A Bargaining Model of Tax Competition

Seungjin Han John Leach

Department of Economics, McMaster University

1 December 2005

Abstract

This paper develops a model in which competing governments of-fer financial incentives to individual firms to induce the firms tolocate within their jurisdictions. Equilibrium is described underthree specifications of the supplementary taxes. There is no mis-allocation of capital under two of these specifications, and theremight or might not be capital misallocation under the third. Thisresult contrasts strongly with that of the standard tax compe-tition model, which does not allow governments to treat firmsindividually. That model almost always finds that competitionamong governments leads to the misallocation of capital.

1 Introduction

The tax competition literature assumes that the economy is divided intoautonomous regions, and that capital can move freely between the regions.Its objective is to determine the rates at which the regions will tax profits,and the impact of the profits tax on resource allocation. It finds that thetax rates generally vary across regions, and that the variation in tax ratesleads to an inefficient allocation of capital: the low tax regions use too muchcapital and the high tax regions use too little. A limitation of this literatureis that it assumes that a government can only attract capital by reducingthe rate at which it taxes profits. In reality governments attach so much

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Government AnnualCompany Location Investment Aid Output

Honda Lincoln, AL 825 248 270Hyundai Montgomery, AL 1000 252 300Nissan Canton, MS 1400 360 400Toyota Princeton, IN 1900 117 400Toyota San Antonio, TX 800 133 150Toyota Woodstock, ON 625 100 100

Source: The Globe and Mail, Toronto, 1 July 2005. Investment andaid are in millions of US dollars; output is in thousands of vehicles.

Table 1: Assembly Plants Built by Asian Auto Makers in North Americasince 1998

importance to new capital investment that they will often make substantialfinancial concessions to get it. Table 1 shows the concessions recently givento Asian auto makers building assembly plants in North America. Theseconcessions have been as high as 30% of the new investment.

A government hoping to induce a major corporation to locate facili-ties within its jurisdiction will often offer a package of financial incentives.Smaller corporations tend to be less generously treated than larger corpo-rations, and very small firmsmom and pop grocery storescan expect nospecial treatment. Nevertheless, if competition among governments is to beproperly understood, the focus must be on the overall financial package, andthe way in which that package varies from firm to firm. A model of this sortis described here. A key element of the model is that capital is embodied inheterogeneous firms. Each firm is mobile, and each firms productivity variesfrom region to region. The firms differ in the way that their productivityvaries across regions. Each firm receives an offer from the government ofevery region, and the firm locates in the region in which its after-tax profitswould be highest. The government uses its tax revenue to provide a publicgood to the regions citizens.

Casual observation shows that governments are prepared to negotiatewith some firms but not with others, so this model is an abstraction. Thestandard tax competition model, in which no-one gets a special deal, is alsoan abstraction. Our view is that these two models constitute polar cases

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in the study of tax competition. Reality lies somewhere between them, butalmost all research has been concentrated on one of the two poles. Our hopeis that models like this one will ultimately lead to a more balanced view oftax competition.

Our basic model assumes that each government can levy a tax on theprofits of firms located within its jurisdiction, and that the government canalso levy a lump-sum tax on the incomes of the citizenry. Under theseassumptions, the standard tax competition model predicts optimal publicgoods provision but misallocation of capital. This finding was first presentedby Hamada (1966). Hamadas model has been expanded in many differentways, but its core result has remained largely unchanged. The bargainingmodel, by contrast, predicts both optimal public goods provision and optimalallocation of capital. The resource misallocation that has been the focus ofthe tax competition literature is simply not there.

Since the existence of such a broadly based lump-sum tax might be viewedwith some scepticism, we consider two alternative assumptions. The first isthat the lump-sum tax strikes only wages. Capital is again correctly al-located, but optimal provision of the public good is no longer guaranteed.Public goods are optimally provided if the constraint on lump-sum taxationis not binding, and they are underprovided if it is binding. The second isWilsons (1999) assumption that the profits tax is the only tax.1 Wilsonshows that, under this assumption, the standard tax competition model pre-dicts both underprovision of the public good and misallocation of capital.The bargaining model is more agnostic. There is underprovision and capitalmisallocation if the typical firms output would not fall greatly if it movedfrom its best location to its second best location; and there is optimal pro-vision and optimal capital allocation if this move causes the typical firmsoutput to fall dramatically.

An important feature of these results is that they are derived from a gen-eral equilibrium model, and hence can be directly compared to those of thestandard fixed-rate model of tax competition. A number of earlier papershave offered explanations of the tax breaks given to mobile firms, but thesepapers have described the negotiations between a single firm and one or twogovernments. Doyle and van Wijnbergen (1994) examine the intertemporal

1Our assumption is actually slightly different from Wilsons, in that we assume thata government that raises too much revenue through the profits tax can return the excessrevenue to the citizens through a negative lump-sum tax. Wilson requires an exact matchbetween revenue and public goods expenditure.

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structure of a firms tax payments. They note that a mobile firm has greaterbargaining power than a firm that has already incurred the sunk costs asso-ciated with locating in a particular region. They argue that mobile firms willuse their extra bargaining power to extract concessions. Bond and Samuelson(1986) present an alternative explanation of the same phenomenon: a regioncan offer a tax holiday to a mobile firm to signal that firms that locate thereexperience high productivity. The firm will willingly pay higher tax rates inlater periods because it is very productive, and these high tax rates allow thegovernment to recover the cost of the initial tax holiday. A low-productivityregion could not offer the same incentive: firms that located there wouldrelocate when they found that they had low productivity, so that the regionwould be unable to recover the cost of the initial tax holiday. King, McAfeeand Welling (1993) allow the firm to negotiate simultaneously with two gov-ernments, and add a stochastic element to the regional productivities. Blackand Hoyt (1989), by contrast, present a static model in which the subsidies tomobile firms undo the distortionary effects of average cost pricing of publiclyprovided services.

Section 2 of this paper sets out an economy in which firms earn locationalrents. Section 3 describes the Pareto optimal allocations. Section 4 describesthe bargaining model, and derives the major result of the paper: there is nomisallocation of capital when governments bargain with firms. Section 5provides a broader comparison of the bargaining model and the standardmodel. Section 6 examines the role of the lump-sum tax, and section 7contains brief conclusions. All proofs are contained in the appendix.

2 Preliminaries

The economy consists of I regions and a continuum of firms. The regionsare identified by the elements of the set I{1, ..., I}. A firm is characterizedby its ownership structure and by its productivity in the various regions.A firms ownership structure is represented by the vector (1, , I),where i is the fraction of the firm owned by residents of region i. The setof all possible ownership structures is

={ [0, 1]I :Ii=1 i = 1}

A firms productivity in region i is governed by the parameter i R+,and the firms productivity in each of the regions is described by the vector

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(1, , I). The set of all possible productivity vectors is RI+.The distribution of firms is represented by an atomless and -finite measurespace (,B, F ). Here, is the sample space of firms, B is a -algebraover the sample space, and F (X) denotes the measure of firms in any set Xcontained in B. It is assumed that is a compact set and that

F ( ) = F (Int( )) > 0where Int( ) denotes the interior set of .2

Each firm locates and produces in one of the regions, or in none of them.The firms output when it locates in region i is denoted yi. It is determinedby i and by ni, the quantity of labour employed by the firm.

yi =

(1

1 )i n

1i 0 < < 1

The total quantity of labour available in region i is fixed and equal to Ni.Let Li be the set of firms that locate in region i, and let thedistribution of firms across the economy be L {L1, , LI}. Let themapping n : R+ describe the quantity of labour employed by firmsof each type.

The residents of each region consume two goods, a private good anda public good. One unit of output can be transformed into one unit ofeither good. Let ci R+ be the aggregate quantity of the private goodin region i, and let gi R+ be the aggregate quantity of the public good.The social preferences of region i are represented by a social welfare functionsi, which is assumed to be strictly concave, strictly increasing, and twicecontinuously differentiable in (ci, gi). Let the vectors c (c1, , cI) andg (g1, , gI) describe the aggregate quantities of the private and publicgoods in the economy as a whole.

