Innovation and Manufacturing Offshoring with
Fully Endogenous Productivity GrowthColin Davis
Ken-ichi Hashimoto

October 2016
Discussion Paper No.1636

GRADUATE SCHOOL OF ECONOMICS
KOBE UNIVERSITYROKKO, KOBE, JAPANInnovation and Manufacturing Offshoringwith Fully Endogenous Productivity GrowthColin Davis
Doshisha University∗

Ken-ichi Hashimoto
Kobe University†

October 2016

Abstract
This paper investigates the relationship between net offshoring patterns forinnovation and manufacturing and fully endogenous productivity growth in a
two-country model. The occupational choice of skill-differentiated workers into
low-skilled employment in production and high-skilled employment in innovation determines labor market allocations, and perfect investment mobility allows
firms to shift innovation and manufacturing independently between countries.
These mechanisms generate a tension between access to technical knowledge andlow-cost high-skilled labor in the location decision for innovation, which results
in innovation and manufacturing tending to concentrate in the asset-wealthy(asset-poor) country when trade costs are high (low). The model exhibits apositive relationship between innovation costs and the concentration of industry
and innovation, ensuring that a rise in knowledge diffusion between countries
coincides with increases in net offshoring flows in innovation and manufacturingfrom the asset-wealthy country to the asset-poor country, and a faster rate ofproductivity growth, when the asset-wealthy country has larger shares of innovation and production.
JEL Classifications: F12, F43, R11
Keywords: innovation offshoring, manufacturing offshoring, endogenous productivity growth, process innovation, industry location, international trade∗

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: We are grateful for helpful comments from participants of the Rokkodai
Macroeconomics Seminar at Kobe University, the International Economics Seminar at Kyoto University, the RITM Lunch Seminar at Université Paris-Sud, and ETSG 2016, and acknowledge financial
support from JSPS through a Grant-in-Aid for Scientific Research (C). We are responsible for all
remaining errors.

1

Introduction

Although research and development (R&D) has traditionally concentrated in advanced
countries, in recent years firms have begun to shift innovation offshore to emerging
economies. For example, despite the steady position of the United States (US) as thelargest producer of R&D services in terms of gross domestic expenditure (NSF 2016),
over the past decade its trade balance in R&D services has deteriorated, even shifting
to a deficit for several years, as shown in Figure 1.1 This trend appears to be driven
by intra-firm trade between US parent firms and their foreign affiliates, and by growthin the US trade deficit in R&D services with emerging economies such as China andIndia. On average, trade deficits with China and India grew at rates of 26% and 25%
between 2008 and 2014 to become US$2.1 billion and US$2.2 billion (BEA 2015).2
There is a growing empirical literature investigating the links between R&D offshoring, innovation performance, and economic growth. At the firm level, Nieto andRodrı́guez (2011) and Bertrand and Mol (2013) find that offshoring R&D leads to
1

Net Exports in R&D Services (1,000,000 US$)
−2000
0
2000
4000
6000

Annual growth rates in US imports and exports of R&D services averaged 17% and 12% between
1999 and 2013 (BEA 2015).
2
The main hosts for R&D offshoring outside the US and the EU are Brazil, China, India, Russia,
Singapore, and Taiwan (Hausmann et al. 2007; Puga and Trefler 2010; Santos-Paulino et al. 2014).

AffiliatedTotal

Unaffiliated

2000

2005

2010

2015

Year

Figure 1: U.S. Net Exports in R&D Services (BEA 2015)2

a higher propensity for the introduction of new products. Similarly, Rodrı́guez andNieto (2016) document a positive relationship between innovation offshoring and sales
growth. At the aggregate level, D’Agostino et al. (2013) show that OECD regions withfirms that offshore innovation to emerging economies have more patent applications,and Castellani and Peiri (2013) report that European regions with a greater numberof outward oriented R&D investment projects exhibit higher growth rates for laborproductivity. While the empirical literature suggests a positive relationship between
R&D offshoring and innovation-based economic growth, to the best of our knowledge
this relationship has not been modelled formally.
This paper develops an endogenous market structure and endogenous growth framework (Peretto 1996; Aghion and Howitt 1998; Peretto and Connolly 2007; Etro 2009)
to study the relationship between offshoring patterns in innovation and manufacturing and productivity growth. In particular, we extend the two-country model of Davisand Hashimoto (2015) to include an occupational choice for skill-differentiated workers
between low-skilled employment in production and high-skilled employment in R&D.
Firms produce differentiated products for supply to domestic and export markets,and invest in process innovation to reduce future production costs. The free movement of investment allows firms to shift innovation and production separately between
countries with the aim of minimizing costs (Martin and Ottaviano 2001).
The framework captures two factors that have been emphasized in the business
literature when considering the attractiveness of a location for R&D: access to technicalknowledge and the supply of low-cost labor (see, for example, Chung and Yeaple 2008;
Manning et al. 2008; Lewin et al. 2009; Demirbag and Glaister 2010). On the one
hand, knowledge spillovers from production to innovation are local in nature, leading
to greater spillovers in the country hosting a larger share of industry. On the other
hand, the concentration of industry generates greater demand for high-skilled labor
in innovation, pushing up high-skilled wages. These factors generate a tension in the3firm-level location decision for innovation.
At the aggregate level there is a positive circular causality between the location patterns of innovation and production. A higher national share of production strengthens
knowledge spillovers and attracts R&D activity. In turn greater high-skilled employment raises income, increases market size and attracts production. Considering countries with symmetric labor forces, but different levels of asset wealth, we show that thecircular causality results in the concentration of economic activity in the asset-wealthy
country for high trade costs and in the asset-poor country for low trade costs.3
These location patterns lead to three cases for the direction of net offshoring flows.
For high trade costs, the market of the asset-wealthy country is not large enoughto support the innovation and production of all domestically-owned firms, and net
offshoring flows towards the asset-poor country. For intermediate trade costs, however, the asset-wealthy has a large market that allows it to maintain large shares ofinnovation and production, and it therefore receives net offshoring inflows from theasset-poor country. Lastly, for low trade costs, net offshoring flows towards the larger
market of the asset-poor country.
An important feature of the framework is a positive relationship between the unit
cost of process innovation and the geographic concentration of industry, as the benefitof greater knowledge spillovers is offset by the cost of rising high-skilled wages. An
increase in the unit cost of process innovation reduces optimal firm-level employment
in innovation, resulting in lower overall per-period labor costs and higher per-period
profits, and attracting new firms into the market. As such, greater industry concentration coincides with a higher level of market entry and slower rate of productivity
growth.
Focusing on the case for which the asset-wealthy country has greater shares of in3

