Demographics and competition for capital in political
economyTadashi Moritay

Yasuhiro Satoz

Kazuhiro Yamamotox

August 2, 2017

Abstract
We examine possible impacts of demographics on outcomes of competition for
capital in political economy. For this purpose, we develop a multi-region overlapping
generations model wherein public good provision …nanced by capital tax is determinedby majority voting. When a population is growing, younger people represent the
majority, whereas when a population is decreasing, older people represent the majority.
We show that the race to the bottom is likely to emerge as a result of tax competitionin an economy with a growing population whereas the race to the top might emerge
in an economy with a decreasing population.

Keywords: tax competition, demographics, political economy
JEL classi…cation: H41, H73, J10

1

Introduction

This paper investigates the possible impacts of demographics on the results of competition
for capital in political economy. Given the drastic increases in capital ‡ows across countries
This study was conducted as a part of the Project “Spatial Economic Analysis on Trade and Labor
Market Interactions in the System of Cities” undertaken at the Research Institute of Economy, Tradeand Industry (RIETI). This work was also supported by JSPS KAKENHI Grant Number 15H03348,
15H03344, 16H03615, and 17H02519. We thank participants at North American Regional Science Council
and seminars held at University of Tokyo and Tohoku University for their helpful comments.
yFaculty of Economics, Kindai University, e-mail : t-morita@kindai.ac.jpzCorresponding author, Faculty of Economics, University of Tokyo, e-mail : ysato@e.u-tokyo.ac.jpx
Graduate School of Economics, Osaka University, e-mail : yamamoto@econ.osaka-u.ac.jp1and regions, many scholars have analyzed the e¤ects of globalization in the capital market
over the past few decades. One of the most important strands in this …eld is the theoryof capital tax competition, which has a long history dating back at least to Zodrow andMieszkowski (1986) and Wilson (1986).1 Researchers in this strand investigated the roleof governments in attracting capital to their jurisdictions. In standard tax competition
models, governments are benevolent and maximize the representative resident’s welfare.
Nonetheless, they set ine¢ ciently low capital tax rates because capital taxation causes
capital ‡ight, which increases the tax bases in other countries and causes positive …scal
externalities. This result is known as the race to the bottom and has attracted muchattention (see e.g., OECD, 1998).
Around the same time, we observe large di¤erences in demographic structure among
countries and drastic demographic changes in many of them. In fact, when we consider
the old-age dependency ratio, which is the ratio of people older than 65 years of age tothe working-age population, we …nd large di¤erences among countries. For example, the2014 ratios were 9.6 in Mexico, 11.1 in Turkey, 21.6 in the United States, 27.7 in Spain,
34.4 in Italy, and 41.9 in Japan.2 Similarly, the median ages in 2010 were 26.6 in Mexico,28.3 in Turkey, 36.9 in the United States, 40.1 in Spain, 46.4 in Italy, and 47.8 in Japan.3
Moreover, we also observe drastic changes in these …gures: the old-age dependency ratios
and median ages of OECD countries rose from 13.7 to 24.2 between 1960 and 2014 andfrom 28.9 to 45.4 between 1950 and 2010, respectively. These facts imply that in political
economy, decisive voters are younger generations in countries such as Mexico and Turkey,
whereas they are older generations in countries such as Italy and Japan. Moreover, they
are getting older in OECD countries. Put plainly, decisive voters and hence, the objectivesof governments might change over time and place. This would, in turn, a¤ect the outcomesof inter-governmental competition for capital.
In this paper, we ask how demographics are related to outcomes of governments’policy competition for capital and …rms. To be more speci…c, we ask whether the race tothe bottom, which is believed to prevail in the world, emerges under any demographic
1

For surveys on this strand, see Wilson (1999), Wilson and Wildasin (2004) and Zodrow (2003) among

others.
2
World Bank, http://data.worldbank.org/indicator/SP.POP.DPND.OL
3OECD,
http://www.oecd-ilibrary.org/education/trends-shaping-education-2013/median-age-goingup-into-the-next-century_trends_edu-2013-graph43-en

2

structure.
To analyze this issue, we develop a capital tax competition model involving the overlapping generations structure wherein policies are determined by majority voting. In ourmodel, all individuals live for two periods, young and old, and the population grows with anexogenous constant growth rate. If the population growth rate is positive, then the young
individuals represent the majority because their population size is larger than that of old
individuals. In contrast, if the population growth rate is negative, then the old individuals
represent the majority. We consider multiple countries and each country’s government
supplies public goods and …nances them with capital tax. A government chooses the levelof public good provision and capital tax rate to maximize the utility of the individuals
representing the majority.
In our model, young individuals supply labor to …rms, which produce private consumption goods with labor and capital. The wage income of young individuals increases withthe capital inputs in the country, whereas the savings income of old individuals increases
with the rate of return on savings. Young individuals consume private and public goods
and save the wage income for old-age consumption. Old individuals consume private andpublic goods. Governments at a certain period ignore the e¤ects of taxes and public goods
on instantaneous utility of minority, because the subjective of government is to maximize
the majority’s utility only. When the decisive voter is in the young (old) generation,
the government disregards the utility of old (young) individuals. Hence, capital taxation
causes inter-generational externalities. The government disregards the e¤ects of capitaltaxation on the bene…ts of public good provision to the minority generation irrespectiveof the majority’s generation, which we call the Public-good externality. The sign of the
Public-good externality is ambiguous because a rise in the capital tax rate increases thetax revenues from one unit of capital but causes capital out‡ows to decrease the tax base.
In addition, when old individuals represent the majority, the government ignores the negative e¤ect of capital taxation on wages of young individuals, which is caused by capitalout‡ows. We call this the Gerontocracy externality, which makes a government have ahigher incentive to tax capital in an economy with a decreasing population than in an
economy with a growing population.
Moreover, our model has three types of inter-country externalities caused by competition for capital. First, because we assume free mobility of capital between countries,

3

capital taxation in one country causes capital ‡ight to other countries and increases theirtax bases, which is the Fiscal externality, as is standard in models of capital tax competition. Second, the capital ‡ight also raises the wage rates in the other countries, which wecall the Wage externality. Finally, capital taxation decreases the rate of return on savings,
which reduces the savings income of old individuals in the other countries. We call thise¤ect the Interest-rate externality. The Fiscal and Wage externalities are positive whereas
the Interest-rate externality is negative.
The relative signi…cance of positive and negative externalities characterizes the (in)e¢ ciencyof the equilibrium capital tax rate. When individuals strongly prefer public good consumption, sum of positive externalities overwhelms the sum of negative externalities, whichyields an ine¢ ciently low capital tax rate, i.e., we observe the race to the bottom. When
individuals do not strongly prefer public good consumption, the capital tax rate is inef…ciently high, which we call the race to the top. Furthermore, because an economy witha decreasing population has the Interest-rate and Gerontocracy externalities as negative
externalities but an economy with a growing population has only the Interest-rate externality a negative externality, the race to the top is more likely to emerge in the former
than in the latter.
We also consider asymmetric countries. When the population is growing or decreasing
in all countries, we can obtain qualitatively similar results with the case of symmetriccountries. When the population is growing in some countries and decreasing in othercountries, we show a possibility that capital tax rates are ine¢ ciently low in countries witha growing population and ine¢ ciently high in countries with a decreasing population.
Our results has the following strong policy implications: Demographic structure matters for the outcomes of tax competition. Although the race to the bottom has prevailed
as a popular outcome of tax competition in the past few decades during which population
has increased in many countries, we might observe the race to the top as aging would
proceed in many countries. In fact, Keen and Konrad (2013) mentioned the minimum
excise duty rate adopted by the European Union to limit downward spiralling of rate bytax competition. However, if aging proceeds, policy debates should refer to the possibilityof the race to the top and coordinated tax decreases.
Several existing papers have investigated capital tax competition in political economy
models. Persson and Tabellini (1992), Borck (2003), Lockwood and Makris (2006), Grazz-

4ini and van Ypersele (2003), Fuest and Huber (2001), and Ihori and Yang (2009) assumed
that individuals have di¤erent endowments of labor and capital, and that capital tax rates
are determined by the political economy process.4 These papers commonly showed that if
the decisive voter’s capital endowment is smaller than the average, then the equilibrium
capital tax rate tends to be ine¢ ciently high. Our paper also studies the tax competition
in a political economy model. However, in our model, individuals are not heterogeneouswith respect to endowments. Alternatively, we consider a di¤erence between generations
by using an overlapping generations model. Put di¤erently, we focus on inter-generational
political con‡icts, whereas existing studies focus on political con‡icts among individuals
with heterogeneous endowments.
We also refer to existing studies that used overlapping generations models in political
economy to investigate a macro economy. Alesina and Rodrik (1994) and Persson andTabellini (1994a, b) constructed overlapping generation models wherein individuals haveheterogeneous endowments of labor and capital, and a median voter chooses the capitalincome tax rate. In these models, when the median voter has more capital endowments,
the equilibrium capital tax rate becomes lower, which raises the equilibrium growth rate.
These papers analyzed the closed economy models and focused on the e¤ects of income
distribution on growth rates. Mateos-Planas (2010) analyzed the e¤ects of demographics
on the mix of tax rates on households’labor and capital income by using a median voter
model. He focused on the quantitative e¤ects of increases in young population on thecapital income tax rate in a close economy. Our paper also constructs an overlapping
generation model in political economy. However, we consider an open economy wherein
capital is mobile among countries and focus on the e¤ects of demographics on the resultsof policy competition.
Moreover, our analysis regarding asymmetric countries relates to studies of asymmetrictax competition, which have considered di¤erences in many aspects between regions andcountries. In particular, whereas previous studies focused on regional characteristics anddisparities in population size, technology, preferences, labor market, and initial endowment, we add a new and signi…cant view to asymmetric tax competition by considering
4

Montén and Thum (2010) considered the e¤ects of demographics on the inter-generational con‡ict

under tax competition. Their focus is on to derive conditions under which gerontocracy improves the
young generation’s welfare. Hence, e¢ ciency issues of tax competition are out of the scope of their paper.

