Discussion Papers In Economics
And Business

Dynamic Political Economy of Redistribution Policy:
The Role of Education Costs

Ryo Arawatari and Tetsuo Ono
Discussion Paper 07-31-Rev.

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANDynamic Political Economy of Redistribution Policy:
The Role of Education Costs

Ryo Arawatari and Tetsuo Ono
Discussion Paper 07-31-Rev.

November 2007

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Dynamic Political Economy of Redistribution Policy:
The Role of Education Costs∗
Ryo Arawatari† and Tetsuo Ono‡
November 2007

Abstract
This paper focuses on how education costs affect the political determinationof redistribution policy via individual decision-making on education. For cases ofhigh costs, there are multiple equilibria: the high-tax equilibrium featured by theminority of highly educated individuals and a large size of the government, and the
low-tax equilibrium featured by the majority of highly educated individuals and asmall size of the government. For cases of low costs, there is a unique equilibriumfeatured by the majority of highly educated individuals and a large size of thegovernment.
Keywords: Markov perfect equilibrium; Dynamic political economy; Redistribution policy; Education costs
JEL Classification: D72; D78; E62∗

This paper was formally circulated under the title “Markov Perfect Political Equilibrium with Social Security: The Role of Education Cost.” We would like to thank Hideki Konishi, Ryuichi Tanaka
and participants at the Seminar on Economic Theory and Econometrics, Osaka Prefecture University,
February 2007, the Macroeconomics Conference for Young Professionals, Osaka, March 2007, Japanese
Economic Association Annual Spring Meeting at Osaka Gakuin University, June 2007, Osaka University
Forum at Groningen, June 2007, Far Eastern Meeting of the Econometric Society at Taipei, July 2007,
and European Meeting of the Econometric Society at Budapest, August 2007 for their useful comments
and suggestions. Arawatari acknowledges 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.17730131) and the 21st Century
COE Program (Osaka University). All remaining errors are ours.
†
Graduate School of Economics, Osaka University, 1-7, Machikaneyama, Toyonaka, Osaka 560-0043,
Japan. E-mail address: ege001ar@mail2.econ.osaka-u.ac.jp‡
Corresponding Author. Graduate School of Economics, Osaka University, 1-7, Machikaneyama, Toyonaka, Osaka 560-0043, Japan. E-mail address: tono@econ.osaka-u.ac.jp

1

Introduction

For the past few decades, there has been growing concern about the political process ofthe determination of redistribution policy in macroeconomics (see Galasso and Profeta,
2002, and the references therein). Several earlier papers analyzed this issue by assuming
myopic voting behavior (for example, Verbon and Verhoeven, 1992; Alesina and Rodrik,
1994; Meijdam and Verbon, 1996) or once-and-for-all voting (for example, Boadway and
Wildasin, 1989; Bertola, 1993; Casamatta et al., 2000). In other words, they investigated
the issue under the environment of the absence of dynamic interaction between redistribution policy and individual decision-making. However, as represented by the pay-as-you-go
public pension, redistribution policy has a dynamic or an intergenerational aspect in nature, implying that there is a feedback mechanism between redistribution policy andindividual decision-making. Expectations about future redistribution policy affect thecurrent individuals’ decision-making on saving or educational investment, which, in turn,has an effect on the future distribution of income and, thus, on the future voting behavior
over redistribution policy. Therefore, there is a need to incorporate this mechanism into
the analysis of political economy of redistribution.
Recently, Hassler et al. (2003) and Hassler et al. (2007) have provided politicoeconomic frameworks that incorporate this feedback mechanism, and have shown thatthe mechanism results in multiple political equilibria: the pro-welfare state and the antiwelfare state. In the pro-welfare state, expectations of higher future redistribution leadto lower educational investments and, thus, to a lower proportion of highly educated individuals. This implies a larger size of low-income individuals, which, in turn, increases
future demand for redistribution, resulting in a higher tax rate and a larger size of thegovernment. In the antiwelfare state, expectations of lower future redistribution lead tohigher educational investments and, thus, to a higher proportion of highly educated individuals. This implies a larger size of high-income individuals, which, in turn, decreases
future demand for redistribution, resulting in a lower tax rate and a smaller size of the1government.1
Their two predictions provide explanations for the empirical observations in many
OECD countries. Figure 1 presents a scatter plot of graduation rates of tertiary education and the ratios of public social expenditure to GDP in 2003 for OECD countries.2 Wetake the graduation rate as a proxy variable of the proportion of highly educated individuals and the ratio of public social expenditure to GDP as a proxy variable of the sizeof the government. The prediction of the pro-welfare state is consistent with empiricalobservations in Continental European countries like Austria, France, Germany and Italy,
and the prediction of the antiwelfare state is consistent with observations in Australia,
Iceland, Ireland, Japan, the United Kingdom and the United States. Therefore, the theory of Hassler et al. (2003) and Hassler et al. (2007) contributes to the understanding of
the difference with respect to welfare programs among some OECD countries that sharesimilar political backgrounds. However, their model does not fully explain the empiricalobservations in Nordic countries (Denmark, Finland, Norway, and Sweden) that experience higher graduation rates and larger sizes of governments among OECD countries.
The aim of this paper is to provide a framework that provides the third prediction, which
was not shown by Hassler et al. (2003) and Hassler et al. (2007).
[Figure 1 around here.]
For the purpose of analysis, we focus on education costs in the model of Hassler et al.
1

The difference between Hassler et al. (2003) and Hassler et al. (2007) is that Hassler et al. (2003)
assume age-independent taxation whereas Hassler et al. (2007) assume age-dependent taxation. The ageindependent taxation creates a dynamic connection between the taxation of the old and the educational
investment of the young, which produces the persistence effect of the state variable, i.e., the proportionof the highly educated individuals. In contrast, the age-dependent taxation breaks this link and, thus,
produces a stationary property of equilibrium that makes the analysis more tractable. However, theassumption of age-dependent taxation still keeps the property of the feedback mechanism on which we
focus here.
2
The data on graduation rates of tertiary education (type A) are based on OECD (2005). The tertiary
type A course is largely theory-based and designed to provide sufficient qualifications for entry to advanced
research programs and professions with high skill requirements. There is another course (type B) thatis shorter than the type A and it focuses on practical, technical or occupational skills for direct entry
into the labor market. We here focus on tertiary type A education from the viewpoint of suitability forour analysis. The data on public social expenditure to GDP are based on OECD (2007). Public social
expenditure comprises cash benefits, direct in-kind provision of goods and services, and tax breaks withsocial purposes.

