Discussion Papers In Economics
And Business

Demographics and tax competition in political economy

Tadashi Morita, Yasuhiro Sato and Kazuhiro Yamamoto

Discussion Paper 16-13

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANDemographics and tax competition in political economy

Tadashi Morita, Yasuhiro Sato and Kazuhiro Yamamoto

Discussion Paper 16-13

May 2016

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANDemographics and tax competition in political economy
Tadashi Moritay

Yasuhiro Satoz

Kazuhiro Yamamotox

May 10, 2016

Abstract
We examine possible impacts of demographics on outcomes of capital tax competition in political economy. For this purpose, we develop an overlapping generations
model wherein public good provision …nanced by capital tax is determined by 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 thatthe race to the bottom is likely to emerge in the population growing economy whereas
the race to the top might emerge in the population decreasing economy.

JEL Classi…cation: H20, J11,
Keywords: tax competition, majority voter, …scal externality, political externality

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, Trade andIndustry (RIETI). This work was also supported by JSPS KAKENHI Grant Number 15H03348, 15H03344
and 16H03615. We thank seminar participants at 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

1

Introduction

This paper investigates the possible impacts of demographics on the results of capitaltax competition in political economy. Given the drastic increases in capital ‡ows acrosscountries and regions, many scholars have analyzed the e¤ects of globalization in thecapital market over the past few decades. One of the most important strands in this…eld is the theory of capital tax competition, which has a long history dating back at
least to Zodrow and Mieszkowski (1986) and Wilson (1986).1 Researchers in this strand
investigated the role of 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 capitaltaxation causes capital ‡ight, which increases the tax base in other countries and causes
positive …scal externalities. This result is known as the race to the bottom and has attracted
much attention (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, if we consider atthe 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 capital tax competition.
To analyze the e¤ects of demographics on capital tax competition, we develop a capitaltax competition model involving the overlapping generations structure wherein policies are
determined by majority voting. In our model, all individuals live for two periods, young
and old, and the population grows with an exogenous 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 themwith capital tax. A government chooses the level of 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 and1

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

public goods. Governments at a certain period cannot commit to future policies whenmaximizing the majority’s utility. This brings about a discrepancy between the government’s objective and the life-time utility of individuals. Hence, capital taxation causesinter-generational externalities, which we call the political externality. Such an externality
is positive when the individual’s preference for public good consumption is strong andnegative when it is weak in both population growing and decreasing economies.
Moreover, because we assume free mobility of capital between countries, capital taxation causes a positive externality, which we call the …scal externality, as is standard inmodels of capital tax competition. Moreover, because young individuals’ wage income
increases with capital inputs in the country whereas old individuals savings income does
not depend on them, young individuals prefer a lower capital tax rate than old individuals.
Therefore, a government has a stronger incentive to o¤er tax-cuts on capital and this results in a stronger …scal externality in a population growing economy than in a population
decreasing economy.
The relative signi…cance of political and …scal externalities characterizes the (in)e¢ ciencyof the equilibrium capital tax rate. When individuals strongly prefer public good consumption, the political externality becomes positive. Combined with positive …scal externality,
it yields 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 political externality becomes negative, yielding the possibility of an ine¢ ciently high capital tax rate, which wecall the race to the top. Furthermore, because the …scal externality is weaker in a population decreasing economy than in a population growing economy, we can show that theformer is more likely to exhibit the race to the top than 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 population
growing countries and ine¢ ciently high in population decreasing countries.
Several existing papers have investigated capital tax competition in political economy
models. Persson and Tabellini (1992), Borck (2003), Lockwood and Makris (2006), Grazzini 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. 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 high. This political economy e¤ect sometimes overwhelms thetax competition e¤ect, which implies that the equilibrium capital tax rates become inef…ciently high. Our paper also studies the tax competition in a political economy model.
However, in our model, individuals are not heterogeneous with 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‡ict, 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.

3

These papers analyzed the closed economy models and focused on the e¤ects of income
distribution on growth rates. Our paper also constructs an overlapping generation modelin political economy. However, we consider an open economy wherein capital is mobile
among countries and focus on the e¤ects of demographics on the results of capital tax
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 consideringinternational di¤erences in demographic structure.4
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 periods. At the …rst (young) period, an individual works to earn 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/he exits the economy. We
call a cohort of individuals who are young at time t as generation t. 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 growthrate, n, is exogenous, and in the baseline model, we assume symmetric countries so thatn is common to all countries. Hence, we have Lit+1 = (1 + n)Lit .

