DP2011-11

Anti-agglomeration Subsidies with
Heterogeneous Firms
Toshihiro OKUBO

March 16, 2011

Anti-agglomeration Subsidies with Heterogeneous Firms
Toshihiro Okubo 1
Kobe University
January 2011

ABSTRACTThis paper studies anti-agglomeration subsidies in a core-periphery setting when firms
are heterogeneous in labour productivity, focusing on the effects of relocation subsidy on
firm location in various tax-financing schemes (local versus global). We discuss how
subsidy can enhance welfare in periphery. As a result we find that subsidy proportional
to profits can induce the relocation of high productivity firms and that a subsidy canincrease welfare in periphery. Concerning tax-financing schemes, local tax financing
scheme has an optimal level of subsidy.
JEL F21,H32, H25.
Keywords: heterogeneous firms, anti-agglomeration relocation subsidy, global/local
tax-financing scheme.

1. INTRODUCTION
Low income per-capita and low economic growth in undeveloped peripheral regions within acountry is a serious concern in regional policy. To solve this problem, governments often pursue ananti-agglomeration strategy that induces firms to relocate from core to undeveloped peripheral areas

1

Kobe University, Research Institute for Economics and Business Administration. 2-1 Rokkoudai Nada Kobe Hyogo,657-8501 Japan. E-mail: okubo@rieb.kobe-u.ac.jp I would like to thank Henry Overman, Mike Artis, Matthiew Crozet,
Bellone Flora, Francis Kramarz, Vincent Rebeyrol, Frederic Robert-Nicoud and Reinier de Jong, Ben Morton, an editor
and two anonymous referees for their helpful comments and suggestions. Also I would like to thank participants at theWorkshop on Economics: Heterogeneity of Firms, Productivity and Trade (Granada, Spain on November 18th 2006).

2and creates industrial clusters. One of the main policy programs is the relocation subsidy. 2 For
example, the EU Structural Funds have influenced the relocation of industries to peripheral regionsof the EU in the last decade. 3 Many previous studies empirically test the impact of Structural Funds
on regional growth using regional level data. Some find negative impacts, however some others
observe a positive or statistically insignificant effect (Mohl and Hagen, 2010). 4 Much closer to ourpaper, there are a few empirical studies on the impact of the Structural Funds on industrial location
patterns. (Midelfart-Knarvik and Overman, 2002 and Midelfart-Knarvik et al.,2000). 5 Furthermore,
one of the recent advancements in relocation subsidy studies is to use micro data and investigatessome impacts of a regional subsidy on firm performance and location patterns based onheterogeneous firm productivity. A few current works analyse regional subsidy programs using
micro firm/plant level data (Martin et al., 2010 and Okubo and Tomiura, 2010). They examine theimpact of relocation subsidy programs on productivity and location patterns in terms of firm
productivity. 6 To link with the emerging firm-level empirical research on this issue, our paper uses
an economic geography model combined with firm heterogeneity. Our paper has two aims, firstly to2

Relocation subsidy in our paper includes not only direct subsidy in which relocated firms receive subsidy but also

indirect subsidy in which public infrastructure in specific areas is developed, local employment is subsidised or taxation
is deducted. Even indirect subsidy can reduce cost and give incentive to operate production in periphery.
3

EU structural funds have several objectives. One of the biggest spending is to support economic development in theless developed regions. In reality, the funds prohibit direct subsidy to relocated firms and intend to promote capital
investment in periphery through indirect way such as the regional development of infrastructure.
4

The divergence of the result is due to different research designs in methodology and data samples. See Table 1 in Mohl

and Hagen (2010) for an excellent survey of main results and methodology in previous studies on this issue.
5

Midelfart-Knarvik and Overman (2002) find that European Structural Funds have some effects on industry relocation,in particular a slightly positive impact for R&D intensive industries in specific regions.
6

Martin et al. (2010) studies a French subsidy program and Okubo and Tomiura (2010) studies case of Japan. The

Industrial Relocation Subsidy started in the 1970s Japan is a kind of direct subsidy, just as in our model. Once firms
relocate from core to periphery, they receive a certain amount of subsidy. See Okubo and Tomiura (2010) for moredetails.

3

provide theoretical predictions on the impact of a regional subsidy on firm relocation and secondly,
to discuss which type of regional subsidy results in a successful outcome in the periphery.
An economic geography model with firm heterogeneity is not entirely new. Baldwin andOkubo (2006) use a model that marries the footloose capital (FC) model of Martin and Rogers(1995) and the heterogeneous-firm-trade model of Melitz (2003). 7 One section in their paper
provides a theoretical prediction for the impact of a specific-relocation (per-firm) subsidy. It showsthat a fixed amount of relocation subsidy per-firm would always attract the least productive firms
since they have the lowest opportunity cost of leaving the agglomerated region. If their theoreticalprediction is true, regional policies should unintentionally widen the productivity gap between thecore and the periphery. 8 However, their explanation is the first step in explaining the mechanism of
relocation subsidies in a simple model. In this paper, we aim to show a contrasting consequence of
relocation subsidies, whereby the subsidy reduces the productivity gap and enhances the welfare in
periphery. Keeping the basic framework of the economic geography model of Baldwin and Okubo(2006) we integrate schemes of different types of subsidy, including a tax financing scheme. In
particular, while Baldwin and Okubo (2006) employ a per-firm relocation subsidy and discuss thechange of location caused by the subsidy without welfare analysis, we consider a relocation subsidy
proportional to the firm’s profits (or productivity), taking account of various tax financing schemes
and then find optimal policy scheme based on welfare. 9 As in Baldwin and Okubo (2006), the FC
model is employed due to it being the simplest tractable model, i.e. it has analytically solvable
7

