Competing for firms under agglomeration: Policy timing and welfare

Regional Science and Urban Economics 49 (2014) 48–57

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Competing for firms under agglomeration: Policy timing and welfare☆ Michiel Gerritse ⁎ Faculty of Economics and Business, University of Groningen, Nettelbosje 2, 9747 AE Groningen, The Netherlands

a r t i c l e

i n f o

Article history: Received 19 June 2013 Received in revised form 10 August 2014 Accepted 12 August 2014 Available online 23 August 2014 JEL classification: R38 R50 R53 F13

a b s t r a c t This article studies government tax competition for firms under agglomeration effects. Agglomeration forces avert races to the bottom. They also eliminate the need to harmonize policy if large regions set policy first. However, if regions set policy at the same time, harmonization can still improve welfare. The case against harmonization thus rests on the assumed timing of policy-formation, not on agglomeration itself. Tax floors, an often advocated alternative to harmonization may not form Pareto-improvements: that depend on the effects of local policy outside the own region. © 2014 Elsevier B.V. All rights reserved.

Keywords: Policy competition Timing of policy Agglomeration Spatial general equilibrium

1. Introduction A fear of “races to the bottom” has led many economists to advocate tax harmonization. Mobile firms locate where taxes are low, so governments set low taxes to attract firms. Governments do not take into account that lower taxes imply tax revenue losses elsewhere, so suboptimally low tax rates and foregone provision of public goods follow (see Wilson and Wildasin, 2004, or Zodrow and Mieszkowski, 1986). Increasing globalization and integration of regions and countries causes concern that governments dress down policies to attract economic activity. Accordingly, movements for fair or just taxation propagate the international harmonization of tax and industrial policy. However, with the advent of New Economic Geography, the recommendation to harmonize was revised (Baldwin and Krugman, 2004). Trade costs and imperfect competition give rise to agglomeration rents for firms (i.e., higher operating returns in larger markets), which make firms willing to pay higher taxes to locate in larger regions. Reducing taxes does not necessarily attract firms, because firms accept higher tax rates in large locations to cluster with other firms. Therefore, races to the bottom do not occur; rather, governments in large markets can set

☆ This article was written while affiliated to VU University Amsterdam. Financial support of Platform31 is gratefully acknowledged. ⁎ Tel.: +31 50 363 5995. E-mail address: [email protected].

http://dx.doi.org/10.1016/j.regsciurbeco.2014.08.004 0166-0462/© 2014 Elsevier B.V. All rights reserved.

high taxes (e.g. Jofre-Monseny, 2013).1 If such agglomerations occur, harmonization “always harms at least one nation” (Baldwin and Krugman, 2004), because the optimal rate of taxation is not the same across countries, and harmonization makes at least one country select a non-optimal tax rate. Instead of harmonization, the policy prescription is to set a tax floor: this prevents small regions from undercutting tax rates and so allows large regions to set still higher taxes. Following these results, a case can be made against harmonization in agglomerated economies: central action might not be required in low-tax countries in the EU, for instance. However, New Economic Geography (NEG) models typically introduced changes other than agglomeration to models without agglomeration forces. Firstly, along with the agglomeration rent, the large region typically has a first-mover advantage: the large region selects and commits to a tax rate before the small region does. In models without agglomeration, such a strategic advantage of setting policy first does not usually feature. Instead, no-agglomeration models typically study simultaneous policy-making, in which no country can commit to a tax rate beforehand. Secondly, in (NEG) models with increasing returns to scale, policies are often assumed to have no effect on the real economy: taxes are levied on firms and disappear from the accounts (Baldwin and Krugman, 2004, for instance). This is in contrast to most models without

1 The specific setting of this insight was originally a New Economic Geography model, but the results hold for agglomeration in general (Krogstrup, 2008).

M. Gerritse / Regional Science and Urban Economics 49 (2014) 48–57

agglomeration, in which governments transform tax revenue into public services (see, e.g., Wilson and Wildasin, 2004).2 These two simplifications in agglomeration models make the analysis of tax competition under scale effects more plain. However, they also obscure whether it is agglomeration effects or other changes with respect to the original tax competition model that accounts for the reversed policy conclusions. As a main contribution, this article shows that the agglomeration effects from the New Economic Geography do not make a case against harmonization — policy harmonization can still improve welfare if such agglomeration effects are present. The desirability of policy harmonization depends on the dynamics of the policy competition game. If large regions are Stackelberg leaders (i.e. set policies first), they commit to maintaining their agglomeration, and small regions never try to attract firms. If governments set taxes simultaneously, there is no such commitment device, and small regions set low tax rates with some probability because they can attract firms. Moreover, low tax rates from the small region force down the large region's tax rates, which may benefit consumers in the small region. The effect of firm taxes on firm entry in our Dixit Stiglitz model is an example of such a policy with spillovers: the small region's low tax decreases the tax rate in a neighboring large region, which increases the number of available firm varieties to consumers in the small region. In our model, if smaller regions attempt to attract firms, policy harmonization can form a Pareto improvement. Our results thus suggest that the timing and not agglomeration effects of NEG-based policy competition model account for whether harmonization is desirable or not. The policy conclusions therefore depend on whether large regions have timing advantages in policy formation. If governments choose their own timing, it is not likely that large economies form policy first (Kempf and Rota-Graziosi, 2010). Whether simultaneous or sequential policy-setting is more realistic probably depends on numerous factors, including the institutional arrangements and political cycles in different governments. The objective of this article is not to show which timing structure is more realistic. Rather, it shows that timing is responsible for changes in policy implications, which also occurs in the tax competition models without agglomeration effects (Gordon, 1992; Wang, 1999). The article is set up as follows. Section 2 briefly positions the contribution. We lay out the model of the economy in Section 3, where we treat government policy as given. This sketches how the economy responds to changes in government policy. In Section 4, we endogenize government policy, and study the solutions to the strategic situation that arises in this economy. Section 5 concludes. 2. Local policy competition: policy effects and interaction Baldwin and Krugman (2004) first examined international competition in tax rates when there are agglomeration rents. In their new economic geography model, factor rewards are higher in the large, core region than in the smaller peripheral region. As a result, the mobile factor does not leave the core if it is taxed to some extent. Baldwin and Krugman assume a first-mover (Stackelberg) advantage for the large, core region. They argue that the core's government exploits its firstmover advantage and agglomeration premium in factor rewards to set a higher tax rate than the peripheral government. The tax rate is low enough, on the other hand, to deter the peripheral government from undercutting the core's tax rate and attracting firms. Effectively, the core region sets a limit tax. Baldwin and Krugman “conjecture that [their] results hold in a broad range of models” next to the footloose entrepreneur model they use. A number of modifications have indeed confirmed their intuition. For instance, under incomplete agglomeration (Borck and Pflüger, 2006), two-factor models (Kind et al., 2000) and 2 Other real general equilibrium effects may occur, for instance on the wages of immobile workers (Wilson, 2005).

