Industry Concentration, Knowledge Diffusion, andEconomic Growth Without Scale Effects
Colin Davis
Doshisha University∗

Ken-ichi Hashimoto
Kobe University†

February, 2014
Abstract
This paper develops a two region model of trade to study the relationshipbetween geographic patterns of industry and economic growth without scale effects. With transport costs, imperfect knowledge diffusion, and perfect capital
mobility, firms locate production, process innovation, and product development
independently in their lowest cost regions, leading to the partial concentrationof production and the full agglomeration of innovation in the region with thelargest market. A rise in industry concentration increases knowledge spillovers
from production to innovation, resulting in a fall or a rise in the level of market
entry depending on whether productivity increases more for process innovationor for product development. As a result, the rate of economic growth may riseor fall, depending on the effects of industry concentration on market entry.
Key Words: Industry Concentration, Industry Share, Knowledge Diffusion, Productivity Growth, Scale EffectJEL Classifications: F43; O30; O40; R12∗

The Institute for 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: We are grateful for helpful comments from T. Iwasako, T. Nakamura, andparticipants of a seminar held at Kobe University and the spring meeting of the Japanese Societyof International Economics. We acknowledge financial support from JSPS though a Grant-in-Aid
for Scientific Research (C) and a Grant-in-Aid for Young Scientists (B). We are responsible for all
remaining errors.

1

Introduction

In recent years there has been considerable interest in understanding the implicationsof the geographic distribution of industrial activity for patterns of economic growth atthe local, regional, and international levels. Indeed, a general consensus that industry concentration promotes economic growth appears to have developed within thetheoretical literature of the new economic geography (Baldwin and Martin 2004).
The empirical evidence is mixed, however. For example, Bosker (2007) and Gardiner et al. (2011) report a negative relationship between a number of measuresof industry concentration and GDP growth for several levels of agglomeration using
empirical data. In addition, Abdel-Rahman et al. (2006) find that the growth ofurban areas affects economic growth negatively in developing countries. In contrast,
investigating cross-country data, Brülhart and Sbergami (2009) conclude that the relationship between industry concentration and GDP growth depends on a country’slevel of economic development, while Braunerhjelm and Borgman (2004) find thatindustry concentration has a positive effect on labor productivity growth in Sweden.
One possible source of the discrepancy between these mixed empirical results andthe general consensus of the new economic geography literature is the close connection
between a scale effect, whereby growth is positively linked with the size of the labor
force, and the positive relationship between industry concentration and growth derived
by standard models. In this paper we re-examine the relationship between industry
concentration and economic growth using a novel approach that shifts the focus fromaggregate research and development (R&D) activity to innovation at the level ofindividual product lines thereby sterilizing the scale effect.
More specifically, building on Peretto and Connolly (2007), we develop a two region model of trade and endogenous growth that focuses on the production, process
innovation, and product development of manufacturing firms. In a world characterized by perfect capital mobility, transport costs, and imperfect knowledge diffusion,
1

firms are free to locate these activities independently across regions with the objective of raising profits on the margin (Martin and Rogers 1995; Martin and Ottaviano
1999, 2001). As a result, geographic patterns of production and R&D are determined
endogenously, with a concentration of production and the full agglomeration of process innovation and product development in the larger market of the two regions, asmeasured by household expenditure.
A key feature of the endogenous market structure and endogenous growth framework (Smulders and van de Klundert 1995; Peretto 1996, Aghion and Howitt 1998;
Dinopoulus and Thompson 1998; Peretto and Connolly 2007; Etro 2009) adopted inthis paper is a fixed operating cost incurred on individual product lines that drives
the rate of market entry down to the rate of growth in market size, or the population
growth rate. As such, the level of entry relative to market size is constant, but closely
related to the distribution of economic activity. In particular, a rise in industry concentration lowers R&D costs, through greater knowledge spillovers from production
to innovation in the larger region, with two effects. The first is a product development
effect, whereby a fall in the cost of creating a new product design raises the levelof market entry. The second is a process innovation effect, whereby a fall in laborcosts raises firm-level employment in process innovation, decreasing operating profits
and lowering the level of market entry. Overall, the relationship between the level ofmarket entry and industry concentration depends on which effect dominates.
The distribution of economic activity also has important implications for the rateof economic growth. While the constant rate of population growth determines thepace of variety expansion, the rate of productivity growth is determined by firm scale,and is therefore intrinsically linked with the level of market entry. Accordingly, an
increase in industry concentration has a direct positive effect on productivity growth,through a rise in the labor productivity of process innovation, but may have a negative
or a positive indirect effect on productivity growth, depending on whether the level of2market entry rises or falls. In general, we find a positive relationship between industry
concentration and economic growth when the process innovation effect dominates, anda convex relationship when the product development effect dominates. Our framework
therefore produces mixed results with respect to the effects of industry concentration
on economic growth.
This paper contributes to the theoretical literature investigating the relationshipbetween geography and economic growth using key elements of the variety-expansion
model of endogenous growth (Grossman and Helpman 1991).1 Within this literature,
our paper is most closely related to Martin and Ottaviano (1999, 2001) in that theyalso assume “footloose” production and product development–firms locate these activities independently in the lowest cost regions.2 These studies find that agglomeration
economies promote economic growth by raising the productivity of labor in R&D whenknowledge spillovers are local in scope.3 A key aspect of the variety-expansion models
adopted in this literature, however, is the existence of a scale, which is problematicin that the empirical evidence does not support a significant relationship between
economic growth and population size (Jones 1995a; Dinopoulos and Thompson 1999;
Barro and Sala-i-Martin 2004; Laincz and Peretto 2006).
In a recent paper, Mittini and Parello (2011) extend the model of Martin andOttaviano (1999) to correct for the scale effect by introducing population growth anddiminishing returns to knowledge according to Jones (1995b). With this modification, the model produces semi-endogenous growth, and the rate of variety expansion
determined proportionately with the population growth rate. Accordingly, while the
distribution of industry is important for the level of product variety, it has no effect1