2(2007) and extend their model by generalizing the education cost function. Hassler et al.
(2007) assumed that the cost of investment in education e, which can be interpreted as
the disutility of educational effort, is given by (e)2 , while we assume that it is given byη · (e)2 where η > 0. For cases of high values of η, with Hassler et al. (2007) (η = 1) as a
special case, individuals have a weak incentive to invest in education, which results in asimilar equilibrium characterization to Hassler et al. (2007): the economy attains the prowelfare state or the antiwelfare state depending on expectations by individuals. However,for cases of low values of η, individuals have a strong incentive to invest in educationeven if they expect a high future tax rate and, thus, a low return from the investment.
Therefore, for cases of low education costs, the economy attains a state featured by alarge proportion of highly educated individuals and a large size of the government. The
generalization of the education cost function creates the third prediction, which was notpresented by Hassler et al. (2003) and Hassler et al. (2007).
The organization of this paper is as follows. Section 2 presents the model. Section 3
characterizes the political equilibrium and explores empirical implications of our result.
Section 4 characterizes the Ramsey allocation to evaluate the efficiency of the politicalequilibrium. Section 5 provides concluding remarks.

2

The Model

The model is based on that developed by Hassler et al. (2007). Time is discrete anddenoted by t = 1, 2, · · · . The economy consists of a continuum of agents living for two
periods. Each generation has a unit mass. Agents are, at birth, of two types: high ability
and low ability, in proportions µ(> 1/2) and 1 − µ(≤ 1/2), respectively. The role of theassumption µ > 1/2 is discussed in Section 3.
High-ability agents can affect their prospects in life by an educational investment.
In particular, they either become rich or poor, and by undertaking a costly investment
can increase the probability e of becoming rich. The cost of investment, which can beinterpreted as the disutility of educational effort, is given by η · (e)2 , η > 0: Hassler et3al. (2007) considered the special case of η = 1. Low-ability agents make no investment
choice: irrespective of their private actions, they are deemed to be in poverty. Rich agents
earn a high wage, normalized to unity, in both periods, whereas poor agents earn a low
wage, normalized to zero.
The government provides transfers (s) financed by taxes levied on the rich. The tax
rates are age dependent, τ o for the old and τ y for the young.3 The tax rates are determined
before the young agents decide on their investment. The government budget balances inevery period.4
There is no storage technology in this economy. Each individual uses up his/herendowments within a period. Therefore, the expected utility of agents alive at time t isgiven as follows:
Vtos = 1 − τto + st ,Vtou = Vtol = st ,o
Vty = et · {(1 − τty ) + β(1 − τt+1)} + (st + βst+1 ) − η(et )2 ,Vtyl = (st + βst+1 ),where Vtos , Vtou , and Vtol denote the utility of the old who were successful in youth, theold who were unsuccessful in youth, and old low-ability types in period t, and Vty and
Vtyl denote expected utility of young high-ability and young low-ability agents born inperiod t. Note that Vty is computed prior to individual success or failure. The parameterβ ∈ (0, 1) is the discount factor.

3

We can alternatively assume age-independent taxation (Hassler et al., 2003) that creates a dynamic
connection between the taxation of the old and the investment of the young (see Hassler et al. (2007)
Section 6 for a discussion). Because our focus is on the role of education costs, we follow the assumptionof age-dependent taxes in Hassler et al. (2007).
4
Hassler et al. (2007) assumed that the government provides transfers (s) as well as public goods(g). We abstract from public goods provision in our analysis for simplicity of exposition. However, thissimplification does not qualitatively affect the result shown below.

4Given these preferences, the optimal investment of young high-ability agents is:oe∗ (τty , τt+1)=

1o
(1 + β − τty − βτt+1
).
2η

(1)

Because high-ability agents are identical ex ante, agents of the same cohort choose thesame investment, implying that the proportion of old poor among high-ability agents inperiod t + 1 is given by:out+1 ≡ 1 − e∗ (τty , τt+1)=

1o(2η − (1 + β) + τty + βτt+1).
2η

Thus, the proportion of old poor among high-ability agents in period t + 1, ut+1 , depends
on the income tax rate levied on the young rich in period t, τty , and the discounted tax
orate levied on the old rich in period t + 1, βτt+1.5The government runs a balanced budget in every period, implying that the budget
can be expressed as:o
2st = µ(1 − ut )τto + µe∗ (τty , τt+1
)τtyo
= W (ut , τto ) + Z(τty , τt+1
),

(2)

o
where W (ut , τto ) ≡ µ(1 − ut )τto is the tax revenue financed by the old, and Z(τty , τt+1)≡oµe∗ (τty , τt+1
)τty is the tax revenue financed by the young.

3

Political Equilibrium

This section characterizes a political equilibrium where agents vote on taxation, period
by period. Section 3.1 provides the definition of political equilibrium based on the concept of Markov perfect equilibrium with majority voting. Sections 3.2 and 3.3 provide
characterization of political equilibrium classified according to the pattern of taxation on
5

When we characterize a political equilibrium (defined below) in the next section, we check whether
or not the levels of e∗ and u are given within the range (0, 1).

5

the old, and discuss the role of education costs in determining the properties of politicalequilibrium. Section 3.4 discusses empirical implications of our model.

3.1

Definition of Political Equilibrium

Following Hassler et al. (2007), it is assumed that agents vote on current taxes at the
beginning of each period but that only the old vote.6 The respective utility of the old richand the old poor can be written as:¢
1 ¡o· W (ut , τto ) + Z(τty , τt+1
) ,
2
¢
1 ¡
ol
o
= Vt = · W (ut , τto ) + Z(τty , τt+1
) .
2

Vtos = 1 − τto +Vtou

Whatever the level of ut , the old rich would never tax themselves to finance transfers
because the marginal cost of taxation is greater than the marginal benefit of taxation:∂Vtos /∂τto = −1 + µ(1 − ut )/2 < 0. However, the old poor prefer taxation on the old
because they pay no tax but benefit from transfer: ∂Vtou /∂τto = µ(1 − ut )/2 > 0. Thus,
the rich prefer a zero tax rate on the old, while the poor prefer 100% taxation on the old.
As for the choice of τty , the old agents have the same political aim irrespective of theirotype: they wish to maximize the revenue from the young, Z(τty , τt+1).

