Tax reform, delocation and heterogeneous firms:
Base widening and rate lowering ruleRichard E. Baldwin
Graduate Institute, Geneva and CEPR
andToshihiro Okubo
Kobe University, RIEB1
May 2008; Second draft:July 2006, First draft:October 2005,

ABSTRACT

We model international tax competition allowing for agglomeration forces andheterogeneous firms. This provides a new perspective since a tax schedules havedifferent effects on the international relocation decision of small and large firms(large firms are endogenously more sensitive to tax competition) and thesedecisions affect industry productivity in addition to the usual effects. The modelallows us to study rate-lowering base-widening reforms. We show it is generally
possible to design such a reforms that raises revenue without losing firms.
JEL H32, P16.
Keywords: tax reform, heterogeneous firms, agglomeration forces.

1.

INTRODUCTION

International tax competition has been an important concern for decades, but the heightened
mobility of firms and technology – spurred in part by the rapid improvements in the quality
and price of international communication – has drawn renewed interest.
From the 1980s onward, one common response has been to cut corporate tax rates but widen
the base to which it is applied. The United States, Austria, France, Finland, Germany, Canada
and the United Kingdom have sharply cut the rates and widened the base of their respective
corporate tax regimes between 1980 and 2003.2 Some small European countries such as
Belgium, the Netherlands and Greece have reduced tax rates with an unchanged tax base.3

1

We thank seminar participants at CES-Ifo (26 October 2005), and Kobe (10 May 2008), as well as Dany
Jaimov for excellent editorial assistance. The first draft was written while Okubo was a PhD student at theGraduate Institute supported by NSF Grant No. 100012-105675.
2
The statutory tax rates have substantially reduced in many countries: the United States from 50 % in 1980 to 39% in 2003, Australia from 50% to 30%, Austria from 61%(in 1982) to 34% (in 2003), Canada from 45% to36%,
France from 50% to 35%, UK from 52% to 30%, Germany from 62% to 40%, Ireland from 45% to 13%, andJapan from 53% to 41%. The present discounted value of depreciation allowances has widened in somecountries: UK from 100% (1980) to 73% (2003), Canada from 94% to 73%, France from 81% to 77%, GermanyTax and heterogeneous firms

2

It has been argued that this sort of reform mitigates distortions and spreads the tax burden
more evenly (UK HM Treasury, 2003). Such reforms may also increase tax revenue. As
Devereux, Griffith and Klemm (2002) show, in many OECD countries the effective corporate
tax rates have fallen and tax bases (measured by the present discounted value of allowances
available in plant and machinery) have substantially widened, while at the same time tax
revenue collected has risen.
It is also commonly asserted that such reforms are necessary to prevent the largest firms fromleaving, a point that is easy to see. A broader base and lower rate has a larger impact on theeffective rate of large firms, so the reforms’ impact on international relocation decisions
varies according to corporate size, boosting the reforming nation’s attractiveness for large
firms more than for small firms.
The purpose of our paper is to formally study international tax competition in the presence ofsignificant agglomeration economies and firm heterogeneity. The former allows us toconsider a situation where big economies maintain higher taxes in equilibrium (Ludema andWooton 1998, Kind, Midelfart-Knarvik and Schjelderup 2000, Andersson and Forslid 2003,
and Baldwin and Krugman 2004). The later allows us to consider the differential impact oftax reforms by firm size and to examine the firm-level impact of tax reform – an issue that has
recently been highlighted by empirical studies on the firm-size distribution of tax payments,
which typical finds that tax is paid by quite a small number of firms and the payers are
typically large firms (Baer, 2002; Shome, 2004; Auriol and Warlters, 2005).4
At the theoretically level, our framework opens the door to extensions of the international tax
competition literature. In particular, it allows us to consider the implications of tax reforms(such as the rate-lowering-base-widening) that alter the effective tax rate paid be firms ofdifferent size/profitability. Obviously, such reforms cannot be fully explored theoretically inthe classic international tax competition model which assumes homogenous firms. In
particular, we show that allowing for heterogeneity permits a given tax scheme to have a
different effect on the relocation decision of small and big firms, with the biggest firms being
endogenously more likely to relocate in reaction to high-taxes.
The inclusion of firm heterogeneity permits three extensions of the theoretical analysis ofinternational tax competition. First, it allows the model to capture the very real-world concernthat large/profitable firms find it easier to escape taxation. Second, it allows us to consider therevenue implications of reforms that tilt the size-tax-burden profile in a setting where firms
can relocate to avoid taxation and some do so in equilibrium. This creates the smooth tradeoff between higher tax rates and keeping firms at home. Third, since firm-size is associatedwith firm-level productivity in our model (following that by-now standard approach in Melitz
2003), tax reform has an impact on the average productivity of firms in each nation. In

from 76% to 71% and Australia from 71% to 66%. On the other hand, some small nations have kept the samerates over the twenty year period: 75% in Belgium, 78% in Switzerland and 73% in the Netherlands. Note thatthe present discounted value is calculated for an investment in machinery and plant. See the data resource fordetail: http://www.ifs.org.uk/corptax/internationaltax.zip.
3
Ireland has widened its tax base by reducing deductions while leaving rates unchanged, leading to increased
tax revenue over the decade.
4
For instance, as shown in Baer (2002) 0.4% of taxpayers account for 61% of total domestic tax collection in
Kenya and 57% in Colombia. According to Shome (2004), large taxpayers account for 80-90 percent of the tax
revenue in Asian and Latin American countries. To reflect this phenomenon, an attempt to widen the profit tax
base is one of the most possible ways of raising tax revenue in developing countries. A narrow tax base comesfrom higher opportunity costs and entry costs for small firms. Auriol and Warlters (2005) found that a 1%
increase of the entry sunk cost increases the informal sector by 14% and suggested that reducing market entry
fees in developing countries could enlarge their tax base.

