Title

The importance of government commitment in
attracting firms: A dynamic analysis of tax
competition in an agglomeration economy

Author(s)Kato, Hayato

Citation

European Economic Review. 74 P.57-P.78

Issue Date 2015-02
Text Version author
URL

http://hdl.handle.net/11094/81781

DOI

10.1016/j.euroecorev.2014.11.008

rights

©2014. This manuscript version is made available
under the CC-BY-NC-ND 4.0 license

Note

Osaka University Knowledge Archive : OUKA
https://ir.library.osaka-u.ac.jp/
Osaka University

The Importance of Government Commitment in
Attracting Firms: A Dynamic Analysis of Tax
Competition in an Agglomeration Economy
Hayato Kato ∗First draft: January, 2014
This version: October, 2014

Abstract
Agglomeration tendencies of industrial ﬁrms signiﬁcantly aﬀect the nature of tax competition. This paper analyzes tax competition between two countries over an inﬁnite time horizon
in an economy with trade costs and internationally mobile industrial ﬁrms. Most of the previous studies on tax competition in the ‘new economic geography’ framework employ static
models. In this study, two governments dynamically compete with each other to attract ﬁrms
through their choices of taxes and subsidies. It is shown that the commitment of the governments to their policies is crucial in determining the distribution of ﬁrms in the long run.
Speciﬁcally, if governments ﬁnd each others’ tax policies credible, then one country will attract all the ﬁrms when trade costs are low enough to make agglomeration forces dominant.
If policies are not credible, both countries may attract an equal share of ﬁrms even when trade
costs are low, as the lack of commitment by governments acts as a dispersion force.
Keywords: Economic geography; Tax competition; Forward-looking behavior; Diﬀerential
game; Commitment
JEL classiﬁcation: F15; F22; H20; H30∗ Graduate School of Economics, Hitotsubashi University, Naka 2-1, Kunitachi, Tokyo, 186-8601 Japan. E-

mail address: ed132002@g.hit-u.ac.jp

1

1 Introduction
Since the 1990s, low corporate tax rates have been a main driver for attracting foreign
direct investment (FDI) and thereby promoting economic growth in some European countries
such as Luxembourg and Ireland. Ireland is particularly noteworthy, as it drastically reduced
its corporate income tax rate from 45% to 12.5% from 1998 to 2003. This rate is much
lower than that of other EU15 countries, which was 25% in 2012. Due to this reduction,
Ireland succeeded in hosting several multinational enterprises including Hewlett-Packard andIntel, and achieved a massive inﬂow of FDI. The geometric mean of the ratio of the net
inﬂow of FDI to GDP in Ireland was 15.8% from 1998 to 2003, far higher than that ofother EU15 countries: 3.3%(data are from UNCTAD). The success of Ireland seems to havefanned the fear of ﬁerce tax competition among European countries, which had led to theinternational pressure on Ireland to raise its tax rate. France and Germany, for example, haverequested Ireland to raise its taxes in return for its bailout rescue since 2010 (see, e.g. Mitchell(2009); Stewart (2011)). However, Ireland conﬁrmed its commitment to the 12.5% tax rateand has since maintained this rate. Through this commitment, Ireland has tried to establish a
reputation that it will continue to keep its tax rate low among rival countries.
The objective of this paper is to investigate how government commitment aﬀects the location of industrial ﬁrms as a result of tax competition. In a symmetric two-country economy
with agglomeration forces, I model the strategic interactions between two governments as an
inﬁnite-horizon dynamic game. The governments maximize their life-time payoﬀs throughtaxes and subsidies while considering the migration process of myopic ﬁrms. Assuming thegovernments care mostly about future payoﬀs and that they will agree on Pareto eﬃcient
locations, I examine two forms of commitment governments may make: full commitment
and no commitment. The former corresponds to the open-loop Nash equilibrium while thelatter corresponds the Markov-perfect Nash equilibrium. In the full commitment case, bothgovernments announce a tax schedule over the entire horizon at the outset of the game andnever change it. In the no commitment case, however, they choose their tax rate at each point
in time by observing the current distribution of ﬁrms.
The results of tax competition dramatically change depending on whether or not commitment is possible. I ﬁnd that tax competition with full commitment leads to the core-periphery
conﬁguration of ﬁrms when trade costs are low enough for the concentration of ﬁrms to sup2

ply an ample tax base. This result can be explained by the fact that the governments face
the situation known as ‘battle of sexes’. The dispersed conﬁguration where both the governments share ﬁrms evenly is too costly for them because they must provide negative taxes(or subsidies) to prevent ﬁrms with strong agglomeration motives from clustering into one
country. Of course the core position where all ﬁrms are clustered is the most desirable forboth governments. However, the peripheral position where no ﬁrms are located is better thanthe dispersed conﬁguration because the government in the periphery does not have to subsidize ﬁrms and avoid the intense competition. Either country may become the core so that
self-fulﬁlling expectations of both governments select the long-run outcome; the country thatsucceeds in convincing the counterpart will eventually attract all the ﬁrms. I also ﬁnd thatthe tax competition without commitment may result in the dispersed conﬁguration of ﬁrms
even when trade costs are so low that beneﬁts from agglomeration are large. Compared to thefull-commitment case, the governments have more incentive to raise their tax rate because
they know that the loss of ﬁrms from raising their own tax rate will be partially diminishedby the reaction (i.e., tax increase) of the other. Both governments may set a higher tax rate
while sharing ﬁrms evenly so that the dispersed pattern may be more desirable than the corestatus.
With these results, I draw the implication that, in the contemporary environment of deepening economic integration, eﬀective tax competition relies critically on a government’s commitment to its policies and its ability to convince foreign rivals that they will be implemented.
This may explain the success of Ireland. But, at the same time, I also emphasize the weakness of commitment strategies. Credible policies imply that governments promise to keep
their original policies even if they are aware that the policies are no longer optimal. Thus,
such rigid policies are vulnerable to unexpected events such as sudden changes in the industrial location or policy changes in foreign rivals. When policymakers decide attitudes toward
policies, it is worth recognizing the positive and negative aspects of such rigorous commitment.

1.1 Related literatureThis paper draws from the literature on tax competition in an agglomeration economy. The‘new economic geography’ formulates an imperfect competition trade model with mobile factors and trade costs, which is the so called ‘core-periphery model’. This model demonstrates
3

that as economic integration proceeds, which is modeled as falling trade costs, internationallymobile ﬁrms are more likely to cluster in one country.*1 Based on the core-periphery model,
some researchers have studied the eﬀects on ﬁrm locations of taxes strategically chosen by
governments. Earlier contributions include Ludema and Wooton (2000); Kind et al. (2000);
Andersson and Forslid (2003); Baldwin and Krugman (2004). The main ﬁndings of the literature on agglomeration and tax competition are the superiority of the core country and a ‘race
to the top’ (Baldwin and Krugman (2004); Borck and Pﬂüger (2006), among others); theability of the core country to charge a higher tax rate, and the inverted U-shaped relationshipof the tax diﬀerential in terms of trade costs between the core and periphery.*2 For example,
Baldwin and Krugman (2004) consider a Stackelberg game played by the core (leader) andperipheral country (follower), and show that, when trade costs are low, the industrial cluster
brings about to the core a taxable rent (‘agglomeration rent’). They also show that a decline
in trade costs ﬁrst increases and then decreases the tax gap. Borck and Pﬂüger (2006) conﬁrm
the result in a core-periphery model allowing for partial agglomeration. These results contrast
with the conclusion of the standard neoclassical tax competition model that tax competition
results in a ‘race to the bottom’.*3 The existence of a taxable agglomeration rent is also supported by some empirical research (e.g., Charlot and Paty (2007) for French data; Coulibaly(2008) and Brülhart et al. (2012) for Swiss data).
The conclusions of the previous studies crucially depend on the initial industrial locations.
To avoid the diﬃculty of discontinuities in the reaction functions, which is a by-product of
the core-periphery model, most existing studies assume that one country is the core from the
beginning. Because policy implementation and ﬁrm migration are allowed to occur after theannouncement of the policy in these models, the number of ﬁrms located in each country
at the beginning of the game plays a decisive role in determining the equilibrium outcome.
The advantages of the agglomerated position are thus assumed, and the strategic aspect ofcompetition over mobile factors is not completely captured. Also, since the policy scheduleconsists of one tax rate, the eﬀect of the government commitment to the policy cannot be
fully examined.
*1 See Fujita and Thisse (2013) for a recent survey. Baldwin et al. (2003) focus more on policy issues including

tax competition.
*2 Recent studies such as the work of Ottaviano and Ypersele (2005), which allows for the diﬀerence in sizeof immobile factors, and the work of Baldwin and Okubo (2014), which introduces ﬁrm heterogeneity inproductivity, also observe the superiority of the core country.
*3 See Keen and Konrad (2012) for a comprehensive survey on tax competition in the neoclassical framework.