Ekholm and Hakkala (2007) develop a static general equilibrium model in which production
requires an intermediate R&D input that can be located separately from production, and show thatvarious location patterns are possible depending on the level of trade costs.

4

novation and production, we investigate the effects of an improvement in international
knowledge diffusion, and find that net offshoring flows from the asset-wealthy country
to the asset-poor country rise as firms offshore innovation to the asset-poor country totake advantage of lower high-skilled wages. As a consequence, the increased dispersionof innovation and production away from the asset-wealthy country leads to lower unit
costs for process innovation and a higher rate of productivity growth.
Our paper is closely related to the international trade literature that builds on thetwo-country variety-expansion model of Grossman and Helpman (1991) to consider theeffects of manufacturing offshoring on innovation-based growth (Martin and Ottaviano
1999, 2001; Gao 2005, 2007; Naghavi and Ottaviano 2009a; 2009b). Within the modelsof this literature, it is common for innovation to agglomerate fully in the advanced
country either by assumption or as a result of localized knowledge spillovers. As such,these models are not viable for the study of how R&D offshoring affects economic
growth. Davis (2013) attempts to investigate the trade-off between localized knowledge
spillovers and high-skilled wage costs in the R&D location choice for firms by extending
Martin and Ottaviano (1999) to include an occupational choice for skill-differentiatedworkers. The link between industry location and economic growth is cut, however,leaving no relationship between economic growth and offshoring patterns. Thus, thekey contribution of our paper is the introduction of a framework that allows for atheoretical study of the positive link between R&D offshoring and innovation-based
economic growth that has been documented in the empirical literature.
The outline of the paper is as follows. Section 2 introduces the model and Section 3
characterizes industry and innovation location patterns. In Section 4 we investigate thedirections of net offshoring flows, and in Section 5 we consider the relationships between
offshoring patterns, market entry, and productivity growth. Section 6 concludes.

5

2

The Model

Two countries, home and foreign, potentially employ labor in four activities: traditional production, manufacturing, process innovation, and market entry. Home andforeign have symmetric labor endowments, and although there is no internationalmigration, skill-differentiated workers choose between low-skilled employment in production and high-skilled employment in innovation within each country. We focus on
the home country while introducing the model. The setup for foreign is analogous,
however, with an asterisk denoting variables associated with foreign.
2.1

Household Preferences

Dynastic households in each country select optimal expenditure-saving paths with theaim of maximizing utility over an infinite time horizon. The lifetime utility of therepresentative household in the home country is

U(t) =Z ∞e−ρ(τ −t) (α ln CX (τ ) + (1 − α) ln CY (τ )) dτ,(1)t

where the consumptions of a composite of manufacturing goods and a traditional good
are CX and CY , the subjective discount rate is ρ, and α ∈ (0, 1). The manufacturingcomposite is

CX (t) =Z N (t)ci (t)

σ−1
σ

di +

Z N ∗ (t)cj (t)

σ−1
σ

0

0

dj

σ
! σ−1

,

(2)

where ci and cj are the demands for manufacturing varieties i and j of the N and N ∗varieties produced in home and foreign, and σ > 1 is the elasticity of substitution.
Lifetime utility is maximized subject to a national flow budget constraint:Ḃ(t) = r(t)B(t) + I(t) − E(t),6(3)

where E is household expenditure, I is labor income, B is asset wealth, r is theinterest rate, and a dot over a variable denotes time differentiation. The solution to
the household’s utility maximization problem is the optimal expenditure-saving pathdescribed by the Euler condition: Ė/E = Ė ∗ /E ∗ = r − ρ, with equal access to aninternational financial market ensuring a common interest rate on asset wealth, andcommon motions for household expenditure. To simplify notation, we suppress timearguments when possible.
With identical investment opportunities, the asset wealths of home and foreign
accumulate at the same rate (Ḃ/B = Ḃ ∗ /B ∗ ), and national shares of asset wealth areconstant across time. Summing across the flow budget constraints (3) to obtain world
expenditure EW ≡ E + E ∗ as a function of world asset wealth BW ≡ B + B ∗ , andusing the result to substitute the net return to investment (r − ḂW /BW ) out of thenational flow budget constraint yields home expenditure as follows:

E = I + b(EW − IW ),(4)with IW ≡ I + I ∗ and national shares of asset wealth determined by initial levels ofasset wealth: b ≡ B/BW and b∗ ≡ B ∗ /BW .
At each moment in time, households allocate constant shares of expenditure to thepurchase of the manufacturing composite and the traditional good: PX CX = αE andPY CY = (1 − α)E, where PX is a price index for manufacturing goods and PY is the
traditional good price in home. The price index over manufacturing goods is

PX =

Z N
0

pXi

1−σ

di +

Z N∗
0

(ζp∗Xj )1−σ dj

1
 1−σ

,

(5)

where pXi and p∗Xj are the prices of goods produced in home and foreign. Iceberg trade
costs are incurred on international shipments with a shipment of ζ > 1 units required
for every unit sold in an export market. Viewing (5) as the unit expenditure function
7

over manufacturing goods, Shephard’s Lemma yields the home country demands forhome and foreign produced varieties:

σ−1
ci = αp−σ
Xi PX E,

cj = α(ζpXj )−σ PXσ−1 E.

