − μ)wτ
1−μProof. There exists a steady-state level of u satisfying ut+1 = φ3 (ut ) if uH (τ ) < ū3 ⇔(1 − γ)/μ + γ < w. With the condition of ū3 < 1 ⇔ (w − 1)/(1 − μ)w < τ , we obtain(14).
As illustrated in Panel (c) of Figure 3, given the properties of φ3 (·) < 0 and φ3 (1) = 1,
the graph of ut+1 = φ3 (ut ) crosses the ut+1 = ut line from below at ut = ū3 , implyingthat the steady-state equilibrium with ut = ū3 is unstable. The level of ū3 is calculated
by setting ut = ut+1 = ū3 in (13). The level of s̄3 is calculated by substituting u = ū3
into the government budget constraint.

The area marked by (L3) in Figure 4 illustrates the set of parameters (w, τ ) satisfying
the condition (14). The condition requires that the wage of the skilled should be higher
than (1 − γ)/μ + γ, ensuring the existence of a tax rate τ ∈ (0, 1) satisfying condition(14). In addition, the condition (14) requires that the redistributive tax rate should behigher than (w − 1)/(1 − μ)w.
To understand the requirement for the tax rate, recall the probability of success for= w(1 − τ ) + st . A higher redistributive tax has two
the rich-born individuals, erichtopposing eﬀects on the probability of success. One is a negative eﬀect via the after-tax
income w(1 − τ ), and the other is a positive eﬀect via redistribution st . Because the sizeof the redistribution st depends positively on the number of rich old, the positive eﬀect
via redistribution is greater than the negative eﬀect via the after-tax income when thenumber of poor at the beginning of a period, ut , is smaller than some critical level. In
this case, there is a ﬁnancially unconstrained equilibrium as demonstrated in Proposition
1. In contrast, when the number of poor old is larger than some critical level, the positiveeﬀect via redistribution is less than the negative eﬀect via the after-tax income. In thiscase, there is an equilibrium where both the rich and the poor are ﬁnancially constrained.
11

The result implies that the distribution of the poor at the beginning of a period is crucial
for the determination of the probability of success and the distribution of the poor in thenext period.
Apart from the steady-state equilibrium demonstrated in Lemma 3, there is anothersteady-state equilibrium where all individuals are ﬁnancially constrained. This is theequilibrium with u = 1 where all individuals are poor and no redistribution occurs inequilibrium. We call this state the poverty trap. The following lemma shows the condition
for the existence and (in)stability of the poverty trap.
Lemma 4. There exists a steady-state equilibrium with u = 1 (i.e., the poverty trap)
for any w ∈ [0, 1] and τ ∈ [0, 1]. This equilibrium is stable if τ < (1 − γ)/μw orτ > (w − 1)/w(1 − μ); it is unstable if τ ∈ [(1 − γ)/μw, (w − 1)/w(1 − μ)].
Proof. Given that the function φi (·)(i = 1, 2, 3) satisﬁes φ1 (1) = 1 − γμ, φ2 (1) = 1,
and φ3 (1) = 1, there exists a steady state with u = 1 if φ2 (1) = 1 or φ3 (1) = 1 is feasible;that is, if uL (τ ) < 1 ⇔ μwτ < 2. This condition always holds under assumptions 1 and 2.
The steady-state equilibrium with u = 1 is stable if the slope of φi (·)(i = 2, 3) isless than one at u = 1; that is, if φ2 (1) < 1 ⇔ τ < (1 − γ)/μw or φ3 (1) < 1 ⇔ τ >(w − 1)/w(1 − μ). If both of them fail to hold, the steady-state equilibrium with u = 1 isunstable.



The areas marked by (L4S ) and (L4U ) in Figure 4 indicate the sets of parameters(w, τ ) satisfying the conditions for the existence of stable and unstable steady states withu = 1, respectively. From the ﬁgure, we can ﬁnd that the economy with the higher wage
inequality is less likely to attain the stable steady-state equilibrium with u = 1 and thus
is more likely to avoid the poverty trap. This is because, given the redistributive tax rate,a higher wage inequality implies a larger transfer from the rich to the poor, which enables
the poor to invest more in education.

