ISSN (Print) 0473-453X

Discussion Paper No. 1066

ISSN (Online) 2435-0982

IMPORT COMPETITION
AND INDUSTRY LOCATIONIN A SMALL-COUNTRY MODELOF PRODUCTIVITY GROWTH

Colin Davis
Ken-ichi Hashimoto

October 2019

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

Import Competition and Industry Location
in a Small-Country Model of Productivity GrowthColin Davis∗

Ken-ichi Hashimoto †October 2019

Abstract
We study the effects of import competition on industry locations patterns in a small
open economy with two regions. Domestic productivity growth converges to the international rate through firm-level investment in process innovation. With firms locatingproduction and innovation in their lowest cost locations, the concentration of industry
in the larger region is linked with firm-level innovation through an import competition
effect that is increasing in the market share of imported goods and the productivity differential of domestic firms with the rest of the world. We show that increased importcompetition, through either a larger number of imported goods or a faster international
rate of productivity growth, leads to greater industry concentration by reducing domestic
market entry and decreasing the relative productivity of domestic firms. We also consider the implications of improved regional and international economic integration.
JEL Classifications: F12; F43; O40; R12
Keywords: Import competition; Industry concentration; Endogenous productivity growth∗

The Institute for the Liberal Arts, Doshisha University, Karasuma-Higashi-iru, Imadegawa-dori, Kamigyo,
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.
Acknowledgements: This work was completed in part while Davis was visiting RITM at the University ofParis-Sud. He thanks them for their hospitality and support. We also acknowledge financial support from JSPS
through Grants-in-Aid for Scientific Research (S) and (C): grant number 15H05728 and 19K01638. The usual
disclaimer applies.

1 Introduction
Foreign competition poses a key policy concern for developed and developing countries alikeas national governments attempt to resolve regional imbalances in the location patterns ofdomestic industry while promoting innovation-based economic growth. Indeed, increased
trade openness has been found to have important implications for the geographic concentration of industry (Brülhart 2011; Rodrı́guez-Pose 2012; Ezcurra and Rodrı́guez-Pose 2013).
In addition, empirical evidence suggests that greater competition from an increased range ofand lower prices for imported goods has a direct influence on firm-level investment in research and development (R&D) (Bloom et al. 2017; Autor et al. 2019). Although a broad
literature has studied the potential effects of industry location patterns on innovation-based
economic growth (Baldwin and Martin 2004; Desmet and Rossi-Hansberg 2009, 2014), theeffects of innovation activity on domestic industry location patterns are less well-understood.
In this paper, we highlight the potential for foreign competition to influence industry location
patterns through its effect on firm-level investment in process innovation.
In particular, we extend the theoretical framework of Davis and Hashimoto (2016) tostudy the implications of increased import competition for industry location patterns in asmall open country with two regions. Adopting an endogenous growth and endogenous market structure framework (Aghion and Howitt 1998; Laincz and Peretto 2006; Etro 2009;
Peretto 2018), the model focuses on a monopolistically competitive sector in which firms
compete with imported goods as they supply the households of each region and invest in process innovation with the aim of lowering future production costs (Smulders and van de Klundert 1995; Peretto 1996). Characterizing long-run equilibrium in terms of the domestic levelof market entry and the productivity differential between domestic firms and internationalfirms, we derive sufficient conditions for the convergence of the domestic rate of productivity
growth rate with the international productivity growth rate.
Production and innovation are independently located in the regions with the lowest respective costs: firms undertake each activity in a single region and need not locate production2and innovation together (Martin and Rogers 1995; Martin and Ottaviano 1999; Davis andHashimoto 2016). Interregional trade costs and imperfect knowledge diffusion result in thepartial concentration of production and the full concentration of innovation in the region withthe larger market, as measured by population size. The regional concentration of productionis then linked with firm-level innovation through an import competition effect that induces
firms to shift production to the larger region to mitigate the negative profit effect associatedwith imported goods. This import competition effect is magnified by a fall in international
trade costs, a rise in the market share of imported goods, or a fall in the productivity of domestic firms relative to the international firms producing the goods imported from the rest of
the world.
We use the framework to consider how improved economic integration and increased
import competition affect domestic market entry, the productivity differential of domestic
firms with international firms, and the regional concentration of industry. First, we showthat improvements in regional and international knowledge diffusion both reduce innovation
costs, raising profit and inducing domestic firms to enter the market. Domestic productivity
growth temporarily rises above the international productivity growth rate over the transition toa new long-run equilibrium, resulting in a smaller productivity differential with internationalfirms. The import competition effect then reduces the concentration of industry in the region
with the larger market.
Turning next to trade integration, on the one hand, we find that a fall in interregional trade
costs increases firm profitability, causing firms to enter the market, and temporarily accelerating domestic productivity growth as firm-level investment in process innovation rises. As
such, the productivity differential of domestic firms with international firms contracts. The
home market effect dominates the import competition effect, however, generating an increasein the concentration of industry in the larger region. A fall in international trade costs, on theother hand, reduces firm profitability, inducing domestic firms to exit the market. As theincreased market share of imported goods leads to a fall in firm-level investment in process3innovation, but potentially raises knowledge spillovers, the regional strength of knowledge
diffusion determines whether productivity growth temporarily accelerates or decelerates over
the transition to a new equilibrium. Productivity growth temporarily accelerates (decelerates), and the productivity differential with international firms falls (rises) for strong (weak)regional knowledge diffusion. Regardless of the change in the productivity differential, we
find that overall the import competition effect always leads to greater industry concentration.
Lastly, addressing the effects of increased import competition, we investigate the implications of either an expansion in the number of imported goods, or an increase in the given
productivity growth rate for international firms. In both cases, profits fall, and domestic firms
exit the market. In the case of an increase in the number of imported goods, the domestic rateof productivity growth falls over the transition to the new long-run equilibrium. In the caseof an increase in the international productivity growth rate, the domestic rate of productivity
growth accelerates to match the international rate. In both cases the productivity of domestic
firms necessarily falls relative to international firms. Consequently, the import competition
effect ensures a rise in the larger region’s share of industry.
Our framework builds on the theoretical literature examining the effects of trade integration on the spatial concentration of industry.1 Krugman and Livas Elizondo (1996) introduce
a small open economy framework in which labor is mobile between two regions. Economies
of scale, market size, and domestic transport costs pull labor and industry towards a single region, while congestion costs push industry to disperse regionally. Trade liberalization
weakens the pull factors, resulting in a dispersed industry location pattern. Paluzie (2000) andMonfort and Nicolini (2000) adapt the framework by removing congestion costs and addinga regionally immobile production factor that is employed in a constant returns to scale sector. Similar to the model developed in this paper, the immobile production factor tends todisperse industry by creating exogenous regional demand. This adaption reverses the resultof Krugman and Livas Elizondo (1996) with trade liberalization inducing greater industry
1