(6)

The demand conditions for foreign households are analogous.
2.2

Occupational Choice

Home and foreign have equal masses of workers (Z = Z ∗ ) with heterogeneous skill
levels z that follow continuous uniform distributions with support [0, 1]. Workers arefree to choose between employment in innovation or production. A worker employedin production supplies one unit of low-skilled labor, regardless of skill level, earning
the low-skilled wage rate wL . A worker employed in innovation supplies z units ofhigh-skilled labor, earning income zwH , where wH is the high-skilled wage rate.
National labor markets are competitive with all firms paying the same low-skilled
and high-skilled wages. When there is positive employment in production and innovation, one marginal worker potentially earns the same incomes from low-skilled andhigh-skilled employment, and is therefore indifferent between employment type. The
skill level of this marginal worker equals the relative wage rate z = ω = wL /wH ,and separates the labor force into workers with skill levels z ∈ [0, ω] who choose
employment in production, and workers with skill levels z ∈ [ω, 1] who choose employment in innovation. This national labor allocation results in the following effective low-skilled and high-skilled labor supplies for the home country: L = ωZ and
H = (1 − ω 2 )Z/2. Expected national labor income, conditional on employment levels,is therefore I(ω) = wL L + wH H = wL (1 + ω 2)Z/(2ω).

8

2.3

Traditional Production

Traditional firms employ one unit of low-skilled labor with a constant returns to scaletechnology to produce one unit of output for supply to an international market characterized by free trade. Setting low-skilled labor as the model numeraire, we assume thatthe world demand for traditional goods is large enough that both countries produce
traditional goods at all moments in time. The traditional good price and low-skilled
wage rates then equalize across countries, PY = PY∗ = wL = wL∗ = 1, and the homeand foreign demands for traditional goods determine the world demand for low-skilledlabor in traditional production:

LY + L∗Y = (1 − α)EW .

2.4

(7)

Manufacturing

The manufacturing sector is monopolistically competitive (Dixit and Stiglitz 1977),
with each firm producing a single unique product for supply to domestic and export
markets. In addition to production, each firm employs high-skilled labor (hI ) in process
innovation and low-skilled labor (lF ) in firm management. The production technology
of firm i with production located in home isxi = θγ lXi ,(8)where xi and lXi are firm-level output and low-skilled employment in production, θ isfirm-level productivity, and γ ∈ (0, 1) is the productivity elasticity of output. Althougheach firm employs a unique production technology, we assume that productivity levels
are symmetric across firms (θ = θ∗ ), regardless of the location of production.
Firms maximize operating profit (π = px − lX ) by setting price equal to a constantmarkup over unit cost: pX = p∗X = σ/((σ − 1)θγ ), where we now suppress the firm9

index i. Equating supply with the demands from home and foreign households (6),x = c + ζc∗ , optimal operating profit for a firm with production located in home is
lX
αp1−σ
π=
= X
σ−1
σ



E

ϕE ∗
+
PX1−σ PX1−σ
∗



,

(9)

where ϕ ≡ ζ 1−σ ∈ (0, 1) describes the freeness of trade between countries with ϕ = 0
indicating prohibitively high trade costs and ϕ = 1 indicating perfectly free trade.
Firms are free to shift production between countries, with the aim of maximizing
operating profit (Martin and Ottaviano 2001). Therefore, when there are active manufacturing sectors in both countries, operating profit is the same for all firms, regardlessof production location; that is, π = π ∗ . Combining the price indices (5) with operating
profit (9), we solve for the share of firms locating production in home assX ≡E − ϕE ∗b − ϕb∗ (1 + ϕ)(b∗ I − bI ∗ )
N
=
=
+
,
NW
(1 − ϕ)EW
1−ϕ(1 − ϕ)EW(10)

where NW ≡ N + N ∗ . Substituting the manufacturing shares of each country backinto operating profit yields firm-level employment in production for all locations:lX =α(σ − 1)EW
,
σNW

(11)

where we have used the production function (8).
2.5Process Innovation

Firms invest in process innovation with the aim of lowering unit production costs. The
evolution of productivity for a firm with process innovation located in home follows

θ̇ = kθhI ,

10

(12)

where hI is firm-level high-skilled employment and kθ is labor productivity in process
innovation. Following the in-house process innovation literature (Smulders and van deKlundert 1995; Peretto 1996; Peretto and Connolly 2007), firm-level R&D exhibits anintertemporal knowledge spillover through which the technical knowledge created bycurrent innovation activity improves the labor productivity of future innovation efforts.
Specifically, the productivity coefficient θ represents the current stock of technicalknowledge, and k determines the strength of intertemporal knowledge spillovers fromthe stock of knowledge into current innovation activity.
There is a large body of empirical research documenting both the localized natureof knowledge spillovers and the international scope for knowledge diffusion (Bottazi
and Peri 2007; Mancusi 2008; Coe et al. 2009; Ang and Madsen 2013).4 Adapting the
setup of Baldwin and Forslid (2000), we capture the geographic nature of knowledge
diffusion with the following specification for the strength of knowledge spillovers fromproduction into process innovation located in home:k = sX + δ(1 − sX ),(13)

where the localized nature of knowledge spillovers is regulated by the degree of knowledge diffusion δ ∈ (0, 1). Therefore, the labor productivity of high-skilled workers
in innovation is determined as the weighted-average productivity of the observablestock of knowledge, with a stronger weighting for production technologies employedin proximity to the innovation department of the firm.
Accounting for process innovation costs (wH hI ) and fixed per-period management
costs (lF ), total per-period profit is Π = π − wH hI − lF and firm value equals the4

See Keller (2004) for a survey of the various channels through which knowledge spillovers arise.

