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econstor Make Your Pubication Visibe A Service of Wirtschaft Centre zbwleibniz-informationszentrum Economics Hoyt, Wiiam H. Working Paper The Assignment and Division of the Tax Base in a System of Hierarchica Governments CESifo Working Paper, No. 5801 Provided in Cooperation with: Ifo Institute Leibniz Institute for Economic Research at the University of Munich Suggested Citation: Hoyt, Wiiam H. 2016 : The Assignment and Division of the Tax Base in a System of Hierarchica Governments, CESifo Working Paper, No. 5801 This Version is avaiabe at: http://hd.hande.net/10419/130443 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentiche oder kommerziee Zwecke verviefätigen, öffentich aussteen, öffentich zugängich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen insbesondere CC-Lizenzen zur Verfügung gestet haben soten, geten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your persona and schoary purposes. You are not to copy documents for pubic or commercia purposes, to exhibit the documents pubicy, to make them pubicy avaiabe on the internet, or to distribute or otherwise use the documents in pubic. If the documents have been made avaiabe under an Open Content Licence especiay Creative Commons Licences, you may exercise further usage rights as specified in the indicated icence. www.econstor.eu

The Assignment and Division of the Tax Base in a System of Hierarchica Governments Wiiam H. Hoyt CESIFO WORKING PAPER NO. 5801 CATEGORY 1: PUBLIC FINANCE MARCH 2016 An eectronic version of the paper may be downoaded from the SSRN website: www.ssrn.com from the RePEc website: www.repec.org from the CESifo website: Twww.CESifo-group.org/wpT ISSN 2364-1428

CESifo Working Paper No. 5801 The Assignment and Division of the Tax Base in a System of Hierarchica Governments Abstract Vertica externaities, changes in one eve of government s poicies that affect the budget of another eve of government, may ead to non-optima government poicies. These externaities are associated with tax bases that are shared or co-occupied by two eves of government. Here I consider whether the co-occupancy of tax bases is desirabe. I examine the optima extent of the tax bases of a ower eve of government oca and a higher eve state. I find that it is optima to have co-occupancy in the absence of other corrective poicies if commodities in tax base are substitutes. Further, if the state government can differentiay tax the co-occupied segment of the tax base and the segment it aone taxes it wi obtain the second-best outcome obtained with other poicy instruments such as intergovernmenta grants. Finay, if there are horizonta externaities generated by cross-border shopping, there is sti reason to co-occupy the tax base if commodities are substitutes. As we, oca governments shoud have those commodities with the owest cross-border shopping costs in their tax base. JEL-Codes: H200, H710, H730, R120, R280, R410. Keywords: fisca competition, vertica externaities, tax base co-occupancy. Wiiam H. Hoyt Department of Economics Gatton Coege of Business and Economics University of Kentucky USA Lexington, KY 40506 whoyt@uky.edu This Version: February 2016 Thanks to David Agrawa, Robert Inman, participants at the 2015 Nationa Tax Association meetings, and the University of Kentucky Department of Economics workshop.

1 Introduction Whie the concept of a horizonta fisca externaity arising from tax competition among governments at the same eve has been the topic of numerous papers in for more than thirty years, the focus on vertica fisca externaities received ater attention. Among the eary theoretica contributions were Johnson 1988 and Fowers 1988 and continued with Dahby 1996, 2008, Boadway and Keen 1996, Boadway, Marchand and Vigneaut 1998. More recent contributions incude Boadway, Cuff and Marchand 2003, Keen 1998, Keen and Kotsogiannis 2002, 2003, 2004, Hoyt 2001, Dahby and Wison 2003, Wrede 1996, 2000, Keen and Kotsogiannis 2002, and Wison and Janeba 2005 among others. As the name vertica impies, these externaities arise between governments at different eves, for exampe, between state and oca governments or federa and state governments. In this case the focus is on the overap in the tax bases of two eves of government. An exampe from Dahby 1996 is the excise tax paced on cigarettes by both the federa and state governments in the United States. When choosing its tax rate, each state presumaby ony considers the tax s impact on its own revenues and ignores the impact on the revenues of other states and the federa government. As a resut of an increase in the state s tax rate, other states tax revenues wi increase because of cross-border shopping a horizonta externaity and federa tax revenues wi be reduced because of the reduction in the cigarettes purchases, part of their tax base a vertica externaity. Because of these impacts on the revenues of other governments, the cost of funds perceived by the state differs from the socia cost of the funds. Whie the horizonta fisca externaity is positive, the vertica externaity is negative as increases in the state tax reduce federa revenues. Because the state government ignores this negative externaity, it wi overtax cigarettes. A number of studies have considered poicies by the higher eve of government to correct for the vertica externaities created by taxes imposed by the ower eve of government. Corrective poicies incude separating the tax bases of the two eves of government Fowers 1988; increasing the number of ower-eve governments Keen 1995; Keen and Kotsogiannis 2004; and providing intergovernmenta grants Dahby 1996; Boadway and Keen 1996; Boadway, Marchand and Vigneaut 1998; and Foche and Madies 2002. Whie eary studies focused on vertica externaities in a singe market, research has extended to consider the impacts of co-occupancy in a mutipe market incuding studies by Dahby 1996, Keen 1998, Hoyt 2001, Dahby, Mintz and Wison 2000, and Dahby 2001, and Dahby and Wison 2003. Whie I aso examine tax poicies in a hierarchica system of governments, I depart from previous studies in a severa respects. First I consider vertica externaities with mutipe tax bases. Specificay, I consider a arge number a 2