5international di¤erences in demographic structure.5
This paper is structured as follows. Section 2 presents the baseline framework. Section
3 provides the various e¢ ciency properties of our model. Section 4 extends the baseline
framework by considering asymmetric countries. Section 5 concludes.

2

Baseline framework

Consider an overlapping generations model wherein time is discrete and each individual
lives for two (young and old) periods. At the …rst (young) period, an individual works toearn wage income, consumes, and saves, whereas at the second (old) period, she/he does
not work and spends her/his savings to consume. At the end of the old period, she/heexits the economy. We call a cohort of individuals who are young at time t as generationt. This economy has M countries, and each country i (i = 1; ::M ) has a population of size
Lit + Lit 1 , where Lit represents the population size of generation t. We assume that the
population growth rate, n, is exogenous, and in the baseline model, we assume symmetriccountries so that n is common to all countries. Hence, we have Lit+1 = (1 + n)Lit . In
order to keep population positive, we assume that n >

2.1

1.

Individuals

Individuals obtain utility from private good consumption, c, and public good consumption,
g. We specify the utility function as follows:
Uit = uiyt + uiot+1 ;
where

(1)

(2 (0; 1)) is the time discount rate. The subscripts y and o represent the young

and old periods, respectively. uiy is the utility from consumption at the young period incountry i and uio is that from consumption at the old period in country i. We assume
that uij (j = y; o) is given by
uijt = ln cijt +where

ln git ;is a positive constant that represents the preference for public good consumption.

Budget constraints are given by
wit = ciyt + sit ;5(2)(1 + rit+1 )sit = ciot+1 ;See Ogawa et al (2016) for a brief survey of the existing studies on asymmetric tax competition.

6

where w, s, and r are the wage income, savings, and rate of return on savings, respectively.
We assume that individuals are price-takers. At period t, an individual in country i
and generation t inelastically supplies her/his labor endowments, of which amounts arenormalized to one, to earn wit , and chooses ciyt and sit given all prices. At period t + 1,
she/he receives (1+rit+1 )sit and chooses ciot+1 given all prices. We assume perfect foresight
regarding individual’s expectation on rit+1 . Standard life-time utility maximization yields
ciyt =

2.2

wit
;
1+

sit =

wit
;
1+

ciot+1 =1+(1 + rit+1 )wit :

(3)

Firms

Firms produce the numéraire using labor and capital under constant returns to scale. The
numéraire can be traded with no cost between countries. We assume perfectly competitive
goods, labor, and capital markets. We employ a Cobb-Douglas production function:
yit = Lit Kit1
where y is the output level,;(2 (0; 1)) is a positive constant representing the labor sharein production, and L and K are labor and capital inputs, respectively. Letting k denotethe capital per capita (capital-labor ratio, = K=L), pro…t maximization yields
1
wit = kit
;

where

kit =

1
rit + it

1=

;

(4)

represents the capital tax rate. As is standard in capital tax competition models,

capital taxation decreases the capital per capita (@kit =@ it < 0).

2.3

Market clearing conditions

In this paper, we assume that individuals are immobile between countries, implying thatthe labor market is local, whereas capital is freely mobile, implying that the capital market
is global. Hence, the labor market clearing condition in country i is given by
Lit = Lit :The global capital market clearing condition is given by
M
X
i=1

Kit =

M
X
i=1

7

sit 1 Lit 1 :Because capital is assumed to be freely mobile among countries, the rate of return on
savings becomes common to all countries (rit = rt , 8i). Then, the capital market clearing
condition can be written as
M
X
i=1

2.4

1
rt + it

1=

Lit =

1+

M
X
i=1

1
rt 1 + it 1

(1)=

Lit 1 :

(5)

Governments

In each country, a government uses capital tax revenues to …nance public good provision.
We assume that policies are determined by majority voting: the capital tax rate,

it , andthe level of public good provision, git , are determined so that they maximize the utilityof the majority at period t.6 Hence, when the population size of generation t is larger
than that of generation t

1 (Lit > Lit 1 ), the government at period t chooses

it andgit that maximize Uit . When the opposite holds true (Lit < Lit 1 ), it chooses

it andgit that maximize uiot .7 When deciding on

it and git , governments regard past variables(wit 1 ), other countries’policies ( jt and gjt ), prices determined in the global market (rtand rt+1 ), and own future policy (git+1 ) as given.8 We make the last assumption because
governments make decisions at each period, which implies that they cannot commit to
future decisions. We assume perfect foresight regarding government’s expectations on
git+1 , and rt+1 . The following …gure summarizes the structure of the model.
[Figure 1 around here]
6

In our setting wherein only two types of individuals exist, this corresponds to maximize the utility ofthe median voter.
7
When Lit = Lit 1 , we assume that the government chooses to maximize Uit or uiot with equal
probability. If it chooses to maximize Uit , then the results are the same as those in the case of Lit > Lit 1 ;if it chooses to maximize uiot , then the results are the same as those in the case of Lit < Lit 1 . For thesake of expositional simplicity, we omit the case of Lit = Lit 1 .
8
We assume that a government regards global prices as given for analytical simplicity. Such an assumption would be appropriate when many countries exist. Even if we do not assume this, the government’s
incentive to tax capital is di¤erent between an economy with a growing population and an economy witha decreasing population. Hence, the economy would have equilibrium ine¢ ciencies that depend on demographics as will be shown in this paper.

8

2.4.1

Economy with a growing population

We start from the case of n > 0, which implies the population is increasing. In this case,because Lit > Lit 1 , the young individuals represent the majority of the population andthe government maximizes Uit . Plugging (2), (3) and the government budget constraint
git =it Kit into (1), we obtain

Uit =
+
where

ln it

ln(1 + rt+1 ) +is de…ned as(1 + )(1ln(rt + it )ln

ln git+1 +)ln(rt + it )(6)

ln Lit + ;)1= +(1+ ) ln (1(1+ ) ln(1+ )+ ln(1)(1For the later use, we de…ne M Wiyt as the di¤erentiation of (6) with respect to
M Wiyt1

1(rt + it )it

The …rst term of M Wiyt represents the e¤ect of
second term of M Wyt denotes the e¤ect of

)= :
it :

(1 + )(1):(rt + it )it on the supply of public goods and theit on the wage rate. The …rst order conditionof the government’s maximization is given by M Wiyt = 0, yielding
it =

2.4.2

(1 +

+ )(1

)

rt :

(7)

Economy with a decreasing population

Next, we consider the case of n < 0, which implies that the population is decreasing. In
this case, because Lit < Lit+1 , old individuals represent the majority and the government
maximizes uiot . Plugging (3) and git =uiot =
where

ln it

it Kit into (2), we obtainln(rt + it ) + ln(1 + rt ) +ln Lit + ln wit 1 + ;(8)is de…ned as
ln

+

1+

ln(1

)1= :

Again, for the later use, we de…ne M Wiot as the di¤erentiation of (8) with respect to
M Wiot

1
it

it :

1
:
(rt + it )The …rst order condition of the government’s maximization is given by M Wiot = 0, yielding
it =

1
9

rt :

(9)

Equations (7) and (9) imply that for a given rate of return on savings, an economy witha decreasing population has a higher capital tax rate than an economy with a growing
population. Because the government regards the global price, rt , as given, capital taxationa¤ects the utility through changes in capital tax revenues,

it Kit , and changes in wage

rate, wit . Although the government recognizes the former e¤ect in both economies, itdisregards the latter e¤ect in an economy with a decreasing population. Moreover, as
shown in (4), capital taxation, by decreasing capital per capita, kit , lowers wit and Uit
because capital and labor are complementary in production. Hence, the government has
a stronger incentive to tax capital in an economy with a decreasing population than in an
economy with a growing population.9

2.5

Transitional dynamics

In the baseline model, we assume symmetric countries, which implies that all countries havethe same capital holdings at period 0, the same population size, and the same population
growth rate, implying that Lit = Ljt (i 6= j) for all t. From the capital demand (4), andthe capital market clearing condition (5), the sequence of capital per capita, k, can be
written as
kt =

k1 ;
(1 + )(1 + n) t 1(10)

regardless of population growth rate. Note here that kt is common to all countries. Moreover, (3) and (4) result in common consumption levels, i.e., ciyt = cyt and ciot = cot ,
8i.