Tax and heterogeneous firms

3particular, a rate-lowering-base-widening reform tends to bring the most productive firms‘back home’ and thus can raise average industry productivity as well as raise tax revenue.
The inclusion of heterogeneous firms is not entirely new to the tax literature, since it has been
already analysed by the important papers of Burbidge, Cuff and Leach (2004, 2006). Their
model, however, is quite different from ours in that the former can be thought of as anextension of the basic tax competition model (perfect competition, immobile labour andmobile capital) which assumes that firm productivity differences are both firm-specific andlocation specific. In our model, the source of firm heterogeneity is the exogenous distributionof firm-level productivity – due, for example, to firm-specific assets such as organisational
capital, intellectual property, or tacit technological knowledge. In Burbidge et al., some firms
have a comparative advantage in one country, while the other firms have it in the other. As aresult, tax rates could be higher in one nation without driving out all firms – even with perfectcompetition. In their model, the firm-level-region-specific productivity differences create aquasi-rent that can be taxed up to a point without firms relocating away from the higher tax.
This leads to spatial sorting whereby firms locate to exploit their own locational comparativeadvantage.
The focus of Burbidge, Cuff and Leach (2006) is the study of tax regimes and the provision of
public goods, rather than tax reforms and firm location with trade costs as in our model. A
related paper is Haufler and Schjelderup (2000) which is a theoretical study concerning
optimal tax systems in the presence of profit shifting in foreign direct investment (FDI) via
transfer pricing. They suggest that the optimal tax reform is to reduce tax rates so as toprevent firms from shifting their profits to foreign nations when FDI is introduced.
Our paper is organised in six sections. The next introduces the application of the basic model.
Section 3 studies the impact of taxation on firm relocation. Section 4 explores implications oftax reform, and Section 5 considers the impact of globalisation (i.e. free trade). The lastsection provides our concluding remarks.

2.

THE HETEROGENEOUS MOBILE FIRMS MODEL

This section introduces the basic economic model with internationally mobile heterogeneousfirms. It is best thought of as a marriage of the Meltiz (2003) model and the ‘footloose capital’model of Martin and Rogers (1995).

2.1. The footloose capital model with heterogeneous firms
We work with a world consisting of two nations (North and South) and two sectors(manufacturing and the numeraire sector). Before turning to the equations, we discuss thebasic intuition behind the forces that affect the location of industry in the absence of taxes.
Most of the basic forces in the model are not directly related to the heterogeneity of firms.
The manufacturing section is marked by Dixit-Stiglitz competition, increasing returns at thefirm-level and trade costs. As is well-known from the international trade and economic
geography literature, this combination of assumptions generates both agglomeration anddispersion forces. The agglomeration force stems from the fact that firms want to locate in the
big market (other things equal) to reduce their trade costs. This agglomeration effect iscountered by a dispersion force known as the ‘local competition’ effect. That is, while
locating in the big market allows firms to save on trade costs, the presence of many firms also
implies tougher competition. Since firm want to be far from their competitor (other things

Tax and heterogeneous firms

4equal), this is a dispersion force.5 The location equilibrium is marked by an international
division of firms that just balances the agglomeration and dispersion forces.
Firm heterogeneity introduces new effects since the balance of agglomeration and dispersion
forces varies according to firm size. The ultimate source of firm-level heterogeneity in ourmodel is firm-level differences in marginal cost (productivity), which implies that firms withlow marginal costs charge a lower price and thus sell more and earn higher operating profits.
Since different firms sell different amounts, the balance of agglomeration and dispersion
forces varies by firm size. In particular, the trade cost saving aspect of big-market location isespecially attractive to big firms that sell a lot. Likewise, the competition-protection aspect oflocation in the small market is especially attractive to firms that sell little. The thrust of this isthat large firms tend to agglomerate preferentially in the large region. In other words, theequilibrium tends towards a spatial separation of firms by size with the big market tending tohave a disproportionate share of large, highly productive firms.
This feature of the model is the key to our novel tax analysis, since it means that changes inthe tax gap between the big and small markets will lead to changes in the spatial segmentation
by firm size. We now turn to introducing the model more formally.
2.1.1

Basic assumptions

The manufacturing sector (‘industry’) consists of firms that each produce a differentiatedvariety and compete in a monopolistic competition setting. The tastes of the representative
consumer in each region are quasi-linear:(1)(

