I. INTRODUCTION

The influence of national tax policy on geographic patterns of industry continues to be a key concern among policy makers at local, regional, and national levels, as seen, for example, in the recent Organisation for Economic Cooperation and Development proposal for reforms to national corporate tax policy (The Economist 2015). The attention on corporate tax rates has in part been driven by the European Commission's investigations into a number of preferential tax settlements completed between EU member countries and multinational firms (The Economist 2016). One aspect of the debate on corporate tax reform has emphasized the pro-growth effects of low corporate tax rates as a component of trade policy. Indeed, there is strong empirical evidence suggesting that corporate tax rates are an important factor in the location decisions of firms (Brulhart, Jametti, and Schmidheiny 2012; Feld and Heckemeyer 2011; Kammas 2011). Given these policy trends, it is important to consider how economic growth is affected by shifts in production between locations, as firms respond to changes in corporate tax differentials.

This paper attempts to address this question by studying the effects of corporate taxes on industry location in an endogenous market structure and endogenous growth framework (Aghion and Howitt 1998; Etro 2009; Laincz and Peretto 2006; Peretto 1996; Smulders and van de Klundert 1995). (1) Specifically, we extend the two-country model of Davis and Hashimoto (2016), within which monopolistically competitive firms produce differentiated products for supply to their domestic and export markets, and employ labor in process innovation with the aim of reducing unit production costs. Labor productivity in process innovation is determined by the weighted average productivity of the production techniques visible to a firm, with a stronger weighting for production located in proximity to the firm's innovation activity. As a by-product of current innovation efforts, knowledge accumulates within the production technology of each firm, reducing future innovation costs and potentially generating endogenous productivity growth.

The imperfect nature of knowledge diffusion between countries leads to lower innovation costs for the country hosting the largest share of production, thereby linking the location patterns of industry and innovation activity. Perfect capital mobility allows firms to locate production and innovation independently in the countries with the lowest costs. Accordingly, with symmetric labor endowments, trade costs ensure that a larger share of firms locate production in the country with the lowest corporate tax rate, following the home market effect of the New Trade Theory (NTT) literature (Krugman 1980; Martin and Rogers 1995). As a consequence, with imperfect knowledge diffusion, all firms locate process innovation in the low-tax country in order to take advantage of its low innovation costs.

We use the framework to consider the effects of changes in national tax policy on market entry and productivity growth. An increase in the international corporate tax differential raises the production share of the low-tax country, and thereby affects market entry and investment in innovation through two channels. First, industry-level profits are convex in the corporate tax differential, as the shift in the location of production toward the low-tax country both intensifies competition and lowers the industry-level cost of transporting goods between countries. The negative competition effect dominates and industry profits fall for a small tax differential, while the positive trade cost effect dominates and profits rise for a large tax differential, a standard result of the New Economic Geography (NEG) literature (Baldwin et al. 2003). The second channel is a knowledge spillover effect, through which greater industry concentration in the low-tax country improves labor productivity in innovation, prompting firms to increase employment in process innovation. Then, because firm-level employment in process innovation represents a fixed cost at each moment in time, firms tend to exit the market, causing the firm-scale of production to expand.

Therefore, although an increase in the corporate tax differential accelerates productivity growth as firm-level innovation employment rises, adjustments in the level of market entry depend on whether the profit and knowledge spillover effects are aligned or opposed. On the one hand, if the initial tax differential is small, the profit and the knowledge spillover effects are both negative and market entry falls. On the other hand, if the initial tax differential is large, the profit effect is positive and the relationship between corporate tax policy and market entry is generally ambiguous.

We also study the relationship between corporate tax policy and national welfare levels. Changes in the corporate tax differential affect welfare levels through three channels. First, process innovation leads to falling prices that generate a common rate of growth in national welfare across countries. Second, national welfare is linked with market entry through the preference of households for greater product variety: the love of variety effect (Helpman and Krugman 1985). The third channel is associated with adjustments in the average price of manufacturing goods in each country. A rise in the corporate tax differential increases the concentration of production in the low-tax country, causing the average price of manufacturing goods to fall in the low-tax country and to rise in the high-tax country. Comparing the strengths of the three channels, we show analytically that in our framework increased industry concentration resulting from a larger tax differential raises the welfare of the low-tax country, but may raise or lower the welfare of the high-tax country.

The empirical literature tends to support a negative relationship between corporate taxes and economic growth, as reported by, for example, Kneller, Bleaney, and Gemmell (1999), Lee and Gordon (2005), Arnold et al. (2011), and Gemmell, Kneller, and Sanz (2011, 2014). However, Angelopoulos, Economides, and Kammas (2007) report a positive relationship between corporate taxes and economic growth, while Ojede and Yamarik (2012), Xing (2012), and Arachi, Bucci, and Casarico (2015) find that tax policy has no significant influence on growth. With the empirical literature generally concluding that corporate taxes adversely affect growth, our contribution is a theoretical framework that emphasizes the importance of considering the firm location decision, when estimating the size of the effects of corporate taxes on economic growth, as discussed in Gemmell, Kneller, and Sanz (2014).

There is a broad theoretical literature studying the relationship between corporate taxes and economic growth. Within the endogenous market structure and endogenous growth literature, (2) Peretto (2003a, 2007) considers the effects of fiscal policy in closed economy models, and finds that when research and development (R&D) costs are fully expendable, productivity growth is positively related with corporate taxes. In particular, with positive entry costs, an increase in the corporate tax rate lowers firm profits, reducing the incentive to enter the market and shifting labor from product development to process innovation. This mechanism is neutralized in this paper with the assumption of zero entry costs, allowing us to focus on the corporate tax effects resulting from adjustments in knowledge spillovers as industry shifts locations between countries in an open economy framework. Iwamoto and Shibata (2008) and Palomba (2008) also develop open economy models using overlapping generations frameworks to study the effects of capital income tax rates on the movement of capital and the rate of capital accumulation, and find that lowering tax rates with the aim of attracting capital may have a negative effect on economic growth. Both of these models exhibit scale effects, however, with a positive relationship between the size of the labor force and the rate of economic growth. In this paper, we develop an open economy framework that allows for firm mobility, and study the relationship between corporate taxes and growth without the bias of scale effects.