4This paper diﬀers from the previous studies in that it considers tax competition in a fully
dynamic context so as to treat agglomeration forces endogenously. By analyzing an inﬁnitehorizon game, I avoid the diﬃculty of discontinuities and allow for all possible initial conditions in order to not be restricted to the core-periphery conﬁguration. Moreover, by utilizing
the diﬀerential game theory, I can analyze the impact of diﬀerent forms of government commitment, which is not addressed in the previous studies. I show that, contrary to previous
research, the core-periphery pattern may not emerge as a result of tax competition between
governments without commitment. Even when the core-periphery pattern emerges in the
full-commitment case, initially being the core is of little importance, as expectations of governments instead determine which country will be the core. However, the ‘race to the top’
result on the relationship between the core’s tax rate and trade costs still holds whenever the
core-periphery pattern emerges.
This paper is not the ﬁrst to incorporate the forward-looking agent into the new economic
geography framework (see, e.g.,Baldwin (2001); Ottaviano (2001); Ottaviano et al. (2002)
Section 6; Oyama (2009a,b)).*4 These studies deal only with the forward-looking behavior
of industrial ﬁrms*5 and do not model governments explicitly. In contrast, the present study
allows only governments to be forward-looking agents. This paper and the studies just mentioned should be seen as complements because both try to investigate the role of far-sighted
behavior but of diﬀerent agents. In spite of this diﬀerence, one of the results I obtain that‘expectations matter’ for equilibrium selection can be found in Baldwin (2001), Ottaviano(2001) and Ottaviano et al. (2002).
Dynamic aspects and commitment issues have been addressed in the literature of the standard tax competition without agglomeration forces (see, e.g., Janeba (2000); Keen and Konrad (2012), Section 2.3 and the references therein). The closest study in that ﬁeld to this paper
is Han et al. (2014). They consider an inﬁnite-horizon tax game between two countries ofunequal size. An asymmetric form of government commitment, which is not addressed inthis paper, is assumed and the larger country adopts open-loop strategies while the smaller*4 The ﬁrst three consider the Krugman (1991) type of migration dynamics, where ﬁrms can move freely by

paying costs determined by the current ﬂow of migrants. The last one adopts Matsuyama (1991) type of
migration dynamics, where only ﬁrms which receive an opportunity to revise their location choice can move
without any costs. This paper and most of the studies in the new economic geography are in line with the
myopic version of the Krugman (1991)’s dynamics.
*5 Oyama (2009a) discusses tax policy and points out that agglomeration rents should be much smaller becauseof the possibility of self-fulﬁlling coordinated migration, but the paper does not explicitly model governments.

5

country adopts Markov-perfect strategies. They show that, if mobility of ﬁrms is high and thefuture payoﬀ is important (as in this paper), the smaller country will lose all the ﬁrms in thelong run. The result that a government without commitment will be the periphery is similar
to one of the results I obtain. However, the model I present describes not only the conﬂict
between governments but also the interaction among governments and ﬁrms with agglomeration motives. When considering the world economy in which FDI is motivated mainly by
reducing transport costs and exploiting economies of scale (see, e.g., Markusen and Maskus(2002)), examining tax competition in the core-periphery model may be more relevant today
than in the neoclassical world with constant returns to scale and perfect competition.
The rest of the paper proceeds as follows. The next section develops a simple generalequilibrium model that induces agglomeration forces. Section 3 introduces governments andformulates a dynamic tax competition. The two equilibrium concepts in dynamic games,
the open-loop Nash equilibrium (OLNE) and the Markov-perfect Nash equilibrium (MPNE),
are also explained. The assumptions imposed on the analysis are also discussed. Section 4
presents the results whereby I characterize the stable steady state both under OLNE and under
MPNE. Section 5 concludes.

2 The model
In this section, I consider the Ludema and Wooton (2000) and Thisse (2010) version of thecore-periphery model. The model may be viewed as the ‘minimalist framework’ of economic
geography and has several distinctive features that make the analysis simple. First, the model
follows the ‘footloose entrepreneur’ setting as in Forslid and Ottaviano (2003) whereby the
manufacturing industry needs internationally immobile workers for variable inputs and mobile entrepreneurs for ﬁxed inputs. Second, consumers’ preferences are described by thequadratic quasi-linear utility function as in Ottaviano et al. (2002). Third, contrary to most of
the core-periphery models, manufacturing ﬁrms produce a homogeneous product and engage
in Cournot competition. Due to these simpliﬁcations, closed-form solutions can be obtainedwith a reduced number of parameters, though qualitative results are quite similar to the linear
version of the core-periphery model by Ottaviano et al. (2002).

6

2.1 The basic setup
In this model, the economy consists of two countries (home, h, and foreign, f ), two homogeneous goods (agricultural and manufactured goods), and two factors of production (workers
and entrepreneurs). Total mass of workers in the world is I > 1, and that of entrepreneurs is
L > 1. While workers are equally distributed between countries, the 100 × λ percent of theentrepreneurs reside in country h where λ ∈ [0, 1]. The total population of country i ∈ {h, f }becomes S i = I/2 + λi L, where λh ≡ λ and λ f ≡ 1 − λ.
The two types of factors are diﬀerent in their place of employment and in their mobility.
Each entrepreneur can move between countries and provides one unit of headquarter service with the manufacturing sector in the country where she resides, while each worker isimmobile and provides one unit of labor service with the agricultural sector.
The agricultural sector is characterized by perfect competition and constant returns-to-scaletechnology. The goods are freely traded internationally without any trade costs. The price of
the agricultural good is chosen as a numéraire, and only labor service by workers are required
as variable inputs. The manufacturing sector, on the other hand, is characterized by imperfectcompetition and increasing returns to scale technology. The shipment of one unit of the
manufactured good incurs an additional τ units of trade costs. To start a business, a potential
ﬁrm in the sector hires one entrepreneur as a ﬁxed input, but needs no variable inputs.

2.2 Demand Side
Every individual in country h has the same preferences, and their utility function takes thefollowing quasi-linear quadratic form:*6
"
!#
!
Qa
1 Qh
Qh
uh = 1 −
+ h,
2 Sh
Sh
Sh

(1)

where Qh (Qh /S h , respectively) is the total (per-capita, resp.) demand for the manufactured
good and Qah (Qah /S h , resp.) is total (per-capita, resp.) demand for the agricultural good.
Aggregating across individuals, total utility in the home country is given by the following:
"
!#
1 Qh
Uh = S h uh = 1 −Qh + Qah .
(2)
2 Sh*6 In what follows, I concentrate on h. Symmetric expressions hold for f .

7

The national budget constraint requires that the expenditure on the two goods equals the
remuneration of production factors and the endowment of the numéraire good. I focus onlyaon the case where Q is suﬃciently large for the numéraire good to be consumed. Letting piabe the price of the manufactured good, Yi the total reward for factors,*7 and Qh the endowmentof numéraire good, the national budget constraint can be written as follows:aph Qh + Qah = Yh + Qh .

(3)

The representative agent in country h maximizes aggregate utility (2) subject to the national
budget constraint (3) by choosing the total demand for the industrial good, Qh . Optimization
yields a linear-demand function for the manufactured product independent of income level:
ph = 1 −Qh.
Sh

2.3 Supply SideFor the agricultural sector, perfect competition and costless trade lead to the equalizationof price between countries. One unit of labor by worker produces one unit of the agricultural
good so that its price denoted by pai and the wage of a worker denoted by wai in both countries
are equalized to unity: pah = paf = wah = waf = 1.
For the manufacturing sector, ﬁrms located in i provide their goods to both domestic andforeign markets. Since an entrepreneur is needed to launch one industrial ﬁrm, country h has
λL ﬁrms and country f has (1 − λ)L ﬁrms. The operating proﬁt of a ﬁrm in i can be written
as follows:
πh = ph qhh + (p f − τ)qh f ,(4)where qi j denotes the amount of production by a ﬁrm in i, sold in j (i, j ∈ {h, f }). For theindustrial good market to clear, it must hold that:Qh = λLqhh + (1 − λ)Lq f h .

*7 As will be clear in the following discussion, suppose wa is the wage for a worker and w for a entrepreneur,

the reward to a worker is wah and the reward to a entrepreneur is wh . Total reward to the factors in h becomes
Yh = wah I/2 + wh λh L.

8Due to the separation of markets, ﬁrms can choose diﬀerent quantities for each market
given the inverse demand functions they face. As a result of Cournot competition, the market
outcome is determined by a Nash equilibrium:q∗hh = S h p∗h ,q∗f h = S f (p∗f − τ),

where
p∗h =

1 + (1 − λ)Lτ
.
L+1

p∗f =

1 + λLτ.
L+1

The increase in the share of domestic ﬁrms and decline in trade costs make the domestic pricego down.
For international trade to occur, trade costs must be low enough for the mill price p∗f − τ tobe positive whatever the distribution of ﬁrms may be:Assumption 1. τ < τtrade ≡

1
.
L+1

This assumption holds throughout the analysis.
By substituting equilibrium prices and quantities into (4), the equilibrium operating proﬁt
can be calculated as follows:π∗h = S h (p∗h )2 + S f (p∗f − τ)2 , [1 + (1 − λ)Lτ]2  I [1 − {1 + (1 − λ)L} τ]2
I+.
+ λL ·
+
(1
−
λ)L
·
=
2
2
(L + 1)2(L + 1)2
Under Assumption 1, falling trade costs decrease the proﬁts from the domestic market but
increase those from the foreign market. Allowing for free entry and exit, the operating proﬁtof a ﬁrm goes to the entrepreneur so that it becomes equal to the reward. Letting wi be thereward to an entrepreneur in i, it holds that π∗i = wi . Since there is one entrepreneur for every
manufacturing ﬁrms, I refer to them interchangeably in what follows.
The utility of an entrepreneur in h evaluated at the equilibrium prices and quantities isexpressed asu∗h = CS h + wh + qa .
awhere qa = Qh /S h is the endowment for individuals*8 and CS h is the individual consumer*8 qa is supposed to be suﬃciently large for taxes, which will be introduced in the next section, to be paid.

Further, qa is assumed to be identical between countries so that it does not aﬀect international diﬀerences inindirect utility.

9

surplus:
CS h =

L2[1 − (1 − λ)τ]2 .
2(L + 1)2

CS h is increasing in the domestic ﬁrm’s share, but is decreasing in trade costs as long asAssumption 1 holds. An entrepreneur, as a consumer, beneﬁts from a lower price due to the
expansion of the domestic ﬁrm’s share.

2.4 Location equilibrium
The gap of the indirect utility, which determines the entrepreneurs’ location choice, can be
expressed as∆u∗ (λ) ≡ u∗h − u∗f= CS h + wh − (CS f + w f )
!
1
=Z λ−
,
2
where
Z=

L[2L2 + (2I + 5)L + 2(I + 1)] ∗τ(τ − τ),(L + 1)2τ∗ =2(3L + 2)
2L2 + (2I + 5)L + 2(I + 1).