49

governments that pursue welfare of immobile workers instead of tax revenue (Ludema and Wooton, 2000), economically large regions still exploit firms' benefits associated with co-location. The results of the model of Baldwin and Krugman change when governments do not consume tax revenue, but spend their budget in the economy. Taxes may finance local public goods or production inputs that attract people or firms (Keen and Marchand, 1997). In particular, Brakman et al. (2002) show that increasing returns in public goods production or productivity-enhancing public investment may foster agglomeration. Likewise, Commendatore et al. (2008) show that productive public expenditure may attract more firms, but the tax that finances expenditure reduces demand, making the outcome of government policy ambiguous on balance. The observation that government expenditure has spatial implications is in line with the general equilibrium nature of new economic geography models. In the current model, the impact of government policy on local industry is felt in nearby regions as the number of imported varieties grows when nearby governments promote firm entry. Although less sophisticated, the entry effects of tax in our model tie in with arguments that government policy affects the organization of industries (Davies and Eckel, 2010; Pflüger and Suedekum, 2013). In particular, policy spillovers affect the desirability of tax floors. If restrictions-like tax floors allow large regions to reduce firm entry with higher taxes, the small region suffers from the restriction. The tax floor is therefore not a Pareto improvement. Regions interact in policy-setting, apart from policy spillovers. In the model of Baldwin and Krugman, the larger region has a firstmover advantage in addition to the advantage stemming from its size. The Stackelberg timing allows the larger region to credibly select a limit tax. The limit tax discourages the smaller region from setting a tax that could attract firms. Ludema and Wooton (2000) use a parallel requirement: the core needs to re-emerge as the core after the policies have been set in a stable equilibrium. The Stackelberg formulation circumvents the problem that there is no pure strategy Nash equilibrium in an agglomerated (“lumpy”) economy. The best responses of the two governments never coincide. If the peripheral regions attempt to attract firms, the core competes intensely. Facing this competition, the peripheral region prefers not to compete for firms. If the peripheral region does not attract firms, however, intense competition is not optimal for the core region. Essentially, a situation of “rock–paper–scissors” emerges: strategies are never mutually best responses. The Stackelberg advantage of the large region simplifies the model considerably, because it generates a limit tax as a (pure-strategy) subgame perfect equilibrium. Timing advantages change the nature of policy competition, whether there are agglomeration forces or not. In a commodity tax model, Wang (1999) shows that Stackelberg leadership in the tax competition casts doubt on the desirability of harmonization: the smaller region may be harmed by harmonization. The model, however assumes size differences from the outset, so agglomeration rents and lumpy firm movement play no role in the policy decisions. In a related paper, Gordon (1992) takes up the question why mobile capital is taxed at all in open economies. While this goes unexplained when governments set taxes simultaneously, Gordon shows that Stackelberg leadership of capital-exporting countries could explain positive tax rates. While related to the current article, Gordon studies a very different policy (capital taxation under a double-taxation convention) without making welfare recommendations, and does not study the role of size. Kempf and Rota-Graziosi (2010) study what tax setting sequence emerges if countries can choose their policy's timing in a capital tax competition model. They argue that smaller countries face higher tax base elasticities, which makes them set taxes before large countries. Instead, Ogawa (2013) shows that the government opt to choose taxes simultaneously, if the mobile capital is owned by residents instead of non-residents. Either result suggests that a Stackelberg leadership for the large country is arbitrary, and that the welfare conclusions change when allowing countries to set taxes sequentially. As tax competition models with agglomeration

50

M. Gerritse / Regional Science and Urban Economics 49 (2014) 48–57

typically also introduce a Stackelberg leadership, it is worthwhile to disentangle the role of first-mover advantages and agglomeration forces per se in the welfare conclusions. The current model contributes to the literature that studies the strategic context of policy competition, focusing on competition for firms (a mobile production factor) and not on commodity taxes (no “cross-border shopping” competition as Wang, 1999). Most importantly, compared to these articles, we discuss an economy with agglomeration forces, so that firm movement is lumpy — policymakers need to consider the possibility small tax changes relocate the majority of firms. As a result, economic size is endogenous (not assumed asymmetry via the population density like Wang, 1999, endowments like Gordon, 1992 or exogenous productivity as Kempf and Rota-Graziosi, 2010), which allows studying the consequences of centripetal forces next to timing advantages in the results of Baldwin and Krugman (2004). The sequential and simultaneous policy setting not only describe different institutional contexts, they also have different game theoretical characteristics. A drawback of simultaneous mixed strategies is that their interpretation is less straightforward compared to pure strategies. The mixed-strategy profile is a description of the likelihood of strategies being played, not an observable single pure strategy. On the other hand, the mixed-strategy equilibrium is robust to some criticisms of the Stackelberg form of the tax game. If policy setting is a repeated process, the Stackelberg follower may adopt rational punishing strategies that compromise the one-shot Stackelberg equilibrium (Cruz, 1975; Aoyagi, 1996). Since the mixed strategy equilibrium is a simultaneous subgame-perfect Nash equilibrium, its solution is robust to repetition. Similarly, the Stackelberg solution is sensitive to the number of players, but the mixed-strategy equilibrium is can be extended to multiple players (Fudenberg and Tirole, 1991, p. 98). Finally, and most important in the context of the model, the mixed-strategy equilibrium avoids one conceptual step. In the mixed-strategy equilibrium, the asymmetric position of smaller and larger regions ensures that the larger region has an incentive to set different tax rates than the small regions do. A region's propensity to select competitive strategies is a function of the number of firms in the region at the start of the game. This is in contrast to the sequential structure, where the policy selection follows the Stackelberg advantage, which follows from the regional size. Put differently, if the first-mover advantage was given to the smaller region, the outcomes of the sequential game would no longer be clear-cut. 3. Model setup This section develops a “footloose entrepreneur vertical linkages” (FEVL) model. The FEVL is a variant of the footloose entrepreneur model used in Baldwin and Krugman (2004). However, its agglomeration properties are virtually identical; and the break and sustain points are equal (Forslid and Ottaviano, 2003). It is not surprising, therefore, that Baldwin and Krugman's results hold in the FEVL model, as shown later. The addition of vertical linkages does not change the agglomeration model, but has substantial benefits in not requiring footloose entrepreneurs to be actors in the economy. Essentially the homogenous, immobile population in the vertical linkages model allows studying welfare functions like the “classical” tax competition literature does (Wilson and Wildasin, 2004). We elaborate on the welfare properties of this model in Section 4. This section describes a two-region model, where subscript r refers to variables relevant to only region 1 or 2. The population in both regions is fixed. Consumers derive utility from two types of private goods, food (Ca) and manufactures (captured in Cm), and a government-provided good G. Their utility function is two tiered, with Cobb–Douglas preferences over the three aggregates, and a CES-subutility function over the consumption c (i) of individual manufacturing varieties i: 1−α

U ¼ Ca

α

γ

C m Gr ; C m ¼

Z

n 0

ðσ−1Þ=σ

cðiÞ

σ =ðσ−1Þ di :

ð1Þ

In this utility function, the term 0 b α b 1 measures the preference for manufacturing goods in private consumption and 0 b γ b 1 governs preferences for government-provided goods. The term σ measures the elasticity of substitution between different manufacturing varieties i. Consumers spend their income net of wage taxes t on manufactures and food, which serves as a numeraire.3 The government good is not priced, it is financed from tax revenue. Using pðiÞ as the delivered price of a manufacturing variety, the private budget constraint is: Z Ca þ

n 0

cðiÞpðiÞdi ¼ ð1−t r Þw ¼ Y r ;

ð2Þ

where Yr denotes the per capita disposable income. The first stage of optimization (maximizing the top tier of the utility function) shows that spending on food is a fixed share of income: Ca = (1 − α)Yr. The residn

ual share α of income is spent on manufactures: ∫0 cðiÞpðiÞdi ¼ αY r . Using this constraint, maximizing the utility derived from manufacturing goods Cm yields the standard Dixit Stiglitz demand function for manufacturing varieties: −σ

cðiÞ ¼ pðiÞ

σ −1

αY r P r

Z ; Pr ¼

n 0

1−σ

pðiÞ

1=ð1−σ Þ di :

ð3Þ

Using the demand functions in the utility function gives a (linear transformation of) the indirect utility function: Vr = Gγr Yr/Prα. The net real wage Yr/Prα is higher where many varieties are present because the price index is lower when varieties are not imported. The price index is based on delivered prices, and with pricing based on the numeraire wage, the quantity of goods that needs to be imported determines the price index. Secondly, utility is higher if the government good Gr is provided more, ceteris paribus. Firms have variable and fixed costs of production, giving rise to internal increasing returns to scale. The variable part of production uses labor at inverse productivity am. The fixed costs arise because the firm needs to buy inputs from other firms as a fixed requirement F in production. Following convention (Baldwin et al., 2003, chapter 8), the fixed factor is assembled under a Cobb–Douglas technology that nests agriculture and a CES function of manufacturing varieties with the same parameters as the consumer's utility function.4 Using this technology, the firm's cost function is: α