2.1

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 ;(1)where (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 +ln git :(2)

is a positive constant that represents the preference for public good consumption. Budget
constraints are given by
wit = ciyt + sit ;(1 + rit+1 )sit = ciot+1 ;where wit , sit , 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 in4

See Ogawa et al (2016) for a brief survey of the existing studies on asymmetric tax competition.

4

generation t inelastically supplies her/his labor endowments, which are normalized 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.
We assume perfectly competitive goods, labor, and capital markets. We employ a CobbDouglas 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
;

1

kit =

1=

rit + it;(4)

where 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

Kit =

i=1

M
X

sit 1 Lit 1 :i=1Because capital is assumed to be freely mobile among countries, the interest rate 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, the 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 utility5

of the majority at period t.5 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 .6 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.7 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]

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
Ut =
+
where

ln it

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

ln Lit +is de…ned asln(1 + ) ln(1 + ) +The …rst-order condition regarding
it =

2.4.2

+ (1 + )(1ln(1)1= + (1 + ) ln (1

)(1

)=

:

yields
(1 +

+ )(1

)

rt :

(7)

Economy with a decreasing population

Next, we consider the case of n < 0, which implies that the population is decreasing andaging. In this case, because Lit < Lit+1 , old individuals represent the majority and thegovernment maximizes uiot . Plugging (3) and git = it Kit into (2), we obtainuiot =ln it

ln(rt + it ) + ln(1 + rt ) +ln Lit + ln wit 1 + ;(8)5In our setting wherein only two types of individuals exist, this corresponds to maximize the utility ofthe median voter.
6
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 .
7
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 population growing and decreasing economies. Hence, theeconomy would have equilibrium ine¢ ciencies that depend on demographics as will be shown in this paper.

6whereis de…ned asln

The …rst-order condition regarding

+

1+

ln(1

)1= :

gives
it =

1

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 . The former e¤ect appears in both Uit and uiot whereas the latter appears onlyin Uit . 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, thegovernment has a weaker incentive to tax capital in a population growing economy thanin a population decreasing economy.8

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
(10)
kt =
k1 ;
(1 + )(1 + n) t 1
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 , andkt = kt+1 = k ).9 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
8This 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 the capital endowment of the median voter. In our framework, the medianvoter is young individuals in the population growing economy and old individuals in the population decreasing economy. Because young individuals have no capital, the government has a weaker incentive totax capital in the population growing economy than in the population decreasing economy.
9
Note here that (10) has a unique steady state. Combined with (3) and (4), this implies that there
exists a unique consumption level.

7

capital). We can solve (7) and (11) to derive equilibrium and r in a population growing
economy whereas we can use (9) and (11) to obtain equilibrium and r in a population
decreasing economy.

2.7

Possible externalities

Here, we summarize the possible externalities emerging in our framework. The optimal
tax rate maximizes the life-time utility given by (6), implying that it is identical to thegovernment’s objective function in a population growing economy. However, in the optimal, the e¤ects of capital taxation, t , on the global price, rt , must be considered indetermining t , whereas in the equilibrium, the government treats rt as given. Such adi¤erence causes a distortion.10 More speci…cally, capital ‡ight caused by capital taxation
enlarges the tax base in other countries, resulting in a positive externality. This is standard
in tax competition models and we call it the …scal externality. Note that rt appears in
the term f[ + (1 + )(1)] = g ln(rt + it ) of the government’s objective function (6)in a population growing economy and in the term ( = ) ln(rt + it ) of the government’s
objective function in a population decreasing economy. From this, we know that a given
over-evaluation regarding changes in r + by the government causes smaller changes in theobjective function, i.e., the …scal externality is weaker in a population decreasing economy
than in a population growing economy.
Our framework of political economy yields additional externalities. A government
chooses its current policies to maximize the majority’s utility and cannot commit to future
policies. This implies that the government in a population growing economy does notrecognize the e¤ects of the capital taxation on future variables i.e., terms ln(1 + rt+1 ) +ln git+1 in (6), whereas such e¤ects must be internalized in the optimal. Ignoring thee¤ects on these terms implies that the government ignores the e¤ects on old individual’sutility. Put di¤erently, capital taxation causes an inter-generational externality, which
we call the political externality in a population growing economy. Note here that capitaltaxation decreases r and increases g. Hence, when is su¢ ciently small, the former e¤ect
on the old individual’s utility dominates the latter e¤ect and the political externality
becomes a negative externality, and when is su¢ ciently large, the opposite holds trueand it becomes a positive externality.
In a population decreasing economy, the government’s objective is to maximize theold individual’s utility, (8). Hence, the government ignores its taxation e¤ects on theterms ln it f[ + (1 + )(1)] = g ln(rt + it ) in the young individual’s utility, (6).
Thus, we again observe an inter-generational externality, which is the political externality
in a population decreasing economy. Capital taxation in a particular country, i, cancause a positive externality by increasing ln it for young individuals within country i anda negative externality by increasing rt + jt (j 6= i) for young individuals in the othercountries.11 When is su¢ ciently large, the former e¤ect dominates the latter e¤ect, andthe political externality in a population decreasing economy becomes a positive externality.
When is small, the opposite holds true and the political externality becomes a negative
externality.
10
Even if a government does not regard r as given, it considers the e¤ects of on 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.
11
Because a change in it does not a¤ect rt + it , the negative externality is not relevant for young
individuals within country i.