continuum of commodities to be incuded in either or both of eves of governments tax bases. The consideration of mutipe commodities enabes us to address the question of centra interest to this paper how shoud the tax base be aocated between the two eves of government? Vertica fisca externaities act in both directions state taxes affect oca revenues and oca taxes affect state revenues. Fowers 1988, Wrede 1996, and Keen and Kotsogiannis 2002, for exampe, assume that both eves of government ignore the vertica externaity imposed on the other eve of government when setting tax poicies. This eads to excessive taxation at both eves of government. Here, I consider both the case in which the state government considers the impact of its tax poicies on oca revenues as we as the case in which it does not. Figure 1 iustrates the fisca externaity imposed by an increase of a oca tax on state government revenues. In addition to having mutipe tax bases, different eves of governments rey on very different sources of revenue. As Tabe 1 suggestions, whie there is ony imited overap or co-occupancy in sources of revenue of state and oca governments in the United States, for exampe, there is ikey to be a strong ink between their aternative tax bases. Changes in a major source of state revenue such as the persona income tax wi undoubtedy affect revenues from the property tax, a major oca source of revenue. In contrast, there is much more apparent co-occupancy of the federa and state tax bases primariy because the persona income tax and, to a esser extent, the corporate income tax are major sources of tax revenue for both eves of government. Thus, whie vertica fisca externaities wi amost certainy arise in a co-occupied tax base, it does not foow that eiminating co-occupancy eiminates fisca externaities, an idea that may underie the recommendation by some to eiminate cooccupancy. In fact, eiminating co-occupancy may change the fisca externaity from being negative to being positive if the commodities in the two tax bases are substitutes. This, in turn, woud suggest under-taxation rather than over-taxation in the framework deveoped here. The issue addressed here, what eve of government shoud tax what goods or services or inputs is referred to in the federaism iterature as the assignment probem. In a surprisingy sma iterature, the best known discussion of the appropriate assignment of the tax base in a system of hierarchica governments may be found in Musgrave 1983 with nice summaries in Musgrave and Musgrave 1989, Oates 1994, Keen 1998 and Dahby 2001. Whie Musgrave provided some genera guideines for assigning tax bases based on the easticity of aternative tax bases, he does not discuss how vertica fisca externaities might affect assignment. Keen 1998 does devote some discussion and anaysis to cooccupancy and assignment by addressing the question of whether it is better to co-occupy 3

an ineastic tax base or a more eastic tax base. Dahby 2001 raises severa concerns with Musgrave s rues for assignment incuding the issue of co-occupancy. Whie not presenting any forma mode, Dahby 2001, by highighting the genera interdependency of tax bases, raises questions simiar to those I address here. In very different contexts both Haufer and Lufesmann 2015 and Kotsogiannis and Raimondos 2015 consider optima cooccupancy. However, in both cases, the co-occupancy corrects for horizonta externaities: in the case of Haufer and Lufesmann 2015 these are associated with capita taxation by asymmetric countries and in the case of Kotsogiannis and Raimondos 2015 countries evy taxes to change the terms of trade. An extensive iterature on the possibe efficacy of co-occupancy has addressed issues reated to horizonta fisca externaities and cross-jurisdictiona trade distortions as a resut of ower-eve government tax poicies. Perhaps the most extensive discussion of the tax assignment issue has been reated to the VAT taxation and the issue of source versus destination taxation. Contributions to this iterature incude Bird 2000; Bird and Gendron 1998, 2000; Keen 2000 and Keen and Smith 2000. Recent reated contributions to notion of higher-eve government taxation correcting for inefficiencies associated with ower-eve government poicies incude Haufer and Lufesmann 2015 and Kotsogiannis and Raimondos 2015. Here I address the assignment question using a very different framework from those in either Musgrave 1983 or Keen 1998 but simiar in many ways to the framework impicit in Dahby 2001. Rather than considering the type of tax base that shoud be taxed by different eves of government, I first consider how to divide a uniform tax base among two eves of government and whether co-occupancy is desirabe or not. Whie this very simpe framework means that I ignore many of the issues associated with tax assignment such as geography, benefit taxation, and cross-border shopping discussed in, for exampe, Bird 2000 and McLure 2001 it aows for focus on the question of whether the existence of vertica fisca externaities might, as suggested by Fowers 1988 and Dahby 2001 among others, ead to the concusion that there shoud be no or very imited co-occupancy among tax bases. Even when the eimination of co-occupancy may be optima, it does not, in genera, eiminate vertica fisca externaities. As a consequence, even in the absence of co-occupancy the tax rates of the two eves of government wi not be optimay set. If the commodities in the tax base are gross substitutes, eiminating co-occupancy resuts in a positive fisca externaity, meaning that tax rates wi become too ow. Here, because both the tax rates and tax bases of governments are poicy instruments, I need to distinguish between fisca externaities associated with changes in a government s tax rate and one associated with changes in its tax base. Whie the division of the tax base obviousy infuences the vertica externaities 4

associated with the tax rates, the extent and direction of the fisca externaities associated with tax increases and those associated with increases in tax bases can be quite different. In fact, I find that in the case in which commodities are gross substitutes, co-occupancy, at east to some extent, is optima; when commodities are gross compements, it is unikey that co-occupancy is optima. The basic mode, found in Section 2, is of a continuum of commodities with identica demands aong the ines, for exampe, of Dixit and Stigitz 1977, Yitzhaki 1979, and Wison 1989. In this section I motivate the optima poicies with regard to both the tax rate and base by considering the poicy undertaken by a singe, centra government that finances two pubic services with separate taxes. I then consider the aternative extreme the choices of tax rates and tax bases made by the two eves of government state and oca when they choose them independenty. As I assume that commodities are identica both governments wi set uniform tax rates on their respective bases. 1 In Section 3 the optima tax bases from their perspective for the two eves of government is considered. In this section I first consider the question of how to divide the tax base between the two eves of government in the absence of any overap. I then consider whether and under what conditions, woud co-occupancy be sociay optima. As we, I aow for the possibiity that the state government can set different tax rates on the base that it aone taxes and the base that it shares co-occupies with the oca government. Finay, in Section 4 I extend the mode to generate a horizonta externaity. In this case, aong the ines of Agrawa 2012 and Niesen 2001, for exampe, I aow for cross-border shopping. In addition to generating a horizonta externaity, as I aow for differences in the costs of cross-border shopping among commodities, I can aso address the question of what commodities might be incuded in the state and oca tax bases rather than simpy the extent of the two bases and whether co-occupancy is optima. Section 5 concudes. 2 Tax Choices with Independent Governments 2.1 A Simpe Mode I consider an economy with a singe state government and n oca governments with each ocaity having a singe, identica resident. Each government provides a pubic service to its residents with g s being the eve provided by the state government and g j, j 1,..., n the 1 Further assumptions regarding the easticity of the demands of the products are necessary for this resut as wi be seen ater. Aternativey, that there are uniform tax rates across commodities can aso be considered an assumption refecting most state and oca saes tax in the United States and VAT systems esewhere as discussed in?. 5