TC ¼ am qðiÞw þ FP r ;

ð4Þ

where the costs of the fixed factor increase in the number of varieties that need to be imported. Each firm also supplies to other firms so it is neither upstream nor downstream, but rather horizontally linked. Note that the name of the model, Footloose Entrepreneurs with Vertical Linkages is somewhat misleading in this case: entrepreneurs collect no reward, the costs of the fixed factor are simply revenue to other firms. There are many firms and they maximize profits. Firms are small, and take the prices of other firms as well as the tax rates in both regions as given. Using the demand curve (Eq. (5)) and cost function (Eq. (4)), the optimal price for the local market is a markup over marginal costs, p(i) = σ/(σ − 1)amw, which is standard in this model of monopolistic competition (Dixit and Stiglitz, 1977). Moreover, mill pricing is optimal so that the price of a product sold in a foreign market is pðiÞ ¼ τpðiÞ. Aggregating the demand functions from both regions, a firm from

3 Using wage as a numeraire requires that the constant returns to scale agricultural sector is active in both regions, we provide the conditions in Appendix A. 4 The minimization of assembly cost is thus dual to the optimization of consumer utility:  α min ∫ q(i)p(i)di + Qa s.t. F ¼ ∫qðiÞðσ −1Þ=σ di Q a1−α . The price of the fixed factor F is σ σ−1

therefore equal to the aggregate consumption price index.

M. Gerritse / Regional Science and Urban Economics 49 (2014) 48–57

region 1 faces the demand curve (a parallel curve exists for a firm in region 2): " −σ

qðiÞ ¼ pðiÞ

α

b P 1−σ 1

þτ

1−σ

# 1−b w E : P 1−σ 2

ð5Þ

A fraction b of the expenditure stems from region 1, and the complement from region 2. Ew is the aggregate expenditure on agriculture and manufacturing goods, i.e. the sum of resident's income and firms' demand for inputs from both locations. Following convention, we normalize the markup to the inverse productivity. Using the pricing equation and the normalization, the manufacturing price indexes for either region (see Eq. (3)) raised to the power 1 − σ can be written as: 1−σ

P1

¼ n1 þ τ

1−σ

1−σ

n2 ; P 2

¼τ

1−σ

n1 þ n2 :

ð6Þ

With free entry and exit, the firms' profits are driven to zero. The fixed markup over marginal cost implies that a constant fraction (1/σ) of revenue are operating profits. The pure profits are the operating profits less fixed costs and taxes. We assume that the government taxes the nominal fixed costs. In equilibrium, the pure profits are equal to zero5:  −α  Π r ¼ pðiÞqr ðiÞ=σ− F P r þ t r ¼ 0:

ð7Þ

Following convention, we assume that firms enter when profits are positive, following the motion equation n˙ r ¼ ∏r (nr non-negative). Lastly, workers freely change jobs between agriculture and manufacturing, we normalize the constant agricultural productivity to one.

 t þ t2 w w þ n1 ðpq1 ðiÞ=σ −Ft 1 Þ þ n2 ðpq1 ðiÞ=σ− Ft 2 Þ: E ¼ N w 1− 1 2 Inserting the demand functions (Eq. (3)), aggregate expenditure simplifies to: w

w

N wð1−ðt 1 þ t 2 Þ=2Þ− F ðn1 t 1 þ n2 t 2 Þ : 1−α=σ

b¼

" # ð1−α=σ ÞðN1 wð1−t 1 Þ−Fn1 t Þ α b 1−σ 1−b n1 ; þ þ τ Nw wð1−t þt σ P 1−σ P 1−σ 2 Þ− F ðn1 t 1 þ n2 t 2 Þ 1 2 1

ð8Þ

Aggregate expenditure is equal to aggregate earnings of workers, less the labor taxes, Nww(t1 + t2)/2, and less firm taxes, F(n1t1 + n2t2) (“footloose entrepreneurs” claim no factor reward — all costs for the 5 The tax is at the firm level, as in the Baldwin and Krugman (2004) model. A tax on the nominal fixed factor implies that the optimal tax rate does not change in the price index. This avoids that tax-setting results are due to asymmetries in the costs of the fixed factor. Alternatively, the tax could be levied on revenue (profits, an alternative tax base, are zero in equilibrium). In the results below, taxation leads to a lower number of firms, but it is not obvious that this occurs under a revenue tax. If production were taxed, the profit function would be Π = pq/σ − FPα − tpq, so that the equilibrium output is q = FPαp−1/(1/σ − t). The direct effect of a tax is to expand production, and therefore reduce firm entry. However, the price p and price index P are now also affected by the tax rate and that general equilibrium effect not analytically solvable in a multiregion case.

2

ð9Þ

where the first term represents region 1's share in after-tax expenditure and the second term reflects the expenditure of firms from region 1. The expenditure share (Eq. (9)) is still a function of b, but it can be solved so that b is a function of region 1's share of firms n1/nw exclusively. The properties of this FEVL model are analyzed in Baldwin et al. (2003, 8.4). However, the current analysis studies core-periphery outcomes where all firms end up in one of the two locations. The coreperiphery patterns that are most relevant to this paper considerably simplify the expressions. 3.2. Tax policy to destabilize agglomerations We focus our discussion of tax competition in the economy on an agglomerated situation: region 1 hosts all firms; and region 2 hosts none.6 The agglomeration, however, is not cast in stone because undercutting the tax rate could allow region 2 to take over firms from region 1. For the equilibrium with all firms in region 1 to be stable, firm profits must be zero in region 1, and it must be unprofitable to set up a firm in region 2: nw = n1, Π1 = 0 and Π2 b 0. These no-profit conditions determine the number of firms under full agglomeration. Using Π1 = 0 and n1/nw = 1, rewriting the profit function for firms in region 1 (Eq. (7)) for the number of firms gives: w

Two further relations define the private equilibrium. The regional share in world expenditure determines demand in either region. Zeroprofit conditions determine the eventual distribution of firms over the regions. The expenditure originating from one region is the sum of inhabitants' expenditure for consumption, and firms' expenditure on inputs. By the zero-profit condition (Eq. (7)), the amount spent by a single manufacturing firm is equal to operating profits less any taxes: p(i)q(i)/σ − Ft. The aggregate expenditure of firms and inhabitants from one region is E = Nw(1 − t) + n(p(i)q(i)/σ − Ft). Summing over the two regions gives world expenditure:

E ¼

fixed factor are spent on other goods). Dividing the expenditure originating in region 1 by the world expenditure gives region 1's share of world expenditure, b:

n ¼

3.1. Equilibrium in the private sector

51

 w ðσ −1Þ=ðσ−1þα Þ αN wð1−t þt 2 Þ−σ Ft 1 n1 : F=ðσ −α Þ 1

2

ð10Þ

The number of firms is related to consumer's expenditure on manufacturing goods, and the taxes levied on firms (which enter with the inverse revenue to fixed costs ratio, σ). If the government of the agglomeration (region 1 in this case) raises taxes on firms, the equilibrium number of firms reduces: firm taxes act directly as an entry barrier, and they reduce aggregate demand, indirectly making entry less profitable. Given a tax rate in region 1, the equilibrium number of firms allows studying what tax rate in region 2 would make firm operations profitable in region 2. Given the agglomeration forces in this model, if one firm can operate profitably in region 2 at a low tax, then all firms will find it more profitable to move to region 2. Once a firm moves to region 2, demand from region 2 increases because that firm acquires inputs. Similarly, the cost of acquiring inputs for firms in region 2 falls, because more input suppliers are located in region 2. Under agglomeration in region 1, the profit function for a firm in region 2 is Π2 ¼

 α=n1  1−σ 1−σ α − Fτ n1 −Ft 2 : τ E1 þ E2 =τ σ α 1−σ

Rewriting for profits to be non-negative in region 2 gives that region 2 can take over the agglomeration from region 1 if:

t2 b

α σ

  τ 1−σ E1 þ E2 =τ1−σ − Fτα n1

1−σ−α 1−σ

Fn1

:

ð11Þ

The amounts of expenditure E, and the number of firms in the agglomeration n1 are a function of the tax rate of the agglomeration, t1. The tax of region 2 that marginally satisfies the above condition for profits in region 2 is the “break tax” t∗2, which is just low enough to attract firms. The break tax generally needs to be smaller than the tax 6 An equilibrium without agglomeration also exists, but to study the effects of agglomeration, we focus on agglomerated equilibria in the tax competition game.

