On the desirability and feasibility of partial co-operation in

fiscal competition∗

Ajay Sharma†

Indira Gandhi Institute of Development Research, Mumbai(India)

June 17, 2014

Extended Abstract

This paper studies the desirability and feasibility of partial co-operation among regional gov-

ernments in fiscal competition. We analyze the implication of partial co-operation in a three

players (i.e. regional authorities/government) strategic interaction game to attract foreign in-

vestment using capital taxation and productivity enhancing public investment as their strategies.

We demonstrate that there are incentives to have partial co-operation in such competition for all

the regions i.e. both cooperating and non-cooperating. This result is new and can be attributed

to the inclusion of public investment in the strategy space of the competing regions. This not

only reduces the race-to-the-bottom in tax rates but also in turn increases the tax revenue for all

the regions. So, all the regions have incentive from the existence of partial co-operation. But the

caveat is that, cooperating regions have lower welfare and tax revenue when compared to non-

cooperating region in the partial co-operation situation (though sufficiently high welfare and tax

revenue outcome as compared to non-cooperative outcomes). This situation creates a prisoner

∗Preliminary draft, please do not quote.†Corresponding Address: Ajay Sharma, Indira Gandhi Institute of Development Research

(IGIDR), Gen A. K. Vaidya Marg, Goregaon (East), Mumbai-400065, India. e-mail: ajays@igidr.ac.in;

ajaysharma87@gmail.com

1

dilemma scenario, for all the regions, due to symmetric nature of gains and losses. We show that

even though all the regions have incentive to enter into partial co-operation due to higher welfare

and tax revenue level, there is a dilemma of who will enter into the partial co-operation first, be-

cause all the other regions will get higher benefit from waiting. In the Nash equilibrium outcome,

none of the regions will form a partial coalition. Thus, in the case of symmetric regions, it is

desirable to have partial co-operation but it is not the feasible outcome. To analyze the nature of

feasible equilibrium of partial co-operation in fiscal competition, we introduce asymmetry across

the regions in terms of their level of productivity of capital. In case of asymmetry, the gains and

losses of the high and low productive regions differ in case of being part of a coalition or from

being a non-cooperative region. In the equilibrium, we find that, if the regional asymmetry is

large enough, then it is desirable as well as feasible for the high productivity region to enter into

partial co-operation with any other region. On the other hand, for the low productive region, it is

always optimal and desirable to become part of a coalition with the high productive region. We

also show that for the low productivity regions it is not feasible and desirable to enter into partial

co-operation if the non-cooperating region has higher productivity of capital. We demonstrate

that, in the asymmetric regions, partial co-operation can be feasible as well as desirable under

certain condition, as compared to the symmetric region case. This paper enriches the literature

by characterizing the nature and existence of partial co-operation in a multi-strategy fiscal com-

petition. Some of the possible implications of this study are that existence of partial co-operation

can be beneficial for all the regions whether part of co-operation or not. Such co-operation out-

comes reduces the race-to-the-bottom in tax competition and thus providing revenue sources for

the regions.

Keywords: Mobile capital, Tax competition, Public investment, Collusion, Co-operation,

Social welfare.

JEL Classification: F21, H25, R50, H40, D60

2

1 Introduction

In the literature on fiscal competition for mobile capital, the role of capital taxation is well

recognized along with productivity enhancing public investment. But there are very few

papers which consider these strategies in a unified setting. In the recent time, Kotsogiannis

and Serfes (2010), Pieretti and Zanaj (2011), Dembour and Wauthy (2009), Hindriks et al.

(2008), Pal and Sharma (2013), Pal and Sharma (2011) and Hauptmeier et al. (2012) are

some recent studies that have enlarged the strategy space of literature on fiscal competition

by including more than one strategy in the standard models. But these studies are mainly

concerned about the nature of competition and outcome of these models and how race to

the bottom in tax rates is controlled in these settings. They have ignored one interesting

aspect of regional co-operation in multi-strategy fiscal competition games. Given that

there are real world examples of tax treaties as well combined public investment decision

making at sub-national level by governments in a federation, it seems very interesting to

understand the impact of co-operation on the capital tax rates, public investment choices

as well as foreign capital allocation in a region.

This paper studies the incentive to co-operate in a multi-strategy simultaneous game

in tax rate and public investment. We analyze the implication of partial co-operation in a

three players (i.e. regional authorities/government) strategic interaction game to attract

foreign investment using capital taxation and productivity enhancing public investment

as their strategies. We demonstrate that there are incentives to have partial co-operation

in such a game for all the players i.e.both co-operating and non co-operating regions, but

co-operating regions have lower welfare level (though sufficiently high welfare outcome

as compared to non cooperative game outcomes) as compared to non co-operating re-

gion. This creates a situation where all the regions have incentive to enter into partial

co-operation but there is dilemma of who will enter into the co-operation first. In the case

of symmetric regions, it is not feasible to decide which regions will opt for co-operation in

the fiscal competition. Therefore, we consider the case where regions are asymmetric in

terms of productivity of capital. We find that based on the level of productivity differences

3

across regions, various equilibrium conditions arise which are discussed in the relevant

sections.

