1-s2.0-S0094119006000854-main.pdf

Journal of Urban Economics 62 (2007) 76102www.elsevier.com/locate/jue

Corporate profit tax, capital mobility, and formulaapportionment

Santiago M. PintoDepartment of Economics, West Virginia University, 412 Business and Economics Building, P.O. Box 6025,

Morgantown, WV 26506-6025, USAReceived 6 December 2005; revised 17 August 2006

Available online 6 October 2006

Abstract

The paper develops an analytical framework in which regional governments strategically determine thestructure of the corporate profit tax system when an apportionment formula determines the proportion ofthe firms income subject to regional taxation. The conclusions can be summarized as follows: (i) Regionalgovernments subsidize capital through the corporate tax system. (ii) Tax rates become higher and the portionof capital costs that can be deducted from taxable income becomes smaller as the formula weighs moreproduction shares. (iii) The regionally provided good may be below or above the efficient level. (iv) Theextent of the distortion depends on the particular formula put into practice. (v) Regional governments strictlyprefer a formula that exclusively weighs the production proportion to any other alternative. 2006 Elsevier Inc. All rights reserved.

JEL classification: F2; H2; H7; R5Keywords: Corporate profit tax; Formula apportionment; Capital mobility

1. Introduction

If a corporation has business activities established in multiple jurisdictions, regions, or coun-tries,1 then the local authority can levy a tax on income generated in that location. However,measuring income earned within each region raises a difficult conceptual problem. For instance,the current system of corporate taxation in the European Union requires firms to maintain dif-ferent accounts for its activities in each country where it operates (separate accounting). The US

E-mail address: [email protected] These three terms will be used indistinctively along the paper.0094-1190/$ see front matter 2006 Elsevier Inc. All rights reserved.doi:10.1016/j.jue.2006.08.008

S.M. Pinto / Journal of Urban Economics 62 (2007) 76102 77

and Canada, on the other hand, have adopted a system of formula apportionment (FA) to allocateincome across states. FA, as used in the US, asserts that the proportion of a multi-regional firmsincome earned in a given state is a weighted average of the proportion of the firms total sales,property, and payroll in that state. Thus, the firms activities in a specific region is approximatedby the share of these factors in the region, so the firm is not required to keep different accounts.Specifically, let I denote the set of states where the firm operates. The tax due by a firm to statei I is T i = t i ii , where t i is state is tax rate, i is the share of total profits that are subjectto taxation in state i according to the formula selected by that state, and i represents the firmstaxable profits as defined by state is tax law.2 The share i is defined by

i = miK(

kiiI ki

)+ miF

(f iiI f i

)+ miW

(wiiI wi

)

= miKiK + miFiF + miWiW (1)where ki , f i , and wi are property, sales, and payroll in state i, respectively; miq is the weightgiven to factor q = K,W,F in the apportionment formula in state i, such that 0miq 1, andmiK +miF +miW = 1; and iq is the share of factor q in state i. Table 1 shows the weights miqchosen by different states in the US in 2003. It is clear from the table that states do not followthe same principle when choosing the apportionment method.3

Even though this method of apportionment is relatively easy to administer, it creates verycomplicated incentive effects. On one hand, firms operating in different regions react to differentformulas by changing the allocation of property, sales and workers across regions. On the otherhand, given that the tax policy chosen by different regional governments affects residents of otherstates, some kind of strategic interaction can be expected.

An additional problem arises when regions are allowed to choose their own FA systems.If they all adopt the same formula, exactly 100 percent of a corporations income will beapportioned across states.4 Non-uniformity, however, can result in more or less than 100 per-cent of a corporations income being subject to state income tax.5 In an effort to encouragetax uniformity across jurisdictions in the US, the Multistate Tax Compact (1967) establishedthat the three factors considered in the apportionment formula are to be weighted equally(miK = miF = miW = 1/3 for all regions i). In spite of this, most states have recently devi-ated from the uniform apportionment formula and moved towards a greater weight on the salesportion of the corporate income tax, as shown in Table 1. It has been claimed that by manipulat-ing the formula in this way, officials can offer tax breaks that help the economic development ofthe region. However, if more states pass such legislation, other states will be compelled to do the

2 Regional governments may use different rules to define tax bases according to their tax codes. The present paper willconsider one way in which tax bases may differ across regions: the proportion of capital costs that can be deducted fromthe corporate taxable income.

3 Mintz and Smart [19] provide a good description of the subnational corporate tax system in Canada. In this country,provinces use the same method of allocating income across jurisdictions. Specifically, the general formula is given bythe sum of payroll and sales shares in a province divided by two.

4 In other words, i = 1.5 However, if the definition of taxable income differs across states, i.e., i is not the same for all i, then it may not betrue that 100 percent of the firms income is apportioned when all states use the same apportionment formula. We willcome back to this issue later when we introduce the theoretical model.

78 S.M. Pinto / Journal of Urban Economics 62 (2007) 76102

Table 1Apportionment formulas for manufacturing businesses (as of January 1, 2006)STATE Formula STATE FormulaALABAMA 3 Factor NEBRASKA FALASKA 3 Factor NEVADA No State Income TaxARIZONA Double weighted F NEW HAMPSHIRE Double weighted FARKANSAS Double weighted F NEW JERSEY Double weighted FCALIFORNIA Double weighted F NEW MEXICO Double weighted FCOLORADO 3 Factor / F and K NEW YORK 60% F, 20% K, 20% WCONNECTICUT Double weighted F / F NORTH CAROLINA Double weighted FDELAWARE 3 Factor NORTH DAKOTA 3 FactorFLORIDA Double weighted F OHIO 60% F, 20% K, 20% WGEORGIA 80% F, 10% K, 10% W OKLAHOMA 3 FactorHAWAII 3 Factor OREGON FIDAHO Double weighted F PENNSYLVANIA 60% F, 20% K, 20% WILLINOIS F RHODE ISLAND Double weighted FINDIANA Double weighted F SOUTH CAROLINA Double weighted F / FIOWA F SOUTH DAKOTA No State Income TaxKANSAS 3 Factor TENNESSEE Double weighted FKENTUCKY F TEXAS FLOUISIANA Double weighted F UTAH 3 Factor / Double weighted FMAINE Double weighted F VERMONT Double weighted FMARYLAND Double weighted F / F VIRGINIA Double weighted FMASSACHUSETTS Double weighted F / F WASHINGTON No State Income TaxMICHIGAN 90% F, 5% K, 5% W WEST VIRGINIA Double weighted FMINNESOTA 75% F, 12.5% K, 12.5% W WISCONSIN 60% F, 20% K, 20% WMISSISSIPPI Separate Accounting / 3 Factor WYOMING No State Income TaxMISSOURI 3 Factor / F DIST. OF COLUMBIA 3 FactorMONTANA 3 Factor

Note. F: sales; K: property; W: payroll. A slash separating two formulas indicates taxpayer option.Source. Federation of Tax Administrators (FTA) [5] .

same, initiating a race to the bottom, in which all states end up imposing the same (lower) taxliability.6

The present paper develops a theoretical framework, built on the analysis of Gordon and Wil-son [8], that focuses on the issues described above. In our setup, regional governments face theproblem of deciding the structure of their corporate profit tax system in a context of strategicinteraction, when firms simultaneously operate in many regions at the same time, and profits areregionally apportioned using a FA system. The tax revenue is devoted to finance the provision ofa regional good.7 Much of the tax incidence literature,8 adopts the interpretation that a corporateprofit tax is ultimately a tax on capital in the corporate sector. The argument essentially developsaround the fact that taxable profits may differ from actual economic profits. The ways in whichthese two concepts differ can be attributed to many different and complicated factors.9 In this

6 See, for example, Edminston [4] and Gavin [6].7 It is assumed that there are no other taxes available. The results will not be affected if the corporate tax revenue is

devoted to finance only a fixed proportion of the regional good. The paper focuses on alternatives that employ capital (orproperty), sales (or production) shares, or a linear combination of the two, i.e., miW is assumed to be equal to zero forall regions.

8 See, for example, Hargerger [9], Mieszkowski [17] and McLure [14].9 These factors depend, among other things, on the details of the tax law, on the market interest rate, etc.


paper, such discrepancy is explicitly considered by assuming that the tax base (i.e., the corporatetaxable income) includes both pure profits plus a portion of capital costs.10 Our setup, however,also considers the possibility of capital subsidization through the corporate tax system. The latterwill take place if the portion of capital expenditures that can be deducted from taxable incomeexceeds the true capital costs. Furthermore, it is assumed that regional tax authorities can imple-ment different policies in terms of the portion of capital costs that can be deducted from taxableincome and the portion that is computed as part of the tax base. Consequently, regions have somecontrol over the size of the taxable income.