The previous study on the integration of the economic geography model and heterogeneous-firm-trade model is Okubo(2009).
8

In another example of theoretical work on the unintended effects of regional policy, Dupont and Martin (2006) showthat regional policies, especially subsidies to poor regions proportional to firms’ profits that are financed by national
taxes, increase cross-regional income inequality within a country. Their argument rests on the observation that the effectof the subsidy spills over into rich regions, where many owners of capital (who are the beneficiaries of subsidised profits)
reside.
9
A relocation subsidy proportional to firm profits and size could be more natural than per-firm subsidy, since large andproductive firms would be expected to receive more subsidy and to be more likely to benefit from a relocation subsidy
than smaller firms.

4equilibrium since there is no circular causality, and no backward nor forward linkages. The FC
model’s assumptions are innocuous to our analysis because our focus is on firm relocation driven bysubsidy rather than labour migration. 10
Our paper makes two main contributions to regional science literature. First, we study the impact of a
relocation subsidy in more depth, extending an idea of Baldwin and Okubo (2006) by integrating
heterogeneous-firm trade model à la Melitz (2003) to an economic geography model . In regional
policy literature, this paper sheds light on how important firm heterogeneity is and how spatialsorting is driven by relocation subsidy. 11 Second, the discussion of tax financing schemes (local or
global tax financing scheme) and welfare analysis allow for more precise policy analysis. We canfind the optimal relocation subsidy that can relocate high productivity firms to the periphery and canenhance welfare in the periphery.
As a result of analysis, we find that 1) a proportional relocation subsidy attracts the higher
productivity firms to the periphery, 2) the periphery has an incentive to provide a proportional
subsidy policy financed by a local tax and 3) while a centralised economy chooses zero-subsidy to
avoid tax payment, a decentralised economy leads the periphery to choose the optimal level ofsubsidy, resulting in enhanced welfare in the periphery. These results are either in contrast with or
missing in the per-firm relocation subsidy as proposed in Baldwin and Okubo (2006).
There are five sections in this paper. The next section introduces the basic model and explores theequilibrium. Section 3 examines how subsidy rates are determined and discusses both the10

Since our model is aimed at studying the impact of subsidy on firm location patterns with various tax financing

schemes, which affect per-capita income and welfare in periphery, we use the FC model. If we allowed for labour
migration, then the model would combine a heterogeneous labour market and so make it more complex, to the extent that
it would become impossible to solve the equilibrium and welfare analytically due to circular causality. What is more, thetiming of taxation and subsidy might also affect expectations of the migrants. This is out of our scope however is a spacefor future research.
11

Previous studies on regional policies presume firms are homogeneous (as in the standard economic geography model).

See Baldwin et al. (2003), Dupont and Martin (2006) and Ostbye (2010).

5

tax-financing of subsidies and socially optimal firm location. Section 4 provides some further
discussion. The last section presents our concluding remarks.

2. THE BASIC MODEL2.1. The footloose capital model with heterogeneous firms
Our model works with two regions, two sectors and two factors of production. The regions – referred
to as the North (core) and the South (periphery) – have identical technology, tastes, and openness totrade, but they differ in their market size (the North has larger demand by convention). The two
sectors are the M-sector (manufactures) and the A-sector (a numeraire sector). The numeraire sector
produces a homogenous good using labour subject to constant returns, perfect competition andcostless trade. The M-sector produces a differentiated variety of products using labour as well as
capital with increasing returns to scale, there is Dixit-Stiglitz monopolistic competition and iceberg
trade costs. Specifically, firms’ marginal costs are flat with respect to the scale of production andinternational trade is subject to iceberg trade costs whereby some goods “melt away” duringtransportation.
We have two factors, capital and labour. Capital can move between the North and the South, butlabour is immobile. We assume that each individual is not mobile and owns an endowment of labourand capital. All capital is owned by individuals so while capital is mobile, capital owners(individuals) are not (this cuts out complications that would otherwise arise from backward linkages).
We can exogenously give total units of labour (Lw) and capital (Kw) in the world as well as sL and sK,
which denote the Northern share of labour and capital endowments at the initial equilibrium
respectively. Initial endowment ratios equate shares of capital and labour, sL=sK, although capital ismobile and firm location is endogenously determined in the long-run equilibrium. Capital returns are
repatriated to the origin. For clarity’s sake, each individual in the North and the South has a capitalendowment consisting of a portfolio comprising a common share of stock of all the world’s firms