Most of these models discussed in the literature analyze the non-cooperative aspects of

tax competition. But a stream of tax competition literature also analyzes the co-operative

tax competition games, considering the impact of partial and full co-operation and har-

monization of the taxes across regions on the inter-regional competition. Considering the

focus of the paper on co-operation in a multi-strategy fiscal game, we discuss only the

co-operative tax competition literature 1.

Konrad and Schjelderup (2002) discuss the implication for the partial harmonization of

the taxes in a subset of countries competing with each other. Using the model developed in

the seminal paper of Wildasin (1988), they show that it is feasible for a subset of countries to

make coalition and charge higher taxes in Nash equilibrium provided the strategic variables

(taxes) are complements, leading to higher welfare and better provision of public good for

the countries in the subset of countries forming coalition. In a similar framework, Conconi

et al. (2008) show the implication of tax harmonization using two types of taxes (on capital

and labour income) imposed simultaneously. The paper assumes a policy commitment

problem in the framework. They analyze three situations and show that compared to

non co-operative situation, full harmonization may not be welfare improving because there

will be hundred percent taxation and no investment will be done, all the revenue will

be generated by the labor income tax which will also suffer due to reduced investment.

Moreover there will be incentive to deviate for the member countries when there is no

credible control mechanism. Finally they show that partial harmonization is desirable over

no and full tax harmonization in this framework. Haufler and Wooton (2006) analyze the

impact of partial co-ordination or union in the framework developed earlier ( Haufler and

Wooton (1999)) which is comparatively less restrictive considering regional asymmetries

in size, tariffs and product market competition. The countries compete in profit taxes

and subsidies considering the spillover of Foreign Direct Investment over union and non-

1See Wilson 1999, Wildasin 2005 for detailed discussion on non co-operative fiscal competition games.

4

union countries. They produce in the host country and sell in all the countries involving

trade cost. So based on the trade cost and impact of spillover, tax rate in the union

(i.e. coordinating countries) may or may not increase. This result is different from the

earlier result which show that in case of co-ordination tax rate increases in the coordinating

countries. Even if this model provides some enriching results. There are certain loops in

the model concerning commitment issues relating to the profit taxation and the impact of

the co-ordination relating uncertainty on the location choice of the firm.

Most of the articles discussed above show that partial tax co-operation is feasible as well

as beneficial for the coordinating regions. On the contrary, Marchand et al. (2003) assume

that, in a model where both labour and capital are perfectly mobile and land is the only

immobile factor of production, all three factors of production are taxed. They show that, in

a symmetric regions case where labour and capital are substitute, a minimum tax imposed

on capital income (as a co-ordination policy) leads to welfare worsening outcomes and

co-ordination becomes undesirable for both the regions. Kehoe (1989) considers the case

of benevolent governments under the time inconsistency problem and shows that any type

pf policy co-operation among them is welfare reducing and therefore undesirable. Instead

of a single and centralized entity making the decision, it is better to have decentralized

decision making process, where regional entities decide their optimal strategy.

As is evident from relevant literature, there are mixed results supporting as well as re-

jecting partial tax co-opertion as a feasible and beneficial outcome for co-operating regions

as well as non co-operating regions. One thing that is clear is that full co-operation is

neither feasible nor a stable outcome for all regions. Given this, most of the papers in the

literature consider the case where only taxation (including capital, labour and land rent)

are part of the strategic variables used by the competing or co-operating regions in the fiscal

competition. Moreover, most of these papers consider the case of symmetric regions which

is very restrictive in nature. We fill these gaps in the literature. In this paper, we consider

the model where regions compete in terms of capital taxation and public investment choice.

Further, we also assume that regions are asymmetric in terms of productivity of capital

which affects the overall production function and firm efficiency across the regions.

5

The rest of the paper is organized as follows. Section 2 outlines the theoretical model.

Section 3 shows the outcomes in case of non co-operative fiscal competition among regions.

Section 4 considers the scenario where regions partial co-operate in the fiscal competition

game. Section 5 considers the asymmetric regions case and different outcomes in the partial

co-ordination scenarios. Section 6 concludes.

