Trade integration and regional disparity in a model of scale-invariant growth

Regional Science and Urban Economics 41 (2011) 20–31

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Regional Science and Urban Economics j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / r e g e c

Trade integration and regional disparity in a model of scale-invariant growth Antonio Minniti a,⁎, Carmelo Pierpaolo Parello b a b

University of Bologna, Italy Sapienza Università di Roma, Dipartimento di Economia Pubblica, Via del Castro Laurenziano, 9, I-00161, Rome, Italy

a r t i c l e

i n f o

Article history: Received 23 July 2009 Received in revised form 20 July 2010 Accepted 21 July 2010 Available online 29 July 2010 JEL classification: F12 R12 D31 O31

a b s t r a c t This paper explores the relation between trade integration, economic growth and regional inequality in a two-region model of endogenous industry location and scale-invariant growth. We depart from recent contributions of New Economic Geography in that our model does not exhibit the “strong” scale effect in R&D which is inconsistent with time-series evidence from advanced OECD economies. In contrast with previous research, we find that, when R&D spillovers are localized, trade integration affects economic growth only in the short-run; the sign of this (temporary) effect depends on whether intertemporal knowledge spillovers in R&D are positive or negative. We show that this result has important implications for the relation between trade integration and regional income disparity. © 2010 Elsevier B.V. All rights reserved.

Keywords: Regions Trade integration Regional disparity Growth Scale effects

1. Introduction New Economic Geography (NEG) and New Growth Theory (NGT) deal with issues that are strictly related. The first field analyzes the mechanisms that are responsible for the spatial concentration of economic activities. The second field, instead, explains the fundamental determinants of economic growth and analyzes how new economic activities arise as a result of technological innovations. Since these topics are strictly interconnected, scholars have tried to merge the two bodies of literature by linking economic growth to geography. The basic framework of monopolistic competition à la Dixit and Stiglitz (1977), which encompasses both the literature of NEG and that of NGT, has been extensively used in various studies. Among such contributions, Englmann and Walz (1995) and Walz (1996) incorporated the centripetal forces highlighted by Krugman (1991a,b) and Venables (1996) in a two-region growth framework. The authors showed that, with preference for diversity in imperfectly tradable intermediates and inter-regional factor mobility, linkages between intermediate and final good producers could lead to a core-periphery pattern, with production and innovation activities agglomerated in one region. Martin and Ottaviano (1999), Baldwin and Forslid (2000) and Baldwin et al. (2003, ⁎ Corresponding author. Facoltà di Economia, Piazza Scaravilli n. 2, 40126, Bologna, Italy. Tel.: + 39 051 2098486. E-mail address: [email protected] (A. Minniti). 0166-0462/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.regsciurbeco.2010.07.003

ch. 7) showed that a similar process of clustering could arise when consumers love variety of final goods that are not perfectly tradable. All of these papers usually build on models of growth with brand proliferation along the lines of Romer (1990) and Grossman and Helpman (1991, ch. 3) — hereinafter called RGH models. A common feature of these R&D-based endogenous growth models is that the rate of economic growth increases with the amount of resources devoted to the research sector, with the size of an economy determining its rate of growth. This effect, termed by Jones (2005) as the “strong” scale effect, is frequently seen as a major shortcoming of the first-generation models of endogenous innovation. In fact, as shown by Backus et al. (1992) and Jones (1995a), empirical evidence from industrialized countries does not support the strong scale effect in R&D.1 Although in recent contributions of NGT the strong scale effect has been removed, NEG models rely on theoretical frameworks that are empirically inconsistent as they are still exposed to the “Jones critique”. In this paper, we revisit the relation between trade integration, 1 Empirically, this effect has two implications. First, large economies should grow faster than small ones. Second, population growth should be accompanied by faster income growth. Jones (1995a) showed that these predictions are at odds with the historical record of the OECD countries, in which dramatic increases in population were accompanied by relatively constant growth rates. During the last decade, considerable effort has been devoted to eliminating the strong scale effect; a variety of non-scale growth models have been proposed, including Jones (1995b), Kortum (1997), Segerstrom (1998) and Howitt (1999).

A. Minniti, C.P. Parello / Regional Science and Urban Economics 41 (2011) 20–31

industrial location and growth in the light of the recent debate on scale effects. Specifically, we depart from recent contributions of NEG in that we explore the growth and welfare effects of trade integration in a tworegion model of scale-invariant growth. With respect to the NEG literature, our paper differs in at least three respects. First, our model assumes that the rate of population growth in the two regions is positive (instead of being zero). Second, in order to avoid the counter-factual empirical implications of the first-generation endogenous growth models, we assume that the creation of new products takes place at a decreasing rate as in Jones (1995b). Third, our model belongs to the so-called second-generation endogenous growth models (see Jones, 1999) and exhibits a “weak” scale effect in per capita production, meaning that a larger economy has a higher per capita production than a smaller one. This property of the model is in line with recent empirical studies by Alcalá and Ciccone (2004) and Antony (2005, 2006) that support this prediction, respectively at the country and regional levels. In a spatial model of endogenous growth with perfect capital mobility, we find that getting rid of the strong scale effect delivers new and interesting outcomes. In particular, the steady-state growth rate of the world economy does not depend on either trade integration or the geographic location of economic activities. Trade integration has only a temporary impact on economic growth when R&D spillovers are localized; the sign of this effect, however, depends on the strength of intertemporal knowledge spillovers in R&D. When spillovers are positive, meaning that the productivity of researchers in creating new knowledge increases in the stock of knowledge, we find that trade integration spurs temporary economic growth. The opposite occurs when spillovers are negative. These results are in sharp contrast with the NEG-growth literature, whose main finding is that trade integration as well as geography permanently affect economic growth when R&D spillovers are localized (see, e.g., Martin and Ottaviano, 1999). Our model exhibits a different steady-state scenario, because of decreasing returns in the R&D sector. Existing models linking geography to growth only allow for constant-returns-to-scale, and implicitly assume that intertemporal knowledge spillovers in R&D are quite strong. Removing the strong scale effect has also important implications for the relation between trade integration and income disparity across regions. In fact, in NEG models economic growth acts as a channel through which the geography of production affects regional disparity: industrial agglomeration fostering growth is ultimately detrimental to the value of capital. This occurs because faster entry of new firms leads to more competition and harms capital owners. As a consequence, regions with a different capital equipment tend to be more similar to each other and income inequality shrinks. Such a result does not hold in our model: since geography does not affect growth in the long run, the value of capital and the level of income inequality across regions do not change when economies become more integrated. This prediction finds empirical support in a recent work by Bouvet (2007); the author, studying the determinants of regional income inequality in the EU, shows that trade integration did not have any significant effect on interregional disparity during the period 1997–2003. Our work is strictly related to the recent literature of New Trade Theory (NTT) dealing with international trade and endogenous growth. This strand of literature, combining a two-country setting with R&D-based growth, addresses trade policy issues and presents many analogies with NEG contributions.2 Among this literature, it is worth mentioning two contributions that share some similarities with 2 For instance, both approaches usually feature monopolistic competition and international knowledge spillovers in a two-region/country set-up. Despite these similarities, there are some important differences in the objects of study. While the NEG literature explains important spatial issues trying to connect trade with location, the main focus of new trade theorists is not the analysis of the relation between agglomeration and growth. Moreover, NTT models do not contain circular causality mechanisms, which are important determinants of industrial clustering in the NEGgrowth literature (see, e.g., Fujita and Thisse, 2003; Yamamoto, 2003).