6(portfolio assumption). Thus, by sharing all firms’ profits over the world as stock holder, the capital
reward that each individual receives is the average operating profits of all firms in the world. Due tothe portfolio assumption and the quasi-linear utility function (as shown below), it always follows
sE=sL, where sE=E/Ew is the North’s share of the total income in the world (Ew), normalised, i.e. Ew=1.
Hence, we have sL=sK=sE.
Basic set up
The tastes of the representative consumer in each region are given by the quasi-linear utility
function:
1 /( 1-1/σ )1-1/σ
U = µ ln C M + C A , C M ≡  ∫ ci di 

 i∈Θ

,

1 > µ > 0, σ > 1

where CM represents consumption of the composite of M-sector varieties, CA stands for consumptionof the A-sector good, μ denotes the expenditure share of the composite of M-sector varieties. σ>1 isthe constant elasticity of substitution between any two M-sector varieties and Θ is the set of allvarieties produced. 12 We note that quasi-linear utility function excludes an income effect.
On the supply side, firm-level heterogeneity stems from differences between firms’ marginal costs.
Although all the Dixit-Stiglitz varieties enter consumers’ preferences symmetrically, the cost of
producing each variety is different. A typical (Northern) firm j’s cost function reads:
TC j = π j + wa j x jwhere π j is the firm j’s fixed cost and thus denotes operating profit (capital return), which is due tothe requirement of one unit of capital, and wa j x j is firm j’s variable cost, which is driven by labourcosts. xj is j’s output, aj is unit labour requirement, w is wage. Importantly, firms have heterogeneous

12

This set of varieties is pre-determined by endowments because each variety requires one unit of capital and the world

capital stock is fixed, normalised to unity.

7

unit labour requirements. This means that the unit labour requirements are variety-specific and thus
firm-specific. The distribution of firm-level efficiency is part of each region’s endowment. The
specific assumption is that the distribution of the a’s is subject to the Pareto distribution, which
cumulative density function is defined as:(1)

G[a ] = (

aρ
a0

ρ

),

1 ≡ a 0 ≥ a ≥ 0, ρ ≥ 1

Here a0 is the scale parameter (highest possible marginal cost) and ρ is the shape parameter. A smallρ means more dispersed in ‘a’, that is, firms are more heterogeneous. We can normalise a0=1. The
cumulative density function of a’s can be depicted in Figure 1. We initially consider full
agglomeration in the North with small trade costs (the condition for full agglomeration is specified
below). The distribution in the North is given as G[a], where the total mass of firms is 1 in the Northand 0 in the South. The total mass of firms is unity and the total capital endowment in the world isthus also unity, Kw=1. Note that our model assumes away entry and exit in the M-sector.

Mass of varieties

North (Core: All firms)
G[a]

1

0

1

‘a’

Mass of varieties
South (Periphery: No firms)
1

‘a’
0

1

Figure 1: Endowed distribution of capital and the mass of firms (at initial equilibrium).

8

2.2.

Intermediate results

Results for the numeraire sector where one has costless trade and perfect competition are well known.
Constant returns, perfect competition and zero trade costs equalise nominal wage rates across theregions. We choose units of the numeraire good such that pA=1=w=w* (where * stands for theSouth’s value). This means that all differences in M-firms’ marginal costs are due to differences intheir unit labour requirements, a’s, so we can refer to the a’s as the marginal cost without ambiguity.
The M-sector is marked by all the usual Dixit-Stiglitz results. Firms’ prices are a constant mark-up of
their marginal selling costs. In the local market, these marginal costs entail only production costs as
there are no internal transportation costs. The price in the export market includes iceberg trade costs,t≥1, marked up by the constant Dixit-Stiglitz mark-up. The prices are written as p j =p jt =a jt
1 −1/ σaj
1 −1/ σand.

The standard CES demand function for a variety-j produced and sold in the North can be written as
c j = ( p j ) −σ

µE
m

;

m≡∫

i∈Θ

pi

1−σ

di + ∫h∈Θ*

φ ph 1−σ dh ,σ > 1 ≥ φ ≡ t 1−σ ≥ 0where cj is variety j’s consumption and pj is variety-j’s producer price (which equals its consumer
price since it is produced locally), E denotes total income in the North. The first term in thedefinition of m reflects the prices of goods that are produced in the North. The second term reflects
the imported varieties whose producer prices are ph; Θ and Θ* are the sets of goods produced in the
North and the South, respectively. We note that m falls as the prices of individual varieties rise. A
parameter that plays a critical role in our paper is φ, which we refer to as the free-ness of trade. When
iceberg trade costs are prohibitive (t=∞), the freeness of trade is zero(φ=0), while when there are noiceberg trade costs (t=1), the freeness of trade is unity (φ=1).

92.3. Initial Equilibrium—Full agglomeration
Since agglomeration produces rent which holds firms in their own agglomeration region when trade
costs become smaller (we call critical trade costs creating rents as “sustain point”), this rent calls forvarious sorts of public policy incentives with regards to shifting firm location (Baldwin et al. 2003).
To exclude the marginal impact of relocation subsidies and focus on the substantial impact of
relocation subsidy on firm location, we start with full agglomeration in the North. For this reason, we
regard full agglomeration as an initial stable equilibrium. In other words, we start by considering
small trade costs so that all firms in the North choose to remain there, i.e. φ above the sustain point
given as φ CP =1 − sE, where we have introduced sE as shorthand for the North’s share of worldsE

expenditure (we adopt the convention that the North is bigger so sE >½). 13
Utility maximisation generates the familiar CES demand functions in the manufactures sector. These,
together with the standard Dixit-Stiglitz monopolistic competition assumptions on market structure,
imply ‘mill pricing’ is optimal and operating profit earned by a typical firm in a typical market is 1/σtimes firm-level revenue. 14 Accordingly, operating profit realised by a firm in the North is:



(2)

a

)1−s

µ
p [a] =  s E 1 − 1 / s
1−s
s
p di



(

∫

i

i

+ (1 − s E )a)1−s 
1 −1/ s.
*1−s∫i pi di φ(

The first term in the bracket is the value of firm-specific sales in the Northern market. The second
term shows the firm’s export sales, which is similar except the firm’s price includes the iceberg trade
cost raised to 1-σ and the denominator involves p*’s, the prices of all goods in the Southern market.