eve provided by ocaity j. Both pubic services are produced with constant costs with the cost of providing g s to the n ocaities equa to ng s and the cost of providing the oca pubic service in ocaity j equa to g j. Whie there are n independent ocaities, as each oca government has the same poicy objectives, in equiibrium a ocaities choose the same poicies. Then given this symmetry, I denote oca poicies by the subscript. To further simpify the anaysis, I aso assume that the number of ocaities is arge enough so that no individua ocaity considers the impacts its poicies have on state revenues. In addition to the pubic services, residents aso consume private commodities. Foowing Dixit and Stigitz 1977, Yitzhaki 1979, and Wison 1989, I consider a continuum of these private commodities identified on the interva 0,K. Whie the interva of commodities is 0,K ony the interva 0,1 is subject to taxation by either the state or oca governments. 2 As my interest is in how to aocate the tax base between the two eves of government, ike Dixit and Stigitz 1977 I assume identica demand functions over the set of commodities. By this I mean that when the prices of two commodities are identica, the quantity demanded is the same for both. The utiity function can be represented by U ˆK 0 U x xqk, Q dk + U g + U s g s 2.1 The gross of tax price of commodity k, xk, is denoted by qk with the net of tax prices for a K commodities equa to unity. The term Q qkdk is an index of a commodity prices. 0 Unike Dixit and Stigitz 1977, I assume a reativey genera form of the utiity function. As the demand function for commodities are identica then when the price of commodity i and commodity j are the same, their demands are the same. I denote the derivatives of the demand equations by x 11 xk and x qk 21 xk Q and, what wi be used when characterizing tax rates, the percentage change in demands, x 11 xk 1 and x qk xk 21 xk 1 that I treat Q xk as constant in the anaysis. For my purposes, an important impication of having identica commodities is that the optima tax structure is extremey simpe a commodities shoud be taxed equay. As both the oca and state governments assess uniform commodity taxes to finance their pubic services, the gross price of each commodity depends on whether it is part of the tax base for the oca, state, or both governments. Locaities tax the set of commodities on the interva 0, k whie the set taxed by the state government is on the interva ks, 1. Let k, k s, and k s denote, respectivey, the ength of interva taxed ony be the oca government, 2 Whie we may think of the commodity space as a straight ine or circe, ike Dixit and Stigitz 1977 the distance between any two commodities has no bearing on the reationship between them, that is, the degree to which they are substitutes or compements. This is seen in the formuation of the price index. 6

ony by the state government, and by both eves of government. Then the gross of tax price for the commodities can be summarized by qk 1 + τ, 1 + τ s, 1 + τ + τ s, k 0, min k, k s k max k, k s, 1 2.2 k k s, k if, k > k s where τ and τ s denote the oca and state tax rates respectivey. 3 Figure 2 iustrates the division of the tax base with no overap no co-occupancy and with overap co-occupancy. Then the indirect utiity function can be expressed as 4 V ˆ K τ, τ s, k s, k V qk, Q dk + U s g s τ, τ k, k s, k + U g τ, τ k, k s, k. 2.3 0 2.1.1 Government Objectives The objective function for oca governments is given by W ˆ K τ, τ s, k, k s V qk, Qdk + U g τ, τ s, k, k s 0 2.4 where the eve of the oca pubic service is impicity defined by the state and oca tax rates and bases, g τ, τ s, k, k s. As oca governments are assumed to ignore the impact of their poicies on state revenues, state pubic services are not incuded as an argument in the oca government s wefare functions. 3 With these taxes the price index is Q ks 1 + τ 0 dk + k k s 1 + τ + τ s dk + 1 k 1 + τ s dk + K 1 dk K + k τ + k s τ s + k s τ + τ s k 0 1 + τ dk + k s k dk + 1 k s 1 + τ s dk + K 1 dk K + k τ + k s τ s. k > k s k < k s. ˆ 1 0 4 where the term V qk, Q dk k 0 U x xqk 1 + τ, Q dk + k s k U x xqk 1, Q dk + 1 k s U x xqk 1 + τ s, Q dk + K ks 0 U x xqk 1 + τ, Q dk + k s k + 1 k s U x xqk 1 + τ s, Q dk + K 1 U x xqk 1, Q dk, k < k s U x xqk 1 + τ + τ s, Q dk 1 U x xqk 1, Q dk, k > k s 7

For the state government consider the objective, W ˆ K s τ, τ s, k, k s V qk, Qdk + U s g s τ, τ s, k s, k + αu g τ, τ s, k s, k 2.5 0 where α 0, 1. The parameter α denotes the extent that the state government considers the impacts of its poicies on the oca government revenues. If α 1 the state government fuy considers the impacts of its poicies on oca revenues and the wefare of its residents; at the other extreme α 0 the state ignores the impacts of its poicies on oca revenues and pubic services. My interest wi be focused on these two extreme cases. 2.1.2 Government Budget Constraints The state budget constraint is given by ng s nτ s k s x s + k s x s nτ s X s and the oca budget constraint is given by g τ k x + k s x s τ X where x s, x, and x s denote the demand for commodities subject to the state tax ony, to the oca tax ony, and to both taxes, respectivey. Critica to understanding the tax rates chosen by the two eves of government and the optima tax bases for them is understanding the impacts of changes in their tax rates and bases on their revenues. These impacts, summarized beow, are derived in Appendix A.1. dg j dτ j dg i X j 1 + τ j x 11 + k j + k s x 21 and k dτ j τ i X s x s i k s +k i x i x 11 + k j + k s x 21, i, j, s; i j 2.6 where x i k sx s +k i x i, i, s. Note that the sign and magnitude of the fisca externaities, k s +k i dg i dτ j, i, j, s; i j depend on both the overap in the tax based k s and cross-price effects x 21. Whie the focus of the iterature on fisca externaities has been on tax rates, I am aso interested in the impacts of changes in tax bases of the two eves of government. Then the impacts of increases in the tax base on revenues are dg j x τ dk j x z j j x j + τ k s + k j x 21 and dg i dk j τ j τ i x i D x i x j x 11 + k s + k i x 21, i, j, s; i j, 2.7 where D 0 1 and z js if k < > k s, j, s. 5 As is the case with tax rates, changes in the tax bases of the two eves of government affect tax revenues of the other eve of government. The impact of the expansion of the tax base by one eve of government on the other eves tax revenue depends on whether the two tax bases overap and the cross-price 5 An increase in the state tax base means a decrease in k s so rather than dgs dk s and dg. dk s and dg dk s 2.7 is for dgs dk s 8