Fig. 1 shows the relationship between Z and trade costs. Parameter values are L = 1 andI = 5 so that I have τtrade = 0.5.

Fig. 1. Slope of the migration equation Z.
10

Z captures a ﬁrm’s incentive to agglomerate (disperse, resp.) if it is positive (negative,
resp.). Agglomeration forces reach the peak (Z) when trade costs are intermediate (τ∗ /2),
while they become smallest (Z) when trade costs are highest (τtrade ). The intuition behind theinverted U-shaped relationship between τ and Z is easy to grasp when considering the two
extreme cases.*9 When τ = τtrade and the costs of shipping industrial goods are prohibitively
high, agglomeration of ﬁrms in one country is harmful because they earn little from theforeign market and operate solely in the domestic market. However, when τ = 0 and trade iscompletely free, agglomeration is not necessary because the location choice does not matter.
When trade costs are intermediate, agglomeration is proﬁtable. The positive utility diﬀerence
between the core and periphery (∆u∗ (λ = 1) = Z/2) is the so called ‘agglomeration rent’.
Following the tradition of the core-periphery model, I consider the myopic behavior ofﬁrms. That is, entrepreneurs, the owners of ﬁrms, relocate to the country that aﬀords thehigher current utility Therefore, the movement of ﬁrms is governed by a simple migration
equation λ̇ ≡ dλ/dt = ∆u∗ (λ). I simply assume that the motion stops if the economy hits
either of the boundaries λ ∈ {0, 1}. This restriction will be removed in the next section.
When τ > τ∗ , the slope of the migration equation Z is negative and the dispersed distribution λ = 1/2 is stable, while when τ < τ∗ , the slope is positive and the agglomerated
distributions λ ∈ {0, 1} are stable. For the dispersed location to emerge, I impose the ‘noblack-hole’ condition:
Assumption 2. τ∗ < τtrade ,
p
i
1h
⇒L<
2I − 5 + (2I − 5)2 + 32(I − 1) .
8
The condition simply states that, to prevent agglomeration forces from being too strong, thenumber of entrepreneurs should be small compared to that of workers.
The following lemma is the central ﬁnding of the core-periphery model:
Lemma. If trade costs are high (τ > τ∗ ), the symmetric conﬁguration is the only stableone. If trade costs are low (τ < τ∗ ), the core-periphery structure is the only stable one.
*9 Note that the diﬀerence of consumer surplus is increasing in τ as long as Assumption 1 holds: CS h (λ =1) − CS f (λ = 1) = L2 τ(1 − τ)/2(L + 1)2 . Note also that the wage gap has an inverted U-shaped relationship interms of τ as long as I > 1, Assumption 1 and 2 hold: wh (λ = 1) − w f (λ = 1) = Lτ[2 − (L + I + 1)τ]/(L + 1).

11

In this linear model, the ‘break point’ at which the symmetric state is no longer stable coincides with the ‘sustain point’ at which the concentrated states become stable.

3 Tax competition over an inﬁnite horizon
Having presented the base model, I now introduce taxes and a government for each country
that plays tax competition over an inﬁnite time horizon. This section also explains equilibrium concepts in the dynamic game and includes a few additional assumptions.

3.1 The governments’ problem
Following Ludema and Wooton (2000) and Borck and Pﬂüger (2006), taxes are levied ina lump-sum manner. The government in country i ∈ {h, f } imposes lump sum taxes on eachdomestic ﬁrm so that the migration equation for ﬁrms is modiﬁed as follows:λ̇= (u∗h − T h ) − (u∗f − T f )
γ
!
1
=Z λ−
− (T h − T f ), λ ∈ [0, 1],2
where γ is a positive constant that represents the migration speed, and T i denotes the taxes
in i if it is positive or subsidies if negative. Unlike the previous section, hereafter, I do not
require λ̇ to be zero when λ hits a boundary. The governments redistribute their tax revenues
to the domestic immobile workers as numéraire goods so that the governments’ budgets arebalanced at each point in time.
Taking into account the ﬁrms’ incentive to relocate, both governments try to maximize thefollowing discounted sum of the instantaneous payoﬀ Wi :Z ∞max
exp(−ρt)Wi (λ(t), T i (t))dt ≡ Ji ,{T i (t)}∞
0
t=0



λ̇(t)/γ = Z[λ(t) − 1/2] − [T h (t) − T f (t)],s.t. 

λ(t) ∈ [0, 1], λ(0) = λ0 given,(5)

where ρ > 0 is a discount rate. Following Baldwin and Krugman (2004) and Borck andPﬂüger (2006), I adopt a simple instantaneous objective function:k
Wi (λ(t), T i (t)) ≡ Lλi (t)T i (t) − T i2 (t),212where λh ≡ λ and λ f ≡ 1 − λ, and k > 0 is a parameter. The ﬁrst term on the right hand sideis the total tax revenue for country i and the second term is the quadratic loss from taxation.
This simple instantaneous objective function captures the basic conﬂicts governments face:
they seek to raise more tax revenue while maintaining a low tax rate. It is regarded as a
reduced-form objective function that either selﬁsh or benevolent governments adopt (Baldwin
and Krugman (2004), Section 5). From a diﬀerent point of view, the second term can beconsidered the administrative cost incurred in collecting taxes (see, e.g. Kenny and Winer(2006) for empirical evidence). The second term may also be interpreted as the costs oftax changes which are analogous to adjustment costs of price (e.g., Rotemberg (1982)) andinvestment (e.g., Summers (1981)). Since the model has a long time dimension, it would bereasonable to assume that an excessive tax increase or cut within a short period would be too
costly and that governments prefer tax smoothing.

3.2 Equilibrium concepts: Is commitment credible?
This subsection introduces two equilibrium concepts, the open-loop Nash equilibrium(OLNE) and the Markov-perfect Nash equilibrium (MPNE). Intuitively speaking, OLNEs
consider situations in which both governments can make a precommitment; they stick to theirstrategies announced at the beginning of the game. Conversely, MLNEs assume such a precommitment away, so both governments may deviate from their predetermined tax policies.
Precommitment to tax schedules is possible in two coses.*10 One case is that the governments
cannot observe the time path of the distribution of ﬁrms (except for the initial one) nor thetax path of the rival country. The other case is that the governments can observe the state andother player’s actions but they can precommit to their future policies with the aid of commitment device such as national legislations with compelling power. In the context of tax
competition, I adopt the latter explanation why precommitment is possible.*11*10 The following explanation is based on Reynolds (1987) Section 3.
*11 It does not seem plausible that the actual governments adopt the open-loop strategies, but there are somecountries that appear to do this. In particular, as discussed in Introduction, Ireland is a good example. A
government report released in 2013 (Department of Finance of Ireland (2013)) ﬁrmly asserts the commitment
toward its tax rate: “Our competitive taxation system is, of course, an element of the Irish package and weremain committed to our competitive 12.5% tax rate.”(p.3)

133.2.1 Open-loop Nash equilibrium

In this model, an open-loop strategy for country i is a tax policy T i which is a functionof time and the boundaries of the state variable λ ∈ {0, 1}. A pair of open-loop strategies{T ho (t, λ ∈ {0, 1}), T of (t, λ ∈ {0, 1})} forms an OLNE such that
Ji (T io , T oj ; λ) ≥ Ji (T i , T oj ; λ), for every open-loop strategy T i (t, λ ∈ {0, 1}), i, j ∈ {h, f }; i , j,where Ji (·, ·) is a maximized value of the objective function deﬁned in (5). At an OLNE,
the payoﬀs of both governments are maximized and neither has an incentive to deviate fromits open-loop strategies, given the open-loop strategy of the other. To solve for OLNEs, theoptimal control method can be applied. I formulate the current-value Hamiltonian for thehome government Hh and the Lagrangian Lh as follows:
"
!
#
k 2
1
Hh = LλT h − T h + µh γ Z λ −− (T h − T f ) ,22
andLh = Hh (λ, T h , T f ) + νh1 λ + νh2 (1 − λ),where µh is the co-state variable associated with the state variable, and νh1 and νh2 are theLagrange multipliers on the boundaries of the state variable. The optimal control problem
here is somewhat diﬀerent from the standard problem in that the state variable λ should lie
in between [0, 1]. In order to deal with the state variable constraints, I employ ‘Informal
theorem’ 4.1 in Hartl et al. (1995) (see, e.g., Oyama (2009b) for applications in economics).
Given an open-loop strategy of the foreign government T f , an open-loop strategy of the homegovernment T h is characterized by the following necessary conditions for optimality:
!
1
λ̇
=Z λ−
− (T h − T f ), λ ∈ [0, 1],γ2∂Hh= Lλ − kT h − γµh ,∂T h∂Hhρµh − µ̇h == LT h + γZµh ,∂λlim exp(−ρt)µh = 0,
t→∞






>
0
if
λ(t)
=
0
> 0 if λ(t) = 1

2
,
,
ν
(t)
νh1 (t) h= 0 if λ(t) < 1
= 0 if λ(t) > 0
14

(6-1)
(6-2)
(6-3)
(6-4)
(6-5)

and for any time s in a boundary interval and for any contact time s,*12 µh (·) may have a
discontinuity governed by the following jump conditions :µh (s− ) − µh (s+ ) = η1h (s) − η2h (s),






