GCOE Discussion Paper Series
Global COE Program

Human Behavior and Socioeconomic Dynamics
Discussion Paper No.188

Solution Concept for Intergenerational Conflict:
the Role of Intergenerational Bargaining

Yusuke Kinai

April 2011

GCOE Secretariat
Graduate School of Economics

OSAKA UNIVERSITY

1-7 Machikaneyama, Toyonaka, Osaka, 560-0043, Japan

Solution Concept for Intergenerational Conflict:
the Role of Intergenerational Bargaining∗
Yusuke Kinai†
Graduate School of Economics, Osaka University

March 30, 2011
Abstract
This paper specifically examines intergenerational conflict and analyzes an overlapping generations
model of public goods provision from the viewpoint of time-consistency. Public goods are financed
through labor-income and capital-income taxation, thereby distorting savings and the labor supply.
Taxes redistribute income across generations in the form of public goods. Under such a situation, there
emerge dual intergenerational conflicts: the first is related to the amount of public goods and the second
is the tax burden. We then contrast the politico-economic equilibrium with commitment allocation,
and analyze the sources of conflict and time-inconsistency, and attempt to resolve such a conflict byintroducing the concept of ‘intergenerational bargaining’. Our main findings are the following. First,
taxation derived using Lagrange method fails to be time-consistent. Second, depending on bargainingpower, taxation based on intergenerational bargaining can be time-consistent. Third, we portray theproperties of taxation and public goods provision rules based on intergenerational bargaining.

Keywords : Dual Intergenerational Conflicts; Intergenerational Bargaining;Time-inconsistency;
Modified Public Goods Provision Rule.

JEL Classification : E61; E62; H41.

∗ The author thanks Masatsugu Tsuji and Kazuo Mino for their encouragement and helpful suggestions at various
stages of this research. Moerover, he is grateful for the comments from Shigeki Kunieda, Noriyuki Yanagawa,Takashi Oshio, Shingo Ishiguro, Koichi Futagami, Midori Hirokawa, Ryuichi Tanaka, Ryo Arawatari, Kayoko
Kobayashi, Kojun Hamada, Koichiro Sawano, Wataru Kobayashi, Fumitoshi Moriya, and Takuro Yamashita on
the previous version of this paper. I also thank the seminar participants at the meeting of the Japanese Economic Association, CTW Summer Conference, Osaka University, Kinki University, the applied microeconomics conferencein Hakone, and PET Conference held in Hanoi for their helpful comments. Any comments and reasonable critiques
are welcome. Of course, all remaining errors remain the sole responsibility of the author.
† Correspondence to: Graduate School of Economics, Osaka University, 1-7, Toyonaka, Osaka, 560-0043, Japan
E-mail : ykinai@js8.so-net.ne.jp

1 Introduction
Traditionally, in analyzing economic policies such as fiscal or monetary policy, economists
have assumed that the government is a monolithic organization that is intended mainly tomaximize social welfare or the economic growth rate. In this regard, the following points
should be emphasized: First, various entities (voters, bureaucrats, representatives, organizations, etc.) that are involved in policy determination foster conflicts. Therefore, it is impossible for the government to organize a policy based on only one situation or position. In otherwords, the government cannot avoid determining a policy that incorporates implications of
numerous opinions. Second, even if a policy were derived that maximizes social welfare or
the growth rate, carrying out such a policy with certainty would be difficult: it is hard tocommit to such a policy1) . As a result, we cannot regard actual governments as monolithic
organizations, as many economists have assumed. Similarly, looking at the voter side, various conflicts exist: young vs. old, unemployed vs. job-holders, rich vs. poor, and so forth2) .
Given the existence of such conflicts, “political compromise” is inevitable; it is difficult tomaintain the government commitment entirely. As countermeasures for such a situation, we
find that it is necessary ex post to carry out some kind of coordination policy to attenuate
inefficiencies that result from discretionary policy.
In light of the existence of such conflicts, we find that normative model analysis, in which
the government is assumed to be benevolent in the sense that they maximize social welfare,is inadequate to determine policy implications because such a model cannot encompass thenature of the current situation. Therefore, this paper presents analysis of conflicts of interest
among voters, especially intergenerational conflict. In addition, as a problem corresponding
to such a conflict, it is necessary to consider the time-inconsistency problem3) because boththis problem and such conflicts are concurrently related to government commitment. For that

1) Some conflicts exist even within actual governments.

As just one example, the Ministry of Finance andthe Ministry of Health, Labor and Welfare in Japan have frequent conflicts. These two organizations seek
different objectives: the former seeks to decrease the deficit or debt, whereas the latter is responsible forpromoting a social security system, even if it is very costly. Consequently, a contraposition of duties pertains
between the two organizations.
2) As an example of conflicts among voters, we can indicate conflicts surrounding issues of social security,
particularly pension systems.
3) For pioneering studies on this problem, see Kydland and Prescott (1977), Fischer (1980), and Chari, Kehoeand Prescott (1989).

1

reason, this paper describes the investigation of intergenerational conflict in connection witha time-inconsistency problem.
In turn, let us explain the relationship between this paper and past studies. We use a
two-period version of an overlapping generations (OLG) mode and limit our survey to paststudies that have used a similar model. In the context of intergenerational conflicts4) , manystudies center their scope on intertemporal conflict, as explained below. Here, we regard theintertemporal conflict case, in which a transfer from young people to elderly persons is donein each period, as in a pay-as-you-go type pension system. This can also be interpreted as a
one-sided transfer case. In this regime, intergenerational conflict arises with the result thatthe burden that young people should bear varies depending on when they are born.
Among such studies that posit a one-sided transfer case, some studies such as Casamatta,
Cremer and Pestieau (2000) (pension, transfer from elderly persons to young people) andHoltz-Eakin, Lovely and Tosun (2004) (education, a transfer from elderly persons to young
people) address policy decisions based on majority voting, whereas Azariadis and Galasso(2002) investigates how cooperative solutions are attained, with specific attention to policy
decision based on majority voting in combination with veto power. As another method ofpolicy decision, Grossman and Helpman (1998) introduces lobbying between households
and a myopic government. It is true that numerous studies have specifically addressed intergenerational transfer. However, as already described, we can categorize such studies asintertemporal conflict cases despite their associated methods of policy decision5) .
On the other hand, to the best of our knowledge, few studies have analyzed intratemporal
conflict, even though redistributive policies include two-sided transfers as well as one-sided
transfers6) . Among such studies, Bassetto (2008) expands past studies described above into a4) For a survey, see Breyer (1994), Galasso and Profeta (2002), and de Walque (2005). Breyer reviews precedentstudies that specifically examined the assumptions of models and classifies the models accordingly. Somestudies use the median voter theorem, based on the assumption of single-peaked utility, in determining policy.
Related to this issue, see also Galasso and Profeta (2002) and Walque (2005).
5) For studies that incorporate intragenerational conflict as well as intergenerational conflict, we can point out
Tabellini (1991, 2000), who examines substitution between social security (i.e. pension) and debt payment,using the altruistic model. That study shows that decisions reached by the older generation are not necessarily disadvantageous to young people, depending on the strength of the bequest motive. Lambertini andAzariadis (2003) specifically evaluates income redistribution from young people to elderly persons and compares open-loop equilibria (corresponding to the case with commitment) and closed-loop equilibria (withoutcommitment) under the situation in which inequality pertains with regard to productive ability. They showthat the amount of intergenerational and intragenerational transfer is determined according to the degree of
inequality.
6) Here, we view a two-sided transfer as the case in which a transfer from young people to elderly persons and2