52

M. Gerritse / Regional Science and Urban Economics 49 (2014) 48–57

prevalent in region 1, because it needs to compensate a firm i) for moving away from its buyers and ii) for moving away from its input suppliers (these are found back in the two respective terms in the numerator of condition 11). Following the notation of Baldwin and Krugman (2004), the break tax for region 2 can be written as a function Ω of region 1's tax rate: t∗2 = Ω(t1). Similarly, the condition t2 N Ω(t1) is a “no delocation condition”, because no firm has incentive to relocate if the condition is satisfied: prospective profits are higher in region 1 than in region 2. 4. The tax competition game Given the economy's response to different policy rates, we study strategic behavior by governments. Defining government behavior, however, requires a specification of what goals governments pursue. Following Baldwin and Krugman (2004), we first investigate an ad hoc objective function. The results of the original Baldwin and Krugman model are replicated exactly in this model, because the government is insensitive to whether agglomeration effects are caused by agglomeration forces via mobile entrepreneurs, or via vertical linkages between firms. In the footloose entrepreneur model without vertical linkages, an exact (social) welfare function has two important obstacles. First, footloose entrepreneurs and immobile workers are not the same, so the welfare function needs to specify the relative importance of entrepreneurs and immobile workers. Second, as entrepreneurs are mobile, peripheral governments need to take the welfare of its prospective entrepreneurs into account. In choosing payoffs, peripheral governments might therefore make policy choices based on potential inhabitants that are still outside their region. The ad hoc government preferences about tax revenue circumvent these obstacles: they do not specify actors in the economy, so do not need to address mobility and inequality. The vertical linkages as an agglomeration mechanism solve these issues. While agglomeration properties are alike, the costs of the fixed factor accrue to other firms in the vertical linkages model, rather than to a second, mobile type of worker. Therefore, there is no need to address intraregional heterogeneity. As a result the vertical linkages model enables a social welfare function like most other papers in the tax competition literature (see, e.g. Wilson and Wildasin, 2004). 4.1. The approximated welfare function Baldwin and Krugman use an ad hoc objective function to “avoid lengthy asides on political economy” and for solvability. They argue that governments like tax revenue but dislike high tax rates, and propose an approximation: 1 2 W ¼ G− t : 2 In this government objective or payoff function, tax revenue finances government goods G. Like the welfare function, the objective function is concave in the tax rate, which is required to solve the game. If the large region or core's policy choice precedes the peripheral region's policy choice, a subgame-perfect (Stackelberg) equilibrium emerges. The large region sets its tax rate such that the small region is

indifferent between setting the best tax rate that induces no firm to move location, or setting a tax rate that relocates the agglomeration. Suppose that the optimal tax rate for the periphery that preserves the agglomeration is t2lo, where lo refers to “local optimal”. Then, the large region 1 selects t1 such that: t2lo ∗ 0 − (1/2)(t2lo)2 = t∗2nw − (1/2)(t∗2)2, where t∗2 is a function of t1. We shall call this the “competitive” tax rate, t1c . The competitive tax rate is potentially different from the tax rate that region 1 would set if the agglomeration could not move, t1lo (the “local optimal” tax rate for region 1). Table 1 summarizes the payoffs of playing these different tax rates in a payoff matrix. In the payoff matrix, the local optimal tax rate t1lo is region 1's best response if region 2 plays t2lo. For region 1, welfare is higher if the tax rate choice when hosting the agglomeration is not restricted by region 2's competitive tax rate; t1lo(n1 + N1) − (t1lo)2/2 N t1c(n1 + N1) − (t1c)2/2. If this were not true, region 1 preferred competitive tax rates irrespective of region 2's policy, so there would be no tax competition. Conversely the competitive rate t1c is a best response if region 2 plays a break tax t∗2. This reflects that region 1 rather keeps the agglomeration than getting rid of it, which is also a requirement for competition for firms to occur. For region 2, the best response to t1lo is the break tax: it wants to take over the agglomeration. If region 1 is competitive, a local optimal tax rate yields the highest payoffs. There is no coincidence of best responses in the payoff matrix, so there is no pure-strategy Nash equilibrium. Endowing the large region with a first-mover advantage resolves that issue. By backward induction, the large region will not choose tlo 1 because that will trigger a break tax rate from region 2, shifting the agglomeration. Instead, the large region chooses a competitive tax rate that deters break tax rates from the small region. The first-mover advantage thus precludes firms moving to another region. Even if there is no pure-strategy Nash equilibrium, a Nash equilibrium still exists in this game albeit in mixed strategies (Nash, 1950). The mixed-strategy equilibrium assigns probabilities of being played to each non-dominated strategy. The mixed strategy is harder to interpret, but it does not have to assume a first-mover advantage to solve the game. Instead, it allows simultaneous policy choices. Therefore, other strategic incentives surface: the smaller region 2 is not presented with a done deal, and therefore acts less lethargically. In the simultaneous mixed-strategy equilibrium, both governments play either strategy with positive probability. Intuitively, if the large region's government always played the same strategy, say t1lo, then a break tax would become the dominant strategy for the small region. However, if the large region is certain that the small region sets break tax rates, it would select a competitive tax rate instead of a local optimal tax rate. Effectively, the game is in equilibrium if governments are indifferent between setting different tax rates. If playing one specific tax rate paid off better than another, it would always be selected, triggering different responses. In the equilibrium where governments set taxes simultaneously, the large region ensures that the smaller region is indifferent between a local optimal tax rate t2lo and a break tax rate t∗2. This equilibrium requirement is the same as the requirement for subgame-perfection in the sequential (Stackelberg) game: the large region eliminates the small region's incentive to compete for the agglomeration. For the large region, playing t1c yields a payoff of t1c(n1 + N1) − (t1c)2/2, irrespective of the tax rate of region 2.

Table 1 Payoffs under the ad hoc government objective. Region 2

Region 1

tlo 1 tc1

tlo 2

t∗2

 2 lo 2 lo t lo =2, N2tlo 2 − (t2 ) /2 1 ðn1 þ N 1 Þ− t 1  2 lo tc1n1 − (tc1)2/2, N 2 t lo =2 2 − t2

  2 lo 2  =2 tlo 1 N1 − (t1 ) /2, t 2 ðn2 þ N 2 Þ− t 2  2 t c1 ðn1 þ N 1 Þ− t c1 =2, t∗2N2 − (t∗2)2/2

M. Gerritse / Regional Science and Urban Economics 49 (2014) 48–57 Table 2 Payoff matrix under the welfare function. Region 2

Region 1

tlo 1

tc1

tlo 2

t∗2

h  iγ  γ lo ð1−t lo1 Þ N þ Fnlo ð1−t lo2 Þ Nt lo 1 t1 2 ,  α=ð1−σ Þ  α=ð1−σ Þ lo nlo τ α n1 1  γ