See Brülhart (2011) for a survey of the theoretical and empirical literatures considering the relationshipbetween trade openness and spatial concentration.

4

concentration. Subsequent research by Brülhart et al. (2004) and Crozet and Koenig (2004)
has shown that trade liberalization leads to the spatial concentration of industry in regionswith better access to export markets. In addition, Behrens et al. (2007) demonstrate that therelationship between trade openness and industry location patterns depends on the size of domestic transport costs when firms have variable markups. As the studies introduced above do
not consider innovation or economic growth, our results extend the literature by demonstrating that import competition and trade liberalization influence industry concentration throughtheir effects on firm-level investment in process innovation.
The paper proceeds as follows. In Section 2 we introduce our small open economy modelof import competition and industry concentration. Section 3 characterizes the long-run equilibrium and derives the conditions for saddlepath stability. We study the effects of improvements in regional and international economic integration in Section 4, and then consider the
implications of increased import competition in Section 5. We conclude in Section 6.

2 The Model
Consider a small open economy with two distinct regions, and three types of economic activity: traditional production, manufacturing, and process innovation. Labor is mobile acrossactivities, but not between regions or with the rest of the world. The traditional sector produces a numeraire good for a competitive international market characterized by free trade.
Adopting a unit-coefficient constant-returns-to-scale technology, and assuming that both regions are active in traditional production, we normalize the traditional good price to unity.
As such, the wage rate equals one in both regions.

52.1 Households
The demand side of the model consists of dynastic households with symmetric regional preferences. The lifetime utility of the representative household residing in region i is

Ui =

Z ∞

e−ρt (α log CXi (t) + (1 − α) log CY i (t)) dt,(1)0

where CXi and CY i are the quantities consumed of a composite of manufacturing goods andthe traditional good, ρ is the subjective discount rate, and α ∈ (0, 1). We suppress the timeargument (t), where possible, in order to simplify notation.
The manufacturing composite is a constant elasticity of substitution (CES) aggregator
across the manufacturing varieties produced in three locations:

CXi =

Z Ni

ci (m)

σ−1
σ

0

dm +

Z Nj

cj (m)

0

σ−1
σ

dm +

Z N∗
0

c∗ (m)

σ−1
σ

σ
 σ−1
dm
,

(2)

with c(m) denoting the household consumption of a representative variety m from the masses
of product varieties produced in region i (Ni ) and region j (Nj ). The total mass of domestically produced varieties is N ≡ N1 + N2 . Consumption of a representative variety m fromthe mass of imported varieties (N ∗ ) is denoted by c∗ (m), where variables associated with therest of the world are indicated with an asterisk. The elasticity of substitution between any
given pair of varieties is σ > 1.
The household’s utility maximization problem is solved in three stages. First, the household sets an optimal expenditure path with the objective of maximizing lifetime utility (1)subject to the flow budget constraint V̇i = rVi + Li − Ei , where r is the interest rate, Vi is asset wealth, Li is the labor force (labor income) in region i, and time differentiation is denotedwith a dot over a variable. While households do not have access to international investment
assets, a nationally integrated financial market ensures that the interest rate equalizes acrossregions. As a result, the solution to the household’s optimization problem features common
expenditure dynamics for the households residing in each region: Ė1 /E1 = Ė2 /E2 = r − ρ.
6Second, at each moment in time, households allocate constant shares of expenditure to
the manufacturing composite and traditional goods: CXi = αEi /PXi and CY i = (1 − α)Ei .
The price index associated with the manufacturing composite is

PXi =

Z Ni

pi (m)

1−σ

0

dm +

Z Nj

(τ pj (m))

1−σ

dm +

Z N∗
0

0

(τ p (m))
∗ ∗

1−σ

1
 1−σ
,
dm

(3)

where pi and pj are the prices of goods produced in regions i and j, and p∗ is the price ofgoods imported from the rest of the world. Iceberg trade costs are incurred on shipments
between regions and on imports, but at different rates. Specifically, τ > 1 units must be
shipped for each unit sold in regional trade, while τ ∗ > 1 units must be shipped for each unit
imported and sold to households residing in either region 1 or region 2. We assume that trade
costs are lower for regional shipments than for international shipments; that is, τ < τ ∗ .
Third, given the constant expenditure share allocated to the manufacturing composite,
applying Shephard’s Lemma to the price index yields the instantaneous demand functions forthe household in region i

σ−1
ci = p−σi PXi αEi ,σ−1cj = (τ pj )−σ PXi
αEi ,

σ−1
c∗ = (τ ∗ p∗ )−σ PXi
αEi ,

(4)

for varieties produced in regions i and j, and for varieties imported from the rest of the world.