two-sided transfer case by incorporating public goods in an OLG model7) . In this approach,an intergenerational conflict arises depending on how the cost of public goods is assigned
to young people and the older generation. Bassetto focuses on this point by incorporating
public goods that serve in the role of intergenerational transfer (redistribution), and illustrates
the possibility of multiple equilibria using numerical simulation8) . Although the analysis
in the present paper resemble those of Bassetto (2008) in the respect that an intratemporal
conflict is emphasized, it differs absolutely from his analysis in other respects, as explainedlater.
In many of these studies, policy-making is assumed to be based on the median voter theorem or majority voting, which have the common disadvantage of abandoning an underdog
or minority group9) . Different from those studies, we elucidate intergenerational conflict andintroduce the concept of ‘intergenerational bargaining’, as a solution to such a conflict, expanding the idea of Besley and Ghatak (2001). In this regard, some noteworthy points are
the following: First, the government raises funds through taxation to procure public goods.
Then the government redistributes tax revenues in the form of public goods, which affects
the utility of both young and old generations10) . Second, the younger generation has an incentive to decrease labor-income taxation, whereas elderly persons also have an incentive todecrease capital-income taxation because the labor supply is also endogenous in our model.
In other words, an intergenerational conflict arises at this stage. Moreover, each generation
decreases the tax burden without careful consideration, engendering a situation in which theamounts of public goods provision decrease. As a result, both generations decrease utility,
creating a trade-off between utility derived through the use of public goods and the disutility
that arises from the tax burden. Intergenerational conflict arises in terms of the tax burden.
its reverse are implemented during the same period.
7) For another study, see Renström (1996), and Pirttilä and Tuomala (2001), which incorporate public goods

into the utility function in an OLG model. Renström derives optimal taxation, whereas Pirttilä and Tuomala
derives the public goods provision rule using Mirrlees’ approach. Moreover, Kaas (2003) introduces public
investment into an OLG model. However, those studies neglect the time-inconsistency problem.
8) Nevertheless, that result depends strongly on the specification of utility function. Moreover, this point also
differs from our paper.
9) As another criticism of these means of policy-making, see Grossman and Helpman (1998, section 1). Thosestudies point out that this approach ignores the aspect of the parliamentary system, that is, coalition of voters.
Moreover, the median voter theorem has some restrictions. For instance, this theorem is applicable only undera one-dimensional policy space.
10) From another perspective, we can regard this kind of public good as “intergenerational public goods”, according to the term used in Sandler and Smith (1976). For further examination of this point, see remark
1.

3Moreover, regarding the amounts of public goods, an intergenerational conflict also arises
because the amounts of public goods that young and older generations demand generally do
not coincide. In other words, dual intergenerational conflicts emerge. Under such a trade-off,or because of intergenerational conflicts, how do we determine the public goods provision
rule and taxation?11) To answer such a question, this paper offers a new solution concept:‘intergenerational bargaining’12) .
Our main findings are summarized as follows: First, taxation derived by the Lagrange
method fails to be time-consistent. Second, depending on the value of bargaining power,such taxation based on intergenerational bargaining can be time-consistent. Third, we showedproperties of taxation and the modified public goods provision rule based on intergenerational
bargaining, which differs from Bassetto (2008).
The organization of this paper is as follows: section 2 describes the model and section 3
derives optimal taxation under the assumption that the government has commitment technology, and investigates the properties of such taxation. In section 4, we define the equilibrium
concept and derive the modified public goods provision rule and taxation through intergenerational bargaining. Section 5 presents conclusions of this paper and future problems that
require analysis.

2 The model
Our model resembles that of Renström (1996). We consider an overlapping generations economy with an endogenous labor supply in which the time horizons are infinite and discrete, as
indexed by t = 0, 1, 2, · · · . This economy has no uncertainty. We then present a brief description of the respective behaviors of households, firms and the government. Figure 1 depicts an
intuitive structure of this model along with flows of goods and capital.

11) As another study that analyzes the public goods provision rule in an OLG model, we can point out Batina(1990). However, Batina expands the Samuelson rule under the assumption that public goods affect only theutility in the young period.
12) As other studies that incorporate intergenerational bargaining, see Bassetto (2008) and Renström (2002).
However, the former highlights the existence of equilibria indeterminacy and the model of the latter does
not depict the situation of dual intergenerational conflict. Moreover, neither of the two papers highlight the
time-inconsistency problem, as our paper does.

4t − 1 periodt + 1 periodt periodt + 2 periodτt Rt st−1

st−1
F(k̂t , lt )gt

ηt wt ltst
Flow of Goods and Capitalτt+1 Rt+1 st

F(k̂t+1 , lt+1 )

gt+1

ηt+1 wt+1 lt+1

External EffectFigure 1 Structure of the model; External Effect and Flow of Goods and Capital

2.1 Preferences, Technology, and Policy
2.1.1

Production

First, we describe firms’ behavior. We assume that factor markets are perfectly competitive and that firms maximize their profits. Labor and capital stock are used for production;
production technology yields constant returns to scale. Therefore, production functions are
expressed as Yt = F(Kt , Lt ) : ℜ2+ → ℜ+ , where Kt , Lt , and Yt respectively represent capitalstock, labor, and output in aggregate terms. For definitions of capital stock and output perdcapita, lt and k̂t respectively denote LNtt and KNtt . The maximization process yields
Rt =∂ F(k̂t , lt )∂ F(k̂t , lt )≡ Fk̂ (k̂t , lt ), wt =≡ Fl (k̂t , lt ).
∂ lt
∂ k̂t

In those equations, Rt and wt respectively indicate the rental rate of capital and the real wage
rate.
2.1.2

Government

Next, let us describe government behavior. The government determines taxation and public
expenditure to maximize social welfare. Some assumptions must be made. First, the government finances its expenditure with a tax on capital income and labor income; it is used topurchase public goods. Second, they cannot make use of lump-sum taxes and a consumption
tax. Therefore, the government’s budget constraint is expressed asRt bt + gt = bt+1 + Rt τt st−1 + wt lt ηt ,5(1)

where bt denotes the national debt.
2.1.3

Households

Finally, Households live two periods in a closed-economy with neither population growth
nor a bequest motive. Each generation consists of only a single agent. Dynasties derive
utility from public goods as well as consumption and leisure. For simplicity, preference ofthe dynasty’s cohort that remain alive at t period is described by the following additively
separable function:U(ct , dt+1 , 1 − lt , gt , gt+1 ) = u(ct ) + h(gt ) + v(1 − lt ) +1{u(dt+1 ) + h(gt+1 )},
1+ρ

(2)

where ct , dt , lt , gt and ρ respectively denote consumption in the younger period and old,the labor supply, the amount of public goods at period t, and the rate of time preference.
We assume that the public goods at period t depreciate entirely at the end of that period.
Furthermore, each function is twice differentiable, strictly increasing, strictly concave, andeach satisfies the Inada condition. Two kinds of tax exist: a tax on labor income (η ), and a tax
on saving (τ ). Households that form at date t divide after-tax labor-income into consumption(ct ) and saving (st ), and consume (dt+1 ) as after-tax savings when they are old. Consequently,
the young-period and old-period budget constraints are shown respectively as
ct + st = (1 − ηt )wt lt , dt+1 = (1 − τt+1 )Rt+1 st .