The analysis is carried out in two steps. In the first step, the equilibrium tax structure fordifferent exogenously given FA systems is derived. This means that each region chooses the cor-porate tax rate and the corporate tax base (basically determined by the proportion of capital coststhat can be deducted from taxable income), but takes the FA system as exogenous. In this way,we are able to compare equilibrium tax structures and evaluate the effect on economic welfare ofusing different formulas. In the second step, we also allow regional governments to select theirown formulas in a non-cooperative way and derive the FA system chosen in equilibrium.11

The model focuses the analysis on the conditions arising at a symmetric equilibrium. Theapproach is justified on the grounds that our goal is to explain the general trend, as observed forexample in the US, to use formulas that rely less on capital shares and more on output shares.12It is outside the scope of our paper to explain how the specific characteristics of a state, region,or country determine the choice of a particular formula.13

The paper addresses a number of policy-related issues. First, the model shows the trade-offfaced by regional governments deciding between different tax instruments (tax rate levels, pro-portion of capital costs that can be deducted from taxable income, and the choice of a specificformula). Second, it allows us to determine whether tax competition in terms of FA systemsactually lead to a race to the bottom as observed in the property tax competition literature.14Finally, it provides a conceptual framework that can be used, for instance, to evaluate the recentshift towards a FA system predominantly based on sales shares, as verified in the US.

A few papers have formally studied the implications of using apportionment formulas to al-locate taxable income across jurisdictions. In general, given the complexity of the issue understudy, it has been extremely difficult for the literature in this area to provide clear and unam-biguous conclusions. The earlier papers by McLure [15,16], Goolsbee and Maydew [7], andMieszkowski and Zodrow [18] first elucidated that formally apportionment mostly transforms thestate corporate income tax into three separate taxes on the factors in the apportionment formula.Gordon and Wilson [8] examine the response of firms to a system of formula apportionment,restricting attention to cases in which all states use the same system, with different corporate taxrates. Most of their analysis is concerned with the component of the tax tied to the allocation

10 This same approach is followed by Gordon and Wilson [8].11 The results obtained in the first step greatly simplifies the analysis of the second step.12 Since late 1970s, more than 60 percent of the states in the US deviated from the uniform formula recommended bythe Multistate Tax Compact of 1967. In general, states tend to adopt a formula that weighs the sales proportion moreheavily, as illustrated by Table 1.13 We are aware that by following this approach we somewhat limit the scope of our conclusions. However, we believethat it is justified in terms of the goals of the paper. In addition, as it will become clear later, dropping this restriction willnot only make the analysis inconclusive but also untractable. It is also worth mentioning that the most important resultsin the standard tax competition literature (i.e., the issue of underprovision of public goods) holds under symmetry, and

may not be true in general.14 See, for example, Brueckner and Saavedra [2].


of property (i.e., miF = miW = 0 for all regions i). The setup of our theoretical model is alsorelated to Pethig and Wagener [22], Kolmar and Wagener [13], and Mintz and Smart [19].15 Allthese papers, however, assume that the formulas are exogenously given and do not consider theendogenous choice of tax bases by regional tax authorities. Anand and Sansing [1] provide anexplanation for why states may choose different apportionment methods, even though aggregatesocial welfare is maximized when states use the same formulas. They demonstrate analyticallythat importing states have incentives to increase sales factor weights, while exporting stateshave incentives to reduce the weights on productive factors. The results are explained by thefact that they assume that some inputs are completely immobile. One limitation of their analysisis that they assume that tax rates and tax bases are not only exogenously given but also equalacross regions.16 Our paper, on the other hand, endogenizes the choice of tax rates, tax bases,and formulas in a context of strategic competition.

The organization of the paper is as follows. Section 2 derives the optimal corporate tax policywhen a central government is in charge of designing the corporate tax system. The corporateprofit tax base for a central government is the sum of the firms profits in each region, so a FAsystem to allocate profits across regions is not required in this case. The framework developed inthis section will serve as our benchmark case. In Section 3, we derive the equilibrium corporatetax policy under different FA systems, and in Section 4, we allow regional governments to choosetheir own formulas. Finally, Section 5 concludes.

2. The benchmark case: central government intervention

Before analyzing the effects of a FA system and to fully understand its implications, we derivethe optimal corporate tax policy when a central government is in charge of designing the corpo-rate tax system. This framework will serve as our benchmark case. Consider an economy withtwo completely identical regions, A and B .17 An homogeneous consumption good xi , whichserves as the numraire, is locally produced by a multi-regional firm, i.e., a firm that operatesin both regions. The regions constitute a common market. There is a representative immobileconsumer-investor in each jurisdiction that owns a proportion i of the firm. Capital is the onlyvariable factor, it is in perfectly elastic supply, and can be rented at an exogenous rate r .18 Themulti-regional firm can shift capital across regions. Output is determined by the production func-tion f (ki), which is identical across regions, and satisfies fk > 0, fkk < 0.19 Gross profits ineach region i are given by i = f (ki) rki , total output (or total sales) by f (kA) + f (kB),and total gross profits by = A + B = f (kA) + f (kB) rk, where k = kA + kB . In thissection, we assume that the multi-regional firm pays profit taxes to a central government, who isin charge of designing the structure of the corporate profit tax system. The central government

15 Mintz and Smart [19] focuses on FA systems that exclusively use capital or property shares, too.16 There is also a growing literature that compares tax spillovers generated by national corporate tax systems whichrely on separate accounting (SA) or FA to determine tax bases (see, for instance, Nielsen et al. [21] and Sorensen [24]).However, these papers, as some of the papers mentioned before, assume that the tax system is exogenously given andconcerns a FA system that only employs capital or property shares.17 Since we assume that regions are identical, restricting the number to two locations does not affect the conclusions ofthe paper. The results will remain the same if we considered many identical regions.18 The main results will not affected if capital is fixed as in the tax competition models. This assumption is introducedhere to simplify the analysis.

19 It is assumed that there is an implicit fixed factor that gives rise to pure profits in each region. We can think of thisfactor as being entrepreneurial services.


determines the corporate tax base and tax rate, and divides the total tax revenue among the tworegions. Note that the central government is not required to determine the amount of profits ac-crued in each region as corporate taxes are based on total profits. Thus, a formula apportionmentis not necessary in this case.

Consumers derive utility from the consumption of x and from a regional good publiclyprovided gi , which is financed by the proportion of tax revenue assigned to region i. All con-sumers have identical preferences represented by a strictly quasi-concave and strictly increasingcontinuous utility function U(x,g). The utility function also satisfies limh0 Uh = , whereUh U(x,g)/h for h = x,g. Uih indicates the marginal utility of good j for a consumer re-siding in i. The firms after tax profits are entirely distributed among consumers according to theownership shares, and constitute the only source of income.

Standard corporate profit tax models assume that taxable corporate profits differ from eco-nomic profits in many different and complicated ways. Following Gordon and Wilson [8], weassume that this discrepancy is basically determined by the proportion of capital expendituresthat can be deducted from taxable income. Specifically, the tax base is the value of output minusa fixed share of the true capital cost.20 Since total capital costs are rk, the firm is allowed todeduct rk from its total revenue. Thus, taxable income as computed by the central authority, , and total economic profits, , are related in the following way:

= f (kA)+ f (kB) rk = + (1 )rk. (2)The deductibility of capital expenses for tax purposes can either be above or below its true costs.If a tax system allows only incomplete deduction of capital costs, then < 1, which means thata positive tax is also levied on capital. If = 1, the corporate tax falls only on pure profits, andwhen > 1 capital is subsidized, i.e., firms can deduct more than the true capital expenditures rkfrom taxable income. Denoting by t the corporate profit tax rate, taxes paid by the firm areT = t , so the firms total net profits are N = T , or

N = [f (kA)+ f (kB) rk](1 t) t (1 )rk= (1 t) t (1 )rk. (3)

The model can be represented as a game where the central government moves first and the firmreacts after observing the governments choices. Specifically, events take place sequentially asfollows: (i) the central government announces the tax policy, i.e., it announces the corporate taxrate (t) and tax base (essentially determined by ); (ii) the firm observes the tax policy andchooses the amount of capital that will be assigned to each region; (iii) payoffs are realized. Thenext sections characterize the sub-game perfect Nash equilibrium of this game.