easticities impacts among commodities, reative to their own price easticities when the two tax bases overap, increases in the oca state tax base wi aways decrease the state oca tax base. 2.2 Centraized Soution Before considering the tax poicies chosen when governments are making independent poicy decisions, I first consider the choices of a singe government providing both services. Whie the obvious poicy woud be to impose a uniform rate on the entire tax base to finance both services, to provide a contrast to the poicy choices of independent governments, consider the choices when the singe government finances the two services, g and g s, by two separate taxes, τ and τ s on two distinct, though possiby overapping tax bases. The government s probem, then, is Max τ, τ s, k, k s W τ, τ s, k, k s ˆ 1 0 V qk, Qdk + U s g s τ, τ s, k s, k + U g τ, τ s, k s, k The first order conditions with respect to the state tax rate can be expressed as 2.8 W τ MRS s 1 + MRS s τ s x 11 + k s + k s x 21 X +MRS τ k s x s 0 2.9 X s k +k s x x 11 + k s + k s x 21 and with respect to the oca tax rate by W τs MRS 1 + MRS τ x 11 + k + k s x 21 X +MRS s τ s k s x s 0 2.10 s X k s+k s x s x 11 + k + k s x 21 V g where MRS j j V, j, s is the margina rate of substitution between pubic service j y and private consumption. 6 The derivations of 2.9 and 2.10 as we as those for other first order conditions and proofs of propositions in the remainder of the section are found in Appendix A.2. Rather than stating the first order conditions for the tax bases, two characteristics of the soution shoud be evident: the margina rates of substitution of the two goods shoud be equa MRS MRS s and the tax rate on a commodities shoud be the same. Then given ony two distinct tax rates, to have equa rates on a commodities, either both eves of government tax the entire base k s 1 or there is no overap k s 0. When commodities are either gross substitutes or compements ˆx 21 0 it is not possibe 6 I suppress V y from a expressions of first order conditions as it has no bearing on the anaysis. 9

to tax a commodities with no overap k s 1 k, have equa tax rates τ τ s, and equa margina rates of substitution, MRS MRS s. Note that when the two tax bases competey overap, the taxes sti have negative fisca externaities, that is, increases in one of the tax rates wi reduce tax revenues from the other tax. Proposition 1. When a centra government finances the oca pubic service g from revenues of the oca tax τ on the oca tax base X and finances the state pubic service g s from revenues of the state tax τ s on the state tax base X s the optima poicy is to appy both taxes to the entire tax base k 1 and k s 0. With this poicy, the margina rate of substitution is equa for both services MRS MRS s and a commodities are taxed equay. Again, the resut is not surprising given that the centra government internaizes the cross-tax impacts on revenues. However, it does suggest that distinct taxes and overapping tax bases are not necessariy a concern. 2.3 Fisca Externaities and Tax Rates As discussed, the co-occupancy of tax bases generates negative fisca externaities and, as a resut, may ead to over-provision of pubic services when the externaities are not internaized in the poicy decisions of the government. Less noted in the iterature are the fisca externaities associated with taxation of bases that are not co-occupied. If the two tax bases are substitutes then it is a positive externaity; if compements, a negative fisca externaity. I begin, then, by iustrating the nature of the fisca externaities associated with taxes in the context of this mode. First I focus on the tax rates each government chooses when facing a restricted tax base. Next I consider the possibiity that governments can choose both their tax rates and bases. 2.3.1 Loca Tax Poicy Each ocaity maximizes its residents wefare by choosing its tax rate given the tax rates and bases of the other ocaities and the state government. Its choice of tax rate aso depends on the extent of its tax base. Formay the oca government s probem is to choose τ to sove 2.4. Then in Nash equiibrium, the first order condition for the oca tax rate τ can be expressed as W τ 1 + MRS 1 + τ x 11 + k + k s x 21 0 or MRS 1 1+τ x 11+k +k s x 21 1 D L 2.11 10

where τ is the equiibrium oca tax rate. The margina externa cost MEC τ associate with the tax of a singe ocaity is The MEC τ MEC τ 1 n τ smrs s X s ks k s + k s xs x 11 + k s + k x 21 x s 2.12 is simpy the wefare vaue associated with an increase or decrease in eve of the state pubic good. From 2.12 it is apparent that MEC τ is negative positive if k s xs > < x 21 2.13 k s + k s k s + k x s x 11 If x 21 < 0, that is, the goods in the tax base are compements, MEC τ if goods in the tax base are substitutes x 21 > 0 then the sign of MEC τ wi be negative; depends on the extent of the overap in the two tax bases and the ratio of the cross-price and own-price impacts on demand, x 21 x 11. 2.3.2 State Tax Poicy In the Nash equiibrium the first order condition for the state tax rate τ s can be expressed as W s τ s +αmrs τ 1 + MRS s 1 + τs x 11 + k + k s x 21 X k s X s k s +k MRS s 1 αmrs τ x s x x 11 + k s + k s x 21 X k s x s x Xs k s +k x 11 +k s+k s x 21 D S 0 or 2.14 where τs is the equiibrium state tax rate and D s 1 + τs x 11 + k s + k s x 21. As seen in 2.14 if α > 0 the state considers the effects of its tax rate on oca revenues. The impact of the state tax on the oca tax base is anaogous to that of the oca government. However, as the state government may internaize some of the impact of its tax rate on oca revenues, the extent, though not the sign, of the externa cost associated with an increase in the state tax depends on the vaue of α with MEC τs 1 α MRS τ X These resuts are summarized beow: Proposition 2. a If α 1 MEC j > < 0 if k s k s +k j ks k s + k xs ˆx 11 + k s + k s ˆx 21. 2.15 x x s x j ˆx 11 > < ˆx 21, j, i, s, i j; 11