(3)

Each household is assumed to maximize his or her lifetime utility (2) by choosing a consumption plan and a level of labor supply, subject to the budget constraints of Eq. (3). Taking
into consideration the budget constraint of the government, Eq. (1), the first order conditions
are expressible as
1{R̃t+1 u′ (dt+1 ) + Rt+1 τt+1 h′ (gt+1 )} = 0, and1+ρ
Ul = w̃t u′ (ct ) − v′ (1 − lt ) + wt ηt h′ (gt ) = 0,

Us = −u′ (ct ) +

(4a)
(4b)

where R̃t+1 ≡ (1 − τt+1 )Rt+1 and w̃t ≡ (1 − ηt )wt . By solving these two equations, we obtain6

a savings and labor supply function as follows13) :st = s(w̃t , R̃t+1 ; gt ), lt = l(w̃t , R̃t+1 ; gt ).
■Remark 1(5)

Here, as for the FOCs, Eqs. (4), the criticism might arise that the terms h(gt )and h(gt+1 ) should be treated as given because gt represents public goods and does not affect both generations’ behavior. In our setting, we calculate FOCs in this manner because
the government expenditure is dependent upon taxation, which affects households’ behavior.
Note that this calculation is based on the assumption that there is only single agent in eachgeneration. From this viewpoint, we might regard public service rather than public goods asused by young and older people who are alive in the same period. Using the term in Sandler
and Smith (1976), we can call these kinds of public services “intergenerational public goods”(IPG).14)

2.2 Market Equilibrium
We then formulate equilibrium conditions for each market. Because k̂t ≡ KNtt = KLtt · NLtt = kt lt ,where N is population, we can write this condition as follows:
1. Commodity market
ct + dt + gt + kt+1 lt+1 = Rt kt lt + wt lt(6)st − bt+1 = kt+1 lt+1 (= k̂t+1 )(7)

2. Capital market

3. Labor market
It is assumed that, in this economy, full employment is founded, meaning that a supplydemand balance is attained. That is, ltd denotes labor demand.
ltd = lt

(8)13) To close the model, we must describe the behavior of the initial old. As in other periods, the initial old agent

solvesmax {u(d1 ) + h(g1 )}s.t. d1 = (1 − τ1 )R1 s0 ,given R̃1 , w̃o , s0 , and g0 .

14) Rangel (2005) cites environmental preservation or R&D as an example of IPG. Unlike this paper, he regards

IPGs as goods that affect the utility of the next generation as well as the current generation.

7

2.3 Timing of Decisions
Finally, let us summarize the sequence of decision-making (or political process) as follows:∞ according to theStage 1. At period 0, the government determines the tax policy {τt∗ , ηt∗ }t=0

tax sequence derived by the Lagrange Method. Note that this taxation is ex ante
∞ is also determined.
optimal. Similarly, {gt∗ }t=0Stage 2. At the tth period, a new generation is born.
Stage 3. Based on gt∗ determined at stage 1, the elderly and the young who are alive in thetth period start bargaining over the amount of public goods. Consequently, a newamount of public goods, ḡt is determined.
Stage 4. After this kind of bargaining, taxation is also determined as {τ̄t , η̄t } in the same
period through bargaining.
Stage 5. The t + 1th generation is newly born.
After that, Stages 2–5 are repeated.

3 Dynamic Optimal Allocation
3.1 Benchmark
In this section, as a benchmark, we derive the dynamic optimal allocation (Ramsey allocation)
under the assumption that the government implicitly has a commitment device, following
traditional optimal taxation theory15) . This taxation corresponds to Stage 1, as described in
the 2.3 subsection. In this stage, the choice set of the planner is the set of sequences of taxes∞ , in other words, {R̃ , w̃ , g }∞ . In this section, we assume
and public goods: {τt , ηt , gt }t=0tt t t=0that public goods entail no congestion effect, so that both the younger and older generations
can make use of public goods. Next, we define the social welfare function as
{
}
∞
1
1
t
SW ≡ ∑ β u(ct ) + v(1 − lt ) + h(gt ) +{u(dt+1 ) + h(gt+1 )} +{u(d0 )+h(g0 )},
1+ρ
1+ρ
t=0

15) In other words, this concept is identical to the open-loop solution.

8

where β (< 1) stands for the social discount factor. Then, the constraints are summarized asfollows:16)
1. First-order conditions: These equations can be interpreted as households’ response
functions.
U = −u′ (c ) + 1 {R̃ u′ (d ) + (R − R̃ )h′ (g )} = 0,
s
t
t+1
t+1
t+1
t+1
t+1
1+ρ

Ul = w̃t u′ (ct ) − v′ (1 − lt ) + (wt − w̃t )h′ (gt ) = 0(4’)

2. Commodity market
ct + dt + gt + kt+1 lt+1 = Rt kt lt + wt lt(6)

3. Government budget constraint
Based on the fact that F(·) is homogeneous of degree one, we can transform Eq. (1)
as:17)
gt = F(k̂t , lt ) − R̃t kt lt − w̃t lt + kt lt + bt+1 − R̃t bt(1’)

More formally, the problem of the government can be rewritten as
max{w̃t ,R̃t ,bt ,gt }SW

s.t. Eqs. (1’), (4’), (6), given k̂0 > 0.

(IPG )

Setting {µ1t , µ2t , κt , λt } as the Lagrange multiplier, let L be the Lagrange function as[∞
L = ∑ β t {u(ct ) + v(1 − lt ) + h(gt ) +t=0

1{u(dt+1 ) + h(gt+1 )}1+ρ

1{R̃t+1 u′ (dt+1 ) + (Rt+1 − R̃t+1 )h′ (gt+1 )}}1+ρ
′+ µ2t {w̃t u (ct ) − v′ (1 − lt ) + (wt − w̃t )h′ (gt )}]{}+ κt gt − F(k̂t , lt ) + R̃t kt lt + w̃t lt − bt+1 + R̃t bt + λt (Rt kt lt + wt lt − ct − dt − gt − kt+1 lt+1 ) .
+ µ1t {−u′ (ct ) +Note that κt and λt are interpreted respectively as the marginal social value and themarginal rate of substitution. Moreover, µ1t and µ2t can be interpreted respectively as the16) If we take the borrowing constraints into consideration, another condition, k ≥ 0, is added, but we ignore itt

in this paper.
17) This equation is derived in the following manner.

gt=

Rt τt st−1 + wt lt ηt + bt+1 − Rt bt
Rt τt (kt lt + bt ) + wt lt ηt + bt+1 − R̃t bt (... eq.(7))=

F(k̂t , lt ) − R̃t kt lt − w̃t lt + bt+1 − R̃t bt=

9

sensitivity for the saving and labor supply. Noting that state variables are kt and bt , we canderive the first-order conditions (and the transversality condition) as follows18) :
)
(
1 wt
wt
′
′
w̃t : µ2t {u (ct ) +( − 1)h (gt )} = κt − λt lt
1 + ρ w̃t
w̃t
R̃t

:

bt

:

lt

:

k̂t

:

gt

:

µ1,t−1 1 ′1 Rt( − 1)µ1,t−1 h′ (gt+1 ) +u (dt+1 ) + κt (kt lt + bt )1 + ρ R̃t
β 1+ρRt
+ λt kt lt = 0
R̃tκt−1−κt R̃t +=0β−v′ (1 − lt ) + µ2t v′′ (1 − lt ) + κt (R̃t kt + w̃t − kt )λt−1
kt+1 = 0+ λt (Rt kt + wt ) −β
λt−1
κt (−Rt + R̃t ) + λt Rt −=0β
1 1h′ (gt ) +{h′ (gt+1 ) + (Rt − R̃t+1 )µ1,t−1 h′′ (gt+1 )}β 1+ρ
+ µ2t (1 − w̃t )h′′ (gt ) + κt − λt = 0

(9a)

(9b)
(9c)

(9d)
(9e)

(9f)

Finally, the transversality condition islim β t λt kt = 0.

t→∞18) For the first-order condition with respect to w̃ , we use the fact thatt

(1 − ηt )wt . The same calculation applies for R̃t+1 .