11

presented discounted value of the expected future profit stream:

V (t) =

Z ∞

Rτ

′

′

e− t (r(τ )+λ)dτ Π(τ )dτ,

(14)

t

where λ > 0 is an instantaneous default rate that indicates the probability that a
firm-specific shock forces the firm to exit the market (Baldwin 1999).
Each firm sets its level of high-skilled employment in process innovation to maximize firm value subject to (12). We solve this optimization problem with the following
current-value Hamiltonian function: F = Π + pI kθhI , where pI is the internal price ofa unit mass of new innovations developed by a firm in home over the time interval dt,
and each firm perceives the price indices (5) as constant when considering the impact ofchanges in its price on profits, given its small market share. The first order conditions
for optimization provide the following static and dynamic efficiency conditions:

pI =

wH
,
kθ

pI (r + λ) − ṗI =∂π(σ − 1)γπ
=
.
∂θ
θ

(15)

The internal price of process innovation captures two key factors in the location decision for R&D that have been emphasized in the literature: the observable stock oftechnical knowledge and the cost of employing high-skilled labor (Chung and Yeaple
2008; Manning et al. 2008; Lewin et al. 2009; Demirbag and Glaister 2010).
Firms are free to shift their innovation actives between countries, ensuring a common internal price for new process innovations when there is innovation located in both
home and foreign; that is, pI = p∗I . We combine this condition with ω = 1/wH to obtain the home production share required to equate the price of new process innovations
across countries:sX =∗ω ∗ − δωwH − δwH=,∗(1 − δ)(wH + wH)
(1 − δ)(ω + ω ∗ )

12

(16)

where we have used (13) and (15). Similar to the framework of Ekholm and Hakkala(2007), the degree of knowledge diffusion regulates the range of relative high-skilled∗wages over which dispersed location patterns are feasible; that is, wH /wH∈ (δ, 1/δ)is required for sX ∈ (0, 1). Substituting (16) back into (13) yields the equilibrium
strength of knowledge spillovers into innovation in home: k = (1 + δ)ω ∗ /(ω + ω ∗ ).
2.6

Market Entry

In order to focus on firm-level investment in process innovation, we consider a simple
setup for market entry with new firms employing low-skilled labor as they prepare
to enter the manufacturing industry. Following Etro (2004), Peretto and Connelly(2007), and Peretto and Valente (2015), we assume that the cost of market entry
is proportionate to the value of production after market entry, generating a positive
relationship between entry costs and market size. Specifically, the costs of market
entry in home and foreign are V = pX x and V ∗ = p∗X x∗ , and are equal when bothcountries have positive shares of manufacturing, since pX x = p∗X x∗ .
The value of a new firm equals the present value of the future profit stream earned
after entering the manufacturing sector (14). Free entry drives firm value down to thecost of market entry:

V = pX x =

αEW
,
NW

(17)

where we have used (9) and (11). The time derivative of (14) combined with (17)yields the following no-arbitrage condition for investment in market entry (Grossman
and Helpman 1991):(r + λ)V − V̇ = π − wH hI − lF .

(18)

The investment conditions (15) and (18) imply that ωk = ω ∗ k ∗ , hI /ω = h∗I /ω ∗, and13

khI = k ∗ h∗I when firms locate innovation in home and foreign.
Aggregating across countries, we obtain the following differential equation to describe the dynamics of market entry and exit:ṄW =LN + L∗N(LN + L∗N )NW− λNW− λNW =
pX x
αEW

(19)

where we have used (8), (9) and (11), and LN and L∗N are the national levels of lowskilled labor employed in market entry. At each moment in time λNW firms default
and are forced to exit the market.

3Long-run Location Patterns

We now derive the long-run industry and innovation location patterns consistent withequilibrium in the labor and investment markets. In order to simplify the analysis
we consider the level of market entry relative to market size: n ≡ NW /(αEW ). In
addition, without loss of generality, we assume that the home country has a greater
share of asset wealth (b ≥ 1/2) for the remainder of the paper.
3.1

National Labor Allocations

We first solve for a condition that determines the national labor allocations associatedwith common prices for goods pX = p∗X and innovations pI = p∗I . Using the flow budget
constraint for total expenditure (3) with BW = NW V = αEW and ĖW /EW = r − ρ,
we find that total labor income is determined proportionately with total expenditure:
IW = (1 − αρ)EW . Then, equating (10) and (16) and reorganizing the result yields
ω ∗ − δωαρ(b − ϕb∗ ) (1 − αρ)(I − ϕI ∗ )
=
+
,
1−ϕ
(1 − ϕ)IW(1 − δ)(ω + ω ∗)

14

(20)

which we refer to as the share locus, as it indicates the relative wage combinations
consistent with national labor allocations and production shares that ensure equalized
operating profits and innovation costs across countries at all moments in time.
Second, we derive the dynamics associated with national labor allocations. Common motions for household expenditures (Ė/E = Ė ∗ /E ∗ ) result in constant production shares (ṡX = 0) and common motions for relative wages (ω̇/ω = ω̇ ∗ /ω ∗ ).
Combining (15) with IW = (1 − αρ)EW , hI /ω = (H/ω + H ∗/ω ∗ )/NW , and ĖW /EW =−((H/ω + H ∗ /ω ∗)/IW )(ω̇/ω) = r − ρ, we obtain the following motion for the evolutionof relative wages:(nH − n)(ρ + λ)IWω̇=,
ω(L + L∗ )nnH =(σ − 1)γωk (1 − αρ)(H/ω + H ∗ /ω ∗ )ωk−,σ(ρ + λ)α(ρ + λ)IW