U = μ ln CM + C A , CM ≡ ∫i∈Θci 1-1/ σ di)1/( 1-1/ σ ),σ >1> μ > 0

where CM and CA are, respectively, consumption of the composite of M-sector varieties andconsumption of the numeraire-sector (good A). Here, σ > 1 denotes the constant elasticity of
substitution between any two M-sector varieties, μ is the expenditure share of manufacturing
goods, and Θ is the set of all varieties consumed.
Heterogeneity in our model stems only from differences on the supply side. Each
manufacturing firm requires a unit of capital as its fixed cost and uses only labour in thevariable costs. However, firms have heterogeneous unit labour requirements. The marginal
production cost for firm i is given by the wage rate ‘w’ times its firm-specific unit labour
coefficient, denoted as ‘ai’. It may help to think of ‘capital’ as knowledge capital, i.e. eachfirm’s blueprint for producing its unique variety. The ‘ai’ is associated with the firm’s
blueprint, so it is unaffected by the firm’s location choice (i.e. the unit labour input coefficientis firm-specific, not market-specific). Since all Dixit-Stiglitz varieties enter (1) symmetrically,
we see that heterogeneity enters only on the supply side.
Each nation’s endowment of labour and capital is fixed, as are all of the firm-level ai’s. To be
concrete and to keep the analysis tractable, we assume each nation’s distribution of ai’s isdescribed by the Pareto distribution:
ρ

(2)

⎛ a ⎞
G[a] = ⎜ ⎟ ;⎝ a0 ⎠

ρ ≥ 1 ≡ a0 ≥ a ≥ 0

5That is, firms can use international trade costs as a partial shield from competitors located in the big market bylocating in the small market.

Tax and heterogeneous firms

5Here ρ is shape parameter and a0 is the scale parameter, i.e. the highest possible a; wenormalise a0 to unity by choice of units.
Capital is internationally mobile while labour is not. Since each firm is associated with aparticular unit of capital (a blueprint), capital mobility is synonymous with firm mobility.
Trade in manufactures is subject to ‘iceberg’ trade costs in the sense that firms must ship τ > 1
units of their good in order to sell one unit in the other nation.
Since the main focus of the analysis is on the interaction between taxes and the location ofindustry, we make the numeraire sector as simple as possible. Specifically, we assume it ismarked by constant returns, perfect competition, costless trade and that it uses only labour.
The thrust of our analysis concerns the impact of taxes on firm migration. Since we do notwant to conflate technology-driven effects on location with those of taxes, we assume theG[a] is identical for the two regions. Moreover, to avoid capital movement that is driven byunequal capital-labour ratios, we assume that the nations have identical capital-labour ratioseven though North is bigger, i.e. North has proportionally more of both L and K, so nations
differ only in size.

Figure 1 shows the distribution of a’s in North and South. The distribution in the North isG[a] K; in the South it is G[a] K*, where K and K* are the North’s and South’s endowment of
capital (i.e. mass of blueprints/varieties). This means that the total mass of firms in the Northand South are K and K*, respectively.6

KG[a]frequency
1

K*G[a]G[a]0a0‘a’ (marginal
costs)

Figure 1: Endowed distribution of capital and marginal costs in North and South.
The wage rate in each nation is set in a competitive labour market, but the reward to eachfirm’s unit of capital is determined by the firm’s Ricardian rent, i.e. its operating profit. As aconsequence, different firms will earn different rewards on their capital in equilibrium. As in
Melitz (2003), the most efficient firms will earn the highest reward on their capital. Melitz(2003) shows that the aggregate level of capital can be endogenised such that the averagereward to capital equals the discount rate, but allowing for this would unduly complicate ourmodel. Instead, we take the nations’ capital stocks and G[a] as part of the nations’endowments.
2.1.2

Intermediate results

The presence of the simplified numeraire sector facilitates the general equilibrium analysis
substantially. Constant returns, perfect competition and zero trade costs equalise nominal
6

Since we take the range of varieties to be continuous, we speak of the ‘mass’ of firms with a particularmarginal cost. We assume that the mass is the same for every level of marginal cost (this is demonstrated in
Melitz (2003) as the outcome of an endogenous entry/exit process).

Tax and heterogeneous firms

6wage rates across nations and we choose the units of labour such that w=w*=1.7 This meansthat all differences in manufacturing firms’ marginal costs boil down to differences in theira’s. Moreover, firm relocation does not affect wages paid, so factor costs are never an issue infirms’ location decisions.
As is well-known, Dixit-Stiglitz monopolistic competition implies that the profit-maximisingproducer price of a typical firm with marginal cost aj is:

pj =

aj
1 − 1/ σand that ‘mill pricing’ is optimal, so the price of variety-j in the other market is just τ times
the producer price pj.
Utility maximisation generates the familiar CES demand functions in the manufactures
sector.8 For example, the demand for variety j in the North market is:(3)

p j −σ B ;E
B ≡ 1−σ ,P

P1−σ ≡ ∫i∈Θ

pi1-σ diwhere B can be thought of as the “per-firm demand” that firms take as given under DixitStiglitz competition; P is the usual CES price indices in the Northern market (Θ is the set of
all varieties consumed).
A second well-known property of Dixit-Stiglitz competition is that operating profit of firm-j
equals 1/σ times the firm’s revenue.9,10 The firm-specific revenue of a typical North-based
firm in the Northern market is just the consumption given by (3) times that firm-specificprice. Using similar calculations for operating profit earned on Southern-market sales, thefirm-specific operating profits for a North-based firm is:π [ p ] = p1−σ ( B + φ B * ) ;0 ≤ φ ≡ τ 1−σ ≤ 1~~where B * is the Southern market version of B in (3) and φ is the parameter that gauges the‘freeness’ of trade (recalling that 1-σ < 0, φ ranges from zero when iceberg trade costs areprohibitive, i.e. τ = ∞, to unity when the trade costs are zero, i.e. τ = 1).