13

See Martin and Rogers (1995) for the derivation of the sustain point in the FC model.

14

A typical first order condition is p(1-1/σ)=wa; rearranging, the operating profit, (p-wa)c, equals pc/σ.

10

The features of (2) that matter for our results are 1) all firms earn a positive operating profit, 2) the
profitability of firms increases with firm-level efficiency, and 3) the most efficient firms, i.e. firms
with low marginal costs, are the largest in the sense that they sell the most.
The operating profit as a function the firm’s ‘a’ for a firm based in the North can be rewritten as:
(3)

σ
∆

π [a] = a1−σ  E +φ (1 − σ E )  µ∆* σthe ∆ ’s are the denominators of the North’s and the South’s CES demand functions in the initial
1equilibrium, i.e. full agglomeration in the North: ∆ ≡ ∫ a1−σ dG[a ] = λ0where λ ≡1∆* ≡ φ ∫ a1−σ dG[a ] = λφ ,0

ρ> 0, ρ > 1 . We assume 1-σ +ρ > 0 (which ensures the integrals converge).
1−σ + ρ2.4. Relocation subsidy and Deviation Tendency
A relocation subsidy is employed so as to induce some firms to relocate to the periphery. Once a firm
relocates from core to periphery, the firm receives the subsidy proportional to its profit. Specificallywhile the operating profit in the North can be kept as (3), a proportional subsidy, u, is added to theoperating profit in the South if a single firm relocates from the North to the South. Operating profit
in the South as a function of the relocated firm’s ‘a’ can be written as:(4)

 φs Eπ * [a] = a 1−s  ∆+1 − sE  µ (1 + u )∆*  swhere u denotes the rates of a relocation subsidy. 15
Starting with all firms in the North, the change in operating profit for an atomistic firm moving fromthe North to the small South would be:(5)15Note that the subsidy in this paper, which adds to the profits of the relocated firms rather than costs, never affects themass of varieties and marginal costs and thus never influences prices.

11

π [a ] − π * [a ] = a

1−s


 sE
 µ a1−s µ  (1 − s E )  s E (1 − s E ) 
1 − sE 1 −  φs E + + φ
− φ
+

(
1
+
u
)
(1 + u ) =
sλ  
φ 
∆*
∆* 
 ∆
∆
s


Figure 2 plots (5) in terms of a. As long as the above equation is positive, no firms have incentive to
move to the South. Above a certain level of subsidy, u min , in contrast, the above equation switchesfrom positive to negative. The most efficient firms are those most likely to relocate to the small
region, as shown in Figure 2. Note that the minimum subsidy, u min , such that at least one firms

1 − sE 
move to the South, is satisfied as 1 −  φsE +(1 + u min ) = 0 which is derived from (5). On theφ other hand the subsidy rates to induce all firms to move to the South are given as
u max =sEφ

+ φ (1 − sE ) − 1 .

π −π *

Non-subsidy (u=0)

Proportional subsidy (u>umin)a1Figure 2: Relocation tendencies with a proportional subsidy.

Result 1: The first firms to respond to a relocation subsidy will be the most efficient firms. The
minimum subsidy rate is u min =1− 1 , which can induce the most efficient firms to
1 − sE
sEφ +

φ

12

move to the small region. The maximum rate is u max =sEφ+ φ (1 − sE ) − 1 which stimulates all

firms to move to the small region.
Note that minimum and maximum rates are independent of firm heterogeneity (λ, ρ), and amaximum rate is finite. 16
Without relocation subsidies, there is no capital movement from core to periphery. This is a result ofsmall market size and the lack of a cost advantage in the periphery (the production costs are the sameas in the core however there is smaller demand in the periphery). However, once substantial
subsidies proportional to firm size are introduced, the situation changes: capital moves from core to
periphery even with small trade costs. High productivity firms can receive more subsidy than the low
productivity ones, implying that the most efficient firms can obtain the largest subsidy, and viceversa. Since the most efficient firms are most sensitive in their location and profits, they are the firstones to relocate to the South benefiting from the largest amount of subsidy.

2.5. Locational equilibrium
Based on relocation tendencies, the locational equilibrium is determined to equalise profit between
two regions. The profit gap can be written as(6)

π [a ] − π * [a ] = a1−s(

∆ = λ φa R

16
(1 − s E )  s E (1 − s E ) µ  sE + φ+− φ(1 + u )
∆* 
∆*
s∆
 ∆


1−σ + ρ+ (1 − a R

1−σ + ρ)),(∆* = λ a R

1−σ + ρ+ φ (1 − a R

1−σ + ρ))The minimum rate of u is always positive. From the full agglomeration condition in the initial equilibrium(non-subsidy): π [a ] − π * [a ] = µ 1 − φσ − 1 − σ E  > 0 ,
E
σ 
φ 
thus: u min =1−1 > 0 .
1− sE
s Eφ +φ

1 − (φs E +1 − sEφ) > 0 can be derived. Thus, the minimum subsidy is13where a R is the cut-off level of efficiency below which all firms locate in periphery (‘R’ is amnemonic for relocation cut-off). The cut-off a R is determined by π [a R ] − π * [a R ] = 0 and canbe solved analytically:
1 /(1−σ + ρ ) 1 − φχ(7) a R =  (1 − φ )(1 + χ ) 
where

χ≡

(1 − φ (1 + u )) s E. u is positively correlated with a R , which means that a higher rate of(1 + u − φ ) 1 − s E

subsidy promotes firm relocation. Figure 3 plots the (Northern) production share, denoted as sP, andfreeness of trade (φ). 17

Figure 3: Production Shares.

