Tax competition in a simple model with heterogeneous firms: How larger markets reduce profit taxes 1

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1 Tax competition in a simple model with heterogeneous firms: How larger markets reduce profit taxes 1 Andreas Haufler 2 University of Munich and CESifo Frank Stähler 3 University of Tübingen and CESifo Final version, May 2012 fothcoming in: International Economic Review 1 Paper presented at the meeting of the Association of Public Economic Theory in Istanbul, at the European Trade Study Group meeting in Rome, and at conferences at CESifo and the Max Planck Institute in Munich. We thank Johannes Becker, Ron Davies, Carsten Eckel, Clemens Fuest, Kai Konrad, Sebastian Krautheim, Ferdinand Mittermaier, Gareth Myles, Michael Pflüger, Evelyn Ribi, Jens Südekum and Ian Wooton for many helpful comments. We also wish to thank two anonymous referees for their detailed and constructive suggestions. Haufler acknowledges financial support from the German Research Foundation (Grant No. HA 3195/8-1) 2 Seminar for Economic Policy, Akademiestr. 1/II, D Munich, Germany. Phone: , fax: , Andreas.Haufler@lrz.uni-muenchen.de 3 Department of Economics, University of Tübingen, Mohlstr. 36 (V4), D Tübingen, Germany. Phone: , fax: , frank.staehler@unituebingen.de

2 Abstract We set up a simple two-country model of tax competition where firms with different productivity decide in which location to produce and sell output. In this model a unique, asymmetric Nash equilibrium is shown to exist, provided that countries are sufficiently different with respect to their exogenous market size. Sorting of firms occurs in equilibrium, as the smaller country levies the lower tax rate and attracts the low-cost firms. A simultaneous expansion of both markets that raises the profitability of firms intensifies tax competition and causes both countries to reduce their tax rates, despite higher corporate tax bases. Keywords: tax competition, heterogeneous firms, imperfect competition JEL Classification: H25, H73, F21

3 1 Introduction The taxation of corporate profits is an ongoing and controversial policy issue in virtually all developed countries. On the one hand, the globalization of production and the worldwide competition for capital are making it increasingly difficult to enforce high tax rates on corporate profits. On the other hand, an effective corporate income tax is viewed by many as a critical element in maintaining an overall tax system that is accepted as fair and equitable by a majority of citizens. In this respect a widespread dissatisfaction with the current distribution of tax burdens arises from the fact that corporate profits have generally risen over the past two decades, whereas the tax rates on these profits have simultaneously fallen. 1 In the light of these controversies, it is essential to have a precise understanding of how the decisions made by mobile, incorporated firms adjust to tax policy parameters, and how these adjustments in turn affect corporate tax reforms. These issues have been addressed in a large theoretical and empirical literature. 2 Nevertheless several important issues remain that have not been explained by the existing literature in a satisfactory way. In this paper we focus on two of these puzzles. First, there has long been evidence that the before-tax profits of firms are substantially higher in low-tax countries, even though tax deductions are also lower in these countries as compared to their high-tax neighbors (see Hines, 1999). More recently, Becker et al. (2010) show, for a large sample of European multinationals, that high-profitability affiliates systematically cluster in low-tax countries, whereas low-profitability firms locate in high-tax countries. Clearly this pattern is not consistent with conventional arbitrage considerations, which equalize the after-tax return to capital in different countries. A second empirical puzzle is that the rates of the corporate income tax (CIT) have fallen significantly and almost universally over the last two decades, yet at the same time corporate tax revenue, as a fraction of GDP, has increased. This is summarized 1 An exemplary policy discussion is presently under way in the United States. Business associations lobby for lower tax rates on the grounds that the U.S. maintains one of the highest corporate tax rates in the developed world. At the same time, public interest organizations speak up in favor of higher effective tax rates on corporations, arguing that many highly profitable corporations are able to avoid most of their tax payments through international tax planning. See The trouble with tax reform (The Economist, February 4, 2011) and Citizens for Tax Justice (2011). 2 Wilson (1999) and Fuest et al. (2005) provide thorough surveys of the theoretical literature. The earlier empirical literature is summarized by de Mooij and Ederveen (2003). 1

4 Table 1: Corporate taxation in selected OECD countries statutory effective ave- CIT revenue tax rate a rage tax rate b (% of GDP) large countries (population > 20 million) Australia Canada France Germany Italy Japan Spain United Kingdom United States large countries c small countries (population < 20 million) Austria Belgium Finland Greece Ireland Netherlands Norway Portugal e 2.8 Sweden Switzerland small countries c all countries c total OECD d The table lists all countries for which effective average tax rates are available since a including local taxes b base case: real discount rate: 10%, inflation rate: 3.5%, depreciation rate: 12.25%, rate of economic rent: 10% (financial return: 20%) c weighted average in sample, countries weighted by GDP in 2005 d unweighted average of all OECD countries; see OECD (2008) e 1990 Sources: Devereux et al. (2002); OECD (2008), Table 12 OECD (2009). 2