(4)

Four features of (4) play important roles in the subsequent analysis. First, all firms earn
positive operating profit in equilibrium (this is the reward to capital, i.e. Ricardian rent).
Second, since (1-σ) < 0, the most efficient firms – i.e. those with relatively low marginal cost
and thus with low relative prices – are the most profitable. Third, a North firm that finds it
optimal to charge a producer price of p when it is located in the North would find it optimal tocharge the same producer price if it relocated to the South (due to wage equalisation and theconstant Dixit-Stiglitz mark-up). Thus its operating profit when located in the South is:(5)7

π *[ p] = p1−σ (φ B + B * )This holds for all possible equilibriums only if the size difference between the nations is not too great; inparticular, the small South needs to be big enough to accommodate all industry and still have some labour
leftover to employ in the numeraire sector.
8
Individual demand for a typical variety j is c(j)=p(j)-σμ/Δ, where Δ≡ ∫p(i)1-σdi and the integral is over all
available varieties, μ is expenditure on all varieties.
9
Given the iceberg trade cost, the consumer/producer price gap exactly offsets the gap between the quantity ofgoods shipped and the quantity of goods consumed. This is a well-known feature of Dixit-Stiglitz competition.
10
A typical first order condition is p(1-1/σ)=wa; rearranging, the operating profit, (p-wa)c, equals pc/σ.

Tax and heterogeneous firms

7

The difference is that being South-based, the firm adds the iceberg trade cost to its price in the
Northern market but not in its price in the Southern market. Fourth, comparing (4) and (5), it~~
is clear that a firm’s profit depends upon its location as long as B and B * are not identical.

2.2. Locational equilibrium with capital mobility but no taxes
Firms’ locational responses to taxes are at the heart of the model, so it is useful to consider
relocation tendencies in the absence of taxes. The key to a firm’s location decision is the firmlevel difference between operating profit earned when the firm is located in the North versus
the South. Without taxes this is:p1−σ (1 − φ )( B − B * ) / σ

π [ p] − π *[ p] =(6)

Plainly the sign of the gap turns on whether the per-firm demand in the Northern market, i.e.
~~
B , is bigger than the per-firm demand in the Southern market, B * . These, in turn, depend
upon the location of firms (trade costs implies that competition is somewhat localised). In thesituation at hand, no firms have moved yet so the mass of firms located in the North and~South are K and K* respectively. To calculate that P’s and thus the B ’s, we change variablesof integration so that the Northern CES price index integral is:
σ −1

⎛ 1⎞
= ⎜1 − ⎟
⎝ σ⎠

1−σ

P

∫ ( Ka
1

1-σ

a =0

)

+ φ K *a1-σ dG[a ]Using (2) to solve the Northern integral and its Southern counterpart, we get:
σ −1

1−σ

(7) P

⎛ 1⎞
= ⎜1 − ⎟⎝ σ⎠λ ( K + φ K ) , (P )
*

* 1−σ

σ −1

⎛ 1⎞
= ⎜1 − ⎟⎝ σ⎠λ (φ K + K * ) ;

λ≡

ρ
>0
1- σ + ρ

where λ is a collection of parameters that is positive assuming a regularity condition, namely(1-σ+ρ) > 0, that ensures that the integrals converge.
To sign the profit gap in (6), we use (7) and the fact that North is a scaled up version of South,
so it has a share s of both world expenditure and world capital. Thus:
B 1 + φ s /(1 − s )≡> 1;B * 1 + φ (1 − s ) / s

s≡

E
K
=
E + E* K + K *The inequality holds as long as the North is bigger, i.e. s > ½. Thus, in the initial situation
where no firms have yet moved, the per-firm demand is larger in the big Northern market, i.e.
B > B * .11 Intuition for this result (which is well known in trade theory) is simple. If E is 10%
bigger in terms of expenditure than E* and there are 10% more firms located in the North,then the per-firm expenditure would be equal if there were no international trade. Trade evensout the differences in competition so although competition is somewhat tougher in the North,
it is less than 10% tougher so per-firm demand in larger in the North with trade but immobile
firms.
To study relocation, we start from the initial situation without delocation, and allow
capital/firm mobility. The trade literature has explored this issue extensively in the context of
11

Note that our assumption that the North is bigger, but is endowed with the same capital-labour ratio, meansthat E/K=E/K* (recall that E is proportional to the number of consumers in a market and this is equal to thenumber of labourers and thus the labour stock in each region). Consequently, we can use (7) to rewrite B as
(E/λK)/(1+φK/K*) and B* as (E*/λK*)/(φ+K/K*). Since 0<φ<1 and K>K* we see that B>B*. The fact that
B>B* without relocation is, of course, the basic economics driving the Home Market Effect (Krugman 1980).