83Equilibrium and its e¢ ciency

3.1

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 capital, rt , into consideration in addition to other local variables. Moreover, because rt + t is determined bypast variables, the government now considers that any increases in are exactly o¤set bydecreases in r (@rt =@ t = 1).
In a population growing economy, Uit given by (6) is an increasing function of t(@Uit =@ t = = t > 0) under consideration of @rt =@ t = 1. Therefore, the government
sets its capital tax rate as high as possible (as long as rt 0), which, combined with (11),results in12
(1 + n)(1 + )(1
)
im
:
(12)
y =
The superscript im represents the case of capital immobility. In a population decreasing economy, the …rst-order condition for the maximization of uiot with respect to t(@uiot =@ t = = t 1=(1 + rt ) = 0) yields
imo =(1 + roim ):(13)

Comparing @Uit =@ t with @uiot =@ t , we know that the government in a population decreasing economy sets a lower tax rate than in a population growing economy, if capitalis immobile between countries. Because changes in rate of return on savings absorb thecapital tax e¤ects (@rt =@ t = 1), capital taxation does not a¤ect capital per capita, kit ,and hence, the wage rate, wit . The government considers such a relationship, implyingthat a government in a population growing economy cares only about the level of public
good provision in maximizing Uit whereas a government in a population decreasing economy considers decreases in returns from savings when maximizing uiot as well. Thus, agovernment in a population decreasing economy is more tentative in taxing capital thana government in a population growing economy if capital is immobile.13
With tax rates in hand, we can examine the e¢ ciency properties of these tax rates bylooking at the e¤ects of a coordinated increase in over time. Such e¤ects can be derived
by plugging t = t+1 = into the individual’s life time utility (6), taking its derivativewith respect to , and evaluating it at yim or at oim . Plugging t = t+1 = into (11), we
can see that the return on savings does not depend on time: rt = rt+1 = r. Equation (6)12We obtain (11) from (5) by setting M = 1.
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.
13

9

then becomes
Ut = (1 + ) ln

(1 + )(1 ++ (1 + ) ln Lt +)ln(1 + n) +ln(r + ) +ln(1 + r)+ (1 + ) ln(1)1= :Di¤erentiating U with respect to , we obtain@Ut=@
By evaluating (14) at=(1 + )
1+r

:

(14)

im
y , we obtain

@Ut
@

=
= yim

(1 + n)(1)

1 :Because cannot be higher than yim because of the non-negative constraint of r, weknow that in a population growing economy, the equilibrium tax rate is optimal if thepreference for public good consumption is su¢ ciently large (i.e.,(1 + n)(1)= ),and ine¢ ciently high otherwise (i.e., < (1 + n)(1)= ). By evaluating (14) at = oim ,we obtain@Ut@=(1 + )(1 + roim )

=

1
> 0:
1 + roim= oim

1 + roim

Hence, we observe an ine¢ ciently low capital tax rate in a population decreasing economy.
Proposition 1 Suppose immobility of capital between countries. Then, the capital tax
rate is optimal or ine¢ ciently high in a population growing economy whereas it isine¢ ciently low in a population decreasing economy.
When capital is immobile between countries, no …scal externality exists and therefore
only the political externality is relevant. In a population growing economy, the politicalexternality is positive when the preference for public good consumption, , is large andnegative when is small. However, when is large, both the optimal and equilibrium
requires the capital tax rate to be the maximum possible rate, implying that the resulting
equilibrium tax rate becomes identical to the optimal one. When is small, the politicalexternality is negative, yielding an ine¢ ciently high capital tax rate in equilibrium. In a
population decreasing economy, only the positive political externality is relevant, making
the equilibrium tax rate ine¢ ciently low.