See Baldwin and Martin (2004) and Baldwin et al. (2004) for surveys of this literature.
See Duranton and Puga (2001) for evidence that firms undertake production and innovation
activities at independent locations.
3
Two papers extending the standard model to allow for a negative relationship between industry concentration and the rate of innovation are Acceturo (2010), which adds congestion costs to
R&D activity, and Cerina and Mureddu (2012), which considers endogenous expenditure shares for
manufacturing goods. However, both papers feature scale effects.
23

on the long-run rate of economic growth. Moreover, semi-endogenous growth is generally not supported by the empirical evidence (Barro and Sala-i-Martin 2004; Laincz
and Peretto 2006; Ha and Howitt 2007). In contrast, the model developed in thispaper allows for the endogenous determination of both the geographic distribution of
economic activity and the rate of economic growth, with both semi-endogenous andfully endogenous components, while removing the scale effect. Further, the endogenous market structure and endogenous growth framework is now supported by a large
body of empirical evidence (Zachariadis 2003, 2004; Laincz and Peretto 2006; Ha andHowitt 2007; Madsen 2008, 2010; Madsen et al. 2010a; Madsen et al. 2010b).
The remainder of the paper is organized as follows. Section 2 introduces a two region model of trade, variety expansion, and productivity growth without scale effects,
and examines the equilibrium location patterns of production, process innovation, andproduct development. Then, in Section 3 we study the effects of changes in industry
concentration on the level of product variety and the rate of productivity growth,and investigate the implications of greater regional integration through a decrease intransport costs or an increase in the degree of knowledge diffusion between regions.
We also briefly discuss the effects of greater integration on regional welfare levels.
Section 4 provides brief concluding remarks.

2

The ModelThis section introduces a two region, North (N) and South (S), model of trade, productivity growth, and variety expansion. There are four type of economic activity:
traditional production (Y ), manufacturing (X), process innovation (I), and productdevelopment (R). The traditional sector produces a homogeneous good for sale ina perfectly competitive market characterized by free trade. The manufacturing sector, on the other hand, consists of monopolistically competitive firms that produce
differentiated product varieties for sale in a market that features transport costs on
4

shipments between regions. Productivity growth arises as a result of process innovation undertaken by incumbent manufacturing firms with the objective of lowering
production costs. The product development sector creates new product designs forfirms entering the manufacturing sector. Production, process innovation, and productdevelopment are all footloose in nature, and can therefore be located independently
across regions, regardless of the home region of the associated firm.
Labor, the sole factor of production, is supplied inelastically by households. We
assume that regional labor endowments are equal and grow at the same rate λ, andaccordingly regional labor endowments are L(t) = LN (t) = LS (t) = eλt at time t,with initial labor endowments normalized to one and subscripts N and S denoting
variables associated with the North and South. There is perfect labor mobility acrosssectors, but no migration between regions. We allow, however, for differences in asset
wealth and assume greater initial asset wealth for the North, BN > BS .
2.1

Households

The demand side of the economy consists of dynastic households that maximize utility
over an infinite time horizon. We adopt a standard formulation for this intertemporalmaximization problem (Barro and Sala-i-Martin 2004), and define the lifetime utilityof a household in region i as follows:

Ui =

Z ∞

e−(ρ−λ)t (α ln CXi (t) + (1 − α) ln CY i (t)) dt,

i ∈ {N, S},(1)0

where CXi (t) and CY i (t) denote household consumption of a manufacturing composite
and a traditional good, ρ is the subjective discount rate (ρ > λ), and α ∈ (0, 1) is theconstant share of expenditure allocated to manufacturing goods.
Each household chooses an expenditure-saving path with the objective of maxi-

5mizing (1) subject to the following flow budget constraint for time t:Ḃi (t) = (ri (t) − λ)Bi (t) + wi (t) − Ei (t),(2)

where Ei (t) is household expenditure, ri (t) is the interest rate, wi (t) are the wage
rate, Bi (t) is asset wealth, and a dot denotes time differentiation. The solution to theintertemporal optimization problem is the Euler equation:Ėi (t)= ri (t) − ρ.
Ei (t)(3)

With perfect capital mobility, interest rates equalize across regions (rN = rS = r)leading to a common motion for household expenditure: ĖN /EN = ĖS /ES = r − ρ.4
We suppress time notation where possible for the remainder of the paper.
With equal subjective discount rates and population growth rates, the integrated
financial market provides equivalent investment opportunities for residents in bothregions, ḂN /BN = ḂS /BS , ensuring constant regional shares of asset wealth. As such,the greater initial asset wealth of Northern households results in higher household
expenditure in the North. Specifically, summing the flow budget constraints (2) to
describe average household expenditure E ≡ (EN + ES )/2 as a function of averageasset household wealth B ≡ (BN + BS )/2, and using the result to substitute the net
return to investment (r − λ − Ḃ/B) out of the regional flow budget constraints gives

Ei = wi + bi (2E − wi − wj ),i, j ∈ {N, S}; i 6= j,(4)where bi ≡ Bi /(BN + BS ) is region i’s share of asset wealth. In the next section,
we show that wage rates are equalized through free trade in the traditional good(wN = wS ). Therefore, the greater asset wealth of the North (bN > 1/2) leads to4

We assume that initially there is no borrowing or lending between regions, as Northern andSouthern households have the same rate of time preference.