10

dwt
1
wt
=
because w̃t ≡=d w̃t1 − ηtw̃t

At the steady state, setting k̂t = k̂t+1 = · · · = k̂ for instance,
(
1 w
w )
′
(9a) : µ2 {u (c) +( − 1)h(g)} = κ − λ l
1 + ρ w̃
w̃
1 R
µ1
R
(9b) :( − 1)µ1 h′ (g) + u′ (d) + κ (kl + b) + λ kl = 0
1 + ρ R̃βR̃(9c) : −β κ R̃ + κ = 0(9d): −v′ (1 − l) + µ2 v′′ (1 − l) + κ (R̃k + w̃ − k) + λt (Rk + wt ) −(9e): κ (−R + R̃) + λ R −(9f): h′ (g) +

(4’)

(10a)
(10b)
(10c)

λ
k=0
β

λ
=0
β

(10d)
(10e)

1 1
{h′ (g) + (1 − R̃)µ1 h′′ (g)} + µ2 (1 − w̃)h′′ (g) + κ − λ =(10f)
0
β 1+ρ
1
: −u′ (c) +{R̃u′ (d) + (R − R̃)µ1 h′ (g)} = 0
(10g)
1+ρ

(4’) :

w̃u′ (c) − v′ (1 − l) + (1 − w̃) − h′ (g) = 0(10h)(1’) :

g − F(k̂, l) + R̃kl + w̃l − kl − b + R̃b = 0

(10i)

(6) :

Rkl + wl − c − d − g ≥ 0(10j)

From these equations, we obtain the following set of solution paths:∗∞, lt∗ , R̃t∗ , bt∗ , w̃t∗ , kt∗ , gt∗ , µ1t∗ , µ2t∗ , κt∗ , λt∗ }t=0.
{ct∗ , dt+1

In this paper, we designate solutions derived by Lagrangian method as “commitment solutions.” From these equations, we obtainτt∗ = 1 −dt
ct + k̂t+1, ηt∗ = 1 −.
wt
Rt k̂t+1(11)

We then discuss some features of the Ramsey allocation in the following subsection.

3.2 Characterization of Commitment Solution
In this subsection, we characterize the equilibrium tax sequence, derived in the above way.
Here, we show the following claim.
Proposition 1 The commitment solution, derived by Lagrange method, has the following
properties:
1. This taxation fails to be time-consistent.
11

2. This taxation presents intergenerational inequality of the amount of public goods.
Proof See appendix A.1.

■Remark 2

For the first result, we have three remarks. First, this result differs from Okuno

and Yakita (1983)19) . Their model assumes a naive expectation in relation to the amountof public goods provision, which means that the amount of public provision at period t
equals that at t + 1 period. Therefore, the time-inconsistency problem does not occur intheir model. Second, the fact that the open-loop solution generally fails to be time-consistentis well known. However, Xie (1997) reported that there is a possibility that even an openloop solution is time-consistent, so it is necessary to check whether the above solution istime-inconsistent or not. Third, intuitively, time-inconsistency occurs in this model for thefollowing two reasons.
1. Cost allocation of public goods varies depending on when households are born.
2. Existence of the state variable, emphasized in this paper as physical capital.
The first point says that the source of time-inconsistency is conflict of interests, as indicatedin Drazen (2000, pp.110–113). Furthermore, the second is more important. The property ofvariables changes from ‘control variables’ to a ‘state variable’ after some time has passed.
In other words, such variables can be treated as a control variable until some period; subsequently, this can be treated merely as a state variable. In other words, this shows that theelasticity of a tax-base ex ante differs from that ex post.
■Remark 3

As for the second result, it is necessary to define a situation that has no inter-

generational conflict. In this paper, a situation with intergenerational conflict is considered
as follows. The amount of provided public goods differs from the level that each generation
desires. In other words, taxation maximizes the sum of utility of young people and elderly
persons as a fair policy. That is, at t period, a certain kind of taxation, {τt , ηt , gt } does notagree with {τt∗ , ηt∗ , gt∗ }, which maximizes the social welfare. Formally, we must show thatthe above commitment solution differs from that defined as follows:19) Moreover, in the model of Okuno and Yakita (1983), the labor supply is not endogenous, which differs fromour model.

12Defnition 1 (Equilibrium with Intragenerational Equal Treatment) The set of allocation
with intragenerational equal treatment satisfies the following three conditions:(i) Together with the situation in which both the generations’ utility and firms’ profits are
maximized,(ii) The allocation satisfies the following:
max U(ct , gt , 1 − lt ) +U(dt , gt ){w̃t ,R̃t }s.t. Eqs. (1), (3).
(iii) The following markets clear.
Commodity Market: Eq. (6), Capital Market: Eq. (7), Labor Market: Eq. (8)For a more detailed proof, see the second half of A.1.

4 Intergenerational Bargaining
In the previous section, we have analyzed the taxation and public goods provision rule under
the assumption of no congestion effect and the government has a commitment device. We
have shown that taxation derived by the Lagrange method, which is determined at the initialpoint, fails to be time-consistent. Consequently, the government has an incentive to change
the policy at some point20) . In this section, we introduce the concept of ‘intergenerational
bargaining’, which is an alternative to Lagrange method or majority voting. In what follows,
first, we define the equilibrium concept (i.e. politico-economic equilibrium). Second, we
formulate a problem in which older and younger generations should solve and derive themodified public goods provision rule. Finally, we examine the properties of the modified
public goods provision rule and explain the relationship using the Samuelson rule.

4.1 Equilibrium Definition
According to Krusell, Quadrini and Rios-Rull (1997), let us define the equilibrium concept(politico-economic equilibrium21) ).
20) One might infer that the government should carry out a no-commitment policy. However, it is not desirable

to do so in terms of credibility. Furthermore, it is extremely difficult to change the policy repeatedly becauseof the cost of policy-change. Third, this way of method does not show how the government should change
the policy in concrete form.
21) This concept corresponds to so-called Markov-perfect equilibrium. These conditions are dependent on therelationship between the t and t + 1 period. Therefore, this concept meets the Markov property.

13Defnition 2 (Politico-Economic Equilibrium)

A politico-economic equilibrium is a

∞ that accords with the following.
sequence {w̃t , lt , ct , dt , R̃t+1 , gt }t=0
∞ , each agent (young or elderly) determines the policy(i) Given the sequence {τt∗ , ηt∗ , gt∗ }t=1

variables that maximize their individual utility. That is, the optimal policy variables
meet the following maximization problem:• Young people:{
max u(ct ) + v(1 − lt ) + h(gt ) +{• Elderly persons:max

1{u(dt+1 ) + h(gt+1 )}
1+ρ

}

}
1
{u(dt ) + h(gt )}1+ρ

Based on the solutions of such problems, the tax policy by which both generations arealive in the same period as that in which demand is determined. Moreover, the policy{gs }∞
s=t+1 , which is also achieved through the above method, is repeated after t + 1period.
(ii) Based on the bargaining solutions, the government determines the policy.
(iii) Finally, the following markets clear.
Commodity Market: Eq. (6), Capital Market: Eq. (7), Labor Market: Eq. (8)This equilibrium concept is superior to the Ramsey allocation discussed in the previous
section in the following two respects: First, by its definition of Markov property, the above
equilibrium meets the property of time-consistency and neither young nor elder agents havean incentive to deviate from the bargaining solution because it is based on the solution ofutility maximization problem. Second, compared with the Ramsey allocation from the viewpoint of intergenerational fairness, this equilibrium does not engender much intergenerational
inequality because this taxation is based on the behavior of living generations in the same period. As for this equilibrium concept, note that the government is assumed to be myopic in
the sense that they ignore the long-run effect and emphasize only the effect of a single period. This situation can be interpreted as a one-period commitment. Hereafter, we search the
public goods provision rule and the taxation that meets the above properties.