(21)

where nH is the steady-state level of market entry consistent with optimal investmentin process innovation and equilibrium in the high-skilled labor market.
Third, we derive an expression to describe the evolution of market entry. Substituting (7), (11), and (19) with IW = (1 − αρ)EW into the low-skilled labor market
clearing condition L + L∗ = LY + L∗Y + NW lX + NW lF + LN + L∗N , we have
(H/ω + H ∗ /ω ∗ ) ω̇
(1 − αρ)(L + L∗ ) σ − αλṅ= (nL − n)lF +, nL =−− , (22)nIW
ωαlF IW
ασlF
lF
where nL is the steady-state level of market entry consistent with optimal investment
in market entry and equilibrium in the low-skilled labor market. Equating (21) and(22) yields a second steady-state condition (ω̇ = ṅ = 0) for national labor market
allocations that we refer to as the investment locus: nH = nL .
Together the investment and share loci determine long-run labor allocations. The
investment locus is depicted by curve aa in Figure 2 and has a strictly negative slope(see Appendix A). As illustrated by the curve bb, the share locus has two hyperbolic
branches with asymptotes implicitly defined by ω = ω ∗ and (δ −ϕ + (1 −δϕ)ωω ∗ + (1 −15

Figure 2: Equilibrium Labor Allocations
ω∗
1

b

ω∗
1

1/δ a
b

1/δ a

b

b
a
δ
b

a
δ

0

1

ω

0 b

(a) ϕ < ϕB1ω(b) ϕ > ϕB

These stylized labor allocation patterns can be reproduced numerically using α = 0.95,
σ = 2.5, γ = 0.5, ρ = 0.01, λ = 0.01, lF = 0.05, b = 0.75, δ = 0.45, and ϕB = 0.89, forboth panels, and with ϕ = 0.75 for Panel (a) and ϕ = 0.95 for Panel (b).

δ)(1 + ϕ)(b∗ − bωω ∗ )αρ = 0 and a vertical transverse axis for b > b∗ .5 The long-run
equilibria associated with the intersection between the investment locus and the upperand lower branches of the share locus are shown in Figures 2a and 2b. In Appendix
A, we investigate the local dynamics of the model and derive two sufficient conditions
for the saddle-path stability of a long-run equilibrium with dispersed industry andinnovation: first that ωklF > ρ + λ and second that the share locus have a greater
slope than the investment locus.6 We limit our analysis to steady states that satisfy
these conditions.
5

Using the expressions introduced in Appendix A, the share locus can be rewritten as follows:ω(1 + ωω ∗ )(b − b∗ )(1 − δ)(1 + ϕ)αρ(ω ∗ − ω)(δ − ϕ + (1 − δϕ)ωω ∗ + (1 − δ)(1 + ϕ)(b∗ − bωω ∗ )αρ)= 1.

The denominator describes the asymptotes that arise for b > b∗ .
6
Numerical simulations suggest that these sufficient conditions are actually necessary conditions
for a saddle-path stable equilibrium, although we are unable to show this result analytically.

16

3.2

Industry and Innovation Location Patterns

National shares of industry are described by (16). In Figure 2, the full concentrationof industry occurs along the ω ∗ /ω = 1/δ dashed line (sX = 1), and in foreign along theω ∗ /ω = δ dashed line (sX = 0). Similarly, denoting the numbers of firms locating process innovation in home and foreign by m = H/hI and m∗ = H ∗ /h∗I , with high-skilledlabor employed solely in process innovation, the home share of innovation output is

sI ≡

wH H
mθ̇
(1 − ω 2 )ω ∗==,∗wH H + wH
H∗(1 − ω 2)ω ∗ + (1 − ω ∗ 2 )ωmθ̇ + m∗ θ̇∗(23)

where we have used khI = k ∗ h∗I and hI /ω = h∗I /ω ∗ . National shares of innovation
activity are described by the investment locus in Figure 2, with innovation fully concentrated in home at point a (sI = 1), and innovation fully concentrated in foreign atpoint a∗ (sI = 0). The shaded areas indicate national labor allocations that are notconsistent with labor market equilibrium.
We characterize industry and innovation location patterns according to the levelof trade costs, and obtain the following proposition.
Proposition 1 (i) If ϕ < ϕB , the asset-wealthy country has larger shares of industry
and innovation, with the full concentration of industry for ϕ ∈ (ϕX , ϕB ) and the fullconcentration of innovation for ϕ ∈ (ϕI , ϕB ). (ii) If ϕ > ϕB , the asset-poor country
has larger shares of industry and innovation, with the full concentration of industry
for ϕ ∈ (ϕB , ϕ∗X ) and the full concentration of innovation for ϕ ∈ (ϕB , ϕ∗I ), whereϕB ≡δ + ωω ∗ + (1 − δ)(b∗ − ωω ∗b)αρ,
1 + δωω ∗ − (1 − δ)(b∗ − ωω ∗b)αρwith positive values for both the numerator and denominator since 1 > αρ. Note that
ϕI < ϕX and ϕ∗I > ϕ∗X for δ < δ, but ϕI ≥ ϕX and ϕ∗I ≤ ϕ∗X for δ ≥ δ.
Proof: See Appendix B.