2.6

Steady-state

We focus on steady-state equilibrium, wherein the level of individual’s consumption, c,
and capital per capita, k, are constant over time (cyt = cyt+1 = cy , cot = cot+1 = co , and9This result is similar to the results shown in the existing studies of tax competition with the medianvoter principle cited in the Introduction, wherein the incentive of governments to impose tax on capitalincreases with the decrease in capital endowments of the median voter. In our framework, the medianvoter is young individuals in an economy with a growing population and old individuals in an economy
with a decreasing population. Because young individuals have no capital, the government has a weaker
incentive to tax capital in an economy with a growing population than in an economy with a decreasing
population.

10kt = kt+1 = k ).10 Then, from (4), we readily know that rt + t = rt+1 + t+1 . Using thisand (4), (10) can be rewritten as
rt + t =(1 + n)(1 + )(1

)

:

(11)

The higher the population growth rate, the smaller the capital per capita becomes, which
results in higher marginal productivity of capital (i.e., higher gross rate of return on
capital). We can solve (7) and (11) to derive equilibrium

and r in an economy with agrowing population whereas we can use (9) and (11) to obtain equilibrium

and r in an

economy with a decreasing population.

3

Equilibrium and its e¢ ciency

Because our model is an overlapping generations model, the equilibrium savings rate is not
optimal because of the dynamic ine¢ ciency (Blanchard and Fischer, 1989: and Romer,2011).11 Moreover, capital taxation in the presence of capital mobility and political process
also cause externalities. Hence, the equilibrium capital tax rate departs from the optimal
one. In this section, we characterize the (in)e¢ ciency of capital tax rate. Following
Blanchard and Fischer (1989), we employ the Benthamite social welfare function as thewelfare criterion:
W =

M
X

uio0 Lo0 +1
Xt

Uit Lt

t=0

i=1

!

:

(12)

In order to ensure the existence of optimal tax rate, we assume thatwhich implies that (1 + n) < 1. Using (3), (4), and git =
Uit =
+
10

log it +log it+1

ln(1 + rt+1 ) +ln Lit +ln(rt+1 + it+1 )ln Lit+1 +

+

P1

i=1

t L < 1,t

it Kit , Uit becomes+ (1 + )(1ln(1)

ln(rt + it ))1= :Note here that (10) has a unique steady state. Combined with (3) and (4), this implies that there

exists a unique consumption level.
11
In Appendix A, we examine the …rst best case, and show that our model has dynamic ine¢ ciency. Notefurther that from (3), (4) and (11), the savings rate is independent of the capital tax rate in the steadystate. Thus, the dynamic ine¢ ciency cannot be internalized by capital taxation in our model. Hence,
when we discuss the e¤ciency of equilibrium tax rate, the e¢ cient allocation is the second-best optimum.

11

Then, by di¤erentiating W with respect toit , we obtain the e¤ects of capital taxation on

the welfare as follows.
1@W= (1 + n)M Wiyt + M Wiott1
@ it
Lt 1 R
where

M
X( i1t +

i3t ) ;

i2t

(13)

i=1

ilt (l = 1; 2; 3) is de…ned as[ + (1 + n)] @rt;(rt + it ) @ it

i1t

i2t

(1 + n)(1 + )(1(rt + it )
i3t

) @rt
;
@ it

@rt
:
1 + rt @ it

The …rst and second terms of (13) represent the e¤ect of an increase in

it on the young

agent’s welfare and that on the old agent’s welfare in country i at time t. The third term isthe e¤ect of an increase in

it on the social welfare via changes in the world rate of returnon savings, which includes the e¤ects on the foreign (country j (j 6= i)) agents’welfare aswell as those on the domestic (country i) agents’welfare. We discuss the interpretation of
i1t ,

3.1

i2t , andi3t in the next subsection.

Possible externalities

In equilibrium, the government sets its capital tax rate so that M Wiyt = 0 when the population is increasing and M Wiot = 0 when the population is decreasing. However, the optimal tax rate needs to satisfy @W=@ it = 0 where @W=@ it is given by (13). Hence, when the
population is increasing, the government ignores the e¤ects represented by the two terms,
PM
M Wiot andi3t ). When the population is decreasing, it ignores
i=1 ( i1t + i2t
PM
the e¤ects represented by the two terms, (1 + n)M Wiyt andi3t ).
i=1 ( i1t + i2t

Such ignorance causes externalities in our framework.

In maximizing the social welfare, one needs to consider the e¤ects of capital taxation,
it , on the global price, rt . However, in the equilibrium, the government treats rt as given

because there exist a large mass, M , of countries in the world. Such a di¤erence causes
distortions.12 More speci…cally, capital ‡ight caused by capital taxation in a country
12

Even if a government does not regard r as given, it considers the e¤ects ofon its own country’s

welfare. In the optimal, one needs to consider the e¤ects of , through changes in r, on the other countries’
welfare and policies as well. Hence, even in this case, we observe a distortion caused by capital taxation.

12

enlarges the amount of capital in other countries and this capital ‡ight induces three
types of externalities, which we call as the Fiscal externality, the Wage externality, and theInterest-rate externality. The Fiscal externality is a positive externality that the capital‡ight increases the tax bases in other countries, which is standard in tax competition
models. In (13), it is represented by

i1t .

The Wage externality is a positive externalitythat the capital ‡ight raises the wages of young individuals in other countries, which isrepresented byi2t in (13). The Interest-rate externality is a negative externality that thecapital ‡ight implies a lower capital demand, which results in a lower rate of return on
savings and decreases the income level of old individuals. It is represented byi3t in (13).

This is not the end of the story. Our framework yields additional externalities because
the equilibrium policies are chosen though political process (majority voting). A government chooses its current tax rate to maximize the utility level of majority and disregards
the minority’s utility. This implies that the government in an economy with a growing
population disregards the e¤ects of capital taxation on old individuals utility, i.e., M Wiotin (13). Put di¤erently, when the young individuals represent the majority, the government disregards the e¤ect on the bene…ts of public good provision to old individuals. We
call it as the Public-good externality. On the other hand, when the old individuals represent the majority, the government takes no thought of the following two e¤ects of capitaltaxation on young individuals’ utility: One is that it disregards the negative e¤ect on
young individuals through changes in the wage rate, which is represented by the second
term of M Wiyt in (13), which we call as the Gerontocracy externality. The other is the
Public-good externality, i.e., it ignores the e¤ect on the bene…ts of public good provision
to young individuals, which is represented by the …rst term of M Wiyt in (13).
Summarizing above arguments, our framework has …ve externalities associated to capital taxation. The Fiscal and Wage externalities are positive, whereas the Interest-rate
externality is negative. The sign of the Pubic-good externality is ambiguous because it
is not clear whether an increase in the capital tax rate might increases the tax revenues.
However, the sum of the Public-good and Fiscal externalities always becomes positive andis increasing in .13 Finally, the Gerontocracy externality works as a negative externality
in an economy with a decreasing population, whereas it does not exists in the economy
13

In an economy with a growing population, the sum of Public-good and Fiscal externalities is R(1 +n)M Wiot +M Wiot +i1t = (i1t == it ) f1 +it =[(rt + it )]g > 0: In an economy with a decreasing population, it is=[ (rt + it )] > 0. Therefore, it is positive in both economies.

13with a growing population. Hence, the overall negative external e¤ects are stronger in an
economy with a decreasing population than in an economy with a growing population.

3.2

Preliminary analysis: closed economy

Before proceeding to the analysis of capital tax competition, we present the political economy outcomes in the absence of capital mobility. For this purpose, temporarily supposethat capital is immobile among countries. Then, all variables become local variables andthe government takes taxation e¤ects on the rate of return on savings in country i, rit , into
consideration in addition to other local variables. Moreover, because rit + it is determinedby past variables, the government now considers that any increases inby decreases in rit (@rit =@ it =it are exactly o¤set

1).

In an economy with a growing population, Uit given by (6) is an increasing functionof= it > 0) under consideration of @rit =@ it =1.