2.2 Manufacturing
The manufacturing sector produces differentiated product varieties for a domestic market
characterized by Dixit and Stiglitz (1977) monopolistic competition. At each moment in
time, a representative firm incurs variable labor costs in the production of a distinct productvariety (lXi ) and in process innovation (lRi ). In addition, the firm bears fixed labor costs
each period in the management of production and innovation activities (lF ). Although the
production technologies associated with specific product lines are unique, we assume thatlabor productivity is symmetric across domestic firms, while allowing for an international
7

productivity differential with international firms located in the rest of the world.
Following the process innovation literature (Smulders and van de Klundert 1995, Peretto1996), the technology of a firm with production located in region i is

xi = θγ lXi ,(5)where lXi is firm-level employment in production, θ is a firm-specific productivity parameterthat is symmetric across regions, and γ ∈ (0, 1) is the productivity elasticity of output. The
innovation technology of a firm with R&D located in region i isθ̇ = θki lRi ,(6)where lRi is firm-level employment in process innovation, and θki is labor productivity. Under
this specification for R&D activity, technical knowledge is captured by firm-level productivity
and accumulates within the production technology of the firm.
Following the process innovation literature (Smulders and van de Klundert 1995, Peretto1996), current innovation activity improves the labor productivity of future innovation efforts, generating an intertemporal knowledge spillover that potentially leads to endogenous
productivity growth (Romer 1990). The strength of the intertemporal knowledge spilloveris closely connected with the geography of production through a knowledge spillover fromproduction to innovation (Baldwin and Forslid 2000). In particular, we specify the strengthof knowledge spillovers into process innovation located in region i as follows:

ki = si + δsj + δ ∗ s∗ ω,(7)where δ ∈ (0, 1) is the degree of regional knowledge diffusion, δ ∗ ∈ (0, 1) is the degreeof international knowledge diffusion, si ≡ Ni /(N + N ∗ ), sj ≡ Nj /(N + N ∗ ), and s∗ ≡N ∗ /(N + N ∗ ) are the shares of firms with production located in region i, region j, and the8rest of the world, and ω ≡ θ∗ /θ describes the productivity differential between domestic andinternational firms.
Under specification (7), knowledge spillovers are determined as a weighted average ofthe technologies observable by the firm, with the weights decreasing in the distance between
the observed technologies and the location of the R&D department of the firm. We capture
the geographic nature of knowledge spillovers by assuming that δ > δ ∗ , such that domestic
spillovers are greater than international spillovers. This assumption matches with evidence
from the empirical literature of both the localized nature of knowledge spillovers and thepresence of international knowledge diffusion (Bottazi and Peri 2007; Mancusi 2008; Coe etal. 2009; and Ang and Madsen 2013). Importantly, the region with the greatest share of production has the strongest potential for knowledge spillovers from production into innovation.
The total per-period profit of a firm with production located in region i and innovation
located in region j is composed of operating profit on sales (πi = pi xi − lXi ) minus the costof employing labor in innovation (lRj ) and the per-period fixed cost (lF ):Πi = πi − lRj − lF ,(8)

where the firm may or may not locate production and innovation in the same region, following
from the footloose nature of firm organization.
At each moment in time, a representative firm sets its product price (pi ) and employment
in process innovation (lRj ) with the objective of maximizing firm value, which is equal to thepresent value of the future profit stream:

vi =

Z ∞

Rt

′

′

Πi (t)e− 0 r(t )dt dt.

(9)0This optimization problem is solved using a current value Hamiltonian function Hi = πi −lRj − lF + pRj θkj lRj , where pRj captures the internal value of the firm’s technical knowledge
when production is located in region i and innovation is located in region j.
9With CES preferences over product variety, the optimal pricing strategy is a constantmarkup over the unit cost of production, p = σ/((σ−1)θγ ), with symmetric pricing strategies
across regions. Each firm sets its output level to meet the combined demands from each region(xi = ci + τ cj ) generating operating profit on sales equal to
lXi
αp1−σ
i
πi =
=
σ−1
σ

τ 1−σ Ej
Ei
+
1−σ
1−σ
PXi
PXj

!

,

(10)

for a firm with production located in region i.
Firms set their optimal levels of employment in process innovation to satisfy the following
static and dynamic efficiency conditions:

pRj =

1
,
θkj

r = γ(σ − 1)kj πi −

k̇j
θ̇
− .
kj
θ

(11)

Note that firms do not account for changes in either the prices indices (3) or the level ofknowledge spillovers (7), when setting optimal employment for process innovation, given
the small firm-level market shares that arise under monopolistic competition.