b If the state maximizes socia wefare α 1 MEC s 0 c If α 0 then MRS > 1 and MRS s > 1 and MRS s > < MRS if τ s ˆx 11 + k s + k s ˆx 21 < >τ ˆx 11 + k + k s ˆx 21. d If α 0 : i MRS s > < MRS if ii MRS > 1 and if { τ X ˆx 11 + k + k s ˆx 21 α X X s k s s X ˆx 11 + k s + k s ˆx 21 > <τ s ˆx 11 + k s + k s ˆx 21. k s k s +k x s x ˆx 11 + k s + k s ˆx 21 < 0 then MRS s > 1. Part c of the proposition simpy states that when α 0 whichever base has the greatest absoute vaue percentage oss in its tax base from a margina increase in its tax rate wi have the greater MRS. As seen in part d, when α 0 the reationship between the MRS and the changes in tax bases is more compicated as the state considers how changes in its tax rate affect the oca base. If there is a positive negative fisca externaity, then it is possibe for MRS s < > MRS even if τ ˆx 11 + k + k s ˆx 21 < > τ s ˆx 11 + k s + k s ˆx 21. Whie the MRS > 1 for any tax base for the oca government, whether MRS s is greater or ess than one depends on two impacts the magnitude of the decrease in the state tax base due to the tax increase ˆx 11 + k s + k s ˆx 21 X s and the impact on the oca tax base X sˆx 11 + k s + k s X ˆx 21. If the increase in the state tax reduce the oca tax base then it must be the case that MRS s > 1. } 2.4 Fisca Externaities and the Choice of Tax Base In the United States, the choice of tax base, that is what oca governments can tax, is generay at the discretion of state, not oca, governments. Whie this may be the case, it is sti usefu to examine what tax base oca governments woud choose if given the option. Here I begin by considering the probem facing oca and state governments when they can choose both their tax rate and base. 2.4.1 Loca Tax Base Each ocaity maximizes its residents wefare by choosing both its tax rate and tax base given the tax rates and bases of the other ocaities and the state government. As discussed, the optima tax rate is given by 2.11. In the Nash equiibrium the oca wefare-maximizing tax base k satisfies the first order condition, 1 k W 1 k k τ MRS 1 x z + τ MRS X ˆx 21 0, 2.16 where z s, if k > < k s. As MRS 1 > 0 at the optima tax rate, as is evident from 2.16 if ˆx 21 > 0 it must be the case that 2.16 is ony satisfied when k 1 the oca 12

government chooses to tax the entire base. As shown in the Appendix, if ˆx 21 < 0 it is sti optima to tax the entire base. 2.4.2 State Tax Base The optima tax base for the state is characterized by the first order condition, k sw s k s k sτ s MRS s 1 x z + MRS s τs X sˆx 21 +αmrs τ Dˆx 11 + X ˆx 21 0, 2.17 where z s s and D 10 if k > < k s. Using the first order condition for the state tax rate 2.14 we can express 2.17 as k sw s k s k sτ s x s MRS s 1 x z x s 1 MRS s τs ˆx 11 + αmrs τ D k s k s+k s x s x s ˆx 11 0 2.18 If the state ignores any impacts expansion of its tax base has on oca revenues α 0 then, as is the case with oca governments, the state wi choose to tax the entire base k s 0. Less obvious is the case when α 1 and the state fuy considers the impact on oca revenues when choosing of both its tax rate and tax base. The state faces a trade off when expanding its tax base it owers the margina cost of funds associated with any state tax rate but aso reduces oca revenues. Intuitivey, the gain in socia wefare of an increase in a tax base, absent other distorting taxes, is equa to MRSτ ˆx 11. That the impact in 2.18 is x MRS s 1 D z x s 1 MRS s τs ˆx 11 refects the fact that when the two tax bases partiay overap, the addition to the tax base of another commodity x s is ess than the average tax base per commodity in the existing base x s. However, in the Nash equiibrium, the oca government wi tax the entire base, making k s 0. In this case then, it is ceary optima for the state government to tax the entire base as Wk s s τs 2 V y MRS sˆx 11 > 0. Some impications of 2.16 and 2.17 are summarized in the proposition beow. Proposition 3. Assume the oca and state governments independenty choose their tax bases. a In equiibrium both eves of government tax the entire tax. b When the state government maximizes socia wefare α 1 in equiibrium MRS s > MRS ; when α < 1 the reationship between MRS s and MRS is not obvious. That the MRS s > MRS when the state government maximizes socia wefare α 1 is a resut of the fact that the state government considers the impact its taxing decision have on oca revenues whie oca governments do not consider the effects of their taxes on state 13

revenues. The reationship between MRS s and MRS can be obtained by subtracting the first order condition for the oca tax rate, 2.11, from that of the state 2.14 to obtain 1 + 1 α τ MRS s ˆx 11 + ˆx 21 1 + τ s ˆx 11 + ˆx 21 MRS. 2.19 3 Optima Tax Base Division and Co-occupancy As shown in the preceding section, both eves of government wi tax the entire tax base if given the option. Changes in the oca governments tax bases as we as their tax rates wi generate fisca externaities. Here I address the question of what is the socia-wefare maximizing division of the tax base between the two eves of government. I first address the question of how the tax base shoud be divided if there is no co-occupancy. After deriving the optima division of the tax base, the question of whether the state and oca governments shoud share tax bases, that is, whether there shoud be any co-occupancy, is then addressed. If the state government fuy considers the impacts of its tax poicy on oca revenues α 1, if it were aso have the authority to determine the extent of both its and the oca governments tax bases, the choices woud be wefare-maximizing. However, if α < 1, the state government does not fuy consider the impacts of its tax poicy on oca revenues. Then, in this case, if the state government were to determine the tax bases for both eves of government it woud not choose the wefare-maximizing division. As I wish to investigate the wefare-maximizing division of the tax base in the case of α < 1 rather than the state government determining the tax base division, a third-party federa government or panner chooses the tax bases to maximize socia wefare. In addition to having a third party choose the optima division of the tax base, the timing of the division of the tax base, reative to the setting of the state and oca tax rates, aso needs to be determined. My focus wi be on a Nash equiibrium in which the state and oca governments choose their tax rates at the same time as the panner chooses the division of the tax base. In Section 3.3 I discuss a two-stage game in which the federa government chooses the division of the tax base in the first stage of the game and the state and oca governments simutaneousy set their tax rates in the second stage. 3.1 The Optima Division of the Tax Base The optima division of the tax base in the absence of co-occupancy soves Max k W τ, τ s, k ˆ 1 0 V k dk + U s g s τ, τ s, k + αu g τ, τ s, k 3.1 14