We now provide the definition of political equilibrium. The current paper focuses on
stationary Markov-perfect equilibria, where the state of the economy is summarized by
the proportion of the current old poor among high ability agents (ut ).
Definition: A (stationary Markov perfect) political equilibrium is defined as a triplet offunctions {T o , T y , U }, where T o : [0, 1] → [0, 1] and T y : [0, 1] × [0, 1] → [0, 1] 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 the following functional equations hold:6

For discussion about the role of this assumption, see Section IIB in Hassler et al. (2003) and Section
6 in Hassler et al. (2007).

6

1. T o (ut ) = arg maxτto ∈[0,1] W dec (τto , ut ), where:
 1 − τ o + 1 W (ut , τ o ) if ut ≤ 1 − 1/2µ,
t
t
2
dec o
W (τt , ut ) =1W (ut , τto )if ut > 1 − 1/2µ.
2oo
2. U (τty ) = 1 − e∗ (τty , τt+1), with τt+1
= T o (U (τty )).
oo
3. T y = arg maxτty ∈[0,1] Z(τty , τt+1
) subject to τt+1
= T o (U (τty )).

The first equilibrium condition requires that τto maximizes the utility of the decisive
old voter who is either the old rich (if ut ≤ 1 − 1/2µ) or the old poor (if ut > 1 − 1/2µ).
We assume that the rich decide τto in the case of an equal number of rich and poor voters(ut = 1 − 1/2µ).7 The second equilibrium condition implies that all young agents chooseotheir investments optimally, given τty and τt+1
, and that agents hold rational expectations

about future taxes and distributions of types. The third equilibrium condition requiresthat τty maximizes the objective function of the decisive voter.

3.2

The Determination of T o and U

We now solve the three equilibrium conditions recursively. The mapping T o (·) that satisfies equilibrium condition 1 is given by:

T o (ut ) =
 0 if ut ≤ 1 − 1/2µ, 1 if ut > 1 − 1/2µ.

(3)

The decisive voter will set τto = 0 if the rich are the majority: µ(1 − ut ) ≥ 1/2, i.e.,ut ≤ 1 − 1/2µ, whereas it will set τto = 1 if the poor are the majority: µ(1 − ut ) < 1/2,
i.e., ut > 1 − 1/2µ.
Next, we rewrite equilibrium condition 2 by substituting in the optimal investment
7

Recall that we have assumed µ > 1/2 in Section 2. If µ ≤ 1/2, then µ(1 − ut ) < 1/2 always holds:
there is no opportunity for the rich to become the majority of voters. In order to present a case where
the rich are the majority, we have adopted the assumption of µ > 1/2.

7oe∗ (τty , τt+1). This yields the following functional equation:

U (τty ) =1(2η − (1 + β) + τty + βT o (U (τty ))) .
2η

(4)

Because T o (·) ∈ {0, 1} is given by (3), any solution of (4) must be a combination of the twolinear functions: U (τty ) = (2η − (1 + β) + τty ) /2η if T o = 0 and U (τty ) = (2η − 1 + τty ) /2η
if T o = 1.
Under the assumption of rational expectations about future taxes and distributions oftypes, any solution to the functional equation (4) is given by:

U (τty ) =













(2η − (1 + β) + τty )/2η

if τty ≤ 1 − η/µ{(2η − (1 + β) + τty )/2η, (2η − 1 + τty )/2η} if 1 − η/µ < τty ≤ 1 − η/µ + β(2η − 1 + τty )/2ηif 1 − η/µ + β < τty .
(5)ooSuppose that young agents in period t expect τt+1
= 0 (τt+1= 1): the rich (poor) old

are the majority of voters in period t + 1. Under this expectation, young agents choose
their investments as e∗ (τty , 0)(e∗ (τty , 1)). This expectation is rational if µe∗ (τty , 0) ≥ 1/2(if µe∗ (τty , 1) < 1/2), that is, if τty ≤ 1 − η/µ + β (if 1 − η/µ < τty ). Figure 2 illustrates
possible cases of U that satisfy the second equilibrium condition (5). The solid lines show
the graphs of U satisfying equilibrium condition 2. Since our focus is on the educationalcosts, these cases are classified according to the size of η/µ.
[Figure 2 around here.]Suppose that the costs of education are high such that η/µ > 1 (see Panels (a)and (b)). A high η implies that educational investment is costly, thereby giving thehigh-skilled individuals a disincentive to invest in education. Thus, a high η results
in a low probability of being successful, and the majority in the next period can bepoor ∀τty ∈ [0, 1]. Alternatively, suppose that the costs of education are low such thatη/µ ∈ (0, 1] (see Panels (c) and (d)). The high-skilled individuals have a strong incentive
to invest in education, which results in a high probability of being successful. Thus, the8majority in the next period can be rich for τty ∈ [0, min(1 − η/µ + β, 1)]. In particular, ifτty ≤ 1 − η/µ, the majority in the next period are always the rich.
Hassler et al. (2007) ignored the cases displayed in Panels (c) and (d) by assuming a
specific form of education cost function with η = 1. If η = 1, there is no τty ∈ [0, 1] thatsatisfies τty ≤ 1 − η/µ: there is no case in which the majority, in the next period, arealways the rich. Their analysis was therefore limited to the cases illustrated in Panels (a)and (b). The current paper extends the model of Hassler et al. (2007) by generalizing
the cost function of education, and shows that in the cases presented in Panels (c) and(d), there is a political equilibrium whose outcome is qualitatively different from Hassler
et al. (2007).
As depicted in Figure 2, 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, let us define the critical rate of τty : θ ∈ (max(0, 1 −η/µ), min(1 − η/µ + β, 1)]. The rate θ, which depends on the expectations of agents, isthe highest tax rate that yields a majority of old rich in the next period. For τty > θ, themajority is the poor. However, for τty ≤ θ, the majority in the next period is either therich or the poor depending on expectations by agents. Thus, the equilibrium features amonotonic or a nonmonotonic function U. Without loss of generality, we hereafter focus
our attention on equilibria featuring a monotonic function U (Figure 3).8 The function
U is thus given by:

U (τty ) = (2η − (1 + β) + τ y ) /2η if τ y ≤ θ,tt(2η − 1 + τty ) /2η

if τty > θ.