The remainder of the paper proceeds as follows. In Section II, we introduce our theoretical framework and investigate the effects of national corporate tax rates on the location patterns of production and innovation. Then, we consider how corporate taxes influence productivity growth, market entry, and national welfare, and provide simple numerical examples for the predictions of the framework. The paper concludes in Section III.

II. THE MODEL

This section extends Davis and Hashimoto (2016) to consider the implications of national corporate tax policy for patterns of industrial activity, market entry, and productivity growth. The model consists of two countries, home and foreign, that potentially employ labor in three activities: traditional production, manufacturing, and process innovation. The home and foreign labor supplies, L and L*, are mobile between sectors, but not between countries, with an asterisk denoting variables associated with foreign. We focus on home as we introduce the model setup.

A. Households

The demand side of the economy is made up of the dynastic households residing in each country. These households choose optimal saving-expenditure paths over an infinite time horizon with the aim of maximizing lifetime utility, which takes the following constant intertemporal elasticity of substitution form:

(1)

U = [[integral].sup.[infinity].sub.0] [e.sup.-[rho]t] [([C.sub.X][(t).sup.[alpha]][C.sup.Y][(t).sup.1-[alpha]]) .sup.1-[xi]] - 1/1 - [xi] dt,

where [C.sub.X](t) and [C.sub.Y](t) are household consumptions of a manufacturing composite and traditional goods at time t, respectively, [rho] is the subjective discount rate, 1/[xi] is the intertemporal elasticity of substitution, and [alpha] [member of] (0, 1). Lifetime utility is maximized subject to the following flow budget constraint:

[Angstrom](t) = r(t)A(t) + w(t) + T(t) -[P.sub.X](t)[C.sub.X](t)-[P.sub.Y](t)[C.sub.Y](t),

where A(t) is asset wealth, r(t) is the interest rate, w{t) is the wage rate, T(t) is a lump-sum transfer from government to households, [P.sub.X](t) is the price index associated with the manufacturing composite, [P.sub.Y](t) is the price of traditional goods, and a dot over a variable indicates differentiation with respect to time.

The households of home and foreign have equal access to an international financial market, leading to a common interest rate across countries (3)

(2)

[??]/E = r - [rho]/[xi] + ([xi] - 1)/[xi] [??]/P,

where P is the aggregate price index and E = [P.sub.X][C.sub.X] + [P.sub.Y][C.sub.Y] is household expenditure.

The per-period demands for the manufacturing composite and the traditional good are [C.sub.X] = [alpha]E/[P.sub.X] and [C.sub.Y] = (1 - [alpha])E/[P.sub.Y], and accordingly the aggregate price index in home is P = [([P.sub.X]/[alpha]).sub.[alpha]]([P.sub.Y]/[(1 - [alpha])).sup.1-[alpha]]. The manufacturing composite, and its price index, take a constant elasticity of substitution (CES) form:

(3) [C.sub.X] = [([[integral].sup.N.sub.0] [c.sup.[sigma]-1/[sigma].sub.i] di).sup.[sigma]/[sigma]-1] [P.sub.X] = [([[integral].sup.N.sub.0] [p.sup.1-[sigma].sub.i]di).sup.1/1-[sigma]]

where the mass of product varieties available (N [equivalent to] n + n*) equals the sum of varieties (n) produced in home and varieties (n) produced in foreign, [c.sub.i] and [p.sub.i] are the household consumption and price of variety i, respectively, and the CES between any pair of varieties is [sigma] > 1. Given the constant level of household expenditure allocated to manufacturing goods, the home household demands for a domestically supplied variety i and an imported variety j are

(4) [c.sub.i] = [alpha][p.sup.-[sigma].sub.i] [P.sup.[sigma]-1.sub.X]E, [c.sub.j] = [alpha] ([tau][p*.sub.j]) [P.sup.[sigma]-1.sub.X]E,

where [tau] > 1 is an iceberg trade cost, under which [tau] additional units must be shipped for every unit sold in an export market (Samuelson 1954). We also derive analogous demand conditions for foreign households.

B. Production

The traditional good sector employs labor with a unit coefficient technology that exhibits constant returns to scale. We suppose that the share of traditional goods in household expenditure is large enough to ensure that both countries produce traditional goods. Thus, with free trade in a competitive international market, the price of traditional goods and the wage rate are common across home and foreign. Setting the traditional good as the model numeraire, we have [P.sub.Y] = [P*.sub.Y] = w = w* = 1.

The manufacturing sector features Dixit and Stiglitz (1977) monopolistic competition, with each firm supplying a single unique product variety. While there are no costs associated with the development of new product designs, every period firms incur labor costs in the management ([l.sub.F]) and implementation of innovation ([l.sub.R]), which are fixed with respect to production. The production technology of a firm with production located in home is

(5) x = [[theta].sup.[gamma]][l.sub.X],

where x and [l.sub.X] are output and employment in production, [theta] is a firm-level productivity coefficient, and [gamma] > 0 is the output elasticity of productivity. Although each firm employs a unique production technique, we assume that productivity is symmetric across all firms in all locations ([theta] = [theta]*).