2.3 Market Entry
Per-period profit determines the domestic level of entry into the manufacturing sector (s ≡N/(N +N ∗ )). In particular, we suppose that firm value responds correctly to market entry andexit through adjustments in profit (∂Π/∂s) in the spirit of Novshek and Sonnenchein (1987).2
The time derivative of firm value (9) yields a no-arbitrage condition that equates the return tomarket entry with the risk free interest rate: rv = Π+ v̇. With no costs incurred in the creationof new product designs, new firms enter when firm value is positive (v > 0), decreasing
profit and reducing firm value. Alternatively, firms exit when firm value is negative (v < 0),increasing profit and raising firm value. The adjustment process is immediate (v̇ = 0), and2

See Smulders and van de Klundert (1995), Peretto (1996), and van de Klundert and Smulders (1997) for a
full discussion of the dynamics that arise in this class of growth models.

10

leads to an equilibrium level of market entry that satisfies v = 0. As such, we have thefollowing free market entry and exit condition for the manufacturing sector:

lXi = (σ − 1)(lRj + lF ).

(12)

With firm value driven to zero, domestic asset wealth also falls to zero (V = Nv = 0). The
household’s flow budget constraint then confirms that expenditure equals labor income ineach region Ei = Li , and thus r = ρ at all moments in time.
2.4 Location Patterns
Firms shift production and innovation freely and independently between regions, at no cost,with the aim of maximizing firm value by minimizing both production and innovation costs(Martin and Rogers 1995; Martin and Ottaviano 1999; Davis and Hashimoto 2016). In thissection, we show that the domestic share of manufacturing firms (s) and the productivity
differential (ω) play key roles in the determination of regional location patterns.
Looking first at the location pattern for production, when both regions have positive production shares, location arbitrage leads to the equalization of operating profit across regions:
π1 = π2 . We suppose that imported varieties are also priced using a constant markup overunit production costs: p∗ = σ/((σ − 1)θ∗ γ ), where we normalize the common wage rate in
the rest of the world to unity. Substituting (10) into the location-arbitrage condition (π1 = π2 )yields regional shares of domestic production:(Li − ϕLj ) (Li − Lj )ϕ∗ s∗ ω γ(σ−1)
si
=
+
,
s
(1 − ϕ)L(1 − ϕ)Ls(13)

where L ≡ L1 + L2 is the national labor force, s ≡ N/(N + N ∗ ) is the share of productvarieties produced domestically, and s∗ ≡ N ∗ /(N + N ∗ ) is the share of product varieties
imported from the rest of the world. The level of regional trade costs is measured by ϕ ≡τ 1−σ ∈ (0, 1), with dϕ/dτ < 0: there is no regional trade for ϕ = 0 and regional trade is
11free for ϕ = 1. In a similar manner, ϕ∗ ≡ τ ∗ 1−σ ∈ (0, 1) indexes the level of trade costs onimports.
We use (si /s) as a measure of the domestic level of industry concentration. There are
two mechanisms determining industry concentration. The first term on the righthand sideof (13) describes a home market effect (Krugman 1980) whereby a greater share of firms
locate production in proximity to the market of the larger region. The second term captures
an import competition effect, through which a rise in the share of imported varieties (s∗ )induces firms to shift production towards the larger region in order to mitigate the effects ofgreater import competition. The import competition effect is increasing in the technology gap
with the rest of the world (ω ≡ θ∗ /θ). Hereafter, we assume without loss of generality that
region 1 has the larger population and thus the greater production share; that is, L1 > L2 ands1 > s2 . In addition, because we are interested in location patterns that feature non-negative
production shares for both regions, we focus on equilibria that satisfy (Li − ϕLj )s + (Li −Lj )ϕ∗ s∗ ω γ(σ−1) ≥ 0 in order to ensure that si ≥ 0.
Turning next to the location pattern for innovation, given the current state of technology(θ), firms shift their innovation activities to the region with the lowest cost of process innovation (pRi ). Because region 1 has a larger production share (s1 > s2 ), it supports stronger
knowledge spillovers from production into innovation. As a result, all firms locate process
innovation in region 1 in order to take advantage of lower innovation costs (pR1 < pR2 ). With
innovation fully concentrated in region 1, the no-arbitrage condition for investment in process
innovation becomesk̇ρ = γklX − klR − ,k(14)where we now drop the subscript on the strength of knowledge spillovers (k1 = k) with theunderstanding that while there are positive shares of firms locating production in each region,
all firms locate process innovation in region 1.
The equilibrium strength of knowledge spillovers is obtained by substituting regional
12

production shares (13) into (7):k(ω, s) =(1 − δ)((L1 − ϕL2 )s + (L1 − L2 )ϕ∗ s∗ ω γ(σ−1) )+ δs + δ ∗ s∗ ω.
(1 − ϕ)L(15)

A rise in the domestic share of firms (s) has three opposing effects on the strength of knowledge spillovers. First, as shown by the first term on the righthand side, while an increasein the larger region’s share of firms (s1 ) improves knowledge spillovers from production to
innovation, from (13) the direction of the adjustment in s1 after an increase in s depends onwhether the home market effect or the import competition effect dominates. The second termshows the direct improvement in domestic knowledge spillovers resulting from a rise in s.
The third term describes how a fall in the imported share of varieties (s∗ ) leads to a deterioration in international knowledge spillovers. The overall balance of these effects depends
on regional and international knowledge diffusion, regional and international trade costs, andthe size of the productivity differential. In contrast, an increase in the productivity differential
itself always leads to stronger knowledge spillovers.
Finally, with a common level of operating profit for all domestic firms, regardless of thelocation of production, the equilibrium scale of production becomeslX (ω, s) =α(σ − 1)(1 + ϕ)Ls∗σ((1 + ϕ)s + 2ϕ∗ s∗ ω γ(σ−1) )N ∗(16)

for all firms, where we have used regional production shares (13) with (10). An increasein the domestic share of firms (s) reduces the scale of production as the industry becomes
more competitive. Similarly, an increase in the productivity differential leads to a smaller
production scale as the price of imports falls relative to the price of domestic products.