21

our work. Gustafsson and Segerstrom (2010a,b) are models of international trade with expanding product variety in which disembodied knowledge generates imperfect international spillovers in research as in Grossman and Helpman (1991, Ch. 9). We differ from these papers in many respects. As concerns Gustafsson and Segerstrom (2010a), our model features North–South trade rather than North–North trade, which allows us to deal with regional disparity. In Gustafsson and Segerstrom (2010b) the South can imitate northern varieties thanks to weaker patent protection in the southern region. In our paper, instead, innovation occurs only in the North although the assumption of perfect capital mobility allows the accumulation of product varieties in the South. The paper is organized as follows. The next section sets up the model for the case of localized spillovers. Section 3 solves for the balanced growth path and analyzes the steady-state properties of the model. Here, we analyze the geography-growth link and explore the relation between trade integration and regional disparity. Section 4 extends the analysis to the case of globalized spillovers. Finally, Section 5 concludes. 2. The model 2.1. Basic set-up There are two locations, North and South, which trade with each other. They are identical except for their initial level of non-labor wealth, K(0) in the North and K ⁎(0) in the South; the North is initially richer than the South so that K(0) N K ⁎(0). In each of the two regions, there is a fixed measure of households that provide labor services in exchange for wage payments. Each member of a household has an infinite life horizon and is endowed with one unit of labor that is inelastically supplied. In both economies, the size of each household, measured by the number of its members, grows exponentially at the exogenous rate gL N 0. The market for labor is perfectly competitive in each region. Labor is perfectly mobile within an economy and is paid the wage rate w per unit of labor supplied. In what follows, we describe the economy only in the North as the South is almost symmetric. An asterisk refers to variables of the South. 2.2. Consumption In both locations households share identical preferences. The representative household maximizes the following discounted lifetime utility: U=

∫

∞

0

−ðρ−gL Þt

e

h i 1−α α log Y ðt Þ Dðt Þ dt;

ð1Þ

where ρ N gL is the subjective discount rate, Y is a homogeneous “traditional” good that is taken as numéraire and D is a composite “modern” good which, following the framework of Dixit and Stiglitz (1977), is made up of a large number of differentiated products, that is: " Dðt Þ =

∫

N ðt Þ

0

1−1 = σ

Dω ðt Þ

#1 = ð1−1 = σ Þ with σ N 1:

dω

ð2Þ

In Eq. (2), N is the total number of varieties produced in the two regions, Dω is the quantity consumed of variety ω and σ is the elasticity of substitution between any two products. The value of expenditure E is: Eðt Þ =

∫

i∈n

pi ðt ÞDi ðt Þdi +

∫

j∈n⁎

τp⁎j ðt ÞDj ðt Þdj + Y ðt Þ;

where pi and p⁎j are the prices in the North and South, respectively. While there are no transaction costs on the traditional good, trade in the modern good is inhibited by frictional trade barriers, which are

22

A. Minniti, C.P. Parello / Regional Science and Urban Economics 41 (2011) 20–31

modeled as iceberg transport costs à la Samuelson: for one unit of the differentiated good to reach the other region, τ ≥ 1 units must be shipped. The set of firms in the North and South is endogenous, being equal respectively to n and n⁎ (with n + n⁎ = N). Utility maximization yields that a constant fraction α of expenditure E is devoted to the consumption of the modern good D and the complementary share 1 − α to the consumption of the traditional good Y. Moreover, the individual demands of domestically-produced and imported varieties are respectively equal to:

Di ðt Þ =

pi ðt Þ−σ αEðt Þ ; P ðt Þ1−σ 

where P ðt Þ≡

∫

i∈n

Dj ðt Þ =

pi ðt Þ1−σ di +

τ−σ p⁎j ðt Þ−σ αEðt Þ P ðt Þ1−σ

∫

j∈n⁎

;

ð3Þ

1 = ð1−σ Þ δp⁎j ðt Þ1−σ dj is the CES

price index associated with Eq. (2) and δ ≡ τ1 − σ ∈ (0, 1] is a measure of the freeness of trade, which increases as τ falls and is equal to one when trade is costless (τ = 1). Every household faces an intertemporal optimization problem by maximizing lifetime utility (Eq. (1)) under the usual intertemporal budget constraint A˙ ðt Þ = wðt Þ + r ðt ÞAðt Þ−Eðt Þ−gL Aðt Þ, where A(t) is the individual's assets at time t, w(t) is the individual's wage rate at time t, and r(t) is the market interest rate at time t. Taking prices and expenditure as given, dynamic optimization gives the well-known Euler equation: E˙ ðt Þ = r ðt Þ−ρ; Eðt Þ

ð4Þ

which says that consumer expenditure E grows over time if and only if the market interest rate r exceeds the subjective discount rate ρ. 2.3. Production The homogeneous good Y is produced through a constant-returnsto-scale technology using labor (LY) only. By choice of units, one unit of labor is required to produce one unit of this good, that is: Y ðt Þ = LY ðt Þ:

ð5Þ

The Y-good, which is sold at its cost price (equal to the wage rate w), is freely traded both between and within regions.3 Labor is intersectorally mobile and the presence of the constant-returns-to-scale sector ties down the wage rate in each location at each instant. Following recent NEG contributions, we assume that both countries always possess the constant-returns-to-scale sector so that wages are identical in both regions. By choosing the Y-good as numéraire , the common wage is also pinned down to 1. The assumption that traditional goods are produced in both regions, which is often referred to as the Non-Full Specialization (NFS) condition, is a key-feature of most NEG models; technically, it amounts to requiring that no region has enough labor to satisfy the domestic demand for the traditional good. Differentiated varieties produced under monopolistic competition and increasing returns to scale due to fixed and variable costs. More precisely, the production of each variety requires a fixed cost of one unit of capital and β units of labor. The global capital stock Kw determines the total number of varieties available in the two economies. As in equilibrium each good is produced by one and only one firm, Kw also determines the total number of firms, that is: w

K ðt Þ = K ðt Þ + K ⁎ ðt Þ = nðt Þ + n⁎ ðt Þ = N ðt Þ:

3 This assumption is rather standard in NEG models. The interested reader is referred to chapter 7 in Fujita et al. (1999) for further details.

As concerns capital mobility, we let capital owners decide where to locate production. In order to avoid catastrophic agglomeration, we follow Martin and Ottaviano (1999) and assume that capital is freely mobile even though capital owners are not; profits, thus, are repatriated in the region where capital is owned. This, in turn, implies that the share of firms located in the North, denoted by sn, may differ from the share of capital owned by the North, denoted by sk.4 2.4. Product markets Since there exists a continuum of firms, each producer chooses a profit-maximizing price taking aggregate expenditure and other firms' prices as given. Both northern and southern producers face the same marginal production cost β and the same constant elasticity of demand σ. Thus, regardless of their location, firms choose the following pair of profit-maximizing prices: p = p⁎ =

σβ ; σ−1

ð6Þ

which is the standard markup of price over marginal cost. By substituting for the price, a firm's profits write as: π=

x β; σ−1

ð7Þ

with x being the size of the firm. In equilibrium demands have to be equal to supplies at home and abroad; thus, the flows of sales of northern and southern firms can be written respectively as: !   σ−1 E E⁎ δ + ⁎ x = αL ; βσ n + n⁎ δ n + nδ !   ⁎ σ−1 E Eδ + : x⁎ = αL βσ n + n⁎ δ n⁎ + nδ

ð8Þ

Once a patent is granted, the entrepreneur receives monopoly rights on the new variety and can choose to relocate freely production facilities across regions. Since there are no relocation costs on capital, the entrepreneur will repatriate profits if she decides not to locate manufacturing in the region of residence. In deciding where to set up their manufacturing facilities, entrepreneurs compare profits they would make in the two regions. In equilibrium profits have to be the same in both economies so that entrepreneurs will be indifferent where to locate production.5 Absent any wage divergence between regions, the assumption of free mobility of capital prevents the occurrence of catastrophic agglomeration in the North. Before proceeding with the analysis, it is worth spending some words on the equalization of wages between regions. This feature of the model, which derives from the NFS condition coupled with the assumption that the traditional good is freely traded, distinguishes NEG contributions from other modeling approaches, such as that of

4 The model's dynamics strictly depend on the assumption of capital mobility. In the paper, capital is mobile even though capital owners are not. This implies that the share of firms located in the North, sn ≡ n / N, may not coincide with the share of capital owned by the North, sK ≡ K / Kw. As we will see shortly, sn is endogenous and determined by an arbitrage condition that says that the location of firms is in equilibrium when profits are equalized in the two regions. Because of capital mobility, the decision to accumulate capital is identical in both locations so that the share of capital owned by the North, sK, is time-invariant and entirely determined by the initial distribution of capital ownership between the two regions, K(0) / K ⁎(0). 5 The assumption of perfect capital mobility has important implications for firms' location decision. In our model, as in Martin and Ottaviano (1999), capital owned by residents of a region can be located in the other region. The arbitrage condition requires the equalization of profits across locations so that firms are indifferent between the two regions. By pinning down the equilibrium location of firms, this arbitrage condition leaves no incentive for relocation at the margin.