With CES preferences over product variety, firms maximize profit by setting price equal to p = p* = [eta]/[[theta].sup.[gamma]], where [eta] [equivalent to] [sigma]/([sigma] - 1) > 1 is the constant markup and l/[[theta].sup.[gamma]] is the unit production cost. Matching supply with the demands from the home and foreign markets, home-based production is x = [c.sub.i]L + [tau][c*.sub.i]L* , where [tau] > 1 units must be produced for every unit sold in the export market. Together with Equation (4), this condition yields optimal operating profit on sales for a firm with production located in home:

(6)

[pi] = px - [l.sub.x] = ([eta] - 1)[l.sub.x] = [alpha]([eta] - 1)[p.sup.1/(1-[eta])].sub.i]/[eta] (EL/[P.sup.1/(1-[eta])/X] + [phi]E*L*/[P*1/(1-[eta])/X]),

where [phi] = [[tau].sup.1/(1 - [eta])] describes the freeness of trade.

C. Process Innovation

Manufacturing firms invest in process innovation with the aim of reducing production costs. Each period a representative firm employs [l.sub.F] fixed units of labor in the management of innovation and [l.sub.R] units of labor in process innovation, with the evolution of firm productivity governed by

(7) [??] = k[theta][l.sub.R],

where k[theta] is labor productivity in innovation. The stock of technical knowledge is contained within the production processes of individual firms, and is therefore captured by [theta]. As a result, knowledge accumulates as a by-product of process innovation, generating an intertemporal knowledge spillover through which current innovation efforts reduce future R&D costs (Peretto 1996; Smulders and van de Klundert 1995).

The strength of intertemporal knowledge spillovers depends on the average productivity of production technologies observable to the firm, with a stronger weight given to production located in proximity to the R&D department of the firm. Specifically, adapting the specification of Baldwin and Forslid (2000), we assume that knowledge spillovers from production to innovation diminish with distance:

(8) k = s + [delta]s*,

where s [equivalent to] n/N and s* [equivalent to] n*/N are the shares of firms with production located in home and foreign, and [delta] [member of] (0, 1) is the degree of knowledge diffusion between countries. There is broad empirical evidence supporting the localized nature of knowledge spillovers (Bottazzi and Peri 2003; Mancusi 2008; Thompson 2006), given that technical knowledge tends to include both codifiable aspects that are easily transmitted across large distances and tacit aspects that are only conveyed through face-to-face communication (Keller 2004).

We consider territorial tax systems in which the source of production, rather than the point of sale, is used for taxation (IMF 2014). As such, identifying firms by where they locate production, the net per-period profits of a home firm with innovation located in either home ([[PI].sub.H]) or foreign ([[PI].sub.F]) are, respectively

(9) [[PI].sub.H] = (1-z)([pi] - [l.sub.R] - [l.sub.F]), [[PI].sub.F] = (1 - z)[pi] - (1 - z*) ([l*.sub.R] + [l.sub.F]),

where z [member of] (0, 1) and z* [member of] (0, 1) are the corporate tax rates set on per-period profits in each country. (4) With this specification, process innovation is subsidized through a full tax exemption when innovation and production are located in the same country, and through a partial tax exemption when they are located in different countries. In order to ensure that firms are not subsidized for shifting innovation activity out of the country, we assume that the forfeited subsidy associated with the partial tax exemption is equal to min{0, z* - z} when home firms locate innovation in foreign.

Firm value equals the present discounted value of net per-period profits, and therefore depends on the location of innovation. The potential firm values associated with the net-period profits described for a home firm in Equation (9) are

(10) [mathematical expression not reproducible]

Firm value is maximized subject to Equation (7) with an optimal employment level and location choice for process innovation. We solve this optimization problem for a home firm using the following current value Hamiltonian functions: [H.sub.H] = [[PI].sub.H] + [mu]k[theta][l.sub.R] for innovation located in home and [H.sub.F] = [[PI].sub.F] + [mu]*k*[theta][l*.sub.R] for innovation located in foreign, with [mu] and [mu]* describing the current shadow values of the firm's stock of technical knowledge in each case. Combining the first-order conditions for each case, for example, [partial derivative[].sub.HH]/[partial derivative][l.sub.R] = 0 and [partial derivative][H.sub.H]/ [partial derivative][theta] = r[mu] - [??] for innovation undertaken in home, leads to the following respective no-arbitrage conditions for optimal investment in process innovation when innovation is located in either home or foreign:

(11) r [greater than or equal to] [R.sub.H] [equivalent to] [gamma]k[pi]/[eta] - 1 - [??]/k - k[l.sub.R],

(12) r [greater than or equal to] [R.sub.F] [equivalent to] (1 - z)[gamma]k*[pi]/(1 - z*)([eta] - 1) - [??]*/k* k*[l*.sub.R],

where we have used Equation (7), and [R.sub.H] and [R.sub.F] denote the internal rates of return to investment in process innovation located in home and foreign by home firms. We assume that, given their small market shares, firms disregard the impact of their innovation efforts on both the composite price indices and knowledge spillovers to rival firms when setting their optimal employment levels in process innovation. The internal rate of return to investment in process innovation equals the risk-free interest rate when firms exhibit productivity growth. Thus, home firms select the R&D location that offers the highest internal rate of return and only the no-arbitrage condition for the selected R&D location binds.

D. National Labor Markets

In the following sections, we show that free market entry and the free international movement of innovation and production lead to the full concentration of R&D and a greater production share for the country with the larger after-tax market. In preparation for the analysis of location patterns, following Smulders and van de Klundert (1995), this section investigates the labor market dynamics associated with a given level of market entry and fixed masses of firms locating production in each country. We obtain the following lemma for labor market stability under the assumption that all firms locate innovation in the home country.

LEMMA 1. National labor markets jump immediately to equilibrium for a given level of market entry and fixed masses of firms locating production in each country.

Proof. See Appendix A.