20 The value of will depend on the details of the tax rules and tax laws, and it will have the potential to influencethe firms behavior. Governments can manipulate by choosing the amount of investment tax credits, depreciationmethods, capital subsidies, and other tax concessions. We assume that captures the dynamic effects of these tax policyvariables on capital accumulation, even though our setup is essentially static. Hines [11,12] provides a critical surveyof the literature that studies behavioral responses by multinational firms to different tax rules. The general approachfollowed by almost every study is to consider the economic effect of the tax system as a whole. For instance, capitaldoes not only respond to differences in tax rates across regions, but is sensitive to differences in tax rules that affect the

definition of taxable income as well. Haufler and Schjelderup [10] also describes how different tax policies employed bynational governments can ultimately affect the value of .


2.1. The firms problem

At the second stage of the game, the firm chooses the amount of capital that is allocated toeach region. The firms problem consists of maximizing net profits N with respect to ki for agiven tax policy. From the first-order conditions, the following expression is obtained

f ik r =t (1 )r(1 t) (4)

for i = A,B , which implicitly defines ki as a function of t,, and r .21 Any value of differentfrom one would induce the firm to choose an inefficient amount of capital. It is straightforwardto derive the firms reaction to changes in t and :

ki

t= (1 )r

(1 t)2f ikk, (5)

ki

= tr

(1 t)f ikk. (6)

An increase in the tax rate decreases the amount of capital in both regions if < 1, attractscapital into the regions if > 1, and does not have an impact on capital if = 1. On the otherhand, for 0 < t < 1, a higher value of will create an inflow of capital into both regions.

2.2. The central governments problem

In the first period, the central tax authority determines the values of t , , gA, and gB thatmaximize the utilitarian welfare function W = UA + UB , where Ui U(xi, gi), anticipatingthe firms reaction.22 In this case, the regional distribution of profits is irrelevant for the centralgovernment because the corporate tax is based on the firms total profits. As explained before,the firms after tax profits are distributed among consumers according to the ownership shares,which means that xi = iN . We assume that i = 1/2 for i = A,B , so the firm is exclusiveproperty of the residents of the two regions.23 Given that net profits are equally divided betweenregions, then xA = xB . The values of gA and gB satisfy the budget constraint gA + gB = t .Throughout the analysis, we will only consider interior solutions for the corporate tax rates, i.e.,tax rates that satisfy 0 < t < 1, and we require to be non-negative.

An interesting implication of this model is that, in equilibrium, is set equal to one. Thisresult simply restates the traditional argument in favor of a complete exemption of investmentfrom corporate taxation. Note that after substituting = 1 into (4), we obtain that fk = r . Hence,the optimal amount of capital inputs ki chosen by the firm will not be affected by the corporatetax, or, equivalently, investment decisions are not distorted. Furthermore, when = 1, the mar-ginal benefit of one additional unit of gi (in terms of xi ) is equal to one, the marginal cost of gi

21 The supraindex i in f i , f ik

, and f ikk

indicates that the corresponding expression is evaluated at ki .22 By choosing a simple utilitarian welfare function, we ignore any redistributive objective that could be attained throughthe fiscal policy.23 It may be the case that A + B < 1, i.e., the firm also belongs to residents of other regions. However, if this is true,

an additional effect comes into play that has to do with tax exporting. In this paper, we ignore such possibility and assumethat A = B = 1/2.


(in terms of xi ). In other words, MRSi = 1, where MRSi = Uig/Uix , so the outcome attained inthe centralized solution is efficient. The following proposition summarizes this finding.24

Proposition 1. When a central government designs the corporate tax system, firms are allowedto deduct all capital expenditures from taxable income, i.e., is set equal to one, and investmentdecisions are not distorted. As a result, the provision of the regional goods gA and gB is efficient.

The next sections show that this result no longer holds when decentralized tax authorities (orregional governments) determine their own corporate profit tax structure through an apportion-ment formula.

3. Strategic determination of corporate tax rates and tax bases for a given FA system

We now assume that the multi-regional firm pays profit taxes to each regional governmentwhere it operates.25 Both regions adopt a FA method to calculate the share of the firms activitiesin each jurisdiction. These shares determine the distribution of tax revenues across regions.

In this section, we assume that regions implement an exogenously given formula and strategi-cally choose tax rates and tax bases. In this way we can examine how different FA systems affectthe choice of these tax variables.26 Our interest is on FA tax regimes that consider sales, property,or a combination of the two factors as proxies of the firms activities in each state.27 In Section 4,we relax the assumption of exogenously given FA systems and derive the equilibrium formula,tax rate, and tax base chosen by regional governments in a context of strategic interaction.

Formally, the FA system can be described as follows. Let A denote the share of the firmsactivities in region A as determined by the apportionment formula employed by the tax authorityin that region, and B the corresponding share in region B .28 Following the notation used inEq. (1), i , for i = A,B , is given by

i miKiK + miFiF , (7)where 0miK , miF 1, and miK + miF = 1,29 and

iK ki

k, iF f (k

i)

[f (kA) + f (kB)] .

If region i exclusively uses capital (or property) shares in the apportionment formula, thenmiK = 1, and i iK . If it only employs production shares, then miK = 0 and i iF .When plants produce for the local markets where they are located, outputs and sales are equiva-

24 The formal proof is relegated to Appendix A.1.25 The model developed in this section is built on Gordon and Wilson [8], but is also related to the work of Pethig andWagener [22], Kolmar and Wagener [13], and Nielsen et al. [21]. However, our results are somewhat different. We willexplain the differences as we move along.26 By following this approach we can evaluate the effect on economic welfare from choosing different formulas.27 We do not consider formulas that use payroll shares, i.e., miW = 0 for all regions i. Wellisch [25], in a differentframework, assumes that labor is immobile, so jurisdictions will end up using payroll as the only apportionment factor.28 In fact, i is a function, i.e., i(miK,miF , kA, kB). For notational simplicity, we will write i i iK iF A B (m ,m ,k , k ) hereafter.

29 From now on, we characterize formulas by their values of mAK .


lent, so iF also represents sales shares.30 A formula that gives positive weights to both factorsis represented by 0 < mAK < 1.

As in the previous section, we assume that a proportion of capital expenditures can be de-ducted from taxable income. However, it is now assumed that this proportion may differ byregion.31 Each regional government allows the firm to deduct a proportion i of its true to-tal capital costs in region i, rki . Given that the firm operates in regions A and B , then it isallowed to deduct a proportion AKA + BKB of total capital expenditures rk. Denot-ing by t i region is statutory corporate profit tax rate, then the firm faces the effective tax ratet tA A + tB B .32 The firms total tax burden is given by

T = t[f (kA)+ f (kB) rk]= t[ + (1 )rk]= t, (8)

where is the tax base. After-tax profits N = T areN = (1 t ) t (1 )rk. (9)

At this point it is worth emphasizing that regions do not necessarily adopt formulas of the sametype. When this is true, more or less than 100 percent of the firms total profits may be subjectto taxation. In other words, A + B may not be equal to one.33 In fact, using the previousdefinitions,

A + B = 1 + (AK AF )(mAK mBK). (10)Note that if regional governments choose the same apportionment formula, then A + B = 1.This implication will become relevant when we analyze the outcome of a symmetric equilibrium.

As in the previous section, actions and events take place sequentially as follows: (i) regionalgovernments simultaneously announce their tax policy for a given FA system, i.e. they announcethe corporate tax rate (t i ) and tax base (determined by i ); (ii) the firm observes the tax policychosen by each state and decides the amount of capital to employ in regions A and B; finally,(iii) gi and xi are determined, and payoffs are realized.

3.1. Firms profit maximization

We begin by solving the firms problem for a given tax policy {t i ,i,miK}, i = A,B , in thesecond stage of the game. We will then compare the impact of a change in t i and i on kA, kBand k under the special FAs mentioned before.