A. Minniti, C.P. Parello / Regional Science and Urban Economics 41 (2011) 20–31

NTT, where wages are not necessarily identical across locations.6 In these papers, when agglomeration in one region is high enough, the northern modern sector can become so large to draw all the labor away from the traditional sector; this, in turn, drives the northern wage rate up. The rise of the wage rate acts as a dispersion force which mitigates agglomeration, thus counterbalancing the lower cost of research in the North.7 Returning to the model, we have that the equalization of profits in equilibrium implies that x = x⁎. Imposing such a condition yields the proportion of firms located in the North: sn ≡

  n 1 ð1 + δÞ 1 = + sE − ; N 2 ð1−δÞ 2

ð9Þ

with sE ≡ E / (E + E ⁎) being the northern share of total expenditure. As can be seen, the share of firms producing in the northern region is more than proportional to the North's share of world expenditure. This is the so-called home market effect arising from the interaction of transport costs and increasing returns (Krugman, 1980). Finally, by using Eq. (9) into Eq. (8), one can express the size of the firm in both the South and the North as: x = x⁎ = αL

  σ−1 Ew ; βσ N

ð10Þ

where Ew ≡ E + E ⁎ denotes the world expenditure.

Firms must devote resources to R&D in order to accumulate knowledge capital. As in Romer (1990), knowledge capital is a quasipublic good, freely available to all potential innovators. In the North, it takes bI units of labor to create one unit of capital. We assume that bI can change over time due to knowledge spillovers, that is: bI ðt Þ =

1 ; K w ðt Þϕ Aðsn Þ

R&D and productivity growth. In particular, data on patenting suggest that intertemporal knowledge spillovers, far from being positive and strong (ϕ = 1), may actually be negative (ϕ b 0).8 In the present paper, therefore, we relax the assumption of ϕ = 1 in order to remove the strong scale effect that characterizes all of the first-generation R&Ddriven endogenous growth models. In particular, we follow Jones (1995b) by using an R&D technology with ϕ b 1; such a strategy is sufficient to generate a balanced growth path that is consistent with increasing labor force.9 Since it is less costly to engage in R&D in the location where there are more firms, the R&D activity is entirely undertaken in the location with an initial higher stock of capital (and firms). Given perfect competition in R&D, North attracts all of the research laboratories, so that R&D activity is carried out entirely in the northern location. To develop a new product, a representative northern firm ω must devote bI units of labor to R&D. Thus, the rate at which firm ω discovers new products — i.e., accumulates capital stock — writes as: L ðt Þ N˙ ω ðt Þ = K˙ ω ðt Þ = Iω ; bI ðt Þ where N˙ ω is the flow of new varieties generated by an R&D project employing LIω units of labor for an interval of time dt. Summing over the research laboratories, the aggregate rate at which the North develops new products is given by: L ðt Þ w N˙ ðt Þ = K˙ ðt Þ = I ; bI ðt Þ

2.5. Innovation

ð11Þ

where Aðsn Þ ≡ ½sn + λð1−sn Þϕ , ϕ b 1 measures the strength of intertemporal knowledge spillovers and λ∈½0; 1 is a parameter governing the internalization of technology spillovers. When R&D spillovers are localized, the R&D cost in a region is a function of the location of firms. In the presence of globalized spillovers, instead, the marginal cost of an innovation would depend on the total stock of existing capital and, therefore, it would be identical in both regions. In the main part of the paper, we will assume that spillovers are localized, so that λ is strictly lower than one. In a final section, we will discuss the case of global spillovers by solving the model for λ = 1. If ϕ N 0, the marginal cost of innovating over a given interval of time is a decreasing function of the knowledge stocks in the two regions; this is often referred to as the “standing on the shoulders” effect. For ϕ b 0, the cost of creating new knowledge increases as the two stocks of knowledge get larger; this is the so-called “fishing out” effect. In models of growth and economic geography based on the RGH framework, it is usually assumed that ϕ = 1. However, as shown by Jones (1995b), ϕ = 1 represents an arbitrary degree of increasing returns and is inconsistent with a broad range of time-series data on

6 See for instance Glass and Saggi (2002), Parello (2008), Dinopoulos and Segerstrom (2010) and Gustafsson and Segerstrom (2010b) for North–South growth models with wage inequality. 7 As said in the Introduction, this is not the unique difference between the two bodies of literature. Other distinctive aspects of the NEG approach with respect to that of NTT are the absence of any imitation activity, the focus on a horizontally-orientated innovation scheme and the presence of perfect protection of intellectual property rights in the South.

23

ð12Þ

where LI ðt Þ = ∑ω LIω ðt Þ is the total amount of labor employed by northern R&D laboratories. Capital is freely mobile and both regions accumulate capital at the same rate. In fact, the grant of a patent in the North does not prevent its use in the South. Manufacturing firms choose to locate and produce in the region that provides the highest level of profits, so that the owners of capital repatriate firms' operating profits in their region of residence. A geographical equilibrium is then achieved when profits are equalized across regions. Coupled with the assumption of a perfect financial market where agents can freely lend and borrow, this implies that individuals face the same incentives to invest in the two regions.10 To simplify the notation we omit, henceforth, the time index from variables except for the index 0 referring to the initial values of variables. 2.6. Stock market There exists a safe asset, bearing an interest rate r in units of the numéraire. Because of free capital movements between the two regions, consumer expenditure has to grow at the same rate in the North and in the South, that is E˙ = E = E˙ ⁎ = E ⁎ . The owners of a firm earn profits π d t during the time interval dt and realize the capital gain ˙vdt, where v is the present discounted value of all future

8 The strong scale effect is at odds with empirical evidence. For instance, Jones (1995a) has shown that there has been no upward trend in productivity growth rates for the U.S., France, Germany and Japan since 1960 in spite of substantial increases in population size and R&D employment. 9 As we will see shortly, if one adopts the RGH formulation and assumes ϕ = 1, no balanced growth path exists with a growing population. 10 This is the reason why no boundary equilibrium (with respect to firms' locations) exists. Under the assumption of perfect capital mobility, in fact, profits have to be the same in both regions; this, in turn, implies that the value of capital is identical in the North and in the South. Since agents have the same incentives to invest in the two locations, both regions accumulate capital at the same rate and any initial distribution of capital is stable. Given the absence of both backward and forward linkages, no “catastrophic agglomeration” can occur in our model.

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A. Minniti, C.P. Parello / Regional Science and Urban Economics 41 (2011) 20–31

operating profits. The total return on equity claims must equal the risk-free market interest rate r, that is: ˙ = rvdt: πdt + vdt This equation represents the usual arbitrage condition on the stock market. Since the R&D sector is characterized by free-entry and perfect competition, the value of a firm v is equal to the marginal cost of inventing a new variety bI. Thus, using Eq. (7) to substitute for π and Eq. (10) to substitute for x, the no-arbitrage condition for the stock market can be written as: z=

α Aðsn ÞEw = σ ; r− b˙ = b I

ð13Þ

The equilibrium of the model can be summarized by the following three equations: the no-arbitrage Eq. (13), the full-employment condition (14) and the positive relation existing between sn and sE (Eq. (9)). The three endogenous variables are the relative R&D difficulty z, the steady-state rate of innovation g, and the world consumption expenditure Ew. Let “∼” denote steady-state values. From Eq. (14) we know that a steady-state with a constant growth rate of varieties only exists if both the endogenous variables, Ew and z, are constant over time. Then, we log-differentiate the identity z ≡ N1 − ϕ / L with respect to time and we get: z˙ = ð1−ϕÞg−gL : z