Tax and heterogeneous firms

8homogenous firms. There, the received wisdom is called the Home Market Effect (e.g. see
Krugman 1980, and Davis and Weinstein, 1999, 2003), which notes that some of the firms
will relocate from the small South market to the big North market. However, as firms shift tothe big market, they produce a counterbalancing shift in local competition. The Northernmarket becomes more competitive and the South market less competitive. Relocation goes on
until the operating profit gap is pushed to zero, i.e. B = B * .
When firms are heterogeneous as in our model, an additional question arises: Which firms
relocate first? The key is to note that large firms sell a great deal more than small firms, solarge firms are most interested in reducing trade costs. More formally, the profit gap in (6) isgreater for more efficient firms that charge a lower price and thus sell more. Following theusual logic, the Southern firms with the most to gain move first, i.e. the largest, more efficientSouth firms are the first to relocate to the big Northern market.12 The relocation ends when B
equals B * and all firms are just indifferent to their equilibrium location, but with a range ofthe most efficient Southern firms having moved to the North. Figure 2 shows the equilibrium
distribution of firm efficiencies by market allowing for capital mobility.
(K+K*)G[a]KG[a]North (big)
‘a’

0

a0

aR

K*G[a]South (small)
‘a’

0

aR

a0

Figure 2: Geographic distribution of firm efficiency with free delocation

What we see from the diagram is that free capital/firm mobility results in the North having adisproportionate share of the world’s industry, and a disproportionate share of the world’s
most productive firms (i.e. those with low marginal costs).
Formally, the range of firms that move northward is [0…aR] where aR is the thresholdmarginal cost defined by:(8)

1=B [aR ]B *[a ]R

Note that the B ’s depend upon the E’s and the P’s. While the E’s are invariant to firm
relocation, the P’s adjust. For example using the mill-pricing and the definition of the North’s
CES price index:
P1−σ = (1 − 1/ σ ){K ∫ a dG[a] + K {∫ a dG[a] + φ ∫ a dG[a]}

σ −1

1

0

1−σ

*

aR

0

1−σ

1

1−σ

aR

12
For details see the analysis in Baldwin and Okubo (2006a,b).The basic idea is that if there are quadratic
adjustment costs or other forms of congestion, then the firms with the most to gain would leave first.

Tax and heterogeneous firms

9Here the three integrals reflect, respectively, the local prices of Northern firms, the local
prices of Southern firms that are now based in the North, and the prices of South-based firms
exporting to the North. Using (2) to solve the integrals, (8) becomes:
1=

(
) ;
E / (φ ( K + K a ) + K (1 − a ) )E / K + K *aRα + φ K * (1 − aRα )
*

α

*

α

*

R

α ≡ 1−σ + ρ > 0R

Solving this tells us that aR equals:
(9)

aRα =

2φ
1
(s − )(1 − φ )(1 − s )2where ‘s’ is the North’s endowment share of world expenditure and capital/firms.
From (9) it is clear that for sufficiently free trade, i.e. φ near 1, aR will be 1, which means that
all firms will have left the South (recall that a = a0 = 1 is the maximum marginal cost). To
keep the analysis interesting, we restrict our investigations to φ’s less than the level offreeness that would result in all firms in the North. Simple calculation reveals that thisthreshold φ is (2s − 1) /{(1 − φ )(1 − s)} <1.
We turn now to including capital taxation.

3.

CAPITAL TAXATION AND EQUILIBRIUM LOCATION OF INDUSTRY

Large highly industrialised nations typically have higher tax rates than smaller poor, lessindustrialised nations. This section introduces capital taxation that reflects this outcome andsets the stage for consideration of the impact of tax reform. To simplify, the Southern capitaltax rate is zero so the Northern tax rate can be thought of as the tax difference.
The simple tax scheme we consider involves a flat capital-income tax rate t that is applied tocapital-income beyond a deductible, D. The tax applies to all firms located in the North
regardless of their nationality and the tax is collected on worldwide capital income. Note thateach firm is associated with a unit of capital and capital’s reward is the firm’s operating
profit, so we can think of our tax as a highly simplified corporate income tax.
Taking account of t and D, the tax paid by a typical firm with marginal costs a that is locatedin the North becomes:(10)

tax[a] = max {(π [a] − D ) t , 0}Plainly the tax paid is increasing in the size of the firm (i.e. decreasing in its marginal cost, a);a sufficiently small firm (profits less than D) pays no tax at all. The implied marginal andaverage tax rates are illustrated in Figure 3.
Given this tax scheme, firms choose their location based on their after-tax income, so thelocation condition with taxation will be:(11)⎧π [a ] − π *[a ] ,0=⎨⎩π [a ] − tax[a ] − π *[a ] ,if π [a ] < D
if π [a ] ≥ D

Small firms (those with operating profits below D) compare operating profit without regard totax while larger firms consider tax when choosing their location.

Tax and heterogeneous firms

10

rate
Marginal rate ‘t’
Average rate

0

Firm
operatingprofit (π)

D

Figure 3: Marginal and average tax rates by firm size.

