Patterns of Technology, Industry Concentration, andProductivity Growth Without Scale Effects
Colin Davis

Ken-ichi Hashimoto

Doshisha University∗

Kobe University†

June, 2011
Abstract
This paper investigates the relationship between geographic patterns of industrial activity and endogenous growth in a two region model of trade thatexhibits no scale effect. The in-house process innovation of manufacturing firms
drives productivity growth and is closely associated with firm-level scales of
production and relative levels of accessible technical knowledge. Focusing on
long-run industry shares and a cross-region productivity gap, we find that dispersed equilibria with positive industry shares for both regions always produce
higher growth rates than core-periphery equilibria with all industry locating
in one region. Moreover, the highest growth rate arises in a symmetric steadystate that features no productivity gap and equal shares of industry leading
to the conclusion that the geographic concentration of industry has a negative
impact on overall growth. Convergence towards a dispersed equilibrium, however, is contingent on the levels of inter-regional transport costs and knowledge
dispersion. Finally, we explore the implications of greater economic integration arising from reduced transport costs and greater knowledge dispersion forpatterns of industry and productivity, and for regional welfare levels within adispersed equilibrium.
Key Words: Industry Concentration, Industry Share, Productivity Gap, Productivity Growth, Scale EffectJEL Classifications: F43; O30; O40; R12∗

Institute for International Education, Doshisha University, Karasuma-Higashi-iru, Imadegawadori, Kamigyo-ku, Kyoto, Japan, 602-8580, cdavis@mail.doshisha.ac.jp.
†
Graduate School of Economics, Kobe University, 2-1 Rokkodai, Nada, Kobe, Japan, 657-8501,
hashimoto@econ.kobe-u.ac.jp.

1

Introduction

Even the most casual observer of economic geography will recognize that the distribution of industrial activity is uneven at local, regional, and international levels. A
distinctly more subtle issue, however, relates to unravelling the relationship between
these patterns of industrial concentration and economic growth. Although a broadly
historical perspective generally leads to the prediction that a higher concentration ofindustry supports a greater rate of economic growth (Baldwin et al. 2001), the results obtained by recent empirical studies are mixed. For example, Braunerhjelm andBorgman (2004) report a positive relationship between industry concentration and labor productivity growth in Sweden, while Brülhart and Sbergami (2009) investigate
cross-country data that suggests the relationship between industry concentration and
GDP growth depends on a country’s level of economic development. Bosker (2007),on the other hand, finds that on average European regions with a denser spread of
employment tend to experience slower rates of growth in GDP. In addition, Gardineret al. (2010) report a negative relationship between a number of measures of industry
concentration and GDP growth for several levels of agglomeration using European
regional data. These mixed results are difficult to interpret as the existing theoreticalmodels of the “new economic geography” literature predict a positive relationshipbetween agglomeration and growth (Baldwin and Martin, 2004).
In this paper, we introduce a novel theoretical approach that supports a negative
relationship between the geographic concentration of industry and aggregate economic
growth. Building upon the in-house process innovation literature established by Smulders and van de Klundert (1995) and Peretto (1996), we develop a two region model oftrade in which the distribution of manufacturing activity and productivity growth are
determined endogenously by the innovation activities of monopolistically competitive
manufacturing firms, and then explore the mechanisms through which growth effects
the geography of industry and in turn the implications of this geography for growth.
1Our model produces two types of long-run equilibrium: a core-periphery equilibrium
in which all industry, and hence all productivity growth, occurs in one region, and adispersed equilibrium in which industrial activity and productivity growth is spread
across regions. Investigating the dynamics around a dispersed equilibrium, we findthat the long-run pattern of industry is determined by the balance between a destabilizing competition effect and stabilizing productivity effect, the relative strengths ofwhich depend on the level of transport costs and the degree of knowledge dispersion
between regions. In particular, convergence in industry shares, the level of productivity, and the rate of productivity growth to a dispersed equilibrium is contingent
on the levels of transport costs and knowledge dispersion between regions. Ascertaining which pattern of industrial activity the economy converges to in the long run
is important as an equal dispersion of industry across regions is found to yield thehighest rate of productivity growth but the lowest level of product variety. As such
our framework lends support to empirical studies such as Bosker (2007) and Gardineret al. (2010) that find a negative correspondence between the density of economic
activity and the rate of economic growth.
Our paper is closely related with studies in the “new economic geography” literature that emphasize key elements of the variety-expansion model of innovation-based
endogenous growth (Grossman and Helpman, 1991). These studies tend to find that
agglomeration economies and growth are reinforcing processes, that is, a higher concentration of industry tends to promote economic growth (Baldwin and Martin, 2004).
A key feature of the models adopted in this literature, however, is a scale effect inwhich the rate of growth is positively correlated with the labor endowment of theeconomy. Indeed, the scale effect appears to play a central role in the relationshipbetween industry concentration and the endogenous pace of growth. The existenceof the scale effect, however, has generally been rejected by empirical studies (Lainez
and Peretto, 2006) indicating the need for a reassessment of this relationship in a2

framework that corrects for the scale effect. To this end, Minniti and Parello (2011)
introduce population growth and decreasing returns in research and development(R&D) following Jones (1995) and investigate the relationship between trade integration and scale-invariant economic growth.1 Interestingly, under this modification of
the variety-expansion model, long-run growth is proportionate to population growthand determined independently of the level of industry concentration. In contrast, thein-house process innovation framework that we adopt in this paper shifts the focus
from R&D activity at the aggregate level towards innovation at the level of individual
product lines thereby removing the effects of population size on firm-level incentives
for R&D and allowing for the endogenous determination of both the spatial distribution of industry and the long-run rate of productivity growth without the bias
generated by scale effects.
A convenient feature of our framework is that the long-run equilibrium can becharacterized completely in terms of the shares of manufacturing activity locating ineach region and a productivity differential between regions. Examining stable dispersed equilibria, we find that the larger region always has a greater share of industry
and a higher relative productivity, and thus that the model is consistent with the wellestablished empirical result that firms are more productive in regions with a greater
density of economic activity (Melo et al. 2009). We also investigate the impacts ofgreater economic integration between regions resulting from lower transport costs andgreater knowledge dissemination. First, a decrease in the level of inter-regional transport costs increases both the share of industry and the relative productivity of thelarger, more advanced region indicating that it attains a higher rate of productivity
growth over the short-run transition. The increased concentration of industry in thelarger region has a negative impact, however, on the overall long-run rate of productiv1

See Futagami and Iwaisako (2007) for more detail on variety expansion models of scale-invariant
endogenous growth. In addition, Segerstrom (1998) and Li (2003) develop rising product quality
models of scale-invariant endogenous growth.

3

ity growth. On the other hand, an increase in the degree of inter-regional knowledge
dissemination raises both the share of industry and the relative productivity of thesmaller region providing credence to agglomeration theories based on localized knowledge spillovers. Although the relative pace of productivity growth rises for the lessadvanced region in the short run, the impact on overall long-run productivity growthis ambiguous owing to the fact that greater knowledge dissemination simultaneously
raises the productivity of labor in innovation and lowers firm-level employment ininnovation as an increase in the total number of incumbent firms crowds out R&D
investment.
The remainder of this paper proceeds as follows. In Section 2 we present our basicmodel of endogenous productivity growth. Section 3 then examines the characteristics of core-periphery equilibria with the manufacturing industry concentrated in
one region. The features of dispersed equilibria are then investigated in Section 4.
We investigate the effects of several key variables on patterns of industry trade andlocation, and the rate of productivity growth in Section 5. Implications for regional
welfare are then discussed in Section 6. Finally, we provide concluding remarks in
Section 7. All calculations are provided in the Appendix.