I

where z ≡ N1 − ϕ / L is a new endogenous variable measuring the relative R&D difficulty at time t. Variable z is not new in the growth literature.11 It captures the idea that pursuing innovations becomes increasingly complex as time goes by. In fact, when ϕ b 0, researchers become less productive in creating new knowledge as the stock of knowledge increases over time. When ϕ∈ð0; 1Þ, instead, innovations make researchers more productive; however, these productivity gains occur only at a decreasing rate. In both cases, producing future innovations requires more R&D labor as time goes by. 2.7. Labor market Labor supply is equal to 2 L at the aggregate level. As concerns labor demands, in the modern sector each manufacturing firm employs βx units of labor independently of its location. By using Eq. (10) and aggregating over industries, labor demand in the two manufacturing sectors amounts to LM = αLðσ−1ÞEw = σ. Moreover, since agents allocate a share 1 − α of their individual expenditure to the consumption of the traditional good, the demand of labor in the traditional sector as a whole amounts to LY = L(Y + Y ⁎) = L(1 − α)Ew. Finally, labor demand in the R&D sector equals LI = N˙ bI . At the world level, the labor market clearing condition, 2L = LI + LY + LM, writes as: ˙ + Lð1−αÞEw + αL ðσ−1Þ Ew ; 2L = Nb I σ

Since z is constant in the steady state, its growth rate must be zero, that is ż / z = 0. This, in turn, implies that the steady-state innovation rate is exogenous and equal to: g˜ =

gL : 1−ϕ

ð15Þ

Eq. (15) establishes that the steady-state rate of innovation g˜ is proportional to the growth rate of population gL, where the factor of proportionality depends on the degree of returns to scale in the R&D sector. As in Jones (1995b), the parameter restriction ϕ b 1 is needed to ensure that the steady-state rate of growth is positive and finite (given that population growth is positive). Let us now analyze the world consumption expenditure Ew. In a steady-state equilibrium nominal consumption expenditures are w constant over time, that is E˙ = E˙ = E˙ ⁎ = 0. By using the Euler Eq. (4), this requires the worldwide interest rate to be equal to the subjective discount rate r = ρ. This result implies that both the regional distribution of income sE and the proportion of firms located in the North sn are constant over time, which also means that A(sn) is constant in the steady-state. By inserting r = ρ into Eq. (13) and using the fact that b˙ I = bI = −ϕgL = ð1−ϕÞ, the no-arbitrage equation becomes: w

which can be simplified as: −1

2 = gzAðsn Þ

+

ðσ−αÞ w E ; σ

z=

α Aðsn ÞE = σ : ρ + ϕgL = ð1−ϕÞ

ð16Þ

ð14Þ

where g ≡ N˙ = N denotes the growth rate of varieties. According to Eq. (14), the world labor supply can be used either in the manufacturing sectors or in the innovation sector. It also implies that a steady state with a constant growth rate of varieties only exists if both Ew and z are constant over time. 3. The steady-state equilibrium In this section we restrict our attention to studying the steadystate equilibrium of the model. The latter is defined as follows: Definition 1. In the steady-state equilibrium, ðiÞ all endogenous variables grow at a constant (not necessarily identical) rate, ðiiÞ the rate of innovation attains a strictly positive value and is constant over time (i.e., ġ = 0) ðiiiÞ industry location is stable (i.e., s˙n = 0), ðivÞ the regional distribution of income is constant over time (i.e., s˙E = 0).

Finally, to close the model, we need a side condition describing the steady-state resource constraint of the economy. Plugging Eq. (15) into Eq. (14), the steady-state full-employment condition reads: −1

2 = zAðsn Þ

σ−α gL w E : + σ 1−ϕ

ð17Þ

Given sn, Eqs. (16) and (17) form a system of two equations in two unknowns: the world consumption expenditure Ew and the relative R&D difficulty z. It is easy to verify that Eq. (16) is upward sloping in ðz; Ew Þ space, Eq. (17) is downward-sloping in ðz; Ew Þ space, and they have a unique intersection in the strictly positive orthant given by:   ϕg 2σ ρ + L 1−ϕ w  E˜ =  ϕgL −αðρ−gL Þ σ ρ+ 1−ϕ

∧

z˜ =

 σ ρ+

2αAð s˜n Þ  : ϕgL −αðρ−gL Þ

1−ϕ

ð18Þ w

11 The concept of relative R&D difficulty was introduced in the growth literature by Segerstrom (1998).

Summarizing, z˜ and E˜ represent, respectively, the level of the world consumption expenditure and relative R&D difficulty compatible with

A. Minniti, C.P. Parello / Regional Science and Urban Economics 41 (2011) 20–31

25

a constant rate of innovation equal to g˜ and a stable distribution of production location between the two regions, s˜n : Then, we can establish the following Proposition:

In equilibrium v = bI, so that Nv / L = NbI / L = zA(sn) − 1. Using Eq. (18) to get rid of z, we write per capita expenditures in the North and South respectively as:

Proposition 1. In the presence of local knowledge spillovers, there exists a unique steady-state equilibrium as that identified by Definition 1 in which both regional inequality sE and the proportion of firms located in the North sn are constant over time, and where the innovation rate of the economy is completely pinned down by the growth rate of population gL and the parameter governing intertemporal knowledge spillovers ϕ.

E˜ = 1 + ðρ−gL Þ

The economic intuition behind Proposition 1 may be provided as follows. Because labor supply L grows at the constant exogenous rate gL and consists of workers employed in either R&D or production, the share of R&D labor in the labor force LI / L is constant over time if and only if R&D employment grows at the rate gL as well. By using Eq. (12) and the definition of bI, the share of workers employed in R&D writes as: LI gz = : Aðsn Þ L In a balanced growth path with LI / L, g, and sn constant, the relative R&D difficulty z has to be constant as well. This implies that the steady-state rate of innovation ˜g is necessarily equal to Eq. (15).12 Before concluding, we make two useful remarks. First, as already mentioned above, the R&D technology used in this paper differs from that used in existing models of growth and geography where the intertemporal knowledge spillover parameter ϕ is typically assumed to be equal to 1. It is worth noticing that Eq. (15) cannot apply if ϕ = 1; in that case, in fact, the denominator of Eq. (15) would explode and no balanced growth path would exist because L is growing. Under the assumption ϕ = 1, the growth rate of product variety is proportional to the R&D employment level (see Eqs. (11) and (12)) and the model exhibits the strong form of scale effects.13 Second, in our scaleinvariant growth set-up, any change in the exogenous parameters of the model leads only to temporary changes in the growth rate of the economy. This result stems from the semi-endogenous nature of the model according to which only the R&D investment is endogenous, not the rate of innovation.14

12 In Appendix A we show that the model's solution is saddle-path stable and that the phase diagram associated with the steady-state equilibrium has two solution branches which can be calculated explicitly. 13 The direct implication of assuming ϕ = 1 in the production function for new ideas is that a growing number of researchers cause the growth rate of the economy to grow exponentially. This prediction, which is a common feature to R&D-based growth models of first generation, is undoubtedly wrong. 14 The term “semi-endogenous growth” was first introduced in the literature by Jones (1995b). Examples of such models are those developed by Kortum (1997) and Segerstrom (1998).

ð19Þ

and ⁎ E˜ = 1 + ðρ−gL Þ

2αð1−sk Þ : α g˜ + ðσ−αÞðρ + ϕ g˜ Þ

ð20Þ

w We can easily obtain s˜E by dividing E˜ in Eq. (19) by E˜ in Eq. (18), that is:

s˜E =

1 αð2sk −1Þðρ−gL Þ + : 2 2σ ðρ + ϕ g˜ Þ

ð21Þ

The northern share of total expenditure s˜E provides a measure of regional disparity in nominal incomes. As one can easily notice, s˜E is larger than 1/2 since the North is initially wealthier (sk N 1/2). According to Eq. (21), the expenditure share in the North depends ˜ This occurs because faster entry of negatively on the growth rate g. new firms (higher growth) translates into more competition so that the value of holding capital is inversely related to the rate of growth. As the North is relatively rich in capital, the level of capital income declines more in the North than in the South and, consequently, the northern share of total expenditure gets smaller. Knowing s˜E , we can look at the distribution of production across the two regions. Plugging s˜E from Eq. (21) into Eq. (9) gives the proportion of firms located in the North, that is: s˜n =