17

The Northern production share, sP, is defined as the share of aggregated production of Northern firms fraction of total

production total production.

14

3. GLOBAL VS LOCAL TAX-FINANCING SCHEME
This section provides explicit consideration of tax schemes used to finance the relocation subsidy.
The subsidy is to relocate to the South as in the last section. We compare two contrasting cases,
global and local tax-financing schemes. In the first scheme central government levies a tax on
individual in both regions at equal rate. In the latter one, local government can raise tax locally thatthen subsidises firms from the North to relocate. We make two further assumptions, the subsidy isfinanced by per-capita taxation and the government budget is always balanced.

3.1. Centralisation and global tax-financed subsidy
In the first scheme, we introduce the global tax to finance relocation. Since the central government isassumed to have the authority to implement a subsidy policy, it can be considered to be thecoordinator between core and periphery interests, attempting to reduce the wealth gap. Taxation is
global and per-capita tax must be the same in both regions, T=T*. With a solution of a R in (7) for acertain u, tax rates are written as:(8) T = T * =µλ  s E 1 − s E  1−s + ρ 18
+ * a R
u.
φ
∆ 
s  ∆

The measurement of welfare uses the per-capita indirect utility function (v),(9)

v = ln(Y − T ) +

µ
σ −1

ln ∆

and

v * = ln(Y * − T * ) +

µ
σ −1

ln ∆*

where Y (Y*) is a per-capita income in the North (South) and T (T*) are per-capita tax in the North(South) to finance subsidies. Since each individual owns one unit of labour and one unit of capital,
income can be specified as Y=Y*=w+ π = w*+ π =1+ π , where π is reward per unit of capital,
which is average operating profit of all firms due to the portfolio assumption. We notice that the tax

18

Note that our utility function is quasi-linear, and so there is no income effect. sE is not influenced by taxation and is

not a function of T and u. The total numbers of individuals and firms (capital) in the world are unity.

15

is refunded as subsidy to relocated firms and then finally returned to all people equally as capital
rewards due to the portfolio assumption, π . Since the increase of the capital reward due to thesubsidy is equally distributed over all people, the tax payment and the relocation subsidy keepincome constant and so just affect firm relocation and thus altering Δ and Δ*. Figure 4 plots welfare
scaled to the same total amount of subsidy. Note that the vertical axis represents per-capita welfare,v[u] in the North, v*[u] in the South, and the horizontal axis depicts subsidy rate, u. Without asubsidy, i.e. u=0, the northern welfare always exceeds the southern one, v[0]>v*[0], where
µ
µ
µ
µ



 19
v[0] =  ln(1 + ) +ln(l )  and v * [0] =  ln(1 + ) +ln(φl )  . Increasing the subsidy from theσσ −1
σ σ −1

minimum levels, i.e. u min , leads to firm relocation from the North to the periphery and thus reduces
the northern per-capita welfare while increasing southern per-capita welfare by changing Δ and Δ*.
At the extreme, all firms move to the South when the highest subsidy ( u max ) is provided, at which
point the periphery’s welfare switches to the initial welfare in the core, and vice versa.
m
m
m
m




ln(l )  .
ln(φl )  and v * u max =  ln(1 + ) +v u max =  ln(1 + ) +
σ
σ −1
σ
σ −1





[

19

]

[

]

This is correspondent to full agglomeration in the standard footloose capital model (non-subsidy equilibrium). Notethat π = µ / σ , ∆ = λ andequilibrium.

∆* = λφ . Note that tax payment, T, and subsidy are null due to non-subsidy

16

welfare v,v*South (Global finance)

North (Local finance)
South (Local finance)
North (Global finance)u*u

Figure 4: Global vs. local tax-financing scheme and per-capita welfare.

Here, we introduce one simple voting system concerning pros and cons of anti-agglomeration
subsidy to highlight characteristics of the global tax-finance scheme. Then, residents’ votes in thecore and periphery perfectly reflect subsidy policy-making of the central government: based on eachindividual’s welfare, v and v*, voting determines the rate of subsidy. Each individual has an equal
voting right. The northern residents prefer a zero-taxation and zero-subsidy policy to keep all firms
in the North, whereas the southern people prefer positive levels of relocation subsidy to attract firms
to the South together with taxation of all people. The North has a larger population than the South.
When voting determines the subsidy rates associated with per-capita taxation in both regions (global
tax-financing scheme), the non-subsidy outcome is an equilibrium, so the core residents’ preferenceof full agglomeration is dominant. Accordingly, a global tax-financing system using centralised
voting could not resolve the gap of per-capita welfare across regions.
Result 2: With centralised taxation, the no subsidy policy is an equilibrium if voting decides
subsidy rates. Therefore, no relocation occurs and the gap of welfare remains between core andperiphery.

17

3.2.