5 in Table 1 for selected OECD countries. The table shows that statutory corporate tax rates have been strongly reduced in virtually all countries in the sample since the mid- 1980s. This downward trend is still clearly visible when using the effective average tax rate, which accounts for the simultaneous broadening of tax bases that has occurred in many countries. 3 Moreover, while the averages of statutory and effective tax rates were almost the same for large and small countries in 1985, tax rates have fallen significantly more among the group of small countries since then. At the same time, corporate tax revenue as a percentage of GDP has risen in all of the smaller OECD countries in the sample, whereas the picture for the larger countries is somewhat more mixed. In the (weighted) OECD average, however, there is a clear increase in corporate tax collections. In the present paper we set up a simple model with heterogeneous firms that offers an explanation for these stylized facts. In this model two asymmetric countries compete for internationally mobile firms with heterogeneous costs. We show that, in equilibrium, firms sort according to their cost structure. Hence low-cost firms locate in the small, low-tax country because, on account of their high profitability, they benefit most from the low tax rate. In contrast, high-cost firms locate in the large, high-tax country, as they are relatively more interested in the larger market. This produces an equilibrium pattern of firm location that corresponds to the first empirical observation. We then study the effects that an expansion of markets has on the equilibrium level of taxes in the two countries. Detailed country studies suggest that such a market expansion, with an associated increase in the aggregate profitability of the incorporated sector, has indeed taken place. 4 We show that corporate tax rates are likely to fall in both countries in response to this exogenous shock, even though the base of the corporation tax is simultaneously enlarged. The reason is that the pivotal firm becomes more profitable and hence reacts more sensitively to tax rate changes, rendering tax competition more aggressive when markets expand. These results are consistent with the second stylized fact above. 3 See Devereux et al. (2002) for an elaboration of this concept and for an overview of the trends in corporate taxation since the 1980s. For a recent summary of these trends, see Auerbach et al. (2010). 4 See Devereux et al. (2004) for the United Kingdom and Becker and Fuest (2010a) for Germany. In the United Kingdom, in particular, this development is closely linked to the expansion of the (until recently) highly profitable financial sector. In both the United States and the United Kingdom the share of corporate tax revenues collected from this sector rose from around 10% in the early 1980s to more than 25% of total CIT revenue in 2003 (Auerbach et al., 2010, Figure 5). 3

6 These findings have several important policy implications. So far, the standard explanation for the clustering of high-profitability firms in low-tax countries has been that multinational enterprises shift income from high-tax to low-tax countries, in order to reduce their worldwide tax payments (e.g. Hines and Rice, 1994; Huizinga and Laeven, 2008). This finding has given rise to tax reforms that were explicitly aimed at curbing the possibilities of multinational firms to engage in profit shifting. 5 Tax practitioners remind us, however, that income shifting is limited by existing rules to trade at arm s-length prices, which cannot be costlessly circumvented. In the United States, for example, one indicator for the limited possibilities of firms to minimize their tax payments is the rising importance of corporate losses that are not deductible from tax. As Auerbach (2007, p. 167) concludes, this development casts some doubt on the importance of tax planning strategies as a vehicle for reducing corporate taxes. Therefore, to the extent that the higher profitability of firms in low-tax countries is not the result of profit shifting, but of the sorting of firms with heterogeneous costs, the emphasis of some recent corporate tax reforms may have been partly misplaced. The co-existence of falling tax rates and rising tax revenues also raises immediate policy issues. Standard optimal tax theory would predict that tax rates should rise when the tax base is enlarged and a given tax increase can therefore generate more additional revenues. Hence the question is whether the observed reductions in corporate tax rates can be in the interest of national revenue (or welfare) maximization, or whether this development must rather be attributed to successful lobbying by businesses. 6 Our analysis shows that, in a setting with heterogeneous firms, rising profits can increase the elasticity of the pivotal firm so strongly as to dominate the simultaneous increase in the tax base. Hence, as a response to rising profitability of firms, reducing the corporate tax rate can indeed be the optimal policy for governments that act in the best 5 One example is the German corporate tax reform of 2008, which has, among other measures, introduced a rigorous ceiling on the tax-deductibility of interest payments, in order to prevent debt shifting within multinational firms. See Homburg (2007) for a critical account of this tax reform. 6 In principle, it is also possible that the increase in the corporate tax base is due to a rising share of companies that has chosen to incorporate, in order to benefit from falling corporate tax rates (as compared to personal income taxes). Empirical studies indicate that this effect is indeed present, but it can explain only a fraction of the observed increase in corporate tax revenues. De Mooij and Nicodème (2008) estimate that increased incorporation has raised the corporate tax-to-gdp ratio by some 0.25 percentage points since the early 1990s, other things being equal. This increase is substantially lower than the average increase in CIT revenues shown in Table 1, even though the latter incorporate the negative effect of falling tax rates. 4

7 interest of their citizens, but compete with each other for mobile capital. An important analytical problem arising in models with interjurisdictional mobility of heterogeneous firms is that governments best response functions are generally not continuous, as each country has an incentive to marginally underbid its neighbor s tax rate and attract the low-cost (and hence high-profit) firms. For this reason, we are not aware of any existing tax competition model with these features, for which the existence of a Nash equilibrium in tax rates has been shown. We therefore introduce several simplifying assumptions that are aimed at keeping the analysis as tractable as possible. Hence we assume that countries compete for a fixed number of firms, which differ exogenously in their costs of producing a homogeneous good. A further simplification in our benchmark model is that each firm produces only one unit of output, in the country of its choice. 7 The simplicity gained from this approach allows us to prove the existence of a unique Nash equilibrium under a well-defined condition that specifies a critical degree of asymmetry in the market size of the competing countries. Our model is related to the recent theoretical literature on trade and firm heterogeneity, starting with Melitz s (2003) model of a monopolistically competitive industry in which firms draw their productivity randomly. We incorporate some of the basic findings of this line of research into the literature on taxation and foreign direct investment. The latter has analyzed the interaction between taxes and firm location in models of industry agglomeration (see Ludema and Wooton, 2000; Kind et al., 2000; Baldwin and Krugman, 2004; Borck and Pflüger, 2006), or in models that explicitly allow for heterogeneous countries, in particular with respect to market size (e.g. Ottaviano and van Ypersele, 2005; Haufler and Wooton, 2010). However, with few recent exceptions, the heterogeneity of firms has been neglected so far in the international tax literature. A first analysis of tax competition in the presence of heterogeneous firms is Burbidge et al. (2006). In this paper each firm s productivity also differs across regions, however. This feature eliminates the sorting of firms on the basis of tax rates only, leading to a smooth trade-off for tax policy that is very different from the one studied here. Closer to our setting is Davies and Eckel (2010). Their framework differs from ours in that they use a model of monopolistic competition and allow for endogenous firm entry. Accordingly the focus of their analysis is on the normative question of whether tax competition distorts the equilibrium number of firms in the industry, whereas our 7 In an extension we show, however, that the basic properties of our simple model carry over to a more general setting where firms output choices are endogenous. 5