1 ð1 + δÞαð2sk −1Þðρ−gL Þ + : 2 2σ ðρ + ϕ g˜ Þð1−δÞ

ð22Þ

Since ∂ s˜n = ∂δ N 0, lower transaction costs are associated with more concentration of firms in the North; intuitively, when transaction costs are low, firms can locate in the largest market, that is the North, and still export to the South. Thus, trade integration (an increase in δ) spurs industrial agglomeration in the northern region. Finally, we determine the two CES price indices of the composite modern good in the North and South. These are equal respectively to: P=

 σ  1 = ð1−σ Þ 1 = ð1−σ Þ β ½ s˜n ð1−δÞ + δ N ; σ−1

ð23Þ

⁎

 σ  1 = ð1−σ Þ 1 = ð1−σ Þ β ½1− s˜n ð1−δÞ N : σ−1

ð24Þ

3.1. Regional inequality We now turn to the issue of regional income inequality. The latter depends not only on the distribution of expenditure and production between the two regions but also on the geographic differences in the cost-of-living. Thus, we initially determine the values of sE and sn in the steady-state equilibrium. Then, we derive the two price indices. Consider first the sE variable. Since asset holdings are stationary, from the intertemporal budget constraint one can write per capita income as per capita labor income (the wage rate 1) plus the income from holding capital; the latter amounts to the value of the firms owned by residents in each region multiplied by the net return, ρ − gL. Thus, per capita expenditures in the North and South can be written respectively as E = 1 + (ρ − gL)skNv / L and E ⁎ = 1 + (ρ − gL)(1 − sk) Nv / L.

2αsk ; α g˜ + ðσ−αÞðρ + ϕ g˜ Þ

P =

As one can see from Eqs. (23) and (24), the price index in the North is lower than the price index in the South. In fact, the majority of firms are located in the North; as northerners import fewer varieties of goods subject to transaction costs, the cost-of-living is lower in the northern region.15 3.2. Productivity growth We are now in the position to measure productivity and solve for how it evolves over time. Since real consumption coincides with real output and labor is the only factor of production, we use per capita real consumption as a measure of productivity. Now, the expansion of product variety is the only source of productivity growth in the model. In order to bring this point out more clearly, we observe that a larger number of varieties 15 The price indices (23) and (24) in the two regions depend on the location of production: an increase in spatial concentration in the North, ˜sn , benefits northerners and hurts southerners.

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increases per capita real consumption in both regions by lowering the two price indices (23) and (24). To determine how per capita real consumption and productivity grow over time, we take one of the two regions, for instance the North, and write per capita real consumption as per capita nominal income divided by the “perfect” consumption price index.16 Given preferences (Eq. (1)) and price normalization, the perfect price index amounts to P α. In steady-state, per capita nominal income E˜ and the share of firms located in the North s˜n are timeinvariant; the price index P, instead, falls along the steady-state ˜ Thus, P falls at the rate growth path since N rises at the growth rate g. g˜ = ð1−σ Þ and productivity grows at the rate α g˜ = ð1−σ Þ. This, in turn, implies that the level of per capita income in the long run is an increasing function of the size of the economy. Since productivity is proportional to the size of the labor force, our model exhibits the weak form of scale effects. This key-feature of R&D growth models of second generation is supported by recent empirical research. Jones (2005) provides an interesting discussion of this issue and cites Alcalá and Ciccone (2004) as providing the best crosscountry evidence in favor of this property.17 At a finer level of geographic detail, Ciccone and Hall (1996), Ciccone (2002) and, more recently, Antony (2005, 2006) find significant evidence in favor of the weak scale effects prediction by using data from European regions and US counties. 3.3. The geography-growth link In models of growth and economic geography based on the RGH framework (such as Martin and Ottaviano, 1999), lower transaction costs permanently increase the growth rate in the presence of local spillovers in R&D.18 In fact, a higher concentration of firms in the location where the R&D activity is entirely agglomerated implies a lower cost of R&D; this, in turn, spurs economic growth. The positive relation between agglomeration in the North and growth is referred to as the geography-growth link. In our scale-invariant growth model, falling transaction costs still lead to more concentration of firms in the northern region but this has no effect on the growth rate of the number of varieties in the long run. However, as anticipated above, trade integration affects temporarily economic growth through its impact on geography. In order to explore this issue, we make use of the relative R&D difficulty z and observe that according to Eq. (18) a change in δ does affect z˜ through its impact on Að s˜n Þ. In particular, if the parameter governing the strength of intertemporal knowledge spillovers ϕ is positive (nega˜ Since tive), an increase in δ leads to a larger (smaller) steady-state z. z ≡ N1 − ϕ / L, the permanent increase (decrease) in z˜ can only occur if the number of varieties N temporarily grows at higher (lower) rates ˜ Thus, we can establish the than the steady-state growth rate g. following: Proposition 2. In the presence of local knowledge spillovers, trade integration (δ ↑) has no long-run effect on the growth rate of product ˜ Moreover, if ϕ b 0, trade integration (δ ↑) does cause a temporary variety g. slowdown in variety growth and a permanent decrease in the number of varieties produced (N ↓ for all t). Instead, if ϕ ∈ (0, 1), trade integration (δ ↑) 16

Using standard terminology, the price index is “perfect” because real income defined with it provides a measure of indirect utility. 17 Controlling for both trade and institutional quality, Alcalá and Ciccone (2004) found that a 10% increase in the size of the workforce in the long run is associated with 2.5% higher GDP per worker. Another influential paper that provides empirical support for the weak scale effect at the country level is represented by Frankel and Romer (1999). 18 Most of these contributions have the property that the location of innovation matters for growth only in the case of local technological spillovers. For instance, Martin and Ottaviano (1999) showed that the geographical location of economic activities does affect economic growth when R&D spillovers are localized. Instead, the impact of geography on the rate of growth is nil when spillovers are global, that is when the North and the South learn equally from an innovation made in any region.

does cause a temporary acceleration in variety growth and a permanent increase in the number of varieties produced (N↑ for all t). A formal proof to Proposition 2 is provided in Appendix A where we investigate in depth the dynamic implications of the model. The intuition behind these results is as follows. Trade integration (through a decrease in the transaction costs) makes it more profitable for firms to become exporters and settle in the North. If intertemporal knowledge spillovers are positive (ϕ ∈ (0, 1)), a higher concentration of firms in the North leads to a lower R&D cost. Since the incentives to innovate increase, the rate of innovation rises temporarily above its steady-state rate g˜ and, consequently, the overall economy experiences a permanent increase in the number of varieties produced; trade integration, thus, promotes variety growth in the short-run by raising the growth rate along the transition path to the new steadystate. The opposite occurs when intertemporal knowledge spillovers are negative (ϕ b 0). In both cases, however, trade integration and geography do not have any long-run effects on growth.19

3.4. Trade integration and regional disparity In this section we examine the effects of trade integration on regional inequality. As observed before, real income disparities across regions depend on differences in nominal incomes and price indices.20 We have already shown that trade integration leaves nominal income disparity among regions unchanged. In our model, therefore, trade integration might affect regional inequality only by changing the cost-of-living in the two regions. At this regard, we identify three separate effects of trade integration on price indices: 1. a direct openness effect arising directly from the decrease in transaction costs: such an effect is positive for residents in North and South; 2. an indirect static effect operating through the change in firm location, s˜n . Lower transaction costs, in fact, lead to more agglomeration in the North: this is beneficial for northerners and detrimental for southerners; 3. an indirect dynamic effect that operates through the change in the number of varieties produced. In fact, an increase in s˜n , affecting temporarily variety growth, does cause a permanent variation in the number of product varieties and, consequently, in the price indices. The sign of such an effect depends on whether intertemporal knowledge spillovers in R&D are positive or negative. In order to disentangle these effects, we look at the response of price indices (in percentage terms) to trade integration. Substituting 1 = ð1−ϕÞ N by ð z˜ LÞ into Eqs. (23) and (24) and using z˜ from Eq. (18), we get: 2 3 ∂ s˜ ð1−δÞ n ∂P = ∂δ 1 4 1− s˜n ϕ ð1−λÞ ∂ s˜n 5 ∂δ ; + + = ð1−ϕÞ Að s˜n Þ1 = ϕ ∂δ P 1−σ s˜n ð1−δÞ + δ s˜n ð1−δÞ + δ

ð25Þ 19 This result is reminiscent of the “policy-invariance” property of semi-endogenous growth models. In such models, in fact, the long-run growth rate depends only on the rate of population growth and on the parameter governing the degree of returns to scale in the R&D sector. Therefore, policy changes do not have any effect on the longrun growth rate. This is in contrast with what occurs in models that exhibit the strong scale effect; in these models, the long-run growth rate is an increasing function of the number of researchers, so that a policy that stimulates R&D investment increases the growth rate. For a more detailed discussion of this issue, see Jones (2005). 20 Throughout the analysis, welfare differences between northerners and southerners are assessed by comparing their real incomes. In fact, dealing with real income is equivalent to performing a welfare analysis.