An allocation is a list (L, c, g, n). An allocation is feasible if

F1. The sets in L are disjoint, and Ii=1Li F2. The mapping n satisfies the condition

Li

ndF = Ni

2These restrictions ensure that in equilibrium, the firms that are indifferent betweentwo (or more) locations constitute a set of measure zero. Imposing these restrictionsallows us to make stronger statements about the differences among equilibria than wouldotherwise be possible.

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for each i.

F3. The vectors c and g satisfy the condition

Ii=1

ci +I

i=1

gi =I

i=1

1

1 (

Li

i n1dF

)Note that this definition of feasibility allows goods produced in one region tobe used to increase the aggregate consumption of another region.

Allocations can differ in ways that do not lead to differences in aggregatesor in social welfare. The following concepts will be used to identify allocationsthat differ only in inconsequential ways.

Definition 1 Any two sets B and B in B are measurably identical ifF ((B B) (B B)) = 0

Definition 2 Any two mappings : R+ and : R+ aremeasurably identical if

F{(, ) : (, ) 6= (, )} = 0

3 Pareto Optimal Allocations

A feasible allocation (L, c, g, n) is Pareto optimal if there does not exist analternative feasible allocation (L, c, g, n) such that si(ci, g

i) si(ci, gi) for

all i and si(ci, g

i) > si(ci, gi) for some i.

A two-step procedure will be used to characterize the Pareto optimalallocations. Condition F3 and the monotonicity of the social welfare functionsimply that any Pareto optimal allocation maximizes the total output of theeconomy. Total output is entirely determined by L and n, so the first stepis to find the conditions under which (L, n) maximizes total output. Thesecond step is to find the restrictions that Pareto optimality places on theallocation of output, that is, on (c, g).

With L given, each regions output is maximized by allocating the avail-able labour so that the marginal product of labour is equalized across firms.That is, under any Pareto optimal allocation, the mapping n : R+satisfies, for each i

i n(, ) =

i n(, ) (1)

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for all (, ) and all (, ) contained in Li.Now consider the collection L. Let k (k1, , kI) be a vector of non-

negative real numbers. For each k, let L(k) {L1(k), , LI(k)} be theunique collection of disjoint sets that satisfies these conditions:

C1. For each i I, (, ) Li(k) implies that kii = max[k11, , kII ].An arbitrary tie-breaking rule determines the placement of firms forwhich the product kii reaches its maximum in more than one region.

C2. Ii=1Li(k) = .

Define the functions

Zi(k) Li(k)

idF for all i I

The function Zi(k) aggregates the productivity factors of the firms that lo-cate in region i under Li(k). It is readily shown that, if labour is allocatedaccording to (1), the output of a firm locating in that region is

yi(i, k) =

(1

1 )i (Ni/Zi(k))

1

and that the regions aggregate output is

Yi(k) =

(1

1 )Zi(k)

N1i

Total output is simply the sum of the regions aggregate outputs.The conditions under which total output is maximized are described by

the following lemma.

Lemma 1 Let k be the solution to the equation system

ki =

(NiZi(k)

)1for all i I (2)

Then:

1. The vector k exists and is unique. Each ki and each Zi(k) is strictly

positive.

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2. If labour is allocated in accordance with (1), total output is maximizedby any L that is measurably identical to L(k).

The intuition behind this lemma is quite simple. If the firms are allocatedacross regions in accordance with L(k), and if the marginal product of labouris equalized across the firms in each region, the output of a single firm thatchose to locate in region i would be

yi =

(1

1 )ik

i

The firms output is maximized by locating in the region in which iki isgreatest,3 which is where the firm does locate under L(k). That is, if everyfirms location is determined by L(k), no firm could raise its output bychanging its location. Any L that is measurably identical to L(k) wouldalso maximize total output. Some firms locate in every region whenevertotal output is maximized.

The allocation of labour can now be more precisely determined. Letn(k) : R+ be the mapping that equalizes the marginal product oflabour across all of the firms in each region, when the firms are distributedin accordance with L(k). That is,

n(k)(, ) = iNi/Zi(k) for all (, ) Li(k) and all i IThe maximization of total output requires n to be measurably identical ton(k).

Let Y be maximal total output, and let R (R1, ..., RI) represent theway in which total output is distributed across regions. The definition ofa feasible allocation assumed that goods produced in one region could beallocated to any region, so the only restriction on R is that

Ii=1

Ri = Y (3)

A unit of the produced good can be converted into one unit of either good,so

ci + gi = Ri for all i I (4)3The firms output in region i depends upon its productivity in that region and the

quantity of labour that would be allocated to it in that region. The latter factor issummarized by ki, which rises with the supply of labour and falls with the number andproductivity of the other firms in the region.

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The optimal policy in region i is to allocate Ri so that the societys marginalrate of substitution between these goods is just equal to the marginal rate oftransformation between them, where the latter rate is fixed at one.

These restrictions on the allocation are both necessary and sufficient forPareto optimality:

Theorem 1 A feasible allocation (L, n, c, g) is Pareto optimal if and only if

P1. L is measurably identical to L(k).

P2. n is measurably identical to n(k).

P3.si(ci, gi)

ci=si(ci, gi)

gifor all i I.

The location of the firms and the allocation of labour across firms aremeasurably identical in every Pareto optimal allocation, but the division ofresources between the regions varies substantially across the Pareto optimalallocations. Indeed, every division of resources that satisfies (3) is part ofsome Pareto optimal allocation.

4 Bargaining over Tax Rates

Governments set the rates at which they will tax profits, but they are pre-pared to offer financial concessions to attract new investment. The combi-nation of fixed rate taxation with firm-specific concessions implies that theeffective rate of taxation varies from firm to firm. The model in this sectionassumes that both firms and governments are only concerned with the ef-fective rate of taxation, and seeks to explain the way in which this tax ratevaries with the characteristics of the firm. It abstracts from the interplay oftax rates and concessions by imagining that each government is able to seta tax rate for each type of firm.

The equilibrium unfolds in two stages:

1. Each government, taking the wage rates as given, chooses the rates(one for every type of firm) at which it will tax profits. The tax rateschosen by the government of region i are represented by the mappingti : R, and the taxes imposed by all governments are givenby t (t1, ..., tI). Each firm, knowing these tax rates and taking the

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wage rates as given, chooses the region in which it will locate and thequantity of labour that it will employ. The wage rates anticipated bythe governments and the firms are the wage rates that ultimately clearthe labour markets. The wage in region i is wi, and the wage rates ofall regions are given by the vector w (w1, ..., wI).

2. Each government chooses the rate at which it will tax the incomes thataccrue to domestic residents. These incomes consists of wages and theresidents share of the after-tax profits of all firms. Since people areassumed to be immobile, this tax is a lump-sum tax. The proceeds ofthe tax, along with the proceeds of the profits tax, are used to provideunits of the public good.

The assumption that governments, like firms, treat the wage rates as exoge-nous is appropriate because each government makes concessions to individualfirms in an attempt to influence their location decisions. Both the firms andthe government believe that the location decision of a single firm will notalter the prevailing wage rate.

The decision-making in stage 2 is very simple. Each regions total re-sources have already been determined. These resources consist of the wagesearned by the regions workers, the governments revenue from the profitstax, and the residents shares of the after-tax profits of all firms. Undercompetitive labour markets, the share of output allocated to wages is always1 , so the total resources of region i are

Ri =

Li

yi[1 (1 i)(1 ti)]dF +j 6=i

(Lj

i(1 tj)yjdF)

(5)

The revenue from the profits tax has already accrued to the government,and the government can acquire additional revenue through the lump-sumtax.4 Since all of the revenue is used to provide units of the public good,choosing the lump-sum tax is equivalent to choosing the quantity of publicgood provided. The governments choice problem is

maxci,gi

si(ci, gi) s.t. ci + gi = Ri (6)

4The lump-sum tax can be negative, so that the government can return some of therevenue from the profits tax to the citizens if that tax raised an excessive amount ofrevenue.

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The maximum value function associated with this problem, Si(Ri), is strictlyincreasing, so the governments objective when it chooses its tax schedule instage 1 should be the maximization of its resources.