(6)

[Figure 3 around here.]8

As shown in Hassler et al. (2007), this restriction does not qualitatively affect the equilibriumoutcome.

9

3.3

The Determination of T y and the Characterization of 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 solutions of U, we introduce the corresponding definitions of political equilibria: a high-tax equilibrium featured by τ o = 1 and a
low-tax equilibrium featured by τ o = 0. In the high-tax equilibrium, agents choose τty to
induce a majority of poor at t + 1; they maximize the tax revenue financed by the youngZ(τty , 1). In the low-tax equilibrium, agents choose τty to induce a majority of rich at t + 1;
they maximize the tax revenue financed by the young Z(τty , 0). We first characterize ahigh-tax equilibrium.
Proposition 1. Suppose that β < (η/µ − 1/2)2 /(1 − η/µ) holds. There exists a setof high-tax equilibria such that∀t, T o (ut ) is given by (3), T y = 1/2, and U is givenby (6). The equilibrium outcome is unique and such that ∀t, τty = 1/2, τto = 1,
ut = 1 − 1/4η, and 2st = 3µ/8η.
Proof. See Appendix.
Suppose that agents choose τty to induce a future majority of poor. In cases illustrated
in Panels (a) and (b) of Figure 2, agents can choose τty ∈ [0, 1] that maximizes Z(τty , 1):1
τty = arg max
Z(τty , 1) = .
y2τt ∈[0,1]
The young agents are taxed on the top of the Laffer curve, conditional on their expecting
100% taxation when old. In the case illustrated in Panel (a), voting that induces a futuremajority of poor is the only option; a high-tax equilibrium is sustained by setting τty = 1/2.
However, in the case illustrated in Panel (b), voting that induces a future majority of richcan be an alternative option for τty ∈ [0, 1 + β − η/µ]. In this case, a high-tax equilibrium
is sustained if Z(1/2, 1) > Z(τty , 0) for all τty inducing a future majority of rich. Z(τty , 0)10

is maximized by setting τty = θ because (i) setting arg maxτty ∈[0,1] Z(τty , 0) = (1 + β)/2
is impossible in the current case, and (ii) θ(≤ 1 + β − η/µ) is the highest tax ratethat induces a future majority of rich. Therefore, a high-tax equilibrium is sustained if
Z(1/2, 1) > Z(θ, 0), i.e., if the expectation parameter θ is set below the critical level:³´pθ < θ̃(β) ≡ 1 + β − β(β + 2) /2. This implies that, for the case illustrated in Panel(b), the existence of the high-tax equilibrium depends on the expectations by agents.
In cases illustrated in Panels (c) and (d) of Figure 2, we still have an option that induces
a future majority of poor by setting τty = 1/2. However, there is an alternative option thatinduces a future majority of rich by choosing τty ∈ [0, min(1 + β − η/µ, 1)]. In particular,
if (1 + β)/2 ≤ 1 − η/µ, agents can definitely induce a future majority of rich by settingτty = arg maxτty ∈[0,1] Z(τty , 0) = (1 + β)/2. Because Z((1 + β)/2, 0) > Z(1/2, 1) holds, ahigh-tax equilibrium is not sustained as an equilibrium. Based on this consideration, we
can conclude that a high-tax equilibrium is sustained if (i) setting τty = (1 + β)/2 is notpossible: (1 + β)/2 > 1 − η/µ, (ii) the expectation θ is low such that a future majority ofpoor will be induced: θ ∈ (1 − η/µ, (1 + β)/2), and (iii) Z(1/2, 1) > Z(θ, 0), i.e., θ < θ̃(β).
The second and the third conditions require that θ is set within the range (1 − η/µ, θ̃(β)).
The condition β < (η/µ − 1/2)2 /(1 − η/µ), which includes the first condition, guarantees
that the range is not empty. Figure 4 illustrates an example of the graph of the tax
revenue from the young, satisfying these three conditions.
[Figure 4 around here.]
Next, we characterize a low-tax equilibrium.
Proposition 2(i) Suppose that β ≤ 1 − 2η/µ holds. There exists a set of low-tax equilibria such that∀t, T o (ut ) is given by (3), T y = (1 + β)/2, and U is given by (6). The equilibriumoutcome is unique and such that ∀t, τty = (1+β)/2, τto = 0, ut = (2η −(1+β)/2)/2η,and 2st = µ(1 + β)2 /8η.
11

(ii) Suppose that β > 1−2η/µ and β ≥ (µ/η)(η/µ−1/2)2 hold. There exists a set of low³oytax equilibria such that ∀t, T (ut ) is given by (3), T = θ ∈ max(1 − η/µ, θ̃(β)),
imin((1 + β)/2, 1 + β − η/µ) , and U is given by (6). The equilibrium outcome isindeterminate and such that ∀t, τty = θ, τto = 0, ut = (2η − (1 + β) + θ)/2η, and2st = µ(1 − θ + β)2 /2η.
Proof. See Appendix.
Suppose that agents choose τty to induce a future majority of rich. In the case illustrated in Panel (a) of Figure 2, such a choice is unavailable for agents. In the caseillustrated in Panel (b), the decisive voter prefers the highest tax rate on the young thatinduces a future majority of rich, τty = θ(≤ 1 + β − η/µ), because the revenue from theyoung Z(τty , 0) is increasing in τty in the range (0, θ). Therefore, the choice of τty = θ underothe expectation of τt+1
= 0 is sustained as an equilibrium if Z(θ, 0) ≥ Z(1/2, 1), i.e., ifθ ≥ θ̃(β). The condition β ≥ (µ/η)(η/µ − 1/2)2 implies that the range [θ̃(β), 1 + β − η/µ]
is not empty.
Next, consider the cases illustrated in Panels (c) and (d) of Figure 2. If 1 − η/µ <(1 + β)/2, i.e., if β > 1 − 2η/µ, agents have an option that induces a future majority ofrich by setting τty = θ. The tax rate on the young depends on the expectations by agents(see Panel (a) of Figure 5). In particular, when the expectation is given by θ = (1 + β)/2,
the young are taxed on the top of the Laffer curve. However, if (1 + β)/2 ≤ 1 − η/µ, i.e.,
if β ≤ 1 − 2η/µ, expectations no longer affect the determination of τty . Voters can chooseτty = (1 + β)/2 and maximize the revenue from the young by taxing on the top of the
Laffer curve (see Panel (b) of Figure 5).
[Figure 5 around here.]
Figure 6 illustrates the parameter conditions derived in Propositions 1 and 2. The
next proposition summarizes the results established so far. The proof of the proposition
is immediate from Figure 6.
[Figure 6 around here.]12