2 The model

Suppose that there are three symmetric2 regions, region-1, region-2 and region-3 competing

for foreign owned mobile investment capital of total amount one, which is exogenously

determined, in order to maximize their respective objectives. Each region decides the tax

rate ti (≥ 0) on mobile capital xi (0 ≤ xi ≤ 1) and the level of public investment gi

(≥ 0), i = 1, 2. Higher tax rate on capital in any region dampens the flow of capital in

that region, but that may lead to higher tax revenue. In contrast, public investment in

any region facilitates production in both the regions and, thus, it enhances productivity

of capital across regions. However, the effect of public investment (gi) in region i on

productivity of capital in ith region is higher than that in the jth region, unless there is

perfect spillover of public investment. The cost to provide public investment gi by region

i is assumed to beg2i2

, i = 1, 2, 3. Following Hindriks et al. (2008), we consider that the

production function of a region i (= 1, 2, 3) is as follows.

Fi(xi, gi) = (γ + gi + θgj)xi −δx2i2, i, j = 1, 2, 3, i 6= j, (2)

where xi is the amount of mobile capital invested in region i,3 γ (> 0) is the technology

parameter, δ (> 0) denotes the rate of decline in the marginal productivity of mobile

capital, and θ (0 ≤ θ ≤ 1) is the spillover effect of public investment in one region to the

other region’s productivity. Higher value of θ denotes higher spillover effect; θ = 1 (θ = 0)

corresponds to the extreme case of perfect (no) spillover. Clearly, regions have symmetric

2This assumption is relaxed in the section 5.3xi can also be interpreted as the share of mobile capital invested in region i.

6

production functions, which are increasing, twice continuously differentiable and concave

in the level of capital. We assume that γ > δ > 1. The first part of the inequality, i.e.,

γ > δ, ensures that marginal productivity of capital is always positive.4 The second part

of the inequality, i.e, δ > 1, ensures existence and stability of interior solutions in all the

cases considered.

Assuming that the capital market is perfectly competitive5 and normalizing the price

of output to be one, we can write the returns to immobile factors of region i as, IRi =

[Fi(.) − xi∂Fi(.)∂xi

] = δ2x2i . Clearly, returns to immobile factors in a region is increasing in

investment of mobile capital in that region, at an increasing rate δ. Since mobile investment

capital is foreign owned, social welfare (SW ) of a region is given by the sum of returns to

immobile factors (IR) and net tax revenue (NT ) of that region:

SWi = IRi +NTi =δ

2x2i + [tixi −

g2i2

], i = 1, 2, 3. (3)

Note that the parameter δ can also be interpreted as the rate of increase in ‘marginal

social welfare’ (∂SWi

∂xi) of a region due to increase in mobile capital in that region. The

above formulation of social welfare function is in line with Kempf and Rota-Graziosi

(2010), Hindriks et al. (2008) and Laussel and Le Breton (1998).6

We consider that a region maximizes its social welfare (SW ) while competing to attract

foreign mobile capital using tax rates and public investment as the strategic variables.

In this paper, we consider two games: game-1: when regions don’t co-operate and decide

their optimal strategies as independent optimal response to other region’s strategies; game-

2: when two regions co-operate and compete with the third region. In game-2, co-operating

regions jointly determine their tax rate and public investment by optimizing joint social

welfare of both regions.

The stages of the game-1 are as follows.

4Note that, in absence of any tax and public investment, if full amount of mobile capital is invested in

any one of the two regions, marginal productivity of capital in that region is equal to γ − δ.5It implies that capital is paid according to its marginal productivity6For further justifications of the objective functions of regions see Laussel and Le Breton (1998).

7

Stage 1: Region-1, region-2 and region-3 are engaged in simultaneous move fiscal

competition in terms of both tax rate and public investment.

Stage 2: Owners of mobile capital decide how much to invest in which region.

The stages of the game-2 are as follows.

Stage 1: Regions decide whether to enter into co-operation for fiscal competition

with other region.

Stage 2: Region-1 and region-2 jointly compete with region-3 in simultaneous

move fiscal competition in terms of both tax rate and public investment.

Stage 3: Owners of mobile capital decide how much to invest in which region.

3 Non co-operative fiscal competition game

In game-1 i.e. non co-operative fiscal competition game, we start from the second stage

by noting that the allocation of capital between the two regions depends on productivity

of capital and tax rate of each region. Since, we assume that capital market is perfectly

competitive, marginal return to capital net of tax in region i is F′i,xi

(xi, gi)− ti, i = 1, 2, 3,

if region i gets xi (0 ≤ xi ≤ 1) amount of mobile capital. It implies that we must have

F′i,xi

(xi, gi)− ti > 0, for region i to get xi amount of mobile capital, considering region i in

isolation. Note that, for any given allocation of capital, if net marginal returns to capital

differ between regions, reallocation of capital takes place from the region with lower net

return to the other region. Therefore, to rule out the possibility of arbitrage, we need to

have the allocation such that net marginal return to capital is same in both the regions.