8 analysis aims to explain existing trends in corporate tax policy. Another difference is that the model of Davies and Eckel is considerably more complex than ours. As a result, the authors are not able to establish sufficient conditions for the existence of a purestrategy Nash equilibrium in tax rates. Krautheim and Schmidt-Eisenlohr (2011) also present a model of tax competition with firm heterogeneity, but their model features a large country and a tax haven, where no production takes place in the latter. Hence the focus is on the competition for book profits shifted to the tax haven, rather than on the competition for the location of internationally mobile firms. Finally, some recent studies address tax policy issues in settings that involve the sorting of heterogeneous firms in the presence of international tax differentials (Becker and Fuest, 2007; Baldwin and Okubo, 2009). These studies are not cast in a tax competition framework, however. 8 The paper is organized as follows. In Section 2 we describe our basic tax competition model. Section 3 asks under which conditions a Nash equilibrium in pure strategies exists in this model. Section 4 analyzes the effects of an increase in the market size of one or both countries on tax rates and tax revenue in each country. Section 5 summarizes the results of several model extensions, which have been carried out with the help of numerical simulations. Section 6 concludes. 2 The model 2.1 Firms We consider a region of two countries i {1, 2} in which two goods are produced. Our focus lies on the market for a homogeneous and non-tradable good x, which is served by a total of N firms. Entry to the x-industry is restricted because one unit of a specific factor ( capital ) is needed to produce at all, and the supply of this factor is fixed at N. 9 Importantly, the N firms differ in their costs of production. Specifically we assume that costs are drawn randomly and independently from a uniform distribution with c [c, c]. These costs reflect the firm-specific employment of labor that is needed to 8 A separate strand in the recent literature focusing on the policy implications of firm heterogeneity analyzes the optimal subsidization of market entry, either for foreign firms (Chor, 2009), or for domestic entrepreneurs (Pflüger and Südekum, 2009). 9 In our benchmark model it is immaterial how this factor endowment is distributed between residents of countries 1 and 2. The factor owners could also be located in a third country (the rest of the world). 6

9 produce one unit of output, irrespective of where the output is produced. Each firm, identified by its unit cost c, decides in which country to settle and produce for the local market. Locations differ with respect to their market size, in a way described in more detail below. Firms decide on their location, knowing both countries markets and their own costs, and forming rational expectations about the location of their rivals. Due to restricted entry to sector x, all firms in this sector will make positive profits in equilibrium. 10 The model is closed by the presence of a second, numeraire sector which produces a tradable good under perfectly competitive conditions and using labor only. In each country one unit of labor is required per unit of the numeraire good; hence free trade in this good equalizes wages across countries at unity. As no profits are generated in the numeraire sector, it remains in the background throughout our analysis. Aggregate labor supplies in each country are exogenously given and labor is immobile across countries. The aim of our analysis is to establish conditions under which a Nash equilibrium in taxes exists in a model of heterogeneous firms, and to determine the effects that market enlargement has on tax rates and tax revenues in each country. These effects are derived in a two-stage game where governments first determine their tax policy and firms then decide on their location. To keep our analysis as simple as possible, we focus only on the location decision of firms in our benchmark model. Hence we assume that, irrespective of its costs, each firm produces exactly one unit of output in equilibrium. Even though output choices are fixed, it is still true in our benchmark model that the lowest-cost firms are the most valuable from the perspective of host countries, in the sense that attracting them yields the largest gain in tax base. Hence the model retains the essential qualitative characteristics of firm heterogeneity for tax policy decisions. In Section 5.1 we analyze an extended version of our benchmark model where firms output choices are endogenized, and we show that the basic properties of our benchmark model carry over to this more general setting. We parameterize the attractiveness of a location by A i and assume, with no loss of generality, that country 1 is the more attractive region so that A 1 A 2. In our setting, where good x is a non-tradable good, A i can simply be interpreted as an indicator of local market size. Hence we refer to country 1 and country 2 as the large and the small market, respectively. We assume that the (inverse) demand function for 10 In Section 5 we discuss the extension to endogenous market structures. 7