Write (5) as

Ri =

ridF

where ri is a particular firms contribution to region is resources:

ri =

{[1 (1 i)(1 ti)]yi if the firm locates in ii(1 tj)yj if the firm locates in j (j 6= i) (7)

The governments ultimate objective is to maximize its total resources, butit achieves this end if it maximizes the resources contributed by each firm.

The equilibrium in stage 1 consists of taxes t, wage rates w, locations Land a labour allocation n such that:

1. No firm that chooses to locate in region i would have greater after-taxprofits if it located elsewhere.

2. For each type (, ) , no government i (i I) that can increasethe resources ri contributed by firms of that type by deviating from theequilibrium tax rate ti(, ) while every other government j adheres toits equilibrium tax rate tj(, ).

3. Given w and L, the labour allocation n assigns the profit-maximizingquantity of labour to firms of each type.

4. The wage vector w clear each regions labour market.

The equilibrium is derived in two steps. The first step characterizes t, L andn for an arbitrary w. The second step shows that w exists and is unique.

The firms, knowing the wage rates, can anticipate the quantity of labourthat they would hire if they located in a given region. Specifically, the quan-tity of labour demanded by a firm that locates in i is

ni(i, wi) = argmaxn

[(1

1 )i n

1 win]= i

(1

wi

)1/

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The firm is also able to anticipate its output and after-tax profits in thatregion. These are, respectively,

yi(i, wi) =

(1

1 )i

(1

wi

)(1)/(8)

pii(i, wi, ti) = (1 ti)yi(i, wi)The location chosen by any given firm is completely determined by its antic-ipated output in the various regions:

Lemma 2 Assume that governments can tax the incomes of domestic resi-dents in a lump-sum fashion. Assume that a firm of some given type (, )believes that it can hire as much labour as it wants at the wages contained inw. Assume that this firm would produce its highest output if it located if regioni and that it would produce its next highest output if it located in region h.Let t(, ) be the vector of equilibrium tax rates. Then, in any equilibrium,

1. The firm locates in a region in which it attains maximal output.

2. The firms after-tax profits satisfy the conditions

yh pii(i, wi, ti ) yipij(j, wj, t

j) pii(i, wi, ti ) for all j 6= i

pij(j, wj, tj) = pii(i, wi, t

i ) for some j 6= i

3. The tax rate offered by region i satisfies the condition

1 1 ti (, ) 1

yhyi

The firm attains its highest after-tax profits in region i and chooses tolocate there, but some other region offers the firm the same after-tax profits.(Otherwise, region i would increase its tax rate.) The matching offer mightor might not be made by region h; but in either case, it must be sufficientlyhigh that region h has no incentive to improve its offer enough to inducethe firm to locate there. Region h has no such incentive if pii is at least asgreat as yh, and this consideration provides the lower bound on pii. Similarly,

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region i has no incentive to decrease its offer, driving the firm away, if pii isno greater than yi.

There are multiple equilibria associated with any wage vector w, and thetax rate actually paid by a firm of a given type varies across equilibria. Thelowest possible tax rate is always so low that pii is equal to yi., implying thatthe firm is being subsidized by an amount equal to labours share of output.The highest possible tax rate depends upon the firms next best output. Ifyh is nearly as large as yi, the firms lowest possible after-tax profits arenearly equal to yi, again implying a very high level of subsidization. Onthe other hand, if yh is nearly zeroif the firm has no credible alternativelocationsthe firms lowest possible after-tax profits are nearly zero.

The market-clearing wage vector exists and is unique. Once this vectoris known, the remaining properties of equilibrium follow directly from theprevious lemma.

Theorem 2 If the governments can tax the incomes of domestic residentsin a lump-sum fashion, an equilibrium exists. The market-clearing wages arethe same in every equilibrium, and L and n are measurably identical to L(k)and n(k). Every equilibrium allocation is Pareto optimal.

The bargaining model shows that tax competition does not necessarilylead to a misallocation of resources. The assumption about the scope of thelump-sum tax is important but not decisive. Section 6 relaxes this assump-tion.

5 Discussion

Models of tax competition typically assume that each government taxes everyfirms profits at the same rate. Each government sets its tax rate to maximizethe welfare of its own region, taking the other governments tax rates as given.An equilibrium is characterized by a tax profile such that no government canraise the welfare of its own region by unilaterally deviating from the taxprofile.

This section briefly sets out a two-region fixed-rate model that incorpo-rates the assumptions set out in section 2. It is almost identical to the modeldescribed by Burbidge, Cuff and Leach (2005, henceforth BCL), and a num-ber of their results are presented here without proof. The implications of thefixed-rate and bargaining models are then compared.

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Let the scalar ti be the tax rate paid by every firm that locates in region i,and let t1 and t2 be given. Let the allocation of firms across regions be givenby L(k) for some vector k (k1, k2). Then every firm locates in the regionin which its after-tax profits are greatest when k satisfies the conditions

ki = (1 ti)(

NiZi(k)

)1i = 1, 2 (9)

Let k(t1, t2) be the solution to this equation system. For any given pair of

tax rates, Li(k) is the set of firms that locate in region i and Yi(k) is theaggregate output of that region.

In a two-region economy the allocation of firms is determined by k2/k1, with an increase in shifting firms from region 1 to region 2. Fur-thermore, is related to the tax rates in a very simple fashion. Combiningthe two equations in (9) and using the linear homogeneity of Zi(k) gives

(Z2(1, )

Z1(1, )

)1=

(N2N1

)1(10)

where

1 t21 t1

Then:

1. Comparing (9) with (2) shows that k(0, 0) is equal to k. By (10), takes the same value whenever the two tax rates are equal, implying

=k2(t, t)

k1(t, t)=k2(0, 0)

k1(0, 0)=k2k1

Thus, total output is maximized whenever t1 and t2 are equal.

2. Since Z2 is non-decreasing in k2 and Z1 is non-increasing in k2, thereis a unique associated with each . An increase in causes to rise.

That is, an increase in one regions tax rate reduces that regions output andraises the other regions output. Total output rises (falls) if the increase inthe tax rate pushes towards (away from) one.

Assume, as in the bargaining model, that each government uses a lump-sum tax to optimally divide the regions resources between the public and

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private goods. Government is objective when it chooses ti is then the max-imization of its resources Ri, as defined by (5). An equilibrium is a pair(t1, t

2) that satisfies the conditions

Riti

= 0 i = 1, 2

5.1 Pre-Commitment

The standard model of tax competition imagines that, when each govern-ment chooses its tax rate, it anticipates both the division of the firms be-tween regions and the impact of this division upon wage rates. That is, itcommits itself to a particular policy with full knowledge of the ultimate con-sequences of that decision. A bargaining model with the same propertiescould be constructed. Government i would choose a non-linear tax scheduleti : R. The equilibrium would consist of a list of tax schedules suchthat no government, correctly anticipating the movement of firms and the ad-justment of the market-clearing wage rates, could increase its own resourcesby unilaterally changing its own tax schedule.

It is not evident, however, that this kind of model would accurately reflectthe behaviour of governments. Suppose that the Minister of Industry forsome government meets the President of General Motors, who tells him,Wed like to build our new plant here, but theres always Tennessee... Amodel that sets out non-linear tax schedules imagines that the governmentsresponse to this kind of initiative is pre-determined. The Minister, hand-cuffed by policy, can do no more than explain his governments tax policy tothe company president. Our belief is that the interaction between the twowould not be so mechanical, that the Minister would make every possibleeffort to influence the location of the new plant. The Minister would negotiatewith the President until they concluded a deal or decided that no deal waspossible.

The Minister would also find that General Motors is not the only companythat comes calling. Our model imagines that he treats each new opportunityin the same way, with negotiations continuing until a deal is made or a dealbecomes impossible. The offer made to each firm is based on that firmseconomic potential, rather than being part of an overarching strategy. Thatpotential is evaluated using a given wage rate, because any one firms locationhas no impact on the regional wage rates.