6

higher expenditure for Northern households (EN > ES ).
At each moment in time, households allocate constant shares of expenditure to
traditional goods and the manufacturing composite:

PXi CXi = αEi ,PY i CY i = (1 − α)Ei ,(5)where PY i is the traditional good price, and PXi is the price index associated withthe manufacturing composite. In particular, the manufacturing composite and priceindex take the following forms:

CXi =

Z N
0

ci (ω)

σ−1
σ

dω

σ
 σ−1

,

PXi =

Z N
0

pi (ω)

1−σ

dω

1
 1−σ

,

(6)

where pi (ω) and ci (ω) are the price and household demand in region i for productvariety ω, N = NN + NS is the total mass of product varieties available, and σ > 1 isthe elasticity of substitution between any two varieties.
Regarding the composite price index (6) as the household’s unit expenditure function for manufacturing goods, household demand for a representative variety in region
i can be obtained using Shephard’s Lemma:σ−1ci (ω) = αpi (ω)−σ PXi
Ei .

2.2

(7)

Traditional Production

Traditional firms employ a constant returns to scale technology, with one unit of labor
required for each unit of output, and market competition ensures that the price of a
traditional good equals the wage rate. As there are no transport costs associated withtrade between regions, prices and wage rates are the same for the North and the South.
The traditional good is set as the model numeraire: PY N = PY S = wN = wS = 1.5
5

Wage rate equalization requires that both regions produce traditional goods in equilibrium.

7

2.3

Manufacturing

Manufacturing firms produce horizontally differentiated product varieties and compete
according to monopolistic competition (Dixit and Stiglitz 1977). In addition to thecosts of market entry associated with product development, firms face a fixed perperiod labor cost (f ) related to product marketing and production management, andincur iceberg costs on the transport of goods between regions, whereby τ units must
be shipped for every unit sold in the export market (Samuelson 1954).
A firm ω, with production located in region i, employs labor lXi (ω) with thefollowing production technology:xi (ω) = θi (ω)γ lXi (ω),(8)

where xi (ω) is output, θi (ω) is a firm-specific productivity coefficient, and γ ∈ (0, 1)
is the productivity elasticity of output (Peretto and Connolly 2007). While each firm
employs a production technique that is unique to its product line, the productivity levels (θ) associated with production techniques are symmetric across all firms
regardless of the location of production. As such, we now suppress the firm index ω.
Firms maximize profit on sales using the constant markup over unit cost pricing rule associated with monopolistic competition, and with symmetric productivity
levels, similar pricing rules are adopted by all firms, regardless of where production is located. Specifically, the price of goods sold in their region of production
is p = σ/((σ − 1)θγ ), and the price of goods sold outside their region of production is p∗ = τ p, with an asterisk indicating variables associated with units that are
consumed outside the region where they were manufactured. Equating household demand (7) with firm supply (8), and substituting the result with the pricing rules into8

πi = pxi − lXi , optimal profit on sales for a firm with production located in region i isp1−σ αL
πi =
σ

ϕEj
Ei
1−σ +
1−σ
PXi
PXj

!

,

i, j ∈ {N, S}; i 6= j,(9)where we have used xi = (ci + τ c∗j )L and p∗ = τ p, and ϕ ≡ τ 1−σ describes the freenessof trade: ϕ = 0 implies prohibitive trade costs and ϕ = 1 indicates free trade.
2.4

Process Innovation

Incumbent manufacturing firms invest in process innovation with the aim of raising
firm value through productivity improvements that lower production costs and raise
profit on sales (9). A firm with its R&D department located in region i employs laborlIi in process innovation, and firm-level productivity evolves according toθ̇ = ki θlIi ,(10)where ki θ captures knowledge spillovers from production to R&D. Technical knowledge
on production techniques accumulates within the firm as a by-product of process
innovation. We adopt the level of productivity as a proxy for the stock of technicalknowledge, and increases in θ therefore raise the labor productivity of future process
innovation, potentially generating perpetual growth in long-run equilibrium.
Following the process innovation framework developed by Smulders and van deKlundert (1995) and Peretto (1996), we model knowledge spillovers into process innovation as a function of the weighted average productivity of technical knowledge
observable by the R&D department of the firm:ki θ = (si + δ(1 − si ))θ,(11)

where si ≡ Ni /N is the share of firms with production located in region i. Although9productivity is symmetric across firms, each firm’s production technology is uniqueand comprises technical knowledge that includes both codifiable aspects which areconveyed easily across large distances and tacit aspects which are only transferredthrough face-to-face communication (Keller 2004). The degree of knowledge diffusionδ ∈ (0, 1) captures the imperfect nature of knowledge spillovers: δ = 0 indicatesthat knowledge spillovers are completely local in scope and δ = 1 indicates perfectknowledge spillovers between regions.6
The total per-period profit of a firm equals operating profit on sales less the costof investment in process innovation and the per-period fixed labor cost:Πi = πh − lIi − f,h ∈ {N, S}.