where, given no co-occupancy, k k and k s 1 k. Then the optima division of the tax base, k, that satisfies the first order condition for 3.1 can be expressed as τ τ s 1 + MRS 1 + τ k ˆx 21 x a 1 + MRS s 1 + τ s k sˆx 21 x s c + MRS sτ s k s x sˆx 21 b + MRS τ k x ˆx 21 d 3.2 where, again, τ and τ s are the equiibrium tax rates. Derivation of 3.2 and other equations as we as proofs of propositions in this section are found in Appendix A.3. The expansion of the oca tax base and contraction of the state tax base directy increases oca revenue by τ x and indirecty by affecting the price of x k, now taxed at a rate of τ rather than τ s. As we, the tax has a direct impact through its impact on the price of x k. These impacts of found in term a of 3.2. Term b is the impact of adding x k to the oca tax base on state tax revenues. Then the optima division of the tax base must be such that the impact on utiity of the expansion of the oca tax base is exacty offset by the impact of the equa reduction of the state tax base, terms c and d of 3.2. Rearranging terms in 3.2 yieds MRS 1 τ x MRS s 1 τ s x s + a τ τ s MRS τ k x + MRS s τ s k s x s ˆx 21 0. b 3.3 Examination of equation 3.3 provides for additiona characterization of the optima division of the tax base. From 3.3 we can see that if commodities are substitutes ˆx 21 > 0 the sign of term b is the sign of τ τ s. Then if, for exampe, τ > τ s it must be the case that term a is negative, requiring MRS < MRS s. If τ.5 > <τ s.5 then W k k.5 > < 0 and the optima division of the tax base must be k > <.5. With ˆx 21 > 0 and k > <.5 3.3 can ony be satisfied when the optima tax rates are τ > <τ s. When commodities are compements or α 0, the reationship between the reative tax rates and the reative MRS is indeterminate with τ > τ s it is possibe to have MRS < MRS s or MRS > MRS s and satisfy 3.3. This being the case, it is possibe for 3.3 to be satisfied with no obvious reationship between the equiibrium tax rates τ, τ s, the associate MRS, and the reative tax rates when the base is eveny spit. Proposition 4. In the absence of co-occupancy, the optima division k of the tax base can be characterized by the foowing: a If α 0 and i τ.5 > <τ s.5 then k > <.5; ii ˆx 21 > 0 and τ.5 > <τ s.5 15

then τ k > <τ s k and MRS k < >MRS s k.b If α 1 and i ˆx 21 > 0 and τ.5 τ s.5 then k >.5 and ; ii ˆx 21 < 0 and τ.5 τ s.5 then k <.5; iii ˆx 21 > 0 then τ k < τ s k ; and iv ˆx 21 < 0 then τ k > τ s k. c At k, MRS τ k MRS s τ s k s. d At k : i if ˆx 21 0, a margina change in either the oca or state tax rate has no impact on socia wefare; ii if ˆx 21 > < 0 and α 0, an increase decrease in the oca or state tax rate wi increase socia wefare; iii if ˆx 21 > < 0, and α 1 an increase decrease in the oca tax rate wi increase socia wefare whie an increase in the state tax rate wi have no impact on socia wefare. Proof of the proposition is found in Appendix A.3.2. Part c of the proposition, MRS τ k MRS s τ s k s, is important when considering whether co-occupancy is optima. Part d provides reationships between the division of the tax base and the fisca externaity from increases in the tax rates. As suggested by the proposition, the division of the tax base wi not eiminate the fisca externaities associated with the two tax rates when commodities have non-zero cross-price easticities. This can easiy be seen by differentiating the socia wefare function with respect to the tax rate of a singe ocaity, W τ 1 n MRS sτ s 1 k k ˆx 21 3.4 In the case of gross substitutes, it may change the fisca externaity from being negative with co-occupancy to being positive with no co-occupancy. This, in turn, means that taxes aso change from being too high to being too ow, that is, beow the wefare maximizing rates. 3.2 Optima Co-Occupancy of Tax Bases That the tax rates and margina rates of substitutions for the pubic services are not equa for the two eves of government and tax rate increases or decreases can enhance socia wefare suggests at east the possibiity that co-occupancy may be desirabe. Beow I consider the possibiity of co-occupancy and under what conditions it may be sociay optima. The probem facing the panner is Maximize k, k s W k, k s, τ s, τ ˆ 1 0 V qkdk + U s g s k, k s, τ s, τ + U g k, k s, τ s, τ 3.5 16

Then the first order conditions with the respect to the tax bases when k s expressed as > 0 can be 1 k W k 1 k Vy τ k s W ks k s V y τs x s x s 1 + MRS 1 + τ k s + k x x s ˆx 21 + MRS s τs ˆx 11 + k s + k s xs x s ˆx 21 1 + MRS s + MRS τ 1 + τ s k s + k s xs x s ˆx 21 ˆx 11 + k + k s x x s ˆx 21 0 and 3.6 0. 3.7 To evauate the impact of an increase in the oca tax base on socia wefare when there is no co-occupancy k k s, we can subtract W τ τ k s 0 to obtain W k 0 2.11 from 3.6 and evauate with τ x s MRS τ MRS s τs ˆx 11 + MRS τ x k 1 + MRS s τs x s k s ˆx 21. x s x s 3.8 Anaogousy, subtracting the first order condition for the state tax rate 2.14 from 2.17 and evauating at k s 0 gives W ks τs ks x s 0 + MRS s τ s k s x s x s 1 MRS s τs MRS τ ˆx 11 + MRS τ k x x s αk s x x s ˆx 21. 3.9 From 3.8-3.9, resuts about the optima co-occupancy of the tax base can be obtained. These resuts are summarized in the proposition beow. Proposition 5. Let τ, τ s, and k represent the vaues of the tax rates and the division of the tax base in the absence of co-occupancy that satisfy 2.11, 2.14, and 3.3. Then: a Necessary conditions for the optima division of the tax base to be such that some share of it is tax by both eves of government co-occupied is that either 3.8 or 3.9 are positive. b If ˆx 21 > 0 either W k > 0 or W ks 0 ks > 0 and it is optima to co-occupy some ks 0 share of the tax base. c If ˆx 21 < 0 then a sufficient condition for it not to be optima to co-occupy the tax base is τ x s τs x s + k s τ τ s x x s s x s 1 ˆx 21 < 0 and τs x s τ x + k τ τ s x x s x 1 ˆx 21 < 0. d Letting K and K s denote the optima division of the tax base. Then if if it optima to have co-occupancy either: i One or both governments wi tax the entire base K 1 and K s 0 k s 1, K 1 and K s > 0 k s < 1, k > 0, k s 0, or K < 1 and K s 0 k s < 1, k 0, k s > 0. or ii each government taxes ony part of the base 0 < K s < K < 1. In this case, at the optima division of the tax base, 17