   γ ð1−tlo2 Þ Nt lo 2 ð1−t c1 Þ N þ Fnc1 t c1 ,  c α=ð1−σ Þ  c α=ð1−σ Þ τ α n1 n1

 γ

  γ ð1−t lo1 Þ Nt lo 1 ð1−tc2 Þ N þ Fnc2 t c2   α=ð1−σ Þ ,  c α=ð1−σ Þ τ α n2 n2

  γ

c γ 1−t c Nt 2 ð1−t c1 Þ N þ Fnc1 t c1 , ð  c2 Þα=ð1−σ  c α=ð1−σ Þ Þ n1 τ α n1

Whether harmonization is desirable, depends on the exact payoffs. Potentially, harmonizing at tlo 2 could Pareto-improve welfare, if that tax rate yields higher welfare than the equilibrium outcome for region 1 playing the competitive tax rate tc1. Whether the harmonized tax rate improves welfare in region 1 (and what the exact probabilities of each playing each tax rate are), depends on the welfare function specification. In that respect, working with an ad hoc welfare function could affect the policy conclusions. Moreover, the ad hoc welfare function possibly ignores important welfare aspects, such as policy spillovers and welfare effects of trade costs. 4.2. Policy under the exact welfare function Earlier tax competition models showed that fiscal externalities reduce average utility (Wilson and Wildasin, 2004). In the footlooseentrepreneur vertical linkages (FEVL) model, average utility is a plausible welfare function, because governments (out of benevolence or reelection chances) care about inhabitants that are homogenous and immobile. Using the indirect utility function as the welfare criterion, the task of each government is to raise taxes and provide public services. We assume the government uses the same technology as before: the supply of the government good is equal to the tax revenue. Hosting the agglomeration generally improves local welfare because it generates a larger tax base. A second benefit overlooked in ad hoc welfare functions is that the local presence of firms reduces prices for local inhabitants. In many tax competition models, firm presence has local benefits, for instance by driving up local wages (e.g. inflows of capital improve labor productivity in Wilson, 2005). In solvable NEG-type agglomeration models, wage are typically assumed equal across regions, and the ad hoc objective functions assume no benefit of hosting more firms. The welfare function in FEVL models recovers that insight through a consumer price effect. The indirect utility function (see Eq. (1)) can be written as an affine transformation of: γ

V¼

½ðN þ FnÞt  : Pα

ð12Þ

With agriculture as the numeraire, the wage and agricultural prices equal to one. The local price index is n1/(1 − σ) if all firms are in the region, and τ an1/(1 − σ) for inhabitants outside the agglomeration. The number of firms depends on the tax rate in the agglomeration (see Eq. (10)). Any tax revenue from local firms Fnt translates into a higher level of public services and is therefore good news for local inhabitants. In the following, we use shorthand notation nc1 to denote the number of firms in an agglomeration in region 1, under a competitive tax rate t1c (superscript lo refers to the local optimal tax rate). Under a first-mover advantage, the large region makes the smaller region indifferent between setting a “local optimal” tax or a break tax. Using the indirect utility function for welfare, the small region has no incentive to compete if the welfare under local optimal rates is at least as high as welfare under a break tax:    lo γ    lo c  c γ 1−t 2 Nt 2 1−t c2 N þ Fn2 t 2 ≥ : ð13Þ     c α=ð1−σ Þ c α=ð1−σ Þ τ α n1 n2

53

The number of firms under an agglomeration in the large region is n1(t1), and the number of firms following a break tax is nc2(= n2(t2∗ )). Region 2's welfare if not an agglomeration (the left-hand side of the inequality) discounts the number of firm varieties with τ α, which reflects that all manufacturing consumption needs to be imported. Rewriting region 2's condition for not preferring break tax rates gives: "

nc2

τ 1−σ nc1

#

α 1−σ

≥

" #γ 1−t c2 ðn2 ðt 2 Þ þ N2 Þt 2 : 1−t lo N2 t lo 2 2

ð14Þ

The right-hand side of condition (Eq. (14)) lists the tax revenue benefits from attracting firms (and consequently, providing public services). The left-hand side reflects the inverse price index if region 2 does not take over the agglomeration, relative to the price index if it does take over the agglomeration. Lower taxes in region 1 increase the number of firms in region 1, and lower the break tax t∗2. Therefore, lower taxes in region 1 discourage region 2 from attempting to attract firms. Second, reduced transport costs captured in lower τ make it more likely that the condition for region 2 not competing is met. Similarly, trade integration reduces the tax gap required for firms to relocate (t∗2 reduces, given t1). Therefore the conclusions of Baldwin and Krugman re-emerge: the core (region 1) sets a tax rate that deters the periphery from attempting to attract firms; and the equilibrium tax gap falls in the trade integration between the regions. The welfare function allows comparing payoffs of different tax pairs. The optimal mixed strategy is the result of the different payoffs; so unlike the ad hoc case, the welfare results and the equilibrium strategies are consistent. Table 2 provides an overview of the welfare levels that result from the potential tax pairs. Again, no Nash equilibrium in pure strategies exists. Fig. 1 provides a graphical representation of the strategic situation (with many potential tax rates). The first panel, (a), shows how the welfare function of region 1 depends on its own tax rate and on region 2's tax rate. All curves show a discontinuity at the break tax: with a smaller tax (left of the discontinuity), region 1 keeps the agglomeration, otherwise the agglomeration moves to region 2. If region 2's tax rate is lower, the maximum tax rate for region 1 to keep the agglomeration in region 1 is lower too: lower opponent taxes restrict the options to keep the agglomeration. If taxes in region 2 are high (e.g. 0.4 in this example), keeping the agglomeration is optimal. For lower taxes in region 2, region 1 i) needs to set lower taxes to keep the agglomeration and ii) loses less in case firms relocate: low taxes in region 2 ensure high firm variety. Hence, for lower taxes in region 2, it becomes attractive to set high taxes in region 1. Panel (b) depicts the reaction curves (or best response curves) corresponding to the situation. Both regions set high tax rates if the opponent has very low tax rates and low tax rates if opponent tax rates are high. For intermediate ranges, the government chooses a break tax along the break tax line (gray line). Note that along the break tax line, it is optimal to slightly undercut the break tax, so the reaction curves do not coincide at the break tax line. As a result, the reaction curves do not coincide anywhere: there is no pure strategy Nash equilibrium in this game. In the mixed strategy equilibrium, both governments chose probabilities of playing either tax rate such that for the other government, expected payoffs are the same for each tax rate. If expected payoffs were not equal, at least one government has an incentive to always play the tax rate that yields higher payoffs. The game has a mixed Nash equilibrium, because both the number of players and the number of strategies are finite (Nash, 1950). Using p as the probability that region 1 chooses a competitive tax rate t1c , region 2 is indifferent if:       lo γ lo γ Nt 2 Nt 2 1−t lo 1−t lo 2 2 ð1−pÞ  α=ð1−σ Þ þ p  c α=ð1−σ Þ lo τ a n1 τa n1      c γ c  c γ 1−t c2 N þ Fn2 t 2 1−t c Nt 2 ¼ ð1−pÞ þ p  c2α=ð1−σ :  c α=ð1−σ Þ Þ n2 τ a n1

54

M. Gerritse / Regional Science and Urban Economics 49 (2014) 48–57

(a)

(b)

Fig. 1. Payoffs and reaction curves.