10 good x is linear in each country and given by p i = A i n i. Here p i is the price of the homogeneous good x in country i and n i is the number of firms, and hence total production in i. The gross profits of a firm with costs c locating in country i are then given by A i n i c. Each country levies a proportional profit tax at rate t i on the gross profits earned by the firms in the x-industry that locate within its jurisdiction. 11 All (labor) costs of production are deductible from the profit tax base. Hence π i (c) (A i n i c)(1 t i ) gives the net-of-tax profit of a firm with cost c in country i. Firms locate in the country in which the expected net-of-tax profit is larger. We denote by ĉ the costs of the firm that is just indifferent between locating in country 1 or in country 2. If firms do not all locate in the same country (that is, if c < ĉ < c) the following arbitrage condition must hold for this firm: (A 1 n 1 ĉ)(1 t 1 ) = (A 2 n 2 ĉ)(1 t 2 ), n 1 + n 2 = N. (1) Through the arbitrage condition (1) the pivotal firm with costs ĉ is just compensated for the higher tax in, say, country 1, either by a larger market size in country 1, or by a smaller number of competitors. To see how changes in production costs affect firms location choice, differentiating π i (c) with respect to c gives π i c = (1 t i) i {1, 2}. (2) Hence a given increase in costs will lead to a smaller reduction in net profits in the high-tax country. The reason is that (labor) costs are deductible from the corporate tax base and this deduction is more valuable, the higher is the tax rate. Together with the arbitrage condition of the pivotal firm [eq. (1)], this implies that all firms with costs c > ĉ will locate in the high-tax country, whereas all firms with costs c < ĉ will prefer to locate in the low-tax region. To ensure that sorting by firms according to their cost type occurs in the location equilibrium, we assume that the total number of firms N is sufficiently large Our treatment implies that the residence country of the investor exempts foreign-earned profit income from tax. This is true for most OECD countries. One of the few exceptions is the United States which is, however, also contemplating a switch to the exemption method. See Becker and Fuest (2010b) for a recent discussion and analysis. 12 A sufficiently large N implies that the relocation of an individual firm will only have a small 8

11 With the high-tax country attracting the high-cost firms, and given our assumption that costs are uniformly distributed in the interval [c, c], the critical cost level ĉ determines the (expected) number of firms that locate in each country by n i = c ĉ c c N, n j = ĉ c c c N, i j, t i > t j. (3) From (3) and the arbitrage condition (1), we can derive the critical cost level ĉ as a function of the exogenous parameters and the endogenous tax rates t i. This depends on which country chooses the lower tax rate and thus attracts the low-cost firms. Denoting by a subscript I (II) the regime where t 1 > t 2 (t 1 < t 2 ) and assuming interior solutions we get 13 ĉ I = (A 2 + cφ)(1 t 2 ) (A 1 N cφ)(1 t 1 ) Θ I if t 1 > t 2, (4a) where ĉ II = (A 1 + cφ)(1 t 1 ) (A 2 N cφ)(1 t 2 ) Θ II if t 1 < t 2, (4b) Θ I (φ + 1)(1 t 2 ) + (φ 1)(1 t 1 ) > 0, (5a) Θ II (φ + 1)(1 t 1 ) + (φ 1)(1 t 2 ) > 0, (5b) and φ N (c c) 1. (6) In expression (6) we thus assume that there is a minimum density of firms, given the cost spread between the firms with the highest and those with the lowest costs of production. This condition is sufficient to ensure that Θ I and Θ II are always positive. Our further analysis is based on the following definition. Definition: Let ĉ be the critical value of costs for the firm that is indifferent between locating in country 1 or in country 2. In Regime I, country 1 chooses the higher tax rate (t 1 > t 2 ) and ĉ is given by (4a). In Regime II, country 2 chooses the higher tax rate (t 2 > t 1 ) and ĉ is given by (4b). effect on the gross profits to be earned in each market. This eliminates the existence of (additional) location equilibria where location patterns are independent of costs. We are grateful to Gareth Myles for pointing this out to us. 13 In Section 3 we discuss in detail under which conditions interior solutions will occur in equilibrium. 9

12 Equations (3) and (4a) (4b) determine the number of firms that locate in each country, as a function of both tax rates. Differentiating the regime-specific values of ĉ with respect to t 1 and t 2 gives 14 I = (1 t 2) [(A 1 N cφ)(φ + 1) + (A 2 + cφ)(φ 1)] t 1 Θ 2 I I = (1 t 1) t 2 (1 t 2 ) I t 1 < 0; II t 1 = (1 t 2) [(A 1 + cφ)(φ 1) + (A 2 N cφ)(φ + 1)] Θ 2 II II = (1 t 1) t 2 (1 t 2 ) II t 1 > 0. > 0, < 0, (7a) (7b) In Regime I, where country 1 is the high-tax region, a rise in this country s tax rate will raise ĉ I. Since country 2 hosts all firms with cost levels between c and ĉ in this regime, this implies a rising number of firms in country 2 and accordingly a fall in n 1. An increase in t 2 will instead reduce ĉ I and thus increases the number of firms in the high-tax country 1. In Regime II the signs of both derivatives are reversed. 2.2 Governments Our analysis is based on the assumption that countries set taxes non-cooperatively before firms decide on their location. 15 Moreover, we assume that the objective of each government is to maximize (expected) profit tax revenue in the x-industry. This assumption implies that governments value tax revenue highly in comparison to consumer and producer surplus. One common explanation is that governments are of a Leviathan type and are therefore mostly interested in tax revenue. Alternatively, governments could be politically forced by the working population to maximize revenue from the corporate income tax, for example because capital is perceived to be gaining from globalization, whereas labor is losing. Whatever its underpinnings, revenue maximization is a frequent assumption in the tax competition literature To sign (7a), note that (A 1 N cφ)(φ + 1) + (A 2 + cφ)(φ 1) = (A 1 + A 2 2N 2c) φ + (A 1 A 2 ) + N(φ 1). Since φ 1 and A 1 > A 2, a sufficient condition for this to be positive is condition (10), which is introduced and discussed below. Eq. (7b) can be signed analogously. 15 For evidence that OECD countries compete over corporate taxes, see Devereux et al. (2008). 16 In Section 5 we discuss the alternative case of welfare-maximizing governments. 10