(12)

A firm invests lIi in process innovation with the objective of maximizing firm value,
R t′
R∞
Vi (t) = t Πi (t′ )e− t r(i)di dt′ , subject to the technological constraint (10). We solve

this optimization problem using the following current value Hamiltonian function:
Hi = Πi + µi kiθlIi , where µi denotes the current shadow value of an improvement in
the technology of the firm. As each firm perceives itself as small relative to the overallmarket, firms ignore the effects of their R&D investments on the price indices andknowledge spillovers, when maximizing firm value.
The solution to the intertemporal profit maximization problem is captured bya static efficiency condition µi = 1/(kiθ) that equates the value of a marginal improvement in technology with the marginal cost of process innovation, and a dynamic
efficiency condition ∂πh /∂θ = rµi − µ̇i that equates the internal rate of return toprocess innovation with the rate of return on a risk free asset.7 Combined, these6

The imperfect nature of knowledge spillovers has been well documented by a number of empiricalstudies, for example, Jaffe et al. (1993), Mancusi (2008), and Coe et al. (2009). Our formulation forimperfect spillovers is adapted from Baldwin and Forslid (2000).
7
The solution to the intertemporal
optimization problem must also satisfy the following tranverRt′′
sality condition limt→∞ e− 0 r(t )dt µi (t)θ(t) = 0.

10conditions yield a no-arbitrage condition for investment in process innovation:r ≥ (σ − 1)γπh ki −k̇i θ̇
− .
ki θ

(13)

This condition binds when process innovation occurs in region i.
2.5

Product Development

Manufacturing firms purchase product designs, and associated production processes,
from the competitive product development sector, and enter the market with thecurrent productivity level of incumbent firms. The overall creation of new productdesigns follows

Ṅ =



kN N
2EL



lRN +



kS N
2EL



lRS ,

(14)

where lRi is regional employment in product development. Setting the stock of knowledge related to product design equal to the number of product varieties that have been
invented to date (N), the creation of a new product designed adds to the existing stockof knowledge, improving the labor productivity of future product development (Romer
1990). Again, however, intertemporal knowledge spillovers are imperfect as they diminish with distance (ki = si + δ(1 − si) < 1). In addition, product development costs
are increasing in the overall market size (2EL = EN L + ES L), as the costs associatedwith product role out are higher for larger markets (Peretto and Connolly 2007; Etro
2009; Dinopoulos and Unel 2011).
Free entry into the product development sector drives the value of a product design
down to its development cost. Therefore, in region i

Vi =

2EL
.
ki N

11

(15)

when labor is employed in product development in region i. Since the value of a
R t′R∞product design equals the value of a manufacturing firm Vi (t) = t Πi (t′ )e− t r(i)di dt′ ,the time derivative of Vi (t) yields a no-arbitrage condition for investment in productdevelopment (Romer 1990; Grossman and Helpman 1991):r≥Πi V̇i
+ ,
Vi
Vi

(16)

which ensures that the rate of return to investment in a product design equals therisk-free interest rate when product development occurs in region i.
2.6

Locations of Production and Innovation

This section characterizes the equilibrium location patterns of production, process
innovation, and product development. Following Martin and Rogers (1995) and Martin and Ottaviano (1999, 2001), manufacturing firms shift their production and R&D
activities independently between regions at zero cost; the locations of production andproduct development need not coincide.
As firms relocate production with the aim of increasing profit on the margin, when
production occurs in both the North and the South, profit on sales equalizes acrossregions, πN = πS . Therefore, with symmetry, combining (6) and (9) with the pricing
rules, we solve for the Northern share of production assN (E; ϕ) =EN − ϕES(1 − ϕ) + 2(bN − ϕbS )(E − 1)=,
(1 − ϕ)(EN + ES )2(1 − ϕ)E

(17)

where we have used (4). Note that with bN > 1/2, we have EN /ES > 1 and sN > 1/2.
The positive relationship between sN and E in (17) describes a home market effect,
whereby a greater share of firms locate production in proximity to the larger Northernmarket in order to minimize transport costs (Krugman 1980). With its greater shareof asset wealth, the North receives a larger share of the investment income generated
12

by total household expenditure, allowing it to maintain a higher level of expenditure.
Substituting (17) and the pricing rules into (9) gives

πN = πS =

α 2EL
.
σ N

(18)

With profit on sales equalized across regions, we can now show that all productdevelopment activity takes place in the North, as its higher share of production (sN >1/2) generates larger knowledge spillovers from production to innovation, kN > kS ,which allows for lower innovation costs than the South. In particular, an examinationof (15) shows that the cost of product designs is lower in the North (VN < VS ),ensuring that no product development occurs in the South, i.e., lRS = 0.
Next, we consider the dynamics associated with household expenditure and theNorthern share of production. Using the flow budget constraint for average household
expenditure E = 1 + B(r − λ − Ḃ/B), the asset market clearing condition B =NVN /(2L) = E/kN , and the Euler condition Ė/E = r − ρ together with (17), we
derive the motion for North’s production share askNṡN =(1 − δ)(E(sN ) − 1)kN
− (ρ − λ) ,E(sN )(19)

where average household expenditure is determined as a function of sN from (17).8
In Appendix A, we evaluate the dynamics of the Northern production share aroundṡN = 0, and find that ∂ ṡN /∂sN > 0. Therefore, production shares and average
household expenditure jump immediately to their steady-state levels, implying from
(3) that r = ρ. Setting (19) equal to zero, the North’s share of production and average
household expenditure are determined through the system described by (17) andsN (E; δ, λ) =8δ(ρ − λ)E−.
(1 − δ)(E − 1) 1 − δ(20)

In particular, rearranging (17) yields E(sN ) = (1 + ϕ)(2bN − 1)/(2(bN − ϕbS − (1 − ϕ)sN )).