3.2

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 froma population growing economy. From (7) and (11), we obtain the equilibrium capital tax
rate as follows:y =(1 + n)(1 + )(1[ + (1 + )(1
10

)
:
)]

(15)

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

(16)

Similarly to the previous section, we examine the equilibrium e¢ ciency properties bylooking at the e¤ects of a coordinated increase in among countries and over time. Suche¤ects can be derived by plugging t = t+1 =into (6), taking its derivative withrespect to , and evaluating it at y . Plugging t = t+1 = into (11), we can see thatrt = rt+1 = r and @r=@ = 1. Di¤erentiating U with respect to , we again obtain (14).
By evaluating (14) at = y (see also (11)), we know that
@Ut
@
where

=
= y

(1 + )y (1 + ry )h) (1 + )2 (1 + n)

(1

(1 + n) (1)+

Hence, we can see that

1

1
"

@Ut
sgn
@

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

= y

#

)

)]

;

(17)

n (3 + ) +
> 0:

i

;

= sgn [ ] :Because we readily know that@= (1) > 0;@
implying that is linearly increasing in . Therefore, we have a unique e that satis…es= 0, and > 0 (resp. < 0) for > e (resp. < e). Such e is given bye(1 + n + 2 ) +n (3 + )(1 + )2 (1 + n)

:

(18)

Therefore, (17) is positive if and only if> e:Proposition 2 Capital tax competition in a population growing economy results in an
ine¢ ciently low (resp. high) capital tax rate if and only if the preference for public
good consumption, , is larger than e (resp. smaller than e).
In a population growing economy, we have the positive …scal externality and positive(resp. negative) political externality when is large (resp. small). If is su¢ ciently large
to satisfy > e, then the overall externality becomes positive, and capital tax competition
yields an ine¢ ciently low capital tax rate, i.e., we observe the race to the bottom. In
contrast, when the opposite holds true ( < e), the negative political externality makes
the capital tax rate ine¢ ciently high, and this externality dominate the …scal externality,
resulting in an ine¢ ciently high capital tax rate, which we call the race to the top.
We can state Proposition 3.2 in a di¤erent way by focusing on the labor share in
production, . The de…nition of e shows that e 0 holds true ife, where e is de…nedas(1 + )2 (1 + n)e:
1 + n + [2 + n (3 + )]11

Corollary 1 Capital tax competition in a population growing economy results in an inef…ciently low capital tax rate, ife.
A su¢ ciently small implies higher signi…cance of capital in production. This implies
that the …scal externality is prominent and the tax competition is likely to result in an
ine¢ ciently low capital tax rate.

3.3

Economy with a decreasing population

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

o =

ro =

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

)2

;

(19)
:

To examine the e¢ ciency of equilibrium, we evaluate (14) atresulting in
@Ut
@

=
= o

[
(1

=

o

(see also (11)),(1 + n) ]:) (1 + n)This is positive if and only ifwhere b is de…ned as

b

> b;

(1 + n):(20)

Proposition 3 Capital tax competition in a population decreasing economy 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 (resp. smaller than b).
In a closed economy, the equilibrium capital tax rate becomes ine¢ ciently low. However, when capital is mobile, we have the negative political externality. When is suf…ciently small, its e¤ect becomes prominent, yielding the possibility of the race to thetop.14

3.4

Possibility of the race to the top

Next, we question when an ine¢ ciently high capital tax rate, that is, the race to the top, islikely to emerge. Let us start from a population growing economy. Simply di¤erentiatinge, which is the threshold value in this economy and given by (18), with respect to n, we, where we de…ne asreadily know that @ e=@n > 0 if and only if
2++1:(1 + )214

We can observe that b > 0 if and only if n > 1, which always holds true. Hence, in a population
decreasing economy, we always have the possibility of an ine¢ ciently high capital tax rate.

12

Thus, when the labor share is large (i.e.,), a larger population growth rate increases
the threshold value, e, and when the labor share is small (i.e., < ), it decreases e.
Moreover, because we can verify that ejn=0 < 0, we observe e < 0 and hence the race to, e < 0 holds true if
e (seethe bottom whenever < . Even in the case ofCorollary 1). Such a possibility can emerge because we can show that < e.15
In addition, we can readily see that @ =@ < 0 and j =1 = 0:75, implying thate>
0:75. Karabarbounis and Neiman (2014) estimated the global labor share andshowed that it has exhibited a relatively steady downward trend from 0:64 to 0:59 during
the past several decades. Their paper also presented the labor share for OECD and nonOECD countries, which showed that the labor share has been under 0:7 in most cases.
Thus, it is reasonable to say that the labor share, , is smaller than 0:75, implying thatin the real world, we observe that e < 0 i.e., the …scal externality is large and capital tax
competition always results in the race to the bottom in a population growing economy.
Let us move to a population decreasing economy. By di¤erentiating b, which is thethreshold value in this economy and given by (20), with respect to n, we can see that
@ b=@n > 0 always holds true. In a population decreasing economy, as the population
growth rate rises, a high capital tax rate causes larger welfare losses of young individuals.
Therefore, the higher the population growth rate, the more likely it is that the capital tax
rate is ine¢ ciently high. In addition, we can see that b > 0 always holds true becausen > 1, i.e., there always exists a case wherein the capital tax rate is ine¢ ciently high ina population decreasing economy.
If we compare a population growing economy with a population decreasing economy,
we observe a possibility that a government in a population growing economy sets anine¢ ciently low tax rate whereas a government in a population decreasing economy mightset an ine¢ ciently high tax rate. This can be shown in relation to Corollary 1: when < e,we know that e < 0 holds true and we observe the race to the bottom in a population
growing economy. However, because e, and hence b do not depend on , a government ina population decreasing economy imposes an ine¢ ciently high tax rate on capital if andonly if < b.16