Consider the case of a general formula where the share of the firms activities in A are givenby A, and the shares in B by B . The values of kA and kB chosen by the firm result from the

30 Defining the weight in this way is consistent with a tax at the origin. In addition, given that in our setting regions areidentical, exports and imports will be zero in a symmetric equilibrium.31 We depart from Gordon and Wilson [8] and Pethig and Wagener [22] in that we assume that governments canmanipulate the deduction policy, i.e., they have some control over the corporate tax base.32 For notational simplicity, and t denote the functions (A,B, kA, kB) and t (tA, tB , A, B), respectively.33 This last statement holds when A = B = 1, i.e., regions use the same definitions of taxable income and the tax

falls on pure profits. If regions adopt different criteria when defining the corporate tax base, then it is no longer true that100 percent of the firms income is apportioned.


maximization of (9) with respect to kA and kB taking the tax system as given. From the first-orderconditions we obtain

f ik r =t (1 )r(1 t )

t rk

(1 t )

ki+

(1 t ) t

ki. (11)

Condition (11) states that the level of capital in region i responds to three different effects. First, itdepends on the proportion of capital costs that can be deducted from taxable income, representedby the first term on the right-hand side of (11). Ignoring all other effects, the marginal cost of anadditional unit of capital is [(1t)/(1 t )]r , which is the same for both regions. In equilibrium,the marginal return on capital would be higher than r if < 1, and would be lower than rif > 1. Only when both regions allow full deduction of capital expenditures, i.e., i = 1, fori = A,B , this effect vanishes. Second, by changing the amount of capital allocated to region i thefirm can alter the value of . This effect is captured by the second term in (11), which basicallydepends on /ki , given by

ki= (i j )iK

ki, for i, j = A,B and i = j. (12)

If i > j , the firm will tend to allocate a disproportionately large amount of capital in region i(and a smaller amount of capital in region j ), which eventually increases , and, consequently,net profits. Of course, the second term vanishes if both regions follow the same deduction policy.Finally, the third term on the right-hand side of (11) states that the firm is able to influence theeffective tax rate t by changing ki through t/ki , represented by

t

ki= tA

A

ki+ tB

B

ki, for i, j = A,B and i = j. (13)

The impact of ki on t depends on the regional tax rates and on the effect of ki on A and B . Inaddition, the previous expressions depend on mAK and mBK , so the formulas implemented bythe regions also affect t . Specifically,

i

ki= miK

iK

ki+ (1 miK)iF

ki, (14)

j

ki=

[mjK

iK

ki+ (1 mjK)iF

ki

], (15)

where

iK

ki= k

j

k2> 0, (16)

iF

ki= f

jf ik

[f A + f B ]2 > 0, (17)

for i, j = A,B , and i = j .34 Note that if both regions adopt the same formulas, i/ki =( j /ki) > 0, so (13) becomes

t

ki= (t i tj ) i

ki, for i, j = A,B and i = j. (18)34 Equation (15) holds because Aq + Bq = 1, for q = K,F .


Then, as long as the regions impose different tax rates, the firm can manipulate the effectivetax rate by changing the distribution of capital across regions. In this case, the effective tax rateincreases with ki if t i > tj , and decreases if t i < tj . In conclusion, the marginal productivity ofcapital in region i will be equal to r only when A = B = 1, tA = tB , and regions adopt thesame formula.35

The comparative static results with respect to tax rates are hard to obtain partly because thefirm does not only influence the tax base, but also because it has some control over the effec-tive tax rate as explained above. In addition, regions may be employing formulas that weighcapital and output differently. For this reason, in this section we only derive the firms responseto a change in tax instruments at a symmetric equilibrium. At a symmetric equilibrium, thetax variables are (ex-post) identical. In other words, the two regions choose the same tax rates(t = tA = tB = t), tax bases ( = A = B = ), and formulas (mAK = mBK ).36 Under theseconditions, kA = kB = k/2 and A = B = 1/2 for every formula, and f Akk = f Bkk = fkk .

Differentiating (11) with respect to tA and evaluating the corresponding expressions at a sym-metric equilibrium, we obtain37

kA

tA= 1

(1 t)fkk[ A(1 )r

(1 t) + A

kA

], (19)

kB

tA= 1

(1 t)fkk[ A(1 )r

(1 t) + B

kA

]. (20)

Similar expressions can be derived for tB .38 Two simultaneous effects take place when region Araises the tax rate, which may potentially have conflicting effects on kA and kB . On one hand,the response of capital to tA depends on the proportion of capital expenses that are subject tocorporate taxation, . This effect is represented by the first term within the square brackets of(19) and (20) and is identical for both kA and kB . A higher tA raises the tax burden on capital if(1 ) > 0, keeping capital away from both jurisdictions. Alternatively, a higher tA increasesthe subsidy on capital if (1 ) < 0, attracting capital into both regions. On the other hand, achange in tA induces the firm to reallocate capital from the region with the highest tax burden(region A) to the one with lowest (region B).39 This effect is captured by the second term withinthe square brackets of (19) and (20). Since A goes up and B goes down with higher levelsof kA, then kA should decline and kB should rise with higher levels of tA. Furthermore, giventhat at a symmetric equilibrium B/kA = A/kA, this second term establishes that theeffect of tA on kA is equal, in absolute value, to the effect of tA on kB .

From (19) and (20), it follows thatk

tA= (1 )r

(1 t)2fkk , (21)

35 The last two terms on the right-hand side of (11) are not present in the centralized situation considered in Section 2.36 The fact that the two regions are assumed to be completely identical reinforces the idea of focusing on a symmetricequilibrium. We also follow this approach because we will later concentrate the analysis on the conditions arising at asymmetric equilibrium.37 The derivation of the following expressions is shown in Appendix A.2.38 Given that our focus is on a symmetric equilibrium, region A we will be considered the representative region hereafter.39 This process can also be explained by observing (18). A slight increase in tA (from a symmetric equilibrium) induces

the firm to lower the amount of kA and increase the amount of kB because, in this way, it reduces the effective tax rateimposed on its profits.


which means that total capital inflows as a result of an change in tA depend exclusively on thevalue of . In other words, k decreases with tA if (1 ) > 0, in which case the marginal returnon capital in the regions is less than r , decreases if (1 ) < 0, given that the marginal returnon capital exceeds r , and does not change if = 1.

From (19) and (20), it becomes clear that the signs and magnitudes of kA/tA and kB/tAdepend, among other things, on the derivatives of A and B with respect to kA and kB .These expressions, as explained earlier, also depend on the particular formula chosen by theregions. Thus, working with specific FA systems will allow us to derive additional results for(19) and (20). Consider again region A. Evaluated at a symmetric equilibrium, expressions (16)and (17), respectively, become

AK

kA= 1

2k, (22)

AF

kA= fk

4f, (23)

where f = f A = f B , which means that A

kA= mAK 1

2k+ (1 mAK) fk

4f. (24)

In order to determine how A/kA changes with mAK , we would need to compare expressions(22) and (23).40 However, this can only be done at the same values of t i and i , in which casethe amount of capital in the regions do not depend on the particular apportionment formulaimplemented.41 Suppose, for the moment, that we maintain the values of t i and i fixed at givenlevels. As production exhibits decreasing returns to scale, then average product in each region,f/(k/2), is always greater than marginal product, fk , or f/(k/2) > fk . Rearranging, we obtainAK/kA > AF /kA. Consequently, as the formula gives higher weight to capital shares, A/kA will get larger.

We are now able to derive additional results for (19) and (20). In first place, the next proposi-tion examines the influence of on the response of capital to changes in tA.42

Proposition 2. The following comparative static results hold in a symmetric equilibrium for allFA systems:

(i) If < 1, then kA/tA < 0, kB/tA > 0, and k/tA < 0.(ii) If = 1, then kA/tA = (kB/tA) < 0, and k/tA = 0.

(iii) If > 1, then kA/tA 0, kB/tA > 0, and k/tA > 0.

Similar expressions can be obtained for changes in tB .

40 Again, given that at a symmetric equilibrium A/kA = B/kB = B/kA , then we can just study theimpact of mAK on A/kA .41 From Eq. (11) it is clear that at a symmetric equilibrium where = 1, the first term on the right-hand side becomest (1 )r/(1 t), and the second term vanishes. Thus, if the values of t and are the same for all FA systems, then kAand kB are also the same. However, it is important to recognize that the equilibrium values of t and are not necessarilythe same under different FA systems. The next section precisely studies the relationship between the tax variables and

the formulas.42 See Appendix A.3 for a complete derivation of the results.


The proposition states that kB/tA is always positive and kA/tA is, in general, negative.However, if > 1 and the first term in (19) dominates the second one, kA/tA will be positive.Additionally, notice that when = 1, kA/tA < 0 and kB/tA > 0. Even though total capitalremains constant in this case, some capital reallocation across regions takes place. This last resultcontrasts with the one obtained in Section 2, where we found that when a central governmentchooses the tax policy, ki/t = 0, i = A,B , when = 1.