2

The Model

Consider a two region general equilibrium model of trade and endogenous growth with
two sectors: traditional production (Y ) and manufacturing (X). The traditional sector is a standard constant returns to scale industry that is characterized by free trade
and allows for a common wage rate in both regions. Firms in the manufacturingsector, on the other hand, compete according to monopolistic competition and incur
iceberg transport costs on inter-regional transactions. Economic growth is driven by
the in-house process innovation of manufacturing firms which leads to improvements
in production technology. Labor (L), the sole factor of production, is supplied in4

elastically by households and, although there is no inter-regional migration, moves
freely between the traditional and manufacturing sectors in each region. We refer tothe two regions as the North and the South, and use an asterisk to indicate variables
associated with the Southern region. In the following description of the model wefocus on the North but analogous conditions can be obtained for the South.
2.1

Households

The demand side of the model is composed of dynastic households that optimize
utility over an infinite time horizon by selecting optimal allocations of income flows
across saving and expenditure. The aggregated preferences of households residing in
the North are described by the following utility function:

U=

Z ∞

e−ρt [α ln CX (t) + (1 − α) ln CY (t)] dt,(1)0

where CX (t) and CY (t) are the respective consumptions at time t of a manufacturingcomposite and the traditional good, α ∈ (0, 1) is the expenditure share for manufacturing goods, and ρ is the subjective discount rate. The manufacturing composite
takes the following form:

CX =

Z n

ci

σ−1
σ

di +

Z n∗
0

0

σ−1
σ

cj

dj

σ
 σ−1

,

(2)

where ci is the demand for variety i of the mass n varieties produced in the North, cj
is the demand for variety j of the mass n∗ varieties produced in the South, and σ > 1is the elasticity of substitution across varieties.
Households maximize lifetime utility subject to the following intertemporal budget
constraint:
Z ∞0R− 0t r(s)dse

E(t)dt ≤

Z ∞
0

5

Rt

e− 0 r(s)ds w(t)Ldt + B(0),where E(t) is aggregate household expenditure, w(t) is the wage rate, L is the labor
force, r(t) is the nominal interest rate, and B(0) is initial asset holdings in the Northern region.2 Intertemporal utility maximization yields the standard Ramsey saving
rule described by the following Euler condition:Ė(t)= r(t) − ρ,
E(t)

(3)

where a dot over a variable indicates differentiation with respect to time. We assume
perfect capital mobility between regions and it follows that the North and the South
share a common nominal interest rate and motions for household expenditure: Ė/E =Ė ∗ /E ∗ = r − ρ. Time notation is suppressed for the remainder of the paper unlessrequired.
A constant share of household expenditure is allocated to the traditional good andthe manufacturing composite each period:

PX CX = αE,

PY CY = (1 − α)E,

(4)where PX is the price index corresponding to the manufacturing composite and PY is
the price of traditional goods in the Northern region. In particular, the price index is

PX =

Z n
0

pi

1−σ

di +

Z n∗
0

(τ p∗j )1−σ dj1/(1−σ),(5)where pi is the price of variety i produced in the North, p∗j is the price of variety j
produced in the South, and τ > 1 is an iceberg transport cost, under which τ units
must be shipped for every unit sold in the export market (Samuelson, 1951).
Regarding the price index (5) as the household’s unit expenditure function for
manufacturing goods, the total Northern demands for representative varieties i and2

The value of asset holdings (B) will be driven to zero by free entry in the manufacturing sector.
See Section 2.5 for more details.

6

j respectively produced in the North and the South can be obtained by using Sherphard’s Lemma on (5) and aggregating across households:
σ−1
ci = αp−σ
i PX E,

2.2

cj = α(τ p∗j )−σ PXσ−1 E.

(6)

Traditional Production

The traditional sector produces a homogeneous good for a perfectly competitive interregional market that is characterized by free trade. Firms in both regions produce
goods using a similar constant returns to scale technology whereby one unit of output
requires the employment of one unit labor. Under this production technology theprice of the traditional good equals the wage rate and, with free trade and incomplete
specialization of traditional production, the wage rate is common across regions.3 We
set the traditional good as the model numeraire and the traditional good price andthe wage rate both equal one, PY = PY∗ = w = 1, at all times.
2.3

Manufacturing

The manufacturing sector features firms that produce horizontally differentiated products and complete according to monopolistic competition (Dixit and Stiglitz, 1977).
At any moment in time, there are n firms in the North and n∗ firms in the South,
each producing a unique variety and investing in process innovation with the objective of reducing future costs of production. Firms enter the market freely withnegligible costs of product development and, with active process innovation, begin
their operations with the same level of productivity as extant firms based in the same
region. Once a firm has entered the market it incurs a per-period fixed labor cost(lF ) that corresponds with product marketing and the management of production
and innovation activities.
3

In particular, we assume that the traditional share of household expenditure (1−α) is sufficiently
large to ensure that there is always an active traditional sector in each region.

7

Manufacturing firms employ a constant returns to scale production technologythat is symmetric within a region but may differ across regions. Specifically, theproduction function for a representative firm based in the North isx = θlX ,(7)

where θ > 0 is a firm-specific productivity coefficient, and lX is firm-level employment
in production. As indicated above, all firms based in the same region have the samelevel of productivity, but we allow for productivity differences across regions (θ 6= θ∗ ).
A firm faces demand for its product from both the domestic and the export markets. With iceberg transport costs, however, firms must produce additional units forsales to the export market in order to cover units lost in transit. Thus, referring
to (6), a Northern firm produces x = ci + τ c∗i units in order to meet the combined
demands of the Northern and Southern markets. As is well known, under monopolistic competition each firm maximizes operating profit on sales by setting price equal
to a constant markup over unit cost. The price set by a Northern firm is thereforep = σ/(σ − 1)θ. Combining this pricing rule with the demand functions (6) andthe production function (7), optimal operating profit on sales can be obtained for a
Northern firm as
αp1−σ PXσ−1 E + ϕPX∗ σ−1 E ∗
,
π=
σ

(8)

where ϕ ≡ τ 1−σ is a measure of the freeness of trade with ϕ = 0 indicating prohibitively high transport costs and ϕ = 1 indicating perfectly free trade. With a large
number of firms, the small market share of any single firm leads it to ignore the termin brackets in (8) when evaluating the impact of changes in price on operating profit
on sales.

82.4

Process Innovation

Manufacturing firms invest in in-house process innovation with the objective of maximizing firm value. Improvements in technology reduce production costs and lead tolower prices allowing firms to increase their market shares and raise profits on sales.
Returning to the production function (7), we consider process innovations thatgenerate improvements in productivity through increases in θ. In particular, theproductivity of a representative Northern firm evolves according toθ̇ = KlR ,(9)where K is labor productivity in in-house process innovation, and lR is firm-level
R&D employment.
Following the in-house process innovation literature (Smulders and van de Klundert, 1995; Peretto, 1996), we model the productivity of labor in in-house innovation
as a function of the weighted average productivity of technologies observable by thefirm:

K = sθ + δs∗ θ∗ ,(10)

where s ≡ n/N and s∗ ≡ n∗ /N are respectively the shares of industry located in the
North and the South, and N ≡ n + n∗ is the total number of firms. The exogenous
parameter δ ∈ (0, 1) regulates the inter-regional scope of knowledge dissemination
between firms.4
Implicit in the specification of (10) is the assumption that a firm’s production technology is a culmination of the accumulation of both codifiable and tacit knowledge
4

In order to focus on the implications of cross-region knowledge spillovers for industry locationand growth within a simple framework, we have assumed that the number of firms is large enough toensure that the influence of a single firm’s technology on labor productivity in in-house innovationis negligible.