Proposition 4 Suppose that < e. Then, capital tax competition always yields an ine¢ ciently low tax rate in a population growing economy. However, it results in an
ine¢ ciently high tax rate in a population decreasing economy if and only if < b.
As shown in Section 2.4.2, when capital is mobile, the government’s incentive to tax
capital is stronger in a population decreasing economy than in a population growing economy because of a weaker …scal externality. This generates a higher possibility of the race
to the top in a population decreasing economy than in a population growing economy,
which comes to the surface when and are su¢ ciently small.
Figure 2 summarizes the results of Proposition 4. Figure 2 (a) represents the casewherein we always observe the race to the bottom in a population growing economy(i.e., when n > 0) whereas the race to the top might emerge in a population decreasing
15

Because @e=@n < 0, we know that e > limn!1 e. Noticing that
lim e =

n!1

(1 + )2;
1 + (3 + )simple comparisons yield that < limn!1 e < e.
16
Note that in deriving e, we need to use n in a population growing economy, whereas we need to use n
in a population decreasing economy in deriving b for a given .

13

economy (i.e., n < 0). Figure 2 (b) depicts the case wherein we have both possibilities inboth population growing and decreasing economies.
[Figure 2 around here]
Figure 2 (a) describes the case wherein < e. When is small, the elasticity of thewage rate to the capital tax rate is large, and the …scal externality is large. In such a case,the government always sets an ine¢ ciently low tax rate. In Figure 2 (b), we haveeand @ e=@n > 0. Hence, e becomes positive when n is su¢ ciently large. In this case,the elasticity of wage to the capital tax rate is small and the …scal externality is small.
Therefore, we might observe the race to the top in a population growing economy.
When < e, we know that @ e=@n < 0 holds true, i.e., capital tax competition ina population growing economy always results in the race to the bottom and a higher
population growth rate makes the tax rate less likely to be ine¢ ciently high. In contrast,
we know that @ b=@n > 0 holds true in a population decreasing economy. Therefore, when
the labor share, , is su¢ ciently small, if we increase the population growth rate, n, froma negative to a positive value, then the possibility of the race to the top …rst increases
then drops to zero when n reaches zero as described in Figure 2 (a).
Such discontinuity in the relationship between the population growth rate and thepossibility of the race to the top re‡ects the discontinuity in the relationship between the
population growth rate and the equilibrium tax rate. As we can see from (15) and (19),the equilibrium tax rates, y and o , rise with the population growth rate, n. Note thatwhen n < 0, we observe o whereas when n > 0, we observe y . Hence, as n decreases
from a positive value to a negative value, we have a discontinuous change from y to o
at n = 0. Moreover, if is su¢ ciently small, then o jn=0 > y n=0 holds true, implyingthat the equilibrium tax rate discontinuously increases at n = 0. Hence, if we compare
a population growing economy with a population decreasing economy, the latter is more
likely to exhibit the race to the top than the former.

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
rt + ht

1
rt + lt
(1

)=

1=

Llt

Lht
+
1 + nh(21)
1
rt + lt

(1

)=

#
Llt
;
1 + nlwhere P
Lht and Llt are total
sizes in each group of countries and de…ned as
PMpopulation
Mh
l
Lht
L
and
L
L
.
it
jt
lt
i=1
j=1
In this extended setting, we have the following three cases: (i) all countries have a
growing populations (i.e., nh > nl > 0), (ii) all countries experience population decreases
14

(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.