Moreover, for feasibility of such arbitrage-proof allocation of capital, x1 + x2 + x3 ≤ 1

must be satisfied. We consider that entire amount of mobile capital is allocated between

region-1, region-2 and region-3: x1+x2+x3 = 1. In other words, we rule out the possibility

of mobile capital to remain idle. Therefore, the arbitrage-proof equilibrium allocation of

mobile capital, for any given levels of public investments and tax rates, between the regions

8

is given by

F′

1,x1(x1, g1)− t1 = F

′

2,x2(x2, g2)− t2 = F

′

3,x3(x3, g3)− t3 > 0, (4)

and x1 + x2 + x3 = 1. (5)

From (4) and (5), we get the equilibrium investment of mobile capital in region-1, region-2

and region-3 given the levels of public investments and tax rates, as follows.

x1 =1

3+

(1− θ)(2g1 − g2 − g3) + t2 − 2t1 + t33δ

, (6)

x2 =1

3+

(1− θ)(2g2 − g3 − g1) + t3 − 2t2 + t13δ

, (7)

x2 =1

3+

(1− θ)(2g3 − g1 − g2) + t1 − 2t3 + t23δ

, (8)

Clearly, the allocation of mobile capital depends on each region’s tax rate as well as public

investment. Increase in tax rate of one region negatively (positively) affects the flow of

mobile capital in that (the other) region(s): ∂xi∂ti

< 0 and∂xj∂ti

> 0; i, j = 1, 2, 3, i 6= j.

In contrast, increase in public investment in one region increases(decreases) capital flow in

that (the other) region, unless there is perfect spillover of public investment: ∂xi∂gi

> 0 and

∂xj∂gi

< 0, unless θ = 1; i, j = 1, 2, 3, i 6= j.

Coming to the first stage of game-1, we substitute the values of x1, x2, x3 in the so-

cial welfare function of the respective regions and maximize the social welfare function,

SWi(i = 1, 2, 3 using tax rates, ti and level of public investment, gi as their strategies:

arg maxti,gi SWi(ti, gi) where i = 1, 2, 3. By solving the problem we get reaction function

for tax rates and public investment as follows:

ti =1

8(δ + (1− θ)(2gi − gj − gk) + tj + tk) , (9)

gi =2(1− θ) (δ − (1− θ)(gj + gk) + ti + tj + tk)

9δ − 4(1− θ)2, (10)

here, i, j, k = 1, 2, 3; i 6= j 6= k

From these reaction functions we can see that tax rates in a region is positively (negatively)

related to tax rates (level of public investment) in other regions; whereas level of public

investment is negatively (positively) affected by public investments (tax rates) in other

9

regions, provided that there is no perfect spillover effect of public investment. From reaction

functions we get that, ∂ti∂tj

> 0; i 6= j = 1, 2, 3, indicating that tax rates are strategic

complements and there is a bertrand type of competition in tax rates. On the other

hand, in case of public investment, we observe ∂gi∂gj

< 0; i 6= j = 1, 2, 3, stating that public

investment levels are strategic substitute and there is Cournot type competition in public

investment strategy space. So regions are engaged in fiscal competition, with two strategic

variables, of which one is strategic complement and other is strategic substitute. This

leads to the final outcome to be of mixed nature, given the parametric restrictions, in

terms of productivity of capital in a region (δ) and public investment spillover effect (θ).

The equilibrium outcomes in game-1 i.e. non co-operative fiscal competition are:

SWi =1

18

(2δ − (θ − 1)2

)xi =

1

3

ti =δ

6

gi =1− θ

3

here i = 1, 2, 3.

Due to symmetric size, all the regions get same level of foreign capital, charge same

tax rates, provide same level of public investment and get same welfare level. It is to be

noted that level of public investment, tax rates, capital allocation as well as welfare level

of a region in a three region fiscal competition is lower as compared to two region fiscal

competition7.

Lemma 1: Increasing the number of regions in a multi-strategy fiscal competition for

foreign owned mobile capital, leads to decrease in the social welfare of all the regions. Tax

rates and public investment level also decreases with increase in the number of competing

regions.

7In a symmetric two region fiscal competition with tax rates and public investment, equilibrium out-

comes are: SWi = 18

(3δ − (θ − 1)2

), xi = 1

2 , ti = δ2 , gi = 1−θ

2 for i = 1, 2

10

Taking this into mind, next we move on to the situation of partial co-operation in this

fiscal competition game, where two of the three regions opt to jointly decide their tax rate

and public investment level to compete with the third region.

4 Partial co-operation in fiscal competition game

Coming to game-2 where two regions decide to co-operate in the first stage of the game,

and jointly compete with the third region, which is not part of the coalition, in the second

stage. Followed by the decision of firms to allocate capital based on the tax rates and

public investment made by the regions. Using the backward induction method, we first

solve the third stage of the model, i.e. capital allocation, given the tax rates and public

investment level decision of the regions. The capital allocation equations are identical to

the equations (6), (7) and (8) from game-1. To solve the second stage of the game, we

substitute the capital allocation values to the social welfare (SW ) equations of the regions.