MRS + MRS s τ sˆx 11 MRS s + MRS τ ˆx 11. 3.10 Proof of part c is found in Appendix A.3.3 with remainder of the Proposition discussed beow. That it is optima to have co-occupancy when the commodities are substitutes ˆx 21 > 0 Part b foows immediatey from 3.8 and 3.9 and part c of Proposition 4 either the first term of 3.8 or 3.9 must be positive. As the second term of both expressions, primariy composed of the impact on the segment of the other eve s tax base that is not co-occupied, must be positive then one or both of 3.8 and 3.9 must be positive. Part d, specificay the condition to be satisfied when each government ony taxes part of the base, foows from 3.6 and 3.7 for an interior soution to satisfy both conditions 3.10 must be satisfied. Note that if MRS s MRS and τ τ s 3.10 is satisfied. More generay, 3.10 suggests that the indirect effects of a margina expansions of the two governments base, the change in tax revenue from the existing bases for both governments cance each other out and the direct effects of expansion, the increase in the benefits from the pubic services for the expanding base MRS, MRS s and the decrease in benefits for the other base MRS τ, MRS s τ s are the ony reevant factors. To provide some further intuition, consider evauating 3.6 at the optima division of the tax base in the case of the state and oca tax rates being equa and, by 3.3, MRS MRS s. Then MRS s τ s MRS τ 0 and 3.8 and 3.9 become W ks k W k k MRS τ x k 1 + MRS s τ x s s k s x s x s MRS s τ s k s xs x s 1 + MRS τ k x x s αk s x x s ˆx 21 and 3.11 ˆx 21. 3.12 As discussed, the state and oca tax rates wi not be equa at the optima division the base if ˆx 21 0. The difference in the two tax rates accounts for the first terms in 3.8 and 3.9 in which there are additiona wefare gains to expanding the base for which the direct impact M RSτ is greatest. Intuitivey, 3.11 is the wefare impact of increases decreases in tax revenue in the segments of both the state and oca tax base that are not co-occupied if ˆx 21 > < 0. This increase decrease in tax revenue is reative to that obtained by expanding the oca tax base whie reducing the state tax base by an equa amount. An anaogous argument can be made for expansion of the state tax base and 3.12. 3.3 Optima Division and Co-Occupancy with Leadership In Sections 3.1 and 3.2 the determination of the optima tax bases is determined in the context of Nash equiibrium in a simutaneous game. In this section, I briefy discuss how 18

the resuts in these sections might change if rather than having simutaneous determination of both the tax bases and tax rates, they are determined in a two-stage game and show that the resuts are quaitativey unchanged. In the first stage of the game tax bases are chosen by the federa government and in the second stage tax rates are determined in a Nash game among the state and oca governments. The tax rates determined in the second stage are characterized by 2.11and 2.14 and are functions of the state and oca tax bases with k s 0 in the absence of co-occupancy. Then in the first stage, the optima division again soves 3.1 but this case the state and oca tax rates are functions of the division of the tax base, τ k and τs k. Then the first order condition is now given by τ 1 + MRS 1 + τ k ˆx 21 x + MRS s τs τ k sx sˆx 21 1 + ˆτ τs 1 + MRS s 1 + τs k sˆx 21 x s + MRS τs τ k x ˆx 21 1 + 1 α ˆτ s. 3.13 where ˆτ j 1 τ j dτ j dk, j, s. As the oca tax rate is chosen to maximize 2.4 without regard to the impact on state revenues, the ony impact that any change in the oca tax rate has on wefare is through its impact on state revenues. Anaogousy, the impact of changes in the state tax rate depend on any impacts depend on the impact on oca revenues and whether the state wefare function incudes oca pubic services as an argument α 0. If α 1 then there is no impact of changes in the state tax on wefare. Further, if ˆτ j < 1, j, s, that is, the change in the tax base eads to ess than a 100 percent change in the tax rates, then the impacts of expansion of the oca tax base on state tax revenues MRS s τ s τ k sx sˆx 21 1 + ˆτ and the impact of contraction of the state tax base on oca revenues MRS τ s τ k x ˆx 21 1 + 1 α ˆτ s are quaitativey the same same sign as in the Nash equiibrium. Then, as in the Nash equiibrium, consider an expansion of the oca tax base and no contraction of the state tax base in the absence of co-occupancy k s 0. Differentiating 3.1 with respect to k gives W k τ x s + MRS τ k x MRS τ MRS sτ s ˆx 11 ˆτ s 1 + MRS s τ s k s x s 1 x s + 1 α τ s τ 1 x s + τ ˆτ ˆx 21 3.14 Anaogousy, subtracting the first order condition for the state tax rate 2.14 from 2.17 and evauating at k s 0 gives. 19