Therefore, the government sets its capital tax rate as high as possible (as long as rit

0),

it

(i.e., M Wiyt

@Uit =@ it =which, combined with (11), results in14
(1 + n)(1 +
im
iy =

)(1

)

:

(14)

The superscript im represents the case of capital immobility.
In an economy with a decreasing population, the government maximizes uiot withrespect toit , and the equilibrium tax rate satis…es M Wiot

@uiot =@ it = = it

1=(1 +

rit ) = 0 and is given by
im
io =

im
(1 + rio):(15)

Comparing M Wiyt with M Wiot , we know that the government in an economy with adecreasing population sets a lower tax rate than in an economy with a growing population,
if capital is immobile between countries. Because changes in rate of return on savings
absorb the capital tax e¤ects (@rit =@ it =1), capital taxation does not a¤ect capital per

capita, kit , and hence, the wage rate, wit . The government considers such a relationship,
implying that a government in an economy with a growing population cares only about thelevel of public good provision in maximizing Uit whereas a government in an economy witha decreasing population considers decreases in returns from savings when maximizing uiot
as well. Thus, a government in an economy with a decreasing population is more tentative
14

We obtain (11) from (5) by setting M = 1.

14

in taxing capital than a government in an economy with a growing population if capitalis immobile.15
With tax rates in hand, we can examine the e¢ ciency properties of these tax rates bylooking at the e¤ects of an increase inon the social welfare (12). Such e¤ects can bederived by di¤erentiating (12) with respect toit , and evaluating it atimiy or atim
io . The

social welfare function (12) in the closed economy is given by
W = uio0 Lo0 +1
Xt

Uit Lt :t=0

By using @rit =@ it =1, we can write its derivative as
1

@W

Lt 1

t 1 @ it= (1 + n)M Wiyt + M Wiot=

(1 + n)+it

Evaluating (16) at it =
"
@W
sgn
@ it
Becauseit

1 + rit

(16)

:

im
iy yields

im
= iy

#

it cannot be higher than= sgn

(1 + )(1(2 + n))(1 + n)1 :imiy because of the non-negative constraint of rit , weknow that in an economy with a growing population, the equilibrium tax rate is (secondbest) optimal if the preference for public good consumption is su¢ ciently large (i.e.,(1 + )(1)(1 + n)= [)(1 + n)= [(2 + n)]), and ine¢ ciently high otherwise (i.e.,< (1 + )(1

im yields(2 + n)]). Similarly, evaluating (16) at it = io
"
#
@W
1+n
sgn
> 0:
= sgn
im
@ it = oim
1 + rio

Hence, we observe an ine¢ ciently low capital tax rate in an economy with a decreasing
population.
15

Mateos-Planas (2010) analyzed the e¤ects of demographics on the mix of tax rates on households’

labor and capital income by using a median voter model, and showed that when the decisive voter changes
from old individuals to young individuals, the capital tax rate increases. He con…rmed the quantitative
relevance of this result by calibrating his model to United States data. Our result on capital tax ratein the capital immobile case is consistent with this. However, he showed that when the proportion of
old individuals decreases while keeping the decisive voter type unaltered, the capital tax decreases, which
is not consistent with our result wherein a higher n and hence, a lower proportion of old individuals
implies a higher capital tax rate. Such a departure would come from the fact we endogenize governments’
expenditure and ignore labor income tax whereas Mateos-Planas (2010) …xed governments’ expenditureand introduced labor income tax.

15

Proposition 1 Suppose immobility of capital between countries. Then, the capital tax rate
is optimal or ine¢ ciently high in an economy with a growing population whereas it
is ine¢ ciently low in an economy with a decreasing population.
In the absence of capital mobility between countries, there exists no externality inducedby tax competition and therefore only the externalities induced by the political process
are relevant.
When the population is decreasing, we observe the Public-good externality, which isrepresented by(1 + n)= it of the right hand side of (16) and is always positive in theclosed economy.16 As a result, the capital tax rate becomes ine¢ ciently low.
In a similar vein, when the population is growing, we observe the positive Public-goodexternality, which is represented by= it of the right hand side of (16). In addition,

we observe the negative Interest-rate externality, which is represented by= (1 + rt ) of

the right hand side of (16). The Public-good externality is stronger than the Interest-rate
externality when the preference for public good consumption,
weaker whensu¢ ciently is small. However, when

, is su¢ ciently large andis su¢ ciently large, individuals

require the capital tax rate to be the maximum possible rate, implying that the resulting
equilibrium tax rate becomes identical to the optimal one. Whenis su¢ ciently small, theInterest-rate externality overwhelms the Public-good externality, yielding an ine¢ ciently
high capital tax rate in equilibrium.
Note here that the interpretations of the Interest-rate externality are di¤erent between the open and closed economies. In the open economy, because each country issu¢ ciently small, each government regards the rate of return on savings as constant. Thismisunderstanding of government is the source of the Interest-rate externality. Hence, theInterest-rate externality is related to tax competition in the open economy. In contrast,in the closed economy, a government considers the e¤ect of capital taxation on the rateof return on savings. Still, when the population is growing, it ignores the e¤ect of capitaltaxation on the capital income of old individuals, which induces the Interest-rate externality. Hence, the Interest-rate externality is related to the political process in the closed
economy.17
16

Because capital taxation does not a¤ect the wage rate in the closed economy, there exists no Geron-tocracy externality.
17
If each government considers the e¤ect of capital taxation on the rate of return from savings in an open
economy, the interpretations of the Interest-rate externality become the same across the two economies.

16

3.3

Economy with a growing population

Now we return to the baseline model wherein capital is mobile between countries, andmove to the e¢ ciency analysis of capital tax competition in political economy. Start froman economy with a growing population. From (7) and (11), we obtain the equilibrium
capital tax rate as follows:y =(1 + n)(1 + )(1[ + (1 + )(1

)
:
)]

(17)

The rate of return on savings becomesry =(1 + n)(1 + )(1)2 (1 + + ):[ + (1 + )(1
)]

(18)

Similarly to the previous section, we examine the equilibrium e¢ ciency properties bylooking at the e¤ects of an increase in

it . By evaluating (13) at

it =

y

(see also (11)),

we know that

where

and

1

@W

Lt 1

t 1 @ it=it = yy (1 + ry )(n + 2)( + 1)(1);[ + (1 + ) (1
)]

(19)

are de…ned as
fn [ (n + 2) + n + 4] + 3g
[1

Hence, we can see that
"

@W
sgn
@ t

(1

= y

) ] + (1#( + 1)(n + 1)(n + 2);)2 + n( + 1)(1

= sgn [ (n + 2)

( + 1)(1)2 > 0:) ]:Therefore, (19) is positive if and only ifwhere e is de…ned as> e;e( + 1)(1(n + 2)):

Proposition 2 Capital tax competition in an economy with a growing population results
in an ine¢ ciently low (resp. high) capital tax rate if and only if the preference forpublic good consumption,, is larger than e (resp. smaller than e).
17In a closed economy with a growing population, the capital tax rate is never ine¢ ciently
low. In an open economy with a growing population, we have the positive Fiscal and Wageexternalities, and the negative Interest-rate externality. Although the sign of the Publicgood externality is ambiguous, the sum of the Fiscal and Public-good externalities isalways positive and is increasing in. Hence, if

is su¢ ciently large to satisfy> e,then the overall externality becomes positive, which results in an ine¢ ciently low capitaltax rate, i.e., capital tax competition in political economy results in the race to the bottom.
In contrast, when the opposite holds true ( < e), the negative Interest-rate externality
dominates positive externalities, resulting in an ine¢ ciently high capital tax rate, i.e., weobserve the race to the top.

3.4

Economy with a decreasing population

Next, consider an economy with a decreasing population. From (9) and (11), we obtain
the equilibrium capital tax rate and rate of return on savings as follows:18

o =

ro =

(1 + n)(1 + )(1)(1 + n)(1 + )(1

)2

;

(20)
:

To examine the e¢ ciency of equilibrium, we evaluate (13) at

it

=

o

(see also (11)),

resulting in
1

@W

Lt 1

t 1 @ it=it = o[ (n + 2)
1
y (1 + ry )( + 1)(1) (n + 1)]:

This is positive if and only ifwhere b is de…ned as

> b;

b

( + 1)(1) (n + 1):(n + 2)(21)

Proposition 3 Capital tax competition in an economy with a decreasing population results
in an ine¢ ciently low (resp. high) capital tax rate if and only if the preference forpublic good consumption,18, is larger than b (resp. smaller than b).

Note that from the assumption of positive population size (n >

18

1),

o

and ro are positive.

In a closed economy with a decreasing population, the race to the bottom always
emerges. However, when capital becomes mobile, we have the negative Gerontocracy
and Interest-rate externalities, yielding the possibility of the race to the top. Whenis su¢ ciently large to satisfy> b, the sum of Fiscal and Public good externalities

becomes large, and we observe the race to the bottom. Whenis su¢ ciently small, theGerontocracy and Interest-rate externalities become prominent, resulting in the race tothe top.