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 (21) to obtain(1 + nw )(1 + )(1)
rt + t =;(22)
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 (22) 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 (22) yield(i)=r(i) =(1 + nw )(1 + ) (1);
[ + (1 + )(1
)]

(23)

(1 + nw )(1 + ) (1)2 (1 + + ):[ + (1 + )(1)]

The superscript (i) represents case (i). We can obtain (23) by replacing n with nw in yand ry (given by (15) and (16)) 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 (17) evaluated at = (i) is positive if and only
if > e(i) .

Proposition 5 Suppose all countries have 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, , is larger than e(i)(resp. smaller than e(i) ).

Case (ii): economy with a decreasing population

When all countries have decreasing population, each government maximizes the old individual’s utility, uiot , and the …rst-order condition yields (9), implying that ht = lt = t .
From (9) and (21), 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

15

;

(24)
:

The superscript (ii) represents case (ii). Again, we can obtain (24) by replacing n with nw
in o and ro (given by (19)) in the baseline framework. Hence, welfare properties become
similar to those obtained in the baseline framework: (14) evaluated at = (ii) is positive
if and only if > b(ii) , where we replace n with nw in b and denote it by b(ii) .

Proposition 6 Suppose all countries have 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): population growing countries v.s. population decreasing countries

In this case, young individuals represent the majority in population growing (group h)countries whereas old individuals do so in population deceasing (group l) countries. Then,the equilibrium tax rates become
ht =
lt =

(1 +
1

+ )(1

)

rt ;

(25)

rt :

Comparing the two tax rates, we know that the equilibrium tax rate is lower in group hcountries than in group l countries. From (25), we can derive+ (1 + )(1(1 + + )(1
1
rt + lt =
rt ;
1

)
rt ;
)

rt + ht =(26)

implying thatrt + ht =+ (1 + )(1
1+ +)(rt + lt ) :We substitute this into (21) to rewrite the capital market equilibrium condition as
1=
(1
) 1=
1
Lht +
Llt
rt + lt
rt + lt
(
(1 )=
(1
)
Lht
+
=
1+
rt 1 + lt 1
1 + nhwhere is de…ned as
and Ht as

(1 +

zt

rt + lt ;(1 + ) (1)

1= (1 + n)=

rt 1 + lt 1

+ ) = [ + (1 + )(1

(1

Ht

(1

1

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

this becomes
zt+1 = Ht1

16

zt1

:

(27)

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 (27), this yields the steady state value of z asz = H (1)=:

Plugging this into (26), 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

+
)=

;

(28)

)=

;

:

We substitute (28) into (14) to obtain
@U
@
@U
@

=

(1 + )(1 +

(iii)

=
(iii)

1

H

= h

= l

+ )

1

1 + (11:

1H

1 + (1;)H)H

From this, we know that @U=@ j = (iii) > @U=@ j = (iii) , resulting in the followinghproposition.17

lProposition 7 If group l countries set an ine¢ ciently low capital tax rate, then so
do group h countries ( @U=@ j = (iii) > @U=@ j = (iii) > 0). If group h counlhtries set an ine¢ ciently high capital tax rate, then so do group l countries ( 0 >@U=@ j = (iii) > @U=@ j = (iii) v). Additionally, there is a possibility that grouphlh countries set an ine¢ ciently low capital tax rate whereas group l countries set anine¢ ciently high capital tax rate ( @U=@ j = (iii) > 0 > @U=@ j = (iii) ).
hlThis result is consistent with that shown in Proposition 4. Put di¤erently, we again…nd that population decreasing countries are more likely to exhibit the race to the top.

4.2

Welfare di¤erence between asymmetric countries

We …nally determine the group of countries that gain from tax competition in political
economy. In so doing, we assume that countries in each group have the same population17
In fact, we have the second case when is large. When is su¢ ciently small, we have the …rst andthe last cases. In fact, if we set = 1=2, the …rst case holds true when is su¢ ciently large and the lastcase holds true when is su¢ ciently small.

17

b kt
size (i.e., Lit = L
is written as

Lkt =Mk ). The indirect utility of an individual in a group k country(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 )) :(29)

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

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 )) :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 (25) that ht < 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 that ht < lt , the second term becomes positive.

5

Summary and discussions

In this paper, we developed an overlapping generations model wherein public good provision …nanced by capital tax 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 provide us
a new viewpoint in policy debates regarding competition for capital. Particularly, because18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.18
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 a population growing economy would …nance its expendituresolely by labor income tax to prevent capital ‡ight, whereas the government in a population
decreasing economy 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 showthat the equilibrium capital tax rate is ine¢ ciently low in a population growing economy
whereas it is so in a population decreasing economy if and only if the preference for public
good consumption is su¢ ciently large, implying the possibility of the race to the top. In
this sense, the introduction 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 a case, the positive externality associated to capital taxation is further strengthened
because public inputs and capital are complementary in production. Consequently, wealways observe the race to the bottom in a population growing economy. Moreover, sinceeach government regards the rate of return on savings as given, it considers that thecapital tax rate does not a¤ect the old individual’s welfare. This induces each government
to set the capital tax rate to maximize the young individual’s welfare even in a population
decreasing economy. Hence, we observe an ine¢ ciently low tax rate in this case as well.
Hence, the introduction of public inputs would make the race to the bottom more likely
to emerge.