13 In Regime I the expected tax revenues of the two countries, denoted by T i, are T I 1 = t 1 φ c ĉ [A 1 φ(c ĉ I ) c] dc, T I 2 = t 2 φ ĉ c [A 2 φ(ĉ I c) c] dc, (8) where the relevant expression for ĉ is given in (4a) and φ is in (6). In order to save on notation, we will use the transformed tax revenue expression ˆT i T i /φ in all derivations that follow. 17 Differentiating country 1 s transformed objective ˆT 1 with respect to t 1 and using the Leibniz integration rule yields the following regime-specific first-order condition: ˆT I 1 t 1 = c ĉ [A 1 φ(c ĉ I ) c] dc t 1 I t 1 [A 1 2φ(c ĉ I ) ĉ I ] = 0. Integrating the first term further gives ˆT [ 1 I = (c ĉ I ) A 1 φ(c ĉ I ) (c + ĉ ] I) t 1 2 t 1 I t 1 [A 1 2φ(c ĉ I ) ĉ I ] = 0. Similarly, the first-order condition for country 2 s government reads ˆT [ 2 I = (ĉ I c) A 2 φ(ĉ I c) (ĉ ] I + c) I + t 2 [A 2 2φ(ĉ I c) ĉ I ] = 0. (9b) t 2 2 t 2 The interpretation of these first-order conditions is straightforward. The first term in (9a) and (9b) gives the increase in tax revenues induced by a higher tax rate at an unchanged tax base. This effect is unambiguously positive. The second terms give the change in the tax base resulting from a small tax increase. Note first from (7a) that in both (9a) and (9b) the second terms have the opposite sign as the squared brackets in these terms. The squared brackets in turn combine two distinct effects. A tax increase induces some firms to relocate to the other country, but the profits of the remaining firms rise due to lower market output and accordingly higher prices. For an interior equilibrium to exist, the first of these effects must dominate the second so that the tax base falls in the country that marginally raises its tax rate. A sufficient condition ensuring this is: (9a) A i 2N c > 0 i. (10) Throughout the following analysis we assume that this condition is met. In verbal terms it states that if all but one firm locate in the same country, attracting the last firm (with the highest cost level c) will still raise aggregate profits in that country. In Regime II the first-order conditions for the two countries optimal tax rates are derived and interpreted analogously. These conditions are given in the appendix. 17 Since φ is a constant, this transformation affects neither the choice of optimal tax rates, nor the signs of derivatives, in which we are interested. 11

14 3 Existence of equilibrium In this section we ask under which conditions a Nash equilibrium in taxes exists in the present model. A fundamental existence problem arises in the presence of heterogeneous firms because country i s payoff is not continuous at the other country s tax rate t j. If country i overbids country j s tax rate, then it will attract the high-cost (and thus low-profit) firms. If instead country i underbids t j, then it attracts the low-cost (highprofit) firms. Hence, each country s tax revenue experiences an upward jump when the tax rate is set marginally lower than in the competing jurisdiction. Accordingly, best response functions can also be discontinuous in our model. These properties require a thorough analysis of equilibrium existence and uniqueness. A natural starting point for our analysis is the case where market conditions in the two countries are identical (i.e., A 1 = A 2 ). In this case it is also natural to focus on a symmetric situation with t 1 = t 2 and ask whether this situation can represent a Nash equilibrium. With identical market conditions and taxes, all firms are indifferent as to their location and each firm will thus locate in each jurisdiction with probability q = 0.5. In a situation where both countries choose the same tax rate the expected tax revenue for each country is thus T i t1 =t 2 = tnq c c (A i Nq c) 1 c c dc = tn 2 [ A i N 2 ] (c + c) 2 i. (11a) In contrast, if country i slightly underbids country j, it will still get half of all firms, but now the low-cost firms will self-select into country i. Hence country i s tax revenue becomes T i ti =t j ε = tn (c+c)/2 c ( A i N 2 c ) 1 c c dc = tn 2 [ A i N 2 (c + c) 2 ] (c c) +. 4 (11b) Comparing (11a) and (11b) shows that profits and tax revenue are unambiguously larger for a country that marginally underbids its neighbor, because sales are the same, but aggregate costs are lower in the low-tax country. Hence for any positive, common tax rate t 1 = t 2 there is an incentive for each country to marginally underbid the other, in order to attract the more profitable firms. Thus t 1 = t 2 > 0, with strictly positive tax rates in both countries, cannot be a Nash equilibrium pair of taxes. A situation with t 1 = t 2 = 0 can also not be an equilibrium, because each country could then gain by setting a positive tax rate and still attract some firms, obtaining strictly positive 12

15 tax revenue. Hence there cannot be a symmetric, pure-strategy Nash equilibrium with t 1 = t 2. This result implies that, unlike in tax competition models with homogeneous firms, a situation of perfect symmetry is not a suitable starting point when firms differ in their productivity levels. 18 In the following we will therefore focus on asymmetric situations and consider in turn the cases where the larger country 1 either has the lower or the higher tax rate than its smaller neighbor. We first ask whether an equilibrium can exist in Regime II. This yields: Proposition 1 There cannot be an interior tax competition equilibrium in Regime II, where the larger country (country 1) has the lower tax rate. Proof: See the appendix. The technical proof for the proposition is relegated to the appendix. The intuition for Proposition 1 is, however, straightforward. In its tax optimum, each country equates the marginal revenue gains and the marginal revenue losses resulting from a small tax increase [see the discussion of (9a) (9b)]. If a Regime II equilibrium with t 1 < t 2 existed, the larger country 1 would clearly have the larger tax base, as it would host more firms and these firms would also be more profitable. Hence the marginal gains from a tax increase would be unambiguously larger for country 1 than for country 2. Moreover, if an interior tax equilibrium existed in Regime II, a marginal tax increase of country 1 would cause those firms to leave the country which have the highest costs among the firms that locate in country 1 and thus are least attractive from the perspective of this country. In contrast, a tax increase by country 2 would create an outflow of the firms which have the lowest costs and hence are most valuable from the perspective of country 2. Hence the marginal costs of a small tax increase would be lower for country 1 as compared to country 2. As country 1 would face higher benefits but lower costs from a tax increase, as compared to country 2, it is impossible in such a situation that the marginal gains and the marginal losses from a tax increase are equated for both countries simultaneously It is seen from (11a) and (11b) that there is no revenue gain from marginally underbidding the neighboring country when c = c and thus the production costs of all firms are identical. This is the reason why symmetric tax equilibria generally exist in models with homogenous firms (e.g. Ottaviano and van Ypersele, 2005; Haufler and Wooton, 2010). 19 Note that this argument does not rely on the specific setup of our benchmark model. Hence it carries over to the case where firms output choices are endogenous (Section 5.1). 13