Here the social welfare for region-1 and region-2 combined will be SW12 = SW1 + SW2,

whereas region-3 will have same welfare level as in game-1. The outcomes for region-1 and

region-2 will be denoted by subscript ”12”. The reaction function for regions will be:

t12 =2

5(δ + (1− θ)(g12 − g3) + t3) , (11)

t3 =1

8(δ + (1− θ)(g3 − g12) + 2t12) , (12)

g12 =(1− θ) (δ − g3(1− θ) + t3 + 2t12)

9δ − (1− θ)2(13)

g3 =2(1− θ) (δ − 2g12(1− θ) + t3 + 2t12)

9δ − 4(1− θ)2, (14)

From the above equations, it is clear that tax rate and public investment reaction

functions of the regions have similar properties as in case of non co-operative game. The

equilibrium outcomes for the partial co-operation game are as follows:

11

Table 1: Comparing outcomes of regions in partial co-operation fiscal competition

Outcomes Co-operating regions (1, 2) Non co-operating region (3) Comparison

Social welfare (SW )(3δ−2(1−θ)2)

2(5δ−(1−θ)2)

72(2δ−(1−θ)2)2(3δ−(1−θ)2)

2

18(2δ−(1−θ)2) SW12 < SW3

Tax Revenue (TR)δ(3δ−2(1−θ)2)

2

18(2δ−(1−θ)2)2δ(3δ−(1−θ)2)

2

18(2δ−(1−θ)2)2 TR12 < TR3

Capital allocation (x) 16

(2− δ

2δ−(1−θ)2

)16

(2 + 2δ

2δ−(1−θ)2

)x12 < x3

Tax rate (t) 13δ(

2− δ2δ−(1−θ)2

)16δ(

1 + δ2δ−(1−θ)2

)t12 > t3

Public Investment (g) (1−θ)(3δ−2(1−θ)26(2δ−(1−θ)2)

(1−θ)(3δ+(1−θ)2)6δ−3(1−θ)2 g12 < g3

First, we discuss the comparative outcomes for the co-operating regions and non co-

operating region as shown in Table 1. Followed by which we will compare the outcomes of

non co-operative fiscal competition game from section 3 with that of partial co-operation

game in this section.

In the partial co-operation fiscal competition game, we find that region-1 and region-2,

which decided their tax rates and public investment level cooperatively, have higher tax

rate and lower public investment level as compared to the region-3 which was not part

of the coalition. We also observe that region-1 and region-2 have lower welfare and tax

revenue level as compared to region-3. This suggests that due to higher tax rate and lower

public investment in cooperating regions, the crowd out effect of foreign capital is higher

that incentives from lower cost of public investment and higher tax revenue per unit of

capital. This result is new.

Proposition 1: In the partial co-operation game, regions which co-operate get lower

12

social welfare and tax revenue outcomes than the non cooperating region. The cooperating

regions also receive less capital allocation than non cooperating region and charge higher

tax rates.

Now given that cooperating regions are having lower outcomes in terms of capital allo-

cation, tax revenue and social welfare level, would it be desirable by these regions to enter

into such co-operation. To answer this question, we compare the outcomes of this section

with that of section 3 i.e. non co-operative fiscal competition outcomes. Comparing the

outcomes of non co-operation game and partial co-operation game (in section 3 and section

4 respectively), we observe that the level of welfare and tax revenue for all the agents is

higher in case of partial co-operation game as compared to non-co-operative game8. Thus,

partial co-operation is pareto improving for all the agents whether co-operating or not.

Moreover, public investment made by the non cooperating agent is higher as compared to

the earlier case (when no region co-operates), and public investment level of cooperating

agents is lower than the earlier case.

Proposition 2: Partial co-operation is beneficial to all the regions (whether part of

coalition or not), when compared with the non co-operative game. All the region get higher

level of social welfare and tax revenue outcomes in partial co-operation game.

Importantly, we also observe that tax rates imposed by cooperating regions as well as

non cooperating region are higher in the case of partial co-operation as compared to non

co-operative outcomes. This suggests that partial co-operation reduces the race-to-the-

bottom in tax competition in the existing framework. This result is new.

Proposition 3: Partial co-operation in a fiscal competition for foreign capital in taxa-

tion and public investment leads to control in the race-to-the-bottom i.e. tax rate of all the

regions are higher in partial co-operation game that non co-operative game.

8We have not shown the detailed comparison but it can be inferred from the equilibrium outcomes of

three regions in the previous section with that of the current section.

13

As shown above the cooperating regions have lower social welfare than non cooperating

region, therefore the question arises, what will induce regions to enter into co-operation

when they know that if the other region enters into co-operation then they are able to get

higher outcomes? This question cannot be answered in case of symmetric regions because

none of the regions have any incentive or disincentive to enter into a co-operation. But

if the regions are asymmetric in terms of their size or productivity then we can clearly

identify which region will be wiling to enter into such co-operation or which region will

prefer to wait.