13

We use this system to obtain the following results:
Proposition 1 The North’s share of production (sN ) is increasing in the freeness of
trade (ϕ), and decreasing in the degree of knowledge diffusion (δ) and the population
growth rate (λ). Average household expenditure (E) is decreasing in ϕ, δ, and λ.
Proof: See Appendix A.
An increase in the freeness of trade strengthens the home market effect, raising
the share of firms with production located in the North. Average household expenditure falls, however, as increased competition in the larger Northern market leads tolower profit on sales, reducing asset income. An increase in the degree of knowledge
diffusion, on the other hand, decreases the cost of market entry, lowering asset valueand household incomes. As a result, average household expenditure and the Northernshare of production both fall. Similarly, a rise in the population growth rate lowers
average household expenditure through a fall in the net return to asset wealth (r − λ),and decreases the North’s share of production.
Finally, with product development fully concentrated in the North, all firms havethe same market value (15) and earn the same per-period profits (12). Therefore, from(18), if process innovation occurs in both regions, employment must be the same for
all firms, regardless of location (lIN = lIS ), as it is determined proportionately withprofit on sales. We can use (9), (10), and (11), however, to rewrite the no-arbitrage
conditions (13) for Northern and Southern process innovation asρ = kN ((σ − 1)γπ − lIN ),ρ > kS ((σ − 1)γπ − lIN ).

(21)

Again, with kN > kS , the larger share of production located in the North attracts all
investment in process innovation, as the no-arbitrage condition is not satisfied for theSouth (lIS = 0). With process innovation fully concentrated in the North, firm-level

14employment in process innovation islIN =α(σ − 1)γ 2EL
ρ
−
,
σ
N
kN

(22)

where we have substituted (18) into the Northern no-arbitrage condition in (21).

3

Product Variety and Productivity GrowthThis section investigates how the geographic concentration of production affects product variety and productivity growth. We also examine the effects of greater economic
integration arising from a fall in transport costs or a rise in knowledge diffusion, anddiscuss the implications for regional welfare.
3.1

Industry Concentration

We begin with product variety. As a positive rate of population growth leads tocontinuous entry into the manufacturing sector, we study the characteristics of theratio of incumbent firms to market size as a measure of the level of product variety,which we define as the number of firms per unit of expenditure, n ≡ N/(2EL), where
total household expenditure (2EL) describes the overall market size.
Substituting (12), (15), (18), and (22) into (16), and reorganizing, the evolutionof the level of product variety can be obtained as follows:ṅ= (νkN − ρ) − (f kN − ρ)n,n

(23)

where ν ≡ α(1 − (σ − 1)γ)/σ ∈ (0, 1) is the marginal industry profit associated withan increase in market size.9 Although population growth generates continuous market
entry, and rising product variety, growth in the level of product variety is driven to9

From (12), (18), and (22), we have ∂(N Π)/∂(2EL) = α/σ − α(σ − 1)γ/σ = ν.

15

Figure 1: The Level of Product Variety
n

n

1

1/2

sN

1

1/2

(a) ν > fsN(b) ν < fzero by the existence of per-period fixed costs, with a stable equilibrium arising for∂ ṅ/∂n = −(f kN − ρ) < 0, as n is a state variable.
We solve for the steady-state level of product variety by setting (23) equal tozero:10n=νkN − ρ,f kN − ρ(24)

where with f kN > ρ as a necessary stability condition, νkN > ρ is also required fora positive level of market entry, with positive productivity growth. Intuitively, a risein marginal industry profit (ν) increases market entry, while a rise in the per-period
fixed cost (f ) lowers market entry. Moreover, n > 1 if ν > f , and n < 1 if ν < f . We
use (24) with kN = sN + δ(1 − sN ) to examine how changes in sN affect n:
Proposition 2 The level of product variety (n) is decreasing in industry concentration (sN ) for ν > f , and increasing in sN for ν < f .
Proof: Partial derivation of (24) yields∂2n2f∂kN ∂n=−,2∂sN(f kN − ρ) ∂sN ∂sN(ν − f )ρ ∂kN
∂n
=−
,
∂sN
(f kN − ρ)2 ∂sN
10While the level of product variety is constant in equilibrium, product variety increases at therate of population growth (Ṅ /N = λ). The dynamics of N are provided in Appendix B.