14

t −1

t +1

t

t +2

bargaining at t

{τt , ηt , gt }bargaining at t + 1{τt+1 , ηt+1 , gt+1 }Figure 2 Schematic of intergenerational bargaining

4.2 Bargaining Problem between Elderly Persons and Young People
In this section, we formulate the concept of intergenerational bargaining. The intuitive structure of this bargaining is depicted in Fig. 2. In this paper, the amount of public goods provision as well as taxation is assumed to be determined by a kind of bargaining, especially
Nash-bargaining. In more detail, we assume that both generations solve their own problems
and determine the amount of public goods through bargaining. Subsequently, the government
determines the public goods provision rule22) .
4.2.1

Optimal Choice of Young IndividualsFirst, the problem for young people is summarized as follows: The younger generation determines the amount of public goods to maximize the lifetime utility: the sum of utilities ofyounger and older people23) . In summary, young people face the following problem:{}1
max u(ct ) + v(1 − lt ) + h(gt ) +{u(dt+1 ) + h(gt+1 )}(IPyoung )gt ,ηt1+ρ
s.t. Eqs. (1’).(3)4.2.2

Optimal Choice of Old Individuals

Next, let us derive the amount of public goods for elderly persons. Note that elderly persons
are myopic in the sense that they consider only their own utility in old age. In this case, we22) This type of bargaining is also adopted by Besley and Coate (2003).
23) True of the old case that is discussed later, we specifically assess the amount of public goods, given the

taxation, {ηt , τt }.

15

can define the amount of public goods as the solution of the following problem:
1{u(dt ) + h(gt )}gt ,τt 1 + ρs.t. Eqs. (1’), (3)

max

4.2.3

(IPold )Determination of the Amount of Public Goods and the Tax Burden through Bargaining

As described in section 1, we do not adopt policy-making based on majority voting but rather
that based on intergenerational bargaining. Especially, we adopt policy-making based on
Nash-type bargaining24) . Although situations in which bargaining does not progress are conceivable, we eliminate such cases from consideration in this paper. Each generation faces thefollowing bargaining problem25) .
Let the bargaining power of young people and elderly persons at t period respectively
represent α , 1 − α (0 < α < 1). Players maximize the following Nash products:• Public goods : gtḡt ≡ arg max (U(ct , dt+1 , 1 − lt , gt , gt+1 ))α · (U(dt , gt ))1−α
gt

(12)

• Tax burden : (τt , ηt )(η̄t , τ̄t ) ≡ arg max = (U(ct , dt+1 , 1 − lt , gt , gt+1 ))α · (U(dt , gt ))1−αηt ,τt(13)

Through transformation into logarithmic form, let us solve the following two problems:max Φ1 = α · ln{U(ct , dt+1 , 1 − lt , gt , gt+1 )} + (1 − α ) · ln{U(dt , gt )}gts.t. (1’), (4’), (6),andmax Φ2 = α · ln{U(ct , dt+1 , 1 − lt , gt , gt+1 )} + (1 − α ) · ln{U(dt , gt )}ηt ,τts.t. (1’), (4’), (6).

(14)

(15)

Now, we are ready to present the following claim:24) As for the Nash Bargaining, see Binmore, Rubinstein and Wolinsky (1986). Note that the outside option ofboth elder generations and young persons is 0 under this setting.
25) Before entering analysis, we explain the role of debt. Taxation through intergenerational bargaining does notnecessarily cover the amount of public goods that is determined by bargaining. We introduce debt to coverthat shortfall.

16

Proposition 2 {τ̄t , η̄t , ḡt } meets the following properties.
1. {τ̄t , η̄t , ḡt } satisfies conditions of the politico-economic equilibrium.
2. Each solution achieved through intergenerational bargaining meets the following
properties.

∂ ḡt
> 0,
∂α

∂ τ̄t
> 0,
∂α

∂ η̄t
<0
∂α

(16)

Proof See appendix A.2.
These second properties inform us that the amount of public goods provision and capitalincome tax increase as the bargaining power of young generation becomes stronger, whilelabor income tax decreases. The intuition behind these results is that, for the relationship between capital-income tax and bargaining power, as the bargaining power, α , becomes larger,
which might imply that the voice of the younger generation becomes larger, they have an
incentive to reduce the burden caused by labor income tax and impose a heavier burden on
elderly persons; the older generation is compelled to bear a bigger burden because of thecapital income tax, and it is the opposite. For the amount of public goods, the intuitive explanation is as follows: Because young persons live two periods and thereby live longer than theelderly group, they have more incentive to make use of public goods than do elderly persons.
Incidentally, is there a possibility that the solution derived by dynamic programming (DP)and the bargaining solution coincide? If so, the time-inconsistency problem is solvable bybargaining26) . The following proposition answers such a question.
Proposition 3 There exists “α ” such that the solution derived by dynamic programmingand that by intergenerational bargaining coincide. In other words, depending on the valueof bargaining power, taxation derived through bargaining can meet the property of timeconsistency.
Proof To prove this claim, we must show that these solutions through intergenerational bargaining coincide with the feedback solution. See also appendix A.3 for greater detail.

26) Note that the DP solution meets the property of time-consistency.

17

4.3 Derivation of the Modified Samuelson Rule through Bargaining
In the previous subsection, we have investigated the properties of policy variables throughbargaining. However, it is unlikely that households can determine all policy variables throughbargaining. In this subsection, we then derive taxation rule based on intergenerational bargaining under the case that only the amount of public goods is determined through bargaining. Hereafter, we define that problem of government taking the discussion in the previous
subsection into consideration. To sum up, the problem for the government is the following.
max SWs.t. Eqs. (1’), (6), (30)

(IPG1 )As in the previous section, we construct the following Lagrangian H , setting {κt , λt , φt }as Lagrange multipliers:[
H = ∑ β {u(ct ) + v(1 − lt ) + h(gt ) +∞1{u(dt+1 ) + h(gt+1 )}
1+ρ
t=0
{
}
+ κt gt − F(k̂t , lt ) + R̃t kt lt + w̃t lt − bt+1 + R̃t bt + λt (Rt kt lt + wt lt − ct − dt − gt − kt+1 lt+1 )h′ (gt ) − u′ (ct ) − v′ (1 − lt )h′ (gt ) − u′ (dt ) ]+ φt {α ·+ (1 − α ) ·}U(ct , dt+1 , 1 − lt , gt , gt+1 )U(dt , gt )t

We then derive the following first-order conditions:{}h′′ (gt )h′′ (gt )′′w̃t : −lt u (ct ) + h (gt ) + φt α
+ (1 − α ){U(ct , dt+1 , 1 − lt , gt , gt+1 )}2{U(dt , gt )}2()
wt= κt − λt lt
(17a)
w̃t
Rt
1 ′
u (dt+1 ) + κt (kt lt + bt ) + λt kt lt
R̃t :
1+ρ
R̃t
{
}
′′
h (gt )
h′′ (gt )+φt α+ (1 − α )=0{U(ct , dt+1 , 1 − lt , gt , gt+1 )}2{U(dt , gt )}2
(17b)
′′
′′
h (gt )
(1 − α )h (gt )gt : h′ (gt ) + κt − λt + φt {α+}=0{U(ct , dt+1 , 1 − lt , gt , gt+1 )}2{U(dt , gt )}2
(17c)
κt−1
bt : −κt R̃t +
=0
(17d)
β
λt−1
=0
(17e)
k̂t : κt (−Rt + R̃t ) + λt Rt −β18