E. Market Entry

With zero costs incurred in the design of new product varieties, net per-period profits determine the level of market entry. (5) When firm value is positive (V > 0), new firms enter the market causing a fall in firm-level market shares and lowering firm value through a fall in per-period profit. Alternatively, when firm value is negative (V < 0), firms exit the market and firm value rises. This process is immediate and leads to two potential sets of free entry conditions for production located in home and foreign:

(13)

[pi] [less than or equal to] [l.sub.R] + [l.sub.F] for [V.sub.H] [less than or equal to] 0, [pi]* [less than or equal to] (1 - z) ([l.sub.R] + [l.sub.F])/(1 - z*) for [V*.sub.H] [less than or equal to] 0,

(14)

[pi] [less than or equal to] (1 - z*) ([l*.sub.R] + [l.sub.F])(1 - z) for [V.sub.F] [less than or equal to] 0, [pi]* [less than or equal to] [l*.sub.R] + [l.sub.F] for [V*.sub.F] [less than or equal to] 0.

In the next section, we show that all innovation concentrates fully in one country. Accordingly, Equation (13) binds if innovation concentrates in home [V.sub.H] = 0 and [V*.sub.H] = 0), and Equation (14) binds if innovation concentrates in foreign ([V.sub.F] = 0 and [V*.sub.F] = 0). Under these conditions, with all firms earning zero profits, corporate tax revenues are zero in both countries (T = 0), and household expenditure equals wage income (E = 1 and E* = 1). Then, as the aggregate price indexes for home and foreign are now written as P = [([[alpha].sup.[alpha]][(l - [alpha]).sup.1-[alpha]]).sup.-1] [(n + [phi]n*).sup.[alpha](1 [eta])][p.sup.[alpha]] and P* = [([[alpha].sup.[alpha]][(l - [alpha]).sup.1-[alpha]]).sup.-1] [([phi]n + n*).sup.[alpha](1-[eta])] [p.sub.[alpha]] from Equation (2) we have r = [rho] + [alpha]([xi] - 1)[gamma]g at all moments in time, where g [equivalent to] [??]/[theta] is the common rate of productivity growth across countries. Hereafter, we assume that the populations of home and foreign are equal (L = L*), and focus on cross-country differences in corporate tax rates.

F. Corporate Taxes and Location Patterns

Free to shift production between countries, at zero cost, manufacturing firms locate production in the country that offers the greatest operating profit, net of corporate taxes, with the aim of maximizing firm value. As such, when manufacturing occurs in both countries, net operating profit on sales equalizes between home- and foreign-based production: (1-z)[pi] = (1 - z*)[pi]*. Substituting Equation (6) into this condition yields the equilibrium share of firms with production located in home:

(15) s ([phi], Z) = (Z - [phi]) - [phi] (1 - [phi]Z)/[(1 - [phi]).sup.2] (1 + Z),

where Z [equivalent to] (1 - z)/(1 - z*) describes the corporate tax differential between home and foreign, and a rise in Z indicates a fall in the corporate tax rate of home relative to that of foreign. In addition, Z [member of]([Z.bar], [bar.Z]), with [Z.bar] = 2[phi])/(1 + [[phi].sup.2]) and [bar.Z] = (1 + [[phi].sup.2]) / (2[phi]), is required for s [member of] (0, 1). The home share of production features a standard home market effect (Krugman 1980), with a greater share of firms locating production in the country with the larger after-tax market. (6)

The effect of a change in the corporate tax differential on the location of production is summarized in the following lemma.

LEMMA 2. An increase in the corporate tax differential (Z) raises the home share of production (s).

Proof. The derivative of Equation (15) with respect to Z yields

ds/dZ = [(1 + [phi]).sup.2]/[(1 - [phi]).sup.2] [(1 + Z).sup.2] > 0.

Intuitively, an increase in the corporate tax differential (a reduction in home's tax rate) makes home more attractive as a location for production, as the relative size of its after-tax market increases. This is a standard result in the NEG literature (Baldwin et al. 2003) that is supported by empirical evidence (Briilhart, Jametti, and Schmidheiny 2012; Feld and Heckemeyer 2011; Kammas 2011). In order to simplify the analysis, we consider long-run equilibria for which home has a lower corporate tax rate and a larger share of manufacturing activity; that is, Z [member of] (l, [bar.Z]) and s [member of] (1/2, 1). Accordingly, we focus on cases for which k [member of] ((1 + [delta])/2, 1) is satisfied.

Next we derive the industry level of operating profit on sales associated with the free movement of production between countries by substituting Equation (15) into Equation (6):

(16)

N[pi] = N[pi]* = [alpha]([eta] - 1)[(1 - [phi]).sup.2](1 + Z)L/[eta](1 - [phi]Z)(Z - [phi]),

where Z [member of] (1, [bar.Z]) ensures positive operating profits since Z [member of] ([phi], l/[phi]). The relationship between industry-level operating profit on sales and the corporate tax differential is summarized in the following lemma.

LEMMA 3. Industry-level operating profit on sales is convex in the corporate tax differential over the range Z [member of] (1, [bar.Z]).

Proof. The derivative of Equation (16) with respect to Z yields

1/N[pi] d(N[pi]/dZ) = (1 + [phi]) (Z - [phi]) - (1 - [phi]Z) (1 + Z)/(Z [phi])(1 - [phi]Z)(1 + Z).

Setting the numerator equal to zero yields the threshold [??] = -1 + (1 + [phi]) / [square root of ([phi])]. Then, we have d(N[pi])/dZ < 0 for Z [member of] (1, [??]), and d(N[pi])/dZ > 0 for Z [member of] ([??], [bar.Z]).