9(Keller, 2004). While codifiable knowledge is easily conveyed across large distances using modern communication technology, the transfer of tacit knowledge tends to requireface-to-face communication. Indeed, the imperfect nature of knowledge spillovers iswell documented in empirical studies such as Jaffe et al. (1993), Mancusi (2008), andCoe et al. (2009). Accordingly, we interpret the parameter δ as describing the level ofknowledge that can be transferred over large distances given the current state of communication technology. For example, when δ = 0 knowledge spillovers are completely
local in scope, and when δ = 1 there is perfect inter-regional knowledge dispersion.5
A firm’s intertemporal optimization problem entails choosing the level of investment in process innovation that maximizes the future flow of profits. The total
per-period profit of a Northern firm equals operating profit on sales minus the costof investment in process innovation and the per-period fixed labor cost lF :

Π = π − lR − lF .

Firms based in the North, therefore, choose lR to maximize V =

(11)
R∞
0

Rt

Π(t)e− 0 r(s)ds dt

subject to the technological constraint (9). This optimization problem can be solved
using the following current value Hamiltonian function: H = Π + µKlR , where µ is
the current shadow value of an improvement in the technology of a Northern firm.
The optimal investment path for process innovation satisfies conditions for both
static and dynamic efficiency in addition to a standard transversality condition.6
Static efficiency requires that a marginal increase in the value of technology equal the
marginal cost of process innovation, that is, µ = 1/K. Dynamic efficiency requiresthat the return to investment in process innovation equal the rate of return on arisk free asset, that is, µr − µ̇ = ∂π/∂θ, where we assume that the market share of5
Baldwin and Forslid (2000) introduce a similar specification for the level of inter-regional knowledge spillovers in a variety expansion model of innovation-based endogenous growth.
6
In the next subsection we show that r = ρ at all moments in time. As a result, the transversality
condition is limt→∞ e−ρt µθ = 0.

10each firm is small enough that it disregards the impact of its innovation efforts on
both the composite price index (PX ) and knowledge spillovers to rival firms (K).7
Combining these efficiency conditions yields the following no-arbitrage condition for
in-house R&D investment:r=(σ − 1)πK K̇
− .
θ
K

(12)

This no-arbitrage condition must bind whenever there is active process innovation.
2.5

Free Entry

Manufacturing firms are free to enter and exit the market. As a result, the numberof firms based in each region adjusts until per-period profits (11) are driven to zero.
With firms earning zero profits, each region’s expenditure equals its labor income andis therefore constant given our choice of the traditional good as the model numeraire,
E = L and E ∗ = L∗ .8 Consequently, from the Euler equation (3), the risk-free interest
rate equals the discount rate and is common across regions at all moments in time,r = ρ.
The relative state of technology arising between firms based in the North and theSouth at any given moment in time is described using

ω≡

θ
,
θ∗

(13)

which we refer to as the productivity gap. Given the pricing rules set by manufacturingfirms, the productivity gap also describes the relative price of Southern produced
varieties, that is ω = p∗ /p.
7
The risk-free asset is simply used as a link between the internal rate of return to in-houseinnovation and the external rate of return to investment in the market for investment funds.
8
More specifically, household wealth equals zero in equilibrium, B = 0, as firm value is driven tozero by free entry, V = 0.

11

The zero profit conditions can be conveniently expressed in terms of firm-level
employment. First, using the demand functions (6) and the production function(7) with the pricing rules yields the optimal levels of employment in production for
Northern and Southern firms respectively as


L
ϕL∗
α(σ − 1)+,lX =σn + ϕω 1−σ n∗ ϕn + ω 1−σ n∗ϕω 1−σ L
ω 1−σ L∗α(σ − 1)
∗
.
+
lX =
σ
n + ϕω 1−σ n∗ ϕn + ω 1−σ n∗

(14)
(15)

Then, setting per-period profits (11) equal to zero and using (14) and (15) with (8),the free market entry conditions can be written as∗∗lX= (σ − 1) (lR+ lF ) ,lX = (σ − 1) (lR + lF ) ,(16)

for the North and the South, respectively.

3

Core-periphery Long-run Equilibrium

In this section we investigate the characteristics of the core-periphery long-run equilibrium for which the manufacturing industry concentrates completely in one region.
In particular, we are interested in state-state equilibria that feature a constant intersectoral allocation of labor in each region. To simplify the exposition we focus on
the case where all manufacturing and innovation activity occurs in the North. In thiscase, although the North continues to produce traditional goods, the South specializes completely in traditional production and imports manufacturing goods from theNorth in return for the traditional good.
To begin with, consider a situation in which the no-arbitrage condition (12) bindsin the North but not in the South. With no Southern employment in process innovation, the technology gap ω increases indefinitely with the North becoming increasingly12

advanced and the South becoming increasingly backward. At first Southern firms will∗
earn positive per-period profits (Π∗ > 0) as they no longer incur the fixed cost lR.

Without process innovation, however, there will be no new market entry in the South,and consequently, the number of Northern firms will rise to absorb these additional∗
profits thereby driving Southern profits to zero, lX= (σ − 1)lF . It then follows thatas Southern employment in production is a decreasing function of ω from (15), therising technology gap steadily reduces the Southern share of manufacturing firms thatcan be supported by the zero profit condition. As a result, the Northern share of the
manufacturing industry converges to one (s = 1).
With all manufacturing and innovation activity occurring in the North, the zeroprofit condition (16) can be used with (9) and (10) to rewrite the binding no-arbitrage
condition (12) for Northern firms as ρ = (σ − 2)lX /(σ − 1) + lF . In addition, setting∗lX= 0, and using n∗ = 0 with the pricing rules for manufacturing firms in (14) yields

the product market clearing condition nlX = α(σ − 1)(L + L∗ )/σ. Combining thesetwo conditions, we find that the equilibrium number of firms isNC =α(σ − 2)(L + L∗ ),σ(ρ − lF )(17)

where the subscript C denotes a concentrated equilibrium, and NC = n as all manufacturing firms are located in the North. Not surprisingly, an increase in the overallmarket size (L + L∗ ) increases the number of firms (NC ). On the other hand, anincrease in lF , or similarly, a decrease in ρ, lowers the level of employment in production (lX ) that satisfies the no-arbitrage condition and thus raises the number of firms
through the product market clearing condition. These results parallel those obtainedby Smulders and van de Klundert (1995) for a closed economy. Finally, a positive
level of market entry requires that the return to process innovation (ρ) cover the
per-period fixed costs (lF ). To ensure both positive market entry and productivity

13

growth, we assume that ρ > (σ − 1)lF for the remainder of the paper.9
The long-run rate of productivity growth associated with the core-periphery equilibrium can now be obtained using (8), (9), and (10) with (12) and (17):gC =α(σ − 1)(L + L∗ )
ρ − (σ − 1)lF
−ρ=
.
σNC
σ−2

(18)