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5.2 The Terms of Trade Effect

A fundamental question in tax competition is whether the profits tax willbe positive or negative. In any fixed-rate model the answer to this questionhinges upon Hamadas (1966) terms of trade effect. Consider an economywith two regions, and let J be the net transfer of after-tax profits from region2 to region 1:

J (

L2

1(1 t2)y2dF L1

2(1 t1)y1dF)

Hamada showed that, if capital is homogeneous, the sign of each regions taxrate is entirely determined by the sign of J :

sign(J) = sign(t1) = sign(t2)If J is positive, so that region 1 is a net recipient of after-tax profits, region 1will choose to subsidize capital. Its subsidization of capital will drive up theeconomy-wide after-tax return to capital, and since region 1 is a net recipientof profits, the increased return to capital will be beneficial to it. Region 2will tax capital to drive down capitals economy-wide after-tax return, as alower return to capital is beneficial to a region that pays out more after-taxprofits than it receives.

The BCL model differs from the standard tax competition model in thatit assumes that each unit of capital is embodied in a firm, and that the firmscan earn locational rents. The existence of these locational rents pushes upthe rates at which profits are taxed. Each government can capture rents thatwould otherwise accrue to foreigners by levying a positive profits tax, andthe governments desire to capture these rents might ultimately determinethe signs of the tax rates. Indeed, BCL show that the equilibrium tax rateis positive in any symmetric economy in which there is a positive measureof firms for which 1 and 2 are not equal. The terms of trade effect is stillpresent in this model, but in a weaker form:

sign(J) = sign(t2 t1)That is, the terms of trade effect determines the relative sizes of the tax ratesbut not their signs.

By contrast, nothing resembling the terms of trade effect arises in thebargaining model. The governments offer to any given firm is not dependent

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upon the success or failure of the offers made to other firms, and hence cannotdepend upon the equilibrium net transfer of profits.

The importance of the terms of trade effect in fixed-rate models impliesthat the distribution of ownership plays a critical role in determining theequilibrium tax rates. In the symmetric equilibrium in the BCL model, forexample, increasing each regions ownership of the firms that are relativelyproductive in that region always leads to a decline in the equilibrium taxrate. (The part of after-tax profits that accrues to foreigners is smaller, sothere is less incentive to tax profits.) Since the terms of trade effect hasno role in the bargaining model, it is not surprising that the distribution ofownership has no impact on the equilibrium tax rates in that model.

If the terms of trade effect does not determine the signs of the tax ratesin the bargaining model, what does? Lemma 3 shows that each firms equi-librium after-tax profits could be as high as its maximal output or as lowas its second highest output. If the firms after-tax profits are equal to itsmaximal output, the firm is receiving a subsidy equal to the wage incomethat it generates. However, if its after-tax profits are equal to its secondhighest output, the firm is subsidized if

yh > yi

where yi and yh are its highest and second highest outputs, and it is taxed if

yh < yi

The equilibrium tax rate is certainly negative if the first inequality holds,but the equilibrium tax rate can be either positive or negative if the secondinequality holds.

5.3 The Role of Locational Rents

Locational rents play a decisive role in both the bargaining model and theBCL model. BCL show that, in a symmetric equilibrium, the tax rate ispositive whenever locational rents exist. Eliminating the locational rents, bysqueezing the distribution F so that 1 and 2 converge for every firm, drivesthe tax rate to zero.

Eliminating the locational rents in the bargaining model can have evenmore extreme implications for the tax rate, but to derive these implications,the manner in which the distribution is altered must be exactly specified.

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Suppose that there are n regions, and for any given firm, let i and h bethe regions in which the firms output would be highest and second highest.Imagine that each firms is altered by replacing h with

h h + (1 )iki /khwhere is a constant between 0 and 1. That is, there is no change toany firms productivity in its best location, but there is an improvement inevery firms productivity in its second best location. (Recall that yi > yhimplies ik

i > hk

h.) Altering the distribution in this fashion will be called

a compression of the locational rents.Compressing the locational rents does not induce any firm to switch re-

gions, so there is no change in any regions aggregate productivity Zj or itswage rate wj. However, every firms second highest output rises toward itshighest output, partly because h rises and partly because nh(h, wh) rises.

Successive compressions of the locational rents drive yh toward yi forevery firm. Since yh and yi are the firms minimum and maximum after-taxprofits, successive compressions raise the firms minimum after-tax profitstoward its maximum after-tax profits. In the limit, when each firms highestand second highest outputs are arbitrarily close, the governments have nobargaining power and must offer each firm a subsidy equal to labours shareof the firms output.

In the BCL model, squeezing the locational rents from the economy re-duces the tax rate to zero. In the bargaining model, squeezing the locationalrents causes every firm to be subsidized. In the limit, every firms after-taxprofits are equal to its output.

6 The Lump-Sum Tax

It has been assumed that the government is able to levy lump-sum taxeson wages and on the domestic residents share of the after-tax profits of allfirms. The finding that the bargaining equilibrium gives rise to a Paretooptimal allocation relies on this assumption. A bargaining equilibrium inwhich every firm receives a subsidy equal to its wage bill is possible, and inthat equilibrium, the government must appropriate all of the residents wageincome to pay the required subsidies. Once this transfer has been made,the residents income consists only of their share of after-tax profits. Thegovernment is only able to provide a positive quantity of the public good if it

18

is able to tax these profits. In light of this observation, what are the welfareimplications of reducing the scope of the lump-sum tax?

One might also speculate that the scope of the lump-sum tax influencesthe nature of the bargains between governments and firms. If a governmentwere unable to impose a lump-sum tax on the domestic residents share ofafter-tax profits, no public goods would be provided in a region in whichevery firm received a subsidy equal to its wage bill. Would an equilibriumwith such large subsidies continue to exist?

These questions are examined here by eliminating the lump-sum tax in astepwise fashion. It is assumed first that only wages can be taxed in a lump-sum fashion, and then that lump-sum taxation is impossible. These changeshave quite different (and perhaps surprising) effects on the equilibrium.

If the scope of the lump-sum tax is limited, some of the regions resourcescannot be appropriated by the government and therefore cannot be allocatedto the provision of the public good. Let Rgi be the part of region is resourcesthat can be allocated to the public good, and let Rci be the part that cannotbe allocated to it. The bargaining equilibrium examined below involves twostages. The first stage determines each firms location, use of labour, andtax rate. The vectors Rc (Rc1, ..., RcI) and Rg (Rg1, ..., RgI) are implied bythese values. The second stage determines c and g.

The governments optimization problem in the second stage is

maxci,gi

Si = si(ci, gi)

s.t. ci + gi = Rci +R

gi

gi RgiLet the solution to this problem be the functions ci(R

ci , R

gi ) and gi(R

ci , R

gi ),

and let the associated maximum value function be Si(Rci , R

gi ). By the enve-

lope theorem,SiRci

=si(ci(R

ci , R

gi ), gi(R

ci , R

gi ))

ci(11)

SiRgi

=si(ci(R

ci , R

gi ), gi(R

ci , R

gi ))

gi(12)

These partial derivatives are equal if the inequality constraint (in the opti-mization problem above) is not binding, and are unequal only if the inequal-ity constraint is binding. Let the marginal rate of substitution MRSi be the

19

value of a unit of public goods, measured in units of private good, evaluatedat the optimum:

MRSi si(ci(Rci , R

gi ), gi(R

ci , R

gi ))

gi

/si(ci(R

ci , R

gi ), gi(R

ci , R

gi ))

ci

Then, (11) and (12) imply that MRSi is also the value of another unit of Rgi

measured in units of Rci :

MRSi =SiRgi

/SiRci

Since both partial derivatives are continuous functions of Rci and Rgi , MRSi

is a continuous function of the same variables:

MRSi = i(Rci , R

gi )

By construction, each element of the vector MRS (MRS1, ...,MRSI) isbounded below by 1. It is assumed henceforth that each element of MRS isbounded above by a positive finite number b.