16

where ∂kN /∂sN = 1 − δ > 0.
An increase in the Northern share of production affects the level of market entry
through two mechanisms: a product development effect and a process innovation effect.
The product development effect increases the level of market entry as greater industry
concentration raises labor productivity in product development through an increasein knowledge spillovers from production to R&D. The process innovation effect, on
the other hand, decreases the level of market entry, as a rise in the labor productivityof R&D induces firms to employ more labor in process innovation, causing per-period
profits to fall. The balance of these opposing mechanisms is determined by ν and f ,with increased industry concentration lowering the level of product variety for ν > f ,as illustrated in Figure 1a, and raising it for ν < f , as shown in Figure 1b.
The relationship between industry concentration and the level of product variety
has interesting implications for productivity growth. Combining (10), (22), and (24),the long-run rate of productivity growth isg≡θ̇αγ(σ − 1)kN
αγ(σ − 1)(f kN − ρ)kN
=
−ρ=
− ρ.
θ
σn
σ(νkN − ρ)(25)

We assume that n < γα(σ − 1)kN /(σρ), in order to ensure active investment in process
innovation, with a positive rate of productivity growth (Peretto and Connolly 2007),and use (25) to consider the effects of changes in sN on g:
Proposition 3 The rate of productivity growth (g) is increasing in industry concentration (sN ) for ν > f , and convex in sN for ν < f , with a minimum at

sN =

(1 +

p

(f − ν)/f )ρ
δ−.
(1 − δ)ν
1−δ

17Figure 2: Productivity Growth
g

g

1

1/2

sN

1/2

(a) ν > fsN(b) ν < f1

sN

Proof: Partial derivation of (25) gives
∂g
=
∂sN



(g + ρ)(ν − f )ρg+ρ+kN
(νkN − ρ)(f kN − ρ)



∂kN
∂2g
2(g + ρ)(ν − f )ρ2 (∂kN /∂sN )2,=−,∂sN ∂s2N(νkN − ρ)(f kN − ρ)2 kN

where ∂kN /∂sN = 1 − δ > 0.
Once again, a rise in industry concentration increases the level of knowledge
spillovers from production to innovation, with two resulting effects on productivity
growth. The first is the direct effect of improved labor productivity in process innovation, which leads to an increase in the rate of productivity growth. The second is
the effect of adjustments in the level of market entry. Recalling Proposition 2, whenν > f , the level of product variety falls, and the effect of lower market entry matcheswith the direct effect of improved labor productivity in process innovation, causing
the rate of productivity growth to increase, as depicted in Figure 2a. Alternatively,
when ν < f the level of market entry increases. Then, as illustrated in Figure 2b, theoverall impact on productivity growth is negative for low levels of industry concentration (sN < sN ), where the negative effect of greater market entry is stronger thanthe direct positive effect of improved labor productivity. For high levels of industry
concentration (sN > sN ), however, the overall impact is positive, as the positive labor18

productivity effect dominates.
Before completing this section, we consider how changes in the population size
and the population growth rate affect productivity growth. Returning to (25), we seethat productivity growth is not biased by a scale effect since increases in the overalllabor endowment (2L) are fully absorbed by a rise in product variety (N) that leaves
the number of firms per unit of household expenditure (n) unchanged. Productivity
growth is also not directly affected by changes in the population growth rate (λ).
Reviewing Proposition 1, however, an increase in the population growth rate reduces
industry concentration, and thus may raise or lower productivity growth dependingon the relationship between industry concentration and the level of product variety.
Proposition 4 An increase in the population growth rate (λ) raises the level of product variety (n) and decreases the rate of productivity growth (g) for ν > f , and lowers
n but may raise or lower g for ν < f .
Proof: From (24) and (25),(ν − f )ρ ∂kN dsN
dn
=−
,
dλ
(f kN − ρ)2 ∂sN dλ

dg
=
dλ



(g + ρ)(ν − f )ρg+ρ+kN
(νkN − ρ)(f kN − ρ)∂kN dsN,∂sN dλwhere ∂kN /∂sN = 1 − δ > 0, and dsN /dλ < 0 from Proposition 1.
Consider a decrease in the population growth rate that raises the net return on
asset wealth, increasing household income. With a larger share of asset wealth, theincome effect is stronger for the North and its share of production increases. The
consequent impacts on market entry and productivity growth then depend on theprocess innovation and product development effects. If the process innovation effectdominates, the level of market entry falls and the rate of productivity growth rises,
as indicated by the arrows in Figures 1a and 2a. Therefore, when ν > f the economy
features a negative relationship between productivity growth and population growth,through adjustments the level of in industry concentration. In contrast, when ν < f19

and the product development effect dominates, the increase in industry concentration
raises the level of market entry, as shown by the arrow in Figure 1b. In this case,the productivity growth rate falls for sN < sN , but rises for sN > sN , as depicted in
Figure 2b, indicating that the economy may exhibit a positive or a negative relationship between productivity growth and population growth, given the current level ofindustry concentration.11
3.2

Regional Integration

We now briefly consider the effects of greater regional integration resulting from eithera decrease in transport costs or an increase in the degree of knowledge diffusion.
Beginning with trade liberalization, we obtain the following results:
Proposition 5 An increase in the freeness of trade (ϕ) lowers the level of productvariety (n) and increases the rate of productivity growth (g) for ν > f , and raises n
but may raise or lower g for ν < f .
Proof: From (24) and (25),dn(ν − f )ρ ∂kN dsN=−,
dϕ(f kN − ρ)2 ∂sN dϕ

dg
=
dϕ



g+ρ
(g + ρ)(ν − f )ρ+kN
(νkN − ρ)(f kN − ρ)∂kN dsN,∂sN dϕwhere ∂kN /∂sN = 1 − δ > 0, and dsN /dϕ > 0 from Proposition 1.
The effects of an increase in the freeness of trade are the same as those for a decreasein the population growth rate. Invoking Proposition 1, a fall in transport costs raises
the share of production located in the North. The resulting rise in knowledge spillovers
from production to innovation then leads to a decrease in the level of market entry
and an increase in the productivity growth rate for ν > f . Alternatively, for ν < f
the level of product variety rises, while the rate of productivity growth falls as theNorthern production share increases, until a threshold level of industry concentration
11

Many empirical studies support a negative relationship between population growth and productivity growth (Ahitur 2001; Headey and Hodge 2009; Herzer et al. 2012; Strulik et al. 2013).