16 From Proposition 1 we know that a Nash equilibrium in pure strategies, if it exists at all, can only arise in Regime I, with the larger country having the higher tax rate. It can be shown that a Regime I equilibrium will indeed exist when A 1 A 2 + N and hence country 1 has a sufficiently large advantage over country 2 with respect to its market size. From (4a) this implies that country 1 would attract all firms, if tax rates were equal in the two countries (i.e., if ĉ I = c holds for t 1 = t 2 ). To establish this result we need to focus on situations in which the own effects of tax rates on marginal tax revenues dominate the cross effects. This is formalized by assuming stability of the equilibrium so that the determinant of the Jacobian matrix J = 2 ˆT I 1 t ˆT I 2 t ˆT I 1 t 1 t 2 2 ˆT I 2 t 1 t 2 > 0 is strictly positive. Given this assumption, we are able to prove both existence and uniqueness of the equilibrium, as summarized in the following proposition: Proposition 2 Assuming that J > 0, there exists a unique Nash equilibrium in pure strategies in Regime I with t 1 > t 2 0, if country 1 is sufficiently large relative to country 2, i.e., A 1 A 2 + N. Proof: See the appendix. To provide an intuition for this proof, we start from a situation where country 2 s tax rate is zero and country 1 s tax rate is at the highest possible level, denoted t 1, that is consistent with attracting all firms to this country. If, at t 1, country 1 s marginal tax revenues are negative [see eq. (9a)], then t 1 > t 2 = 0 must be a Nash equilibrium because a deviation from t 1 in either direction leads to tax revenue losses for country 1, whereas country 2 cannot improve upon the outcome of zero tax revenue. 20 If, in contrast, country 1 s marginal tax revenues are positive at t 1, then it will want to raise its tax rate above t 1. Once t 1 has been increased such that ĉ I > c, country 2 is also able to attract some firms. Hence it has an incentive to raise its own tax rate above zero while still underbidding country 1 s tax rate, in order to maintain a positive tax base. In this case a mutually optimal set of tax rates with t 1 > t 2 will exist, which leads to an interior equilibrium with a positive number of firms in each country. It is also straightforward to derive the conditions under which the Nash equilibrium in our model is an interior solution. For this purpose we compare ĉ I in (4a) with c and c, 20 Strictly speaking, country 2 is indifferent between all tax rates t 2 0, as its tax base is zero. 14

17 respectively. This yields A 2 (1 t 2 ) (A 1 N)(1 t 1 ) > (t 1 t 2 ) c ĉ I > c, (12a) (A 2 N)(1 t 2 ) A 1 (1 t 1 ) < (t 1 t 2 ) c ĉ I < c. (12b) Condition (12b) must be fulfilled in any Nash equilibrium with A 1 A 2 + N, as it is the condition for the large country 1 to host any firms. By setting t 1 sufficiently close to t 2, the large country can ensure that the left-hand side of (12b) turns negative, thus guaranteeing an interior equilibrium in Regime I. Moreover this policy will strictly dominate any tax policy of country 1 where it holds no firms and receives zero revenue. In contrast, condition (12a) may be violated when A 1 >> A 2 + N. In this case it is possible that all firms cluster in the large country 1. In the special case A 1 = A 2 + N, however, condition (12a) reduces to A 2 > c, which is always met [see eq. (1)]. Hence, for A 1 = A 2 + N the unique Nash equilibrium is always an interior solution. Proposition 2 is our first central result, establishing sufficient conditions for the existence of a unique Nash equilibrium in our simple tax competition model with heterogeneous firms. In this equilibrium the larger country is able to levy the higher tax rate. 21 Proposition 3 in Davies and Eckel (2010) has a similar flavor, but these authors are not able to ensure the existence of a tax competition equilibrium in their model. Clearly, this is the result of the simpler structure of the model used here. Intuitively, the sufficient condition A 1 A 2 + N implies that country 2 cannot secure a positive tax base when tax rates are equal, and hence country 1 cannot increase its tax base discretely by marginally underbidding country 2. Therefore both countries best response functions are continuous in this case. In the location equilibrium with taxes, each firm then trades off the net location advantage of country 1 (accounting for the larger number of competitors in this country) against the tax advantage of country 2. Note, finally, that a Nash equilibrium may also exist if A 1 [A 2, A 2 + N], that is, country 1 is larger than country 2 but it will not host all firms when tax rates are equal. The difficulty that arises in this case is that country 2 can secure a positive tax base with a strictly positive tax rate. This makes it potentially interesting for country 1 to underbid country 2 s tax rate and attract all firms. Hence, if A 1 A 2 < N, any candidate equilibrium must be immune against such an underbidding strategy by 21 The result that the large country levies the higher tax rate is familiar from the literature on capital tax competition (Bucovetsky, 1991; Wilson, 1991). The novel element in a model with heterogeneous firms is that the large country, by imposing the higher tax, also attracts the low-profitability firms. 15