References[1] Alesina A., and D. Rodrik, 1994, Distributive politics and economic growth, Quarterly
Journal of Economics 109, 465-490.
[2] Borck, R., 2003, Tax competition and the choice of tax structure in a majority voting
model, Journal of Urban Economics, 54, 173-180.
[3] Fuest, C. and B. Huber, 2001, Tax competition and tax coordination in a medianvoter model, Public Choice, 107, 97-113.
[4] Grazzini, L. and T. van Yprersele, 2003, Fiscal coordination and political competition,
Journal of Public Economic Theory, 5:2, 305-325.
[5] 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.
[6] Karabarbounis, L. and B. Neiman, 2014, The global decline of the labor share, Quarterly Journal of Economics 129, 61-103.
[7] Lockwood, B. and M. Makris, 2006, Tax incidence, majority voting and capital market
integration, Journal of Public Economics, 90, 1007-1025.
18

Formal analyses on these extensions are available in online appendixes.

19[8] Mateos-Planas, X., 2010, Demographics and the politics of capital taxation in a lifecycle economy, American Economic Review 100, 337-363.
[9] OECD, 1998, Harmful Tax Competition: An Emerging Global Issue.
[10] OECD,
http://www.oecd-ilibrary.org/education/trends-shaping-education2013/median-age-going-up-into-the-next-century_trends_edu-2013-graph43-en[11] Ogawa, H., Y. Sato, and T. Tamai, 2016, Who gains from capital market integration: Tax competition between unionized and non-unionized countries, forthcoming
in Canadian Journal of Economics.
[12] Persson, T. and G. Tabellini, 1992, The politics of 1992: …scal policy and European
integration, Review of Economic Studies 59, 689-701.
[13] Persson, T. and G. Tabellini, 1994a, Representative democracy and capital taxation,
Journal of Public Economics, 55, 53-70.
[14] Persson, T. and G. Tabellini, 1994b, Is inequality harmful for growth? American
Economics Review, 84, 600-621.
[15] 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.
[16] Razin, A., E. Sadka, and P. Swagel, 2002, The aging population and the size of thewelfare state, Journal of Political Economy 110, 900-918.
[17] Sato, M. 2003, Tax competition, rent-seeking and …scal decentralization, European
Economic Review 47, 19-40.
[18] Wilson, J.D., 1986, A theory of inter-regional tax competition, Journal of Urban
Economics, 19, 296-315.
[19] Wilson, J.D., 1999, Theories of tax competition, National Tax Journal 52, 269-304.
[20] Wilson, J.D. and D.E. Wildasin, 2004, Capital tax competition: bane or boon, Journalof Public Economics 88, 1065-1091.
[21] World Bank, http://data.worldbank.org/indicator/SP.POP.DPND.OL[22] Zodrow, R.G. and P. Mieszkowski, 1986, Pigou, Tiebout, property taxation, and theunderprovision of local public goods, Journal of Urban Economics, 19, 356-370.
[23] Zodrow, G.R., 2003, Tax competition and tax coordination in the European Union,
International Tax and Public Finance 10, 651-671.

20

Online appendices (not for publication)
Appendix A: 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 :(A1)
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 )=
1=
1
1
git = it
+ it
Lit
(A2)
rit + itrit + it
=

(1

)(1

)= L [
it it(rit + it ) + (1) it ]

1=(rt + it ):Substituting (4), (A1), and (A2) into (6), we obtainUit =ln [ it (rt + it ) + (1) it ]+ (1 + )(1)

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

it )

(A3)
+

ln(1

+

ln

(1

)=

)
1+

(1

Lit + (1 + ) ln (1(1 + rt+1 ) +))=ln(1 + )ln git+1 :Economy with a growing population
In the economy with growing population, the country i0 s government at period t chooses it
and it to maximize (A3) while regarding rt and future variables as given. The …rst-orderconditions yield
0
0
> 0.
and
yit = 0yit =1+ +Hence, governments have no incentive to tax on mobile capital. Still, we can show thatsuch zero-tax rate on capital is excessively low and a coordinated increase in capital tax
rate can improve welfare. To see this, substitute it = it+1 = and it = it+1 = into(A3) to obtain
Uit = (1 + ) ln [ (r + ) + ]
+

ln(1

)(1

)=

+ (1 + )(1

)