Proposition 3(i) If β < (µ/η)(η/µ − 1/2)2 , there exists a unique high-tax equilibrium as in Proposition
1.
(ii) If (µ/η)(η/µ − 1/2)2 ≤ β < (η/µ − 1/2)2 /(1 − η/µ), the equilibrium is indeterminate.
There exist both a high-tax equilibrium as in Proposition 1 and a set of low-tax
equilibria as in Proposition 2(ii).
(iii) If β > 1 − 2η/µ and β ≥ (η/µ − 1/2)2 /(1 − η/µ), there exists a set of low-tax
equilibria as in Proposition 2(ii).
(iv) If β ≤ 1 − 2η/µ, there exists a unique low-tax equilibrium as in Proposition 2(i).
As illustrated in Figure 5, given µ ∈ (1/2, 1], the range of η/µ is limited to [1, 2) if
η = 1. By assuming the cost function of education with η = 1, Hassler et al. (2007)
focused on this range and showed two patterns of political equilibria, as in Proposition
3(i) and 3(ii). Contrary to their analysis, however, we focus on a wider range of η/µ byassuming a generalized cost function. Under this extended framework, we can derive twopatterns of political equilibria, as in Proposition 3(iii) and 3(iv), which were not shown
by Hassler et al. (2007).
The mechanism behind our main finding is intuitive and can be understood as follows.
First, suppose that the investment cost is high such that η/µ > 1. In this case, it is lessattractive for agents to invest in education. When agents attach a low value to the futuresuch that β < (µ/η)(η/µ − 1/2)2 , there is a future majority of poor irrespective of policy.
However, when agents attach a high value to the future, such that β ≥ (µ/η)(η/µ − 1/2)2 ,agents can undertake strategic political behavior to induce a future majority of rich. The
equilibrium is indeterminate in this case.
Second, suppose that the investment cost is low such that η/µ ≤ 1. In this case,
it is more attractive for agents to invest in education. In particular, when the cost is
sufficiently low such that β ≥ (η/µ − 1/2)2 /(1 − η/µ), agents have an incentive to invest13

in education even if they expect a high future tax rate and, thus, a low return from theinvestment. Then, it becomes possible to tax heavily on the young without compromising
the future political equilibrium. Therefore, in this case, there is a future majority of richirrespective of policy.

3.4

Empirical Implications

The results established in Propositions 1–3 show that the level of government spending andthe distribution of rich and poor are affected by the parameter representing educationalcost, η. To investigate the empirical implications of η, consider two levels of η, ηH andηL (< ηH ). Suppose that given ηH , there exist both a low-tax equilibrium and a high-tax
equilibrium, as in Proposition 3(ii). Suppose also that, given ηL , there exists a set of lowtax equilibria as in Proposition 3(iii) or a unique low-tax equilibrium as in Proposition
3(iv).
The equilibrium level of government spending differs across political equilibria. Consider first the case of η = ηH that leads to multiple equilibria. In the high-tax equilibrium,
the level of government spending is given by 3µ/8ηH . In the low-tax equilibrium, thelevel, given by µ(1 − θ + β)θ/2ηH , depends on expectations by agents. The lowest levelis µ/8ηH at θ = θ̃(β), which is lower than 3µ/8ηH . The highest level is µ(1 + β)2 /8ηHat θ = (1 + β)/2, which exceeds 3µ/8ηH if β is high. Therefore, the high-tax equilibriumis featured by less or more spending depending on expectations by agents, θ, and thediscount factor, β.
As already shown by Hassler et al. (2007), the case of ηH provides an explanation forthe cross-country difference in the size of the government spending and the distribution ofrich and poor. To provide an empirical viewpoint, we take the entry and graduation rates
of tertiary education as proxy variables of the size of the rich µe∗ . Then, the high-tax
equilibrium corresponds the some Continental European countries like Austria, France,
Germany and Italy, featured with large sizes of governments and low graduation rates; thelow-tax equilibrium corresponds to Australia, Iceland, Ireland, Japan, the United King14

dom and the United States, featured with small sizes of government and high graduation
rates (see Figure 1).
Although the case of ηH can describe the cross-country difference among some OECD
countries, it does not fit the empirical fact of Nordic countries (Denmark, Finland, Norway
and Sweden); they have large sizes of government and high graduation rates as presented
in Figure 1. However, the case of ηL can provide an explanation for the Nordic countries.
In this case, the majority is rich, and the level of government spending is given by 2st =µ(1 − θ + β)2 /2ηL or 2st = µ(1 + β)2 /8ηL , which is larger than that in the high-tax
equilibrium, 2st = 3µ/8ηH , for a small ηL and a large β. Therefore by focusing on
educational costs, we can show the equilibrium that fits the empirical fact of the Nordic
countries, which was not shown by Hassler et al. (2003)and Hassler et al. (2007).
We have shown the existence of equilibrium featured by the majority of rich and alarge size of government by considering the cases of low education costs. We can show
the existence of a qualitatively similar equilibrium by allowing the wage of the successful
agents, w, to differ from unity. A higher wage inequality results in a unique equilibrium
where the majority of rich prefer a large size of government and imposes a high tax
rate on the young (Hassler et al., 2003). However, this case does not fit the empiricalfact of the Nordic countries that are featured by low inequality (Björklund et al., 2002).
Therefore, focusing on education costs rather than wage inequality is a key to explaining
the empirical fact of the Nordic countries in the framework of Hassler et al. (2003) and
Hassler et al. (2007).