>
0
if
λ(s)
=
0

> 0
2
η1h (s) 
,
η
(z)

h


= 0 if λ(s) > 0
= 0

(6-6)
if λ(s) = 1,
if λ(s) < 1(6-7)

for some η1h (z), η2h (z) for each s, where µh (s− ) ≡ limt→s− µh (t) and µh (s+ ) ≡ limt→s+ µh (t).
An open-loop strategy of f , given an open-loop strategy of h, is derived in a similar manner.
I obtain an OLNE as a pair of tax policies that simultaneously satisﬁes both governments’necessary conditions for optimality just described.
OLNEs assume open-loop strategies, where both players announce the action schedule atthe outset of the game and commit themselves to it. Because open-loop strategies depend ontime,*13 the government sticks to the original plan even if it is no longer optimal to do so on
account of the counterpart’s deviations from the original tax policy, or due to shocks that pullthe distribution of ﬁrms oﬀ the predetermined path. Hence, an OLNE is not robust against
any deviations from the equilibrium path.
3.2.2 Markov-perfect Nash equilibrium

A Markov-perfect strategy and a MPNE are deﬁned similarly to the open-loop case. An
Markov-perfect strategy for country i is a tax policy T i which is a function only of the statevariable λ ∈ [0, 1]. An pair of Markov-perfect strategy {T hm (λ(t)), T mf (λ(t))} forms an MPNE
such that
Ji (T im , T mj ; λ) ≥ Ji (T i , T mj ; λ), for every Markov-perfect strategy T i (λ(t)), i, j ∈ {h, f }; i , j,
As in the open-loop case, I obtain an MPNE as a solution to the two governments’ dynamic optimization problems. Necessary conditions for each government’s problem here
are slightly diﬀerent from the ones in the open-loop case. Condition (6-3) is replaced with

*12 An interval [t1 , t2 ] ⊂ [0, ∞) with t1 < t2 is said to be a boundary interval if λ(t) ∈ [0, 1] for all t ∈ [t1 , t2 ]. Aninterval (t1 , t2 ) ⊂ [0, ∞) with t1 < t2 is said to be an interior interval of λ(·) if λ(t) ∈ (0, 1) for all t ∈ (t1 , t2 ). If
the trajectory is in the interior interval just before and just after s, s is said to be a contact time.
*13 The open-loop strategies we consider are slightly diﬀerent from the standard one; the former depends both ontime and the boundaries of the state variable, whereas the latter depends only on time. However, the argument
does not change much.

15

the following condition:ρµh − µ̇h =∂Hh ∂Hh ∂T f+,
∂λ∂T f ∂λ= LT h + γZµh + γµh

(7)
∂T f
.
∂λ

The necessary conditions from (6-1) to (6-7) other than (6-3) are the same. Symmetric expressions hold for the foreign government.
The second term in the right hand side of (7) captures the feedback eﬀect of the counterpartson the change of the state variable. In the Markov-perfect case, the government takes into
account how the other government changes its tax policy by responding to the changes of the
distribution of ﬁrms. In the open-loop case, the government ignores this feedback eﬀect sothat the term does not appear.
Since the government utilizes the counterpart’s decisions at each point in time, a credible
precommitment for implementing the tax schedule determined at the beginning of the gameis no longer possible. Markov strategies that depend only on the state variable in the game
imply that the governments optimally choose their tax schedules at each time instant. Thus,an MPNE is a subgame perfect equilibrium and is robust against any deviations from theequilibrium path.*14
3.2.3 Assumptions

The subsequent analysis relies on several assumptions. First, to shed light on the dynamic
aspect of the analysis, I assume the following:
Assumption 3. ρ/γ is suﬃciently close to zero.
The ratio of the governments’ discount rates to the migration speed ρ/γ is called the ‘degreeof friction’. The behavior of the dynamics does not depend on ρ or γ independently (seeAppendices for details), but on the ratio. This assumes that the governments greatly value thefuture (small ρ) and/or the ﬁrms move quickly when responding to the tax diﬀerential (largeγ). Thus, when deciding tax policies, the governments put more emphasis on the steady states
where movement of ﬁrms stops, than on the short term payoﬀs on the way to the steady states.

*14 Both solution concepts are time consistent, which is a weaker requirement than the subgame perfectness. See

Dockner et al. (2000) Ch. 4.

16Second, I need a criterion to select the resulting outcomes among others because, as will
be seen in the analysis, the model has multiple stable steady states:
Assumption 4. Pareto ineﬃcient steady states for governments (if any) are eliminated.
This reﬁnement of the Nash equilibrium is known as ‘payoﬀ dominance’ (without uncertainty). Since the governments focus solely on the steady-state payoﬀs due to Assumption 3,
the resulting outcomes are determined by the steady-state payoﬀs. If the payoﬀ at a stablesteady state dominates the others in a Pareto sense, both governments agree that the economy
reaches the state.
Finally, for analytical convenience, I restrict the range of parameter k, which captures theintensity of loss from an increasing tax or subsidy rate:
Assumption 5. 0 < k <

(2 −

√
Z

3)L

≈

0.27L
Z

L(3L + 2)2, Z ≡ max Z =.
(L + 1)2 [2L2 + (2I + 5)L + 2(I + 1)]

This assumption eliminates the imaginary root in the linear MPNE (see Appendix B for details).

4 Results
This section characterizes the resulting industrial location both under OLNE and under
MPNE. The time path of the share of industry and taxes are also discussed.

4.1 Open-loop case: Both governments fully commit to their tax schedules
I illustrate OLNEs by resorting to intuitive arguments, which are veriﬁed by the formal
proof in Appendix A. The present model is somewhat complicated because the economy bynature may have multiple steady states, the symmetric state λ = 1/2 and the concentrated
states λ ∈ {0, 1}*15 However, since the future is suﬃciently important for the governments
by Assumption 3, the governments are concerned mainly with the steady-state payoﬀs whenchoosing equilibrium paths. Assumption 4 requires that the steady state conﬁgurations which*15 Since the migration equation is linear with respect to λ, the interior points, other than the symmetric state,cannot be steady states.

17

bring higher payoﬀs to both governments than others are realized in equilibrium. By comparing the payoﬀs between the symmetric state and the concentrated states, the resulting steadystates can be found. The stability of the states are proved in Appendix A.
First, I look at the symmetric steady state, which can be obtained by making λ̇ = µ̇h = µ̇ f =0 and νh1 = νh2 = ν1f = ν2f = 0 in the necessary conditions. Since both governments have an
equal share of ﬁrms at the steady state, the tax rate will be equalized as a result of dynamic
competition:
T ho

!
!
1
L(ρ − γZ)1oo
= Tf λ =
=
.
λ =
2
2
2[kρ + γ(L − kZ)]o

The expression is simpliﬁed by Assumption 3 that the degree of friction ρ/γ is suﬃciently
close to zero:

!
!


1
1
LZ
≥ 0 if Z ≤ 0
T ho λo == T of λo ==.
< 0 if Z > 0222(kZ − L) 
Note that, by Assumption 5, k is small enough for the denominator to always be negative.
The inverted U-shaped relationship between the slope of the migration equation Z and trade
costs τ (Fig. 1) implies that positive (negative, resp.) Z equals high trade costs τ > τ∗ (lowtrade costs τ < τ∗ , resp.).
It can be easily veriﬁed that if trade costs are high (τ > τ∗ ), making the dis-intensive to agglomerate strong (Z < 0), the tax rate is positive and both governments earn positive payoﬀs
at the symmetric steady state. On the other hand, if trade costs are low (τ < τ∗ ), making the
incentive to agglomerate strong (Z > 0), the tax rate is negative and both governments incurloss by providing subsidies to keep the symmetric location stable.
To see why the tax rate responds to trade costs as such, consider the motives of ﬁrms. High
trade costs lessen the proﬁtability of export and make the domestic competition among ﬁrms
tougher so that ﬁrms try to avoid a country with many competitors. Because of the ﬁrms’
preference for diversiﬁed distribution, the governments can impose positive tax. Conversely,
low trade costs reverse the outcome: since low trade costs makes export more proﬁtable,ﬁrms try to locate in a country with many rivals in order to exploit scale economies. The
governments need to subsidize the domestic ﬁrms to prevent them from moving away. Thisis reminiscent of the ‘race to the bottom’ result. Strong incentives for ﬁrms to gather makethe symmetric conﬁguration unattractive for both governments.
Next, I look at the steady state with full agglomeration in the home country (λ = 1). For
18

the concentrated conﬁguration to be stable, the tax diﬀerential should be determined in sucha way that the motion of ﬁrms λ̇ evaluated at λ = 1 is equal to zero and T h − T f = Z/2.
Evidently, the optimal tax rate for country f is zero and it has zero payoﬀ. Therefore, thecore country will set its tax rate at Z/2:
Z,
2
T of (λo = 1) = 0.
T ho (λo = 1) =Both the tax rate and the payoﬀ of country h at the steady state are negative under low agglomeration incentive Z < 0 and equivalently high trade costs τ > τ∗ , whereas they are
positive under high agglomeration incentive Z > 0 and low trade costs τ < τ∗ .
The same reasoning for the symmetric case applies here. High trade costs strengthen the
incentive for ﬁrms to diversify so that the core country has to subsidize them to maintainclustering. Under low trade costs, however, the incentive to concentrate is so strong that thecore earns the taxable agglomeration rent.
Fig. 2-(a) shows the tax rate of h at each steady state for a various level of trade costs.*16
The instantaneous payoﬀ of h also exhibits a similar shape as in Fig. 2-(b). Parameter values
are L = 1, I = 5, and k = 1. By the symmetry of the model, it holds that T ho (λo = 1) =T of (λo = 0), T ho (λo = 0) = T of (λo = 1) and T ho (λo = 1/2) = T of (λo = 1/2). The same relations
hold for the payoﬀs of country f .

*16 Hauﬂer and Wooton (2010) study a static tax game in a location model (‘footloose capital model’) which issimilar to the present model. Their model also obtains a U-shaped relationship with the tax rate and trade
costs in the symmetric equilibrium. But their paper and this paper are quite diﬀerent in that their model
exhibits weaker agglomeration forces and their focus is only on interior equilibria.

19Fig. 2. Country h’s (a) tax rate and (b) payoﬀ at the steady state under OLNE.