17

There is a positive circular causality between the location patterns of production
and process innovation. A higher share of production strengthens knowledge spillovers
and attracts innovation, which in turn raises high-skilled labor income, expands market size, and attracts production. With changes in national shares of asset wealth,however, the initial direction of this circular causality turns on the balance of thethree effects captured by the share locus (Davis 2013). The investment income effectis described by the first term on the LHS of (20), with an increase in relative asset
wealth expanding a country’s relative market size. The labor income effect is represented by the second term on the LHS of (20), with a rise in high-skilled labor income
also increasing national market size. The investment and labor income effects link national shares of industry with market size through the home market effect (Krugman1980), the strength of which is increasing in the freeness of trade. The innovation costeffect is decribed by the RHS of (20), and regulates how fast high-skilled wages canrise as a national market expands before firms begin to relocate process innovationinternationally to reduce the cost of employing high-skilled labor. The innovation costeffect is increasing in knowledge diffusion: stronger knowledge spillovers facilitate afaster rate of high-skilled wage growth.
Consider the national labor market adjustments that occur when a small increase inb expands the home share of production through the investment income effect. Holding
the degree of knowledge diffusion constant, we use the freeness of trade to identify
two cases for the direction of labor market adjustments as the economy returns toequilibrium. First, for ϕ < ϕB , the innovation cost effect dominates the labor income
effect, and the initial shift in location patterns leads to a lower cost for innovation in
home: pI < p∗I . Firms then relocate innovation and production to the home country,∗with wH /wH
increasing at a faster rate than k/k ∗ until process innovation costs areequal again. In this case, the asset-wealthy country becomes the net exporter ofmanufacturing goods and process innovations, and has a larger market with greater

18

high-skilled employment. Second, for ϕ > ϕB , the labor income effect dominates, andthe initial shift in location patterns generates higher innovation costs for home: pI >p∗I . As a result, innovation and production relocate to foreign with k/k ∗ decreasing∗
at a faster rate than wH /wH
until pI = p∗I again. In this case, the asset-poor country

becomes the net exporter of manufacturing goods and process innovations, and has alarger market with greater high-skilled employment.
The degree of knowledge diffusion determines whether the full concentration of production coincides with the full concentration of innovation. If δ < δ, the knowledge
spillover advantage of the larger national market discourages firms from locating innovation in the smaller country, and innovation concentrates fully before industry; thatis, ϕX < ϕI and ϕ∗I > ϕ∗X . In contrast, if δ > δ, a high degree of knowledge reduces
the localized benefits of knowledge spillovers, allowing the smaller country to attract
innovation with its low-cost high-skilled labor, even with industry fully concentrated
in the larger country; that is, ϕX > ϕI and ϕ∗I < ϕ∗X .
3.3

Offshoring Patterns

The direction of net offshoring flows in the manufacturing sector is determined as
the difference between national shares of asset wealth and production (Martin andOttaviano 1999). For the home country we have

SX ≡ b − s X =(b + δb∗ )ω − (b∗ + δb)ω ∗,(1 − δ)(ω + ω ∗ )(24)

where we have used (16). Net offshoring flows from the asset-wealthy home country to
the asset-poor foreign country when SX > 0, and from foreign to home when SX < 0.
Characterizing offshoring patterns using the freeness of trade, we obtain the following:
Proposition 2 Net offshoring in manufacturing flows from the asset-wealthy country
to the asset-poor country for ϕ 6∈ (ϕXO , ϕB ), and from the asset-poor country to the19Figure 3: Equilibrium Offshoring Patterns
SI , SX
b

SX

SI
0

b∗

SI , SX
b

∗ ϕ∗
ϕX
I

SX

SI
SI
SI

SX

SX

ϕXO

0

ϕIO

0

∗
ϕX

ϕI ϕX ϕB

ϕb∗1(a) δ < δ < δ

ϕXO

0

ϕX

ϕB

ϕ
1

(b) δ > δ > δ

These stylized labor allocation patterns can be reproduced numerically using α = 0.95,
σ = 2.5, γ = 0.5, ρ = 0.01, λ = 0.01, lF = 0.05, b = 0.75 for both panels, and with
δ = 0.45 and ϕB = 0.89 for Panel (a), and δ = 0.85 and ϕB = 0.97 for Panel (b).

asset-wealthy country for ϕ ∈ (ϕXO , ϕB ), where ϕXO ∈ (0, ϕX ).
Proof: See Appendix C.
Net offshoring flows for the manufacturing sector are illustrated by the SX curve in
Figure 3, where the vertical axis measures SX ∈ (−b∗ , b) over the range ϕ ∈ (0, 1) indicated by the horizontal axis. There are three cases for the direction of net offshoring.
Starting from a high level of trade costs over the range ϕ ∈ (0, ϕXO ), the investment
income and innovation cost effects dominate the labor income effect ensuring a greater
share of industry for home. The market of home is not sufficiently large to support
the production of all home-owned firms, however, resulting in net offshoring flows
from home to foreign (SX > 0). For a mid-level of trade costs ϕ ∈ (ϕXO , ϕB ), thehome market is sufficiently large to attract the production of a larger share of firms,
including the production of firms with foreign owners, and net offshoring therefore
flows from foreign to home (SX < 0). For a low level of trade costs over the rangeϕ ∈ (ϕB , 1), the labor income effect dominates the investment income and innovation
cost effects, and the larger market of foreign attracts the greatest share of produc20

tion, with a share of home-owned firms also locating production in foreign, thereby
generating net offshoring flows from home to foreign (SX > 0).
The direction of net offshoring flows in innovation is calculated using the difference
between national shares of asset wealth and innovation. For home we have