3.1. Taxation without a deductible
To fix ideas, we first work through the simpler case where D = 0. Recall that the North cancharge a higher tax rate and not lose any of its firms, since the big market is characterised by
agglomeration rents as in Andersson and Forslid (2003).
Formally, the tax rate that prevents all relocation (so the number of firms in each market isfixed by the endowments K and K*) is:(12)

0 = π (1 − t nr ) − π *where tnr is the no-relocation tax rate. The aim is to analyses the trade-offs facing a typicalhigh tax nation, i.e. a nation that can only raise its tax rate at the cost of losing some firms totax-driven relocation. For this reason, we start with a tax rate that is somewhat higher than therate that would lead to no relocation of firms.
Specifically, consider a tax that is slightly higher than tnr namely tnr plus ε. In this case, thepost-tax profit gap π [a ](1 − t nr − ε ) − π *[a ] will be negative and some firms would move to
the South to escape the tax which now exceeds the agglomeration rent in the big Northernmarket. The firms that have most to gain from leaving are the ones that sell the most and thus
earn the greatest profits. To see this, consider what post-tax profit gap firms would face ifnone moved. By definition of tnr, the post-tax profit gap, π [a](1 − t nr − ε ) − π *[a ] , equals−επ [a ] and so it will be negative for all firms. However, it is much more negative for themost efficient/profitable firms. This is why the most efficient firms will leave first. As thesefirms leave, they make the Southern market more competitive and the Northern one less, andthe exodus continues until post-tax profits are re-equalised in the two regions for the marginalfirm.
More formally, all firms with a’s between zero and aL will move to the South to escape thetax, where this threshold is defined by 0 = a1L−σ ( B + φ B* )(1 − t ) − a1L−σ (φ B + B* ) . This can be
written as:
(13)

{

}

0 = a1L−σ B(1 − t − φ ) − B* (1 − (1 − t )φ )

where
B=

E / λσ,K + K a + φ K * (1 − aαL )
* α
L

B* =

E / λσφ K + K a + K * (1 − aαL )

(

* α
L

)

Tax and heterogeneous firms

11

Using the fact that North has a share s of both the world’s K and E, and defining the tax factor
as T ≡ 1-t, we can solve for aR, i.e. the end of the relocation range:( (1 − s) φ − s(1 − φ ) + s ) T − φ s + (s − φ )(1 − s) ;
=
2

aR

α

2

2

2 2

(1 − φ )(1 − s ) ( ( s (1 + φ ) − φ )T − (1 + φ ) s + 1)

T ≡ 1− t

We take this as the starting point of our reform analysis since it reflects the typical situation
where the large market has a tax rate set sufficiently above the small nation rate, so that somefirms have relocated to escape the tax. At this point, the big Northern market faces acontinuous trade off between raising the tax rate and losing more firms. Formally, the rangeof firms that relocate is those with a∈ [0,aR]; this range widens as t increases (i.e. T falls), as
inspection of (13) reveals.

3.2. Taxation with a deductible
Next we introduce a small deductible that alters the location decisions of all firms. Before thedeductible, all firms would have preferred the North – but for the tax. With the deductible,
sufficiently small firms pay no tax in North, so they clearly prefer being in the North. The
range of such firms that escape taxation in either market consists of those with a∈ [aU,1]
where:

D = aU1−σ ( B + φ B* )(14)

where we normalise K+K*=1 without loss of generality and defineB=

E / λσ,αφ aL + 1 − aαL

B* =E * / λσaαL + φ (1 − aαL )Notice that the Ks disappear from the equilibrium B’s since firms separate spatially according
to the level of their efficiency. All firms with a’s less than aL move to the South to escapetaxation. All firms with a’s above this threshold move to the North to take advantage of thelarger market. The firms big enough to be liable for taxation in the North are affected directlyby the deductible and indirectly by the relocation that D induces (at the very least, thedeductible induces all small firms to relocate to the North). For firms big enough to pay tax in
the North, the new post-tax profit gap is π − (π − D)t − π * , which can be written as{}a1−σ B (T − φ ) − (1 − φT ) B* + (1 − T ) D .
It is important to note that firms will now not be indifferent to location even after all
adjustment occurs. The term is curly braces will be negative13 yet the (1-T)D term is positive.
As a consequence, the post-tax profit gap will be strictly negative for firms with very low a’s
but strictly positive for firms with high a’s. The threshold a that divides firms into those that
prefer the North from those that prefer the South is:
(15)

{

}

0 = a1L−σ B(T − φ ) − (1 − φT ) B* − (1 − T ) D

Another way to phrase this outcome is to note that with D, the effective tax rate depends uponfirm-efficiency, with the firm-specific rate rising with the firm’s efficiency level (i.e. rising as
its a falls). The effective-tax-rate for firms with the threshold a is:

Before D > 0 was introduced, (13) indicated that the {B(T-φ)-(1-φT)B*} was zero. Since the deductible
induces some firms to move to the North, B falls and B* rises, so the term in curly braces must be negative.

13

Tax and heterogeneous firms

12

(1 −

(16)

D
)t
π [ aL ]Firms that face an effective rate above this locate in the South since the advantages of
producing in the large North are not sufficient to outweigh the tax. For firms facing effective
rates below this, the North market is attractive despite the taxation.
To summarise, we write:
Result 1: Taxation with a deductible leads to spatial sorting; all firms that are
sufficiently efficient move to the tax-free country while all others concentrate in thehigh-tax nation. The threshold is defined implicitly by (15).