As in the previous section, we obtain the following conditions at the steady state.
{
}
(
w )
h′′ (g)(1 − α )h′′ (g)′′(17a) : −lu (c) + h (g) + κ − λ l + φ α+=0w̃
{U(c, d, 1 − l, g, g)}2
{U(d, g)}2
(18a)
{
}
′′
′′
1 ′
R
h (g)
(1 − α )h (g)=0(17b) :
u (d) + κ (kl + b) + λ kl + φ α+2
1+ρ{U(c, d, 1 − l, g, g)}{U(d, g)}2R̃
α h′′ (g)(1 − α )h′′ (g)′(17c) : h (g) + κ − λ + φ {
+
}=0
(18b)
{U(c, d, 1 − l, g, g)}2
{U(d, g)}2(17d) : −β κ R̃ + κ = 0(18c)
λ(17e) : κ (−R + R̃ − 1) + λ R − = 0(18d)
βWe define the solution by ⋆ as follows.
⋆∞{ct⋆ , dt+1
, lt⋆ , R̃t⋆ , bt⋆ , w̃t⋆ , kt⋆ , gt⋆ , κt⋆ , λt⋆ , φt⋆ }t=0

We refer to the above solution as “modified public goods provision rule”. Then, we show
the following claim:
Proposition 4 The above modified optimal taxation to finance public goods in the long run
is characterized asu′ (c) u′ (d)u′ (c)
b
R
+
=
1
+
µ
− κ (k + ) − λ k.
2′′′
h (g) h (g)
h (g)
l
R̃
■Remark 4(19)

The left side of the equation shown above (19) denotes the sum of the marginal

rate of substitution between consumption and public goods. Regarding the right side of thisequation, we can infer that this result reflects the following channels.
• the cost of public goods distribution, which corresponds to the first term of R.H.S. ofeq.(19).
• Pigou effect – The sum effect of (i) + (ii), which corresponds to the second and third
term of R.H.S. of eq.(19).
(i) the negative effect of labor-income tax – distortion in labor supply(ii) the negative effect of capital-income tax – distortion in saving• the positive effect of public goods, which corresponds to the fourth term of R.H.S. ofeq.(19).

19

Especially, the third effect has a positive effect on the utility of both young and elderly persons. Moreover, we can regard this result in terms of the Samuelson Rule, which means that∑i MRSi = MRT , where MRS and MRT respectively show the marginal rate of substitution
and marginal rate of transformation.
Proof From eq. (18c), we obtainβ R̃ = 1.

Eliminating Φ from eq.(18a) and eq.(18b), we have
(
1 ′
w )
R
lu′ (c) +u (d) = h′ (g) + κ − λ l − κ (kl + b) − λ kl
1+ρ
w̃
R̃
Dividing both sides of the above equation by l × h′ (g), we have(
1 u′ (d)w )bu′ (c)
R+= 1 + κ − λ − κ (k + ) − λ k′′h (g) 1 + ρ h (g)
w̃
l
R̃

(20)

Here, using the following equations, those are the Eqs. (18a)-(18c) as the first order conditions,u′ (c) − {µ1 − µ2 w̃lt }u′′ (c) − λ = 0u′ (d) − λ R̃u′′ (d) − λ = 0(w )′µ2 u (c) = κ − λ l
w̃

(21a)
(21b)
(21c)

We can rewrite eq.(20) asu′ (c)u′ (c) u′ (d)
R
b
+
=
1
+
µ
− κ (k + ) − λ k.
2′′′
h (g) h (g)
h (g)
l
R̃
This concludes the proof.
■Remark 5

1. In relation to this proposition, we can adopt the following perspective. Consideringthat the structure of intergenerational bargaining proceeds after the t +1 period, we canregard this model as having identical structure to an ordinary repeated game. Therefore, we can conclude that the cooperative solution is attainable by applying Kandori(1992)’s discussion. Thereby, the time-consistent solution is obtainable.
2. As discussed in Atkeson, Chari and Kehoe (1999), taxing capital income is not preferable from the viewpoint of efficiency. However, we showed that this way of fundraising of public goods is justifiable from the viewpoint of intergenerational conflict.
20

4.4 Discussion
In this subsection, let us point out some remarks as for intergenerational bargaining proposedin this section.
■the Specific Example of Bargaining Power

Applying the concept of intergenerational bargaining to reality, bargaining power (α ) can be
regarded as the strength of the coalition within the congress in this paper or simply stated, thepolitical power of young agents. More concretely speaking, in the congress (or for instance,
the national diet), the bargaining power corresponds to the ratio of assembly members, towhom young persons give support. If the voter turnout of young persons is high, there is agreater possibility that the assembly members, to whom young persons lend support, can win
an election. Therefore, from another perspective, the bargaining power also corresponds tothe approval rating among young persons for the policy that the government determines.
■How can we discover the optimal value of bargaining power?

On the other hand, proposition 3 says that depending on the value of the bargaining power,
the bargaining solution coincides with the DP solution. However, another question remains
to be discussed as follows: How can we find the optimal value of α that satisfies such aproperty? Our proposal is as follows: If the government seeks to attain such a value of
bargaining power, they should merely adjust the difference of preferences to the tax burden
and the amount of public goods between elderly and young persons by introducing a proper
policy that is supportable by young people.

5 Concluding Remarks
In this paper, we have presented analysis of dynamic optimal taxation and a public goods provision rule under the presence of intergenerational conflict by introducing a solution concept
for such a conflict as an alternative to majority voting or lobbying. What we have shown inthis paper is summarized as the following: First, taxation derived using the Lagrange Method
fails to be time-consistent. Second, depending on bargaining-power, taxation based on intergenerational bargaining can be time-consistent. Third, we investigated properties of taxation
and a public goods provision rule based on intergenerational bargaining.

21

Finally, we conclude this paper by pointing out possible directions for extending its analysis and implications. First, in Nash-bargaining, bargaining power is given exogenously;
for that reason, we are compelled to endogenize it as a next step. Second, considering theexistence of inequality related to ability, income, and labor supply, we must also analyze
intragenerational conflict27) . Finally, with regard to policy-making, we must also consider
other political decision-making modalities including ideology, veto-power, rent-seeking, andso forth.

Appendix A
A.1 Proof of Proposition 1■Proposition 1.1 First, we begin to prove Proposition 1.1. This fashion of proof is basedon that of Kinai (2005). To prove this claim, we must demonstrate that the solution derived using the Lagrange method (open-loop solution) differs from that by dynamic programming (closed-loop solution). Next, we rewrite the maximization problem that the government
should solve.
{∑ β u(ct ) + v(1 − lt ) + h(gt ) +
∞

max

t

{w̃t ,R̃t ,gt } t=01{u(dt+1 ) + h(gt+1 )}1+ρ}Using Eq. (5), the budget constraints of the government can be written as: 28)
bt+1 = −F(s(w̃t−1 , bt ) − bt , l(w̃t , R̃t+1 )) + R̃t {s(w̃t−1 , bt ) − bt } + w̃t l(w̃t , R̃t+1 ) + R̃bt + gt .
(22)Now, let us define qt as follows by differentiating Eq. (22) with respect to bt+1 .
1 − (wt − w̃t )∂ lt ∂ R̃t+1≡ qt∂ R̃t+1 ∂ bt+1

(23)

Hereafter, using qt , we advance our calculation as follows: Using Eq. (5), the budget27) As an existing study that presents a similar viewpoint, see Hassler, Storesletten and Zilibotti (2007).
28) This transformation is based on the fact that we can rewrite s(w̃
t−1 , bt ) as F(s(w̃t−1 , bt )), using the production

function, F(kt , Lt ).

22

constraints of the government can be written as:()∂ bt+1∂ lt∂ lt ∂ R̃t+1
qt ·= lt + (w̃t − wt )+∂ w̃t
∂ w̃t ∂ R̃t+1 ∂ bt+1
∂ bt+1
= st
qt ·
∂ R̃t
∂ bt+1
qt ·
= 1
∂ gt

(24a)
(24b)
(24c)

Next, we obtain the following equations by differentiating Eq. (22) with respect to twostate variables: bt and w̃t−1 .