From (18) we can see that there is no scale effect in the model as an increase inmarket size (L + L∗ ) is neutralized by an increase in the number of manufacturingfirms (NC ).10
Changes in the fixed cost and the discount rate have opposing effects on the rateof productivity growth. An increase in the fixed cost (lF ) has a negative effect on
productivity growth as it raises the number of firms leading to a greater allocationof labor to per-period fixed cost that crowds out employment in process innovation.
In contrast, although an increase in the discount rate (ρ) has both a direct negative
effect and an indirect positive effect, through NC , on the rate of productivity growth,
the indirect effect dominates the direct effect and as a result gC is positively related
to ρ. These results are similar to those found by Smulders and van de Klundert(1995) and, in a different context, Dinopoulos and Thompson (1998), and highlightthe tension between market concentration and productivity growth that arises in thistype of endogenous growth model.
9

Positive levels of market entry and innovation activity also require σ > 2. This requirement for
the elasticity of substitution is supported by the average and median estimates obtained by Broda
and Weinstein (2006) for various levels of industry disaggregation.
10
The core-periphery rate of productivity growth is determined independently of the degree ofinter-regional knowledge spillovers (δ) and the freeness of trade (ϕ). This is a common characteristicof models which assume that innovation and production are tied to the same location, as shown by
a comparison of, for example, Martin and Ottaviano (1999) and Baldwin et al. (2001).

14

4Dispersed Long-run Equilibrium

This section provides a characterization of the dispersed long-run equilibrium that
occurs when both regions have active manufacturing sectors and positive productivity
growth. Once again we are interested in steady states that feature constant labor
allocations in both regions. For the dispersed long-run equilibrium, this requires aconstant productivity gap and constant numbers of firms based in each region.
In order to describe the evolution of the productivity gap we take the time derivative of (13) and obtain the following differential equation:θ̇ θ̇∗KlR K ∗ lR∗ω̇= − ∗ =
−
,
ω
θ θ
θ
θ∗

(19)

where we have used (9). Clearly the evolution of the productivity gap depends on therelative levels of knowledge available to firms based in each region and their respective
levels of investment in process innovation. Setting (19) equal to zero yields∗
KlR
K ∗ lR
=
,
θ
θ∗

(20)

and a constant productivity gap therefore naturally implies equal rates of productivity
growth in the North and the South.
With active process innovation in both regions the investment and entry decisionsof manufacturing firms are determined by the no-arbitrage conditions of both regions(12), and the free entry conditions (16). In Appendix A we show that the dispersed
equilibrium described by these investment and entry conditions can be reduced to asystem of two equations that fully describe the long-run equilibrium in terms of the
productivity gap (ω) and the North’s share of industry (s ≡ n/N).
The first condition requires that the average level of accessible knowledge relative
to firm level productivity be the same for both regions, that is, K/θ = K ∗ /θ∗ . Sub-

15

Figure 1: Symmetric dispersed equilibrium
s

1/δ

1

τ

sK
sR
1/2

0

1/τ

δ

ω

1

stituting (10) and (13) into this condition and rearranging, we solve for the North’s
share of industry assK =1 − δω −1.
2 − δω − δω −1(21)

As illustrated by the sK curve in Figure 1, this condition is strictly increasing in ω as
an increase in the industry share of Northern firms is necessary to offset the benefit
that Southern firms derive from the increase in relative access to knowledge when
the productivity gap rises. This tension between the industry share of the advanced
region and the access of the backward region to relatively advanced knowledge isregulated by the degree of inter-regional knowledge dispersion (δ). Indeed, a close
inspection of (21) indicates that ω ∈ (δ, 1/δ) is required for sK ∈ (0, 1) with equal
access to knowledge in both regions.
The second condition requires that firm-level scales of production equalize across∗regions, lX = lX
. Substituting (14) and (15) into this condition, we once again solve

for the North’s share of industry as

ϕL∗
1
L
+
.
sR =
σ−1
σ−1
1 − ϕωϕ−ω
L + L∗


16(22)

As shown by the sR curve depicted in Figure 1 for the symmetric case where L = L∗ ,this condition is also strictly increasing in ω. The relationship between the productivity gap and industry share described by the sR curve is determined by the link
between firm-level scales of production and innovation that arises with free entry(16). An increase in ω raises the market shares, and therefore the profits, of Northernfirms precipitating an increase in the North’s share of industry. The size of this effectis determined by the freeness of trade with a rise in ϕ increasing the slope of the sR
curve.
Convergence to or divergence away from the dispersed long-run equilibrium illustrated in Figure 1 depends crucially on the level of transport costs and the degree ofinter-regional knowledge spillovers. In Appendix A we approximate the local dynamics around the symmetric equilibrium using a Taylor expansion of (19) evaluated atω = 1 and s = 1/2, and obtain the following proposition:
Proposition 1 (Convergence in the productivity gap): The symmetric dispersed equilibrium is saddle-path stable for ϕ < ϕ̃, whereϕ̃ = (κ − 1) −p(κ − 2)κ,κ=(σ − 1)(1 + δ)(2ρ − (1 + δ)lF ).
δ(2ρ − (σ − 1)(1 + δ)lF )Proof: See Appendix A. 
The threshold ϕ̃ determines the level of transport costs above which firm-level
rates of productivity growth converge and is closely related to the degree of interregional knowledge spillovers as summarized in the following corollary:
Corollary 1 The threshold ϕ̃ is increasing in the degree of inter-regional knowledge
spillovers (δ) for δ < δ̃ and decreasing for δ > δ̃, whereδ̃ =(σ − 1)(2ρ − lF ).
(σ − 1)(2ρ − lF ) − 4ρ17

Proof: See Appendix B. 
A change in the productivity gap affects firm-level productivity growth in the
North and the South both directly and indirectly through adjustments in the numberof firms based in each region. In the symmetric equilibrium, however, the indirect
effects cancel out leaving two opposing direct effects that combine to determine thestability of long-run equilibrium. These direct effects can be identified by substituting(16) into (20), taking the partial derivative with respect to ω, and evaluating the resultat the symmetric equilibrium:∗∂(KlR /θ − K ∗ lR/θ∗ )(1 + δ)2ϕlX∂ ω̇==− δlR .
∂ω ω=1∂ω(1 + ϕ)2
ω=1(23)

A negative sign for this partial derivative implies a stable dispersed equilibrium.
Starting from the symmetric equilibrium, consider the effects of an exogenous
perturbation that increases the productivity gap. The first term on the right-handside of (23) describes the destabilizing competition effect associated with the ensuingincrease in the production scale of Northern firms relative to Southern firms. As
firm-level employment in innovation is determined proportionately with the scale ofproduction through the free entry condition (16), the rate of productivity growth risesin the North and falls in the South causing subsequent increases in the productivity
gap. The strength of this destabilizing force is increasing in both the freeness of
trade (ϕ) and the degree of inter-regional knowledge spillovers (δ). In contrast, thesecond term on the right-hand side of (23) describes the stabilizing productivity effectcorresponding with the rise of Southern labor productivity in innovation (K ∗ ) relative
to the North (K). The rate of productivity growth falls in the North and rises in theSouth causing the productivity gap to fall back to the symmetric equilibrium. The
strength of this stabilizing force depends on the degree of inter-regional knowledge
spillovers (δ). The stability of the symmetric dispersed equilibrium is determined by18

the balance of these opposing forces with convergence in technologies more likely forlower levels of ϕ and intermediates levels of δ.
To complete this section we investigate the relationship between the productivity
gap and the total number of firms and the consequences of this relationship for longrun productivity growth in a more general setting. To facilitate this investigation we
begin by deriving the relative level of accessible knowledge as a function of ω using(10) and (21):KK∗
1 − δ2= ∗ =.
θθ
2 − δω − δω −1(24)