13

3 Long-run Equilibrium
This section characterizes the long-run equilibrium and derives the necessary conditions forsaddlepath stability. We reduce the model to two implicit expressions for the determinationof the steady-state productivity differential (ω) and the domestic share of firms (s). The longrun equilibrium features constant labor allocations across sectors in each region. As discussed
above, under our small country assumption, the number of international firms (N ∗ ) and theinternational rate of productivity growth (g ∗ ≡ θ̇∗ /θ∗ ) are exogenous.
Beginning with the dynamics for the productivity differential (ω), we substitute the freemarket entry and exit condition (12) into the time derivative of ω ≡ θ∗ /θ to obtain thefollowing differential equation:klXω̇= g ∗ − g = g ∗ − klR = g ∗ −+ klF .
ω
σ−1

(17)

Naturally, a constant productivity differential requires that the domestic rate of productivity
growth (g ≡ θ̇/θ) match the international productivity growth rate (g ∗ ). Thus, in long-run
equilibrium we have lX = (σ − 1)(lF + g ∗ /k), capturing a negative relationship between thestrength of knowledge spillovers and the firm scale of production.
Turning next to the domestic share of product variety (s), we combine (8) and (14) torewrite per-period profit as follows:Π=(1 − γ(σ − 1))lXρ ∂k ω̇+ +− lF = 0,σ−1k ∂ω k 2(18)

where we have used the time derivative of (15). With free market entry and exit, the share ofdomestic firms adjusts immediately (ṡ = 0) to drive per-period profit and firm value to zero,
given both the current level and the evolution of the productivity differential. This adjustment
process requires that firm profit respond correctly to market entry and exit (∂Π/∂s < 0), as
discussed in Section 2.3. To ensure that per-period profit (Π) is an increasing function of

14

Figure 1: Market Entry and the productivity differentials1k(ω, s)lX (ω, s)ω0This stylized figure can be produced using the following parameter values: α = 0.75, σ = 5, ρ = 0.03,
γ = 0.15, lF = 0.15, δ = 0.5, δ ∗ = 0.35, ϕ = 0.15, ϕ∗ = 0.1, g ∗ = 0.03, L1 = 300, L2 = 200, andN ∗ = 500. This parameter set yields ω = 0.75, s = 0.36, s1 /s = 0.67, (ds/dω)|lX −(ds/dω)|k = 0.57,
and dΠ/ds =−0.12.

operating profit on sales (π = lX /(σ − 1)), we focus on the case for which 1 > γ(σ − 1).
The first implicit expression for the steady-state determination of ω and s is obtained by
setting ω̇ = 0 in (17) and (18), solving for k, and then equating the result with (15):k(s, ω) = k ≡ρ + (1 − γ(σ − 1))g ∗.
γ(σ − 1)lF

(19)

This expression is illustrated by the k-curve in Figure 1, and indicates the combinations of ωand s for which knowledge spillovers (k) satisfy the conditions for firm-level optimal investment in process innovation and free market entry and exit. An increase in the productivity
differential improves knowledge spillovers, necessitating an adjustment in the domestic levelof market entry. Given the opposing effects of changes in s on knowledge spillovers, as described by (15), however, market entry may rise or fall depending on regional trade costs andthe regional degree of knowledge diffusion. Figure 1 depicts the case for which an increasein the productivity differential causes domestic firms to exit the market. Pairs of ω and s
to the left (right) of the k-curve describe levels of knowledge spillovers below (above) the
steady-state level.
The second implicit expression is obtained by once again setting ω̇ = 0 in (17) and (18),15but now solving for lX , and combining the results with (16) to derive the firm-level scaleof production as a function of the ω and s combinations that are consistent with optimal
investment in process innovation and free market entry and exit:lX (s, ω) = l X ≡(σ − 1)(ρ + g ∗ )lF.
ρ + (1 − γ(σ − 1))g ∗(20)

This condition is depicted by the lX -curve in Figure 1. An increase in the productivity differential intensifies import competition, forcing a share of domestic firms to exit the market
and returning the firm-level scale of production to its long-run level. The firm-level scale ofproduction rises above (falls below) its steady-state level for combinations of ω and s that lie
to the left (the right) of the lX -curve.
The long-run equilibrium is found at the intersection of the k-curve (19) and the lX curve (20). Importantly, the long-run strength of knowledge spillovers and firm-level scale ofproduction are determined independently of trade costs and knowledge diffusion, depending
solely on the exogenous rate of productivity growth, firm-level fixed costs, and demand parameters. Together (17) and (18) describe a dynamic system for ω and s with one differential
equation and one side condition. In the Appendix A we evaluate the local dynamics aroundthe steady state using a Taylor expansion of (17), and obtain the following proposition outlining the necessary conditions for the saddlepath stability of long-run equilibrium describedby the k-curve and the lX -curve:
Proposition 1 A long-run equilibrium with constant values for the productivity differential(ω̇ = 0) and the domestic level of market entry (ṡ = 0) is saddlepath stable for
ds
ds
=
−
dω lX dω k(1 + ϕ)δ ∗ ω + γ(σ − 1)(2k − 1 − δ)ϕ∗ ω γ(σ−1)(1 + ϕ)s + 2ϕ∗ s∗ ω γ(σ−1)