4

Ramsey Allocation

In this section, we first characterize the Ramsey allocation, and then compare it withthe political equilibria presented in Section 3. Following Hassler et al. (2007), we definea Ramsey allocation as a feasible plan chosen by a benevolent social planner who can

15commit to a policy sequence at time 0. The allocation solves the following problem:[βµ(1 − u0 )V os (s0 , τ0o ) + β(µu0 + (1 − µ)H)V ou (s0 , τ0o )
#
∞
X
©
ª
o
+
λt+1 µV y (et , st , st+1 , τty , τt+1
) + (1 − µ)HV y (st , st+1 ) ,max{st ,τty ,τto }∞
t=0

t=0

where λ ∈ [0, 1) is a discount factor, β(> 0) is a weight on the initial generation of old
agents, and H(≥ 1) is a bias in the planner’s preference towards low-ability agents. We
employ the same form of a social welfare function as Hassler et al. (2007) in order toshow how our extension affects the characterization of the Ramsey allocation.
Given the educational investment (1) and the government budget constraint (2), theproblem can be rewritten as a simple static problem:
βµ(1 − u0 )(1 − τ0o ) + (β + λ)Aµ(1 − u0 )τ0o +maxoτ0 ∈[0,1]

L
,
1−λ

(7)

where

L ≡

maxo

{τ y ∈[0,1],τ ∈[0,1]}

£

©
ª
o
λµ (1 − τ y + β(1 − τ o ))e∗ (τ y , τ o ) − η · e∗ (τty , τt+1
)2+(β + λ)Aµτ y e∗ (τ y , τ o ) + λ(β + λ)Aµτ o e∗ (τ y , τ o )] ,where A ≡ (µ + (1 − µ)H)/2 is the planner’s marginal value of expenditures. The problem
implies that after the initial choice of τ0o , the problem reduces to a sequence of identicalstatic optimization problems over τ y and τ o . The next proposition characterizes thesolution of the Ramsey problem.
Proposition 4: The allocation solving the Ramsey problem has

τ0o =



 1 if A > β ,
β+λ

 0

otherwise,

and a constant sequence of taxes, τ y and τ o given by the following:16

1. Suppose that λ < β. Then,½

A − λ/(β + λ)
τ = max 0, (1 + β)
2A − λ/(β + λ)y¾and τ o = 0.

2. Suppose that λ > β. Then,½¾½¾
A(1 − λ) − λ/(β + λ)1 + β A − β/(β + λ)o
τ = max 0,and τ = min·,1 .
2A − λ/(β + λ)β
2A − β/(β + λ)y

3. Suppose that λ = β. Then,τy = τo = 0

if A ≤ 12 τ y + βτ o = (2A−1)(1+β) if A > 1 .
4A−1
2
Proof. See Appendix.
Proposition 4 shows that the parameter η does not affect the determination of tax
rates in the Ramsey allocation. This result depends on the cost function specified by aquadratic form. With this feature in mind, we now summarize the solution to the Ramsey
problem as follows. First, as regards the choice of τ0o , there is no interaction with future
variables. The planner chooses τ0o to maximize the weighted sum of the utilities of theinitial old agents. Thus, the planner sets τ0o = 1 if A > β/(β + λ) and τ0o = 0 otherwise.
Second, the choice of τ y and τ o depends on the magnitude correlation between β and λ.
If λ < β (part 1 of Proposition 4), the tax burden falls on the young; if λ = β (part 3 of
Proposition 4), the distribution of the tax burden is indeterminate; and if λ > β (part 2of Proposition 4), most of the tax burden falls on the old. In the case of λ > β, for a range
of a small A, the nonnegativity constraint on the young binds; for range of a large A, theconstraint of τ o ≤ 1 is binding. In words, the tax burden is increased by the planner’s
higher value to the low-ability agents.
Given the characterization of the Ramsey allocation above, we now evaluate the efficiency of the political equilibria. First, consider the high-tax equilibrium. The high-tax
17equilibrium shows a qualitative similarity to the Ramsey allocation in the case of λ > β,in that the main tax burden falls on the old. However, the tax rates in the high-tax
equilibrium are quantitatively different from those in the Ramsey allocation in the following two points. First, when A is low, the Ramsey planner does not tax the old at the
maximum rate. Second, in the Ramsey allocation with λ > β, the young are subject to atax τ y ≤ (1 − λ)/2, while they are taxed at a 50% rate in the high-tax equilibrium.
Next, consider the low-tax equilibrium featured by no taxation on the old (part 2).
This equilibrium shows a qualitative similarity to the Ramsey allocation in the case of
λ < β, in that the old bear no tax burden. When the political tax rate on the young
depends on the expectations by agents as in Proposition 2(ii) the low-tax equilibriumwith τ y = θ may realize the same tax rates as those in the Ramsey allocation, dependingon the expectation θ. However, when the political tax rate on the young is determinate
as in Proposition 2(i), the low-tax equilibrium with τ y = (1 + β)/2 realizes a higher tax
burden as compared with the Ramsey allocation.9 Therefore, for low levels of η, there isno possibility that the political equilibrium resembles the Ramsey allocation.

5

Conclusion

This paper has extended the model of Hassler et al. (2007) by generalizing the costfunction of education, which is summarized as follows. First, for high costs, with Hassler
et al. (2007) as a special case, individuals have a weak incentive to invest in education,
which results in a similar equilibrium characterization to that found in Hassler et al.
(2007). The model presents multiple equilibria: the high-tax equilibrium (the pro-welfare
state) featured by the minority of highly educated individuals and a large size of thegovernment; and the low-tax equilibrium (the antiwelfare state) featured by the majorityof highly educated individuals and a small size of the government. Which state is realized
9

Assume that λ < β and A > λ/(β + λ): the Ramsey allocation is featured by τ y = (1 + β){A −λ/(β + λ)}/{2A − λ/(β + λ)} and τ o = 0. Assume also that β < 1 − 2η/µ: the political equilibriumis featured by τ y = (1 + β)/2 and τ o = 0 as in Proposition 2(i). By direct calculation, we find that
(1 + β){A − λ/(β + λ)}/{2A − λ/(β + λ)} < (1 + β)/2 is equivalent to A − λ/(β + λ) < A − λ/2(β + λ),which holds for any A, β and λ.

18

depends on expectations by individuals.
Second, for low costs, individuals have a strong incentive to invest in education irrespective of future redistribution policy. In this case, there is a unique equilibrium featuredby the majority of highly educated individuals and the large size of the government, whichfits the empirical facts of the Nordic countries (Denmark, Finland, Norway and Sweden).
Furthermore, this unique equilibrium may require an excess tax burden on the young
compared with the Ramsey allocation.
The analysis in this paper focuses on the characterization of political equilibria undera system of lump-sum general transfers paid to both the rich and the poor; we haveabstracted from targeted transfers to the elderly or to the poor. However, the paper
has provided a framework for understanding the role of education costs that affect thepolitical determination of redistribution policy via individual decisions on education.