This spatial sorting has obvious effects on the average industrial productivity of the twonations. In particular, all the most productive firms have escaped Northern taxes by moving to
the South.
Result 2: The spatial sorting reduces the average productivity of firms in the taxedcountry and raises it in the other nation.
Tax Base. Firms that are sufficiently small pay no tax due to the deductible; the thresholdsize as characterised by the threshold marginal cost aU is defined by (14). Firms that are
sufficiently large pay no taxes since they are located in the South, where the threshold size, as
characterised by the threshold marginal cost aL is defined by (15). The Northern tax base isthus the range of firms with a’s between aL and aU. Tax revenue in the North is:aU

Tax revenue = ∫ (tπ [a] − tD)dG[a]

(17)

aL

where B and B* are defined as in (15).
The location equilibrium and Northern tax base are illustrated in Figure 4.

(K+K*) G[a]North (big)Tax base
‘a’

0

aU

(K+K*)G[a]South (small)

0

‘a’

aL

aU

1

Figure 4: Location Pattern in the Equilibrium.

We next consider tax reforms including the current trends towards lower-rate-wider-base.

4.

WIDER-BASE-LOWER-RATE TAX REFORM

To fix ideas, consider a tax reform that leaves unchanged the effective tax rate facing firms
with a equal to the threshold aL. It is useful to re-write the location condition as:

Tax and heterogeneous firms

13

0 = π [aL ] − π *[aL ] − tax[aL ](18)

The reform is illustrated in Figure 5. Recalling that the tax base is determined by two
thresholds, the threshold aU above which firms do not pay taxes since they earn too little andthe threshold aL below which firms do not pay taxes since they are located in the South, we
see that the Northern tax is only paid by firms that earn profits between D and π[aL]. It is
immediately obvious from the diagram that this reform raises tax revenue without inducing
any firms to relocate to the low-tax nation. More formally, this is obvious from (17) since theaverage tax rate rises on all the firms paying taxes (those with a’s between aL and aU) and it
increases the range of firms paying tax since D falls.

Average rate (post)

rate

Average rate (pre)0

Firm
operatingprofit (π)π [aL]

D’ D

Figure 5: Rate lowering base widening reform.

To summarise:
Result 3: A rate-lowering with base-widening tax reform that keeps the effective rate
constant on the marginal firm always increases tax revenue.

More generally, we consider the location impact of changing the tax rate, t, and the deduction,
D, separately. Inspection of (15) shows that we cannot find a closed form solution for aL, so
the analysis must be by implicit differentiation. Totally differentiating the location condition(15) with respect to aL, t and D:

(19)

⎧⎪
⎛
dB
dB* ⎞ ⎫⎪
aL−σ ⎨(1 − σ ) B(T − φ ) + B* (T φ − 1) + aL ⎜ (T − φ )− (1 − T φ )⎟ ⎬ daLdaL
daL ⎠ ⎭⎪
⎝
⎩⎪

(

)

{

}

+ a1L−σ ( B + φ B* ) − σ D dT−

{σ T } dD = 0

where
dB
E (1 − φ )=aαL −1 > 0,daL λ[(φ − 1)aαL + 1]2

dB*
E * (1 − φ )=−aα −1 < 0daL
λ[(1 − φ )aαL + φ ]2 L

As long as the tax is not too high, so that T-φ>0, the coefficient on daL is positive.14 Again if
the tax is not too high, some firms will be paying tax so we know aL is less than aU, so from
(14) the coefficient on dT must be positive.15 The coefficient on dD is also positive.

The term (1-T)D is positive, so B(T-φ)-(1-φT)B* must be negative if the sum is to add to zero and since σ>1,
the first term of the coefficient is negative; given the signs of dB/daL and dB*/daL, the second terms is also
negative if T>φ.

14

Tax and heterogeneous firms

14

Combining these results on the signs of the coefficients, we have:
daL
>1,
dt

(20)

daL
<1
dD

This says that raising the marginal tax rate will induce additional firms to relocate to theSouth to escape the higher taxes. Raising the deductible has the opposite effect since it lowers
the effective tax rate on the marginal firm. Consequently, a lower marginal rate teamed with alower deductible will attract more efficient/large Southern firms to the North while narrowing
the range of small/inefficient non-tax payers. It is plain therefore that the inflow of southern
efficient firms to the North induced by the tax reform could raise the average efficiency ofNorthern industry.
Average productivity and revenue effects. The most productive firms are in the low-tax
South (Figure 4) so a Northern tax reform that lowers aL – i.e. that encourages some of theSouthern firms to relocate to the North – will have the rather unexpected effect of raising
average productivity in both nations. The reason is that the marginal firm with marginal costaL would be the most efficient firm in the high-tax North when it moves, but would have been
the lowest-productivity firm in the South before it moved. To summarise:
Result 4: Tax reforms that induce relocation into the high-tax region increase average
productivity in both countries.

5.