∂ bt+1∂ bt∂ bt+1qt ·∂ w̃t−1
qt ·∂ st ∂ R̃t+1∂ R̃t ∂ bt[]
∂ st
∂ st ∂ R̃t= (R̃t − Rt )+∂ w̃t−1 ∂ R̃t ∂ w̃t−1= (R̃t − Rt )

(24d)
(24e)

Setting the value function as Ω(kt , gt−1 ), we establish the Bellman Equation as follows.
Ωt (kt , gt−1 ) = max{w̃t ,R̃t ,gt }[{u(w̃t lt − s(w̃t−1 , bt )) + v(1 − lt ) + h(gt )} +1{u(R̃t+1 s(w̃t−1 , bt )) + h(gt+1 )}1+ρ]+ β Ωt+1 (kt+1 , gt )(25)s.t. Eq.(22)
From Eq. (25), we derive the first-order conditions with respect to w̃t , R̃t , and gt .
{()}∂ Ωt (kt+1 , gt+1 )∂ lt∂ lt ∂ R̃t+1′
qt lt u (ct ) + β
lt + (w̃t − wt )+=0
∂ bt+1∂ wt ∂ lt ∂ wt
1 ∂ R̃t+1 ′∂ Ωt+1 (gt+1 , w̃t )u (dt+1 ) + β
qt
st = 01 + ρ ∂ R̃t∂ bt+1∂ R̃t+1 ′1 ′∂ Ωt+1 (gt+1 , w̃t )qt
h (gt+1 )} + β{h (gt ) +
=0
∂ gt
1+ρ
∂ gt+1

23

(26a)
(26b)
(26c)

Using the envelope theorem yields the Benveniste-Scheinkman condition as29)
{
}
∂ Ωt
∂ st
1 ′
∂ st
′
qt ·
u (dt+1 )R̃t+1= qt −u (ct )+∂ bt∂ bt 1 + ρ∂ bt{}∂ Ωt+1∂ st ∂ R̃t+β(R̃t − Rt )
+ Rt
∂ bt
∂ R̃t ∂ bt
{
}
1 ′
∂ Ωt
∂ st ∂ bt∂ st ∂ bt′
qt ·= qt −u (ct )+u (dt+1 )R̃t+1∂ w̃t−1∂ bt ∂ w̃t−1 1 + ρ∂ bt ∂ w̃t−1[]∂ Ωt+1∂ st∂ st ∂ R̃t+β+β(R̃t − Rt )∂ bt+1∂ R̃t∂ R̃t ∂ w̃t−1

(26d)

(26e)

Now we show that the solution derived in this manner differs from that obtained using
Lagrange method. From Eq. (10c), we obtainβ R̃ = 1.

(27)

Substituting Eq. (23), and Eqs. (26a)–(26d) into the above equations yields the following.

∂ l ∂ R̃
(28a)≡q
∂ R̃ ∂ b{()}∂l∂ l ∂ R̃∂ Ω(b, w̃)′(26a) : q · lu (c) + β(28b)l + (w̃ − w)+=0∂b
∂ w ∂ R̃ ∂ b
1 ∂ R̃ ′∂ Ω(b, w̃)∂Ω1 ′(26b) : q ·u (d) + β
u (d) + β
s = 0 ⇐⇒ q
s = 0(28c)1 + ρ ∂ R̃
∂b
1+ρ
∂b
1 ′
∂Ω
(26c) : q{h′ (g) +h (g)} + β
=0
(28d)
1+ρ
∂g
(23)

: 1 − (w − w̃)

{
}
∂Ω
∂s
1 ′
∂s
′
(26d) : q ·u (d)R̃= q −u (c) +∂b∂b 1+ρ∂b{}∂Ω∂ s ∂ R̃+β(R̃ − R)+R∂b
∂ R̃ ∂ b(28e)

From these equations, eliminating q and Ω(·), we obtain
{
}
{
}
∂Ω
∂s
1 ′
∂s
∂s
∂Ω
′
= q −u (c) +u (d)R̃+β(R̃ − R) + R∂b∂b 1+ρ
∂b
∂b
∂b
{
}
∂s
1 ′
∂s
∂Ω
∂s
⇐⇒ q −u′ (c) +u (d)R̃(29)+(1 − β R)( − 1) = 0∂b 1+ρ
∂b∂b∂b
From Eq. (28c),

∂Ω
1
1 ′
u (d).
=− q∂b
βs 1+ρ

29) These equations can be derived through differentiation with respect to two state variables: b and w̃
t
t−1 .

24

By substituting this equation into Eq. (29), we obtain
{
}
{
}
1 ′
∂s
1 1 ′
∂s
′
−u (c) +u (d)R̃u (d) (1 − β R)( − 1) = 0
+ −
1+ρ
∂b
βs 1+ρ
∂b
{
}
{
}
1 ′
1 1 ′
1 1 ′
′
⇐⇒
−u (c) +u (d)R̃ −u (d)(1 − β R) (−1) = −u (d) (1 − β R)1+ρβs 1+ρ
βs 1+ρ
{
}
1
1 ′
1 1 ′
′
u (d) = −u (d) (1 − β R)⇐⇒ u (c) + {R̃ + (1 − β R)}
βs1+ρ
βs 1+ρ
{
}
1 ′
1 1 ′
′
⇐⇒ u (c) + R̃u (d) = −2
u (d) (1 − β R) ̸= 0.
1+ρ
βs 1+ρ
Thereby, we find that this equation does not necessarily coincide with Eq. (10g) because

β R = 1 is not satisfied. We find that Eq. (27) is not necessarily satisfied. This result meansthat the first-order conditions obtained using the Lagrange method do not coincide with thoseby DP: this proposition holds. This concludes the proof.
■Proposition 1.2 We next proceed to prove proposition 1.2. First, we obtained paths of∞ and {η }∞ in the section 3.1. Now, let us
capital-income and labor-income taxes; {τt }t=0t t=0

derive taxation {τt , ηt } that maximizes the sum of the utility of younger and older generation
by solving the following problem:(τ̂t , η̂t ) = max [θ {u(ct ) + v(1 − lt )} + h(gt ) + {u(dt ) + h(gt )}]τt ,ηts.t. Eqs.(3), (1’), (7)(IP1 )In the first-term, the parameter θ is the relative weight that politicians attach to the well-beingof the elderly. In the case of θ = 1, we can regard this allocation as that with intragenerational
equal treatment. By solving the above problem with θ = 1, we can easily show τ̂t ̸= τt∗ ,η̂t ̸= ηt∗ .

A.2 Proof of Proposition 2■Proposition 2.1 To prove this claim, we show that {ḡt , τ̄t , η̄t } meet conditions of politico-

economic equilibrium (i, ii, and iii).
(i) Because {ḡt , τ̄t , η̄t } are also a solution of both generations, this solution meets the first
condition.
(ii) Because this solution is an aggregation, the second condition holds.
(iii) Clearly, each market-clearing condition holds.
25

Therefore, we find that this solution meets the three conditions.
■Proposition 2.2 First, for public goods, let us define Hg as

Hg ≡

∂ Φ1
h′ (gt ) − (v′ (1 − lt ) + u′ (ct ))h′ (gt ) − u′ (dt )=α·+ (1 − α ) ·
= 0.
∂ gt
U(ct , dt+1 , 1 − lt , gt , gt+1 ){u(dt ) + h(gt )}(30)

By applying the implicit function theorem, we obtain the following:∂ Hg ¯¯∂ ḡt= − ∂∂Hα ¯¯
> 0.
g
∂α
other variables constant∂ ḡt