In the next section, we consider the case where the regions are asymmetric in terms of

productivity of mobile capital. We also characterize which regions will prefer to enter into

a coalition and which region would prefer to wait.

5 Asymmetric regions with partial co-operation

Here, we consider that regions are asymmetric in terms of productivity of mobile capital,

in line with Hindriks et al. (2008) and Kempf and Rota-Graziosi (2010). Any of the

regions can be either a high productivity region or a low productivity region. This leads

to four different scenario: in the first case, cooperating regions have higher productivity

of capital than non cooperating region; in the second case, non cooperating region has

higher productivity of capital than other two regions; in third case, regions with higher

productivity of capital co-operates with region having lower productivity of capital and

non cooperating region has lower productivity of capital; and finally in forth case, region

having higher productivity of capital co-operates with region with lower productivity of

capital and non cooperating region has higher productivity of capital. These alternate

scenarios are discussed as follows.

14

Case 1: Co-operating regions have higher productivity than non

co-operating region

The production function of firms in region-1 and region-2 is Fi(xi, gi) = (γ+gi+θgj+δ,ε2

)xi−δx2i2, i = 1, 2; j = 1, 2, 3, i 6= j; and for region-3 is Fi(xi, gi) = (γ+gi+θgj− δ,ε

2)xi− δx2i

2, i =

3, j = 1, 2, i 6= j, where the parameter ε(0 ≤ ε ≤ 1) denotes the extent of asymmetry be-

tween the co-operating regions and non co-operating region. Here, higher is the value of

ε higher is the degree of asymmetry. All the stages of the game are same as in section

4. Solving by backward induction, given the tax rates and level of public investment, at

the stage 3, capital allocation in the regions will be: x1 = δ(1+ε)+2(2g1−g2−g3)(1−θ)−2 t1+t2+t33δ

;

x2 = δ(1+ε)+2(2g2−g3−g1)(1−θ)−2 t2+t3+t13δ

; x3 = δ(1−2ε)+2(2g3−g1−g2)(1−θ)−2 t3+t1+t23δ

. Clearly, we

can say that because of the lower productivity in region 3, there will be lower allocation of

capital given that level of public investment and tax rates are same. Higher is the asym-

metry of productivity of capital, higher will be the differences in the allocation of capital in

cooperating and non cooperating regions. Now, in stage 2 we solve for tax rates and public

investment level in the regions. The equilibrium level of tax rate and public investment

level in the co-operating regions are: t12 =δ(δ(3+2ε)−2(1−θ)2)

6δ−3(1θ)2,g12 =

(1−θ)(δ(3+2ε)−2(1−θ)2)6(2δ−(1−θ)2) ; and

for non co-operating region are: t3 =δ(δ(3−2ε)−(1−θ)2)

6(2δ−(1−θ)2) , g3 =(1−θ)(δ(3−2ε)+(1−θ)2)

6δ−3(1−θ)2 . We can

check that tax rates and public investment level are positive in this case. It is also clear

that higher regional asymmetry leads to higher tax rates and public investment in the

cooperating regions (1 and 2) and lower tax rate and public investment in the non coop-

erating region (3): ∂t12∂ε

> 0; ∂g12∂ε

> 0 and ∂t3∂ε

< 0; ∂g3∂ε

< 0. Further, higher asymmetry

leads to higher welfare level and tax revenue for the co-operating regions; and lower wel-

fare level and tax revenue for the non cooperating region. This implies that in a partial

co-operation game, regional asymmetry (higher productivity of capital in region 1 and 2)

provides improved outcomes for the co-operating regions and lower outcomes for the non

co-operating region.

Proposition 4: Higher is the regional asymmetry, higher (lower) will be the tax rate

15

and public investment level in the higher (lower) productivity region along with higher

(lower) allocation of capital.

Given the outcomes above, still, if the regional asymmetry is below a critical point,

then non cooperating region gets higher welfare level and tax rates than the co-operating

regions and will get the external benefits of the partial co-operation in fiscal competition;

if the asymmetry is beyond that critical point, co-operating regions will get higher welfare

and tax revenue in the equilibrium 9.

In this situation, we can clearly identify that if the regional asymmetry is large, then

regions with higher productivity of capital have incentive to form a coalition and decide

their tax rate and public investment jointly. On contrary, if the regional asymmetry is not

large enough (below critical point), then there is a trade off in entering the co-operation

because on one side, regions with higher productivity will gain in terms of higher welfare

and tax revenue (as compared to the symmetric regions situation) by cooperating, they will

be losing some of the tax revenue and welfare to non cooperating regions because of the

effect of partial co-operation i.e. higher tax rate and lower public investment (as shown

in section 4). In effect, we can say that for regions with higher productivity of capital

it is always beneficial to engage in partial co-operation as compared to regions with low

productivity of capital.