3.4

Dynamic characterization of the equilibrium

Based on the results established so far, this subsection characterizes the motion of the
distribution ut over time, and demonstrates the number of stable steady-state equilibria inthe economy. The results established so far lead to the following dynamic characterizationof the competitive equilibrium.
Proposition 1. The competitive equilibrium is featured by the following properties withregard to the existence of a stable steady-state equilibrium.
(a) There are multiple, stable steady-state equilibria with u = ū1 and u = 1 if τ ≥1−γ(1−μ)w−1and τ > (1−μ)w
;
μw

12

(b) There are multiple, stable steady-state equilibria with u = ū2 and u = 1 if τ ∈w−1, 1−γ(1−μ)(1−μ)w
μw

and w > 1−γ+ γ;μ(c) There exists a unique, stable steady-state equilibrium with u = ū1 if τ ∈1−γ(1−μ)w−1, (1−μ)w
μw(d) There exists a unique, stable steady-state equilibrium with u = ū2 if τ ∈

1−γ
, w−1
μw (1−μ)w

and τ < 1−γ(1−μ);μw(e) There exists a unique, stable steady-state equilibrium with u = 1 if τ < 1−γ.
μw
Proof. The result is immediate from lemmas 1–4 and Figure 4. 
The area marked by x(= a, b, c, d, e) in Figure 5 indicates the set of parameters (w, τ )satisfying the condition in the result x(= a, b, c, d, e) of Proposition 1. The economy attains multiple, stable steady-state equilibria when the wage rate is higher than (1−γ)/μ+γ
and the redistributive tax rate is higher than (w − 1)/(1 − μ)w. The former condition implies a positive eﬀect via redistribution, whereas the latter condition includes two opposing
eﬀects: the negative eﬀect via the after-tax income of the rich and the positive eﬀect via
redistribution. As argued previously, the positive eﬀect is greater (less) than the negativeeﬀect when the number of poor at the beginning of a period is below (above) a criticallevel. Therefore, depending on the initial number of poor u0 , the economy converges to
one of the two stable steady-state equilibria.
[Figure 5 about here.]
Figure 6 illustrates two examples of the dynamic motion of distribution u. Panel (a)illustrates the result of (a) in Proposition 5; i.e., the case of multiple, stable steady-state
equilibria with u = ū1 and u = 1. When the initial distribution u0 is above the criticallevel ū3 , the negative income eﬀect via the after-tax income overcomes the positive eﬀect
via redistribution. The net eﬀect of the redistributive tax is negative: the economy falls
into a poverty trap in the long run. However, when the initial distribution is below ū3 , thenet eﬀect of the redistributive tax is positive: the competitive equilibrium path converges
to the steady state with u = ū1 . At this state, both the rich and the poor are ﬁnanciallyunconstrained and can undertake their preferred level of educational investment. Panel (b)also illustrates the case of multiple, stable steady-state equilibria. However, the propertiesof the nontrap steady-state equilibria in panel (b) are diﬀerent from those in panel (a).
In the case of panel (b), the poor-born individuals suﬀer from the ﬁnancial constraint in
the long run because the redistributive tax rate in the case of panel (b) is lower than thatin the case of panel (a).
[Figure 6 about here.]
13;

The result in Proposition 1 has the following policy implication with regard to thepoverty trap. Policymakers may avoid the economy falling into the poverty trap by setting
a high redistributive tax rate. To conﬁrm this argument, suppose that the wage rate isgiven within the range ((1 − γ)/μ + γ, min(1/μ, 2)). When the redistributive tax rate is
below (1 − γ)/μw, the economy deﬁnitely falls into the poverty trap. However, when theredistributive tax rate is set above (1 − γ)/μw, the stable steady-state equilibrium withu = ū2 emerges: the economy can avoid falling into the poverty trap and converges in astable manner to the equilibrium with u = ū2 , as long as u0 < 1. By raising the tax ratefurther, the steady-state equilibrium shifts from u = ū2 to u = ū1 where both the richand the poor are ﬁnancially unconstrained. The economy can move to the unconstrained
equilibrium.