Now consider the first stage of the equilibrium. The government recog-nizes that any one firms location decision will have no impact upon theregions wage rate, and likewise, it recognizes that that decision will haveno impact upon the relative values of public and private goods in its region.Thus, the first-stage equilibrium consists of the locations L, the labour al-location n, the taxes t, the wages w, and the marginal rates of substitutionMRS such that:

1. Given L and w, no firm could raise its (pre-tax) profits by deviatingfrom n.

2. Given w and t, no firm could raise its after-tax profits by deviatingfrom L.

3. Let i be the increase in region is resources, measured in units of Rci ,

generated by a firm of a particular type. For each (, ), no governmentj can increase its resources j by deviating from t(, ).

4. The labour allocation n clears the labour market in every region.

20

5. If Rc and Rg are the resource vectors implied by L, n and t, thenMRSsatisfies the condition

MRSi = i(Rci , R

gi )

for all i.

The remainder of this section examines the stage 1 equilibrium under alter-native assumptions about lump-sum taxation.

6.1 Only Wages are Subject to Lump-Sum Taxation

Under this assumption, the government can finance the public good fromeither the wage tax or the profits tax, implying

Rgi = (1 )Yi + Li

tiyidF =

Li

(yi pii)dF

Rci =I

j=1

(Lj

ipijdF

)The contribution of a firm of type (, ) to region is resources, measured inunits of Rci , is

i =

{(yi pii)MRSi + ipii if the firm locates in iipij if the firm locates in j (j 6= i)

This equation replaces (7) in the determination of the equilibrium offers, butotherwise, the characterization of equilibrium proceeds in much the samemanner.

Lemma 3 Assume that the wages of domestic residents can be taxed in alump-sum fashion. Assume that a firm of any given type (, ) believes thatit can hire as much labour as it wants at the wages contained in w. Assumethat the firm would produce its highest output if it located in region i and thatit would produce its next highest output if it located in region h. Let t(, )be the vector of equilibrium tax rates. Then results 13 of Lemma 2 hold inany equilibrium.

21

The range of possible after-tax profits for each firm is bounded by thefirms highest and second highest outputs, exactly as it was when all ofincome was subject to the lump-sum tax. The government of region i iswilling to offer the same subsides when MRSi is highwhen subsidizingfirms has a high social opportunity costas when it is low. The problemthat faces government i when MRSi is high is that R

gi is low relative to R

ci .

Since the resources Rgi are derived entirely from firms that locate within theregion, and since any reduction in the after-tax profits offered to the firmwill (in equilibrium) induce the firm to locate elsewhere, the government isunwilling to moderate its offer to the firm.

Theorem 3 Assume that the wages of domestic residents can be taxed in alump-sum fashion. Then an equilibrium exists. The market-clearing wagesare the same in every equilibrium, and L and n are measurably identicalacross equilibria. Also,

1. (L, n) maximizes total output.

2. There exists an equilibrium in which no units of the public good areprovided by any government.

3. An equilibrium that gives rise to a Pareto optimal allocation exists un-der some specifications of the model.

Since the range of offers that can be made to each firm does not changewhen the scope of the lump-sum tax is restricted, neither does the actuallocation of the firm. The collection L is again measurably identical to L(k).Since L is measurably identical across equilibria, w is the same in everyequilibrium and n is measurably identical to n(k) in every equilibrium. Itfollows that total output is maximized even under the more restrictive taxassumption.

Every equilibrium satisfies the Pareto optimality conditions P1 and P2,but there is no guarantee that P3 will be satisfied. Indeed, an equilibriumalways exists in which any or all of the governments provide no public goodsat all. If a government makes the highest possible offer to every firm, all ofthe wages must be appropriated to pay the subsidiesthat is,

Li(k)

tiyidF = (1 )Yi

22

Then Rgi is equal to zero and no public goods can be provided. Nevertheless,there are some specifications of the model under which P3 is also satisfied, sothat the equilibrium allocation is Pareto optimal. Suppose that each govern-ment makes the smallest possible offer. If each firms second highest outputis small relative to its highest output, the government will be able to collecttaxes from every firm, so that Rgi exceeds (1 )Yi. There are specificationsof the social welfare function under which a government equipped with theseresources will be able to provide the optimal quantity of public goods.

These results are the reverse of a common representation of Hamadas(1966) tax competition model. In that model, the existence of a lump-sumwage tax is commonly assumed to ensure the optimal provision of publicgoods, while the terms of trade effect causes capital to be misallocatedacross regions. Our findings are that firms (which embody the availablecapital) are correctly allocated across regions, but that the optimal provisionof public goods is not assured.5

6.2 Eliminating the Lump-Sum Tax

Wilson (1999) studies a fixed-rate tax competition model in which the profitstax is the only available tax. He finds that, if the regions are not identical,the equilibrium tax rates distort both the division of capital between regionsand the division of a regions resources between the private and public goods.If the regions are identical, each region will underprovide the public good.

If the same assumption is imposed in the bargaining model,

Rgi =

Li

tiyidF =

Li

(yi pii)dF

Rci = (1 )Yi +I

j=1

(Lj

ipijdF

)5Arguably, this characterization of the Hamada model does not take seriously the limit

on lump-sum taxation. In that model, one region subsidizes firms if the other region taxesfirms. If the subsidies are sufficiently large or the desire for the public good is sufficientlystrong, the government will not have enough revenue to both subsidize the firms and pro-vide the optimal quantity of public goods. Hence, a more accurate characterization of theHamada model would be that capital is almost always misallocated (it is not misallocatedin knife-edge circumstances, notably when the regions are identical), and that the optimalprovision of public goods is not assured.

23

The value of the resources gained by attracting a single firm to the region is

i =

{(yi pii)MRSi + (1 )yi + ipii if the firm locates in iipij if the firm locates in j (j 6= i)

The locations of the firms varies with the vector MRS under this speci-fication of i.

Lemma 4 Assume that the profits tax is the only available tax. Consider afirm of type (, ), and assume that the firm believes that it can hire as muchlabour as it wants at the wages w. Let t(, ) be the vector of equilibriumtax rates. Define the variables

xj yj(+

1 MRSj

)for all j I

Then, in equilibrium, the firm locates in region i only if

xi = max [x1, ..., xI ]

Furthermore, the firms after-tax profits satisfy the condition

xh pi(i, wi, ti ) xi (13)where

xh = max [x1, ..., xi1, xi+1, .., xI ]

The variable xj is the maximal value of the firms output, measured inpublic goods, conditional on the firm locating in region j. There would beyj units of profits, and each unit of profitswhether or not it is actuallytaken as taxes by the governmentis valued at one unit of public goods.There would also be (1 )yj units of wages. Since wages are not taxable,each unit of wages is valued at one unit of private goods, which is worth only1/MRSj units of public goods.

Total output is maximized when no firm can increase its own outputby moving to another region. When the profits tax is the only tax, thiscondition is satisfied if and only if every region has the same marginal rateof substitution. The only robust equilibrium satisfying this condition wouldbe one in which each region raises enough revenue to provide the optimalquantity of public goods. Such an equilibrium might exist, but does notnecessarily exist.

24

The taxes paid by each firm are (at last) influenced by the governmentsneed for revenue. The maximum value of a firms after-tax profits pii is xi,which is bounded below by yi. Since the firm is neither taxed nor subsidizedwhen pii is equal to yi, a firm that is earning the maximum after-tax profitsis necessarily being subsidized. The subsidy shrinks as the regions marginalrate of substitution rises, but remains positive. A firm that is earning theminimum after-tax profits could be either paying taxes or receiving a subsidy.Specifically, the firm is paying taxes if

yh

(1 +

(1

)1

MRSh

)< yi

The firms tax rate rises as its second best output falls, and it also rises asthe region in which it attains its second best output becomes more desperatefor revenue. The firm is receiving a subsidy if the inequality is reversed. Thefactors that lead to higher tax rates also lead to lower subsidy rates.

Proving existence is more difficult in this model than in the earlier ones,so only a restricted set of equilibriacredible equilibriawill be considered.

Definition 3 Let pi(, ) be the equilibrium after-tax profits of a firm oftype (, ). Let J (, ) I be the set containing the identities of the regionsthat offer after-tax profits of pi(, ) to firms of type (, ). The equilibriumis credible if, for every (, ) and every j J (, ), the resources that regionj would obtain from a firm of type (, ) are at least as great when the firmaccepts region js offer as when it rejects region js offer.