Next, we examine the extent to which different apportionment methods affect capital respon-siveness to tax rates. This effect will have extremely important implications at the moment ofanalyzing the equilibrium values of tA and A for different values of mAK . A few importantresults are summarized in the proposition below.43

Proposition 3. Evaluated at fixed levels of {t i ,i}, the following results hold in a symmetricequilibrium:

(i) If (kA/tA) < 0 when mAK = 0, then (kA/tA) < 0 for all 0 < mAK 1. Additionally, theresponse of kA to a change in tA is smallest (in absolute value) when the formula exclusivelyemploys production shares, i.e.,(kA/tA){mAK=1}

> (kA/tA){0 0 when mAK = 1, then (kA/tA) > 0 when 0mAK < 1. Ad-ditionally, the response of kA to a change in tA is smallest when the formula exclusively employscapital shares, i.e.,(

kA/tA

){mAK=0} >

(kA

/tA

){0 0.

(iii) The impact of a change in tA on kB is largest under a FA system that assigns full weight toproperty shares, i.e.,(

kB/tA

){mAK=0} >

(kB

/tA

){0 0.

Similar results hold for changes in tB .

In the derivation of the previous results, we used the fact that A/kA is smallest whenmAK = 0, and that, a symmetric equilibrium, A/kA = ( B/kA). The proposition statesthat, if kA declines with higher values of tA, capital is least responsive when the formula assignsfull weight to production shares, while if kA increases with tA,44 the response is smallest whenthe formula weighs exclusively capital shares. A higher level of tA, on the other hand, alwaysaffects kB positively and the response is largest when the formula only weighs capital shares.

The comparative static results with respect to A (and B ) are less complicated to derive. Itis straightforward to show that, evaluated at a symmetric equilibrium,

kA

A= tr

(1 t)fkk > 0, and (25)kB

A= 0. (26)

43 Similar results have been obtained by Pethig and Wagener [22] in a somewhat different framework.44 According to Proposition 2, kA/tA may be positive when > 1.


A few remarks are worth emphasizing concerning expressions (25) and (26). First, the resultshold for all apportionment formulas. Second, as higher values of A imply lower taxes on capitalor, eventually, a subsidy if A becomes greater than one, a slight increase in A attracts morecapital to region A. Finally, at a symmetric equilibrium, the change in A exclusively affects kA,which means that total capital increases with A in the same amount as the rise in kA. Similarresults hold for changes in B .

3.2. The governments problem

In this section we derive the equilibrium values of t i and i for a general exogenously givenFA system assuming that regional governments behave strategically.45 Next, we compare theequilibrium values attained under different apportionment formulas. The analysis focuses on asymmetric (sub-game perfect) Nash equilibrium.

Each regional government simultaneously chooses the values of t i and i taking the taxinstruments decided by the other jurisdiction and the apportionment formula as given. Forexample, the local government of region A determines the levels of {tA,A} that maximizeUA U(xA,gA), where gA = tA A , and xA = AN . The first-order conditions for an inte-rior solution are46

UA/tA UAx

(xA

/tA

)+ UAg (gA/tA)= 0, (27)UA

/A UAx

(xA

/A

)+ UAg (gA/A)= 0. (28)These two equations implicitly define tA and A as a function of tB and B and all other pa-rameters. Similar expressions can be obtained for region B . The four conditions determine theequilibrium values of {tA, tB,A,B}. Given that we focus our attention on a symmetric equilib-rium, the analysis can be performed by studying the conditions that characterize the equilibriumin one region, in this case, region A.

Rearranging (27) and (28), we obtain

MRSA = xA/tA

gA/tA= x

A/A

gA/A. (29)

The first equality in (29) states that the marginal benefit of one more unit of gA, determined byMRSA, should be equal to the corresponding marginal cost. The second equality makes explicitthe trade-off faced by the regional authority when deciding between alternative combinations of{tA,A}. In order to increase gA in one unit, tA has to be changed in [1/(gA/tA)]. The cost ofthis change in terms of units of xA is measured by (xA/tA). Thus, (xA/tA)/(gA/tA)is the marginal cost of increasing gA in one unit by changing tA. A similar reasoning holdsfor A. In equilibrium, the marginal cost for each of the two tax instruments should be equal.

The derivatives of xA with respect to tA and A are given by

45 In other words, the values of miK for i = A,B , are exogenously given. In addition, we also assume that the regionsimplement the same formulas, i.e., mAK = mBK . In Section 4 these parameters will become decision variables as well.46 Throughout the paper, we assume that the second-order conditions on the governments problems are satisfied eventhough we recognize that these are not readily apparent. We share the same technical difficulties as with other papers in

the area of tax competition. In our case, the second-order conditions establish that at the equilibrium 2UA/(tA)2 < 0,2UA/(A)2 < 0 and [2UA/(tA)2][2UA/(A)2] [2UA/tAA]2 > 0.


xA

tA= A N

tA= A A < 0, (30)

xA

A= A N

A= AtrkA > 0. (31)

Net profits N are also a function of ki, i = A,B , but, due to the envelope theorem, the derivativesof N with respect to these variables vanish as ki is optimally chosen by the firm.47

The derivatives of gA with respect to tA and A are more complicated to derive. However,evaluated at a symmetric equilibrium, the expressions become

gA

tA= A

[1 + AA + (1 )rk

(1 t) BA

], (32)

gA

A= trkA( A + AA), (33)

where ij = (tj /ki)(ki/tj ) is the elasticity of ki with respect to tj , and = (k/ A)( A/kA). Using (24), it is straightforward to show that = 1 when mAK = 1 and = fkk/2f < 1when mAK = 0.

Equation (29), evaluated at a symmetric equilibrium, can be rewritten as

MRSA = A

1 + AA + [(1 )rk/(1 t)]BA =A

A + AA . (34)

Rearranging the last equality of (34), we obtain

[AA + (1 )rk

(1 t) BA

]= (1 A) AA. (35)

Given that xA/tA < 0 and xA/A > 0, a necessary condition for a solution to (27) and(28) to exist is that, at the equilibrium, gA/tA > 0, and gA/A < 0. The latter conditionalways holds if AA > 0. However, if AA < 0, then A > |AA|.48 Additionally, it can be shownthat if AA is positive, then it should be less than (1 A).49 Thus, it can be concluded that inevery symmetric equilibrium AA (1/2,1/2).

The first outcome that follows from (35) is that when regional governments determine thecorporate tax rate and tax base strategically, they will choose a value of i that is greater thanunity for every FA system. In other words, regional governments will tend to subsidize capital byoffering an excessive amount of capital exemptions. The following proposition states this resultformally.50

47 Note that total corporate taxable income should be positive in equilibrium because this is the only source ofrevenue available to finance the provision of gA .48 Recall that at a symmetric equilibrium i = 1/2, for i = A,B .49 Given that

AA + (1 )rk(1 t)

BA = t (1 t)fkk[(

kA

tA

)2+(

kB

tA

)2]< 0,

and using (35), it follows that (1A) > AA . The latter is always satisfied if AA is negative, but it introduces an upperAAbound on whenever it is positive.

50 The proof is relegated to Appendix A.4.


Proposition 4. Suppose that regional governments choose {t i ,i}, for i = A,B . At symmetricsub-game perfect Nash equilibrium > 1 for every FA system.

The proposition asserts that when regions act in a decentralized way, they end up offeringbenefits to capital in excess of what would be optimal, i.e., = 1. This result can be explainedas follows. In an effort to attract capital into their own regions, local tax authorities increase thelevel of i . However, they do not fully internalize the costs that their decisions impose on otherregions. For instance, as i is changed, it affects the total tax base , but only part of this impactis taken into account by region i. In other words, the cost of increasing i faced by region i islower than the actual social costs. Given that this external effect is not considered by each region,the value of i will tend to be excessively high.

In Section 2, we showed that a central government provides an efficient amount of the regionalgood, i.e., MRSi = 1 for i = A,B . Does this result still hold in the decentralized case? Considerthe last equality on the right-hand side of (34). Remember that at a symmetric equilibrium A =1/2. It was assumed earlier that i = 1/2, i = A,B , i.e., corporate profits are equally dividedamong residents of the two regions. Suppose that at the equilibrium AA < 0. Since AA cannotbe lower than (1/2), then, from (34), it follows that MRSA > 1, or equivalently, the goodprovided by the regional government is underprovided. Now, if at the equilibrium AA > 0, thenthe regional good ends up being overprovided. In effect, if AA > 0, then 1/2 < A + AA < 1(because AA < 1/2), so A < MRSA < 1. The following proposition summarizes the previousfindings.