3.5

Possibility of the race to the top

As we saw, both of the race to the bottom and the race to the top are possible regardlessof the population growth rate when capital is mobile. It would then be interesting to ask
which economy is more likely to face the race to the top. From (21), we can show thatthe race to the top never emerge in an economy with a growing population (i.e., e < 0holds true for n0) when 2

3 + 2 > 0 and 3 2

1 < 0 hold true.19 However,

we can readily know that b is always positive because of the assumption that n >1,

implying that there always exists a possibility of the race to the top in an economy witha decreasing population.
3 + 2 > 0 and 3 2

Proposition 4 When 2

1 < 0, tax competition always resultsin the race to the bottom in an economy with a growing population (i.e., e < 0),and it might result in the race to the top in an economy with a decreasing population(i.e., b > 0).

In this sense, we can say that the race to the top is more likely to emerge when apopulation is decreasing than when it is increasing. As shown in Section 2.4.2, whencapital is mobile, the government’s incentive to tax capital is stronger in an economy
with a decreasing population than in an economy with a growing population because thegovernment in an economy with a decreasing population ignores the e¤ect of the increase intax rate on the current wage rate of young generation. This generates a higher possibilityof the race to the top in an economy with a decreasing population than in an economy
with a growing population.
In fact, if we follow Romer (2011) and Acemoglu (2009) to set the labor share,
to 2=3, we obtain 2
19

3 + 2 > 0 and 3 2

Proof is given in Appendix.

19

1 < 0. Moreover, Karabarbounis

and Neiman (2014) estimated the global labor share and showed thathas exhibited a

relatively steady downward trend from 0:64 to 0:59 during the past several decades. If
0:59

0:64, 3 2that the time discount rate,

1 < 0 holds true.20 In this case, 2

3 + 2 > 0 requires, is larger than 0:115, which is likely to be satis…ed in thereal world. Because most countries have experienced population growth during the past
several decades, we can think that the case of e < 0 is relevant to the real world. However,even when 2

3 + 2 > 0 and 3 2

1 < 0 hold true, b > 0 and there always exists

a possibility of the race to the top if a population becomes decreasing.
Figure 2 illustrates the above arguments.
[Figure 2 around here]

In Figure 2, we set n = 0:1 for an economy with a growing population and n =0:1 foran economy with a decreasing population. As to , we try two alternative values, 0:75
and 0:6. The former is higher than the values observed in Karabarbounis and Neiman(2014) and the latter is close to the observed values. We takein the horizontal axis andin the vertical axis. The shaded areas represent combinations ofandthat resultin the race to the top. Figures 2-(a-1) and 2-(a-2) describe an economy with a growing
population and that with a decreasing population when

= 0:75, respectively. In bothof these …gures, we observe the possibility of the race to the top for lower values ofalthough the shaded area is larger in an economy with a decreasing population than in
an economy with a growing population. When

= 0:6, which is represented by Figures2-(b-1) and 2-(b-2), we observe no possibility of the race to the top in an economy witha growing population (Figure 2-(b-1)) whereas we do observe it in an economy with adecreasing population (Figure 2-(b-2)).
When 2

3 + 2 > 0 and 3 2

1 < 0, and hence, e < 0 hold true, declines in the

population growth rate can shift the economy from the race to the bottom to the race tothe top. Because we know that b > 0 > e in this case, an economy with

2 (0; bjn=0 )experiences the shift from the race to the bottom to the race to the top as the population

growth rate, n, changes from positive to negative. This yields a strong policy implication
given the downward trends in the population growth rate as observed in many developed
countries. Such trends imply that many countries, which have experienced population
20

When 0 << 1+

p
13 =6 ; 0:77, 3 2

1 < 0.

20

growth in the past, will experience population declines as Japan has already come toface.21 Then, we might observe the race to the top, under which coordinated decreases
in the capital tax rate among countries can improve the welfare. This is in contrast to
the standard result in the tax competition literature wherein we observe the race to thebottom and coordinated increases in the capital tax rate can improve the welfare.

4

Asymmetric countries

In this section, we extend the baseline framework by considering asymmetric countries.
Consider two groups of countries (groups h and l), where group h has Mh countries
and group l has Ml countries. We assume that countries in each group have the same
population growth rate. Let nk denote the population growth rate of group k countries(k = h; l). Without loss of generality, we assume that nh > nl . Noticing that rkt + kt =rkt 1 +

kt 1 holds true in the steady state and that the assumption of global capital

market yields rht = rlt = rt , we obtain the capital market clearing condition as1
rt + ht
"

=

1+

1=

Lht +
1

1
rt + lt(1)=

rt + ht

1=

Llt

Lht
+
1 + nh(22)
1
rt + lt

(1

)=

#
Llt
;
1 + nlwhere Lht and Llt are total population sizes in each group of countries and de…ned as
PMh
PMl
Lht
i=1 Lit and Llt
j=1 Ljt .

In this extended setting, we have the following three cases: (i) all countries have a

growing population (i.e., nh > nl > 0), (ii) all countries experience population decreases(i.e., 0 > nh > nl ), and (iii) group h countries have a growing population whereas groupl countries have a decreasing population (i.e., nh > 0 > nl ). In case (i) (resp. case(ii)), young (resp. old) individuals represent the majority in both groups of countries.
In case (iii), young individuals represents the majority in group h countries whereas old
individuals do so in group l countries.
21

The total population in Japan started to decrease in 2016, and will contiously decrease with time.

21

4.1

Equilibrium and its welfare properties

Case (i): economy with a growing population
In this case, each government maximizes the young individual’s utility, Uit , and the …rstorder condition yields (7) for all countries, implying that

ht =

lt =

t.

We substitutethis into (22) to obtainrt + t =(1 + nw )(1 + )(1

)

;

(23)

where nw describes the world population growth rate and is de…ned as
Lht + Llt
Lht =(1 + nh ) + Llt =(1 + nl )nw

1:Note that we obtain (23) if we replace n with nw in (11). Therefore, we obtain theequilibrium tax rate and rate of return on savings very similar to those shown in thebaseline framework. More speci…cally, (7) and (23) yield(i)(1 + nw )(1 + ) (1);
[ + (1 + )(1)]=r(i) =(24)(1 + nw )(1 + ) (1)2 (1 + + ):[ + (1 + )(1)]

The superscript (i) represents case (i). We can obtain (24) by replacing n with nw inyand ry (given by (17) and (18)) in the baseline framework. Hence, welfare properties inthis case are similar to those in the symmetric case. In fact, replace n in e with nw anddenote it by e(i) . Then, we can see that (13) evaluated at
if

=

(i)

is positive if and only

> e(i) .

Proposition 5 Suppose all countries have a growing population but with di¤ erent rates.
Then, capital tax competition results in an ine¢ ciently low (resp. high) capital tax
rate if and only if the preference for public good consumption,(resp. smaller than e(i) ).

, is larger than e(i)Case (ii): economy with a decreasing population
When all countries have a decreasing population, each government maximizes the oldindividual’s utility, uiot , and the …rst-order condition yields (9), implying that22

ht = lt =t . From (9) and (22), we obtain the equilibrium tax rate and rate of return on savings as(ii)=

r(ii) =

(1 + nw )(1 + )(1)(1 + nw )(1 + )(1

)2

;

(25)
:

The superscript (ii) represents case (ii). Again, we can obtain (25) by replacing n with nw
inoand ro (given by (20)) in the baseline framework. Hence, welfare properties again

become similar to those obtained in the baseline framework: (13) evaluated at
is positive if and only if

=

(ii)

> b(ii) , where we replace n with nw in b and denote it by b(ii) .

Proposition 6 Suppose all countries have a decreasing populations but with di¤ erent

rates. Then, capital tax competition results in an ine¢ ciently low (resp. high) capital tax rate if and only if the preference for public good consumption,, is larger

than b(ii) (resp. smaller than b(ii) ).

Case (iii): countries with a growing population v.s. countries with a decreasing
population
In this case, young individuals represent the majority in countries with a growing population (group h countries) whereas old individuals do so in countries with a decreasing
population (group l countries). Then, the equilibrium tax rates become
ht =
lt =

(1 +
1

+ )(1

)

rt ;

(26)

rt :

Comparing the two tax rates, we know that the equilibrium tax rate is lower in group hcountries than in group l countries. From (26), we can derive+ (1 + )(1(1 + + )(1
1
rt + lt =
rt ;
1

)
rt ;
)

rt + ht =implying thatrt + ht =+ (1 + )(1
1+ +)(rt + lt ) :We substitute this into (22) to rewrite the capital market equilibrium condition as

23

(27)

1=
(1
) 1=
1
Lht +
Llt
rt + lt
rt + lt
(
(1 )=
(1
)
Lht
=
+
1+
rt 1 + lt 1
1 + nhwhereis de…ned as

(1 +

(1

1

)=

rt 1 + lt 1

+ ) = [ + (1 + )(1Llt
1 + nl);)]. Furthermore, if we de…ne zt

and Ht aszt

rt + lt ;(1

Ht(1 + ) (1)=

1= (1 + n)

+ Llt =Lht;h) + (1 + nl )Llt =Lht

this becomes
zt+1 = Ht1

zt1

:

(28)