To understand the impact of the credibility assumption, suppose thatfirms of a particular type (, ) locate in region i. Region i offered after-taxprofits of pi(, ) to these firms; but at least one other region must havemade the same offer, since otherwise region i would be able to reduce itsoffer without losing the firm. Credibility requires that every region j thatdid make the same offer does not prefer its offer to be rejected. This conditionis satisfied if and only if

pi(, ) xjCombining this inequality with (13) leads to the following conclusions: thesecond best offer is made by region h, and pi(, ) is equal to xh. Credibilityimplies that each firms after-tax profits are uniquely determined.

25

Theorem 4 Let ai be a lower bound of i(Rci , 0), and define the lower bound

a min[a1, ..., aI ]

If the profits tax is the only tax, a credible equilibrium exists for all sufficientlylarge values of a. Also,

1. In any two credible equilibria with the same MRS, the only differencesare in L and n, and these differences are not measurable. However,MRS might not be the same in every credible equilibrium.

2. (L, n) is equal to (L(k), n(k)) if and only if every region has the samemarginal rate of substitution.

3. A credible equilibrium that gives rise to a Pareto optimal allocationexists under some specifications of the model.

A credible equilibrium exists for sufficiently large values of a, but whetherthere exist multiple equilibria that are distinctly different remains an openquestion. The central (and unresolved) issue is whether all credible equilibriahave the same vector MRS; if they do, the differences across equilibria arenot measurable.

The efficiency properties of a credible equilibrium are also uncertain. Al-most all equilibria fall into one of two categories: either the allocation isPareto optimal, or total output is not maximized and public goods are un-derprovided. The only exceptions are knife-edge equilibria in which everyregion underprovides the public good but has the same marginal rate of sub-stitution, so that total output is maximized. Failure to maximize outputfollows not from the inability of regions to provide the optimal quantity ofpublic goods, but from disparity in their provision of public goods.

The fixed-rate tax competition model does not generate a Pareto optimalallocation, but the bargaining model will sometimes to do so. This differencearises in part because the firms in the bargaining model are less mobile thanthe firms in the fixed-rate model. Capital in the fixed-rate model is trulymobile, in the sense that it can move to any region and is equally productivein every region. Each firm in the bargaining model is mobile in the sense thatit can locate in any region, but it is not necessarily mobile in the sense thatit can move between regions without a significant loss of productivity. Theextent of these productivity losses largely explains the difference in results.

26

If each firms second best option is almost as good as its best option, thegovernments will subsidize some firms and collect only small amounts ofrevenue from others. The net revenue collected by a government is likely tobe so small that public goods will be underprovided. However, if each firmssecond best option is much worse than its best option, each government willbe able to collect taxes from almost every firm. If each regions preferencesfor the public good are not very strong, the net revenues might be largeenough to allow each region to provide the optimal quantity of public goods.Thus, relatively high mobility (in the sense of movement without significantloss) gives rise to allocations that are not Pareto optimal, as in Wilson (1999)while relatively low mobility gives rise to Pareto optimal allocations.

7 Conclusions

The standard model of tax competition assumes that each government taxesevery firms profits at the same rate. Regional differences in productivity orin endowments lead to an equilibrium in which there is a range of tax rates.Resources are misallocated, with the low tax regions using too much capitaland the high tax regions using too little. By contrast, the model set outabove assumes that the governments negotiate separately with every firm.The predictions of the model depend upon the nature of the supplementarytaxes in the economy. If all of income is subject to a lump-sum tax, aPareto optimal allocation is reached; if only wages are subject to a lump-sumtax, there can be underprovision of the public good but capital is optimallyallocated.

There is the potential for very large subsidies to be paid to the firm undereither lump-sum tax. In the most extreme equilibrium, all of the wages paidto the workers in a region are appropriated by the government to subsidizethe firms, and all of the after-tax income is earned as profits. There are onlytwo assumptions under which such an extreme outcome makes sense. Oneis that the residents of the region are identical, with each resident supplyingthe same amount of labour and owning equal shares in each of the firms.The appropriated wages are then simply returned to the workers that earnedthem in a different form, namely after-tax profits. The other assumption isthat the residents are not identicalsome earn mostly wages and others earnmostly profitsbut that there is a hidden system of lump-sum transfers thatreturns the economy to some preferred distribution. While either assumption

27

makes sense of the outcome, both assumptions are very strong. We suspectthat a complete understanding of subsidies requires the abandonment of asocial welfare function in which only aggregates matter.

A Appendix

Let k be the solution to the equation system

ki = i

(1

Zi(k)

)1for all i I (14)

Here, each i is a positive constant. Then k has these properties.

FP1. The vector k exists and is unique, and ki and Zi(k) are strictly pos-

itive for each i I.FP2. An increase in j causes k

j and each ratio k

j/k

i (i I, i 6= j) to rise.

Proof of FP1: Let k be a fixed point of (14), and let s be the vectorsuch that

si =kiIl=1 k

l

for all i I

Then

si =iZi(k

)1Il=1 lZl(k

)1=

iZi(s)1I

l=1 lZl(s)1

(The second equality follows from the observation that each firms locationdepends only upon the relative sizes of the elements of k, and hence thatZi(k

) = Zi(k) for all positive .) Alternatively, s is the fixed point of theequation system

si =iZi(s)

1Il=1 lZl(s)

1 for all i I (15)

Every fixed point of (14) gives rise to a unique fixed point of (15). Now notethat each fixed point of (15) is associated with a unique fixed point of (14).By the linear homogeneity of Zi,

iZi(s)1 = iZi(k

)1 for all i I

28

Since the right-hand side is ki , the value of ki is entirely determined by s

.Thus, k exists and is unique if s exists and is unique. The existence anduniqueness of s are proved in turn:

1. The difficulty of proving the existence of the fixed point of (15) is thatthe right-hand side of (15) is not bounded or not defined if some ki is zero.This problem is circumvented by considering the mapping

qi (s) =i (Zi(s) + )

1Il=1 l (Zl(s) + )

1 for all i I

Define the set

S {s RI+ :

Il=1

sl = 1

}This set is non-empty, compact and convex. For each > 0, the mappingq : S S is well-defined even when some elements of s are zero. Theassumptions on F ensure that q is a continuous mapping from S into S.Taken together, these conditions ensure the existence of a fixed point s =(s1, , sI). Furthermore, the construction of the mapping ensures that siis strictly positive for all i and all > 0. Since s is a fixed-point of q, wehave

si =i (Zi(s

) + )1Il=1 l (Zl(s

) + )1for all i I

Since 0 < si < 1 for all i at each , the positive sequence {s} is bounded.Therefore, there exists a subsequence of {s} that must converge as 0.For simplicity, assume that we choose this convergent subsequence right fromthe start so that {s} itself converges to s as 0.

Now we will show that 0 < si < 1 for all i I. Suppose not. Thenthere exists a non-empty subset D I such that si = 0 for all i D. Thenwe have Zi(s

) = 0 for all i D because every firm will be located in someregion i such that si > 0. Furthermore, Zi(s

) Zi(s) = 0 as 0.lD s

l is expressed by

lDsl =

lD l (Zl(s

) + )1lD l (Zl(s

) + )1 +

l /D l (Zl(s) + )1

Note that s is a fixed point of q, so Zl(s) + is bounded and positive

for all l / D at any . This implies that {l /D (Nl/(Zl(s) + ))1} is a29

bounded and positive sequence. Furthermore,

l /D l (Zl(s) + )1

as 0 because Zl(s) + 0 for all l D as 0. Therefore, wehave

lD s

l 1 as 0. This contradicts

lD s

l = 0. It follows that

0 < si < 1 for all i. Since s is a fixed point of (15) and 0 < si < 1 for all i,

we have Zi(s) > 0 for all i. Therefore, a fixed point k (with ki > 0 for all

i) of (14) exists and Zi(k) = Zi(s) > 0 for all i

2. Assume that s is not a unique fixed point, and let s0 and s1 be twoof the fixed points. Let region a be the region in which the ratio s0i /s

1i is

lowest. This ratio must be smaller than 1. (If it were not, every element ofs0 would be greater than the corresponding element of s1. Since every fixedpoint has the property that

i si = 1, at least one of the two vectors could

not be a fixed point.) Then, for all and all j 6= a,

js1js1a j

s0js0a, (16)

Furthermore, the inequality must be strict for some j. Since a firm locatesin region a if and only if a > j(sj/sa) for all j 6= a, (16) implies thatZa(s

0) < Za(s1). Then, using (15),

s0as1a

=

(Za(s

1)

Za(s0)

)1> 1

which contradicts the initial assumption that s0a/s1a is smaller than 1. Thus,

the fixed point s must be unique. Proof of FP2: This property is proved by demonstrating that any other

outcome leads to a contradiction. If Zj(k) does not rise when j rises, k

j

must rise to satisfy the jth equation in the system. However, if Zj(k) does not

rise, there must be at least one element ki that rises by a greater proportionthan kj . Let k

h be the element that experiences the greatest proportionate

increase. Since kh and Zh(k) both rise, the hth equation in the system

is not satisfied. Thus, Zj(k) must rise in response to the increase in j.