The relative level of accessible knowledge is convex in the productivity gap with aminimum at ω = 1, and therefore plays a key role in the determination of the total
number of firms and the rate of productivity growth through its impact on the costof innovation.
Next, combining (14) and (15), the total number of firms can be obtained as afunction of the scale of production: ND = α(σ − 1)(L + L∗ )/σlX . Then, using (16),the steady-state no-arbitrage condition ρ = [K/θ][(σ − 2)lX /(σ − 1) + lF ] shows that
there is a negative relationship between the scale of production and the relative levelof accessible knowledge. Using these conditions together with (24) yields the following
expression for the long-run number of firms:

ND =α(σ − 2)(1 − δ 2 )(L + L∗ ),σ [(2 − δω − δω −1 ) ρ − (1 − δ 2 )lF ](25)

where the subscript D indicates a dispersed equilibrium.11 The relationship between
the total number of firms and the productivity gap is also U-shaped with a minimumat ω = 1, as illustrated by the ND (ω) curve in Figure 2a. In particular, for ω < 1, an
11

In Proposition 2 we will show that K/θ < 1 in a dispersed equilibrium. As such, our earlier
assumption ρ > (σ − 1)lF is sufficient to ensure positive market entry and productivity growth inboth core-periphery and dispersed equilibria.

19

Figure 2: Productivity growth and the productivity gapgN
ND (ω)gC

NC

gD (ω)
01δ1/δ(a) Total number of Firms

ω

0

1
δ
1/δ
(b) Rate of Productivity Growthωincrease in ω lowers K/θ and raises the fixed cost of innovation leading to a decreasein the number of firms that the market can support under free entry. For ω > 1, on
the other hand, a rise in ω increases K/θ and the number of firms increases. Thus,
the non-monotonic relationship between ω and ND arises from the links between
firm scale, the relative level of accessible knowledge, and the productivity gap. In
addition, we can show that the total number of firms is always smaller in the dispersed
equilibrium than in the core-periphery equilibrium, as seen in Figure 2a.
Finally, combining (9), (16), (24) and (25), the dispersed rate of productivity
growth can be derived as a function of the productivity gap alone:ρ(σ − 1)(1 − δ 2 )lFα(σ − 1)(1 − δ 2 )(L + L∗ )
−ρ=
−
.
gD =
σ(2 − δω − δω −1)ND
σ − 2 (σ − 2)(2 − δω − δω −1 )(26)

As in the core-periphery equilibrium, the long-run rate of productivity growth ispositively related to the discount rate (ρ) and negatively related to the per-period
fixed cost (lF ). The first and second order derivatives of (26) with respect to ω provide
the following lemma:
Lemma 1 (Long-run productivity growth and the productivity gap): The long-run

20

rate of productivity growth is concave in ω with a maximum at ω = 1.
Proof: Simple partial differentiation of (26). This result is a variant of the rationalization effect (Peretto, 2003) recast in termsof the productivity gap. The non-monotonic relationship between long-run productivity growth and the productivity gap stems from the U-shaped relationship between
the relative level of accessible knowledge and the productivity gap shown in (24). For
example, if ω < 1, a rise in ω lowers K/θ. The consequent fall in the total numberof firms (ND ) raises scales of employment in both production and innovation thereby
increasing the productivity growth rate. In contrast, for ω > 1, an increase in ω raises
K/θ and ND rises. The increase in the total number of firms lowers employment ininnovation and the rate of productivity growth falls.
A comparison of the core-periphery rate of productivity growth (gC ) with that ofthe dispersed equilibrium (gD ) leads to the following proposition:
Proposition 2 (Productivity growth comparison): Productivity growth is always higher
in the dispersed equilibrium than in the core-periphery equilibrium.
Proof: Subtracting (18) from (26) givesgD − gC = 1 −1 − δ22 − δω − δω −1




σ−1
lF .
σ−2

This difference is positive for K/θ < 1, which is always the case if the dispersed
equilibrium exists as ω ∈ (δ, 1/δ) is required for sK ∈ (0, 1), as shown in Figure 1. With imperfect inter-regional knowledge dissemination (0 < δ < 1), the relative
level of accessible knowledge is always higher in the core-periphery equilibrium than
it is in the dispersed equilibrium. Although we might expect this to lead to a greaterrate of productivity growth for the core-periphery equilibrium, the rationalization
effect lowers the firm-level scale of innovation activity below that for the dispersed
equilibrium (lRC < lRD ), and thus the dispersed rate of productivity growth dominates
21the core-periphery rate of productivity growth whenever the dispersed equilibrium
exists, that is, for ω ∈ (δ, 1/δ).

5

Patterns of Industry, Productivity, and Growth

In this section we examine the implications of changes in relative market size, the levelof transport costs, and the degree of inter-regional knowledge spillovers for patterns ofindustry location, the productivity gap, and the long-run rate of productivity growththat arise in a stable dispersed equilibrium. As we have seen, the relative slopes ofthe sK and sR curves are closely related to the freeness of trade and the degree ofinter-regional knowledge spillovers. In Appendix C we compare these relative slopes
with the necessary condition for a stable symmetric dispersed equilibrium given in
Proposition 1 and obtain the following lemma:
Lemma 2 (Relative slopes of the sK and sR curves): In the stable dispersed equilibrium the slope of the sK curve is always greater than the slope of the sR curve.
Proof: See Appendix C. 
Therefore, in the following discussion we evaluate comparative statics using the slope
ranking required by Lemma 2 and illustrated in Figure 1.
As the sK curve is determined independently of labor endowments, comparative
statics for the effect of a change in relative market size can be derived using the sR
curve alone. The results are summarized in the following proposition:
Proposition 3 (Relative market size, industry share, and the productivity gap): In a
dispersed equilibrium, an increase in the relative size of the Northern market (L/L∗ )raises both North’s share of industry (s) and the productivity gap (ω).
Proof: From (22),
1+ϕ
dsR
=
> 0,
dL
4(1 − ϕ)L

dsR
1+ϕ
=−
< 0.
∗
dL
4(1 − ϕ)L∗22

With an increase in the relative market size of the North, the sR curve shifts
upwards leading to a rise in the share of firms locating in the North and a higher
productivity gap. This result stems from the home market effect of the trade literature
whereby a larger share of firms locates in the larger market (Krugman, 1980). In ourframework, however, the rise in the North’s share of industry must be offset by anincrease in the productivity gap to ensure equal rates of productivity growth in the
North and the South.
The results of Proposition 3 provide a characterization of the trade patterns thatarise in the dispersed equilibrium. Although both regions continue to produce both
goods, the larger region becomes a net exporter of manufacturing goods and thesmaller region becomes a net exporter of traditional goods. We henceforth adopt theassumption that the North is the larger region (L > L∗ ), and therefore that ω > 1
and s > 1/2 when evaluating comparative statics.
The sR curve can also be used to derive comparative statics for the effects of achange in the freeness of trade (ϕ), as summarized in the following proposition:
Proposition 4 (Transport costs, industry share, and the productivity gap): In a
dispersed equilibrium, for L > L∗ a decrease in transport costs (an increase in ϕ)increases both North’s share of industry (s) and the productivity gap (ω).
Proof: From (22),