4

Inequality, mobility and welfare

Based on the characterization of the equilibrium in the previous section, this section considers the following questions. How does redistributive taxation aﬀect inequality in termsof the after-tax-and-transfer income, intergenerational mobility, and welfare of individuals
in each steady-state equilibrium? Which type of equilibrium is superior from the viewpoint of equality, mobility and welfare? Does a tax-induced shift of the economy from theﬁnancially constrained equilibrium to the ﬁnancially unconstrained equilibrium improveequality and welfare and enhance intergenerational mobility?
To consider these issues, Subsection 4.1 introduces the Gini index measured in termsof the after-tax-and-transfer income and considers how inequality is aﬀected by redistributive taxation. Subsection 4.2 introduces the deﬁnition of intergenerational mobility
and examines the eﬀect of redistributive taxation on mobility. Subsection 4.3 focuses on
the utility of each type of the old and examines the welfare implications of a tax-induced
shift between equilibria.
In what follows, we omit the poverty trap with u = 1 from the analysis and focus on
nontrivial steady-state equilibria (u = ū1 and ū2 ). This is because one of our purposes here
is to consider the transition from the ﬁnancially constrained equilibrium to the ﬁnanciallyunconstrained equilibrium. It is immediately found that the poverty trap involves theworst performance: the Gini coeﬃcient is one, mobility level is zero, and the utility of theold is −∞.

4.1

Inequality

We calculate the Gini coeﬃcient in terms of the after-tax-and-transfer income. In particular, we focus exclusively on the Gini coeﬃcient of the old because the young make no14

economic decisions in the current framework and the income level of the rich (poor) young
is equivalent to that of the rich (poor) old. Panel (a) of Figure 7 summarizes information
about per capita income, the number of rich and poor old, and the total income of the richand poor old. Based on this information, we can illustrate an example of the Lorenz curve(see panel (b) of Figure 7) and calculate the corresponding Gini coeﬃcient. The Gini coeﬃcient is calculated by A/(A + B) where A is the shaded area and B ≡ B1 + B2 + B3
in Panel (b).
[Figure 7 about here.]Direct calculation leads to:Gini =utst
1 + (1−ut )w(1−τ
)

,

(15)

where the numerator indicates the number of poor ut , and the denominator indicates thesize of the redistribution (i.e., income of the poor) st , divided by the aggregate labor
income of the rich, (1 − ut )w(1 − τ ). (The detail of the calculation is provided in the
appendix.) A larger value of the numerator implies a larger number of poor, thereby
resulting in a higher Gini coeﬃcient. In contrast, a larger value of the denominator
implies falling income inequality between the rich and the poor, thereby resulting in alower Gini coeﬃcient.
In the steady-state equilibrium with u = ūi (i = 1, 2), ūi and s̄i are given in lemmai(= 1, 2). Substitution of ūi and s̄i into (15) leads to:Gini =

1−γ
Gini|u=ū1 ≡ (1 − τ ) 1−γ(1−μ)
if u = ū1 ,1−γ
Gini|u=ū2 ≡ (1 − τ ) μwτ

if u = ū2 .

(16)

Given this, we immediately obtain the following result with regard to the eﬀect ofredistributive taxation on inequality.
Proposition 2. A higher redistributive tax rate leads to lower steady-state inequality interms of the after-tax-and-transfer income.
Redistributive taxation aﬀects the Gini coeﬃcient through two factors: the aftertax wage income of the rich (1 − τ )w and the number of poor u. The former factor
is represented by the term (1 − τ ) whereas the latter factor is represented by the termμwτ in (16). The latter factor does not work in the ﬁnancially unconstrained equilibrium(u = ū1 ) because raising the redistributive tax rate increases the size of the redistribution,
but does not aﬀect the educational investments of the rich and the poor: both of themcan choose their preferred level of educational investments in a ﬁnancially unconstrained
15

situation. In contrast, the factor represented by the number of poor does work in theﬁnancially constrained equilibrium (u = ū2 ) because the educational investment of thepoor is now aﬀected by redistribution.
When the economy is initially in a stable steady-state equilibrium with u = ū2 , anincrease in the tax rate results in a lower inequality. A further increase in the tax ratewould shift the economy from the equilibrium with u = ū2 to the equilibrium with u = ū1 .
This shift results in a further decrease in inequality because Gini|u=ū1 < Gini|u=ū2 holds.
Therefore, the model shows a negative correlation between the redistributive tax andinequality, which is consistent with the empirical evidence among OECD countries.