Proposition 5. If regional governments allocate profits for tax purposes using a FA system, thenthe following results hold at a symmetric sub-game perfect Nash equilibrium for every formula:

(i) If AA < 0, the regional good is underprovided, i.e., MRS > 1.(ii) If AA = 0, the provision of the regional good is efficient, i.e., MRS = 1.

(iii) If AA > 0, the regional good is overprovided, i.e., MRS < 1.

Hence, as in the property tax competition literature, strategic competition between regionsmay lead to an outcome that is characterized by underprovision of the good regionally pro-vided.51 However, in our setup, the possibility of overprovision, or even efficient provision, isalso present. In light of the results presented in Proposition 2, the local public good will end upbeing overprovided if i > 1, so that the capital elasticity is positive. Efficiency is attained whenAA = 0, but in the decentralized case, this also requires setting a value of strictly greater thanone.

Based on the conclusions obtained in Section 3.1, in particular, the fact that the effect oncapital from a change in the corporate tax rate is not the same when different formulas are used,it is reasonable to expect that the equilibrium values of t i and i will also differ. Additionally,the choice of the tax variables will have an impact on the provision of the regional good, and,

51 The underprovision result holds in the property tax competition literature only when regions are completely identi-cal. When regions are not identical, a pecuniary effect should be taken into account, which affects regions differently

depending on whether they are net importers or net exporters of capital. See, for instance, Myers [20] and DePater andMyers [3].


consequently, on economic welfare. The following proposition compares the equilibrium valuesfor different FA systems.52

Proposition 6. At a symmetric sub-game perfect Nash equilibrium:

(i) The corporate tax rate is highest (lowest) and the proportion of capital that can be deductedfrom taxable income is lowest (highest) when the FA system employs exclusively productionshares (capital shares), i.e.,

t{mAK=0} > t{0 {0 1.

(ii) The marginal rate of substitution between the regional good and the private good is smallest(highest) when the formula exclusively weighs production (capital) shares, i.e.,

MRS{mAK=1} > MRS{0 0. As before, the benefits from increasing tA are greaterwhen mAK = 0. Proposition 3 states that, in this situation, a higher tA attracts more capitalinto the regions when mAK = 0. Therefore, the regional governments will be induced to imposehigher levels of tA. In addition, Eq. (33) shows that the costs from increasing A are higher whenmAK = 0 (given that AA is larger in this case), which means that the equilibrium value of Awill be lower.54

Combining the results of Propositions 5 and 6, we can conclude that if in equilibriumAA < 0 for all formulas, so the regional good is underprovided, then the degree of the dis-tortion is smallest when the formula exclusively weighs production shares, i.e., MRS{mAK=0} >

52 A complete proof is provided in the Appendix A.5.53 A higher A will decrease gA in a relatively bigger amount when AA is small in absolute, as described by (33).

54 As indicated in the proposition, the previous results can be extended for formulas that give positive weights to capitaland production shares, i.e., 0 < mAK < 1.


MRS{mAK=1} > 1. This last result holds because, at the equilibrium, |AA{mAK=0}| < |AA{mAK=1}|.55Intuitively, given that a lower elasticity is associated with lower capital mobility, the externaleffects of the regional tax policy are reduced when the formula only considers production shares.

On the other hand, if AA is positive in equilibrium, the capital flowing into region A fromother places when tA is increased, more than compensates the amount of capital flowing outof region B . The outcome is then be characterized by an excessive amount of capital flowinginto the regions, which tends to increase the level of gA beyond its optimal level, i.e., gA ends upbeing overprovided. Additionally, the extent to which gA is overprovided depends on the specificformula chosen as before. Using MRS{mAK=1} > MRS{mAK=0} and the fact that when AA > 0,MRS < 1 for all formulas, it follows that MRS{mAK=0} < MRS{mAK=1} < 1. This last result holdsbecause when AA > 0, capital is more responsive to tA when mAK = 0. In other words, a highertax rate attracts less capital into region A when the formula only uses capital shares compared toother formulas, so the extent to which the regional good is overprovided is smaller. The followingproposition summarizes the previous results.

Proposition 7. Consider the outcome of a symmetric sub-game perfect Nash equilibrium.

(i) If the regional good is underprovided when mAK = 0, it will also be underprovided when0 < mAK 1. The distortion is minimized when mAK = 0 and is largest when mAK = 1,i.e.,

MRS{mAK=1} > MRS{0 1.

(ii) If the regional good is overprovided when mAK = 1, then it will also be overprovided when0 mAK < 1. The distortion is minimized when mAK = 1 and is largest when mAK = 0,i.e.,

MRS{mAK=0} < MRS{0 1, as shown by Proposition 5. Given that MRS{mAK=0} >MRS{mAK=1}, by Proposition 6, then MRS{mAK=0} > MRS{mAK=1} > 1. Thus, the departure from the optimal levelof the regional good is even greater when mAK = 1, whenever the good is underprovided. Moreover, by (34), MRS =A/(A + AA), so the inequality can only hold if |AA{mAK=0}| < |

AA{mAK=1}|.

56 So far, we have derived the results in terms of public good provision and efficiency according to the standard practicefollowed by many papers in the tax competition literature. Other papers, have focused on Leviathan-type tax competition(see, for example, Pethig and Wagener [22], Kolmar and Wagener [13], and Nielsen et al. [21]). In these models, regional

governments choose the tax rate that maximizes total tax collection taking as given the actions of other regions. In termsof our setup, the latter implies that region A, for instance, maximizes T A = tAA with respect to tA . However, in our


Proposition 7 can be used, for example, to evaluate the recent shift observed in the US to-wards a FA system that weighs more heavily production shares. In particular, we can study theeffects of this change on the provision of the regional good, and, consequently, on economic wel-fare. Suppose that, initially, all regions implement the uniform FA suggested by the MultistateTax Compact in 1967 for the US. In our model, a uniform FA equally weighs the property andthe output shares, i.e., mAK = 1/2. Hence, MRS{mAK=1} > MRS{mAK=1/2} > MRS{mAK=0}. If,initially, MRS{mAK=1/2} > 1, i.e., the good is underprovided when mAK = 1/2, then a small de-crease in mAK might alleviate the distortion (the degree of underprovision would be reduced). Onthe other hand, a small decrease in mAK from an initial situation characterized by overprovision,i.e., 1 > MRS{mAK=1/2}, would make the problem even worse. Note, in addition, that efficiencymay be attained with mAK = 1/2. In this case, a slight reduction in mAK would unambiguouslylead to overprovision of the regional good. In conclusion, a shift towards a formula that weighsmore heavily production shares may either ameliorate the extent to which the good is underpro-vided, or increase the degree of overprovision. As a result, this process does not necessarily leadsto a race to the bottom where the regional good ends up being underprovided. In fact, the gen-eralized shift in the formula can be viewed as an attempt by regional governments to implementhigher levels of local public goods, regardless of the initial situation.

4. Choosing an apportionment formula

Thus far, we have treated the FAs as exogenously given. In this section, we extend our previousanalysis and allow regional governments to determine the structure of their formulas. Specifi-cally, regional authorities are not only able to determine tax rates and tax bases, but they can alsochoose the formula used to apportion corporate profits. The formulas can assign full weight toproperty shares, production shares, or any linear combination of the two factors.57

As before, the firms total tax burden T is obtained by multiplying the effective tax rate ttimes taxable profits . In general, the effective tax rate t = tA A + tB B can be written as

t = (tAmAK tBmBK)(AK AF )+ (tA tB)AF + tB, (36)where we have used miK +miF = 1 for i = A,B , and Aq +Bq = 1 for q = K,F . The fact thatregional tax authorities can now choose their formulas means that they will be able to determinethe values of miK , i = A,B .