 σ−1
L
L∗
ω
dsR
=
−
> 0,
σ−1
2
σ−1
2
dϕ
(1 − ϕω )(ϕ − ω ) L + L∗
where the term in parentheses is positive for L > L∗ as ω > 1 from Proposition 3. 
Referring to Figure 3a, an improvement in the freeness of trade rotates the sR
curve counter-clockwise to s′R and consequently for L > L∗ the share of industry
based in the North and the productivity gap both rise. A decrease in transport costs
23

Figure 3: Steady-state comparative statics
s

s
sK

s′R

s′K

sR

b

sK

sR

b

6

?
b

b

-

ω



(a) Effect of an increase in ϕω(b) Effect of an increase in δ

strengthens the home market effect since firms have a greater incentive to locate
in the larger market as the cost of exporting to the smaller market has fallen. Onceagain, however, the increase in the larger region’s share of industry must coincide withan increase in the productivity gap in order to maintain equal rates of productivity
growth in the North and the South.
The results summarized in Propositions 3 and 4 have interesting consequences for
economic growth. Referring back to Lemma 1 and Figure 2b, any movement in the
productivity gap away from the symmetric equilibrium lowers the rate of productivity
growth. Accordingly, increases in the relative market size of the North (L/L∗ ) andthe freeness of trade (ϕ) both depress productivity growth.
Next, in order to examine the effect of a change in the degree of inter-regional
knowledge spillovers, we now focus on the sK curve as the sR curve is determined
independently of δ. The results are described in the following proposition:
Proposition 5 (Knowledge spillovers, industry share, and the productivity gap): In a
dispersed equilibrium, for L > L∗ , an increase in the degree of inter-regional knowledge
spillovers (δ) decreases both North’s share of industry (s) and the productivity gap (ω).

24

Figure 4: Effects of a rise in the level of inter-regional knowledge spillovers (δ)
g

′
gD

0

gD

ω

1

Proof: From (21)dsK
(ω 2 − 1)ω −1
=
> 0,
dδ
(1 − δω −1 + 1 − δω)2
where ω > 1 for L > L∗ from Proposition 3. This result is illustrated in Figure 3b. An increase in the degree of inter-regional
knowledge dissemination (δ) rotates the sK curve counter-clockwise to s′K and theNorth’s share of industry and the productivity gap decrease. In particular, the rise inδ raises Southern labor productivity in innovation relative to that of the North andindustry shares adjust to equate the Northern and the Southern rates of productivity
growth.12
The implications of Proposition 5 for long-run productivity growth are ambiguous.
Referring to Figure 4, we can see that an increase in δ has both a direct and an indirect
effect on gD . The negative direct effect is shown by the downward shift in the gD curve′
to gD, and results from an increase in the relative level of accessible knowledge thatlowers innovation costs and raises the total number of firms. The positive indirect
12

An increase in δ also reduces the concentration of industry in variety expansion models ofendogenous growth, but through a different mechanism. Specifically, greater knowledge spillovers
raise the rate of innovation thereby eroding monopolistic profits given the faster rate of entry by
new firms. Lower profit reduces the level of industry concentration (Martin and Ottaviano, 2004).

25

effect dω/dδ < 0 is described by a movement along the gD curve towards ω = 1 andentails a fall in the relative level of accessible knowledge and a subsequent decrease in
the total number of firms. The balance of these direct and indirect effects determines
whether an increase in the degree of inter-regional knowledge dissemination raises
productivity growth or not (see Table 1 in the next section for numerical examples
showing both cases). The ambiguous relationship between δ and gD contrasts withthe results of the variety expansion literature where an increase in the level of interregional knowledge spillovers always has pro-growth effects (Baldwin and Martin,
2004).

6

Welfare Analysis

In this section we briefly discuss the implications of greater market integration forregion welfare levels in the dispersed long-run equilibrium. In particular, we areinterested in ascertaining whether the North and the South benefit from or are hurt byincreases in the freeness of trade and the degree of inter-regional knowledge dispersion.
Continuing to assume a larger labor endowment for the North (L > L∗ ), we can use(1), (2), (5), (6), and (22) to obtain the indirect utilities for Northern and Southernhouseholds respectively as


αgD
α
L(1 − ϕ2 )ND
ln[Aθ(0)α L]
+ 2 ,
+
ln
U0 =
∗
σ−1
ρ
(σ − 1)ρ
(L + L )(1 − ϕω )
ρ


αgD
α
L∗ (1 − ϕ2 )ND
ln[Aθ∗ (0)α L∗ ]
+ 2 ,
+
ln
U0∗ =
∗
1−σ
ρ
(σ − 1)ρ
(L + L )(1 − ϕω )
ρ

(27)
(28)

where θ(0) and θ(0)∗ are the respective initial productivities of Northern and Southernfirms, and A = (α(σ − 1)/σ)α (1 − α)1−α is a constant.

26

Table 1: Welfare Effects
L

δ

ϕ

ω

s

dg
dδ

dg
dϕ

dU0
dδ

dU0
dϕ

dU0∗
dδ

dU0∗
dϕ

1
1.2
1.2
1.2

0.8
0.8
0.85
0.8

0.3
0.3
0.3
0.35

1
1.08
1.04
1.13

0.5
0.66
0.62
0.76

-0.015-0.010-0.0130.009

0-0.005-0.002-0.021

1.3
-1.5
-0.1
-8.4

2.6
-0.1
1.3
-8.5

1.3
2.4
2.0
3.6

2.6
2.2
2.6
0.5

Parameter values are L∗ = 1, α = 0.5, σ = 2.5, lF = 0.01, and ρ = 0.1.
Beginning with the welfare effects of an increase in the freeness of trade, we obtain(ω σ−1 − ϕ) − (1 − ϕω σ−1 )ϕϕω σ−2 dω
1
dND 1 dgD
ρ dU0
=
+
+
+
,
2
σ−1
σ−1
α dϕ
(σ − 1)(1 − ϕ )(1 − ϕω ) (1 − ϕω ) dϕ (σ − 1)ND dϕ
ρ dϕ
ρ dU0∗(1 − ϕω σ−1 ) − (ω σ−1 − ϕ)ϕϕω σ−2 dω
1
dND 1 dgD
=
−
+
+
.
2
σ−1
1−σ
α dϕ
(σ − 1)(1 − ϕ )(ω
− ϕ)(ω− ϕ) dϕ (σ − 1)ND dϕ
ρ dϕ
The first term on the right-hand side captures the impact of lower prices on traded
goods and is positive for both regions. The second term describes the terms of tradeeffect associated with changes in the relative price of imported manufacturing varieties
and is positive for Northern households and negative for Southern households given
the results summarized in Proposition 4. The third and fourth terms capture thewelfare effects associated with changes in the total number of firms (ND ) and the rateof productivity growth (gD ), respectively, and are the same for both Northern andSouthern households. With the positive relationship between ω and ϕ, however, thetotal number of firms rises and the rate of productivity growth falls, and the overall
welfare effect associated with an increase in the freeness of trade, therefore, depends
on parameter values, as indicated by the numerical examples presented in Table 1.
Turning next to the degree of inter-regional knowledge spillovers, an increase in δ
has several opposing welfare effects:ϕω σ−2 dω
1
dND 1 dgD
ρ dU0
=
+
+
,
σ−1
α dδ
(1 − ϕω ) dδ(σ − 1)ND dδρ dδρ dU0∗
ϕω σ−2 dω
1
dND 1 dgD
= − σ−1
+
+
.
α dδ
(ω
− ϕ) dδ(σ − 1)ND dδρ dδ
27In this case, the first term describes the terms of trade effect and is negative for the
North and positive for the South given the result that an increase in δ lowers the
productivity gap from Proposition 5. As before, the remaining terms describe thechanges in total variety (ND ) and productivity growth (gD ) and are common to bothregions. The overall welfare effects of a change in the degree of inter-regional knowledge spillovers once again depend on parameter values as shown by the numericalexamples presented in Table 1.