Decentralisation and local tax-financed subsidy

Now, we suppose the region is a decentralised nation without a central government. Then, since theperiphery has authority to implement the subsidy policy, it has an incentive to provide a subsidy
financed by southern residents (i.e. local tax-financed subsidy).
Figure 4 plots per-capita welfare, in which the southern welfare is hump-shaped due to the tax
collection (“South (Local finance)” in the figure), while the northern welfare, “North (Local
finance)”, has a similar shape as before. Note that we assume in Figure 4 that the North neverprovides a subsidy to prevent relocation. The hump-shaped southern welfare induces the South to set
an optimal subsidy rate, u* that maximises the welfare, v*. 20 In the South, higher rates of subsidy
attract more firms but at the same time increase southern tax payment, T*>0, which leads to a
reduction in disposable income.
The global tax financing scheme, in which tax is equally levied in both regions and the subsidy isequally distributed via capital returns, has no income effect. However, the local tax-financing scheme
is associated with taxation only in the periphery whilst the subsidy is refunded to both regions’people through capital rewards. The North pays no tax, T =0, however receives a part of the subsidy
via the rise of capital rewards by firms who receive the relocation subsidy from the South. The percapita tax, levied only in the South, derives from

(10)

T* =

µλ  s E 1 − s E  a R1−s + ρ u
+ * 
φ
s  ∆
∆  1 − sL

where the total amount of subsidy is divided by the southern population 1- sL (=1- sE, exogenously
given). 21 Thus, a local tax-financed subsidy plays transfers income from the periphery to the core: A
20

See Appendix 1 for proof of the hump-shape curve

21

Note that tax rates are not relevant to sE because of quasi-linear utility function..

18higher level of subsidy reduces disposal income in the periphery (South) and increases income in thecore (North). This income transfer raises northern welfare and decreases welfare in the South. The
tax payments place a heavy burden on the southern people although South can attract some more
Northern firms, and for sufficiently high levels of subsidy, hence local tax, the southern welfare
declines.

Result 3: With decentralised taxation, the periphery will choose to independently subsidise
firms through locally–financed taxes so as to attract the northern firms. However, since theperiphery needs to collect tax locally, there exists an optimal rate of subsidy to maximise
welfare in periphery.

As a comparative statistic, we study how the optimal subsidy rates change when trade costs andexpenditure shares change. Trade liberalisation increases southern welfare for any u*. A negative
correlation between u* and φ suggests that the South attracts more firms as φ decreases (thisincreases southern welfare),boosts u*,

du *< 0 . 22 Turning to market size, ceteris paribus, an increase in sEdφ

du *> 0 . As the northern market is larger, a higher subsidy is necessary to attract firms
ds E

to the South. Finally, while a higher degree of firm heterogeneity increases welfare in the South at u*,
firm heterogeneity never affects u*,du *= 0 due to the neutrality of a 1−σ + ρ from firmdρheterogeneity. 23

22This result is akin to Dupont and Martin (2006), where the mechanism is such that deeper economic integration causes

stronger impact of subsidy on firm relocation due to the home market magnification effect.
23

Given u*, a R 1−σ + ρ is independent of firm heterogeneity and dependent on sE and φ, as shown in (7). Hence, ρ isneutral in u.

19

One of the largest contrasts with global tax finance, which results in zero-subsidy when using avoting system, is that periphery always has incentive to levy a local tax to employ a subsidy policy.
To clarify this, we allow for both regions’ subsidy to be locally financed by each region. Then a‘subsidy war’ could occur to attract the high productivity firms located in the other region with each
local government using a local tax-financed subsidy. Once the South subsidises firms to attract somehigh productivity firms to relocate to the South, the North attempts to subsidise them to keep full
agglomeration as retaliation. Since the North has a wider (per-capita) tax base due to the higher shareof population (so it more tax payers), it always wins this subsidy war. In detail, regardless of thesame wage and capital returns per capita (average returns), tax per capita in the South always needs
to be higher rate than in the North, given the same amount of subsidy, and thus in the end of thesubsidy war southern individual’s (gross) income would be equal to tax payment, and the South
would not be able to increase subsidy rates any more. Hence, the North wins the war. The Nashequilibrium is full agglomeration in the core with no subsidies in either region (u=0) or the subsidy
rates such that the total amount of subsidy is equal to southern total income.

3.3. Social welfare and average productivity
Then we discuss social welfare under the two tax financing schemes. Using (9) the northern/southern welfares are respectively defined as V N = s L v and V S = (1 − s L )v * . Note that sE =sK =sL.
Our interest is world welfare, V W , with respect to subsidy, which is denoted as
V W = s L v + (1 − s L )v * . 24 Since total income in the world is invariant to the subsidy policy, bothtypes of tax-financed subsidy affect only Δ and Δ* in the world welfare function of V W without
influencing the income part due to invariant total world income (perfectly refunded subsidies). The
firm relocation to the South increases Δ* and decreases Δ (as dΔ =-dΔ*). 25 Since the share of24

In this section, we do not consider retaliation nor subsidy wars.