)(1

)=

Lit + (1 + ) ln (1ln(r + ) + (1 + ) ln(1ln(1 + ) +lnwhere r is given by (11). By keeping as …xed, a coordinated increase in
0 as follows:
if evaluated at = 0 and = yit
@Uit
@

=
=0 and

0
= yit

21

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

) (A4)(1 + r):a¤ects (A3)

2 = [(nwhere is de…ned as1)(1 + )(1
2
19
0 and j =1 =, implying that

@Uit
@

=
=0 and

=

0
yit

)

(1 + + )(1 + )(1 + n)>]. We readily know that @ =@ >(1 + + )+(1 + )(1 + n)

2

> 0:

Economy with a decreasing population
In the economy with decreasing population, a government chooses it and it to maximize
uiot while regarding rt and past variables as given. Substituting (4), (A1), and (A2) 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 ) + (11=

(rt + it )) it ]

Lit :Here, we know that uiot is monotonously increasing in it . To avoid the non-existenceof equilibrium rate of return on savings, we assume the upper-bound of labor income tax
2 (0; 1).20 The government determines tax rates as0oit =By keeping
in21

r (1
1
(1

)

0
oit =

and

)

as …xed, a coordinated increase in , evaluated at
@Uit
@

:
=

0
oit and= , results

2

=
0
= oit
and

(1 + )(1 + )

=

+

0
oit

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

)

0
oit )

:

Hence, we readily know this is positive if and only ifwhere e is de…ned ase0> e0 ;
0 + (1 + n)(1 +[ oit)](1 + )(1 + n) (1 + n + n +

0
oit )

> 0:

Therefore, if governments can impose tax on households, capital tax competition underpopulation growth results in an ine¢ ciently low capital tax rate, and capital tax competition under decreasing population results in an ine¢ ciently high (resp. low) capital tax
rate if the household’s preference for public good consumption is su¢ ciently small i.e.,< e0 (resp. large, i.e., > e).
19

Note here that (11) implies that @r=@ = 1.
When = 1, the equilibrium rate of return on savings diverges to in…nity.
21
Note again that (11) implies that @r=@ = 1.
20

22

) it ]

Appendix B: public inputs
Suppose that individuals obtain utility only from private good consumption, c. We specify
the utility function as follows:
Uit = ln ciyt +ln ciot+1 ;(B1)

where (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. We assume that
wit =

where

1
git kit;> . Pro…t maximization yields

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

=(

1=(

1

)

it

)

=(

Lit

rit + it(B2)

it Kit into (B2) to get
)

:

(B3)

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 government
maximizes Uit . Plugging (3), (B2), and the government budget constraint (B3) into (B1),
we obtain
1+
Uit = (1 + )ln it (1 + )ln(rit + it ) + ln(1 + rit+1 )
+

ln

1+

+ ln

h
+ (1 + ) ln (111+

The …rst-order condition regarding)(1

yields
it =

1

rt :

)=

"=(

Lit

")

i

:

(B4)

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 = it Kit into (2), we obtainuiot = ln

1++ 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 (B4).
23

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+

g (1
) (1
rt + t

j=1

)=

M
X

Lit 1 :

j=1

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

)

:

(B5)

Equilibrium and its e¢ ciency
From (B4) and (B5), we obtain the equilibrium capital tax rate in both economies asfollows:y2 =Plugging

t = t+1 =Uit =+(1 + n)(1 + )(1
(1
+ )

)

:

, the indirect utility of an agent becomes as(1 + )

ln 1 +(1 + ) [1ln+ ](1 + n)(1 + )(1

)

h

)(1

ln (1 + ) + (1 + ) ln (1We di¤erentiate (30) withto obtain@Uit(1 + )=@()(1 + n)(1 + )(1

ln

+
)=

ln
"=(

Lit

)

(30)

1+
i
")
:

1
;
1+r

where r is given byr=(1 + n)(1 + )(1):

From this, we know that the welfare monotonically increases with the capital tax rates.
Hence, y2 is lower than the optimum tax rate.

24

capital tax
public goods
rate of return on
savingssavings

capitalfirm

capitalsavings

wage

labor
consumption

consumption

generation
young period

old period

generation

1
young period

Figure 1 : OLG structure of the modelrace to the bottom

race to the bottom

race to the top

10(a) Case ofrace to the bottom
race to the bottom

race to the top

race to the top
01(b) Case of

Figure 2: Possibility of the race to the top