20

Figure 3: An Increase in the Degree of Knowledge Diffusion
g

n

1
1/2
(a) Level of Product Variety

sN

1/2

sN
1
(b) Productivity GrowthsN

has been surpassed, after which the rate of productivity growth rises. These cases areillustrated by the arrows shown in Figures 1 and 2.
Next, we investigate the effects of greater economic integration arising from anincrease in the degree of knowledge diffusion between regions:
Proposition 6 An increase in the degree of knowledge diffusion (δ) lowers the levelof product variety (n) and raises the rate of productivity growth (g) for ν > f , andraises n but may raise or lower g for ν < f .
Proof: From (24) and (25),(ν − f )ρ dkN
dn
=−
,
dδ
(f kN − ρ)2 dδ

dg
=
dδ



(g + ρ)(ν − f )ρg+ρ+kN
(νkN − ρ)(f kN − ρ)



dkN
,
dδ

where dkN /dδ = ∂kN /∂δ+(∂kN /∂sN )(dsN /dδ) > 0, ∂kN /∂δ = 1−sN > 0, ∂kN /∂sN =1 − δ > 0, and dsN /dδ < 0 from Proposition 1.
An increase in the degree of knowledge diffusion has two opposing effects on
knowledge spillovers from production to innovation. From (11), the first is a direct effect (∂kN /∂δ) that increases knowledge spillovers. The second is an indirect
effect that decreases knowledge spillovers through reduced industry concentration(∂kN /∂sN )(dsN /dδ). As shown in Appendix C, however, the direct effect always
21

dominates the indirect effect, and knowledge spillovers increase with a rise in the
degree of knowledge diffusion. Once again, the subsequent results for market entry
and productivity growth depend on the process innovation and product development
effects. When ν > f , the level of market entry decreases, and the rate of productivity
growth increases. When ν < f , the level of market entry increases, and the productivity growth rate decreases for sN < sN , but increases for sN > sN . The secondcase is illustrated in Figure 3, where the direct effect of a greater degree of knowledge
diffusion is captured by shifts in the n and g curves, and the indirect effect of greater
industry concentration is illustrated by movements along the curves.
3.3

Regional Welfare

In this section, we briefly discuss the welfare implications of greater regional integration through reduced transport costs and improved knowledge spillovers. Steady-stateregional welfare levels are obtained by substituting (5), (6), (10), (24), and (25) into
lifetime utility (1):12(ρ − λ)Ui = ln A + 1 +

α
σ−1



α ln((1 + ϕ)n)αln Ei +
+
σ−1
ρ−λ




λ
+ γg ,
σ−1

where A = (α(σ − 1)θ(0)γ /σ)α (1 − α)1−α is a constant. The second and third termson the RHS are the level contributions of current expenditure and the level of productvariety. The fourth term captures the contribution of total factor productivity growth,which derives from semi-endogenous growth in product variety (λ = Ṅ /N) and fully
endogenous productivity growth (g = θ̇/θ).
The opposing level and growth components of regional welfare make a general
analytical analysis intractable. As an alternative, we use simple numerical examples
to discuss the implications of improved regional integration. Figure 4 plots regional
welfare levels against the level of knowledge spillovers, and with equal population
12

Using (17), we find that si + ϕ(1 − si ) = (1 + ϕ)Ei /(2E).

22

Figure 4: Knowledge Spillovers and Welfare
Ui

Ui
UN
UN
US
US
kN

1

1/2

1

1/2

(a) ν > fkN(b) ν < f

For both panels, ρ = 0.05, λ = 0.02, γ = 0.3, σ = 2.1, f = 0.2, ϕ = 0.8, and bN = 0.6. Noting thatν ≡ α(1 − (σ − 1)γ)/σ, in Panel (a) α = 0.75 and ν = 0.24, and in Panel (b) α = 0.35 and ν = 0.11.

levels, the greater asset wealth of the North naturally leads to a higher level of utility
for all levels of knowledge spillovers. Figure 4 suggests that regional welfare levels are
generally increasing in the level of knowledge spillovers when the process innovation
effect dominates ν > f , and are generally convex in knowledge spillovers when the
product development effect dominates ν < f .
Considering now the effects of a reduction in transport costs, we obtain the following result for the impact of greater integration on Northern households:1(ρ − λ)(σ − 1) dUi=+αdϕ(1 + ϕ)

1 dnγ(σ − 1) dg(α + σ − 1) dEi
+
+
αEi
dkN
n dkN
(ρ − λ) dkN



dkN
,
dϕ

where dkN /dϕ > 0. The first term on the RHS captures the direct positive effect oflower prices as a result of reduced transports costs between regions. The second effectcaptures the overall negative impact of freer trade on regional household expenditure.
The third and fourth terms describe the effects of changes in the level of productvariety and the rate of productivity growth, and may be positive or negative. While
the general effect of a fall in transport costs is ambiguous, returning to the numericalexamples presented in Figure 4, regional utility shifts upward and the regions move23

rightward along their utility curves, suggesting that an increase in the freeness of trade
benefits welfare for ν > f , and may hurt or benefit welfare for ν < f .
Next, we examine the effects of greater economic integration stemming from an
improvement in the degree of knowledge diffusion:(ρ − λ)(σ − 1) dUi
=
α
dδ