Following the NEG literature (Baldwin et al. 2003), a decrease in the corporate tax rate of home (a rise in Z) increases the production share of home (see Lemma 1), with two opposing effects on operating profits. First, the shift in production from foreign to home intensifies competition putting downward pressure on operating profits. Second, the industry-level cost of transporting goods from the smaller foreign country to the larger home country falls, inducing an improvement in operating profits. The negative competition effect dominates for Z [member of] (1, Z), while the benefit of lower transport costs dominates for Z [member of] ([??], [bar.Z]).

To conclude this section, we consider the location pattern for process innovation. Firms select the optimal location for innovation through a comparison of the rates of return to process innovation associated with each location. Combining Equations (12) with (13) and (14), the differences in rates of return for home and foreign firms become

[R.sub.H] - [R.sub.F] = (k - k*) ([l.sub.F] - ([eta] - [gamma] - 1) [pi]/[eta] - 1), [R*.sub.H] - [R*.sub.F] = (k - k*) x ([l.sub.F] - ([eta] - [gamma] - 1) (k - Zk*)[pi]*/([pi] - 1) (k - k*)Z),

where we have used [mathematical expression not reproducible]. All firms locate innovation in home if [R.sub.H] > [R.sub.F] and [R*.sub.H] > [R*.sub.F], and all firms locate innovation in foreign if [R.sub.H] < [R.sub.F] and [R*.sub.H] < [R*.sub.F].

Beginning with the location choice for home firms, as home has a lower corporate tax rate (Z [member of] (1, [bar.Z]), the forfeited subsidy associated with shifting process innovation abroad is zero for home firms and the corporate tax differential drops out of the calculation of [R.sub.H] - [R.sub.F]. Therefore, with manufacturing concentrated in home, greater knowledge spillovers from production to innovation (k > k*) ensure that all home firms locate process innovation in home, since [l.sub.F] - ([pi]- [gamma] - 1)[pi]/([eta] - 1) > 0 is required for a positive rate of return to innovation. Turning next to the location choice for foreign firms, since we focus on symmetric equilibria with a common productivity level for all locations ([theta] = [theta]*), we restrict our analysis to long-run equilibria for which foreign firms also locate process innovation in the home country to ensure that all firms have the same rate of productivity growth ([??]/[theta] = [??]*/[theta]*); that is, [R*.sub.H] > [R*.sub.F]. Thus, for the equilibria that we consider, the subsidy forfeited by foreign firms (z - z* < 0) is dominated by the benefit of greater knowledge spillovers (k > k*). This location pattern matches in a stylized manner with the fact that innovation tends to be more geographically concentrated than manufacturing (Carlino and Kerr 2015).

Combining the free entry conditions (Equation (13)), and r = [rho] + [alpha]([xi] - 1)[gamma]g, with the home no-arbitrage condition (Equation (12)), we solve for equilibrium firm-level employment in production and innovation as follows:

(17) [mathematical expression not reproducible]

where [bar.[eta]] = 1 + [gamma]/(1 + [alpha] ([xi] - 1) [gamma]). These conditions dictate the range of knowledge spillovers consistent with both market entry and productivity growth.

LEMMA 4. Positive employment in process innovation requires [eta] > [bar.[eta]] and k [member of] (([eta] - 1)[rho]/ ([gamma][l.sub.F]), 1).

In the standard monopolistic competition model of the NTT literature (Helpman and Krugman 1985), the price-cost markup determines the ratio of variable labor employment in production to fixed labor employment through free market entry and exit. In our framework, the relationship between the markup and the firm-level employment ratio combines with investment conditions to determine the stability of market entry. Using r = [rho] + [alpha]([xi] - 1)[gamma]k[l.sub.R] from Equation (2) and r = [gamma]k[pi]/([eta] - 1) - k[l.sub.R] from Equation (12), we solve for per-period profit as follows: [PI] = ([eta] - [bar.eta]) [pi]/([eta] - 1) + [rho]/(k(1 + [alpha]([xi] - 1)[gamma])) - [l.sub.F]. Stable market entry requires [partial derivative][PI]/[partial derivative]N = (([eta] - [bar.[eta]])/([eta] - 1)) [partial derivative][pi]/ [partial derivative]N = - (([eta] - [bar.[eta]]) / ([eta] - 1)) [pi]/N < 0, where we have used Equation (16), to ensure that firm value responds correctly to market exit and entry (Novshek and Sonnenchein 1987). (7) Therefore, [eta] > [bar.[eta]] is required for stable market entry, and returning to Equation (17) we can see that k [member of] (([eta] - 1)[rho]/[l.sub.F], 1) is necessary for positive firm-level employment in process innovation. Under these conditions, a fall in knowledge spillovers lowers labor productivity in R&D, but raises profit as firms decrease investment in process innovation. Market entry then leads to a smaller firm-level scale of production. For k [less than or equal to] ([eta] - 1)[rho]/([gamma][l.sub.F]) the model reduces to a standard NTT framework with no investment in process innovation. We focus hereafter on long-run equilibria that satisfy Lemma 4.

G. Market Entry and Productivity Growth

We now consider the effects of changes in the corporate tax differential on market entry and productivity growth. Beginning with market entry, we use firm-level employment in production Equation (17) with [pi] = ([eta] - 1)[l.sub.X] and operating profit on sales Equation (16) to derive

(18) N = [alpha] [(1 - [phi]).sup.2] (1 + Z) L/[eta](1 - [phi]Z)(Z - [phi])[l.sub.X].

This expression highlights the inverse relationship between the firm-level scale of production ([l.sub.X]) and market entry (N) that is a common feature of models of the NTT literature (Helpman and Krugman 1985). In addition, market entry is determined proportionally with market size, as measured by population.