In the long run, the total population size of group h countries expands whereas that ofgroup l countries shrinks, implying that limt!1 Llt =Lht = 0. Hence, the steady statevalue of Ht becomes
H = lim Ht =(1 + ) (1t!1) (1 + nh ):Combined with (28), this yields the steady state value of z asz = H (1)=:

Plugging this into (27), we obtain the steady state values of the rate of return on savingsand tax rates as follows:)H (1rt = (1
(iii)
h

=

(iii)
l

= H (1

1+

)=

H (1

+
)=

;

(29)

)=

;

:

We substitute (29) into (13) to obtain
@W
@ ht
@W
@ lt

=

t

(iii)
ht = h=(iii)
lt = lt(1 + nh )t 1
(1 + nl )t 1""

1
1 + H (1)=

1
1 + H (1

)=

24

+
+

(2 + n)(1 +1(2 + n)
1

H

H
#

:

+ )

#

;

From this, we know that @W=@ ht j(iii)
ht = h> @W=@ lt j(iii)
lt = l, resulting in the following

proposition.22
Proposition 7 If group l countries set an ine¢ ciently low capital tax rate, then so dogroup h countries ( @W=@ ht j

> @W=@ lt j(iii)
ht = h(iii)
lt = l> 0). If group h coun-tries set an ine¢ ciently high capital tax rate, then so do group l countries ( 0 >@W=@ ht j(iii)
ht = h> @W=@ lt j(iii)
lt = l). Additionally, there is a possibility thatgroup h countries set an ine¢ ciently low capital tax rate whereas group l countries set
an ine¢ ciently high capital tax rate ( @W=@ ht j(iii)
ht = h> 0 > @W=@ lt j(iii)
lt = l).

This result is consistent with that shown in Proposition 4. Put di¤erently, we again …ndthat countries with a decreasing population are more likely to exhibit the race to the top.
Equation (29) shows that if countries with a decreasing population compete for capitalwith countries with a growing population, the former countries set higher capital tax rates
than the latter countries. This prediction is consistent with di¤erences in the corporate
income taxes across countries. In fact, if we compare the …ve oldest OECD countries
with the …ve youngest OECD countries in terms of the median age, the average corporate
income tax in 2010 is around 30 percent for the former and 26 percent for the latter.2324
Such a di¤erence is robust even if we take the oldest and youngest ten countries. Of course,di¤erences in capital tax rates can arise from other sources. The most prominent example
would be the country size di¤erences, and many papers have shown that large countries
22

In fact, we have the second case when

the last cases. In fact, if we set

is large. Whenis su¢ ciently small, we have the …rst and= 1=2, the …rst case holds true whenis su¢ ciently large and the lastcase holds true when is su¢ ciently small.
23
Data sources are as follows:
median age: OECD, http://www.oecd-ilibrary.org/education/trends-shaping-education-2013/medianage-going-up-into-the-next-century_trends_edu-2013-graph43-en,
population:OECD,

http://www.oecd-ilibrary.org/economics/national-accounts-at-a-glance-

2014/population_na_glance-2014-table105-en,
corporateincome

tax:

OECD,

http://www.oecd-ilibrary.org/taxation/data/corporate-income-tax/corporate-income-tax-rates_7cde787f-en,
GDP: OECD, https://data.oecd.org/gdp/gross-domestic-product-gdp.htm.
Here, we use the combined corporate income tax, which is the sum of the central government’s corporate

24

income tax and the sub-central government’s corporate income tax. In 2010, we have 34 OECD countries,
and Japan, Germany, Italy, and Finland are the …ve oldest countries in terms of the median age whereas
Mexico, Turkey, Israel, Chile, and Ireland are the …ve youngest countries.

25

set higher capital tax rates than smaller countries.25 If we control for size di¤erences interms of population or GDP, we can still …nd higher tax rates for older countries than foryounger countries.

4.2

Utility di¤erence between asymmetric countries

We …nally discuss which group of countries gain from tax competition in political economy.
In so doing, we assume that countries in each group have the same population size (i.e.,b kt
Lit = LLkt =Mk ). The indirect utility of an individual in a group k country is writtenas(1 + )(1 +Ukt = (1 + ) ln kt
b kt ++ (1 + ) ln L
+ (1 + ) ln (1ln(rt + kt ) +ln(1 + nk ) + (1 + ) ln(1)(1In cases (i) and (ii), we know that
Uht

)

)=

ln(1 + ) +ln

1+ln(1 + rt ))1=:

ht = lt , implying thatb ht
Ult = (1 + ) ln Lb lt +ln L(ln(1 + nh )ln(1 + nl )) :(30)

Thus, in these cases, we can decompose the welfare di¤erence into two terms: the …rst term
represents the population size e¤ect and the second term represents the population growthe¤ect. The larger the population size, the more the country attracts capital because of
the complementarity between capital and labor in production, resulting in a larger tax
base. Hence, individuals in a larger country can consume a larger amount of public goods
than those in a smaller country, which makes the welfare of the larger country higher thanthat of the smaller country. Moreover, in a similar vein, a higher population growth rateincreases the individual’s public good consumption at the old period because it implies alarger population size of the next generation. This makes the welfare of group h countries
higher than that of group l countries.
In case (iii), the welfare di¤erence between countries becomes
Uht

25

Ult = (1 + ) (ln ht
b ht+ (1 + ) ln L

ln lt ) +(1 + )(1 +b lt +ln L(ln(1 + nh ))[ln(rt + lt )ln(rt + ht )]ln(1 + nl )) :See e.g., Baldwin and Krugman (2004), Borck and P‡üger (2006), Bucovetsky (1991), Hau‡er andWooton (1999), and Ottaviano and van Ypersele (2005).

26

Thus, in addition to the population size and growth e¤ects, we have two other e¤ects. The…rst-term of the right-hand side of the above equation represents the tax e¤ect. A higher
tax rate increases the tax revenues and individuals’ public good consumption, resulting
in a higher welfare. As we know from (26) thatht <lt , the …rst term is negative.

The

second term represents the capital cost e¤ect. A higher capital tax rate implies a higher
capital cost, reducing capital input per capita. This results in a lower wage rate and lower
welfare. As we know that5ht < lt , the second term becomes positive.

Summary and discussions

In this paper, we developed an overlapping generations model wherein public good provision …nanced by capital taxation is determined by majority voting. When population is
growing (resp. decreasing), young (resp. old) individuals represent the majority, implyingthat the government’s decision depends on the demographic structure. We showed thatyoung individuals su¤er more from capital ‡ight than old individuals, and that the race
to the bottom is more likely to emerge when the population is growing than when it isdecreasing. It is even possible to observe the race to the bottom when the population
is growing whereas the race to the top might emerge when the population is decreasing.
Such dependence on the outcomes of capital tax competition on demographics provides
us a new viewpoint in policy debates regarding competition for capital and …rms. Particularly, because we observe drastic aging in many developed countries, our results indicate
an increasing relevance of the race to the top.
We brie‡y discuss the robustness of our results against two alternative extensions.26
First, suppose that in addition to capital taxation, the government has another instrument
to …nance its expenditure. As an example, we consider labor income tax on households.
Then, the government in an economy with a growing population would …nance its expenditure solely by labor income tax to prevent capital ‡ight, whereas the government in an
economy with a decreasing population would impose a positive tax on capital while trying
to set income tax as high as possible because it cares only about tax revenues. Moreover,
we can show that under reasonable parameter values, the equilibrium capital tax rate isine¢ ciently low in an economy with a growing population whereas it is so in an economy
with a decreasing population if and only if the preference for public good consumption
26

Formal analyses on these extensions are available in online appendices.

27

is su¢ ciently large, implying the possibility of the race to the top. In this sense, theintroduction of income tax does not alter our main results qualitatively.
Second, in our framework, governments provide public goods. Alternatively, we canassume that governments provides public inputs that a¤ect productivity of …rms. In such acase, the positive externality associated to capital taxation is further strengthened because
public inputs and capital are complementary in production. Consequently, we observe therace to the bottom when the public input elasticity of output is su¢ ciently large. Hence,
the introduction of public inputs would make the race to the bottom more likely to emerge
as the public inputs can contribute to production.
We here refer to two possible extensions. First, it would be worth endogenizing thedemographic structure by introducing decisions on the number of children to have. Governments’ behavior would impact, via changes in economic conditions, such decisions.
Then, by endogenizing the fertility rate, we might be able to analyze the long-run mutualinterdependences between policies and demographics. Second, it would also be signi…cantto consider households’mobility between countries. If we introduce households’mobility,
countries need to compete for people as well as capital, which would result in additional
policy externality. These are important topics for future research.