SI ≡ b − s I =bω(1 − ω ∗ 2 ) − b∗ ω ∗ (1 − ω 2 ),(1 − ω 2 )ω ∗ + (1 − ω ∗2 )ω

(25)

where we have used (23). Net offshoring in innovation flows from the asset-wealthy
home country to the asset-poor foreign country when SI > 0, and from foreign tohome when SI < 0.
Proposition 3 When δ < δ, net offshoring in innovation flows from the asset-wealthy
country to the asset-poor country for ϕ 6∈ (ϕIO , ϕB ) and from the asset-poor country
to the asset-wealthy country for ϕ ∈ (ϕIO , ϕB ), where δ < δ and ϕIO ∈ (0, ϕI ). When
δ > δ, net offshoring in innovation always flows from the asset-wealthy country to theasset-poor country.
Proof: See Appendix C.
The pattern of innovation offshoring depends on knowledge diffusion. As shownin Figure 3a, where net offshoring flows for innovation are measured on the verticalaxis (SI ∈ (−b∗ , b)), when knowledge diffusion is relatively low (δ < δ), there are threecases. The first case occurs over the range ϕ ∈ (0, ϕIO ) where home’s greater share ofindustry provides it with a knowledge spillover advantage that allows it to attract alarger share of innovation. The knowledge spillover advantage is not sufficient, however, to prevent a share of home-owned firms from locating innovation in foreign withthe aim of taking advantage of lower high-skilled wages. As a result, net offshoring ininnovation flows from home to foreign (SI > 0). The second case occurs for intermediate trade costs ϕ ∈ (ϕIO , ϕB ) where home’s share of industry generates a knowledge
spillover advantage that is great enough to ensure that all home-owned firms, and a21

share of foreign-owned firms, locate innovation in the home country, despite higher
high-skilled wages. In this case net offshoring flows from foreign to home (SI < 0).
The third case arises for low trade costs ϕ ∈ (ϕB , 1) when foreign has a larger share ofindustry, and all foreign firms and a share of home-owned firms locate innovation in
the foreign country to take advantage of greater knowledge spillovers, generating net
offshoring flows from home to foreign (SI > 0).
In general, marginal improvements in knowledge diffusion have ambiguous effects
on national labor allocations. We can show, however, that for relatively high values ofδ, we have ϕ < ϕB , sI > 0, and sX > 0, with improved knowledge spillovers inducing
a fall in the relative wage rate and reductions in the home shares of innovation andproduction that cause upward shifts in the SX and SI curves in Figure 3. In particular, as shown in Figure 3b, for sufficiently high knowledge diffusion, the knowledge
spillover advantage associated with concentrated industry is relatively weak, creating
an incentive for firms to focus on minimizing the cost of employing high-skilled laborwhen choosing where to locate innovation. As such, process innovation never concentrates fully in one country. Indeed, with perfect knowledge diffusion (δ = 1), exactly
half of all innovation activity is located in each country: SI = b − 1/2 > 0. Therefore,
in the case for δ > δ, net offshoring in innovation always flows from the asset-wealthy
home country to the asset-poor foreign country (SI > 0).

4

Offshoring and Productivity GrowthThis section compares the effects of increased industry concentration on net offshoringflows, market entry, and productivity growth. We combine (9), (11), (12), (15), and(18) to obtain the level of market entry and the rate of productivity growth as follows:n=(ν − ρ − λ)ωk,
ωklF − ρ − λg=(σ − 1)γ(ωklF − ρ − λ)− ρ − λ,σ(ν − ρ − λ)

22

(26)

where ν = (1 − (σ − 1)γ)/σ ∈ (0, 1) and ν > ρ + λ is necessary for a positive levelof market entry, given that we assume ωklF > ρ + λ to ensure saddle-path stability.
Note that productivity growth is not biased by a scale effect as changes in overall
population size (2Z) are fully absorbed by adjustments in the number of firms in themarket (NW ), leaving the level of market entry (n) unchanged.7
We use the expressions in (26) to examine how greater economic integration between countries affects market entry and productivity growth.

A key feature of

the framework is a positive relationship between the unit cost of process innovation(pI = 1/(ωkθ)) and the concentration of innovation and production in either home orforeign, as the benefit of improved knowledge spillovers is outweighed by the cost of
rising high-skilled wages.8 Returning to (26), an increase in the unit cost of process
innovation leads to a greater level of market entry and a faster rate of productivity
growth. Specifically, optimal firm-level employment in process innovation falls, resulting in lower overall per-period labor costs (wH hI ) and higher per-period profits(Π = ν/n − (ρ + λ)/(ωk) − lF ) that draw new firms into the market (dn/d(ωk) < 0).
The decrease in firm-level employment in innovation, however, depresses the rate ofproductivity growth (dg/d(ωk) > 0). Thus, improved economic integration through afall in trade costs or a rise in knowledge diffusion affects market entry and productivity
growth through adjustments in the unit cost of process innovation.
Beginning with a decrease in trade costs we obtain the following proposition.
Proposition 4 A reduction in trade costs increases market entry and dampens productivity growth for ϕ < ϕB , while decreasing market entry and accelerating productivity growth for ϕ > ϕB .
7

There is a large empirical literature concluding that there is no significant relationship between
economic growth and population size (Dinopoulos and Thompson 1999; Barro and Sala-i-Martin
2004; and Laincz and Peretto 2006). The framework presented in this paper corrects for scale effects
by focusing on the innovation associated with the production technologies of individual product lines,
rather than considering R&D at the national level.
8
In general the empirical literature reports mixed results for the effect of industry concentration
on economic growth. See Gardiner et al. (2011) for a literature survey and for evidence supportinga negative relationship between a number of measures of industry concentration and GDP growth.