Vice versa, region 1 is indifferent between playing t1lo and t1c if region 2 chooses its probability q of playing a break tax t2∗ such that:  h  i    lo lo γ lo γ N þ Fn1 t 1 Nt 1 1−t lo 1−t lo 1 1 þq ð1−qÞ  α=ð1−σ Þ   α=ð1−σ Þ lo τ a n2 n1       c  c γ c  c γ 1−t c1 N þ Fn1 t 1 1−t c1 N þ Fn1 t 1 ¼ ð1−qÞ þq :  c α=ð1−σ Þ  c α=ð1−σ Þ n1 n1 These equalities of expected payoff define the mixed strategy profile. Solving for the probabilities of playing competitive tax rates, p and q, gives: p¼

1

;  c  γ h lo γ   γ i t 1 N2 t 2 − t 2   h i 1 þ          γ lo lo γ n2 t 2 N 2 þ n2 t 2 F t 2 −τ−α n1 t 1 N2 t2 τ−α n1

−α 1−σ

−α 1−σ

−α 1−σ

q¼

ð15Þ

1

i :  c    c   c γ   h lo γ n1 t 1 N 1 þ n1 t 1 F t 1 −τ−α n2 t 2 N1 t 1    iγ    1 þ  lo  h     lo lo c c c γ n1 t 1 N 1 þ n1 t 1 F t 1 −n1 t 1 N1 þ n1 t 1 F t 1 −α 1−σ

−α 1−σ

−α 1−σ

−α 1−σ

For both regions, the probability of playing a competitive strategy is positive but smaller than 1. To see that p is smaller than 1, it is useful ∗ to recall that tlo 2 N t2 (in order to convince firms to move, region needs to lower its tax rate to t⁎ from its local optimal rate t lo); and that for region 2, the payoff to a successful break tax rate is higher than local opti      γ N2 þ n2 t 2 F t 2 N mal strategies (which implies that n2 t 2   h iγ  lo lo −α N2 t 2 τ n1 t 1 . The latter condition is required for governments −α 1−σ

−α 1−σ

to want to compete for firms. Similarly, in the expression for q, it iγ       γ   h N1 þ n1 t c1 F t c1 Nτ−α n2 t 2 N1 t lo , must hold that n1 t c1 1 −α 1−σ

−α 1−σ

which states that the large region prefers to defend its agglomeration with a low tax rate instead of having the “local optimal” rate without firms. Similarly, by definition, the large region prefers to retain the agglomeration with a local optimal tax rate tlo 1 instead of the compet  h    iγ c lo lo itive tax rate t1, which ensures that n1 t 1 N 1 þ n1 t lo N 1 F t1  c    c   c γ N1 þ n1 t 1 F t 1 . n1 t 1 While the presence of an agglomeration leads optimal taxes to diverge, the strategies p and q also allow studying whether such divergence has strategic reasons. Probabilities p and q state how often governments deviate from that “local optimal” tax rate. A decrease in transport costs τ leads to a decrease of the probability p that the large region selects a competitive tax rate.7 The core selects competitive −α 1−σ

−α 1−σ

7

This is most easily seen by multiplying the numerator and the denominator in the ex2 3−1  γ  

pression for p with τα, so that p becomes 41 þ

−α 1−σ

γ

 n1 t c1 N2 t lo 2 −t 2   h iγ 5       γ τ α n2 t 2 N2 þ n2 t 2 t 2 −n1 t lo N 2 t lo 1 2 −α 1−σ

−α 1−σ

.

taxes less often under lower transport costs, because it has become less attractive for the periphery to take over the agglomeration — the costs of importing are lower. By contrast, the periphery selects competitive tax rates more often when trade costs decrease. With lower transport costs, the costs of losing the agglomeration are also lower to the core. To ensure that the core does not switch to local optimal tax rates, the periphery needs to increase the core's odds of losing firms when not selecting competitive tax rates. As before, integration (reducing transport costs) gives strategic incentives for the policy gap to decline when governments select tax rates simultaneously.8 If we assume for a moment that the small region 2's competitive tax rate is tc (so that it takes over the agglomeration if the large region selects tlo but not otherwise), we can study the effect of tax-induced firm entry and exit on the policy game. In that case, if the agglomeration is won or held with tc, the corresponding number of firms is nc. If nlo / nc is close to 1 (the difference in firm population under “local optimal” and competitive taxes is small), p is smaller: the core is less competitive.9 The reason is that the returns to a competitive tax rate in terms of firm variety are low; so the core does not need to discourage the periphery from competing. The reverse is true for the small region 2: if nlo is close to nc, q is higher. If higher taxes do not discourage firm entry, the core is more tempted to set high taxes. The periphery is therefore more likely to attract firms with a competitive tax rate. The same effect is brought about by a high σ: a high firm variety gets a lower weight in the welfare function. The simultaneous and sequential game predict different policy.10 The Stackelberg (sequential) game equilibrium is the lower-left cell of Table 2, (t c, t lo). Under simultaneous strategies, this is not an equilibrium, because region 1 prefers to set a higher tax rate t1lo. However, the best response of region 2 to tax t1lo is the break tax t∗2. This threat is useful to region 2: if it cannot take over the agglomeration, it is still better off under tc1 than under t1lo. The reason is that the resulting number of firms is higher under a lower tax rate: n1(t1c) N n1(t1lo). Hence, potentially setting a break tax rate allows region 2 to increase firm variety in the economy. The goods market spillover can therefore provide a motive to set lower taxes. A comment is in order on the outcomes for region 2 in case it attempts to attract the agglomeration but fails. If region 2 makes a failed attempt to attract firms, it sets a low tax rate, but not low enough to attract firms. In our setup, where the firms and workers are taxed, such a failure is costly: given that region 2 fails to attract the agglomeration, a 8 Note that this result is on the strategic incentives, it is feasible that transport cost reductions affect the break tax and equilibrium number of firms in the opposite way. 9 Again, seeing this is the easiest when dividing the fraction in the denominators of p and q by nc. 10 As a reviewer pointed out, there is a third possibility, in which the small region is the Stackelberg leader. In that case, compared to the game with the large region as Stackelberg leader, the same agglomeration pattern can occur but the tax rates are not generally the same. The analysis is available in an unpublished Appendix.

M. Gerritse / Regional Science and Urban Economics 49 (2014) 48–57

a)

b)

55

c)

Fig. 2. Strategy profiles numerically.

local optimal tax would have yielded higher welfare. If workers were not taxed, failing to attract firms is no issue — tax revenue from firms would be zero either way. Therefore, the extension of the tax to workers (that is also present in tax-competition-with-agglomeration models, like Baldwin and Krugman) ensures that competitive efforts are not costless. The cost of competing reflects that the small region commits resources to attracting firms. The expected tax revenue from firms reduces the tax that needs to be levied from citizens. Failing to attract firms, however, the budget for public services is suboptimally low. An alternative means of ensuring that competing for firm is not free is to allow governments to optimize the taxes on citizens separately from the taxes on the mobile factor. In that case, the decreasing marginal returns to public services lead to lower optimal taxes on citizens if there is tax revenue from firms. This requires that taxes are set before firm relocation materializes. Either when optimizing taxes on firms and citizens separately, or when optimizing a single tax rate, failing to attract firms misaligns tax rates. Finally, the analytical results use two representative strategies. Especially for break taxes, however, there is more than one pair of tax rates that classifies as “competitive”. For completeness, we report the numerical results that allow for many strategies, yielding a grid of 625 policy outcomes. We construct the payoff matrix for each government by calculating local welfare for each strategy pair. The discretization is required for a numerical approximation, and it also guarantees the existence of a mixed strategy equilibrium (Dasgupta and Maskin, 1986). The mixed strategy equilibrium is solved using a Lemke and Howson (1964) algorithm in Gambit software. Fig. 2 plots the strategy profiles for the tax competition game with many strategies. The density functions show for each level of tax rates what the equilibrium probability is that a government selects it. The numerical strategy profile mimicks the analytical results. Indeed, not only a single tax rate but also a set of tax rates can be a best response, so more than one tax rate have a positive probability of being played. The taxes that the large region selects are possibly lower,11 but it is possible for the agglomeration to shift between region 1 and region 2. If trade costs decrease, the results for the tax rates chosen are ambiguous. Although a lower p suggests higher equilibrium taxes due to higher competitiveness, other equilibrium relations including the break tax gap change — in this case, the tax gap has increased. A higher σ yields generally higher tax rates, because the loss of firm variety caused by high taxes is given a less grave weight when σ is high.