In order to investigate how changes in the corporate tax differential affect market entry, we take the derivative of Equation (18) with respect to Z to obtain

(19) 1/N dN/dZ = 1/N[pi] d(N[pi])/dZ - 1/[l.sub.X] d[l.sub.X]/dk dk/ds ds/dZ,

where dk/ds = 1 - [delta] > 0 and ds/dZ > 0 from Lemma 1. Changes in the corporate tax differential affect market entry through two channels. The first is the direct profit effect captured by the first term on the right-hand side of Equation (19), and is negative for Z [member of] (1, [??]) and positive for Z [member of] ([??], [bar.Z]), following from Lemma 4.

The second channel is the indirect knowledge spillover effect described by the second term on the right-hand side of Equation (19). The increase in the home share of production, associated with a rise in Z, improves knowledge spillovers from production to innovation, leading to higher labor productivity in process innovation. The rise in labor productivity prompts firms to increase employment in process innovation, raising fixed costs ([l.sub.R] + [l.sub.F]) and inducing firms to exit the market.

A comparison of the profit and knowledge spillover effects yields the following proposition.

PROPOSITION 1. A rise in Z has a negative effect on market entry for Z [member of] (1, [??]), and an ambiguous effect for Z [member of] ([??], [bar.Z]).

The balance between the profit and knowledge spillover effects determines the overall effects of adjustments in the corporate tax differential on market entry. On the one hand, after an increase in the corporate tax differential, for Z < [??] a negative profit effect aligns with the negative knowledge spillover effect, and the level of market entry falls. On the other hand, for Z > [??] the profit effect is positive, and as either effect may dominate, the overall impact on market entry is ambiguous. We present a numerical example that illustrates these cases in Figure 1 below.

Turning next to the equilibrium rate of productivity growth, we combine Equation (15) with firm-level employment in process innovation (Equation (17)) to obtain

(20) g [equivalent to] [??]/[theta] = [gamma]k[l.sub.F] - ([eta] - 1) [rho]/(1 + [alpha] ([xi] - 1)[gamma])([eta] [bar.eta]),

where [bar.[eta]] = 1 + [gamma]/(1 + [alpha]([xi] - 1)[gamma]). From this expression, we can see that productivity growth is not biased by a scale effect, as proportionate increases in the population sizes of home and foreign do not affect national shares of production. Similarly, changes in the corporate tax differential do not affect the rate of productivity growth directly, as the profit effect is fully absorbed by adjustments in the level of market entry.

Changes in the corporate tax differential do influence productivity growth, however, through the knowledge spillover effect. A decrease in the home tax rate (a rise in Z) raises the home share of production, resulting in greater knowledge spillovers from production to innovation. Higher labor productivity then spurs firms to increase employment in process innovation, accelerating productivity growth. (8)

PROPOSITION 2. An increase in the corporate tax differential (Z) accelerates the rate of productivity growth (g).

Proof. Taking the derivative of (20) with respect to Z gives

dg/dZ = [gamma][l.sub.F]/(1 + [alpha] ([xi] - 1)[gamma]) ([eta] - [bar.[eta]]) dk/ds ds/dZ > 0,

where dk/ds = 1 - [delta] > 0 and ds/dZ > 0 from Lemma 1.

We briefly consider the empirical evidence associated with the theoretical links driving the relationship between the corporate tax differential and productivity growth in our framework. First, as discussed with respect to Lemma 1, the empirical evidence presented by Feld and Heckemeyer (2011), Kammas (2011), and Brulhart, Jametti, and Schmidheiny (2012) indicates that firm location is influenced by changes in corporate tax differentials. Second, within our framework, an increase in the concentration of industry in the low-tax country leads to an improvement in knowledge spillovers from industry to innovation. This result is supported by a broad body of empirical literature that documents the localized nature of knowledge spillovers, for example, Bottazzi and Peri (2003), Thompson (2006), and Mancusi (2008). Given that the strength of knowledge spillovers diminishes with distance, higher industry concentration leads to improved knowledge spillovers. Third, as knowledge spillovers improve, and firm-level productivity and employment in innovation rise, the rate of productivity growth increases. The NEG literature generally concludes that industry concentration has a positive effect on economic growth (Baldwin and Martin 2004). The results of the empirical literature, however, are generally ambiguous (Gradiner, Martin, and Tyler 2011).

Considering the overall relationship between the corporate tax rate and productivity growth, the results of our framework match with the empirical literature that tends to find a negative relationship, for example, Kneller, Bleaney, and Gemmell (1999), Lee and Gordon (2005), Arnold et al. (2011), and Gemmell, Kneller, and Sanz (2011, 2014), although Angelopoulos, Economides, and Kammas (2007) find a positive relationship, and Ojede and Yamarik (2012), Xing (2012), and Arachi, Bucci, and Casarico (2015) find no significant relationship at all. With the empirical literature generally concluding that corporate taxes have a negative impact on growth, our framework highlights the importance of considering firm mobility when estimating the size of the effects of corporate taxes on economic growth, as discussed in Gemmell, Kneller, and Sanz (2014).

H. National Welfare and Corporate Tax Rates

This section considers the relationship between national welfare and the corporate tax differential. With the governments of both countries earning zero-tax revenues (T = 0), steady-state welfare levels can be derived using Equations (1), (3), (7), and (15):

[mathematical expression not reproducible]

where [theta](0) = 1 and [A.sub.l] [equivalent to] [([alpha]/[eta]).sup.[alpha](1 - [xi])] [(1 - [alpha]).sup.(1 - [alpha])(1 - [xi])] > 0 is a constant.