¯
lX
> 0,
ω

under ∂Π/∂s < 0.
Proof: See Appendix A.
Consider the effects of a small increase in the productivity differential (ω), starting from16

the long-run equilibrium. The balance of two mechanisms determine whether the productivity differential will return to its steady-state level: a knowledge spillover effect (15) andan innovation scale effect (16). First, the direct effects of a rise in ω are an improvementin knowledge spillovers (∂k/∂ω > 0) and a reduction in the firm-level scale of process innovation (∂lR /∂ω = 1/(σ − 1)∂lX /∂ω < 0). As a result, per-period profit falls under ourassumption that per-period profit is increasing in operating profit (1 > γ(σ − 1)). Next, theindirect effect of the rise in ω is the market exit of a share of domestic firms in response to
the fall in profit (∂Π/∂s < 0). The decrease in s allows the remaining firms to increase thescale of innovation (∂lR /∂s = 1/(σ −1)∂lX /∂s < 0), while causing knowledge spillovers toeither improve or deteriorate, depending on whether the home market effect, the import competition effect, or the direct knowledge spillover effect dominates in (15). These mechanisms
lead to saddlepath stability if the slope of the lX -curve is greater than the slope of the k-curve,(ds/dω)|lX > (ds/dω)|k , with an increase in ω temporarily raising the domestic productivity
growth rate above the international growth rate (g > g ∗ ), causing the productivity differential
to return to its steady state level. The conditions outlined in Proposition 1 ensure this slope
ranking, as depicted in Figure 1.

4 Economic Integration
This section investigates the implications of improved economic integration at the national
and international levels. Beginning with the effects of improvements in regional and international knowledge diffusion, we have the following proposition.
Proposition 2 An improvement in either the regional (δ) or the international (δ ∗ ) degree ofknowledge diffusion induces domestic market entry (s), lowers the productivity differential(ω), and reduces industry concentration (s1 /s).
Proof: See Appendix B.
Improvements in regional and international knowledge diffusion directly increase the17strength of knowledge spillovers, thereby improving labor productivity in process innovation, as described by a leftward shift in the k-curve in Figure 1. Accordingly, the per-periodinnovation cost falls and profit rises, inducing new firms to enter the market and increasing
the domestic share of firms (s). Improvements in knowledge diffusion temporarily acceleratedomestic productivity growth, causing the productivity differential (ω) to fall until firm-level
employment and knowledge spillovers return to their steady-state levels, and domestic productivity growth once again matches the international rate. The rise in market entry and thefall in the productivity differential reduce the strength of the import competition effect in(13), leading to a lower level of industry concentration in the larger region (s1 /s).
Turning next to increased trade integration both domestically and with the rest of theworld, we obtain the following proposition:
Proposition 3 A reduction in regional trade costs (a rise in ϕ) induces market entry (s),lowers the productivity differential (ω), and raises industry concentration (s1 /s). Reducing
international trade costs (a rise in ϕ∗ ) lowers s, but decreases ω when k̄ < (1 + δ)/2 andincreases ω when k̄ > (1 + δ)/2. A rise in ϕ∗ increases s1 /s.
Proof: See Appendix C.
A decrease in regional trade costs raises the firm-level scale of production (16), shifting
the lX -curve to the right in Figure 1. Similarly, for a given productivity differential (ω) andlevel of market entry (s), the home market effect allows the larger region to increase its shareof domestic production, improving knowledge spillovers (15) and shifting the k-curve to theleft. As such, greater firm-level investment in process innovation combines with greater laborproductivity to temporarily accelerate domestic productivity growth above the international
productivity growth rate. The productivity differential falls as the level of market entry rises
over the transition to a new steady state. Adjustments in the regional pattern of industry
location (s1 /s) depend on the balance of the home market effect, which encourages increased
industry concentration toward the larger region, and the import competition effect, throughwhich a smaller productivity differential and a larger level of market entry tend to disperse18

industry across regions. We find that the home market effect always dominates, however,confirming that a reduction in regional trade costs leads to greater industry concentration.
A decrease in international trade costs reduces the firm-level scale of production (16),as the relative price of imported products falls. The negative effect of increased imported
competition is illustrated by a leftward shift in the lX -curve in Figure 1. At the same time,
however, the import competition effect in (13) leads to greater industry concentration, raising
the strength of knowledge spillovers (15) and shifting the k-curve to the left. As a result,profit (18) deteriorates and domestic firms exit the market. The direction of the adjustment in
the productivity differential then depends on the regional degree of knowledge diffusion (δ)
and the steady-state strength of knowledge spillovers (k̄): the positive knowledge spillover
effect dominates and ω increases for k < (1 + δ)/2, but the negative production scale effectdominates and ω decreases for k > (1 + δ)/2. Finally, the fall in international trade costs
and the lower level of market entry combine to strengthen the import competition effect andincrease the level of industry concentration in the larger region (s1 /s), regardless of whether
the productivity differential rises or falls.