18 country 1. We know that underbidding is always profitable when countries are identical (A 1 = A 2 ). However, the more asymmetric countries become, the lower is both the tax rate and the tax base of country 2 in an asymmetric candidate equilibrium. Hence country 1 can only secure a small additional tax base by underbidding country 2, and doing so requires a large drop in country 1 s tax rate. This implies that the incentive for country 1 to underbid its smaller neighbor will monotonically fall as the asymmetry grows. From this discussion we expect that there is a critical difference in market sizes, (A 1 A 2 ) c < N, such that an asymmetric pure-strategy equilibrium with the properties of Proposition 2 exists, once this critical threshold is surpassed Market expansion and tax competition As we have discussed in the introduction, there are several empirical indications that the expansion of highly profitable services, in particular in the banking and finance sector, has contributed to rising corporate profits in many OECD countries during the last decades. In the following we capture this development by an exogenous increase in the size of either one or both markets in our model, as given by the parameters A i. This exogenous market expansion raises the profitability of all firms in equilibrium. Another possible reason for the empirically observed increase in the profitability of the corporate sector is a general reduction in production costs. In principle, this exogenous change could be equally analyzed in our setting, with many similarities to the analysis below, as a downward shift in average cost is equivalent to an upward shift in market size. Since the shift affects the entire distribution of costs, however, the analysis of this case would be considerably more complex than the change in market size studied here. Our theoretical analysis is based on the benchmark case where the market sizes of the two countries are related by A 1 = A 2 + N. Proposition 2 then guarantees the existence of a unique, interior Nash equilibrium where t 1 > t 2 > 0 and ĉ I > c. 23 We then consider small changes in the exogenous model parameters. By a continuity argument, we assume that an equilibrium still exists after the small perturbation of the initial 22 These expectations are confirmed by numerical simulations, which are available upon request. 23 Focusing on A 1 = A 2 + N greatly simplifies the theoretical analysis, because it allows to relate A 1 and A 2 to each other by a strict equality condition. At the end of Section 4.2 and in Section 4.3 we will, however, present simulation results for the more general case A 1 A 2 + N under which a Nash equilibrium in pure strategies has been shown to exist. 16

19 equilibrium has taken place. Our comparative static analysis starts from the optimal tax conditions in Regime I, as given in (9a) (9b). Perturbing this equation system yields the following responses of optimal tax rates to an exogenous parameter change dξ: 24 [ ] dt i dξ = 1 2 ˆTj 2 ˆTi J t 2 j t i ξ + 2 ˆTi 2 ˆTj t i t j t j ξ. (13) Before we compute the different terms in (13), it is helpful to recall that optimal taxation in a setting with only one taxed good generally follows some variant of the inverse elasticity rule. This is particularly transparent when the objective is the maximization of tax revenues, as is the case in our analysis. Denoting the tax base of country i by B i, the (tax rate) elasticity of country i s tax base is defined as η i B i t i t i B i i {1, 2}. (14) Any external shock that causes η to rise will thus tend to reduce the optimal tax rate of country i in equilibrium. The change in η is driven by two main effects. First, η will rise when a small increase in t i causes a larger loss in the tax base of country i. In our setting this marginal tax base loss is simply the profitability of the marginal firm. Other things equal, rising profits of this firm will therefore tend to reduce country i s optimal tax rate. Secondly, η will fall when the tax base of country i rises. Intuitively, a large tax base B i makes it more attractive for country i to raise its tax rate, because more additional revenue can be raised. In the following the optimal tax responses of each country can therefore easily be understood by looking at the effects on country i s tax base on the one hand, and on the profitability of its pivotal firm on the other. We first derive the slope of best response functions, the sign of which is equal to 2 ˆT1 / t 1 t 2. This is ambiguous for country 1: ( 2 ĉ 2 ˆT1 t 1 t 2 = t 1 t 2 t 1 t 2 ) [A 1 2φ(c ĉ) ĉ] t 1 t 1 t 2 (2φ 1) 0. (15a) The ambiguity arises from the effects collected in the round bracket in the first term of (15a). The first term in this bracket is positive, as / t 2 < 0 from (7a). Thus an increase in t 2 raises country 1 s tax base by lowering ĉ. The second term in the round bracket is negative, however, as 2 ĉ t 1 t 2 = 1 Θ 3 {[2 t 1 t 2 + φ(t 1 t 2 )][(A 1 N cφ)(φ + 1) + (A 2 + cφ)(φ 1)]} > From here on, we suppress the regime index, as all expressions refer to Regime I. 17