4.2

Intergenerational mobility

Let Mtup denote the number of poor-born individuals who become rich (skilled) becauseof educational success, and let Mtdown denote the number of rich-born individuals who
become poor (unskilled) because of educational failure. Under this deﬁnition, Mtup and
Mtdown are given by, respectively:
Mtup ≡ ut epoor; Mtdown ≡ (1 − ut )(1 − erich
).
t
t

(17)

Because the number in each generation is unity, Mtup (Mtdown ) also denotes the proportionof the poor-born (rich-born) individuals who become rich (poor) within a generation.
In the steady-state equilibrium, M̄ = M up = M down where M̄ denotes the steady-state
mobility, which we focus on below.
In the steady-state equilibrium with u = ūi (i = 1, 2), ūi and s̄i are given in lemmai(= 1, 2). Substitution of ūi and s̄i into (17) leads to:
⎧
⎨
M̄ =


⎩ M̄ 

(1−γ)γμM̄ u=ū1 ≡ 1−γ(1−μ)
u=ū2

1−γ
(1 − γ)≡ 1 − μwτif u = ū1 ,if u = ū2 .

Given this, we immediately obtain the following result with regard to the eﬀect ofredistribution on intergenerational mobility.
Proposition 3. A higher redistributive tax rate (a) has no eﬀect on mobility in theﬁnancially unconstrained equilibrium with u = ū1 and (b) leads to a higher mobility
level in the ﬁnancially constrained equilibrium with u = ū2 .
The result in proposition 3 implies a nonnegative correlation between mobility and theredistributive tax rate. To understand this implication, suppose that the economy attains
the ﬁnancially constrained steady-state equilibrium with u = ū2 . In this equilibrium, a16higher redistributive tax rate leads to a larger redistribution, thereby resulting in a higher
probability of success for the poor-born individuals (see (2)) and thus a higher mobility
level. A further increase in the tax rate would shift the economy from the ﬁnanciallyconstrained equilibrium with u = ū2 to the ﬁnancially unconstrained equilibrium withu = ū1 . In the latter equilibrium, the probability of success for the poor-born individuals
and that for the rich-born individuals become γ and γμ, respectively, both of which are
independent of the redistributive tax rate.
The key to the nonnegative correlation is ﬁnancial constraints. Although the utility
maximizing level of educational investment is γμ for the poor old, they cannot undertakesuch a high level of investment when their income is below γμ. Their investment islimited up to their after-tax-and-transfer income, that is, the size of the redistribution, s,
thereby resulting in a mobility level that depends on redistributive taxation. If they wereallowed to borrow, they could realize their preferred level of γμ; the mobility level would
be independent of the redistributive taxation.
The results in propositions 2 and 3 have the following prediction for inequality andmobility. Proposition 2 implies a negative correlation between redistributive tax andinequality, and proposition 3 implies a positive correlation between redistributive tax andmobility in the ﬁnancially constrained equilibrium. Therefore, the model predicts that acountry with a higher redistributive tax rate is featured by lower inequality and a higher
mobility level when poor individuals are ﬁnancially constrained.
This prediction is consistent with the empirical ﬁnding that Nordic countries are characterized by high redistributive tax rates, low inequality and high mobility, while Italy,
the United Kingdom and the United States are characterized by low redistributive tax
rates, high inequality and low mobility (OECD, 2006; D’Addio, 2007). Although themodel economy analyzed in this paper is abstract, the ﬁnance-constrained equilibrium
provides one possible explanation for the cross-country diﬀerence in inequality and mobility aﬀected by redistributive taxation.
The empirical evidence generally shows a negative correlation between inequality andmobility among OECD countries, as mentioned above. However, Checchi, Ichino andRustichini (1999) report that when the focus is on the group of countries featured byhigh inequality and low mobility, Italy, compared with the United States, displays lessincome inequality and less intergenerational mobility. This puzzling result cannot only
be explained in terms of redistributive taxation, but can also be explained when thediﬀerence in wage inequality is taken into account in the following way.
Let τ j and wj denote the redistributive tax rate and the wage of the skilled in countryj (= Italy, U S). Suppose that both countries attain an equilibrium with u = ū2 where
only the poor are ﬁnancially constrained. We assume that wItaly /wU S < 1, implying thatwage inequality is higher in the United States than in Italy (Blau and Kahn, 1996); and17

assume that τ U S /τ Italy < 1, implying that the tax rate is lower in the United States thanin Italy (OECD, 2006). Under these assumptions, the Gini coeﬃcients and mobility levelsof both countries are compared as follows:
wItalyτ U S 1 − τ Italy
>
·
,
wU S
τ Italy 1 − τ U S
U S
ItalywItalyτ USM̄ u=ū2 > M̄ u=ū2 ⇔ U S < Italy .
w
τ