A. Minniti, C.P. Parello / Regional Science and Urban Economics 41 (2011) 20–31

2 3 ∂ s˜ ð1−δÞ n ∂P ⁎ = ∂δ 1 4 ϕ ð1−λÞ ∂ s˜n 5 s˜n ∂δ = − + : 1−σ 1− s˜n ð1−δÞ 1− s˜n ð1−δÞ ð1−ϕÞ Að s˜n Þ1 = ϕ ∂δ P⁎

ð26Þ

The first term into brackets is the direct effect, the second term corresponds to the indirect static effect, while the last term is the indirect dynamic effect. The direct effect benefits less the North than the South as the latter imports more goods subject to transport costs (sn N 1/2). The indirect static effect, instead, benefits the North but is harmful to the South. Finally, the indirect dynamic effect, which affects to the same extent both the North and the South, benefits or harms consumers depending on the sign of intertemporal knowledge spillovers in R&D. Since the indirect dynamic effect is the same in the two regions, when comparing changes in regional price indices, one needs to account for the direct and the indirect static effects only. As concerns the North, both these effects are positive. In the South, instead, the direct effect is positive, while the indirect static one is negative. Interestingly, the former effect dominates the latter one and is so pronounced to make the price index in the South vary to the same extent as in the North.21 In order to see this, we compare Eq. (25) with Eq. (26), that is:   2 ∂ s˜n ⁎ 1−δ −ð2 s˜n −1Þ ∂P = ∂δ ∂P = ∂δ ∂δ − = : ⁎ ˜ P ð 1−σ Þ s ½ ð 1−δ Þ + δ½1− s˜n ð1−δÞ P n

ð27Þ

This expression equals zero, as one can verify by using Eq. (9). Thus, a decrease in the transaction costs leads to the same variation in the cost-of-living in the two regions. A direct consequence of this fact is that trade integration does not affect the disparity in real income across regions. Therefore, we can establish the following Proposition: Proposition 3. Trade integration (δ ↑) does not affect regional income disparity. However, it leads to a more uneven distribution of production by increasing firm agglomeration in the larger region ( s˜n ↑). Results in Proposition 3 are partly in contrast with those obtained in previous studies. The reason is as follows. In models of growth and geography based on the RGH set-up, the geography-growth link has important implications for the relation between trade integration and regional disparity. Industrial agglomeration fosters economic growth, which in turn produces a detrimental effect on the value of capital income. Since the North is relatively rich in capital, income declines less in the South than in the North; this, in turn, implies that the northern share of total expenditure shrinks, with the result that regional income inequality decreases. In these models, therefore, nominal income disparity gets smaller and living in the North becomes relatively more costly (the difference

∂P = ∂δ P

⁎

∂P = ∂δ P⁎

−

27

tends to be more agglomerated, regional income inequality does not vary. It is interesting to notice that the prediction of Proposition 3 finds empirical support in the recent work by Bouvet (2007); the author studies the determinants of regional income inequality within 197 NUTS2 European regions during the period 1977–2003.22 In particular, the paper explores the impact of economic integration on regional income disparity through a panel data analysis relating inequality measures to demographic, macroeconomic and policy characteristics. The author separates the effects of EMU from those of the EU trade integration, by proxing the latter with the share of the intra-EU trade in total trade; in most of the specifications, it is shown that trade integration does not have any significant effect on interregional disparity.23 4. Extensions: local vs. global spillovers Thus far, we have discussed the properties of the model for the case of localized spillovers in R&D. In this section, we make some comparative statics by making the degree of localization of technology spillovers, λ, vary; then, we solve the model for the case of globalized spillovers, that is when λ is equal to one. It is worth observing that the steady-state innovation rate does not depend on λ. The reason why changes in λ do not affect long-run growth is not difficult to grasp. In our scale-invariant growth model, in fact, variations in the exogenous parameters lead only to temporary changes in the growth rate of the economy. Moreover, by using Eqs. (15) and (18) to get rid of g˜ and z˜ we get that the share of workers employed in R&D is totally independent on Að s˜n Þ, that is: L˜I = L

 σ ρ+

gL 1−ϕ

2α



ϕgL 1−ϕ

:

−αðρ−gL Þ

Consequently, a change in λ does not affect the steady-state share of workers employed in R&D, L˜I = L. Thus, we can establish the following: Proposition 4. A variation in the parameter measuring the strength of international R&D spillovers λ leads to: ðiÞ a temporary change in the innovation rate g; ðiiÞ no changes in the steady-state share of labor in R&D, L˜I = L: Let us turn to the case of globalized spillovers. When λ = 1, the function Aðsn Þ boils down to one and Eqs (16) and (17) become: z=

αEw = σ ; ρ + ϕgL = ð1−ϕÞ

ð28Þ

2=

σ−α zgL w E : + σ 1−ϕ

ð29Þ

in

Eq. (27) is positive instead of being zero). In our model, economic growth depends on geography in the short-run but not in the long run; consequently, the value of capital income in the steady-state is not affected by the geography of production. Since the northern share of total expenditure s˜E in Eq. (21) does not change when the North

21 This, in turn, implies that the overall effect of a decrease in the transaction costs on the cost-of-living in the two regions and on consumers' welfare depends on the sign of the indirect dynamic effect. In fact, when intertemporal knowledge spillovers in R&D are positive (ϕ N 0), trade integration reduces the cost-of-living and makes consumers better off in the long-run. Instead, when these spillovers are negative (ϕ b 0), trade integration leads to lower price indices and higher consumers' welfare as long as the indirect dynamic effect is not large enough (in absolute value) to compensate for the other two effects.

Eqs. (28) and (29) preserve their original slopes. As a result, there exists a unique steady-state that is characterized by the following pair

22 NUTS (Nomenclature of Territorial Units for Statistics) is a geocode standard developed by the EU for referencing the administrative division of countries for statistical purposes. According to this classification, NUTS0 refers to country level data, while increasing numbers indicate increasing levels of sub-national disaggregation. 23 We believe that the analytical framework we use in the present paper fits well the socio-economic situation in Europe. In the model, labor is immobile within regions but is able to move inside each of the two regions. This assumption seems coherent with the observation that intra-EU labor mobility is quite modest, especially if compared to that of the US. Moreover, free trade is coupled with international capital movements (free capital mobility); capital owners are not mobile and spend all of their income locally in the region where they reside. This feature of the model is also consistent with income repatriation patterns observed in the EU.

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A. Minniti, C.P. Parello / Regional Science and Urban Economics 41 (2011) 20–31

of steady-state values for, respectively, the world consumption expenditure and the relative R&D difficulty:   ϕg 2σ ρ + L 1−ϕ w  E˜ =  ϕgL −αðρ−gL Þ σ ρ+ 1−ϕ

∧

z˜ =

 σ ρ+

2α  : −αðρ−gL Þ

ϕgL 1−ϕ

w One can notice that E˜ is the same as for the case of localized spillovers, whereas z˜ is larger in the presence of globalized spillovers. Interestingly, the relative R&D difficulty z˜ does not depend on industry location s˜n ; this implies that the geography of production does not affect economic growth also in the short-run when spillovers are global. Finally, by looking at Eqs. (25) and (26), one can easily ascertain that the indirect dynamic effect of trade integration on price indices is nil, which means that a decrease in the transaction costs always leads to a reduction in the cost-of-living in the two regions. Moreover, as in Proposition 3, trade integration does not affect regional income disparity. We summarize these results as follows:

Proposition 5. In the presence of globalized knowledge spillovers: ðiÞ there exists a unique steady-state equilibrium as the one identified by Definition 1; ðiiÞ geography does not produce effects on the rate of economic growth either in the long run or in the short-run; (iii) trade integration does not affect regional income disparity.

interesting to re-consider the same issue when there is no capital mobility as in Baldwin (1999) and Baldwin et al. (2001). This will be the object of future research. Acknowledgments Antonio Minniti acknowledges the financial support of the University of Bologna through the Marco Polo program; Carmelo Pierpaolo Parello acknowledges the financial support of Sapienza, University of Rome through the Ateneo Federato program. Appendix A. The economic dynamics In this Appendix we provide a formal proof to Proposition 2. In doing so, we first analyze the dynamic properties of the model and, then perform a comparative dynamics analysis of the steady-state equilibrium when the parameter measuring the freeness of trade, δ, changes. A.1. The dynamic system The market equilibrium of the model is described by Eqs. (13) and (14) in the two endogenous variables Ew ≡ E + E ⁎ and z ≡ N1 − ϕ / L: w

z=

α Aðsn ÞE = σ ; r− b˙I = bI −1

Proposition 5 substantially confirms what we have found in the case of localized spillovers. The only difference with respect to our preceding results is that the growth effect of geography is nil also in the short-run when spillovers are globalized.