To sum up, high trade costs bring the dispersed equilibrium λ = 1/2, while low trade
costs bring the agglomerated equilibrium λ ∈ {0, 1}. When trade costs are high (Z < 0),
the governments share the equal number of ﬁrms while setting a positive tax rate by taking
advantage of the ﬁrms’ anti-agglomeration motive. Since the governments have to provide
subsidies to keep all the ﬁrms in their country, neither of them tries to achieve agglomeration.
In contrast, when trade costs are low (Z > 0), the governments have to set negative tax rates tomaintain the symmetric conﬁguration because of the ﬁrms’ agglomeration motive and instead
they try to become the core. In this case, the governments face a ‘battle of sexes’ situation.
20

Although both governments prefer the core-periphery patterns to the dispersed one, they havea conﬂict of interests over which of them attains agglomeration. As in the battle of the sexes
game, the winner of tax competition cannot be determined in the model. These ﬁndings
reveal that the result of tax competition dramatically changes below and above the threshold,which is called the ‘bifurcation of tax competition’.*17 The diversiﬁed distribution of ﬁrms
emerges when τ > τ∗ , and the concentrated distribution emerges when τ < τ∗ . Thus, I obtainthe following proposition:*18
Proposition 1. The OLNEs are characterized in the following way (except for the nongeneric case: τ = τ∗ or Z = 0)(i) If trade costs are high (τ > τ∗ or Z < 0), the symmetric conﬁguration is the only stablesteady state and the tax rates at the steady state are given by
!
!
1
1
LZ
o
o
o
o
Th λ == Tf λ ==.
2
22(kZ − L)(ii) If trade costs are low (τ < τ∗ or Z > 0), the only stable steady states are in the coreperiphery conﬁguration and the tax rates at the steady states are given by
Case λo = 0 : T ho (λo = 0) = 0, T of (λo = 0) =

Z
,
2

or
Case λo = 1 : T ho (λo = 1) =Z
, T of (λo = 1) = 0.
2

Proof. See Appendix A. It can be readily veriﬁed that the static analysis in the same setting
mimics the results in Proposition 1 (see Section 4.3).
Proposition 1 conﬁrms the ‘race to the top’ result from Baldwin and Krugman (2004) ina dynamic context. When trade costs are low, the core country eventually hosts all the ﬁrms
while setting a positive tax rate. The tax gap between the core and periphery (T ho (λo = 1) −T of (λo = 1) = Z/2) exhibits a inverted U-shaped curve as in the dashed line in Fig. 2-(a). In
contrast to their sequential move game, any initial location is allowed and the initial advantage*17 I am grateful to Richard Baldwin for naming this.
*18 If trade costs happen to be the critical value τ∗ , the slope of the migration equation Z equals zero and any

distribution is stable. But this particular case is unimportant.

21

would disappear in the dynamic model. The country that becomes the core is determinedby ‘expectations’ of governments, not by ‘history’ of industrial locations. An analogous
argument can be found in the studies on the forward-looking behavior of entrepreneurs in the
core-periphery model (Baldwin (2001); Ottaviano (2001); Ottaviano et al. (2002), Section6).*19 Consider why the peripheral country fails to attract ﬁrms even when competition lasts
an inﬁnite time. Even if the peripheral country, say f , took back all the ﬁrms and achieved
the core position (λ = 0, T h = 0 and T f = Z/2), it could not commit itself to keep its tax rate
low enough to make the distribution of ﬁrms stable. f would have an incentive to increase its
tax rate after catching all the ﬁrms. Thus, it could not stop ﬁrms from relocating to the othercountry.*20
Fig. 3 shows the transition path from the initial state to the steady state with high trade
costs (τ = 0.2 in Fig.3-(a)) and low trade costs (τ = 0.4 in Fig. 3-(b)). Initial distributions ofﬁrms are λ0 = 0.7 in (a) and λ0 = 0.3 in (b), Other parameter values are L = 1, I = 5, k = 1
and ρ/γ = 0. In Fig. 3-(a), trade costs are higher than the bifurcation of tax competition(τ = 0.4 > 0.3448 = τ∗ ). Country h, which has more ﬁrms at the beginning, will decrease its
tax rate gradually, while country f , which has fewer ﬁrms, will behave in the opposite way.
In Fig. 3-(b) where trade costs are low enough (τ = 0.2 < 0.3448 = τ∗ ), a more spectacularchange is observed. The prospective core country, h, reduces taxes excessively at the outset
and raises taxes as more ﬁrms relocate in h.

*19 Oyama (2009a,b) also examine the forward-looking expectations of entrepreneurs.

In contrast to the threepapers cited in the text, these two papers obtain the uniqueness of stable steady state in a more general coreperiphery model with asymmetries of market size and trade costs.
*20 Mathematically, the discussion here corresponds to the choice of the Lagrange multipliers ν1 and ν2 on theiiboundaries of the state variable. In steady states, the state and co-state (or control) variables are constant overtime (λ̇ = µ̇h = µ̇ f = 0). The multipliers are uniquely determined so as to attain a steady state. Since themultipliers cannot be chosen in such a way that both the two concentrated states are steady states at the sametime, either one of the steady states is achieved.

22

Fig. 3. Transition path under OLNE with (a) high trade costs and (b) low trade costs.

It is worthwhile to mention the limitation of OLNE. An OLNE is not subgame perfectand is fragile to any unexpected events. Suppose that unanticipated shocks happen at a point
in time and thereby alter the path of λ from that time onwards that both governments haveexpected. Although the original tax schedule that both governments announced at the initialstage of the game is no longer optimal, they will stick to the original tax schedules. In thecase of Ireland discussed in Introduction, its aggressive tax policies have been eﬀective for
attracting FDI at the expense of the ﬂexibility.

23

4.2 Markov-perfect case: Neither government commits to its tax schedule
I turn to the analysis of MPNE. Only the symmetric linear strategy with respect to the statevariable is considered:T h (λ) = aλ + bh ,T f (λ) = a(1 − λ) + b f ,where a, bh and b f are constants that are determined in equilibrium.
As in the preceding analysis, consider the tax rate and payoﬀ both at the symmetric andconcentrated steady states. In the later case, the tax rate is determined so as to keep the coreperiphery structure stable: the core country sets its rate at Z/2, while the peripheral country
sets at zero. Since the smaller Z makes the tax rate lower, the concentrated steady statescannot be realized as long as Z < 0 or τ > τ∗ .
The tax rates at the symmetric steady state are calculated as follows:


∂T f
!
L
Z
−
1
∂(1−λ)
T hm λm =
= h 

i,
∂T
f
2
2 k Z − ∂(1−λ)
−L


!
∂T h
L
Z
−
1
∂λ

i
T mf λm =
= h 
∂T
2
2 k Z− h −L
∂λUnlike the open-loop case, the tax rates reﬂect the feedback of the rival country (∂T h /∂λ and∂T f /∂(1 − λ)) because each government considers how its counterpart changes its tax rate in
response to changes in the distribution of ﬁrms. If these feedback terms were zero, the tax
rates would be the same as those under the open-loop case (see Proposition.1-(i)). It can be
veriﬁed in Appendix B that the feedback eﬀect (∂T h /∂λ = ∂T f /∂(1 − λ) = a) is positive. Thismeans that the tax rate of a country should be directly proportional to the share of ﬁrms inthat country.
The tax rates at the symmetric steady state in the Markov-perfect case is higher than that in
the open-loop case.*21 To see why, consider the situation where both countries stick to theiropen-loop tax schedules predetermined at the beginning until some point in time and country*21 Note that T m (λm = 1/2) − T o (λo = 1/2) = L2 a/[2(L − kZ){L + k(a − Z)}] > 0 as long as a > 0 and Assumptionii

5 holds.

24

h deviates from the schedule by increasing its tax rate. The tax increase of h will cause someﬁrms to move away from there to f . Under full commitment, country f will continue to stickto the predetermined schedule so that such deviation of h will never improve its payoﬀ. In theabsence of commitments, however, h believes that f will respond to this expansion of tax baseby raising its tax rate according to the linear strategy. This consideration is captured by the
feedback eﬀect in the tax rates. The loss of ﬁrms in h by raising its tax rate will be mitigatedto some extent by the reaction of f . Thus, h’s deviation from the open-loop schedule will
increase its overall payoﬀ. If f anticipates this, it is also desirable for f to deviate from theopen-loop schedule.
In the open-loop case, when trade costs are low and the ﬁrms’ incentive to agglomerate isstrong, the tax rates at the symmetric state are negative so that both governments agree on the
core-periphery outcomes. In the Markov-perfect case, however, competition becomes lesssevere because of the lack of commitment and the tax rates at the symmetric state may bringa higher payoﬀ to both governments than those at the position of the core even when agglomeration forces are strong. The Pareto criterion of Assumption 4 implies that the dispersed
conﬁguration emerges in a wider range of trade costs.
Figs. 4 and 5 graphically illustrate the above discussions. Parameter values are L = 1,
I = 5, k = 0.5 in Fig. 4 and k = 1 in Fig. 5. The tax and payoﬀ of country f have amirror image. When parameter k is small and thus tax adjustment costs (−kT i2 /2) are lesssevere, the tax rate and the tax revenue (Lλi T i ) at the symmetric state become higher.*22 In
this case, the symmetric state generates a higher payoﬀ than the agglomeration rent (Fig.
4-(b)) and emerges in equilibrium whatever trade costs may be by Assumption 4. When k
becomes larger and the motive of tax smoothing is strong, the tax rate and the tax revenue
become lower. The symmetric state may become less desirable than the concentrated states(Fig. 5-(b)). The core-periphery patterns are observed in equilibrium when trade costs aremin between the two bifurcations of tax competition (τ ∈ (τm
1 , τ2 )) where the agglomeration

rent is large. At the expense of ﬁrms, the governments’ payoﬀ at the symmetric state under
MPNE is higher than that under OLNE.