If kj does not rise while Zj(k) rises, there must be at least one element

ki that falls by a greater proportion than kj . Let k

h be the element that

experiences the greatest proportionate decline. Since both kh and Zh(k)

fall, the hth equation in the system is not satisfied. Thus, the rise in Zj(k)

must be accompanied by an increase in kj . Let kh be the element of k

thatexperiences the greatest proportionate increase. If its proportionate increaseis at least as great as that of kj , Zh(k

) also rises, so that the hth equation

30

in the system cannot be satisfied. It follows that the proportionate increasein kj must be greater than the proportionate increase in any other elementof k.

Proof of Lemma 1: Since (2) is (14) with i set equal to (Ni)1, the

first part of Lemma 1 follows immediately from FP1. To prove the secondpart of Lemma 1. suppose that the location decision L is not measurablyidentical to L(k). Then, under the location rule L, there exists a compact setMi of firms in region i such that F (Mi) > 0 and kjj > kii for all firms inMi.It will be shown that moving a subset of these firms from region i to regionj raises total output. Let Mi be the set of all subsets of Mi, and identifysome (, ) in the interior of Mi. Define a mapping m : R+ Mi such that(i) m(0) = (, ) , (ii) m(x) m(x) for all x and x such that x < x, and(iii) C = F m is continuous and differentiable at all x R+. The firms inthe set m(x) will be moved from region i to region j; C(x) is their measure.For each m(x), let g(x) be the decline in region is aggregate productivitywhen the firms are moved out of region i, and let h(x) be the increase inregion js aggregate productivity gain when the firms are moved into thatregion. Since C is continuous and differentiable, g and h are continuous anddifferentiable. Assuming that labour is reallocated in accordance with (1),the movement of the firms causes total output to rise by

D(x; k) =1

1 [(Zj(k

) + h(x))N1j (Zi(k) g(x))N1i]

Taking the first-order derivative of D with respect to x and evaluating it atx = 0 gives

D(0; k) =1

1 [h(0)Z1j N

1j g(0)Z1i N1i

]=

1

1 [h(0)kj g(0)ki

]Since

jkj > ik

i

for every firm that is moved between regions, this derivative is positive. Thatis, moving a small but positive measure of firms between the regions raisestotal output. Then L does not maximize total output if L is not measurablyidentical to L(k), where where k is some fixed point of (2). Therefore, it isnecessary condition for the output-maximizing location rule L that L must

31

be measurably identical to L(k), where kis some fixed point of (2). Sincethere exists a unique fixed point k of (2), the necessary condition becomesthe sufficient one for the output-maximizing location rule.

Proof of Theorem 1: P1 follows from Lemma 1. P2 follows from thefacts that the marginal product of labour is equalized across firms withineach region and that L is measurably identical to L(k), so Zi(k) inducedby any L, which is measurably identical to L(k), is the same. Since siis strictly concave, increasing, and twice differentiable, P3 is the neces-sary and sufficient condition for (ci, gi) to maximize si subject to the con-straint (4). Let (ci (Ri), g

i (Ri)) be the solution to this maximization prob-

lem. Since si is strictly concave, strictly increasing, and twice differentiable,si(c

i (Ri), g

i (Ri)) is strictly increasing in Ri. Consequently, shifting resources

from one region to another raises one regions welfare at the expense of theother region. Then any allocation of output that satisfies (3) can be part ofa Pareto optimal allocation.

Proof of Lemma 2: Assume that, in equilibrium, a firm with charac-teristics (, ) chooses to locate in region i, and assume that this firms nextbest offer came from region m. The resources that region i extracts from thefirm rise as the firms after-tax profits fall, so region i will offer the smallestafter-tax profits that induce the firm to locate there.

pii(i, wi, ti ) = pim(m, wm, t

m) (17)

Also, since region i chooses to induce the firm to locate within its boundaries,the firm must contribute more to region is resources by it locating there thanit would by locating in region m:

yi(i, wi) (1 i)pii(i, wi, ti ) ipim(m, wm, tm) (18)

The resources that region m gains from the firm must be at least as greatwhen the firm locates in region i as when it locates in region m:

mpii(i, wi, ti ) ym(m, wm) (1 m)pim(m, wm, tm) (19)

(If this condition did not hold, regionm would be able to make an offer to thefirm that induces the firm to locate in region m and increases that regionsresources.) Conditions (17) and (18) imply

yi pim(m, wm, tm) (20)

32

while (17) and (19) imply

pii(i, wi, ti ) ym (21)

Combining (20) and (21) shows that yi is at least as large as ym. Regions iand m have no profitable deviation if (17), (20) and (21) are satisfied. Nowconsider some other region j. Assume that the region j could attract the firmby offering a tax rate tj such that

pij(j, wj, tj) = pii(i, wi, t

i ) (22)

Then, there exist no profitable deviations for region j (j 6= i) ifyj(j, wj) (1 j)pij(j, wj, tj) jpii(i, wi, ti ) (23)

Using (22), (20) and (23) implies that if the following condition holds, for all(j 6= i)

yi pii(i, wi, ti ) yj (24)then no region has incentive to deviate from its offer. Consequently, no otherregion j (j 6= i) has a profitable deviation if

pii(i, wi, ti ) yh (25)

where yh is the firms second highest output. There is no requirement thatregions h and m are the same region. Conditions (21) and (25) are the sameif they are, and (25) is the tighter constraint if they are not. Thus,

yh pii(i, wi, ti (, )) yi (26)implying

1 1 ti (, ) 1

yhyi

The firms after-tax profits are equal to yi at the lower tax rate and equal toyh at the higher tax rate. By (26), the firm has located in a region in whichit has the greatest possible output.

Proof of Theorem 2: The first step is to show that a vector of market-clearing wages exists, and that this vector is unique. Define the vector k(w) (k1(w), ..., kI(w)), where

ki(w) =

(1

wi

)(1)/for all i I

33

Then, by (8), the firms output in region i is

yi =

(1

1 )iki(w)

Lemma 2 shows that, under any wage vector w, a firm locates in the regionin which its output is highest, so the firms locations are given by L(k(w)).The labour demand of a firm that locates in region i is

ni(i, wi) = i (1/wi)1/ = iki(w)

1/(1)

The aggregate demand for labour is found by integrating over the demandsof the individual firms in the region:

NDi =

(Li(ek(w))

idF

)ki(w)

1/(1) = Zi(k(w))ki(w)1/(1)

The labour market clearing condition equates this demand to the supply oflabour. This condition can be written as

ki(w) =

[Ni

Zi(k(w))

]1A vector of market-clearing wages exists if and only if there exists a vectork(w) such that this condition is satisfied for all I markets. The required

vector k(w) is simply a fixed point of (2). Lemma 1 shows that this fixedpoint exists and is unique, so a vector of market-clearing wages exists andunique. Furthermore, Lemma 2 implies that the locations of the firms andthe distribution of labor across firms are measurably identical respectivelyacross equilibria since the market-clearing wage vector is identical acrossequilibria. Now consider the issue of Pareto optimality. All of the firms ina region equate their marginal products of labour to the market wage rate,so the marginal products of labour are equalized across firms. Consequently,P2 is satisfied. The uniqueness of the fixed point implies that k(w) is the

same as k, so that any L(k(w)) is measurably identical to L(k) and henceP1 is satisfied. Since the governments use their lump-sum taxes to attainan optimal division of their resources between the public and private good,P3 is also satisfied. Therefore, the equilibrium allocation induced by anyequilibrium is Pareto optimal.