GLOBALISATION AND TAX REFORMS

Our model provides a simple and amenable framework for considering a wide range ofinteractions and tax reforms. The previous section analytically proved that a specific, ratelowering-base-widening tax reform would raise tax revenue. Here we examine what happens
to revenue when the tax scheme is unreformed in the face of freer trade (globalisation).
One of the key points in Andersson and Forslid (2003) is that agglomeration forces produced
taxable quasi-rents with the size of the quasi-rents varying with the level of trade freeness in a
hump-shaped manner. The quasi-rents are low when trade was either very closed or very
open, reaching their maximum at intermediate levels of trade freeness. Since the basic
agglomeration forces are in effect in our model, we also see a hump-shaped variation in quasirents, however, in our model firms relocate in reaction to such changes. In relocating, theyalter the tax base and thus tax revenue. The net result is that globalisation – as measured bygreater trade freeness (higher φ) – has a hump-shape impact on tax revenue for a given tax
scheme (i.e. fixed t and D).
Numerical simulation of the tax-revenue impact of freer trade is shown in Figure 6 for aconstant t and D.16 The bottom curve shows the impact for an initial level of D and t. Startingfrom a low level, a rise in trade freeness φ would increase the agglomeration rents in the
North if there were no firm relocation to the Northern market. The incipient profit shift,
however, induces more firms to move to the big, high-tax Northern market, so the net result is
a wider tax-base and higher tax revenue as shown. Specifically, the offsetting relocation

15

1−σ

1−σ

Since aU ( B + φ B ) − σ D = 0 , and aU > a L , then aL ( B + φ B ) − σ D > 0 since σ>1.
**

16
The parameters we choose for the simulation are σ=2, ρ=2, E=0.6, E*=0.4. The initial tax scheme involves
t=0.3 and D=2; the reformed tax scheme involves t=0.2 and D=1.

Tax and heterogeneous firms

15

implies that the level of profitability changes little in the North, so the tax base’s upper
threshold, aU, changes little, but aL falls.17
The rising attractiveness of the North in the face of freer trade, however, begins to fade forlevels of φ beyond the peak of the hump. This is where the agglomeration rents in the Northwould begin to decline if there were no offsetting relocation. As before, the relocationinduced by the incipient change in profitability reduces the tax base and results in lower tax
revenue. The figure shows φ’s up to the point where 1-t = φ. Beyond this point, there is noadvantage to being in the big, high-tax nation for any tax-paying firm, so the revenue drops tozero.
It may be useful to step outside the model and conjecture that the forces at work in our simple
simulation would result in tax rate changes that bear some resemblance to actual events. In
this first phase of globalisation (the φ’s corresponding to the upward sloped part of therevenue curve), the Northern government might decide to raise the tax rate in reaction to its
increased attractiveness. As globalisation proceeds even further, the Northern government
finds it increasingly difficult to keep its tax rates high without losing firms and so decides to
cut rates. Figure 6 also shows the same calculation for a reformed tax structure that involves
smaller D and t (top curve). Note that the upper limit for φ is now higher, but the revenue iseverywhere higher for this wider-base-lower-rate scheme.
Tax Revenue
Small D and Small t (t’’)

Large D and Large t (t’)

φ’ φ”

φ

Figure 6: Globalisation and hump-shaped tax revenue.

One of the crucial features in our model is firm heterogeneity, so we briefly consider the
impact of varying the degree of heterogeneity as measured by ρ. We focus on the impact ofheterogeneity on the link between tax-rate cuts and tax revenue. The numerical results, shownin Figure 7, are generated for the same values as those in footnote 16.
According to the well-known properties of the Pareto distribution, (2), firms become moreheterogeneous as ρ falls. What the diagram shows is that greater heterogeneity increases the
responsiveness of revenue to rate changes. Intuitively, a low ρ means that a higher fraction ofindustry output and profits is concentrated in the hands of the most productive firms. Thus as
the tax rate attracts more firms back to the North, it has a bigger impact on the tax base andthus on revenue. In short, in industries where firms are more heterogeneous, tax reforms are
more effective in the sense of boosting tax revenue.
Freer trade affects both thresholds but has a much large impact on aL since aL depends upon the difference inprofitability in the two nations, while aU depends only upon profitability in the North; see (14) and (15).

17Tax and heterogeneous firms

16

Tax revenue
Small ρLarge ρTax ratet
Tax cut

Figure 7: Firm heterogeneity and tax revenue.

6.

CONCLUSION

This paper proposes a simple model in which agglomeration forces are present and firms areheterogeneous. Both extensions are useful in allowing the international tax competition
literature to consider a broader range of effects than has hitherto been possible. The presenceof agglomeration forces allows consideration of the international trade competition issues
raised in Andersson and Forslid (2003) and Baldwin and Krugman (2004) in the context ofidentical firms. Allowing for heterogeneity permits a given tax scheme to have a different
effect on the relocation decision of small and big firms, with the biggest firms being the most
likely to relocate to escape high-taxes imposed in the big nation.
The model provides a simple and amenable framework for considering a wide range ofinteractions. The policy experiments we conduct in this paper concern 1) the impact of a ratecutting-base-widening reform, and 2) the impact of freer trade (i.e. globalisation) on the tax
competition. The model should also help inform future empirical research concerning theimpact of tax reforms on tax revenue, firm location and average productivity using firm level
data sets.

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