*22 It can be checked that d(am )/dk < 0 and d[T m (λm = 1/2)]/dk < 0.
i
225

Fig. 4. Country h’s (a) tax rate and (b) payoﬀ at the steady state under MPNE with small k.

26

Fig. 5. Country h’s (a) tax rate and (b) payoﬀ at the steady state under MPNE with large k

Therefore, the following proposition is obtained:
Proposition 2. The MPNEs are characterized in the following way (except for the nongeneric cases: τ = τ∗ or Z = 0).
√ √4(i) If 0 < k ≤ (2 − 2 3)L/Z ≈ 0.14L/Z, the symmetric conﬁguration is the only stablesteady state no matter what trade costs may be. The tax rates at the steady state are given by!
!
L(Z − am
1
1
2)
m
m
m
m
Th λ == Tf λ ==,m222[k(Z − a2 ) − L]27wherepi
1 h
L + kZ + (L + kZ)2 − 6kLZ > 0.
3k
√ √4
√
(ii) If (2 − 2 3)L/Z < k < (2 − 3)L/Z, there are two cases:
am
2 =

m
(ii-i) If trade costs are suﬃciently high (τ ≥ τm2 ) or suﬃciently low (τ ≤ τ1 ), the symmetricconﬁguration is the only stable steady state and the tax rates at the steady state are given by!
!
L(Z − am
1
1
2)
m
m
m
m
Th λ == Tf λ ==.
222[k(Z − am2 ) − L]m(ii-ii) If trade costs are intermediate (τm
1 < τ < τ2 ), both the symmetric and concentrated

conﬁgurations are stable steady states and the tax rates at each steady state are given byZ
Case λm = 0 : T hm (λm = 0) = 0, T mf (λm = 0) = ,2!
!
L(Z − am
1
1
1
2)
m
m
m
m
m
Case λ = : T h λ == Tf λ ==,2222[k(Z − am2 ) − L]orZ
Case λm = 1 : T hm (λm = 1) = , T mf (λm = 1) = 0,2
√ √4mwhere τm
2 3)L/k.
1 and τ2 are the solutions of Z = (2 −Proof. See Appendix.
Dynamic tax competition between governments that have no commitment leads to increases in their taxes at the symmetric state and thereby acts as a dispersion force. Especially when the adjustment of tax is not so costly (small k), sharing ﬁrms between countries
is more desirable than collecting all the ﬁrms even when trade costs are intermediate and theagglomeration rent is large. This result is in stark contrast to the results from OLNE and fromthe existing literature that obtain the superiority of the core in a static setting (Baldwin andKrugman (2004); Borck and Pﬂüger (2006)). In the fully dynamic model, the credibility ofpolicies signiﬁcantly changes the result of tax competition.
The implication of MPNE that the lack of commitment gives governments a higher tax rate
may explain the recent corporate tax policies of France. France has been one of the highestcorporate-tax countries in the EU since the mid-2000s, while at the same time it has been oneof the most strong opponent of the world’s lowest corporate tax rate of Ireland and has also
28been one of the biggest supporter of tax harmonization in the EU (see, e.g., Mitchell (2009);
Stewart (2011)). Part of reason for the high tax in France may be that France has urged othercountries to increase their tax rate and thereby expected to lessen losses from the relocationof domestic ﬁrms to low-tax countries. It is also worth noting that the direction of tax policies
in France is sometimes subject to change (e.g., the regime change in 2012), which indicatesthat commitment is not so stringent. This story follows exactly what happens in MPNE.
Not only at the steady state, but also in the interim, the tax rate at the symmetric state under
MPNE is higher than that under OLNE. Fig. 6 shows the transition paths with the same trade
costs τ = 0.4 as in Fig. 3-(a). Initial conditions and other parameter values are also thesame: λ0 = 0.7, L = 1, I = 5, k = 1, and ρ/γ = 0. Unlike the open-loop case, the coreperiphery patterns exhibit no transition dynamics (see Appendix B for details). Because of
the subgame perfectness of MPNE, even amidst unexpected shocks, governments optimallyalter their taxes according to the linear Markov-perfect strategies.

Fig. 6. Transition path under MPNE with the same trade costs as in Fig. 3-(a).

294.3 Discussion: What if ﬁrms are forward looking?
The analysis has so far dealt with the situation where ﬁrms choose their location by looking
at their current payoﬀs (the migration equation (6-1)). This subsection brieﬂy argues that themain results would remain unchanged if ﬁrms are forward looking. Though a full characterization of far-sighted ﬁrms is the outside scope of the paper, I can examine some stylizedcases.
The forward-looking behavior of ﬁrms is already reﬂected to some degree in the migration
equation. The speed of adjustment γ captures to what extent ﬁrms value their future payoﬀs
because the larger γ helps ﬁrms reach the stable steady state more quickly and thus makes
ﬁrms focus only on the long-run outcome.*23 With this interpretation in mind, two cases are
examined: in the one case ﬁrms are very far-sighted (γ = ∞) and in the other case they
discount their future payoﬀs to a certain extent (γ ∈ (0, ∞).).*24
The case of γ = ∞In this case, the ﬁrms reach the stable steady state in a moment andtherefore the governments also care solely about it regardless of their discount rate ρ. The
dynamic tax game reduces to the static game.*25 This means that the governments choose
their tax rate once and it does not matter whether or not they commit. As shown in the formal
analysis in Appendix C, the results are summarized as follows. When trade costs are high,the dispersed conﬁguration is the equilibrium one. When trade costs are low, however, thecore-periphery pattern is the equilibrium one where the country that has more ﬁrms at the
beginning of the game becomes the core. Since there is no transition path, what matters here
is in which state the game starts. In this sense, the initial state acts as an equilibrium selection
device. These location patterns are the same as those in the open-loop case discussed in
Section 4.1 except for the role of the initial state and are also consistent with many of theexisting studies on agglomeration and tax competition (e.g., Baldwin and Krugman (2004);*23 The formal dynamic analysis of the new economic geography model by Oyama (2009b) shows that a suﬃ-

ciently high speed of adjustment and a suﬃciently low future discount rate of ﬁrms give the same result.
*24 The very myopic ﬁrms, which values the today’s payoﬀ most, can be represented as the almost zero speedof adjustments γ = 0, in which case the ﬁrms stay in the initial state. The governments set their tax rate byconsidering only the initial location of ﬁrms.
*25 At the steady state, condition (6-3) for the optimality of the open-loop strategies (or (7) for the optimalityof the Markov-perfect strategies) becomes: (ρ/γ − Z)µh = (L/γ)T h . When γ = ∞, the condition holds forµh = 0. The zero co-state variable means that the current strategies of agents do not aﬀect future payoﬀs andmakes the analysis static.

30

Borck and Pﬂüger (2006)).
The case of γ ∈ (0, ∞)If the governments are very far-sighted (ρ ≃ 0), the degree offriction ρ/γ is suﬃciently close to zero as in Assumption 3 and the results of the main analysis
hold. Consider the situation where the degree of friction is not so small. In this case, it isexpected that the analysis yields similar results to those under the suﬃciently small degree offrictions. It can be veriﬁed that the Markov-perfect strategies gives the governments a larger
incentive to tax at the symmetric steady state than the open-loop strategies.*26 Therefore,
the dispersed conﬁguration may arise in a wider range of trade costs in the Markov-perfectequilibrium than in the open-loop equilibrium as in Proposition 1 and 2.*27 The main results
under Assumption 3 (ρ/γ ≃ 0), in particular, the importance of commitment aspect, would
carry over to a more general case (ρ/γ ∈ (0, ∞)).
In a complete dynamic analysis which rigorously considers forward-looking behavior ofgovernments and ﬁrms, government commitment toward both rival governments and ﬁrms
can be important. Since the expectations of both sides interact, multiple equilibria may occur
for a broad range of parameters. Commitment aspect as well as initial conditions may be
crucial in equilibrium selection. One possible way to deal with such indeterminacy is tointroduce asymmetries of market size and trade costs and utilize ‘potential methods’ as in
Oyama (2009a). Further research is needed in this area.

5 Conclusion
Previous studies on agglomeration and tax competition deal mainly with static situations
that imply that the initial industrial location matters and the core maintains its initially advantageous status. This paper contributes to this literature by introducing fully dynamic strategicinteractions between governments.
This paper has considered OLNE and MPNE, contrasting equilibrium concepts in the sensethat the former presumes full commitment of governments and the latter does not. As a result of tax competition under open-loop strategies, a dispersed spatial conﬁguration emerges
when trade costs are high, while when they are low, an agglomerated conﬁguration appears.
*26 Note that T m (λm = 1/2) − T o (λo = 1/2) = L2 a/[2{L + k(ρ/γ − Z)}{L + k(ρ/γ + a − Z)}] > 0 as long as a > 0iiand Assumption 5 holds.
*27 Unlike the case of ρ/γ ≃ 0, since the governments put emphasis on the way to the steady state, the initial state

would starts to matter in determining which country will be the core.

31However, if the governments adopt Markov-perfect strategies, the dispersed conﬁguration
may be observed regardless of trade costs (note that the agglomerated conﬁguration may also
happen under low trade costs). The conclusion of the superiority of the core in the existing literature still holds in the dynamic model when the governments have full commitment, but theinitial position is no longer important. Instead, the expectations of governments are essential
in determining the core. The result that Markov-perfect strategies insulate the industrial ﬁrms
from locating together especially when agglomeration tendencies are prevailing is simply because a lack of government commitment to their tax policies induces the governments to raise
their tax rates, while an enormous decrease in tax is needed to convince ﬁrms to relocate.
These results suggest that, to become the winner of tax competition, it is necessary fora government to make foreign rivals expect that it is committed to its tax policies. This
may explain why Ireland has succeeded in hosting large multinational enterprises. However,fully credible policies are not free from problems; they do not guarantee optimality once any
unexpected events strike the economy. As an example, it may have been desirable if Ireland
had modiﬁed its tax schedule when it faced ﬁnancial collapse in 2010.