20 By lowering ĉ, the rise in t 2 increases the profitability of the marginal firm in country 1. Hence the effects of an increase in t 2 on the elasticity of country 1 s tax base are counteracting. Accordingly, country 1 s best response function can be upward or downward sloping. In contrast, country 2 s best response is always upward sloping. This is seen from ( ) 2 ˆT2 2 ĉ = + t 2 [A 2 2φ(ĉ c) ĉ] t 2 (2φ + 1) > 0, (15b) t 2 t 1 t 1 t 1 t 2 t 1 t 2 where (7a) is used to sign the effects on the marginal firm and 2 ĉ/( t 1 t 2 ) > 0. An increase in t 1 raises the tax base of country 2 and at the same time lowers the profitability of the marginal firm in country 2. Both of these effects tend to reduce the elasticity of country 2 s tax base, thus causing t 2 to rise. Our analysis shows that the difference in the slopes of the best responses are caused by the opposite effects that changes in t 1 and t 2 have on the profitability of the pivotal firm. This leads to an additional interdependence between the two countries tax policies that is caused only by firm heterogeneity. 4.1 Isolated market expansion in one country We first consider a unilateral increase in the market size of one country. One example for an exogenous increase in market size is population growth. Here we focus on country 2 so that dξ = da Hence we analyze a catching-up process of the region which has the smaller market in our analysis. 26 We analyze how this change affects tax rates in both countries and consider country 1 s tax response first. The impact effect of an increase in country 2 s market size on the tax rate in country 1 can be represented by 27 [ ] 2 ˆT1 2 ĉ = [A 1 2φ(c ĉ) ĉ] + t 1 t 1 (2φ 1) < 0 (16) t 1 A 2 A 2 t 1 A 2 t 1 A 2 25 Our analysis is still based on initial equilibria where the market sizes are related by the condition A 1 = A 2 +N. While this relationship no longer holds, with strict equality, after the marginal increase in A 2 has taken place, the comparative static changes are evaluated at the values in the initial equilibrium. Hence the regularities imposed by the condition A 1 = A 2 + N can still be exploited in this case. 26 A prime example is Ireland, whose population grew by 18% during the decade This compares with an average population growth rate in the OECD of less than 7% during the same period (OECD, 2011, p. 31). 27 By impact effect we mean the direct effect of a parameter change on a variable of country i, before taking into account further (indirect) effects arising from the endogenous tax response of country j. 18

21 where, from (4a) and (7a), = (1 t 2) A 2 Θ > 0, 2 t 1 A 2 > 0. (17) Hence country 1 s tax rate unambiguously falls, upon impact, when A 2 is increased. On the one hand the larger market size of country 2 leads some firms to relocate to this country, thus reducing country 1 s tax base by the first part of the first term in (16). On the other hand, the expansion of country 2 s market increases the profits of the pivotal firm in country 1. This is now for two reasons, the direct effect of the market expansion [the second part of the first term in (16)] and the changing identity of the pivotal firm in country 1 [the second term in (16)]. The impact change in the tax rate of country 2 can be given by [ 2 ˆT2 = (ĉ c) t 2 A 2 + t 2 t 2 1 2φ ] A 2 ] [ 1 (2φ + 1) A 2 where / A 2 is given in (17) and, from (7a), + [ A 2 φ(ĉ c) ĉ + c ] A t 2 2 ĉ t 2 A 2 [A 2 2φ(ĉ c) ĉ] > 0, (18) 2 ĉ t 2 A 2 = (1 t 1)(φ 1) Θ 2 0. (19) The first two terms in (18) give the increase in country 2 s tax base, induced directly by A 2 and also indirectly by the increase in ĉ following the change in A 2. These terms thus tend to raise country 2 s tax rate. The change in the profits of country 2 s pivotal firm is ambiguous, however. The changing identity of this firm tends to reduce the marginal tax base loss in country 2 [the third term, which is positive from (17), (5a), and (7a)], whereas the direct effect of the market expansion works in the opposite direction [the negative fourth effect]. In the appendix, we show that the last effect is dominated by the other three and hence the overall effect in (18) is unambiguously positive. These results are summarized in: Proposition 3 Starting from an initial equilibrium where A 1 = A 2 + N, the impact effect of an isolated market expansion in country 2 reduces the tax rate of country 1, and increases the tax rate of country 2. Proof: See the appendix. 19

22 Note that Proposition 3 only makes a statement about the impact effect of the change in A 2 on optimal tax rates. To this must be added the indirect effects that result from the best response of each country to the initial change in the other country s tax rate. From (15a) and (15b) these indirect effects tend to reduce t 2, whereas their effect on t 1 is ambiguous. In general, the total effects on changes in both countries tax rates can therefore not be signed without imposing further restrictions on the model. For most specifications, however, a market expansion in country 2 is likely to cause the neighboring country 1 s tax rate to fall in equilibrium. One implication of these findings is that the growing market size of small, peripheral countries may impose downward pressure on the tax rates set by their larger neighbors, irrespectively of the tax change in the small country itself. For completeness, and for the use in Section 4.3 below, we also derive the impact effects on tax rates of a unilateral increase in the market size of country 1. This is done in the appendix [equations (A.17) (A.18)]. These derivations show that an increase in A 1 unambiguously reduces the tax rate in country 2 upon impact, but the effect on the tax rate in country 1 is ambiguous. Intuitively, country 2 s tax base shrinks as some firms relocate to country 1 and its marginal firm becomes more profitable, and hence responds more strongly to tax rate changes. Both of these effects tend to reduce t 2. In country 1, however, the tax base increase and the rising profitability of the pivotal firm have counteracting impact effects on t 1 and the net effect cannot be signed unambiguously. 4.2 Market expansion in both countries Our above analysis has shown that a growing market size in one country is likely to cause a tax cut in the competing country, but it need not raise the tax rate of the country whose market size has risen. In the following we consider a simultaneous expansion of both markets, as given by dξ = da 1 = da 2 da. This could either represent an equal absolute change in the population of both countries. Alternatively, it could represent a simultaneous and equal shift in demand in both countries from the numeraire sector (where no profits are made) to the imperfectly competitive sector x. Our analysis starts again from an initial situation where A 1 = A 2 + N, ensuring the existence of an interior tax equilibrium. We perturb the initial equilibrium and show that the exogenous market expansion will reduce tax rates in both countries, despite 20