S
Italy
Gini|Uu=ū
2 > Gini|u=ū2 ⇔Figure 8 illustrates these two conditions in the τ U S /τ Italy − wItaly /wU S space. Whenthe ratios τ U S /τ Italy and wItaly /wU S are given to satisfy the shaded area in Figure 7, Italy
displays less mobility and less inequality compared with the United States. Our analysis
suggests that wage inequality as well as redistributive taxation are the keys to explaining
the diﬀerence in inequality and mobility between Italy and the United States.
[Figure 8 about here.]

4.3

Welfare analysis

The analysis in the previous two subsections shows that raising the redistributive tax
rate is desirable from the viewpoint of equality and mobility in the economy. In thissubsection, we take a further step and examine the welfare implications of redistributive
taxation. In particular, we consider whether raising the redistributive tax rate improves
the utility of each type of individual and thus attains a Pareto improvement.
For the purpose of analysis, we focus on the steady-state indirect utility functions of
the rich old and the poor old, V̄ o,rich and V̄ o,poor , calculated as follows:γμV̄ o,rich u=ū1 ≡ w(1 − τ ) + 1−γ(1−μ)
wτ + γ log γ − γ if u = ū1=V̄V̄ o,rich u=ū2 ≡ w + γ log γ − γ
if u = ū2⎧γμ⎨ V̄ o,poor  1 ≡wτ + γ log γμ − γ if u = ū1
1−γ(1−μ)u=ū
o,poor=V̄⎩
if u = ū2 .
V̄ o,poor  2 ≡ γ log μ wτ − 1−γo,richμu=ūGiven this, we immediately obtain the following result.
Proposition 4. Raising the redistributive tax rate (a) decreases the utility of the rich old
and improves the utility of the poor old in the ﬁnancially unconstrained equilibriumwith u = ū1 and (b) has no eﬀect on the utility of the rich old but improves the utilityof the poor old in the ﬁnancially constrained equilibrium with u = ū2 .
The result in proposition 4 indicates that in the steady-state equilibrium with u = ū1 ,the utility of the rich old is decreasing in τ whereas the utility of the poor old is increasing18

in τ . In contrast, in the steady-state equilibrium with u = ū2 , the utility of the rich old
is independent of τ whereas the utility of the poor old is increasing in τ . In other words,
there is a conﬂict between rich and poor in the ﬁnancially unconstrained equilibrium,
while there is no such conﬂict in the ﬁnancially constrained equilibrium.
The diﬀerence between the two cases comes from the ﬁnancial constraints of the poor.
In the ﬁnancially unconstrained equilibrium, a marginal increase in the tax rate has noeﬀect on the levels of educational investment for the rich and the poor because theirinvestment levels are given by erich = γ and epoor = γμ. A higher tax rate decreases theconsumption of the rich and increases the consumption of the poor, thereby creating a
trade-oﬀ between the consumption of these two groups.
In contrast, in the ﬁnancially constrained equilibrium with u = ū2 , such a trade-oﬀ nolonger exists. To see this, rewrite the utility functions of the rich old and poor old as:V̄ o,rich u=ū2 = w(1 − τ ) +V̄ o,poor 

u=ū2




1−γ
wτ −
+ γ log γ − γ ,

μ



erich
s̄2

= s̄2 + γ log μs̄2 −1× μs̄2 .
μ 
epoor

For the rich, an increase of the tax burden is oﬀset completely by an increase of thetransfer: a marginal change of the tax rate has no eﬀect on the utility of the rich. For
the poor, an increase in the tax rate is equivalent to an increase of resources available foreducational investment, which results in an improvement of utility from education.
The result in proposition 4 implies that raising the tax rate leads to a Pareto improvement in terms of the utility of the old when the economy is initially in the ﬁnanciallyconstrained equilibrium with u = ū2 . However, a further increase may fail to attainan improvement once the economy shifts to the ﬁnancially unconstrained equilibrium.
Therefore, with the results in propositions 2 and 3, we can conclude that raising theredistributive tax rate is beneﬁcial in terms of equality, mobility and welfare as long as
the economy stays in the ﬁnancially constrained equilibrium, but is not desirable fromthe viewpoint of the utility of the rich old when the economy attains the ﬁnanciallyunconstrained equilibrium.