2 = gzAðsn Þ

+

ðA1Þ

ðσ−αÞ w E ; σ

where bI ≡ K w ðt Þ−ϕ Aðsn Þ−1 and: Aðsn Þ ≡ ½sn + λð1−sn Þ

ϕ

5. Conclusions Within the last decade, the merger of the New Economic Geography with the New Growth Theory has led to the emergence of spatial models of endogenous growth; most of these contributions basically add product innovation à la Romer (1990) to Krugman's core-periphery model (Krugman, 1991b). Relying on Romer's (1990) product-variety framework, models of this type are still exposed to the Jones critique in that they exhibit the strong scale effect in R&D. In the present paper we deal with this issue and propose a spatial model of scale-invariant growth to analyse the relation between trade integration, industrial location and growth. In the presence of localized spillovers, we find that trade integration only temporarily changes the rate of economic growth, in contrast with the permanent growth effects found in previous studies. We also show that the direction of the growth effects depends on whether intertemporal knowledge spillovers are positive or negative; theoretical work in the field of New Economic Geography does not allow for any variation in the strength of these spillovers and implicitly assume that they are quite strong. The lack of permanent growth effects has important implications for the relation between trade integration and regional income disparity. In our model, in fact, the level of income inequality across regions does not change when economies tend to be more integrated; this result is also in contrast with previous research on this topic which predicts that trade integration reduces income inequality across regions. In the paper, we have assumed that transaction costs exist only between regions (inter-regional transaction costs); a natural extension of our analysis would be to account for transaction costs also inside regions (intra-regional transaction costs) as in Martin (1999). Moreover, our work, following Martin and Ottaviano (1999), has focused on the case of perfect mobility of capital. It would be

ðA2Þ

∧ sn =

  1 ð1 + δÞ 1 + sE − ; ð1−δÞ 2 2

ðA3Þ

with sE ≡ E / Ew being the relative consumption expenditure of the North. In order to derive the laws of motion of Ew and z, we perform log-differentiation and obtain: w ⁎ ⁎ E˙ E˙ E E˙ E w = w + ⁎ w; EE E E E

ðA4Þ

z˙ = ð1−ϕÞg−gL : z

ðA5Þ

These two differential equations govern the time evolution of the endogenous variables Ew and z. We start by proving two Lemmas that are useful to simplify the economic dynamics. Lemma 1. The relative consumption expenditure of the north does not change smoothly over time, s˙ E = 0: Proof. Log-differentiation of sE gives: w s˙ E E˙ E˙ = − w: E E sE

Plugging Eq. (A4) and simplifying terms yields: ⁎ s˙ E E˙ E˙ E E˙ E⁎ = − w− ⁎ w E EE sE E E ! ⁎ E˙ E˙ = ð1−sE Þ − ⁎ : E E

A. Minniti, C.P. Parello / Regional Science and Urban Economics 41 (2011) 20–31

29

As both regions share the same preferences and capital is freely mobile across locations, in each region consumption expenditure obeys the same Euler equation: ⁎ E˙ E˙ = ⁎ = r−ρ: E E

ðA6Þ

Eq. (A6) which implies that s˙ E = 0 always holds at each instant t. Thus, when an exogenous parameter changes, the relative consumption expenditure of the North adjusts instantaneously towards the new equilibrium. ■ Lemma 2. The proportion of firms located in the North does not change smoothly over time, s˙ n = 0. Fig. 1. Phase diagram.

Proof. From Eq. (A3), log-differentiation of sn gives: s˙ n = sn

  1+δ 2 s˙ E  1−δ  : 1 1+δ s + ð2sE −1Þ E 2

A.2. The steady state

1−δ

Using Lemma 1, one can conclude that s˙ n = 0 always holds, meaning that when structural parameters change, the proportion of firms located in the North adjusts instantaneously towards the new equilibrium. ■ Lemma 2 has an important implication for our dynamic analysis; in fact, it implies that A(sn) adjusts instantaneously to changes in exogenous parameters. Indeed, log-differentiation of Eq. (A3) yields: A˙ ðsn Þ ϕð1−λÞsn s˙ n = = 0: Aðsn Þ sn + λð1−sn Þ sn

αAðsn ÞEw = σ : r + ϕg

  1 α + ϕðσ−αÞ w E −2ϕ ; σ ρAðsn Þ−1

z=

−1

2 = zAðsn Þ

σ−α gL w E : + σ 1−ϕ w

This, in turn, implies that b˙ I = bI = −ϕg so that Eq. (A1) can be written as: z=

w In the steady state z˙ = E˙ = 0. Consequently, the dynamic system Eqs. (A11)–(A12) boils down to:

ðA7Þ

˜ we get the world real consumption Solving for E˜ and z, expenditure and the relative R&D difficulty in the steady-state (see Eq. (18) in the main text):   ϕg 2σ ρ + L 1−ϕ w  E˜ =  ϕgL −αðρ−gL Þ σ ρ+ 1−ϕ

ðA8Þ

Now, using Eqs. (A2) and (A7) we get: g=

σ−α i h 1 w E ; 2− −1 σ zAðsn Þ

  1 α + ϕðσ−αÞ w E −2ϕ : r= σ zAðsn Þ−1

z˜ =

2αAð ˜sn Þ  : ϕg σ ρ + L −αðρ−gL Þ 

1−ϕ

ðA13Þ

Moreover, plugging Eq. (A6) into Eq. (A4) yields: w E˙ = r−ρ: Ew

and

Finally, plugging Eq. (A13) into Eq. (A9) one obtains the steadystate innovation rate: g˜ =

gL : 1−ϕ

ðA9Þ

ðA10Þ

Substituting for g using Eq. (A9) and for r using Eq. (A10), Eqs. (A5) and (A8) can be rewritten, respectively, as: σ−α i z˙ 1−ϕ h w = E −gL ; 2− −1 z σ zAðsn Þ

ðA11Þ

  w E˙ 1 α + ϕðσ−αÞ w E −2ϕ −ρ: w = σ E zAðsn Þ−1

ðA12Þ

Eqs. (A11) and (A12) form a system of two differential equations in z and Ew. Fig. 1 depicts the system's phase diagram which summarizes the transitional dynamics.