4.3. Harmonization and policy prescription If governments set policy sequentially, tax harmonization is generally not a Pareto-improvement. Any tax between the core and periphery's Stackelberg tax rates reduces welfare in at least one region. The periphery's tax rate is a constrained maximum (given its inability to attract firms). By construction, the core can set the peripheral tax rate (firms will not move), but that is not generally optimal. Related literature (Baldwin and Krugman, 2004; Ludema and Wooton, 2000; Kind et al., 2000) conclude that instead of harmonization, a tax floor is welfare improving. That is not necessarily true in the current model. A tax floor, even if it does not affect the local optimal tax of region 2 (tlo) allows the large region to set higher tax rates. The implication of higher core taxes for the small region is that fewer firms enter; so there is a lower variety of goods in the economy. For that reason, a tax floor lowers welfare in the small region — the tax floor takes away the threat that allows the small region to internalize goods market spillovers of the large region's policy. In contrast to the sequential game, harmonization can be a Pareto improvement in the simultaneous, mixed strategy game. To develop this point, we use a harmonized tax rate, t. Harmonization is a Paretoimprovement if both governments choose harmonization over playing the mixed strategy game (so that the payoff under harmonized taxes is higher than the expected payoff of the simultaneous tax-setting game). For the larger region 1, it is straightforward to see that a welfare-improving harmonized tax rate exists. We take advantage of the fact that in a mixed strategy equilibrium, the expected payoff is the same for every pure strategy. For region 1, the expected payoff of       γ N1 þ n1 t c1 F t c1 . The competitive tax rate tc is playing tc is n1 t c1 constrained by the periphery's willingness to compete for the agglomeration, and a harmonized tax rate marginally higher than tc1 yields a higher payoff for the large region. For the smaller region 2, we compare the payoffs under the harmonized tax rate to the expected payoff of setting a competitive tax rate (tc). The harmonized payoff is higher if: −α 1−σ

 c α=ð1−σ Þ  c   c  c γ −α  c α=ð1−σ Þ  c 

c γ ð1−pÞ n2 1−t 2 N þ Fn2 t 2 þ pτ n1 1−t 2 Nt 2    

 −α γ bτ n t 1−t N2 t : −α 1−σ

ð16Þ If region 2 selects tc2, the worst case scenario is that region 1 also selects a competitive tax rate tc. To see when harmonization is beneficial, we consider the required premium z of harmonization over region 2's     1−t c2 worst case scenario (failing to attract firms), so that τ−α n t c1    



c γ γ Nt 2 þ z ¼ τ−α n t 1−t N2 t . The term z reflects the compensation over the worst possible outcome that harmonization −α 1−σ

11

The strategy is based on welfare, not tax revenue. Therefore, if a location has a larger firm population, citizens possibly prefer a lower tax rate as firm taxes pay for part of the government services.

−α 1−σ

56

M. Gerritse / Regional Science and Urban Economics 49 (2014) 48–57

  

γ yields. From region 1's problem, we have that τ−α n t 1−t N2 t ≥    

γ τ−α n t c1 1−t c2 Nt c2 (both sides are discounted by transport costs), so z is positive. Inserting the endogenous probability of region 1 setting competitive tax rates p (Eq. (15)) into condition 16 along with the expression for the harmonized tax rate, and solving for z gives the condition for welfare benefits from harmonization: −α 1−σ

−α 1−σ

!   lo γ



c γ

c γ c  c γ ð1−t lo2 Þ Nt 2 1−t c Nt 2 ð1−tc2 Þ N þ Fn2 t 2 ð1−tc2 Þ Nt 2 − ðα  c2 Þα=ð1−σ −  c α=ð1−σ Þ     Þ c α=ð1−σ Þ c α=ð1−σ Þ τ n1 τ α n1 n2 τ α n1  γ   : lo lo γ

  

 c c γ c γ ð1−tlo2 Þ Nt 2 ð1−t lo2 Þ Nt 2 ð1−tc2 Þ N þ Fn2 t 2 ð1−tc2 Þ Nt 2   þ − −  c α=ð1−σ Þ  c α=ð1−σ Þ  c α=ð1−σ Þ lo α=ð1−σ Þ τα n1 n2 τα n1 τα n1

5. Conclusion

 zN−

ð17Þ

While this expression is long, its result is quite intuitive. The term

  γ

c γ 1−t c Nt 2 ð1−tc2 Þ N þ Fnc2 t c2 − ð  c2 Þα=ð1−σ  c α=ð1−σ Þ Þ n2 τα n1

is the welfare premium of attracting the

firms versus failing to attract them with a competitive tax rate. The term

 γ

γ ð1−tlo2 Þ Nt lo 2 ð1−t c Þ Nt c2  c α=ð1−σ Þ −  c2 α=ð1−σ Þ α α τ n1 τ n1

gray. In all parameterizations, a set of tax rates exists for which both regions prefer to harmonize tax rates, rather than participate in the tax game. This confirms the analytical result. Numerically, the scope for harmonization is marginally smaller if σ is higher. This conforms to intuition: if σ is large, the small region does not care as much about goods market spillovers from the large region's policy.

is the welfare premium of setting a local

optimal tax rate instead of a competitive tax rate when failing to attract firms. Both premia are positive or zero, so the right-hand side of Eq. (17) is non-positive.12 By virtue of the local optimal tax rate, a tax rate marginally larger than the competitive rate improves welfare. Thus, a range of tax rates exists for which both the core and the periphery prefer harmonization over the expected payoff of the tax game. So if the agglomeration does not relocate, why does the periphery prefer the harmonized tax rate over the tax rate tlo that leads to the constrained optimum? Tax harmonization on a tax rate higher than tc increases the expected amount of product varieties that the periphery consumes. Thus, harmonization internalizes a goods market spillover from the large region's policy in the number of firms. If the firm variety policy spillover is larger, region 2 is more likely to prefer harmonization. In the original tax-competition-with-agglomeration models, the number of firms is fixed (by assuming a fixed number of entrepreneurs), precluding such spillovers. When the number of firms is fixed, the periphery is indifferent about the tax rate of the core. As a result, the expected payoff of the mixed strategy game for fixed n is always achieved by setting tlo 2 ; and harmonization is not generally welfare-improving for periphery. Apart from the firm variety spillover, a number of other, unmodeled policy spillovers could occur when tax rates affect wages, prices, demand or productivity. Second, misalignment of tax rates is crucial for the results: it must be costly to deviate from local optimal tax rates if the core is not attracted. If the competitive tax rate can be reneged upon once it has failed, it becomes a dominant strategy to set competitive tax rates and if they fail, change them to local optimal rates. In that case, the harmonization condition becomes trivial: competitive tax rates are always optimal. A sequential, Stackelberg setting rules out the possibility of tax rate misalignments through backward induction of region 1. Under a simultaneous tax setting, however, tax rate misalignment determines the policy conclusions. In a game where multiple tax rates can be chosen, we need to rely on numerical results because there are no closed form solutions. The expected payoff of the policy game can be calculated, however. As defined before, harmonization is a Pareto improvement when under a harmonized tax rate, both regions have higher welfare than expected welfare under the mixed-strategy game. In the strategy profiles of Fig. 2, tax pairs for which harmonization is Pareto-improving are highlighted in 12 c lo lo c lo The denominator is always positive because it is V(tlo 1 , t2) − V(t1 , t2 ) + V(t1, t2 ) − c lo lo V(tc1, tc2), where V(tlo 1 , t2) − V(t1 , t2 ) is the welfare premium of attracting the core and c c V(tc1, tlo 2 ) − V(t1, t2) is the welfare premium of not failing to attract the core. Also note that this inequality holds for nc1 = nc2. However, typically tc2 b tc1: region 2 needs to compensate agglomeration effects in addition to low taxes to attract firms. A successful agglomeration takeover will therefore generally increase the global number of firms: nc1 b nc2. So, condition 17 generally holds if competitive tax rates for region 2 are somewhat lower than that of region 1.