References[1] Acemoglu D., 2009, Introduction to Modern Economic Growth, Princeton University
Press.
[2] Alesina A., and D. Rodrik, 1994, Distributive politics and economic growth, Quarterly
Journal of Economics 109, 465-490.
[3] Baldwin, R.R. and P. Krugman, 2004, Agglomeration, integration, and tax harmonization, European Economic Review 48, 1-28.
[4] Blanchard, O. and Fischer, S., 1989, Lectures on Macroeconomics, The MIT Press.
[5] Borck, R., 2003, Tax competition and the choice of tax structure in a majority voting
model, Journal of Urban Economics, 54, 173-180.
[6] Borck, R. and M. P‡üger, 2006, Agglomeration and tax competition, European Economic Review, 50, 647-668.
28

[7] Bucovetsky, S., 1991, Asymmetric tax competition, Journal of Urban Economics 30,
167-181.
[8] Fuest, C. and B. Huber, 2001, Tax competition and tax coordination in a medianvoter model, Public Choice, 107, 97-113.
[9] Grazzini, L. and T. van Yprersele, 2003, Fiscal coordination and political competition,
Journal of Public Economic Theory, 5:2, 305-325.
[10] Hau‡er, A. and I. Wooton, 1999, Country size and tax competition for foreign direct
investment, Journal of Public Economics 71, 121-139.
[11] Ihori, T. and C.C. Yang, 2009, Interregional tax competition and intraregional political competition: The optimal provision of public goods under representative democracy, Journal of Urban Economics, 66, 210-217.
[12] Karabarbounis, L. and B. Neiman, 2014, The global decline of the labor share, Quarterly Journal of Economics 129, 61-103.
[13] Keen, M. and Konrad, K., 2013, The theory of international tax competition andcoordination, in Handbook of Public Economics, Volume 5 edited by Auerbach, A.,
Chetty, R., Feldstein, M., and Saez, E., North-Holland.
[14] Lockwood, B. and M. Makris, 2006, Tax incidence, majority voting and capital market
integration, Journal of Public Economics, 90, 1007-1025.
[15] Mateos-Planas, X., 2010, Demographics and the politics of capital taxation in a lifecycle economy, American Economic Review 100, 337-363.
[16] Montén, A. and M. Thum, 2010, Ageing municipalities, gerontocracy and …scal competition, European Journal of Political Economy 26, 235-247.
[17] OECD, 1998, Harmful Tax Competition: An Emerging Global Issue.
[18] OECD,

http://www.oecd-ilibrary.org/education/trends-shaping-education-

2013/median-age-going-up-into-the-next-century_trends_edu-2013-graph43-en[19] Ogawa, H., Y. Sato, and T. Tamai, 2016, Who gains from capital market integration:
Tax competition between unionized and non-unionized countries, Canadian Journalof Economics 49, 76-110.
29

[20] Ottaviano, G.I.P. and T. van Ypersele, 2005, Market size and tax competition, Journal of International Economics 67, 25-46.
[21] Persson, T. and G. Tabellini, 1992, The politics of 1992: …scal policy and European
integration, Review of Economic Studies 59, 689-701.
[22] Persson, T. and G. Tabellini, 1994a, Representative democracy and capital taxation,
Journal of Public Economics, 55, 53-70.
[23] Persson, T. and G. Tabellini, 1994b, Is inequality harmful for growth? American
Economics Review, 84, 600-621.
[24] Razin, A., E. Sadka, and P. Swagel, 2004, Capital income taxation under majority voting with aging population, Review of World Economics/Weltwirtschaftliches
Archiv, 140, 476-495.
[25] Razin, A., E. Sadka, and P. Swagel, 2002, The aging population and the size of thewelfare state, Journal of Political Economy 110, 900-918.
[26] Romer, D., 2011, Advanced Macroeconomics (4th edition), Mcgraw-Hill Education.
[27] Wilson, J.D., 1986, A theory of inter-regional tax competition, Journal of Urban
Economics, 19, 296-315.
[28] Wilson, J.D., 1999, Theories of tax competition, National Tax Journal 52, 269-304.
[29] Wilson, J.D. and D.E. Wildasin, 2004, Capital tax competition: bane or boon, Journalof Public Economics 88, 1065-1091.
[30] World Bank, http://data.worldbank.org/indicator/SP.POP.DPND.OL[31] Zodrow, G.R. and P. Mieszkowski, 1986, Pigou, Tiebout, property taxation, and theunderprovision of local public goods, Journal of Urban Economics, 19, 356-370.
[32] Zodrow, G.R., 2003, Tax competition and tax coordination in the European Union,
International Tax and Public Finance 10, 651-671.

30

Appendix
Derivation of the possibility of e < 0 when n

When n = 0, e is

ejn=0 =

From this equation, when 2with respect to n, we obtain

where

@e
=
@n

(31)

[(

1) + 1] + (1)1)2 ng2

1)2 + ( + 1)(

;

(32)

is de…ned as

We di¤erentiate

2(n + 2)22(n + 1)2 +( +1) 2(n + 2)2(n + 1)2)+1) < 0:( +1)(n+1)2

with respect to n to obtain@=@n2( + 1)(1

Substituting n = 0 into)(2+n + n(14 2 (1) +, we can observe

jn=0 =(1)2

1 < 0 hold true, we obtainfor n > 0 because @ =@n < 0 and
When 2)(2
3 + 2):) ) + (1)2 )3 + 2 > 0, ejn=0 < 0 holds true. By di¤erentiating e( + 1)((n + 2)2 f

n2 +If 3 2( + 1)(1
2 (1
((1

0

3 2

1 :

jn=0 < 0. In such a case,

jn=0 < 0.

< 0 holds true

< 0, in turn, implies that @ e=@n < 0.

3 + 2 > 0, ejn=0 < 0 and hence we know that e < 0 for n > 0.

31

Online appendices (not for publication)
Appendix A: First best allocation
In this Appendix, we analyze the …rst best outcome, in which the social planner candirectly choose the amount of young consumption, old consumption, savings, and public
goods, and capital. The social welfare function is
"
M
1
X
X
W =
uio0 Lo0 +t#

Uit Lt ;t=0

i=1

where we assume that the social planner has the same discount rate as individuals. Since
the planner perfectly internalize international externalities, the …rst best optimum with M
symmetric countries is the same as that with M autarkic countries. Then, the objectivefunction of the planner can be written as"

W = M uio0 Lo0 +

1
X

t

#

Uit Lt :t=0

In autarky, the resource constraint is
kt1= (1 + n)kt+1 + cyt +cot+ bt ;
1+n

where bt is the per capita opportunity costs to provide public goods. The …rst orderconditions of the planner’s maximization are
cot :

kt :

cot

(1 + n)h(1 + n)kt1(1 + n)kt+1

cot =(1 + n)

(1 + n)kt1 1(1 + n)kt
bt :(1

cot 1 =(1 + n)

kt1

bt 1

(1 + n)kt+1

cot =(1 + n)bt

i = 0;)kt (1 + n)

(1 + n)kt+1(1 + n)kt1

bt

cot =(1 + n)

bt

= 0;

(2 + n)= 0:bt+Here, we focus our attention on the steady state (k = kt = kt+1 ; c = cot = cot+1 andb = bt = bt+1 ). The …rst order condition can be rewritten as follows:

c

[k 1
k1

k1

(1 + n)(1 + n)k c=(1 + n)
(1 + n) [ (1(1 + n)k)k1]
c=(1 + n)

R(1 + n)k
(1 + n)k c=(1 + n)
32

b

+

k]

k

= 0;

= 0;

(2 + n)(A1)= 0:

Equation (A1) states that the …rst best amount of capital per capita is
k f = [ (1)]1= :(A2)

In contrast, the equilibrium and the second best amount of capital per capita can beobtained from (4) and (11) and is given by27
1=

k =

:

(1 + n)(1 + )(A3)Equations (A2) and (A3) show that k f 6= k :Thus, the equilibrium and the second
best savings level can not coincide with the …rst best level, from which we can con…rmthat the economy in this model brings about dynamic ine¢ ciency as that in Acemoglu(2009), Blanchard and Fischer (1989) and Romer (2011).

Appendix B: Income tax
Suppose now that governments can impose tax on households in addition to capital tax. As
an example, we introduce labor income tax,

it 2 [0; 1], into the baseline model developedin Section 2. Such tax modi…es the individual’s demand (3) as
ciyt =(1it )wit

1+

;

(1

sit =

it )wit1+;

ciot+1 =1+(1 + rit+1 )(1it )wit :(B1)
The budget constraint of the government becomes git =it wit Lit + it Kit = [ it wit + it kit ] Lit ,which, from (4), can be written as"
1
git = itrit + it

1

=

(1

)(1

(1

)=

+ it

)= L [
it it

1=

rit + it(rit + it ) + (1) it ]

1=(rt + it )In the steady state, the per capita capital becomes
k=

1=
(1
)
:
(1 + )(1 + n)This equation implies that
r+
27

=

(1 + )(1 + n)(1(1)

The second best tax rate is obtained by setting (13)= 0.