1 dn
γ(σ − 1) dg(α + σ − 1) dEi
+
+
αEi
dkN
n dkN
(ρ − λ) dkN



dkN
,
dδ

where dkN /dδ > 0. The first term on the RHS captures the overall negative impactof greater knowledge dispersion on regional household expenditure. The second andthird terms describe the effects of changes in the level of product variety and the rateof productivity growth, and may be positive or negative. Again, the overall effect isambiguous. The numerical examples in Figure 4, however, suggest that an increase in
the degree of knowledge dispersion benefits welfare for ν > f , and may hurt or benefit
welfare for ν < f , as the regions move rightward along their utility curves.

4

Concluding Remarks

In this paper, we have investigated the relationship between geographic patterns ofindustrial activity and economic growth in a two region model of trade and endogenous growth that corrects for scale effects. The production, process innovation, andproduct development of monopolistically competitive manufacturing firms assume acentral role in the model and, faced with transport costs, imperfect knowledge diffusion and perfect capital mobility, firms locate these activities independently acrossregions with the objective of minimizing costs. These location decisions lead to thepartial concentration of production and the full agglomeration of innovation in theregion with the larger market, as measured by household expenditure.
Within this framework, we find that the level of market entry is closely related withthe distribution of economic activity. In particular, an increase in the concentration of24

industry raises knowledge spillovers from production to R&D, leading to lower costs
for process innovation and product development. As a result, the level of market entry
decreases if costs fall more for process innovation, and increases if costs fall more for
product development. As firm-level employment in process innovation is closely linked
with firm scale, adjustments in the level of market entry have important implications
for economic growth. Specifically, an increase in industry concentration has a positive
direct effect on productivity growth, through a rise in the labor productivity of process
innovation, but may have a negative or a positive indirect effect on productivity
growth, depending on whether the level of market entry rises or falls. As such, in
general we find a positive relationship between industry concentration and economic
growth when improved knowledge spillovers lead to a greater fall in costs for process
innovation, and a convex relationship when greater knowledge spillovers lead to agreater fall in costs for product development.

Appendix A
We first evaluate the partial derivative of (19) at ṡN = 0 to confirm that sN and E
jump immediately and permanently to their steady-state levels:
2(E − 1)kN(1 − ϕ)kN
∂ ṡN
=
+
> 0,
∂sN
E
(1 − δ)(bN − ϕbS − 2(1 − ϕ)sN )E

where (bN − ϕbS ) − (1 − ϕ)sN > 0 is required for E > 0, from (17). The steady-statecomparative statics presented in Proposition 1 are then derived from (17) and (20):









(bN −ϕBS )−(1−ϕ)sN0 1 − dsN  (1−ϕ)E

=

ρ−λ
E
1
− (1−δ)(E−1)dE(1−δ)(E−1)225(bN −ϕbS )(E−1)(1−ϕ)2 E

0



 dλ


0


.

dϕ


1+sN


− 1−δ
dδ

The determinant of the Jacobian matrix given above is strictly positive: |J| > 0. The
results of Proposition 1 are found using Cramer’s rule:dsN(bN − ϕbS ) − (1 − ϕ)sN
=−
< 0,
dλ
(1 − δ)(1 − ϕ)(E − 1)|J|(ρ − λ)(bN − ϕbS )
dsN
=
> 0,
dϕ
(1 − δ)(1 − ϕ)2 E(E − 1)|J|dsN(1 + sN )((bN − ϕbS ) − (1 − ϕ)sN )
=−
< 0,
dδ
(1 − δ)(1 − ϕ)E|J|

dE
E
=−
< 0,
dλ
(1 − δ)(E − 1)|J|dE(bN − ϕbS )(E − 1)
=−
< 0,
dϕ
(1 − ϕ)2 E|J|dE
1 + sN=−< 0.
dδ(1 − δ)|J|

Appendix B
With positive productivity growth, the aggregate labor market clearing condition isα(σ − 1)2EL+
2L = (1 − α)2EL +σαγ(σ − 1)2EL
ρ
−
σN
kN



N + fN +2EL Ṅ
,kN N

where the RHS captures labor demands from traditional production, manufacturing
production, process innovation, the per-period fixed cost, and product development.
Using kN = (ρ − λ)E/(E − 1) from (20), we obtain the following differential equation
for market entry:(f kN − ρ)NṄ= νkN − ρ −
+ λ,
N
2EL
where ν = α(1 − (σ − 1)γ)/σ ∈ (0, 1). Then, using n ≡ N/(2EL) and (24), the rate ofentry reduces to the rate of population growth in steady-state equilibrium: Ṅ/N = λ.

26

Appendix C
Using the result for dsN /dδ from Appendix A, we can show that an increase in the
degree of knowledge diffusion raises knowledge spillovers:dsN
4(1 − sN )(1 − ϕ)E 2
dkN= (1 − sN ) + (1 − δ)
=
> 0.
dδ
dδ
(1 + ϕ)(2bN − 1)|J|

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