Continuing with our previous approach, first, we study the firms reaction to changes in theformulas; next, we set up the regional governments maximization problem; and finally, we de-rive the equilibrium.

model, the tax authority in A can choose both tA and A . Using the results obtained earlier, it straightforward to showthat similar implications as those derived in Proposition 6 apply here as well. In other words, when regional governmentsplay this Leviathan-type of game, regional governments will end choosing higher corporate tax rates and will allow lowercapital expenditure deductions when they employ a FA that exclusively weighs production (or sales) shares compared toany other formula. A similar result has been derived by Pethig and Wagener [22], but, as mentioned earlier, they constrainthe value of i to 0 from the beginning of their analysis. Pinto and Choi [23] show that if is assumed to be exogenouslygiven and < 1, then the good is underprovided. However, one important conclusion of the present paper is that when is decided in a decentralized way, local governments will end up choosing a value of greater than one. This last result

is consistent with underprovision, overprovision, or even efficient provision of the regional good.57 Even though, we ex-ante allow regions to choose different formulas, we will later focus on a symmetric equilibrium.


4.1. Firms profit maximization

In Section 3.1 we studied the impact of a change in t i and i on capital allocation for anexogenously given apportionment formula. Given that the choice of the particular formula isnow a decision variable, a comparative static analysis with respect to mAK and mBK resultsvery useful. Again, we will evaluate the impact of a change in the formula on capital allocationat a symmetric equilibrium. In addition, we focus on region A as the results are equivalent forregion B . Thus, differentiating (11) with respect to mAK and evaluating the expressions at asymmetric equilibrium, it follows that58

kA

mAK= t (

AK/kA AF /kA)(1 t)fkk < 0, (37)

kB

mAK= t (

AK/kA AF /kA)(1 t)fkk > 0. (38)

As the FA in region A weighs more capital shares (and less production shares), the firm will tendto decrease its capital allocation in A and increase it in B . In fact, these two effects are equal inabsolute value, so a change in mAK merely results in a redistribution of capital across regionswithout altering the total amount of capital, i.e., k/mAK = 0.

4.2. The governments problem

The regional authority in region A now selects the optimal values of tA, A, and mAK whichmaximize the utility of the representative consumer U(xA,gA), such that xA = AN , gA =tA[mAKAK + (1 mAK)AF ] , and 0 mAK 1. The FOCs with respect to tA and aresimilar to the ones derived before (Eqs. (27) and (28), respectively). Now consider the derivativeof U(xA,gA) with respect to mAK :

UA

mAK= UAx

xA

mAK+ UAg

gA

mAK. (39)

Evaluated at a symmetric equilibrium, N/mAK = 0, which implies that x/mAK = 0, so thefirst term on the right-hand side vanishes. This result holds because the decrease in kA originatedby an increase in mAK is offset by an equal increase in kB leaving the after tax profits unaltered.The second term depends on gA/mAK , which, at a symmetric equilibrium, becomes

gA

mAK= 2

[AF

kA+ mAK

(AK

kA

AF

kA

)]t

kA

mAK. (40)

First of all, we established in Section 3.1 that, due to the presence of decreasing returns to scale,(AK/kA AF /kA) > 0. Next, the previous section showed that kA/mAK < 0. As aresult, total taxes collected by the government in A decrease with mAK , i.e., gA/mAK < 0.Combining this last result with the fact that x/mAK = 0, allows us to conclude that utilitystrictly declines as mAK rises, i.e., UA/mAK < 0. Therefore, the tax authority will choosethe lowest possible value, which in this case is mAK = 0 defining a formula that only weighsproduction shares. The following proposition summarizes this finding.58 The expressions are derived in Appendix A.2.


Proposition 8. If regional governments are allowed to choose their own apportionment formula,then at a symmetric sub-game perfect Nash equilibrium they will choose one that fully weighsproduction shares, i.e., mAK = mBK = 0.

Proposition 8 claims that at a symmetric sub-game perfect Nash Equilibrium choosing a for-mula that only weighs production shares dominates all other alternatives. By reducing miK tozero, regional governments will not affect xi and they can unambiguously increase gi , raisingthe utility of the representative consumer as well. Thus, miK = 0 dominates all other alterna-tives. The rest of the analysis follows the work developed in Section 3.2, except that now we cansubstitute A for AF and B for BE .

The previous results differ from the conclusions in Anand and Sansing [1]. In a differentsetup, they suggest that regions have incentives to choose different apportionment formulas in anoncooperative solution even though social welfare is maximized when they all use the same one.This result holds because they assume that some inputs are immobile. Specifically, they considerthat regional output is obtained using an aggregate input which combines labor and capital. Thesupply of this input is upward-sloping.59 In our model, on the other hand, capital is the onlyvariable factor of production, and it is available in perfectly elastic supply. We believe that thisdistinction is crucial when considering apportionment formulas based on capital shares and/orproduction shares.60

The information presented in Table 1 shows a clear shift towards a formula apportionmentsystem that predominantly weighs sales or production shares in the US. Regional heterogeneitymay explain why we do not strictly observe miK = 0 for all regions i. However, in light of theimplications of our model, it seems that the issue of capital mobility has played a decisive role,more important than any other possible factor, at the moment of choosing a FA system.

5. Conclusions

States and countries that finance part of its activities with a corporate profit tax face the prob-lem of measuring corporations tax liabilities to each region. The use of a FA to allocate incomeacross locations introduces very complicated incentive effects for both the firms that operate indifferent jurisdictions and for local governments that shape the structure of the corporate taxsystem. The paper considers a tax competition model with respect to corporate profit taxes, taxbases, and FA systems, and analyzes the consequences, in terms of economic welfare, of adoptingdifferent alternatives.

Despite the complexity of the analysis, the model has been able to establish some unambigu-ous results. First, for every FA system, regional authorities will allow firms to deduct more thanthe optimal amount of capital expenditures from taxable income, i.e., they will tend to subsidizecapital through the corporate tax system. By following this strategy, the regional governmentcan attract more capital into their regions. Second, governments that use a FA system that only

59 Anand and Sansing [1] assume from the outset that the statutory corporate tax rates and tax bases are the same acrossstates. In terms of our notation, this is equivalent to t = tA = tB = t and = A = B = . In addition, they alsoexogenously assume that = 1 and that t < 0.5 so that no more than 100 percent of the firms net profits is taxed. Theiranalysis only focus on the choice of the formula apportionment given the other tax variables.60 Another important difference is that we assume that regional consumers receive a share of the firms total net profits,

so they are directly affected by the tax system implemented by the regions. Hence, regional governments will end upinternalizing part of the costs of their tax policy.


weighs production (or sales) shares choose higher corporate tax rates and allow a smaller portionof capital costs to be deducted from taxable income. This finding is partially explained by thefact that capital is more responsive to a change in the tax rate when the formula weighs moreheavily capital shares. Third, the strategic competition between regional governments leads to anoutcome that is consistent with both underprovision or overprovision of the regionally providedgood. The importance of these distortions depend on the specific formula put into practice. Fi-nally, if regional governments are allowed to design their own formulas, they will choose onethat fully weighs production (or sales) shares.

The paper has some interesting policy implications. For instance, the model can be use toevaluate the recent shift by most states in the US to a FA that gives more importance to production(or sales). According to the conclusions of our work, it is not clear that this process can beconsidered a race to the bottom where strategic competition leads to underprovision of theregionally provided good. In fact, a shift towards a formula that weighs more heavily productionshares is consistent with both underprovision, or even overprovision, of the good. It is possible toexplain the generalized shift in the formula as an attempt by regional governments to implementhigher levels of local public goods.

Acknowledgments

I gratefully acknowledge financial support from the Regional Research Institute, WVU, andfrom the College Research and Library Committee, WVU. I am grateful to Kangoh Lee, Jan K.Brueckner, participants of the conferences organized by the Southern Regional Science Associ-ation and the North American Regional Science Association, and two anonymous referees fortheir helpful comments. Of course, the usual disclaimer applies.