Proposition 5: In a partial co-operation fiscal competition game with productivity

asymmetry: (a)if the cooperating regions have higher productivity of capital than non co-

9Since there are three parameters (θ, δ, ε) which decide the level of welfare for all three regions, it

becomes tedious to decide the conditions under which one region will get higher welfare and other outcomes

than the others. Since we want to identify the effect of the regional asymmetry, we consider that θ == 0

i.e. there is no spillover effect of public investment across regions. We also assume that δ == 2 which is

to satisfy all the second order conditions for the equilibrium to hold. Now, looking at the ε parameter,

we find that if differences in productivity of capital between co-operating and non co-operating regions

is lower than 0.10, then non co-operating region (3) gets higher welfare and tax revenue in the partial

co-operation situation. While if ε > 0.10 then co-operating regions (1 and 2) have higher welfare and tax

revenue than region 3.

16

operating region, then below a critical value of the ε (parameter of regional productivity

asymmetry), cooperating regions have lower welfare and tax revenue than non cooperating

region; (b) if the value of ε is above that critical value, then cooperating regions have higher

welfare level and tax revenue than non cooperating region.

Case 2: Co-operating regions have lower productivity of capital

than non co-operating region

In this case also, stages of the game remain same as in section 4. The production functions

of firms in region-1 and region-2 are as follows:Fi(xi, gi) = (γ + gi + θgj − δ,ε2

)xi − δx2i2, i =

1, 2; j = 1, 2, 3, i 6= j; and for region-3 is Fi(xi, gi) = (γ + gi + θgj + δ,ε2

)xi − δx2i2, i =

3, j = 1, 2, i 6= j. Solving the stages of the game by backward induction, we get the

equilibrium value of tax rate, public investment, capital allocation and welfare level. For

sake of brevity, we are not showing all the calculations and results here. We observe

that:∂t12∂ε

< 0; ∂g12∂ε

< 0 and ∂t3∂ε> 0; ∂g3

∂ε> 0. It shows that regional asymmetry in favour of

non cooperating regions leads to lower tax rate and public investment level for cooperating

regions (1 and 2) and higher tax rate and public investment for non cooperating region

(3). Moreover,welfare level and tax revenue of non cooperating region(3) remains higher

than that of cooperating regions (1 and 2).

So we can say that, in this case, none of the cooperating regions has any incentive to

enter into a co-operation for fiscal competition because non co-operating region always gets

higher welfare level than cooperating regions.

Proposition 6: In fiscal competition with cooperating regions having lower productivity

of capital than non cooperating region, irrespective of the level of regional asymmetry, non

cooperating region always gets higher welfare and tax revenue level than cooperating regions.

17

Case 3: Co-operating region-1 has higher productivity, region-2

has lower productivity and non co-operating region-3 has lower

productivity

Solving this case, we find that: ∂t12∂ε

> 0; ∂g12∂ε

> 0 and ∂t3∂ε

< 0; ∂g3∂ε

< 0. This means

that overall higher regional asymmetry leads to higher tax rate and public investment

in the cooperating regions and lower tax rate and public investment in non cooperating

region. But even though the combined social welfare of co-operative regions increases

with regional asymmetry, there is decrease in the social welfare of the region-2 i.e. low

productivity region. It means that increased social welfare of region-1 is compensating

the decrease in the social welfare of region-2. A similar observation is being made in the

case of tax revenue. Here, we can say that due to one of the cooperating region being low

productive none of the regions has unilateral benefit from entering the co-operation. But

for the region-2 which has lower productivity of capital like region-3, the incentive to enter

into co-operation will depend on the level of regional asymmetry i.e. those level of regional

asymmetry under which the benefits of entering a co-operation with high productive region

will surpass the loss from entering the co-operation in general.

To better illustrate this point, we show some numerical results. Assuming that there

is no spillover effect i.e. θ == 0 and δ == 2 to satisfy all the second order conditions for

the equilibrium to hold. We find that welfare level and tax revenue of the region-1 will be

higher than region-3 if the regional asymmetry i.e. the value of ε > 0.177. Therefore a

region with high productivity of capital will be willing to enter into co-operation with low

productivity region if the level of regional asymmetry is above a critical threshold value.

For region-2 there is no such incentive to enter into the partial co-operation beacuse under

no condition will it get higher welfare than non co-operating region.

Proposition 7: If the regional asymmetry is above a critical value, then high pro-

ductivity of capital region have incentive to engage into partial co-operation with a low

productivity of capital region.