5 Import Competition
In this section, we consider the effects of increased import competition arising from either agreater variety of imports or a faster international rate of productivity growth. Looking firstat an increase in the number of imported products, we derive the following result:
Proposition 4 An increase in the number of imported products (N ∗ ) lowers market entry (s),increases the productivity differential (ω), and raises industry concentration (s1 /s).
Proof: See Appendix D.
Increased import competition through a rise in N ∗ leads to a lower production scale (16) for
domestic firms, shifting the lX -curve to the left in Figure 1. Consequently, firms exit as profit
falls, reducing the level of market entry (s). With a smaller firm size, employment in process
19innovation declines, and the domestic rate of productivity growth temporarily falls below
the international rate of productivity growth over the transition to a new steady state. Thus,
greater import competition expands the technology gap. Together the fall in s and the risein ω strengthen the import competition effect in (13), generating a higher level of industry
concentration (s1 /s) in the larger region.
Lastly, we study the effects of an acceleration in the international rate of productivity
growth. The results are summarized in the following proposition.
Proposition 5 An increase in the international rate of productivity growth (g ∗) lowers market
entry (s), increases the productivity differential (ω), and raises industry concentration (s1 /s).
Proof: See Appendix E.
A rise in the international productivity growth rate (g ∗ ) expands the firm-level scale ofproduction (16) causing a leftward shift in the lX -curve in Figure 1. In addition, knowledge
spillovers (15) strengthen and the k-curve shifts to the right. The domestic rate of productivity growth accelerates to match the international rate of productivity growth as the economy
transitions to a new steady state with a larger productivity differential (ω). Profit falls, reducing the level of market entry (s). Together the fall in s and the rise in ω strengthen the importcompetition effect in (13), increasing the concentration of industry in the larger region (s1 /s).

6 Conclusion
In this paper we introduce a model of import competition and industry location for a small
open economy with two distinct regions that are symmetric with the exception of market
size. In a monopolistically competitive industry, firms compete with imported products fromthe rest of the world as they produce goods for supply to domestic households. Productivity
growth converges to the international rate through firm-level investment in process innovation. With firms free to locate production and innovation independently in the regions withthe lowest respective costs, the concentration of industry is linked with firm-level innovation
20

through an import competition effect that is increasing in the market share of imported goods
and the productivity differential of domestic firms with the rest of the world.
We use the framework to study the effects of improved regional and international economic integration. First, improvements in regional and international knowledge diffusion
lead to a greater domestic level of market entry, a reduced productivity differential between
domestic firms and the rest of the world, and a lower level of industry concentration acrossregions. Second, decreases in regional and international trade costs expand the domestic
level of market entry. While lower regional trade costs reduce the productivity differential,
however, lower trade costs for imported goods may increase or decrease the productivity differential, depending on the strength of regional knowledge diffusion. Decreases in regional
and international trade costs both increase the level of industry concentration.
Turning to the effects of import competition, we find that an increase in the number ofimported products, or an acceleration in the international rate of productivity growth, causes
a contraction in the domestic level of market entry, an increase in the productivity differential,
and an expansion in the concentration of industry in the larger region.
Appendix A (Proposition 1): With the zero profit condition determining the level of market entry associated with a given productivity differential at each moment in time, the total
derivative of (18) evaluated around ω̇ = 0 dictates how s adjusts in response to changes in ω:
ds/dω = −(∂Π/∂ω)/(∂Π/∂s), with
∂Π
=
∂ω
∂Π
=
∂s





1 − γ(σ − 1)
ω
∂k
−
σ−1
(σ − 1)k ∂ω
1 − γ(σ − 1)
ω
∂k
−
σ−1
(σ − 1)k ∂ω




∂lX
−
∂ω
∂lX
−
∂s





ρ

ωlR ∂k
2 +
2
k
k ∂ω
ρ

ωlR ∂k
2 +
2
k
k ∂ω




∂k
,
∂ω
∂k
,
∂s

where firm-level employment is constant in the steady state according to (20), lR = γ(σ −211)g ∗lF /(ρ + (1 − γ(σ − 1))g ∗ ), and from (15) and (16), we have
(1 − δ)(L1 − L2 )ϕ∗ s∗ γ(σ − 1)ω γ(σ−1)−1∂k=+ δ ∗ s∗ > 0,∂ω
(1 − ϕ)L∂k(1 − δ)(L1 − ϕL2 ) + δ(1 − ϕ)(δ − k)L
=
R 0,
∂s
(1 − ϕ)Ls∗2γ(σ − 1)ϕ∗ s∗ ω γ(σ−1)−1 lX
∂lX
=−
< 0,
∂ω
(1 + ϕ)s + 2ϕ∗ s∗ ω γ(σ−1)∂lX(1 + ϕ)lX
=− ∗
< 0.
∂s
s ((1 + ϕ)s + 2ϕ∗ s∗ ω γ(σ−1) )Setting the domestic share of production (s) as a control variable and the productivity
differential (ω) as a state variable, the long-run equilibrium is saddlepath stable when thelinearized system has one positive and one negative eigenvalue. Therefore, a first-order Taylor expansion of (17) around the steady state described by (19) and (20) yields a saddlepath
stable steady state for dω̇/dω = ∂ ω̇/∂ω + (∂ ω̇/∂s)(ds/dω) = γlF Ω/(∂Π/∂s) < 0, where
ds
ds
=
−
Ω=
dω lX dω k(1 + ϕ)δ ∗ ω + γ(σ − 1)(2k̄ − 1 − δ)ϕ∗ ω γ(σ−1)(1 + ϕ)s + 2ϕ∗ s∗ ω γ(σ−1)