19

6

Appendix

6.1

Proof of Proposition 1oSuppose that at time t, agents know that τty = 1/2 and expect that τt+1
= 1. Then,

1 − e∗ (1/2, 1) = ut+1 = 1 − 1/4η = 1 − (1/4µ)(µ/η). Because the condition β < (η/µ −1/2)2 /(1 − η/µ) ensures that η/µ > 1/2 (see Figure 6), ut+1 = 1 − (1/4µ)(µ/η) > 1 − 1/2µoholds for all t. By (3), this implies that τt+1
= 1, fulfilling initial expectations. Therefore,

there exists a high-tax equilibrium 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 note thefollowing properties of the function Z: (i) Z(τty , 0) is concave in τty and is maximized atτty = (1 + β)/2; (ii) Z(τty , 0) > Z(τty , 1). Given these properties, we derive the conditionsothat (i) setting τty = (1 + β)/2 is not available under the expectation τt+1
= 0; (ii) Z(τty , 0)is maximized at τty = θ; and (iii) Z(θ, 0) < Z(1/2, 1) hold. These conditions are givenby (i) (1 + β)/2 ≤ 1 − η/µ, (ii) θ ∈ (max(0, 1 − η/µ), (1 + β)/2), and (iii) θ < θ̃(β),
respectively. The second and third conditions require that θ is set within the range(max(0, 1 − η/µ), θ̃(β)). The condition β < (η/µ − 1/2)2 /(1 − η/µ), which guarantees thatthe set (max(0, 1 − η/µ), θ̃(β)) is not empty, includes the first condition, as illustrated in
Figure 6.

6.2¥

Proof of Proposition 2o(i) Suppose that in period t, agents know that τty = (1 + β)/2 and expect that τt+1
= 0.

Then ut+1 = 1−e∗ ((1+β)/2, 0) = (2η−(1+β)/2) ≤ 1−1/2µ if and only if β ≥ −1+2η/µ.
The condition β ≥ −1 + 2η/µ holds under the assumption of β ≤ 1 − 2η/µ (see Figure 6).
o= 0, fulfilling initial expectations.
By (3), the condition ut+1 ≤ 1 − 1/2µ implies that τt+1

Given that Z(τty , 0) > Z(τty , 1)∀τty ∈ [0, 1] and that arg maxτty ∈[0,1] Z(τty , 0) = (1 + β)/2,
setting τty = (1 + β)/2 is optimal for the decisive voter.
(ii) Under the assumption that U is monotonic, T o (U (θ)) = 0 implies that ∀τty ≤ θ,
T o (U (τty )) = 0 and U (τty ) = (2η − (1 + β) + τty )/2η. Consequently, the relevant payoff
20

function in the range τty ≤ θ is Z(τty , 0). The function Z(τty , 0) is a hump-shaped functionof τty , with a maximum at (1 + β)/2.
Given the first assumption β > 1 − 2η/µ, i.e., 1 − η/µ < (1 + β)/2, we can consider
the following two cases: (a) the case of 1 − η/µ < (1 + β)/2 < 1 + β − η/µ and (b)the case of 1 + β − η/µ ≤ (1 + β)/2. In case (a), Z(τty , 0) is increasing in the range
τty < (1 + β)/2 and decreasing in the range τty ≥ (1 + β)/2. In case (b), Z(τty , 0) is
increasing in the range τty < 1 + β − η/µ. Therefore, the decisive voters prefer τty = θ ∈(1 − η/µ, min(1 + β − η/µ, (1 + β)/2)] to any τty < θ under the assumption of β > 1 − 2η/µ.
An alternative option for the decisive voter is to set τty = 1/2 under the expectationoo= 1. The choice of τty = θ under the expectation of τt+1
= 0 is sustained as anof τt+1

equilibrium if Z(θ, 0) ≥ Z(1/2, 1), i.e., if θ ≥ θ̃(β), whereθ̃(β) ≡1+β−pβ(β + 2)
1+β
<
.
2
2

Therefore, there exists a set of low-tax equilibria featured by T o (ut ) is given by (3), T y = θand U is given by (6) if θ is set within the range:³imax(1 − η/µ, θ̃(β)), min((1 + β)/2, 1 + β − η/µ) .

The condition β ≥ (µ/η)(η/µ − 1/2)2 , which is rewritten as θ̃(β) ≤ 1 + β − η/µ, implyingthat the above range is not empty.

6.3¥

Proof of Proposition 4

It is immediate from (7) that τ0o = 1 if A > β/(β + λ) and τ0o = 0 otherwise. The solutionof the pair (τ y , τ o ) is derived by solving (8). The solution must satisfy the following
first-order conditions:∂L− ξ y + θy = 0,
∂τ y
∂L
:
− ξ o + θo = 0,
o
∂τ
21

τy :

(8)

τo

(9)

where θy and θo are the Kuhn-Tucker multipliers associated with the constraints τ y ≥ 0
and τ o ≥ 0, respectively, whereas ξ y and ξ o are the Kuhn-Tucker multipliers associatedwith the constraints τ y ≤ 1 and τ o ≤ 1, respectively.
Assume, first, that τ o = ξ y = ξ o = θy = 0. Then, we obtain τ y = (1 + β) A−λ/(β+λ) ,2A−λ/(β+λ)
2 θo = µA2 · (1+β)(β−λ)(β+λ) ,2η

2A(β+λ)−λwhere τ y < 1 and θo > 0 as long as β > λ. This establishes part 1.
Assume, next, that τ y = ξ y = ξ o = θo = 0. Then, we obtainA−β/(β+λ)τ o = 1+β
· 2A−β/(β+λ),β

2 θy = µA2 · (1+β)(λ−β)(β+λ) ,2η

β(2A(β+λ)−β)where θy > 0 as long as β < λ. Furthermore, τ o < 1 holds as long as A < β/(1−β)(β +λ).
If A ≥ β/(1 − β)(β + λ), then τ o = 1 and ξ o > 0. However, it remains to be checked
whether τ y = 0 continues to be a solution. Given τ o = 1, we obtain
∂L
1
y
y
−
ξ
+
θ
=
(−λµ(1 − τ y ) + (β + λ)Aµ(1 − λ + 2τ y ) + θy = 0.
y∂τ2ηThe solution features τ y = 0 and θy > 0 if and only if A ≤ λ/(1 − λ)(β + λ). For larger
values of A, the solution instead features θy = 0 andτy =A(1 − λ) − λ/(β + λ).
2A − λ/(β + λ)This establishes part 2.
Finally, consider the case where β = λ. Then, the objective function can be writtenas:L(x) = βµ{(1 + β − x) · e∗ (x) − η · e∗ (x)2 } + 2βAµx · e∗ (x),where x ≡ τ y + βτ o . The first-order condition yields the result in part 3.
22¥