25

λα (φ − 1)α Rd∆== −1 where α=1+ρ-σd∆ * λα (1 − φ )α R α −1

α −1

20

population is smaller in the South (1-sL<0.5), the increase in welfare caused by the rise in Δ* isdominated by the decreases in Δ. Therefore, the subsidy decreases total world welfare. This social
welfare-reducing subsidy boils down to a result in Baldwin et al. (2003): The laissez faire outcome in
the FC model is Pareto optimal, when sE =sK =sL. This implies that a regional subsidy to change firm
location is harmful to the world, so a zero subsidy is socially optimal. In spite of it, zero subsidy
would not be equilibrium in a decentralised state, because the periphery would always attempt to
introduce a local tax-financed subsidy. Therefore, in order to sustain the socially-optimal zerosubsidy, a state should be centralised: the decision making on subsidy policy should be done by acentral government, with a voting mechanism, which should prohibit the local tax financing scheme.
However, we note that zero-subsidy is not always the best policy. For instance, our heterogeneous
firm geography model gives one more aspect, i.e. average productivity. If the objective of thegovernment is to boost average productivity in periphery, an optimal policy has a different nuance.
As in Melitz (2003), average productivity is defined as a frequency weighted mean of individual
productivities, written as ϕ = ∆

1

σ −1

1

and ϕ * = ∆* σ −1 . As above mentioned, since the firm relocation to

the South due to subsidy increases in Δ* and decreases in Δ, average productivity decreases in the
North and increases in the South. If the regional policy is aimed at promoting average productivity in
the South, a positive rate of subsidy is effective. So there are contrasting policy implications forusing a productivity objective rather than a social welfare objective.

4. DISCUSSION
Finally, we discuss our results and derive implications. In terms of social welfare, a policy of
non-subsidy with centralisation is socially optimal. Importantly, however we note that maximisingsocial welfare is not always a sole objective of subsidy policy in the real world. As many previous
empirical studies discuss, regional subsidies are aimed at promoting employment in periphery,

21

boosting average productivity and promoting regional economic growth. If the objective is to attract
firms to periphery for these aims, the subsidy policy is still worthwhile to exercise. When a regional
subsidy is introduced, it should be consider what type relocation subsidy program should beundertaken (e.g. a proportional or a specific (per-firm) subsidy) together with the type oftax-financing scheme used and method of decision making (e.g. local or centralised government).
One aspect is average productivity. It is important to attract productive firms to periphery for the
purpose of boosting average productivity in periphery. In order to do so, a proportional subsidy
should be adopted. A per-firm subsidy, as in Baldwin and Okubo (2006), attracts only unproductive
firms. Thus this indicates that optimal subsidy policies should involve schemes that more productive
firms receive more benefits. For example, deduction of profit/business taxes alongside theconstruction of transport infrastructures could be better at raising productivity than a per-firm
subsidy in which each relocated firm receives a equal amount of subsidy directly.
The other aspect is the centralisation or decentralisation of decision making. Regional interests
conflict. As we have seen in the last section, the worst outcome obtains if a subsidy war occurs. Thus
it is essential that central government coordinates the different regional interests, even if theobjective of the regional policy is to boost the average productivity in that region and subsidy policy
is taken in a decentralised state. The subsidy policy program and taxation for that subsidy should becoordinated by central government. In particular, taxation to fund the subsidy should be global inorder to prevent tax competition and the distortion of disposal income between the core and the
periphery.

22

5. CONCLUDING REMARKSThis paper studies the impact of relocation subsidies in a heterogeneous firms model with acore-periphery structure. Our main findings are 1) a proportional subsidy induces the mostefficient firms to move to the periphery, and 2) non-subsidy is the Nash equilibrium in therepeated game but the periphery in a decentralised state has an incentive to implement local
tax-financed subsidies in the stage game. 3) While centralised economy chooses zero-subsidy,
decentralised economy leads periphery to choose optimal level of subsidy, resulting in enhancing
welfare in periphery.
This paper aims to provide the simplest model of an anti-agglomeration subsidy together with firm
heterogeneity. Of course, this paper cannot cover many important issues and aspects related to the
public policy. Possible extensions of this paper are the many other forms of taxation and thedifferent types of subsidy policies, including transfer mechanisms and heterogeneous households.
In addition, our model is static without free entry and exit of firms. This feature might rule out
interesting dynamic dimensions of subsidisation that may be easily analysed within the same
framework. For instance, firm relocation implies a lower (greater) mass of active firms in the core(periphery). This will generate a dynamic impact on productivity growth that may more thancompensate the sorting effect under subsidies.

REFERENCES

Baldwin, R. and R. Forslid, P. Martin, G. Ottaviano and F. Robert-Nicoud (2003), Economic Geography andPublic Policy, Princeton University Press, Princeton.
Baldwin, R. and T. Okubo (2006), Heterogeneous firms, agglomeration and economic geography: spatial
selection and sorting, Journal of Economic Geography, 6, pp.323-346.

23

Dupont, V. and P. Martin (2006) 'Subsidies to Poor Regions and Inequalities: Some Unpleasant Arithmetic'.
Journal of Economic Geography, vol. 6(2), pages 223-240.
Martin, P, T.Mayer, F. Mayneris (2010) “Public support to clusters A firm level study of French “Local
Productive Systems””, Regional Science and Urban Economics, forthcoming.
Martin,P and C.A Rogers (1995) `Industrial location and public infrastructure' Journal of International
Economics 39: pp335-351.
Melitz, M. (2003). “The impact of trade on intraindustry reallocations and aggregate industry productivity,”
Econometrica, 71, pp 1695-1725.
Midelfart-Knarvik, K.H., H. Overman, S Redding and A Venables (2000) “The Location of European
Industry”, European Commission, D-G for Economic and Financial Affairs, Brussels. Economic
Papers no 142.
Midelfart-Knarvik, K.H. and H.Overman (2002) “Delocation and European integration: is structural spending
justified?” Economic Policy vol. 17 Issue 35, pp. 321-359.
Mohl, P and T. Hagen (2010) “Do EU structural funds promote regional growth? New evidence from various
panel data approaches”, Regional Science and Urban Economics, 40: pp353-365.
Okubo, T (2009) "Trade Liberalisation and Agglomeration with Firm Heterogeneity -Forward and Backward
Linkages" Regional Science and Urban Economics, 39 (5): pp.530-541.
Okubo, T and E. Tomiura (2010). "Industrial Relocation Policy and Heterogeneous Plants Sorted by
Productivity: Evidence from Japan," RIETI Discussion papers 10016.
Ostbye, S (2010) “Regional policy analysis in a simple general equilibrium model with vertical linkages”,
Journal of Regional Science, 50 (3), pp. 756-775.