33

)

#

:

:

Lit

(B2)

Substituting (4), (B1), and (B2) into (6), we obtainUit =ln [ it (rt + it ) + (1

+ (1 + )(1) it ])ln(rt + it ) + (1 + ) ln(1

it )

(B3)
+

ln(1

+

ln

)(1
1+

)=

)(1

)=

uio0 Lo0 +1
X

Lit + (1 + ) ln (1(1 + rt+1 ) +ln(1 + )ln git+1 :The social welfare function is

W IT =

M
X
i=1

"

t

#

Uit Lt ;

t=0

(B4)

which yields
@W IT
=
+
1 + rtLt 1 t 1 @ it
1

(1)(2 + n)
:
(r
+
)
) it
it
it
it + (1Economy with a growing population
In the economy with a growing population, the country i0 s government at period t chooses
it andit to maximize (B3) while regarding rt and future variables as given.

The …rst-

order conditions yield0
yit = 0

and

0
yit =

1+

> 0.

+

Hence, governments have no incentive to tax on mobile capital. Still, we can show thatsuch zero-tax rate on capital is excessively low and an increase in capital tax rate can
improve welfare. By keeping
it = 0 and

@W IT
@ it

it =

it as …xed, an increase in

it a¤ects (B4) if evaluated at0
yit as follows:=
it =0 and

t+1

Lit 1

0
it = yit2+n
1+n+ (1 ++ )(1

If we follow Romer (2011) and Acemoglu (2009) in assuming thatet al. (1999) in assuming that)(1 + n):= 2=3 and Eckstein= 2=3, we observe
@W IT
@ it

> 0:
=0 and0= yit

Under the realistic parameter values, tax competition results in the race to the bottom inan economy with a growing population.

34

Economy with a decreasing population
In the economy with a decreasing population, a government chooses it andit to maximize

uiot while regarding rt and past variables as given. Substituting (4), (B1), and (B2) into
uiot , we obtainuiot = ln
= ln

1+
1+

)(1

(1

)= L [
it it(1 + rit ) (1it 1 ) wit 1 +ln(1 + rit ) (1it 1 ) wit 1 +ln [ it (rit + it ) + (1ln (rt + it )1= +

)(1

ln(1

)=

(rit + it ) + (1(rt + it )1=) it ]

Lit :Here, we know that uiot is monotonously increasing init .

To avoid the non-existenceof equilibrium rate of return on savings, we assume the upper-bound of labor income tax
2 (0; 1).28 The government determines tax rates as0oit =By keeping

1

r (1
(1

as …xed, an increase in

)

0
oit =

and

)

it , evaluated at@W IT
Lt 1 t 1 @ it:0it = oit andit =, results in29

10
it = oit and

it =

(1
) (2 + n)0 ) + (10
(rt + oit
) oit=

1 + rrHence, we readily know this is positive if and only ifwhere e0 is de…ned as> e0 ;e00(1 + n)(1 + ) + oit> 0:
2 (1 + r)(2 + n)

Therefore, if governments can impose tax on households, capital tax competition underdecreasing population results in an ine¢ ciently high (resp. low) capital tax rate if thehousehold’s preference for public good consumption is su¢ ciently small i.e.,large, i.e.,
28
29

> e).

When = 1, the equilibrium rate of return on savings diverges to in…nity.
Note again that (11) implies that @r=@ = 1.

35

< e0 (resp.

) it ]

Appendix C: Public inputs
Suppose that individuals obtain utility only from private good consumption, c. We specify
the utility function as follows:
Uit = ln ciyt +whereln ciot+1 ;(C1)(2 (0; 1)) is the time discount rate. From utility maximization, we can obtain thefollowing equations:ciyt =

wit
;
1+

sit =

wit
;
1+

ciot+1 =1+(1 + rit+1 )wit :Firms produce the numéraire by using labor and capital under constant returns toscale. Here, we assume that public inputs raise productivity of …rms. We employ a
Cobb-Douglas production function:
yit = git Lit Kit1
whereis a positive constant that represents the public input elasticity of output. We

assume that

> . Pro…t maximization yields
1
wit = git kit
;

where

;

gt (1
) 1=
;
rit + itkit =represents the capital tax rate. We substitute git =
git =

=(

1=(

1

)

it

)

it Kit into (C2) to get

=(

Lit

rit + it

(C2)

)

:

(C3)

The social welfare function is
WP =

M
X
i=1

"

uio0 Lo0 +

1
X
t=0

t

#

Uit Lt ;which yields@W P=
Lt 1 t 1 @ it
1+

1 + rit+(1 + n)(1 + ) 1
itln(1 + rit ) + (1 + n) (1 + )ln it(1 + )1+ln(rit + it ) :Economy with a growing population
We start with the case of n > 0, which implies that the population is increasing. In thiscase, because Lit > Lit+1 , young individuals represent the majority and the government36

maximizes Uit . Plugging (3), (C2), and the government budget constraint (C3) into (C1),
we obtain
Uit = (1 + )
+

ln

1+

ln it

(1 + )+ ln

1
1+

The …rst-order condition regarding1+ln(rit + it ) +h+ (1 + ) ln (1ln(1 + rit+1 )

)(1

)=

"=(

Lit

")

i

:

yields
it =

1

rt :

(C4)

Economy with a decreasing population
Next, we consider the case of n < 0, which implies that the population is decreasing. In
this case, because Lit < Lit+1 , old individuals represent the majority and the government
maximizes uiot . Plugging (3) and git =uiot = ln

1+

it Kit into (2), we obtain+ ln(1 + rit ) + ln wit 1 :In this case, the utility of the old individual does not depend on

it , implying that old

agents are indi¤erent to any tax rate. Therefore, policies that maximize Uit are supportedby the majority, and the …rst-order condition of the maximization again yields (C4).
Steady-state
We assumed symmetric countries, which implies that all countries have the same capital
holdings at period 0, the same population size, and the same population growth rate,
implying that Lit = Ljt (i 6= j) for all t. From Lit = Ljt (i 6= j), we obtain cit =cjt = ct and kit = kjt = kt . We focus on the steady-state equilibrium, wherein the levelof individual’s consumption, ct , and capital per capita, kt , are constant over time (ct =ct+1 = c and kt = kt+1 = k ). Then, from (4), we readily know that rt + t = rt+1 + t+1 .
Using this, the capital market clearing condition (5) can be rewritten asMg (1) 1= XLit =grt + t
1+
j=1

g (1
) (1
rt + t)=M
X

Lit 1 :

j=1

From Lit+1 = (1 + n)Lit for all countries, we obtainrt + t =(1 + n)(1 + )(1

37

)

:

(C5)

Equilibrium and its e¢ ciency
From (C4) and (C5), we obtain the equilibrium capital tax rate in both economies asfollows:y2 =An increase init , evaluated at(1 + n)(1 + )(1
(1
+ )

)

:

it = y2 results in@W P
Lt 1 t 1 @ it1=
it = y21( + 1)"(n + 1)
;
(1
+
r)
(1
+ ")
y2

(C6)

where
" [1
The sign of(1) ] + n( + 1)(1

)2

(1

)(

2 + 1):is positive if and only if" > "e;

where
"e

(1

)(

2 + 1)1 (1n( + 1)(1))2:

Thus, when the public input elasticity of output is higher than "e, (C6) becomes positive

and we observe the race to the bottom.

References[1] Acemoglu D., 2009, Introduction to Modern Economic Growth, Princeton University Press.
[2] Blanchard, O. and Fischer, S., 1989, Lectures on Macroeconomics, The MIT Press.
[3] Eckstein, Z., P.S. Mira, and K. Wolpin, 1999, A quantitative analysis of Swedish
fertility dynamics: 1751-1990, Review of Economic Dynamics 2, 137-165.
[4] Romer, D., 2011, Advanced Macroeconomics (4th edition), Mcgraw-Hill Education.

38

capital tax 𝜏𝜏𝑡𝑡

savings 𝑠𝑠𝑡𝑡−1

firm

capital 𝐾𝐾𝑡𝑡labor 𝐿𝐿𝑡𝑡

generation 𝑡𝑡

public goods 𝑔𝑔𝑡𝑡

rate of return on
savings 𝑟𝑟𝑡𝑡
wage 𝑤𝑤𝑡𝑡

consumption 𝑐𝑐𝑦𝑦𝑦𝑦

young periodsavings 𝑠𝑠𝑡𝑡

capital 𝐾𝐾𝑡𝑡+1

consumption 𝑐𝑐𝑜𝑜𝑜𝑜+1
old period

generation 𝑡𝑡 + 1

Figure 1 : OLG structure of the model

young periodAn economy with a growing population

An economy with a decreasing population

(a-1)

(a-2)

(b-1)

(b-2)

𝛾𝛾=0.75𝛾𝛾 = 0.6

Figure 2: Possibility of the race to the top
Notes: We set 𝑛𝑛 = 0.1 for an economy with a growing population and
𝑛𝑛 = −0.1 for an economy with a decreasing population. For each case,we consider two alternative values of 𝛾𝛾 (𝛾𝛾 = 0.75 and 𝛾𝛾 = 0.6). The
shaded areas represent combinations of 𝛼𝛼 and 𝛽𝛽 that result in therace to the top.