7

Concluding Remarks

In this paper we have investigated the relationship between productivity growth andspatial patterns of technology and industrial activity. In particular, we develop a tworegion model of trade and endogenous growth without scale effects in which manufacturing firms invest in in-house process innovation with the objective of reducing
production costs. Using this framework we examine core-periphery equilibria forwhich all industrial activity occurs in one region and dispersed equilibria for which
industrial activity is dispersed across regions, and find that a sufficiently high levelof transport costs and an intermediate level of inter-regional knowledge dispersion
is required for convergence in technologies to a dispersed long-run equilibrium. Determining which type of equilibrium the economy converges to is important as thehighest rate of productivity growth arises in a symmetric equilibrium with no productivity gap and equal shares of industry across regions. As such, a key conclusionof our framework is that geographic concentration of industry has a negative impact
on overall growth.
Considering the effects of greater economic integration, we find that within adispersed long-run equilibrium although an increase in the freeness of trade leads
to increases in both the larger region’s share of industry and its relative level oftechnology, lower transport costs allow for a greater total number of manufacturing28

firms and as a result investment in innovation falls causing a decrease in the rate ofproductivity growth. On the other hand, an increase in the degree of inter-regional
knowledge dispersion decreases the larger region’s share of industry and its relative
level of technology. In this case, the consequences for productivity are ambiguous as
the number of manufacturing firms may rise or fall. Put together, these results haveinteresting implications for the impact of greater economic integration on regional
welfare levels.
The framework presented in this paper is limited in two respects. Firstly, we haveassumed that the production and innovation activities of a firm must occur in the same
location. An interesting extension might include the separation of these activities
across regions. Secondly, we have adopted an assumption of negligible entry costs for
manufacturing firms. The inclusion of product development costs for manufacturingfirms would, however, allow for a richer characterization of the dynamics of entry andexit associated with greater economic integration.

29

Appendix A: Proof of Proposition 1
The local dynamics around the symmetric steady state are determined by the differential equation for the productivity gap (19) and the no-arbitrage conditions for the
North and the South:∗δs∗ K ∗ lR
KlX
− slR −,θK∗K ∗ lX
δsKlR
∗ ∗
ρ=
−
s
l
−
.
R
θ∗
K∗

ρ=

(A1)
(A2)

∗
Respectively adding and subtracting KlR /θ and K ∗ lR
/θ∗ from (A1) and (A2), these

conditions can be rewritten as follows:△1 + △2 = 0,

(A3)△∗1 + △∗2 = 0,

(A4)

where
 ∗ ∗
∗
δs θ
KlR K ∗ lR
△2 =
− ∗
,
θ
θ
K


∗
δsθ
KlR K ∗ lR∗
△2 = −
− ∗
.
θ
θ
K∗


K(lX − lR )
− ρ,
△1 =
θ
∗
∗
K ∗ (lX
− lR
)
△∗1 =
− ρ,
∗
θ

These conditions can be used to obtain sK and sR . First, in the steady-state equilib∗
rium ω̇ = 0 requires KlR /θ = K ∗ lR/θ∗ and therefore △1 = △∗1 = △2 = △∗2 = 0. Next,∗
using (16) to substitute lX and lX
out of △1 = △∗1 , we find that K/θ = K ∗ /θ∗ . This∗∗implies that lR = lR
and thus we have lX = lX
. Finally, the conditions K/θ = K ∗ /θ∗∗and lX = lX
can be solved respectively for (21) and (22).

The no-arbitrage conditions (A3) and (A4) implicitly determine the numbers offirms based in each region for a given level of the productivity gap at each moment∗∗in time. Substituting in lR = lX /(σ − 1) − lF and lR= lX/(σ − 1) − lF , from (16),30

taking the total derivatives of (A3) and (A4), and employing Cramer’s rule yields∂(△∗ +△∗ )∂(△∗ +△∗ )∂(△1 +△2 )1 +△2 )
1
2
1
2
− ∂(△∂ndn∗∂ω∂n∗= ∂(△ +△ ) ∂(△∗ +△∗ ) ∂(△ +△ ) ∂(△∂ω∗ +△∗ ) ,
1
2
1
2
1
2
1
2
dω
−
∗
∗
∂n

∂n

∂n

∂(△∗ +△∗ )∂(△∗ +△∗ )∂(△1 +△2 )1 +△2 )
1
2
1
2
− ∂(△∂ωdn∗∂n,= ∂(△∂n+△ ) ∂(△∂ω∗ +△∗ )∂(△1 +△2 ) ∂(△∗1 +△∗2 )
1
2
1
2
dω
−
∗
∗
∂n

∂n

(A5)

∂n

∂n

(A6)

∂n

where
∂△1
∂△∗1
(1 − δ)(lX − lR ) (σ − 2)(1 + δ)(1 + ϕ2 )lX
=
=
−
,
∂n
∂n∗
2ND
(σ − 1)(1 + ϕ)2 ND∂△1∂△∗1(1 − δ)(lX − lR ) (σ − 2)(1 + δ)ϕ2lX
−
,
=
=−
∗
∂n
∂n
2ND
(σ − 1)(1 + ϕ)2 ND∂△∗1
δ(lX − lR ) (σ − 2)(1 + δ)ϕlX
∂△1
=−
=−
+
,
∂ω
∂ω
2
(1 + ϕ)2∂△2
2δ(1 − ϕ)2 lX∂△∗
2(1 − δ)δlR
∂△2
∂△∗2
−
,
= − ∗2 = − ∗ =
=
∂n
∂n
∂n
∂n
(1 + δ)ND(σ − 1)(1 + ϕ)2 ND
∂△∗2
2δϕlX
δ 2 lR
∂△2
=−
=
−
.
∂ω
∂ω
(1 + ϕ)2 1 + δ∗
Now, setting (A3) and (A4) equal, solving for KlR /θ − K ∗ lR/θ∗ , and substituting

the result into (19), the motion for the productivity gap can be rewritten asω̇(△∗1 − △1 )KK ∗=.
ωδ(sθK + s∗ θ∗ K ∗ )(A7)

We approximate the local dynamics around the symmetric dispersed steady state(ω = 1 and n = n∗ ) using a Taylor expansion of this differential equation:(1 + δ) ∂△2(1 + δ) ∂(△∗1 − △1 ) dn ∂(△∗1 − △1 ) dn∗ ∂(△∗1 − △1 )
∂ ω̇
=
=
+
+
.
∂ω
2δ
∂n
dω
∂n∗
dω
∂ω
δ
∂ω
, this Taylor expansion simplifies toUsing (16) and (25) with lX = α(σ−1)2L
σND



∂ ω̇
(σ − 1)(1 + δ)2ϕ
2ρ − (1 + δ)lF
+ δlF .
=
−δ∂ω
(1 + ϕ)2(σ − 2)(1 + δ)
31

(A8)