Changes in the corporate tax differential affect national welfare levels through three channels:

[mathematical expression not reproducible]

where [A.sub.2] = [alpha]([U.sub.0](1 - [xi])[rho] + 1)/[rho] > 0. The first term on the right-hand side represents a trade cost effect (Baldwin et al. 2003) that captures the impacts of adjustments in the average price of manufacturing goods in each country. For example, a rise in the corporate tax differential increases the concentration of production at home, causing the average price of goods to fall and improving home welfare, given the smaller share of goods priced to include the trade cost. Similarly, the average price of goods consumed in foreign rises, with a negative impact on foreign welfare. The second term is the love of variety effect arising from households' preference for product variety. Given the direct link between product variety and the level of market entry, the love of variety effect may be positive, negative, or ambiguous after a rise in the tax differential, depending on the size of the tax differential (Proposition 1). Finally, the third term describes the growth in national welfare associated with falling prices as firms invest in process innovation (Proposition 2). The balance of these channels can be considered using the price-cost markup, as outlined in the following proposition.

PROPOSITION 3. The welfare effects of an increase in the corporate tax differential (Z) depend on the price-cost markup ([eta]): both countries benefit for [eta] < [[eta].sub.1], and home benefits while foreign is hurt for [eta] > [[eta].sub.1]; where [[eta].sub.1] > [bar.[eta]].

Proof. See Appendix B.

The welfare results outlined in Proposition 3 share both similarities and differences with the results of the standard variety-expansion model of endogenous growth with footloose capital in the NEG literature (Baldwin and Martin 2004). The standard model exhibits three welfare effects after an increase in the concentration of industry: the trade cost effect described above, a strictly positive growth effect resulting from a fall in innovation costs, and a strictly negative wealth effect as the fall in innovation costs lowers firm value. Depending on the strengths of these effects, greater industry concentration may lead to either positive or negative welfare outcomes for both countries. In contrast, in our framework, because free entry drives firm value to zero, there is no wealth effect. Instead, the direction of welfare effects after a rise in the corporate tax differential increases industry concentration in the low-tax country depends on the relative strengths and directions of the market entry and growth effects.

I. Numerical Example

This section investigates the quantitative effects of changes in corporate tax rates on market entry and productivity growth. Our aim is not to provide a full calibration of the model, but rather a simple numerical example of the direction of welfare effects resulting from changes in the tax differential for plausible parameter values.

For the demand parameters, we set [rho] = 0.05, [xi] = 2, [alpha] = 0.7, [sigma] = 5.35, and L=10. The coefficient of relative risk aversion is standard and consistent with the estimates of, for example, Chiappori and Paiella (2011). The elasticity of substitution references values reported by Bernard et al. (2003) and Broda and Weinstein (2006), and generates a price-cost markup equal to [eta] = 1.23. Next, the degree of knowledge diffusion is fixed at [delta] = 0.15 to match the mid-range of estimates provided by Bloom, Schankerman, and van Reenen (2013), and we use [tau] = 1.7 for the level of trade costs following the estimates of Anderson and van Wincoop (2004) and Novy (2013), leading to a value of [phi] = 0.1 for the freeness of trade. We specify the fix cost at [l.sub.F] = 0.01 and the output elasticity of productivity at [gamma] = 0.1 in order to target a benchmark productivity growth rate of g = 0.02 for equal tax rates (Z = 1).

Figure 1 provides numerical plots of the home production share (s), the productivity growth rate (g), market entry (N), and the marginal utilities for home (dU/dZ) and foreign (dU*/dZ) against the corporate tax differential over the range from 1 to [bar.Z] = 5.08, for which s [member of] (0.5, 1). Following Proposition 1, the profit effect is negative for Z < [??] = 2.49 and aligns with the negative knowledge spillover effect causing market entry to fall with an increase in the tax differential. For Z > [??], however, the profit effect is positive and market entry falls if the knowledge spillover effect dominates, but rises if the profit effect dominates. As shown in Proposition 2, an increase in the tax differential accelerates productivity growth. The welfare effects match with the case outlined for [eta] > [[eta].sub.1] in Proposition 3, and a higher corporate tax differential therefore benefits home but hurts foreign.

III. CONCLUSION

In this paper we consider how changes in national tax rates on corporate incomes affect the geographic location of industry, the level of market entry, and fully endogenous productivity growth without scale effects in a two-country model of trade. Economic growth is driven by monopolistically competitive firms that invest in process innovation with the aim of lowering production costs. Faced with imperfect knowledge diffusion and trade costs, firms shift their production and innovation activities between countries in order to take advantage of the lowest cost locations, leading to a greater share of firms locating production and all firms locating innovation in the country with the lowest corporate tax rate and thus the largest after-tax market.

Investigating the relationships between national tax policy, productivity growth, and market entry, we find that the effects of changes in the international corporate tax rate differential depend on the initial levels of relative tax rates. Focusing on the policy of the country with the relatively low tax rate, an increase in the corporate tax differential accelerates productivity growth, but has a negative effect on market entry if the initial tax rate is high and an ambiguous effect on market entry if the initial tax rate is low. Lastly, we show that increases in the corporate tax rate benefit the low-tax rate country, but may benefit or hurt the high-tax rate country.

APPENDIX A

Following Smulders and van de Klundert (1995), we show that national labor markets jump immediately to equilibrium for a given level of market entry and fixed masses of firms with production located in each country. We assume that all R&D activity takes place in home. First, defining g [equivalent to] [??]/[theta] and Z [equivalent to] (1 - z)/(1 - z*), from Equation (12) we have r = [gamma]k[l.sub.X] - g = k[l*.sub.X]/Z - g, which implies [l*.sub.X] = Z[l.sub.x] and [[??].sub.x]/[l.sub.x] = [[??]*.sub.X]/[l*.sub.X]. Second, denoting world labor market aggregates with a superscript "w," since[ .sub.ci]/[c.sub.j] = [[tau].sup.[sigma]] and [c*.sub.j]/[c*.sub.i] = [[tau].sup.[sigma]], we have [P.sub.X][C.sub.X]L + [P*.sub.X][C*.sub.X]L* = [eta][L.sup.w.sub.X], and substituting [L.sup.w.sub.Y] = [alpha][eta]/(1 - [alpha])[L.sup.w.sub.X] and [L.sup.w] = [L.sup.w.sub.Y] + [L.sup.w.sub.X] + N[l.sub.R] + N[l.sub.F] into Equation (7) yields