This article studies the welfare conclusions of policy competition under agglomeration. Agglomeration forces change the nature of the “race to the bottom”, in which governments compete to attract firms with low taxes, hence dressing down their government services. However, agglomeration-based models typically also introduce firstmover advantages. Based on the combination of first-mover advantages and scale effects, agglomeration model show that harmonization is undesirable. But is it the first-mover advantage or the agglomeration effect that makes harmonization undesirable? This article disentangles the strategic incentives of large regions associated with agglomeration forces from the advantages associated with the Stackelberg (first-mover) advantage. It develops a spatial general equilibrium model of agglomeration (a footloose entrepreneur model with vertical linkages), in which government tax firms to supply public services. The game has no simultaneous pure strategy Nash equilibrium, so in addition to the sequential pure strategy Nash equilibrium (i.e. the Stackelberg equilibrium), it develops a simultaneous (mixed strategy) Nash equilibrium. Throughout, the main predictions on government behavior are consistent with earlier findings: large regions set taxes to deter the peripheries from competing for firms, and greater agglomeration advantages increase the policy gap. The welfare conclusions, however, show that the case against harmonization rests on the assumed timing advantages in policy-setting for large regions. If governments set policy at the same time instead of sequentially, there are harmonized policies that both governments would choose over the laissez-faire equilibrium. The intuition is that the credible threat of a large first-mover eliminates all incentives for smaller regions to compete for firms in the Stackelberg equilibrium. This threat is no longer credible under simultaneous moves, so small regions get an incentive to set tax rates that could attract firms. Attempts to take over the agglomeration may explain peripheral low taxes, but low peripheral taxes also act as a disciplining device toward the core. If there are policy externalities, forcing down core taxes with low peripheral taxes increases welfare in the periphery. From policy as chosen in the simultaneous game, policy harmonization can be a Pareto improvement. This is in sharp contrast to the sequential model (and earlier agglomeration based literature) that recommends against harmonizing policies. Earlier literature also suggested a tax floor as a Pareto improvement under sequential policy formation. This article shows that this does not generally hold if there are policy externalities (i.e. free entry in the monopolistically competitive market in this model). Our results imply that policy harmonization may be optimal, even with agglomeration forces. Under agglomeration externalities, the conclusion depends on whether one views policy-making as sequential or simultaneous. It is not our purpose to discuss whether sequential or simultaneous policymaking is more realistic, but to show that introducing first-mover advantages assumes away important strategic advantages. By shedding light on the conditions for policy harmonization to improve welfare, the article contributes to a growing debate over the desirability of harmonizing regional and international policies to attract firms. Appendix A. Condition for incomplete specialization To use the wage rate as a numeraire, it needs to be equal between the two regions. As agriculture is freely traded and workers are mobile between sectors, it suffices that the largest region has at least some

M. Gerritse / Regional Science and Urban Economics 49 (2014) 48–57

agricultural workers. In the large region, the number of workers in manufacturing is n ∗ l, where l is the firm's labor requirement. Using the price indexes for an agglomeration in region 1, the output of a manufacturing firm in region 1 is −σ w

q¼w

E =n1 :

ðA:1Þ

Using the firm technology and the expression for world expenditure, the aggregate employment in manufacturing is    α ðt þ t 2 Þ w −σ am N w 1 þ 1 −n1 t 1 w : 1−α=σ 2 Using the government's budget requirement and wage as a numeraire, the requirement that manufacturing employment is less than total employment can be written as: N1 N

   ασ ðt þ t 2 Þ w N w 1þ 1 −G1 : σ −α−α σ t 1 2

ðA:2Þ

Effectively, the use of the wage as a numeraire implies an upper bound on taxes, as taxes expand the size of manufacturing firms. We assume this inequality to hold in the theoretical results, and choose parameter values and strategy spaces consistent with this requirement. References Aoyagi, M., 1996. Reputation and dynamic Stackelberg leadership in infinitely repeated games. J. Econ. Theory 71 (2), 378–393. Baldwin, R.E., Krugman, P., 2004. Agglomeration, integration and tax harmonisation. Eur. Econ. Rev. 48 (1), 1–23. Baldwin, R., Forslid, R., Martin, P., Ottaviano, G., Robert-Nicoud, F., 2003. Economic Geography and Public Policy. Princeton University Press. Borck, R., Pflüger, M., 2006. Agglomeration and tax competition. Eur. Econ. Rev. 50 (3), 647–668. Brakman, S., Garretsen, J.H., van Marrewijk, C., 2002. Locational Competition and Agglomeration: The Role of Government Spending. CESifo Working Paper No. 775.

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Commendatore, P., Kubin, I., Petraglia, C., 2008. Productive public expenditure in a new economic geography model. Econ. Int. 114, 133–160. Cruz, J.J., 1975. Survey of Nash and Stackelberg Equilibrium Strategies in Dynamic Games. Annals of Economic and Social Measurement. vol. 4, number 2. National Bureau of Economic Research, Inc., pp. 339–344 (NBER Chapters). Dasgupta, P., Maskin, E., 1986. The existence of equilibrium in discontinuous economic games, i: Theory. Rev. Econ. Stud. 53 (1), 1–26. Davies, R., Eckel, C., 2010. Tax competition for heterogeneous firms with endogenous entry. Am. Econ. J. Econ. Pol. 2 (1), 77–102. Dixit, A.K., Stiglitz, J.E., 1977. Monopolistic competition and optimum product diversity. Am. Econ. Rev. 67 (3), 297–308. Forslid, R., Ottaviano, G.I., 2003. An analytically solvable core-periphery model. J. Econ. Geogr. 3 (3), 229–240. Fudenberg, D., Tirole, J., 1991. Game Theory. MIT Press, Cambridge, MA. Gordon, R.H., 1992. Can capital income taxes survive in open economies? J. Financ. 47 (3), 1159–1180. Jofre-Monseny, J., 2013. Is agglomeration taxable? J. Econ. Geogr. 13 (1), 177–201. Keen, M., Marchand, M., 1997. Fiscal competition and the pattern of public spending. J. Public Econ. 66 (1), 33–53. Kempf, H., Rota-Graziosi, G., 2010. Endogenizing leadership in tax competition. J. Public Econ. 94 (9–10), 768–776. Kind, H.J., Knarvik, K.H.M., Schjelderup, G., 2000. Competing for capital in a ‘lumpy’ world. J. Public Econ. 78 (3), 253–274. Krogstrup, S., 2008. Standard tax competition and increasing returns. J. Public Econ. Theory 10 (4), 547–561. Lemke, C.E., Howson Jr., J., 1964. Equilibrium points of bimatrix games. J. Soc. Ind. Appl. Math. 12, 413–423. Ludema, R.D., Wooton, I., 2000. Economic geography and the fiscal effects of regional integration. J. Int. Econ. 52 (2), 331–357. Nash, J.F., 1950. Equilibrium points in n-player games. PNAS 36 (1), 48–49. Ogawa, H., 2013. Further analysis on leadership in tax competition: the role of capital ownership. Int. Tax Public Financ. 20, 474–484. Pflüger, M., Suedekum, J., 2013. Subsidizing firm entry in open economies. J. Public Econ. 97, 258–271. Wang, Y.-Q., 1999. Commodity taxes under fiscal competition: Stackelberg equilibrium and optimality. Am. Econ. Rev. 89 (4), 974–981. Wilson, J.D., 2005. Welfare-improving competition for mobile capital. J. Urban Econ. 57 (1), 1–18. Wilson, J.D., Wildasin, D.E., 2004. Capital tax competition: bane or boon. J. Public Econ. 88 (6), 1065–1091. Zodrow, G.R., Mieszkowski, P., 1986. Pigou, Tiebout, property taxation, and the underprovision of local public goods. J. Urban Econ. 19 (3), 356–370.