5

Concluding remarks

This paper presented a simple theoretical model that analyzes the joint determinationof income inequality and intergenerational mobility aﬀected by redistributive taxation.
Raising the redistributive tax rate relaxes the ﬁnancial constraints of the poor. This19

enhances the educational investment of the poor, thereby resulting in a larger number ofpoor-born individuals who can become rich. Therefore, a higher tax rate leads to a lower
inequality and a higher mobility in the economy when the poor are ﬁnancially constrained.
A further increase in the tax rate shifts the economy to the ﬁnancially unconstrained
equilibrium. In this new equilibrium, raising the tax rate implies a simple income transferfrom the rich to the poor, thereby creating a trade-oﬀ between the rich and the poor interms of consumption, i.e., utility. Therefore, the model implies that raising the tax rate
is beneﬁcial from the viewpoint of equality and mobility, but is not necessarily beneﬁcialfrom the viewpoint of utility.
The current framework is used to provide a potential explanation for the cross-country
diﬀerences in inequality and mobility among OECD countries. Empirical studies showthat a country with a higher redistributive tax rate is featured by lower inequality andhigher mobility. In addition, within a group of low-mobility and high-inequality countries
including Italy and the United States, there is a puzzling result that Italy, compared with
the United States, displays less mobility and less inequality. Our paper gives a simple
explanation for these empirical ﬁndings by focusing on the size of the redistribution as
well as wage inequality.
It should be noted that there are at least two caveats to the analysis. First, for thesimplicity of analysis, the wage of the poor is assumed to be zero. When the assumption
is changed to the poor receiving some positive wages, tax revenues arise from the poor. In
the poverty trap where all parents are poor, some poor-born individuals can become richbecause poor parents can ﬁnance some amount of educational investment by themselves.
However, multiple steady-state equilibria are expected to remain under this extended
framework.
Second, the wage of the rich is assumed to be ﬁxed. When the assumption is changed
so that the wages of the rich and poor are endogenous such that each of them receives
his/her marginal product of labor, the framework can be utilized to analyze the joint
determination of wage inequality and mobility, but at the cost of losing simplicity. Although these extensions are expected to provide additional insights, we leave them for
future work.

20

6

Appendix

Derivation of (15)The Gini coeﬃcient is calculated by Gini = A/(A + B), or:Gini = 1 −(B1 + B2 + B3)
A + (B1 + B2 + B3)= 1 − 2(B1 + B2 + B3),
where the second line comes from A + (B1 + B2 + B3) = 1/2. Given that:1
ut s tB1 = ut,
2 (1 − ut )w(1 − τ ) + st
ut s t,B2 = (1 − ut )(1 − ut )w(1 − τ ) + st
(1 − ut ){w(1 − τ ) + st }1,
B3 = (1 − ut )2
(1 − ut )w(1 − τ ) + st
we obtain:(ut )2 st + 2(1 − ut )ut st + (1 − ut )2 {w(1 − τ ) + st }2(B1 + B2 + B3) =.
(1 − ut )w(1 − τ ) + st
Therefore, we have:(ut )2 st + 2(1 − ut )ut st + (1 − ut )2 {w(1 − τ ) + st }(1 − ut )w(1 − τ ) + st
(1 − ut )w(1 − τ )ut=(1 − ut )w(1 − τ ) + st
ut
=
.
st
1 + (1−ut )w(1−τ)

Gini = 1 −21

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23

Figure 1: The timing of events and the distribution of rich and poor within and acrossgenerations.

24

Figure 2: The probability of success for the rich and the poor.

25

Figure 3: Dynamic motion of the distribution of the poor.

26

Figure 4: Existence and stability of steady-state equilibria.

27

Figure 5: The set of parameters (w, τ ) satisfying conditions in Proposition 1.

28

Figure 6: Examples of the dynamic motion of distribution.

29

Figure 7: Panel (a) summarizes the information about the size of population, per capita
income, and total income. Panel (b) illustrates the Lorenz curve.
30

Figure 8: The shaded area illustrates the case where Italy is less mobile and less unequal
compared to the United States.

31