References[1] Alesina, A., and Rodrik, D., 1994. Distributive Politics and Economic Growth. Quarterly Journal of Economics 109, 465–490.
[2] Bertola, G., 1993. Factor Shares and Savings in Endogenous Growth. American Economic Review 83, 1184–1198.
[3] Björklund, A., Eriksson, T., Jantti, M., Osterbacka, E., and Raaum, O. 2002. Brother
Correlations in Earnings in Denmark, Finland, Norway and Sweden Compared to theUnited States. Journal of Population Economics 15, 757–772.
[4] Boadway, R.W., and Wildasin, D.E., 1989. A Median Voter Model of Social Security.
International Economic Review 30, 307–328.
[5] Casamatta, G., Cremer, H., and Pestieau, P., 2000. The Political Economy of Social
Security. Scandinavian Journal of Economics 102, 503–522.
[6] Galasso, V., and Profeta, P., 2002. The Political Economy of Social Security: A
Survey. European Journal of Political Economy 18, 1–29.
[7] Hassler, J., Rodriguez Mora, J.V., Storesletten, K., and Zilibotti, F., 2003. The
Survival of the Welfare State. American Economic Review 93, 87–112.
[8] Hassler, J., Storesletten, K., and Zilibotti, F., 2007. Democratic Public Good Provision. Journal of Economic Theory 133, 127–151.
[9] Meijdam, L., and Verbon, H.A.A., 1996. Aging and Political Decision Making on
Public Pensions. Journal of Population Economics 9, 141–158.
[10] OECD, 2005. Education at a Glance 2005. OECD, Paris.
[11] OECD, 2007. OECD Factbook 2007 Economic, Environmental and Social Statistics.
OECD, Paris.

23[12] Verbon, H.A.A., and Verhoeven, M.J.M., 1992. Decision Making on Pension
Schemes under Rational Expectations. Journal of Economics (Zeitschrift fuer Nationaloekonomie) 56, 71–97.

24

35
30

SWE

FRA
DEU

DNK

25

AUT
NOR
ITA
POL

HUN

FIN

20

CZE
CHE

ESP

GBR
ISL

15

USA

10

AUS

JPN

SVK

20

30

IRL

40

50

Figure 1: Figure 1 is a scatter plot of graduation rates of tertiary education and the ratios
of public social expenditure to GDP in 2003 for OECD countries. The solid vertical line in
the plot indicates the average graduation rate among OECD countries; the solid horizontal
line indicates the average public social expenditure to GDP among OECD countries.

25

´

´

(b) The case of 1 < ¹ · 1 + ¯(a) The case of 1 + ¯ < ¹
1(2´ ¡ 1 + ¿ty)
2´¢
1 ¡U (¿ty) =2´ ¡ (1 + ¯) + ¿ty
2´

1(2´ ¡ 1 + ¿ty)
2´¢
1 ¡U (¿ty) =2´ ¡ (1 + ¯) + ¿ty
2´U (¿ty ) =U (¿ty ) =´

1+¯¡ ¹´´(c) The case of ¯ < ¹ · 1y(d) The case of ¹ · ¯

1y(2´ ¡ 1 + ¿t )2´¢
1 ¡U (¿ty) =2´ ¡ (1 + ¯) + ¿ty
2´y

U (¿t ) =

´

1¡¹

1
y
(2´ ¡ 1 + ¿t )2´¢
1 ¡U (¿ty) =2´ ¡ (1 + ¯) + ¿ty
2´U (¿t ) =

´

´

1¡¹

1¡¹+¯

Figure 2: The figures represent the equilibrium decision rule ut+1 = U (τty ). The solidlines show the graphs of U satisfying equilibrium condition 2. The panel (a) illustrates
the case of 1 + β < µη ; the panel (b) illustrates the case of 1 < µη ≤ 1 + β; the panel (c)illustrates the case of β < µη ; the panel (d) illustrates the case of µη ≤ β.

26

U (¿ty ) = 21´ (2´ ¡ (1 + ¯) + ¿ty)
U (¿ty ) = 21´ (2´ ¡ 1 + ¿ty)³´´
min 1 ¡ ¹ + ¯; 1´
1¡¹

Figure 3: The figure illustrates on example of a monotonic function U for the case of
η
≤ 1.
µ

27

´
1¡¹

Z

³

´

1
2; 1

> Z(µ; 0)

1+¯
2

Z

³

´

1; 1
2

´
³
´
min 1 ¡ ¹ + ¯; 1

< Z(µ; 0)Figure 4: The solid curves illustrate the available tax revenue from the young, Z, under2ethe assumption of β < (η/µ−1/2), i.e., 1 − µη < θ(β).
1−η/µ

28

´

1¡¹

1+¯
2

³ ´
Z 12 ; 1 > Z(µ; 0)³ ´Z 12 ; 1 < Z(µ; 0)´³´
min 1 ¡ ¹ + ¯; 1

1+¯ 1 ¡ ´
¹
2

´
³
´
min 1 ¡ ¹ + ¯; 1Figure 5: The solid curves illustrate the available tax revenue from the young, Z. The
panel (a) represents the case of β > 1 − 2η/µ, i.e., 1 − η/µ < (1 + β)/2. The panel (b)represents the case of β ≤ 1 − 2η/µ, i.e., 1 − η/µ ≥ (1 + β)/2.

29

³

´
´ 1 2
¹ ¡2

¯= ³

1¡ ¹´

´

´2
³
¹ ´
¯ = ´ ¢ ¹ ¡ 21
9
8

1
4

1
4

´
¹
1
2

´

¯ = 1¡2¢ ¹´¯ = ¡1 + 2 ¢ ¹

Figure 6: The figure displays the set of parameters (β, η, µ) classified according to thecharacterization of political equilibrium in Propositions 1 and 2. The areas (i), (ii), (iii)and (iv) satisfy the parameter conditions given in Proposition 3 (i), (ii), (iii) and (iv),respectively.

30