As the productivity gap is a state variable, we require a negative sign for (A8) for astable dispersed equilibrium. The sign of (A8) can be described as a function of thefreeness of trade (ϕ) using
ϕ2 + 2ϕ (1 − κ) + 1 ≥ 0,

(A9)

F)
> 0. This quadratic equation can be solved for two
where κ = (σ−1)(1+δ)(2ρ−(1+δ)lδ(2ρ−(σ−1)(1+δ)lF )p
thresholds ϕ̃ = (κ − 1) ± (κ − 2)κ. However, since κ − 1 > 1 is required forthese thresholds to take real values, only one of them lies between zero and one:pϕ̃ = (κ − 1) − (κ − 2)κ. For ϕ < ϕ̃, the dispersed equilibrium is saddle path stable,

and for ϕ > ϕ̃ it is unstable, as stated in Proposition 1.
Appendix B: Proof of Corollary 1
First, note that

dϕ̃
κ−1
< 0,
=1− p
dκ
(κ − 2)Kanddκ(σ − 1) [(σ − 1)(1 + δ 2 )lF (2ρ − lF ) − 2ρ(2ρ − (1 − δ 2 )lF )].
=dδδ 2 [2ρ − (σ − 1)(1 + δ)lF ]2(B1)

The sign of ddκϕ̃ dκdepends on the sign of the numerator of (B1) which is determineddδby the sign of the following quadratic equation:4(1 − lF )ρ
2ρ
+1−.
δ + 2δ 1 +(σ − 2)2ρ − (σ − 1)lF[(σ − 2)2ρ − (σ − 1)lF ] lF
2



(B2)

Taking the partial derivative of (B2) with respect to δ and setting the result equal
to zero, we can solve for a threshold value δ̃, below which ddδϕ̃ > 0 and above whichdϕ̃< 0 as stated in Corollary 1.
dδ

32Appendix C: Proof of Lemma 2
First,

dsK− dsdωR
dω



ω=1

δ
− (σ−1)ϕ> 0 implies= 2(1−δ)(1−ϕ)2(σ − 1)(1 − δ)
ϕ − 2ϕ 1 ++ 1 > 0.
δ2

(C1)

Next, subtracting (A9) from (C1) we obtain
2ϕ(σ − 2) [4δρ + (σ − 1)(1 − δ 2 )lF ]> 0,δ [2ρ − (σ − 1)(1 + δ)lF ]for δ, ϕ ∈ (0, 1) as we require 2ρ > (σ − 1)(1 + δ)lF for gD > 0 in the symmetric steadystate. We conclude that the slope of the sK curve is always greater than that of thesR curve for ϕ < ϕ̃ in the symmetric dispersed equilibrium. This completes the proof
of Lemma 2.

33

Acknowledgements: We are grateful for helpful comments from F. Dei, K. Futagami, T. Iwaisako, A. Kitagawa, T. Matsuoka, K. Mino, T. Tabuchi, Y. Tomoda,
K. Yamomoto, M. Yano, and seminar participants of the Macroeconomics Research
Seminar (Osaka University, 2011), the Pacific Economic Review Workshop (Kyoto
University, 2011), the Macroeconomic Research Seminar (Kyoto University, 2011),
and the Japanese Economic Association Spring Meeting (Kumamoto Gakuen University, 2011). All remaining errors are our own.

References[1] Baldwin, R., Forslid R., 2000. The core-periphery model and endogenous
growth: Stabilizing and destabilizing integration. Economica 67, 307-324.
[2] Baldwin, R., Martin, P., Ottaviano, G., 2001. Global income divergence,
trade, and industrialization: the geography of growth and take-offs. Journal of
Economic Growth 6, 5-37.
[3] Baldwin, R., Martin, P., 2004. Agglomeration and Regional Growth. Handbookof Regional and Urban Economics 4, 2671-2711.
[4] Bosker, M., 2007. Growth, agglomeration and convergence: A space-time
analysis for European regions. Spatial Economic Analysis 2, 91-100.
[5] Braunerhjem, P., Borgman, B., 2004. Geographical concentration, entrepreneurship and regional growth: Evidence from regional data in Sweden, 1975-99.
Regional Studies 38, 929-947.
[6] Broda, C., Weinstein, D., 2006. Globalization and the Gains From Variety.
Quarterly Journal of Economics 121, 541-585.
[7] Brülhart, M., Sbergami, F., 2009. Agglomeration and growth: Cross-country
evidence. Journal of Urban Economics 65, 48-63.
[8] Coe, D., Helpman, E., Hoffmaister, A., 2009. International R&D spillovers andinstitutions. European Economic Review 53, 723-741.
[9] Dinopoulos E., and Thompson, P., 1998. Schumpeterian growth without scaleeffects. Journal of Economic Growth 3, 313-335.
[10] Dixit, A., Stiglitz, J. E., 1977. Monopolistic competition and optimum productdiversity. American Economic Review 67, 297-308.
[11] Futagami, K., Iwaisako,T., 2007. Dynamic analysis of patent policy in an
endogenous growth model. Journal of Economic Theory 132, 306-334.
[12] Gardiner, B., Martin, R., Tyler, P., 2010. Does spatial agglomeration increase
national growth? some evidence from Europe. Journal of Economic Geography,
forthcoming, doi:10:1093/jeg/1bq047.
34[13] Grossman, G., Helpman, E., 1991. Innovation and growth in the global economy.
Cambridge: MIT Press.
[14] Jaffe, A. B., Trajtenberg, M., Henderson, R., 1993. Geographical localization ofknowledge spillovers as evidenced by patent citations. The Quarterly Journal of
Economics 108, 577-598.
[15] Jones, C., 1995. R&D-based models of economic growth. The Journal of Political
Economy 103, 759-784.
[16] Keller, W., 2004. International technology diffusion. Journal of Economic
Literature 42, 752-782.
[17] Krugman, P., 1980. Scale economies, product differentiation, and the pattern oftrade. American Economic Review 70, 950-959.
[18] Lainez, C., Peretto, P., 2006. Scale effects in endogenous growth theory: An
error of aggregation not specification. Journal of Economic Growth 11, 263-288.
[19] Li, C., 2003. Endogenous growth without scale effects: Comment. American
Economic Review 93, 1009-1018.
[20] Mancusi, M. L., 2008. International spillovers and absorptive capacity: A
cross-region cross-sector analysis based on patents and citations. Journal ofInternational Economics 76, 155-165.
[21] Martin, P., Ottaviano, G. I., 1999. Growing locations: Industry location in amodel of endogenous growth. European Economic Review 43, 281-302.
[22] Melo, P., Graham, D., Noland, R., 2009. A meta-analysis of estimates of urban
agglomeration economies. Regional Science and Urban Economics 39, 332-342.
[23] Minniti, A., Parello, C., 2011. Trade integration and regional disparity in amodel of scale-invariant growth. Regional Science and Urban Economics 41,
20-31.
[24] Peretto, P., 1996. Sunk costs, market structure, and growth. International
Economic Review 37, 895-923.
[25] Peretto, P., 2003. Endogenous market structure and the growth and welfare
effects of integration. Journal of International Economics 60, 177-201.
[26] Samuelson, P., 1954. The transfer problem and transport costs, II. Economic
Journal 64, 264-289.
[27] Segerstrom, P., 1998. Endogenous growth without scale effects. American
Economic Review 88, 1290-1310.
[28] Smulders, S., van de Klundert, T., 1995. Imperfect competition, concentration,and growth with firm-specific R&D. European Economic Review 39, 139-160.

35