¯
lX
> 0.
ω

Thus, (1 + ϕ)δ ∗ ω + γ(σ − 1)(2k̄ − 1 − δ)ϕ∗ ω γ(σ−1) > 0 becomes a necessary condition forsaddlepath stability, with ∂Π/∂s < 0 required to ensure that firm value responds correctly tomarket entry and exit, as described in Section 2.3.
Appendix B (Proposition 2): Using (15), (16), (19) and (20), we obtain∂k ∂lX 1
dω
=
< 0,
dδ
∂δ ∂s Ωdω∂k ∂lX 1
=
< 0,
dδ ∗
∂δ ∗ ∂s Ωds∂k ∂lX 1=−> 0,dδ∂δ ∂ω Ω
ds∂k ∂lX 1
=
−
> 0,
dδ ∗
∂δ ∗ ∂ω Ω

where we have used ∂k/∂δ = s2 > 0 and ∂k/∂δ ∗ = s∗ ω > 0. Substituting these results into

22

the total derivative of (13) provides the following:(L1 − L2 )ϕ∗ s∗ ω γ(σ−1) γ(σ − 1) dω
1 dsd(s1 /s)
< 0,
=
− ∗
dδ
L(1 − ϕ)s
ω
dδ
ss dδ


d(s1 /s)(L1 − L2 )ϕ∗ s∗ ω γ(σ−1) γ(σ − 1) dω
1 ds
< 0.
=
−
dδ ∗
L(1 − ϕ)sωdδ ∗ ss∗ dδ ∗Appendix C (Proposition 3): Using (15), (16), (19) and (20), we obtain the following:dω∂lX ∂k 1∂k ∂lX
=
−
< 0,
dϕ
∂ϕ ∂s
∂ϕ ∂s Ωdω
(1 + δ − 2k̄)ω γ(σ−1) ¯lX=,
dϕ∗((1 + ϕ)s + 2ϕ∗ s∗ ω γ(σ−1) )Ωds∂k ∂lX 1∂lX ∂k=−> 0,dϕ∂ϕ ∂ω ∂ϕ ∂ω Ωds2δ ∗ s∗2 ω γ(σ−1) ¯lX
=
−
< 0,
dϕ∗
((1 + ϕ)s + 2ϕ∗ s∗ ω γ(σ−1) )Ω

with Ω > 0 from Proposition 1 and ∂k/∂s > 0 by assumption. In addition,(1 − δ)(L1 − L2 )(s + ϕ∗ s∗ ω γ(σ−1) )
∂k
=
> 0,
∂ϕ
(1 − ϕ)2 L
2ϕ∗ s∗ ω γ(σ−1) lX
∂lX
=
> 0,
∂ϕ
(1 + ϕ)((1 + ϕ)s + 2ϕ∗ s∗ ω γ(σ−1) )(1 − δ)(L1 − L2 )s∗ ω γ(σ−1)
∂k
=
> 0,
∂ϕ∗
(1 − ϕ)L∂lX
2s∗ ω γ(σ−1) lX
=
−
< 0.
∂ϕ∗
(1 + ϕ)s + 2ϕ∗ s∗ ω γ(σ−1)
Then, from (7) we have s1 /s = k/((1 − δ)s) − δ/(1 − δ) − δ ∗ s∗ ω/((1 − δ)s), ands∗ ¯lXd(s1 /s)=×dϕ(1 − δ)(1 − ϕ)2 sLΩ(1 − δ)(L1 − L2 ) (1 + ϕ)s + 2γ(σ − 1)ϕ∗ s∗ ω γ(σ−1)+(1 + ϕ)ss∗(1 − ϕ)ϕ∗ s∗ ω γ(σ−1) (1 − δ)(L1 − L2 )s + 2(δ − k̄)sL + 2δ(1 − ϕ)s∗ L> 0,(1 + ϕ)((1 + ϕ)s + 2ϕ∗ s∗ ω γ(σ−1) )sd(s1 /s)(s1 − s2 )δ ∗ s∗ ω γ(σ−1) ¯lX
=
> 0.
dϕ∗
((1 + ϕ)s + 2ϕ∗ s∗ ω γ(σ−1) )s2 Ω

23

Appendix D (Proposition 4): From (15), (16), (19) and (20), we have∂lX ∂k 1
dω
=−
> 0,
∗
dN
∂N ∗ ∂s Ω

ds∂lX ∂k 1
=
< 0,
∗
dN
∂N ∗ ∂ω Ω

where ∂lX /∂N ∗ = −lX /N ∗ < 0. Thus, (13) yields the following:d(s1 /s)(L1 − L2 )ϕ∗ s∗ ω γ(σ−1)=
dN ∗L(1 − ϕ)sγ(σ − 1) dω
1 ds
− ∗
∗
ω
dN
ss dN ∗

> 0.

Appendix E (Proposition 5): Using (15), (16), (19) and (20), we obtain
dω
=
dg ∗



∂k ∂lX∂lX ∂k− ∗∗∂g ∂s ∂g ∂s



1
,
Ω

ds
=
dg ∗



∂lX ∂k∂k ∂lX
− ∗
∗
∂g ∂ω
∂g ∂ω



1
,
Ω

where
∂k
(1 − γ(σ − 1))k
=
> 0,
∗
∂g
ρ + (1 − γ(σ − 1))g ∗∂lXγ(σ − 1)ρl X
=
> 0.
∗
∗
∂g
(ρ + g )(ρ + (1 − γ(σ − 1))g ∗)Combining these results with (13) yields(L1 − L2 )ϕ∗ γ(σ − 1)ω γ(σ−1)−1 ¯lXd(s1 /s)=dg ∗L(1 − ϕ)s2 (ρ + g ∗)Ω(1 − γ(σ − 1))ρδ ∗ s∗ ω
k̄ +
> 0.
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