W ks τs ks x s 0 + MRS s τ s k s x s Then if, as seems reasonabe, the terms 3.14 and MRS s τ s k s x s x 1 x s + τ ˆτ MRS s τs MRS τ ˆx 11 + MRS τ x 1 x s + τ ˆτ + x s k αk s + 1 α τs ˆτ s 3.15 1 x x s + 1 α τ s ˆτ τ s 1 +MRS s τs 1 k s x s x s + τ ˆτ in MRS τ k in 3.15 are positive then with ˆx 21 > 0 MRS τ x s k αk s + 1 α τs ˆτ s it wi be optima to have co-occupancy. As with the case when the tax base and tax rates are simutaneousy determined, when ˆx 21 < 0 co-occupancy may not be optima. ˆx 21. 3.4 Optima Co-Occupancy with Differentia State Taxation Thus far the anaysis has been restricted to considering uniform tax rates across both tax bases. I now consider the possibiity that the state government can set different rates on the shared tax base k s and the base that it aone taxes k s. For the state government to have an incentive to set substantiay different tax rates on the two segments of its base, it must be the case that α 0 it considers the impact of its tax poicies on oca revenues. For simpicity, et α 1. 7 Then as α 1, the choice of the tax bases can be considered that of the state government. Then the objective of the state government is Maximize W 1 k τs s, τs s, k s, τs s, τs s, τ V qkdk + U s g 0 s k, k s, τs s, τs s, τ, k, k s +U 3.16 g k, k s, τs s, τs s, τ where τ s s is the state rate on the co-occupied segment and τ s s is the state rate on the segment it aone taxes. Then the first order conditions for both tax rates are given by and W τ s s W τ s s 1 + MRS s 1 + τs s ˆx 11 + k sˆx 21 +MRSτs s x k s 0, 3.17 s x s ˆx 21 + MRS τ k + k s x x s x 21 1 + MRS s 1 + τ s s ˆx 11 + k sˆx 21 +MRS s τs s x k s s x s ˆx 21 + MRS τ ˆx 11 + k + k s x 0. 3.18 x s ˆx 21 In considering the optima tax base, I assume that there is overap in the two bases k s > 0. 7 That the optima tax rates are identica across the shared and unshared segments of the tax bases when α 0 is an artifact of the assumption of constant easticities. In the absence of this assumption, different optima tax rates might be expected but not of the magnitude of differences that occur when α 1. 20

Then an increase in k means a reduction in the base ony taxed by the state k s and a decrease in k s means a reduction in the base ony taxed by the ocaities k. Then the optima tax bases are determined by 1 k W k MRS 1 τ + MRS s 1 τs s xs MRS s 1 τs s x s + 1 k MRS τ k + k s x + MRS s τ s s k s x s + τs s k s x s ˆx21 τ + τs s τs s 3.19 and k s W ks MRS 1 τ + MRS s 1 τs s xs MRS 1 τ x + 1 k s 0. MRS τ k + k s x + MRS s τ s s k s x s + τs s k s x s ˆx21 τs s 3.20 Consider a possibe soution with MRS s MRS MRS and τ + τs s τs s. In this case, both 3.17 and 3.18 are satisfied ony if k s 0, k 0 the state taxes the entire base. As we, 3.19 wi be satisfied with equaity regardess of the division of the tax base. Then when MRS s MRS MRS and τ + τ s s τ s s, 3.20 becomes 0, k s W ks 1 k s MRS 1 τ + τ s s xs τ x + MRS τ + τs s ˆx21 τs s 3.21 Then with τ + τs s xs τ x > < 0 when τs s > < 0, for W ks > 0 and the state taxes the entire base, it must be the case that when commodities are substitutes compements ˆx 21 > < 0 the state tax on the co-occupied segment of the tax base is positive negative. Thus two soutions appear possibe: 1 the oca government does not tax the entire base k < 1 and MRS MRS s, and τs s + τ τs s ; and 2 both the state and oca governments tax the entire base k 1 with MRS > MRS s. However, to satisfy the first order conditions for both the state 3.18 and the oca tax rates 2.11 in the second soution it must be the case that MRS s > MRS. Thus a contradiction and ony the first soution is feasibe. When MRS MRS s and τs s + τ τs s we can subtract the first order condition for the oca tax rate 2.11 from 3.20 to obtain τ s s ˆx 11 + k sˆx 21 + τ s s k sˆx 21 0. 3.22 Equation 3.22 states that the state government chooses its tax rates and the tax base that the oca government can tax k s so that the oca tax rate generates no fisca externaity. These resuts are summarized in the foowing proposition: Proposition 6. If α 1 and the state can set different tax rates on the shared based k s and the base it aone taxes k s, its optima tax rates and the division of the tax base are 21

such that: a The state wi tax the entire base k s 0 and ocaities wi ony tax some share of the base k < 1 ; b MRS MRS s ; c The combined state and oca tax on the co-occupied share of the tax base equas the state tax rate on the share of the base it aone taxes τ s s + τ τ s s; d The oca tax rate generates no fisca externaity, that is, τ s s ˆx 11 + k sˆx 21 + τ s s k sˆx 21 0; and e The state tax rate on the co-occupied segment is greater ess than zero τs s cross-price easticities are greater ess than zero ˆx 21 > < 0. > < 0 if Whie Hoyt 2001 considers the overapping tax bases and the possibiity of negative tax rates subsidies, it was in the context of exogenousy-determined tax bases. In that case, whether the state government appies a positive negative tax rate on a shared tax base was determined by whether MRS > < MRS s. In this case, the state government is choosing the extent of the overap in tax base the state appies a positive negative tax rate on the shared based when commodities are substitutes compements. 4 Optima Assignment with Horizonta Externaities Whie the framework for my anaysis has been couched in terms of mutipe ocaities, oca tax poicies have ony impacted state tax revenues and not those of other ocaities. That is, it is a framework with vertica externaities but with an absence of horizonta externaities. A number of studies have examined the impications of the existence of both horizonta and vertica externaities. Keen and Kotsogiannis 2002, for exampe, considers a standard mode of capita taxation competition Wison 1986; Zodrow and Mieszkowkski 1986 with two eves of government state and federa with a focus on whether decentraydetermined tax rates are inefficienty high or ow. As discussed in more detai ater, I incorporate fisca externaities into the mode deveoped in Section 2.1 foowing a different approach with the externaity generated by cross-border shopping. I aso introduce some heterogeneity into the nature of the commodities in the tax base, aowing the extent of the horizonta fisca externaity to vary among commodities. My objectives in this section are two-fod. As before, I consider the optima division of the tax base and whether cooccupancy is optima. Second, with this heterogeneity among commodities, the issue which commodities shoud be assigned to which tax base, oca or state, can be addressed as we. This is, in essence, the issue of assignment discussed, for exampe, by Musgrave 1983. 22