Acknowledgement
I deeply acknowledge Jota Ishikawa and Taiji Furusawa for their invaluable advice and continuous encouragement. I am also indebted to Daisuke Oyama whose suggestions greatly improved the analysis. Thanks are also due to Naohito Abe, Richard Baldwin, Shota Fujishima,
Hideshi Itoh, Gang Li, Jim Markusen, Kiminori Matsuyama, Toshihiro Okubo, Tetsuya Saito,
Etsuro Shioji, E.Young Song, Yoichi Sugita, and seminar and meeting participants at Hitotsubashi University, Otaru University of Commerce, University of Toyama, Sogang University
and Toyo University for their helpful comments. Advice on academic writing from Philip
MacLellan was extremely useful. All remaining errors are mine. Financial support fromGrant-in-Aid for JSPS Fellows is gratefully acknowledged.

32

Appendix
Appendix A. Proof of Proposition 1.
First, I derive the system of diﬀerential equations that characterize the economy. Then, I
conﬁrm that the symmetric state is the saddle stable when trade costs are high, and that theconcentrated states are stable when trade costs are low.
By combining the necessary conditions for h, (6-1), (6-2) and (6-3) with the corresponding
conditions for f , the dynamic system is obtained as follows:
!
1
λ̇
=Z λ−
− (T h − T f ),γ
2γLZ
kṪ h = L(2γZ − ρ)λ + k(ρ − γZ)T h + γLT f −+ γ(νh1 − νh2 ),2!
3γZkṪ f = L(ρ − 2γZ)λ + γLT h + k(ρ − γZ)T f + L− ρ − γ(ν1f − ν2f ).
2
To verify that the system depends only on a ratio between ρ and γ, I apply the change ofvariables: λ′ (t) = λ(t/γ), T h′ (t) = T h (t/γ) and T ′f (t) = T f (t/γ). The system can be rewritten
as the following:!
1− (T h′ − T ′f ),λ̇ = Z λ −
2
!
!
ρ
LZ
ρ ′
′
λ +k
− Z T h′ + LT ′f −+ (νh1 − νh2 ),kṪ h = L 2Z −
γ
γ
2
!
!
!
ρ
ρ
3Z ρ
′
′
′
kṪ f = L
− 2Z λ + LT h + k− Z Tf + L−− (ν1f − ν2f ).
γ
γ
2
γ
′

′

Assumption 5 requires that the ratio (the degree of friction) is suﬃciently close to zero:!
1′′
λ̇ = Z λ −− (T h′ − T ′f ),(A.11)
2LZ
kṪ h′ = 2LZλ′ − kZT h′ + LT ′f + (νh1 − νh2 ) −,(A.1h)
2
3LZ
kṪ ′f = −2LZλ′ + LT h′ − kZT ′f − (ν1f − ν2f ) +.
(A.1f)
2
When trade costs are high: τ > τ∗ or Z < 0

In this case, as was discussed in Section 4.1,

the tax rate and the instantaneous payoﬀ of both countries at the symmetric steady state are33

positive and those at the concentrated steady state are non-positive. Thus, only the symmetricstate is achieved in equilibrium. At the interior equilibrium, it holds that νh1 = νh2 = ν1f = ν2f =0. The system can be rewritten in matrix form: ′  λ̇   Z
Ṫ ′   h  =  2LZ/kṪ ′f−2LZ/k

−1
−Z
L/k



λ′ − 1/2

1   T ′ − T o (λo = 1/2)
L/k  h ,−Z T ′f − T o (λo = 1/2)

where the tax rate at the steady state is given by
!
LZ
1
o
o
=
> 0.
T λ =
2
2(kZ − L)
To examine the stability, we check the determinant of the coeﬃcient matrix.
Z
2LZ/k−2LZ/k

−1
−Z
L/k

1
L/k
−Z

=

Z(3L − kZ)(L − kZ).
k2

Under Assumption 5 that k is positive but small enough, the determinant is negative. This
ensures that the matrix has one negative eigenvalue and two positive eigenvalues. Since thesystem has one state variable λ′ and two control variables T h′ and T ′f , the negative determinant
implies the steady state is saddle stable. The transversality conditions limt→∞ exp(−ρt)µh =limt→∞ exp(−ρt)µ f = 0 hold by construction. Since the trajectories of the variables continuously approach the steady state, the jump conditions have no relevance here.
When trade costs are low: τ < τ∗ or Z > 0□

In this case, only the concentrated steadystates are possible because the payoﬀ for both countries at the symmetric steady state is
negative. The proof of the stability is in two steps. First I show that the co-state variablesµ′h (t) = µh (t/γ) and µ′f (t) = µ f (t/γ) satisfy the jump conditions (6-6), (6-7) for h and theircorrespondents for f for some η. Then I conﬁrm the existence of the Lagrange multipliers
for the boundary conditions, νh1 , νh2 , ν1f and ν2f .
By using the necessary conditions, the migration equation can be written as a function of
the co-state variables:
!
!
1
γ
2L
′
′
λ −
+ (µ′h + µ′f ).
λ̇ = Z −k2k
Suppose that s is the time when the economy hits λ′ = 1. The payoﬀ at the symmetricsteady state is negative; therefore, after the economy hits the boundary, it is optimal for both34

countries to keep their position. Thus, I obtain the following:
!
1
2L
γ
′ −
λ̇ (s ) =Z−
+ [µ′h (s− ) + µ′f (s− )] ≥ 0,
2
k
k
!
1
2L
γ
Z−
+ [µ′h (s+ ) + µ′f (s+ )] = 0.
λ̇′ (s+ ) =2kk
The jump conditions require that η1h (s) = η1f (s) = 0, µh (s− ) − µ′h (s+ ) = −η2h (s) and µ f (s− ) −µ f (s+ ) = −η2f (s). The diﬀerence of these equations becomes:γ ′ −[µ (s ) − µ′h (s+ ) + µ′f (s− ) − µ′f (s+ )]k hγ= − [η2h (s) + η2f (s)].
k

0 ≤ λ̇′ (s− ) − λ̇′ (s+ ) =Since η is non-negative, only η2h (s) = η2f (s) = 0 satisﬁes the above condition. Thus, the jumpconditions hold for η1h (s) = η1f (s) = η2h (s) = η2f (s) = 0. λ̇′ (s− ) = λ̇′ (s+ ) = 0 implies that λcontinuously reaches the boundary and remains there. The co-state variables will not change
once the economy hits the boundary, so that the transversality conditions also hold.
From time s onwards, the tax rate is determined so as to satisfy λ̇′ = Z(1−1/2)−(T h′ −T ′f ) =0. Country f , which has no ﬁrms, chooses T of (λo = 1) = 0. Thus country h sets a positive tax
rate T ho (λo = 1) = Z/2.
Finally, I check that the Lagrange multipliers ν satisfy the conditions. The multipliers forthe left boundary are not binding so that I obtain νh1 = ν1f = 0. Those for the right boundaryνh2 and ν2f can be obtained by making Ṫ h′ = Ṫ ′f = 0 and substituting the steady state values
into (A.1h) and (A.1f):
0 = kṪ h′ = 2LZ · 1 − kZ ·⇒ νh2 =

Z(3L − kZ),
2

0 = kṪ ′f = −2LZ · 1 + L ·Z
LZ+ L · 0 + (0 − νh2 ) −
,
2
2

Z
3LZ
− kZ · 0 − (0 − ν2f ) +.
22⇒ ν2f = 0.
νh2 is positive as long as Assumption 5 holds.

□

Appendix B. Proof of Proposition 2.
35

First, I solve for the parameters in the linear Markov strategy, a and b. Second, I compare
the payoﬀ at the three steady states for diﬀerent levels of trade costs and eliminate Pareto
ineﬃcient states (if any) by Assumption 4. Finally, I conﬁrm the stability of the selected
steady states.
Derivation of the parameters a and b in equilibrium

As in Appendix A, from thenecessary conditions both for h and f , the dynamics of the tax rate of h is characterized asfollows:
"
!
#
"
!#
∂T f
∂T f
γLZ
kṪ h = L γ 2Z +−ρ λ+k ρ−γ Z+
T h + γLT f −+ γ(νh1 − νh2 ).
∂λ∂λ2I apply the change of variables: λ(t) = λ′ (t/γ), T h (t) = T h′ (t/γ) and T h (t) = T h′ (t/γ), and let
ρ/γ close to zero:kṪ h′ = L 2Z +
∂T ′f ∂T ′f  λ′ − k Z + T ′ + LT ′ − LZ + ν1 − ν2 .
h
h
h
f
∂λ
∂λ
2

Imposing the linear speciﬁcation of the strategies T h′ = aλ′ + bh and T ′f = a(1 − λ′ ) + b f yields
kṪ h′ = L(2Z − a)λ′ − k(Z − a)(aλ + bh ) + L[a(1 − λ′ ) + b f ] −LZ+ νh1 − νh2 .
2

(B.1)

The motion of T h′ can be written in a diﬀerent way:Ṫ h′ ="!
#
∂T h′ dλ1′′′
=a Z λ −− (aλ + bh ) + {a(1 − λ ) + b f } .
∂λ′ dt2(B.2)

Combining (B.1) with (B.2) yields:


Z
h
i
LZ
2
′
2
3ka − 2(L + kZ)a + 2LZ λ − ka + L + k+ 2bh − b f a + Lb f − kZbh −+ νh1 − νh2 = 0,22
(B.3h)
which is an identity with respect to λ ∈ [0, 1]. For f , an analogous expression can be obtained:


Z

h
i
2
′
2
3ka − 2(L + kZ)a + 2LZ λ − 2ka + L + k+ 2bh − 2b f a − Lbh + kb f Z − 3LZ + ν1f − ν2f = 0.
2
(B.3f)
I solve for a by making the coeﬃcient of λ′ zero:
3ka2 − 2(L + kZ)a + 2LZ = 0,
p
p
i
i
1 h
1 h
2 − 6kLZ , am =2 − 6kLZ .
L
+
kZ
−
L
+
kZ
+
⇒ am
=
(L
+
kZ)
(L
+
kZ)
1
2
3k
3k
36