18

Case 4: Co-operating region-1 has higher productivity, region-2

has lower productivity and non co-operating region-3 has higher

productivity

Solving this case, we observe that: ∂t12∂ε

< 0; ∂g12∂ε

< 0 and ∂t3∂ε> 0; ∂g3

∂ε> 0. Here we can see

that the results are opposite to that of case 3, because the higher productive region-3 is

able to provide for higher public investment and charge higher tax rate. On the other side,

in partial fiscal competition, co-operating regions are losing because one of the cooperating

regions has lower productivity of capital.

A numerical exercise similar to case 3 shows that under no condition, region-2 or region-

1 will get higher social welfare and tax revenue than region-3. Therefore there will be no

possibility of partial co-operation between low and high productivity regions.

Proposition 8: In a three region competition for fiscal competition, if any of the two

regions have higher productivity of capital than third, then possibility of partial co-operation

among high and low productivity regions is not feasible.

6 Conclusion

This paper fills the gap in the literature on co-operation in fiscal competition by consid-

ering the case of two opposite strategic variables- tax rate (strategic complements) and

public investment (strategic substitutes), and regional asymmetry in term of productivity

of capital.

We find that partial co-operation is always desirable over the non co-operative outcomes

in case of symmetric regions. Interestingly, all the regions get higher social welfare and

tax rates in the partial co-operation fiscal competition than non co-operative game, but

19

the cooperating regions have lower social welfare and tax revenue than non co-operative

region. Therefore, none of the regions have any incentive to enter into the partial co-

operation in a fiscal competition if regions are symmetric. Therefore, we characterize the

partial co-operation equilibrium under the regional asymmetry condition. We find that

regions have incentive to co-operate if the non co-operative region has lower productivity

of capital than the co-operative regions. We also show that low productivity regions do

not have any interest to co-operate if the non co-operative region has higher productivity.

Further, a higher productivity region has incentive to enter into a partial co-operation

if the non co-operative region has low productivity of capital and regional asymmetry is

above a critical parameter value.

Thus we can say that even though partial co-operation is desirable by all the regions in

a symmetric regions case; to ensure that regions will enter into partial co-operation there

is a need for regional asymmetry which is large enough to encourage regions to enter into

such partial co-operation. This means that regional asymmetry should be large enough

to compensate for the relative loss of entering into partial co-operation than being a non

co-operative region.

This paper enriches the literature by characterizing the outcomes of partial co-operation

in a multi-strategy fiscal competition. Moreover, we also demonstrate the conditions un-

der which it will be not only desirable to enter into partial co-operation but regions will

preferably enter into such partial co-operation.

Some of the possible extensions of this exercise can be introduction of mobile labour,

endogenizing the decision to co-operate and sequential choice of strategic outcomes i.e. tax

rate and public investment.

20

References

Conconi, P., Perroni, C., and Riezman, R. (2008). Is partial tax harmonization desirable?

Journal of Public Economics, 92(1):254–267.

Dembour, C. and Wauthy, X. (2009). Investment in public infrastructure with spillovers

and tax competition between contiguous regions. Regional Science and Urban Economics,

39(6):679–687.

Haufler, A. and Wooton, I. (1999). Country size and tax competition for foreign direct

investment. Journal of Public Economics, 71(1):121–139.

Haufler, A. and Wooton, I. (2006). The effects of regional tax and subsidy coordination

on foreign direct investment. European Economic Review, 50(2):285–305.

Hauptmeier, S., Mittermaier, F., and Rincke, J. (2012). Fiscal competition over taxes and

public inputs. Regional Science and Urban Economics, 42(3):407–419.

Hindriks, J., Peralta, S., and Weber, S. (2008). Competing in taxes and investment under

fiscal equalization. Journal of Public Economics, 92(12):2392–2402.

Kehoe, P. J. (1989). Policy cooperation among benevolent governments may be undesirable.

The Review of Economic Studies, 56(2):289–296.

Kempf, H. and Rota-Graziosi, G. (2010). Endogenizing leadership in tax competition.

Journal of Public Economics, 94(9-10):768–776.

Kotsogiannis, C. and Serfes, K. (2010). Public goods and tax competition in a two-sided

market. Journal of Public Economic Theory, 12(2):281–321.

Laussel, D. and Le Breton, M. (1998). Existence of nash equilibria in fiscal competition

models. Regional Science and Urban Economics, 28(3):283–296.

Marchand, M., Pestieau, P., and Sato, M. (2003). Can partial fiscal coordination be welfare

worsening?: A model of tax competition. Journal of Urban Economics, 54(3):451–458.

21

Pal, R. and Sharma, A. (2011). Political competition and leadership in tax competition.

Technical report, Indira Gandhi Institute of Development Research, Mumbai, India.

Pal, R. and Sharma, A. (2013). Endogenizing governments’ objectives in tax competition.

Regional Science and Urban Economics, 43(4):570 – 578.

Pieretti, P. and Zanaj, S. (2011). On tax competition, public goods provision and jurisdic-

tions’ size. Journal of International Economics, 84(1):124–130.

22