34

Proof of Lemma 3: The proof of Lemma 3 follows the same steps asthe proof of Lemma 2, except that the government of any region i seeks tomaximize i rather than ri. Here, we present only the proof of the range ofthe equilibrium after-tax profits and the equilibrium location of firms. (18)and (19) are replaced with

[yi(i, wi) pii(i, wi, ti )]MRSi + ipii(i, wi, ti ) ipim(m, wm, tm)and

mpii(i, wi, ti ) [ym(m, wm) pim(m, wm, tm)]MRSm+mpim(m, wm, tm)

where MRSm is positive and bounded above by b.) Since pii(i, wi, ti ) =

pim(m, wm, tm) in equilibrium, these inequalities yields

yi(i, wi) pii(i, wi, ti ) ym(m, wm)Likewise, (23) is replaced with

[yj(j, wj) pij(j, wj, tj)]MRSj + jpij(j, wj, tj) jpii(i, wi, ti ) (27)for all j (j 6= i). This inequality is simplified as

yj(j, wj) pij(j, wj, tj)for all j (j 6= i). if the following condition holds, for all (j 6= i)

yi pii(i, wi, ti ) yjthen no region has incentive to deviate from its offer. Consequently, no otherregion j (j 6= i) has a profitable deviation if

pii(i, wi, ti ) yh

where yh is the firms second highest output. Proof of Theorem 3: The existence of an equilibrium can be proved by

showing the existence of the market clearing wage vector. Since every firm lo-cates into a region where it can attain its highest output, the market-clearingwage vector is unique and the same as the one proved in the correspondingpart of the proof of Theorem 2. It immediately follows that the allocationof firms and the allocation of labour across firms are measurably identical

35

across equilibria when the government can only impose the lump-sum tax onwages as when it can tax all of domestic income, so (L, n) again maximizestotal output. The proof of item 2 follows immediately from the observationthat there exists an equilibrium in which all wage income is used to subsidizethe firms (pii(i, wi, t

i ) = yi), so that the governments are unable to provide

any public good. The proof of item 3 follows from the observation that therecan be an equilibrium in which no firms are subsidized, allowing each govern-ment to provide a quantity of public goods at least as great as the regionstotal wages.

Proof of Lemma 4: Lemma 4 is proved in the same way as Lemma 3,using the revised expression for i.

Proof of Theorem 4: Consider first the existence of a credible equilib-rium. Define the variables

i +1 MRSi

for all i I (28)

and let be the vector (1, ...I). Suppose that, in equilibrium, a firm goesto region i only if

iki jkj for all j ILemma 4 implies that, in equilibrium, a firm goes to region i only if

i

(NiZi(k)

)i j

(NjZj(k)

)j for all j I

Then an equilibrium distribution of firms across regions is described byL(k()), where k() is the solution to the equation system

ki =

(NiZi(k)

)1i for all i I (29)

By FP1, k() exists and is unique for every strictly positive . The function

k() is continuous in its arguments. In any credible equilibrium, Rgi and

Rci are continuous functions of k() and hence continuous functions of

itself. Let these functions be Rgi () and Rci (). By FP1, Zi(k()) is positive

for every strictly positive vector , and hence Rci () is positive for everystrictly positive . Rgi () might be negative for some ; but there is some sufficiently small that Rgi () is positive whenever < i < + . To show

36

this, let be a vector in which i is equal to + ; and let be identical

to except that i is set equal to > 0. By FP2,

ki()

kj()>ki(

)

kj()

for every j other than i, and hence

Li(ki()) Li(ki())

Let the setM() contain all of the elements of Li(ki()) that are not elements

of Li(ki()). Then

Rgi () =

Li(bki())

(yi pii)dF +M()

(yi pii)dF

Consider a firm of some type (, ) contained in M(), and suppose that thisfirms next best offer came from region h. Credibility implies that pii is equalto yhh, and since the firm chose region i over region h,

yhh yi(+ )

The taxes paid by the firm are

yi pii yiNow consider a firm of some type (, ) contained in Li(ki(

)), and supposeagain that the firms next best offer came from region h. For this firm,

yhh yi( )

and the taxes that it pays are

yi pii yiThus,

Rgi ()

Li(bki())

yidF M()

yidF

By FP2,M() shrinks to some non-empty setM as falls, so that the secondterm falls toward zero as falls. The first term is independent of . It follows

37

that Rgi () is positive for all in which smaller than some critical value

.Now consider the mapping

i() = +1

i (Rci (),max[R

gi (), 0])

i I

Define the set

D [+

1 b

, 1

]IThe mapping : D D is continuous and well-defined, so the mapping hasa fixed point .

If Rgi () is non-negative for every i, all of the elements of a credible

equilibrium can be inferred from . The equilibrium vectorMRS is obtainedfrom (28). The equilibrium L is L(k()) or is measurably identical to it. The

equilibrium n is n(k()) or is measurably identical to it. Each regions wageis equal to that regions equilibrium marginal product. Credibility ensuresthat each firms after-tax profits are well-defined, so its tax rate is uniquelydetermined by its output and after-tax profits. Government is revenues areRci (

) and Rgi (), and region is consumption of private and public goods

is ci(Rci (

), Rgi ()) and gi(Rci (

), Rgi ()).

On the other hand, no equilibrium can be inferred from if Rgi () is

negative for some region i, so it is necessary to identify the conditions underwhich every governments revenues are non-negative. Assume that

>1 a

(30)

and recall that, by definition,

1 a

1 i(R

ci , 0)

for all i I and for all Rci 0. Then Rgi () must be non-negative for everyi. Suppose to the contrary that there were some region j for which Rgj (

)were negative. Then

j = +1

j(Rcj(

), 0)< +

38

but by construction, j < + implies Rgj () > 0, contrary to assumption.

Consequently, Rgi () is non-negative for all i I when (30) holds, and (30)

holds if a is sufficiently large.Total output is maximized if and only if there is some positive such

that ki() = ki for all i I. If every element of MRS is the same,

every element of is equal to some , where < 1. Since Zi(k)is linearly homogeneous, and since k() = k when every element of isequal to 1, ki(

) = k so that total output is maximized. Now supposethat total output is maximized, implying k() = k. Evaluating (29) atk determines a unique vector . Since it has already been shown thatk() = k when every element of is equal to , this is the uniquesolution for . Thus, total output maximization implies that every elementof is the same and hence every element of MRS is the same.

The possibility that the equilibrium allocation is Pareto optimal still ex-ists. If each firms second best option is small relative to its best option,every firm will pay taxes in equilibrium. If each regions preferences for thepublic good are not too strong, each governments revenue will be sufficientto provide the optimal quantity of public goods.

References

Black, D. and W. Hoyt, 1989, Bidding for firms, American Economic Review79, 12491256.

Bond, E. and L. Samuelson, 1986, Tax holidays as signals, American Eco-nomic Review 76, 820826.

Burbidge, J., K. Cuff and J. Leach, Tax competition with heterogeneousfirms, forthcoming in the Journal of Public Economics.

Doyle, C. and S. van Wijnbergen, 1994, Taxation of foreign multinationals: asequential bargaining approach to tax holidays, International Tax and PublicFinance 1, 211225.

Hamada, K., 1966, Strategic aspects of taxation on foreign investment in-come, Quarterly Journal of Economics 80, 361-375.

King, I., R. P. McAfee and L. Welling, 1993, Industrial blackmail: dynamictax competition and public investment, Canadian Journal of Economics 26,590608.

Wilson, J., 1999, Theories of tax competition, National Tax Journal 52,269-304.

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