Assumption 5 suggests that both roots are real. I then derive the tax rates at each steady state.
At the symmetric state, it holds that νh1 = νh2 = 0 and bh = b f = b by the symmetry of thesystem. From the identity (B.3), I can solve for b:


Z
NZ
+ 2b − b am −+ Lb − kZb = 0,
− k(am )2 + N + k
2
2
LZ − (kZ − 2L)am⇒ bm =.
6[k(Z − am ) − L]
Thus
T hm

!
!
1
1
1
L(Z − am )mm
λ == Tf λ == am · + bm =.
22
22[k(Z − am ) − L]m

As for the right boundary state, f has no tax base and sets its tax rate at zero. Then,f chooses its tax rate so as to keep the distribution stable (λ̇′ = Z(1 − 1/2) − (T h′ − 0) =Z/2 − T h′ = 0). Parameter b associated with the tax rate at the left boundary are determinedin the following way:Z
2
T mf (λm = 1) = am · 0 + bmp = 0

T hm (λm = 1) = am · 1 + bm
c =Z
− am .
2⇒ bmp = 0.
⇒ bm
c =

□
Ranking of payoﬀs at the steady states

Next, I compare the payoﬀ at the symmetricstate with the state at the right boundary. It is easy to check that the tax rate at λm = 1/2 formm m
am
2 is higher than that for a1 and it is smaller than the unconstrained tax rate (0 < T i (λ =m mmm
1/2; am1 ) < T i (λ = 1/2; a2 ) < L/k = argmaxWi (λ = 1/2)). Thus a2 always gives themhigher payoﬀ than am
1 does. Assumption 4 suggests that a2 should be adopted when themsymmetric equilibrium emerges. Denote Wh (λm = 1/2) ≡ LT h (λm = 1/2; am2 )/2 − k[T h (λ =
2
m
2
*28
1/2; am2 )] /2 and Wh (λ = 1) ≡ LZ/2 − kZ /4. A tedious calculation reveals*28 Note that√√√!
 (kZ − 2L)[5k3 Z 3 − 2k2 (15L + 2 D)Z 2 + 2kL(21L + 8 D)Z − 4L2 (L + D)]
1− Wh λm = 1 =,√√2
8k[5k2 Z 2 − 4k(5L + D)Z + L(17L + 8 D)]√ √4where D ≡ (L + kZ)2 − 6kLZ. The solutions of Wh (λm = 1/2) − Wh (λm = 1) = 0 are Z = (2 ± 2 3)2L/k,
2L/k and 4L/k.

Wh λm =

37

√ √4
!
2 3)L
1
m
m
Z≤
⇒ Wh (λ = 1) ≤ Wh λ =
,
k
2
√ √4
√
!
(2 − 2 3)L(2 − 3)L
1m<Z<⇒ Wh λ =< Wh (λm = 1) .
k
k
2
√ √4
Remembering Wh (λm = 0) = 0, when 0 < k ≤ (2 − 2 3)L/Z, it follows that:
!
!
1
1
m
m
m
m
, Wh (λ = 1) ≤ Wh λ =.
If 0 < τ < τtrade ⇒ Wh (λ = 0) < Wh λ =
2
2
√ √4
√
And when (2 − 2 3)L/Z < k < (2 − 3)L/Z, it follows that:
!
1
m
m
m
m
,
If 0 < τ ≤ τ1 ⇒ Wh (λ = 0) < Wh (λ = 1) ≤ Wh λ =
2
!
1
m
m
m
m
If τ1 < τ < τ2 ⇒ Wh (λ = 0) < Wh λ =< Wh (λm = 1) ,
2
!
!
1
1
m
m
m
m
m
, Wh (λ = 1) ≤ Wh λ =.
If τ2 ≤ τ < τtrade ⇒ Wh (λ = 0) < Wh λ =
2
2
(2 −

The solution of Wh (λm = 1/2) − Wh (λm = 1) = 0 which lies in τ ∈ (0, τtrade ) is Z = (2 −
√ √4
m
m
m
∗
2 3)L/k. Solving this equation for τ gives τm
1 and τ2 . τ2 (> τ1 ) is smaller than τ , whichis the positive solution of Z = 0.

□

Stability of the symmetric state

The boundary conditions are not binding so that I have

νh1 = νh2 = ν1f = ν2f = 0. To see that am
2 guarantees the stability of the symmetric state, I lookat the slope of the migration equation:
!
Z
1
′
′
− (T h′ − T ′f ) = (Z − 2am )λ′ + am − .
λ̇ = Z λ −22
Z − 2am
2 is negative regardless of Z or τ, which ensures stability. The fact that the statevariable converges to a ﬁnite value implies that the co-state variables also do so. Thus thetransversality conditions also hold. The jump conditions trivially holds for η1h = η2h = η1f =η2f = 0.

□Stability of the boundary states

I focus on the situation where the economy hits the rightboundary. I can show that jump conditions are satisﬁed in the same manner as in Appendix
A. Consider the smaller root am
1 ﬁrst. At λ = 1, the conditions for the left boundary are notmmbinding so that I obtain νh1 = ν1f = 0. Substituting am
1 , bc , b p into (B.3h), I can derive the38Lagrange multiplier of h for the right boundary:


Z



LZ
m
m
L+k
am
− νh2 = 0
1 + Lb p − kZbc −2
2!
kZ
Z(3L − kZ)2⇒ νh =− L am.
1 +2
2

2− k(am
1) +

m
+ 2bm
c − bpwhich is positive as long as Z > 0 or τ < τ∗ and Assumption 5 holds. The multiplier of f for
the right boundary is derived from (B.3f):
!#
"
3Z
2
m 2
m
m
m
m
− 2k(a1 ) + L + k+ bc − 2b p am
1 − Lbc + kb p Z − 3LZ − ν f = 02⇒ ν2f = 0.
m
am
1 brings the negative slope of the migration equation (Z − 2a1 < 0) so that there are notransition paths and the economy reaches the boundary in a moment. The transversality
conditions trivially hold.
2
If the larger root am
2 is chosen, νh turns out to be negative so that the agglomerated distri-butions, as well as the dispersed distribution, are never achieved in equilibrium. Therefore,
when the core-periphery patterns emerge in equilibrium, the smaller root am
1 and the associ□mated bmc and b p should be adopted.

Appendix C. Analysis under very far-sighted ﬁrms.
I consider the situation where the migration speed is inﬁnitely high (γ = ∞), i.e., ﬁrms
care only about the long-run payoﬀ. As mentioned in the text, the problem now reduces toan one-shot simultaneous game as in the literature. The game proceeds as follows. First,
each government announces an tax rate simultaneously and non-cooperatively. Then, ﬁrms
relocate to the country that aﬀords a higher payoﬀ in the steady states. The game should be
solved backward. That is, the governments choose their tax rate to maximize their one-shot
payoﬀ while taking into account the steady-state distribution of ﬁrms. The economy without
governments has two stable steady states, the dispersed state when trade costs are high andthe agglomerated state when they are low (see Lemma). I treat these cases separately.
When trade costs are high: τ > τ∗ or Z < 039

In this case, the stable steady state will be λthat satisﬁes λ̇/γ = 0:
!
λ̇
1
=Z λ−
− (T h − T f ) = 0,γ
2
1 Th − T f
⇒λ= +
.
2
Z
By substituting this into the one-shot payoﬀ of each government (Wi = Lλi T i − kT i2 /2, i ∈{h, f }) and diﬀerentiating it with respect to its tax rate, I can obtain the response function ofeach government (h and f respectively):
!
!
2L
1 Tf
− k Th + L
−
= 0,
Z
2
Z
!
!
2L
1 Th
− k Tf + L
−
= 0.
Z
2
Z
I solve these equations and obtain the Nash equilibrium tax rates and the eventual share ofﬁrms.
T hs = T fs =LZ,
2(kZ − L)

λs =

1
.
2

These are the same as in the open-loop case (Proposition 1-(i)).
When trade costs are low: τ < τ∗ or Z > 0

In this case, the stable steady state will beeither λ = 0 or λ = 1. Suppose that ∆u∗ (λ0 ) = Z(λ0 −1/2)−(T h −T f ) ≥ 0 holds given the rival’s
tax rate and the initial state and country h will achieve an agglomeration. Then h will set its
tax rate to the level where ﬁrms’ delocation never occurs, i.e., T h that satisﬁes ∆u∗ (λ0 ) = 0,or it will attain the unconstrained maximum by choosing T h = argmax[LT h − kT h2 /2] =L/2. Similar reasoning applies to the case of country f . The best response function of eachgovernment becomes:

n
o


min
L/k,
Z(λ
−
1/2)
+
T

0
f , Z(λ0 − 1/2) − (T h − T f ) ≥ 0,
Th = 0,
Z(λ0 − 1/2) − (T h − T f ) ≤ −ε.
min {L/k, Z(1/2 − λ0 ) + T h } , Z(λ0 − 1/2) − (T h − T f ) ≤ −ε,
Tf = 0,
Z(λ0 − 1/2) − (T h − T f ) ≥ 0.
where ε is a suﬃciently small positive constant, which excludes the indeterminacy of equilibrium tax rate. Since it always holds that L/k > Z(λ0 − 1/2) under Assumption 5, the Nash40

equilibrium tax rates are given by!
1s
T h = Z λ0 −, T fs = 0,2

λ s = 1.

It is worth noting that the tax rate of the core country T hs is inverted U-shaped in trade costs τ(see the deﬁnition of Z in Section 2.4). The result is so called a ‘race to the top’(Baldwin andKrugman (2004); Borck and Pﬂüger (2006)).

41

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