In Fig. 1, the two curves corresponding to ż = 0 and Ė = 0 intersect at the steady-state. The phase diagram reveals that the dynamic system is saddle-path stable. Along the stable arm, if the economy starts from point X in correspondence of which Ew and z are below the steady-state, both Ew and z rise monotonically along the transitional path. In contrast, starting from point Y at which Ew and z are above the steady-state levels, both variables fall monotonically during the transition to the steady state. A.3. Transitional dynamics To analyze the transitional dynamics, we linearize the differential equation system Eqs. (A11)–(A12) around the steady-state. Thus, we write: 

w E˙ z˙



 w  ˜w = J ⋅ E −E ; z− z˜

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A. Minniti, C.P. Parello / Regional Science and Urban Economics 41 (2011) 20–31

For analytical tractability, following Helpman (1993), we consider the first-order response of ðEw ; zÞ to changes in δ by differentiating (A14) with respect to δ, while neglecting the impact of δ on λ and ω:

where J is a 2 × 2 matrix with the (i, j) element aij being: a11 =

½ρð1−ϕÞ + gL ϕ½αð1−ϕÞ + σϕ N 0; αð1−ϕÞ

  ∂Ew ðt Þ ∂ z˜ λt ∂zðt Þ λt ∂ z˜ = − ωe ; = 1−e : ∂δ ∂δ ∂δ ∂δ

ρσ ½ρð1−ϕÞ + gL ϕ b 0; =− αð1−ϕÞAðsn Þ

a12

a21 = −Aðsn Þ

Let us first consider the case ϕ ∈ (0, 1). Since ∂ z˜ = ∂δ N 0, we have that:

ðσ−αÞð1−ϕÞ b 0; a22 = −gL b 0: σ

Let λ1 and λ2 denote the two eigenvalues of J. Since the determinant of J is: j J j = λ1 λ2 = −

ρð1−ϕÞ + gϕ fρðσ−αÞð1−ϕÞ + gL ½α + ϕðσ−αÞg b 0; αð1−ϕÞ

we have that the two eigenvalues are real and opposite in sign. This, in turn, implies that the dynamic system is saddle-path stable. Solving the characteristic equation j J−Ixj = 0, the two eigenvalues write as: λ1 =

i 1h 1=2 a + a22 + B ; 2 11

λ2 =

∂Ew ðt Þ ∂zðt Þ b 0; N 0: ∂δ ∂δ At t = 0, there is no jump in z as δ increases; in fact, it is easy to check that ∂ z(0) / ∂ δ = 0. However, ∂ Ew(0) / ∂ δ b 0, meaning that there is a downward jump in Ew as δ increases. In the phase diagram, such a downward jump of the initial Ew shows up as a downward shift of the entire saddle-path as illustrated in Fig. 2. This can be ascertained by differentiating Eq. (A15) with respect to δ: w ∂E ðzÞ ∂ z˜ = −ω b 0: ∂δ ∂δ

i 1h 1=2 a + a22 −B ; 2 11

2

where B = ða11 + a22 Þ −4j Jj N 0. Since the two eigenvalues are of opposite sign, it follows that λ1 denotes the positive eigenvalue and λ2 denotes the negative eigenvalue. Thus, the general solution of the linearized system writes as: 

 w      ˜ ω11 λ1 t ω12 λ2 t Ew ðt Þ = B1 e + B2 e + E ; ω21 ω22 zðt Þ z˜

′ where ½ω1i ω2i  is the eigenvector associated to the eigenvalue λi, i = 1,2, and B1 and B2 are two constants to be determined by boundary conditions. Using the initial condition z(0) and the asymptotic boundary ˜ it follows that B1 = 0 and B2 = zð0Þ− z. ˜ Normalcondition zð∞Þ = z, izing ω22 = 1 and writing ω12 ≡ ω and λ2 ≡ λ b 0, we get: w w λt λt E ðt Þ− E˜ = ½zð0Þ− z˜ ωe ; zðt Þ− z˜ = ½zð0Þ− z˜ e :

ðA14Þ

We can easily determine the sign of ω. In fact, by definition, we have: 

a11 a12 a21 a22



   ω ω =λ : 1 1

Solving for ω, we get ω = a12 = ðλ−a11 Þ which is positive as a12 b 0, a11 N 0 and λ b 0. By combining the two equations in (A14), we obtain the equation for the stable saddle-path on the phase diagram:  w  w E ðzÞ = E˜ −ω z˜ + ωz:

Suppose we start from the steady-state I0 corresponding to a certain value of δ and imagine that trade gets freer (δ increases). The equilibrium initial Ew should take a discrete downward jump from I0 w to I1, with the size of the jump given by ∂ Ew(0) / ∂ δ. Since ∂ E˜ = ∂δ = 0 w w and ∂ E (t) / ∂ δ b 0 for t N 0, we have that E (t) remains below the initial level along the transitional path and gradually returns to the initial steady-state level. Moreover, since ∂ z˜ = ∂δ N 0 and ∂ z(t) / ∂ δ N 0 for t N 0, we can say that z(t) rises at each point in time during the transition and takes a higher value in the new steady-state. Let us now turn to the case ϕ b 0. Since ∂ z˜ = ∂δb0, we have that: w

∂E ðt Þ ∂zðt Þ N 0; b 0: ∂δ ∂δ At t = 0, ∂ z(0) / ∂ δ = 0, meaning that there is no jump in z as δ increases. Moreover, since ∂ Ew(0) / ∂ δ N 0, there is an upward jump in Ew as δ increases. In the phase diagram, such an upward jump of the initial Ew is accompanied by an upward shift of the entire saddle-path as illustrated in Fig. 3. To ascertain this point, we just differentiate Eq. (A15) with respect to δ and get: ∂Ew ðzÞ ∂ z˜ = −ω N 0: ∂δ ∂δ

ðA15Þ

Around the steady-state, ∂ Ew(z) / ∂ z = ω N 0 so that we can conclude that the stable saddle-path is upward sloping. A.4. Comparative statics We can now evaluate the impact of a change in δ on the paths of Ew and z. Using (A13), it is straightforward to check that: ∂ E˜ ∂ z˜ 2αϕ½sn + λð1−sn Þϕ−1 ð1−λÞ ∂ s˜n N N   0⇔ϕ 0: = 0; = b ∂δ ∂δ b ∂δ ϕgL σ ρ+ −αðρ−gL Þ w

1−ϕ

Fig. 2. Comparative statics when ϕ ∈ (0, 1).

A. Minniti, C.P. Parello / Regional Science and Urban Economics 41 (2011) 20–31

31

expenditure Ew; ðiiiÞ a temporary fall (or a short-run decrease) in the rate of innovation g. References

Fig. 3. Comparative statics when ϕ b 0.

As before, imagine we start from the steady-state I0 and suppose that δ increases. The equilibrium initial Ew should take a discrete upward jump from I0 to I2, with the size of the jump equal to ∂ Ew(0) / w ∂ δ. Since ∂ E˜ = ∂δ = 0 and ∂ Ew(t) / ∂ δ N 0 for t N 0, Ew(t) remains above the initial level along the transitional path and gradually returns to the initial steady-state level. Furthermore, since ∂ z˜ = ∂δb0 and ∂ z(t) / ∂ δ b 0 for t N 0, z(t) declines at each point in time during the adjustment and takes a lower value in the new steady-state. We are now in the position to analyse the impact of changes in δ on the innovation rate g. According to Eq. (A5), a rise (decline) in z during the adjustment to the new steady-state implies that the ˜ It is innovation rate g goes above (below) the steady-state value g. possible to show that there is an initial over-shooting of the innovation rate g as δ changes. In fact, by differentiating (A9) with respect to δ, we have: h   i σ−α w E Aðsn Þ 2− ∂g ∂z σ =− 2 ∂δ ∂δ z h   i σ−α w ϕ−1 E ½sn + λð1−sn Þ ϕ 2− ð1−λÞ ∂s σ n + z ∂δ   σ−α Aðsn Þ ∂Ew σ − : z ∂δ Since ∂ z(0) / ∂ δ = 0, one can easily prove that: ∂g ð0Þ N N 0 ⇔ ϕ 0: b ∂δ b Thus, when ϕ ∈ (0, 1), an increase in δ implies that g is temporarily ˜ the innovation rate initially higher than the steady-state value g: overshoots by the amount ∂ g(0) / ∂ δ and then gradually converges to ˜ The opposite dynamics occur when the steady-state growth rate g. ϕ b 0. A.5. Summary of results Combining all the results above together, we conclude as follows: When ϕ ∈ (0, 1), an increase in δ leads to (i) a permanent increase (or a long-run increase) in the relative R&D difficulty z; (ii) a temporary fall (or a short-run decrease) in consumption expenditure Ew; ðiiiÞ a temporary increase (or a short-run increase) in the rate of innovation g. In contrast, when ϕ b 0, an increase in δ leads to ðiÞ a permanent decrease (or a long-run decrease) in the relative R&D difficulty z; (ii) a temporary increase (or a short-run increase) in consumption

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