Next, let us turn to the bargaining problem with respect to taxation. Define Hτ and Hη as∂ Φ2−u′ (dt ) + Rt+1 st h′ (gt ) ∂ Rt
Rt+1 st h′ (gt ) ∂ Rt=α
+ (1 − α )
=0
·
·
∂ τt
U(ct , dt+1 , 1 − lt , gt , gt+1 ) ∂ R̃t{u(dt ) + h(gt )} ∂ R̃t
(31a)
′
′
′
∂ Φ2
−u (ct ) + wt lt h (gt )∂ wt
wt lt h (gt ) ∂ wt
=α
Hη ≡
·
+ (1 − α )
·
= 0.
∂ ηt
U(ct , dt+1 , 1 − lt , gt , gt+1 ) ∂ w̃tu(dt ) + h(gt ) ∂ w̃t
(31b)
Hτ ≡

From these equations, by applying the implicit function theorem, we then have the following result:∂ Hτ ¯¯∂ τ̄t= − ∂∂Hα ¯¯
> 0,
τ
∂α
other
variables
constant∂τ∂ Hη ¯¯∂ η̄t= − ∂∂Hα ¯¯
< 0.
η
∂α
other variables constant∂η

A.3 Proof of Proposition 3
We then show that the solutions derived by intergenerational bargaining coincide with thoseobtained by dynamic programming, depending on the value of α . Let us compare Eqs. (30)–(31) with Eqs. (25a)–(25e). Then, let us interlock Eq. (30) and Eq. (25c) and define thatinterlocked equation as F(α ; kt , kt+1 ). That is,F(α ; kt , kt+1 ) ≡ the R.H.S. of eq.(30) − the R.H.S. of eq. (25c).

26

We then obtain the following using the assumption that α is within the interval, (0, 1).
Thereby, we haveF(0) · F(1) < 0.

(32)

Using the Intermediate Value Theorem, we then find that this equation has at least one
solution within the interval, α ∈ (0, 1). Therefore, this policy through bargaining is timeconsistent, depending on the value of α . Regarding the tax burden, {τ̄t , η̄t }, we can obtainthe same conclusion similarly.

References
Atkeson, A., V. Chari, and P. J. Kehoe (1999) “Taxing Capital Income: a Bad Idea” Quarterly Review,Vol. 23 pp. 3–17.
Azariadis, C. and V. Galasso (2002) “Fiscal Constitutions” Journal of Economic Theory, Vol. 103 pp.
255–281.
Bassetto, M. (2008) “Political Economy of Taxation in an Overlapping-Generations Economy” Review
of Economic Dynamics, Vol. 11 pp. 18–43.
Batina, R. G. (1990) “Public Goods and Dynamic Efficiency : The Modified Samuelson Rule” Journalof Public Economics, Vol. 41 pp. 389–400.
Besley, T. and S. Coate (2003) “Centralized versus Decentralized Provision of Local Public Goods: a
Political Economy Approach” Journal of Public Economics, Vol. 87 pp. 2611–2637.
Besley, T. and M. Ghatak (2001) “Government Versus Private Ownership of Public Goods” The Quarterly Journal of Economics, Vol. 116 pp. 1343–1372.
Binmore, K., A. Rubinstein, and A. Wolinsky (1986) “The Nash Bargaining Solution in Economic
Modelling” RAND Journal of Economics, Vol. 17 pp. 176–188.
Breyer, F. (1994) “Political Economy of Intergenerational Redistribution” European Journal of Political
Economy, Vol. 10 pp. 61–84.
Casamatta, G., H. Cremer, and P. Pestieau (2000) “The Political Economy of Social Security” Scandinavian Journal of Economics, Vol. 102 pp. 503–522.
Chari, V. V., P. J. Kehoe, and E. C. Prescott (1989) “Time Consistency and Policy” in R. J. Barro ed.
Modern Business Cycle Theory, Cambridge, Mass: Harvard University Press pp. 265–305.
Drazen, A. (2000) Polical Economy in Macroeconomics, Princeton: Princeton University Press.
Fischer, S. (1980) “Dynamic Inconsistency, Cooperation and the Benevolent Dissembling Government” Journal of Economic Dynamics and Control, Vol. 2 pp. 93–107.
Galasso, V. and P. Profeta (2002) “The political economy of social security: a survey” European Journalof Political Economy, Vol. 18 pp. 1–29, March.
Grossman, G. M. and E. Helpman (1998) “Intergenerational Redistribution with Short-Lived Governments” Economic Journal, Vol. 108 pp. 1299–1329.
Hassler, J., K. Storesletten, and F. Zilibotti (2007) “Democratic public good provision” Journal of
Economic Theory, Vol. 133 pp. 127–151.
Holtz-Eakin, D., M. E. Lovely, and M. S. Tosun (2004) “Generational Conflict, Fiscal Policy, andEconomic Growth” Journal of Macroeconomics, Vol. 26 pp. 1–23.
Kaas, L. (2003) “Productive Government Spending, Growth, and Sequential Voting” European Journalof Political Economy, Vol. 19 pp. 227–246.

27

Kandori, M. (1992) “Repeated Games Played by Overlapping Generations of Players” Review of Economic Studies, Vol. 59 pp. 81–92.
Kinai, Y. (2005) “Analysis of Optimal Taxation in an OLG Model: from the Viewpoint of TimeConsistency” Discussion Papers in Economics and Business 05-04, Osaka University, Graduate
School of Economics and Osaka School of International Public Policy (OSIPP).
Krusell, P., V. Quadrini, and J.-V. Rios-Rull (1997) “Politico-economic Equilibrium and Economic
growth” Journal of Economic Dynamics and Control, Vol. 21 pp. 243–272.
Kydland, F. E. and E. C. Prescott (1977) “Rules Rather Than Discretion: The Inconsistency of Optimal
Plans” Journal of Political Economy, Vol. 85 pp. 473–91.
Lambertini, L. and C. Azariadis (2003) “The Fiscal Politics of Big Governments: Do Coalitions Matter?” in Ravi Kanbur Richard Arnott, Bruce Greenwald and Barry Nalebuff eds. Economics for an
Imperfect World : Essays in Honor of Joseph Stiglitz, Cambridge, Mass: MIT Press pp. 367–386.
Okuno, N. and A. Yakita (1983) “An Optimal Rule on the Supply of Public Goods in a Dynamic
Setting” Public Finance, Vol. 38 pp. 419–428.
Pirttilä, J. and M. Tuomala (2001) “On Optimal Non-linear Taxation and Public Good Provision in an
Overlapping Generations Economy” Journal of Public Economics, Vol. 79 pp. 485–501.
Rangel, A. (2005) “How to Protect Future Generations Using Tax Base Restrictions” American Economic Review, Vol. 95 pp. 314–346.
Renström, T. I. (1996) “Endogenous Taxation: An Overlapping Generations Approach” Economic Journal, Vol. 106 pp. 471–482.
(2002) “Political Institutions and Redistribution” in David K. Levine and William Zame eds.
Proceedings of the 2002 North American Summer Meetings of the Econometric Society: Development, Inequality and Political Economy.
Sandler, T. and V. K. Smith (1976) “Intertemporal and Intergenerational Pareto Efficiency” Journal ofEnvironmental Economics and Management, Vol. 2 pp. 151–159.
Tabellini, G. (1991) “The Politics of Intergenerational Redistribution” Journal of Political Economy,
Vol. 99 pp. 335–357.
(2000) “A Positive Theory of Social Security” Scandinavian Journal of Economics, Vol. 102
pp. 523–45.
de Walque, G. (2005) “Voting on Pensions: A Survey” Journal of Economic Surveys, Vol. 19 pp. 181–
209.
Xie, D. (1997) “On Time Inconsistency: A Technical Issue in Stackelberg Differential Games” Journalof Economic Theory, Vol. 76 pp. 412–430.

28