g = ([L.sup.w]/N - [l.sub.F]) k - (1 + [[alpha].sub.[eta]]/1 - [aplha]) (n + Zn*)k[l.sub.X]/N,

where n, n*, N, and k are constants. Third, common dynamics for household expenditures [??]/E = [??]*/E* = (r - [rho] - [alpha] ([xi] - 1)[gamma]g)/[xi], and the time derivatives of Equations (4) and (5), yield [[??].sub.i]/[c.sub.i] = [[??]*.sub.i]/[c*.sub.i] = [[??].sub.j]/[c.sub.j] = [[??].sub.j]/[c*.sub.j] = [gamma]g + [??]/E and [??]/x = [gamma]g + [??]X/[l.sub.X]. Therefore, [[??].sub.i]/[c.sub.i] = [??]/x implies [??]/E = [[??].sub.X]/[l.sub.X]. Combining these results, we obtain the following motion for firm-level employment in production:

[[??].sub.X]/[l.sub.X] = [gamma]k[l.sub.X] - (1 + [alpha]([xi] - 1)[gamma])g - [rho]/[xi].

As[ .sub.lX] is a control variable and [mathematical expression not reproducible], we find that labor markets jump immediately to equilibrium, as stated in Lemma 1.

APPENDIX B

This appendix provides a proof of Proposition 3. Using Equation (17), the marginal utilities associated with the corporate tax differential are

(B1)

1/[A.sub.3] dU/dZ = [phi] [(1 - [phi]).sup.2] [(1 + Z).sup.2]/(1 - [delta])[(1 + [phi]).sup.2](1 - [phi]Z) + ([eta] - [[eta].sub.bar])[l.sub.R]/([eta] - 1) ([eta] - [??])k[l.sub.X],

(B2)

1/[A.sub.3] dU*/dZ = -[(1 - [phi]).sup.2][(1 + Z).sup.2]/(1 - [delta])[(1 + [phi]).sup.2](Z - [phi]) + ([eta] - [[eta].bar])[l.sub.R]/([eta] - 1) ([eta] - [??])k[l.sub.X],

where [A.sub.3] [equivalent to] [A.sub.2]([eta] - 1)(dk/ds)(ds/dZ) > 0, [[eta].bar] [equivalent to] 1 + [gamma] - (1 + [alpha]([xi] - 1)[gamma])[gamma]k[l.sub.F]/[rho] and [??][equivalent to] 1 + [gamma] - [alpha]([xi] - 1)[[gamma].sup.2]k[l.sub.F]/[rho]. A simple comparison shows that [eta] < [??] and that [??] < [bar.[eta]] for [l.sub.x] > 0. Therefore, dU/dZ > 0. Further, because the second term in Equation (B2) takes large values as [eta] approaches [bar.[eta]], we find that dU*/dZ > 0 for [eta] < [[eta].sub.1], and dU*/dZ < 0 for [eta] > [[eta].sub.1], where [[eta].sub.1] x [bar.[eta]] r] is calculated from dU*/dZ = 0, as outlined in Proposition 3.

ABBREVIATIONS

CES: Constant Elasticity of Substitution

NEG: New Economic Geography

NTT: New Trade Theory

R&D: Research and Development

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COLIN DAVIS and KEN-ICHI HASHIMOTO *

* The authors are grateful for helpful comments from an anonymous referee, Taiji Furusawa, Yoshiyasu Ono, and participants of ETSG 2015. Part of this work was completed while Davis was visiting RITM at the University of Paris-Sud. He thanks them for their generous hospitality and support. The authors also acknowledge financial support from JSPS through Grants-in-Aid for Scientific Research (A) and (C). All remaining errors are their own.

Davis: Professor, The Institute for the Liberal Arts, Doshisha University, Kyoto 602-8580, Japan. Phone +81-75-251-4971, Fax +81-75-251-4971, E-mail cdavis@mail.doshisha.ac.jp

Hashimoto: Associate Professor, Graduate School of Economics, Kobe University, Kobe 657-8501, Japan. Phone +81-78-803-6839, Fax +81-78-803-7289, E-mail hashimoto@econ.kobe-u.ac.jp

doi: 10.1111/ecin.12521

(1.) Empirical support for the endogenous market structure and endogenous growth framework is found in Laincz and Peretto (2006), Ha and Howitt (2007), and Madsen, Ang, and Banerjee (2010).

(2.) Peretto (2003b) and Peretto and Valente (2011) also studied endogenous market structure and endogenous growth models of trade. The former paper considered the effects of economic integration on productivity growth, while the latter paper investigated the effects of resource booms on economic growth when there is international trade in natural resources.

(3.) We assume that initially there is no borrowing or lending between home and foreign as both countries have the same rate of time preference ([rho]).

(4.) Within our framework, z is an average effective tax rate that may reflect a variety of country-specific tax rates (Benassy-Quere, Fontagne, and Lahreche-Revil 2005).

(5.) See Davis and Hashimoto (2015) for a similar frame work with positive entry costs.

(6.) See Crozet and Trionfetti (2008) and Niepmann and Felbermayr (2010) for recent empirical evidence supporting the existence of the home market effect.

(7.) See Smulders and van de Klundert (1995), Peretto (1996), and van de Klundert and Smulders (1997) for detailed dynamic analyses of this class of growth models.

(8.) See Davis and Hashimoto (2016) for a discussion of the effects of greater regional integration resulting from either a fall in trade costs or an improvement in knowledge diffusion.

Caption: FIGURE 1

The Effects of Changes in the Corporate Tax Differential