Appendix A

A.1. Proof of Proposition 1

The maximization of W = UA + UB with respect to t and , subject to t = gA + gB givesthe following first-order conditions:

t : UAx(xA

/t)+ UBx (xB/t)+ [ + t (/t)]= 0, (A.1)

: UAx(xA

/

)+ UBx (xB/)+ t(/) = 0, (A.2)where is the Lagrange multiplier associated with the budget constraint. From the first-orderconditions with respect to gA and gB we obtain UAg = UBg = . Additionally, given that xA = xB ,the latter also implies that in equilibrium total tax revenue should be equally divided betweenregions, i.e., gA = gB . As a result, UAx = UBx . Rearranging (A.1) and (A.2), we can write

MRSi = xA/t + xB/t

+ t (/t) =xA/ + xB/

t(/), (A.3)

for i = A,B , where MRSi = Uig/Uix . The equilibrium combination {t,} is achieved when themarginal benefit of increasing gi , represented by MRSi , is equalized to the marginal cost of

changing t in terms of units of x that must be given up, and the latter is, in turn, equalized to themarginal cost of changing . Using expressions (5) and (6), it is straightforward to obtain


xi/t = i < 0, xi/ = i trk > 0,/t = 2(1 t)fkk

(ki/t

)2< 0, / = rk(1 + ii),

where ii = (t/ki)(ki/t) is the elasticity of capital with respect to t evaluated at the optimalvalues of the choice variables. We can then rewrite (A.3) as:

MRSi = + t (/t) =

11 + ii . (A.4)

It is important to remark that at the equilibrium > t (/t) (otherwise (A.1) would neverhold), and 1 > |ii | (otherwise (A.2) would never hold). Substituting the corresponding expres-sions into (A.4) and simplifying we obtain

(1 )(1 t) = (1 )2kr. (A.5)First of all, notice that = 1 satisfies (A.5). Next, notice that cannot be greater than one,otherwise (A.5) would not be satisfied. Finally, suppose that 0 < 1. Then, Eq. (A.5) becomes(1 t) = kr(1 ), which implies that (1 )kr = t . From the definition of taxableincome (2), we can write (1)kr = . Using the previous result, we obtain that t = ,which implies that N = t = 0 and xi = iN = 0. As a result, Uix tends to infinity and (A.2)would not be satisfied. Hence, 0 < 1 is not a solution.

The last part of the proposition concerns the efficient provision of regional goods. The opti-mal solution entails maximizing UA(xA,gA) + UB(xB,gB) subject to xA + xB + gA + gB =f (kA) + f (kB) rk by choosing {xA,xB,gA,gB, kA, kB}. From the first-order conditions ofthis problem we obtain that MRSi = 1 and f ik = r , for i = A,B .

A.2. Section 3.1: firms profit maximization. Second-order conditions and comparative staticresults

Let F i N/ki and F ikj

F i/kj , for i, j = A,B . The second-order condition for a localmaximum requires that F i

ki< 0, for i = A,B , and |H | = FA

kAFB

kB (FA

kB)2 > 0. At a symmetric

equilibrium tA = tB = t , A = B = ,mAK = mBK , it follows thatF i

ki= (1 t)fkk < 0, |H | =

[(1 t)fkk

]2> 0

given that fkk < 0 and F ikj = 0, i, j = A,B , i = j .The comparative static results with respect to = tA, tB,A,B can be found by differenti-

ating the system of equations F i = 0, i = A,B , with respect to , which yields(FA

kAFA

kB

FBkA

FBkB

)(dkA

dkB

)=

(FA d

FB d

). (A.6)

Applying Cramers rule to (A.6) giveskA

= 1|H |

[FA

kBFB FA FBkB

],

kB

= 1|H |

[FB

kAFA FAkAFB

]. (A.7)

Evaluated at a symmetric equilibrium, the following results hold:

F iki

= Fjkj

, F ikj

= 0, |H | = FAkA

FBkB

,[i

] [j

]

F i

ti= i (1 )r

(1 t) +

ki , F

j

ti= i (1 )r

(1 t) +

ki ,


F ii

= tr, F ji

= 0,

F imiK

= (

iK

ki

iF

ki

)t, F

j

miK=(

iK

ki

iF

ki

)t,

for i, j = A,B , i = j , and = {t i ,i,miK }. Substituting into (A.7), we obtain the correspond-ing comparative static results evaluated at a symmetric equilibrium. In Eq. (20), we use theequality j /ki = ( j /kj ) = ( i/ki), which holds at a symmetric equilibrium.


Most of the results follow directly from (19), (20), and (21). However, to prove that kB/tAis positive for all values of and FA methods requires some additional work. Essentially, thislast result holds because the expression within square brackets in (20) is always negative, i.e.,

A(1 )r(1 t) +

B

kA < 0. (A.8)

In order to show that (A.8) holds, we will be using the fact that B/kA = B/kB = A/kA at a symmetric equilibrium. First, if = 1, then (A.8) holds because B/kB > 0and > 0. Second, if > 1, then both terms on the left-hand side of (A.8) are negative, so theinequality holds. Finally, if < 1, then the first term on the left-hand side of (A.8) is positiveand the second one is negative. It can be shown, though, that the second term always dominatesthe first one. To prove this last result, we rewrite the expression on the left-hand side of (A.8) as

A

k

[2f

(fkk

2f

) (1 )rk

]. (A.9)

In this last step, we used the definition of (given in (8)), the equality (1 )rk/(1 t) =fkk rk, which follows from the first-order conditions (11) (evaluated at a symmetric equilib-rium), and

= k A

A

kA= k

A

B

kB.

Substituting (24) into , it follows that

= mAK + (1 mAK)fkk2f

.

Note that as fkk/2f < 1 due to decreasing returns to scale, then 0 < 1 for all 0mAK 1.Suppose that mAK = 0, so that = fkk/2f . Then, the first term between the square bracketsin (A.9) becomes zero, and the second is negative. Thus, (A.8) holds. Additionally, (A.8) is alsosatisfied when 0 < mAK 1 because > fkk/2f for all these cases.


Proposition 2 establishes that BA > 0 for all values of . The same proposition also ascertainsthat AA < 0 when 1, but AA may be positive or negative when > 1. Considering theseresults, it can be shown that 1 is not a solution to (35). Suppose, on the contrary, that 1

is a solution to (35). Then, the left-hand side of (35) is less than or equal to zero. The right-handside is, however, positive because AA < 0. Therefore, 1 cannot be a solution.



LetUA

tA

(,mAK

)and

UA

A

(,mAK

) (A.10)denote the derivative of UA with respect to tA and A evaluated at = {t,}, for mAK ={m0,m1}, with m1 > m0.61 The strategy of the proof consists of comparing the derivatives(UA/tA)(1,m1) and (UA/A)(1,m1) to (UA/tA)(1,m0) and (UA/A)(1,m0),where 1 = {t1,1} are the equilibrium values of t and (at a symmetric equilibrium) whenmAK = m1.62

First, when mAK = m1, the equilibrium levels of 1 = {t1,1} are implicitly defined by thefirst-order conditions

UA

tA(1,m1) UAx (1,m1)

xA

tA(1,m1)

+ UAg (1,m1)gA

tA(1,m1) = 0, (A.11)

UA

A(1,m1) UAx (1,m1)

xA

A(1,m1)

+ UAg (1,m1)gA

A(1,m1) = 0. (A.12)

1 determines kA and kB according to Eq. (11), and, consequently xA, xB , gA, and gB . Second,similar expressions can be obtained for mAK = m0, and the equilibrium values would be 0 ={t0,0}. Thus, (UA/tA)(0,m0) and (UA/A)(0,m0) are necessarily equal to zero.

The next step is to compare (UA/tA)(1,m0) and (UA/A)(1,m0) to (UA/tA)(1,m1) and (UA/A)(1,m1), where these last two expressions are respectively determined by(A.11) and (A.12). Given that they are all evaluated at 1,63 the following results hold:

(i) UAx (1,m1) = UAx (1,m0), UAg (1,m1) = UAg (1,m0);

(ii) xA

tA(1,m1) = x

A

tA(1,m0) < 0,

xA

A(1,m1) = x

A

A(1,m0) > 0;

(iii) gA

A(1,m1) < g

A

A(1,m0), 0 0,given that (1,m0) < (1,m1).

Substituting (i), (ii), and (iii) into (UA/tA)(1,m0) and (UA/tA)(1,m1), we obtainUA

tA(1,m0) > 0, and

UA

A(1,m0) < 0,

which means 1 does not constitute an equilibrium when mAK = m0. In fact, the equilibriumvalue of tA will be higher (utility increases with tA when mAK = m0 if the expression is eval-uated at 1) and the equilibrium value of A will be lower (utility declines with A) than thecorresponding values of tA and A when mAK = m1. Suppose that m0 = 0. Then, the valueof A will be lowest, and the tax rates will be highest when the formula only weighs productionshares.

In addition, given that a higher tA and a lower A imply lower levels of xA and higher levelsof gA, then the marginal utility of x will increase and the marginal utility of g will decreaseresulting in a lower MRS. Hence, a FA system that relies exclusively on production shares willresult in a lower MRS relative to all other apportionment methods.

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