24

APPENDIX 1. HUMP-SHAPED WELFARE FOR LOCAL TAX-FINCNCE
We prove the hump-shaped curve with respect to u for a southern local tax-financed subsidy.
Specifically, since we cannot solve the welfare function in terms of u in an explicit form, we show
dv*
dv *> 0 and< 0 , which implies at least one hump-shape in the function. The
du u =u min
du u =u max

per-capita welfare is written as v * = ln(Y * − T * ) +µσ −1ln ∆* , where Y*= w*+ π =1+ π ,T=µλ  s 1 − s  1−s + ρ uφ +as  ∆ ∆*  R
1− s

π =

1− s 
1 − s  1−s + ρ
1− s 
µλ  sµλ  sµλ  s1−s + ρ(1 + u ) =)+ + φ *  . Thus we can + φ * a R
 + φ * (1 − a R
∆ 
∆ 
∆ 
s ∆
s ∆
s ∆

and

be rewritten as Y * − T = 1 + π − T =u µλ  s 1 − s 
1−s + ρ+1.
 φ + * 1 − a R
1 − s s  ∆ ∆ 

Using this, we differentiate v* in terms of u:
dv *  dπ d∆ da R dπ d∆* da R 
u 
u
aaR 
µ 1 d∆* da Raa −1 da R




=
+
−
+
1 − a R
 + π  − aa R
1− s 
du  d∆ da R du d∆* da R du 
du 1 − s 1 − s  s − 1 ∆* da R du
*First, we show dv

du u =u min> 0 . At u = umin( a R = 0 ), we can rewrite asµ σ dπµ σdπµ 1− σµ 1 − σ d∆*,,= λα (1 − φ )α αR −1 ,
=
−
=−
=
−
=
−
φ
2
2
*
*2
2
dα R
σ λ d∆
σ ∆
d∆
σ ∆
σ φλ
d∆
d∆* da Rµ,= λa (φ − 1)a aR −1 = −> 0 and π = . Thus, we can derive
da R
da R
du
σ

 dπ d∆* da R dπ d∆* da R dv *da u min 
m 1 d∆* da R
 + π  − aa aR −1 R +=  −+ *du u =u min  d∆ da R du d∆ da R du 
du 1 − s  s − 1 ∆* da R du
1 − s  m d∆* da R
da u min m 
m 1 d∆* da R
 − aa aR −1 R +=  s −+ 2
du 1 − s  s − 1 λφ da R duφ  sλ da R du sλ 


25


maaaR −1 da Rsu min  maaaR −1 da R
2
min
φ
s
s
u
≅  s + (1 − φ ) −
>
−
−
−
(
2
(
1
)(
1
)
)

φ
sλ (1 − s ) dudu
1 − s  sλmaaaR −1 da R
1
= (2 s (1 − φ )(1 − s ) 2 −+ 1)
sλ (1 − s ) dusφ + (1 − s ) / φ> (2 s (1 − φ )(1 − s ) 2 −maaaR −1 da R
1
+ 1)
>0
sλ (1 − s ) du
2 s (1 − s )dv *> 0 always holds.
du u =u min

Similarly, we derive the sufficient conditions fordv *< 0 at u max ( a R = 1 ). We can write as
du u =u maxd∆*µ 1− σµ 1− σ
dπ
d∆
dπ
µ σ
µ σ
,
,
= λα (1 − φ ) ,
φ
φ
−
=
−
=
= λα (φ − 1) , and
=−
=
−
2
*
*2
2
2 2
dα R
σ ∆
σ λ
dα R
d∆
σ ∆
σ φ λ d∆
da R
> 0.
du
(11)

 dπ d∆* da R dπ d∆* da R  u max 

1 
dv *
da u max
m 1 d∆* da R
1 −
 + π  − a R +=  −+ *−du u =u max  d∆ da R du d∆ da R du  1 − s 
du 1 − s 1 − s  s − 1 ∆* da R du
The second term in (11) is always negative. Due to

u max
dπ
dπ
< 0 is satisfied, then the> * , if 1 −
1− s
d∆ d∆

first term is always negative. The negative two first terms indicate that marginal income effect bysubsidy is negative due to tax payment. When the negative marginal income effect (the first and thesecond terms in (11)) is larger than marginal price index effect (the third term in (11)), the impact ofsubsidy on v* is negative. The sufficient condition for this can be derived as

As s is larger, 1 −sφ− 1 − (1 − s )(1 − φ ) > 0 .

dv *
u max
< 0 is more likely to occur. This< 0 becomes more negative anddu u =u max
1− sindicates that smaller population in the South would have a greater burden of tax per-capita, which
results in a greater negative marginal income effect.