23

Proof: See Appendix D.
A decrease in trade costs affects the unit cost of process innovation indirectly
through changes in national shares of production. For ϕ < ϕB , lower trade costs
increase the concentration of industry in home, raising unit innovation costs. Accordingly, optimal firm-level employment in innovation falls, generating lower per-period
labor costs and greater per-period profits that attract new firms into the industry andraises the level of market entry. The decrease in firm-level employment in process innovation, however, depresses the rate of productivity growth. In contrast, for ϕ > ϕB ,lower trade costs reduce industry concentration in foreign leading to lower unit costs
for innovation, a lower level of market entry, and a faster rate of productivity growth.
Next, examining the effects of adjustments in knowledge diffusion we obtain thefollowing result.
Proposition 5 An improvement in the degree of knowledge diffusion reduces market
entry and accelerates productivity growth for ϕ < ϕB , but has ambiguous effects onmarket entry and productivity growth for ϕ > ϕB .
Proof: See Appendix D.
An improvement in knowledge diffusion affects the unit cost of process innovation
both directly through greater knowledge spillovers that raise labor productivity ininnovation and indirectly through adjustments in national shares of production. Thus,on the one hand, as discussed in Section 3, for ϕ < ϕB a sufficient increase in δ
lower the concentration of industry in home, and both the direct and indirect effects
aline to decrease the unit cost of process innovation. The result is greater firm-level
employment in innovation, a lower level of market entry, and accelerated productivity
growth. On the other hand, for ϕ > ϕB , the increase in δ has ambiguous effects
on national shares of production, with ambiguous results for the unit cost of process
innovation, the level of market entry, and the rate of productivity growth.
24

Focusing on the case for which industry and innovation concentrate in the assetwealthy country (ϕ < ϕB ), we now compare the effects of economic integration throughlower trade costs on net offshoring flows and productivity growth. Returning to Figure
3, an increase in ϕ reduces the home shares of innovation and production, thus decreasing SI and SX , and implying that a decrease in net offshoring flows in innovationand manufacturing coincides with a greater level of market entry and a slower rate ofproductivity growth. In contrast, an improvement in knowledge diffusion shifts the
SX and SI curves upwards in Figure 3. Therefore, increases in net offshoring flows ininnovation and manufacturing coincide with a decrease in the level of market entry
and a faster rate of productivity growth. With the evidence presented in Baldwin etal. (2001) suggesting that communication costs have fallen at a faster rate than trade
costs, our results indicate that an improvement in knowledge diffusion is a plausible
explanation for the recent rise in innovation offshoring. In addition, this trend may
coincide with faster productivity growth as the unit cost of process innovation falls.

5

Concluding Remarks

This paper has developed a two-country model to examine the relationship between
net offshoring patterns in innovation and manufacturing and fully endogenous productivity growth. Central to the model, monopolistically competitive firms invest
in process innovation that lowers production costs and drives aggregate productivity
growth. The occupational choice of skill-differentiated workers into low-skilled employment in production and high-skilled employment in innovation determines national
labor allocations, while the free movement of investment allows manufacturing firms
to shift their innovation and production activities freely between countries. These two
mechanisms create a tension between accessing the technical knowledge contained in
production processes and sourcing low-cost high-skilled labor as firms independently
select the optimal locations for innovation and production. A key feature of the model25

is a positive relationship between the unit cost of process innovation and the geographic concentration of industry and innovation as the benefit of greater knowledge
spillovers is offset by the cost of rising high-skilled wages.
Assuming symmetric labor forces, we characterize location patterns according totrade costs. Specifically, while high trade costs lead to a larger market and the concentration of innovation and production in the asset-wealthy country, when trade costs
are low innovation and production concentrate in proximity to the larger market ofthe asset-poor country. Given these location patterns, we use the level of trade costs
to identify three cases for the directions of net offshoring in innovation and manufacturing. For high trade costs, although the asset-wealthy country has greater sharesindustry and innovation, the domestic market is not sufficiently large to attract the innovation and production activities of all firms with domestic owners, and net offshoring
thus flows towards the asset-poor country. For intermediate trade costs, however, net
offshoring flows from the asset-poor country towards the larger market of the assetwealthy country. Finally, for low trade costs, the net offshoring flows towards theasset-poor country as it maintains greater concentrations of industry and innovation.
Focusing on the case for which the asset-wealthy country has greater shares of industry and innovation activity, we investigate the effects of an improvement in knowledge diffusion between countries, and find that net offshoring flows in innovation andmanufacturing from the asset-wealthy country to the asset-poor country increase as
firms offshore innovation to the asset-poor country to take advantage of lower wages
for high-skilled workers. The resulting increased dispersion of industry and innovation
activity away from the asset-wealthy country results in lower unit costs for process
innovation and thus accelerates productivity growth.

26

Appendix A: Saddle-path Stability
With ωk = (1 + δ)ωω ∗/(ω + ω ∗ ), L + L∗ = (ω + ω ∗ )Z, H/ω + H ∗ /ω ∗ = (ω + ω ∗)(1 −ωω ∗)Z/(2ωω ∗) and IW = (ω + ω ∗ )(1 + ωω ∗)Z/(2ωω ∗), the share locus (20) and theinvestment locus (nH = nL ) are written in implicit form respectively as follows:Ω=ω ∗ δ − ϕ + (1 − δϕ)ωω ∗ + (1 − δ)(1 + ϕ)(b − b∗ ωω ∗)αρ−,ωδ − ϕ + (1 − δϕ)ωω ∗ + (1 − δ)(1 + ϕ)(b∗ − bωω ∗ )αρΦ=(σ − 1)γωk (1 − αρ)(1 − ωω ∗)ωk 2(1 − αρ)ωω ∗ σ − α
λ
−
−
+
+ .
∗
∗
σ(ρ + λ)α(1 + ωω )(ρ + λ)α(1 + ωω )lF
ασlF
lF

We use these expressions to obtain the following partial derivatives:∂Ωω∗(1 − αρ)(ω − ω ∗ )2 ω ∗
=− 2+
R 0,
∂ω
ω
αρ(b − b∗ )(1 + ωω ∗)2 ω 2
1(1 − αρ)(ω − ω ∗ )2
∂Ω
=
+
> 0,
∂ω ∗
ω αρ(b − b∗ )(1 + ωω ∗)2 ω(1 − δ)(1 + ϕ)(ω + ω ∗ )(1 + ωω ∗)αρ
∂Ω
=−
R 0,
∂b
(δ − ϕ + (1 − δϕ)ωω ∗ + (1 − δ)(1 + ϕ)(b∗ − bωω ∗ )αρ) ω∂Ω(1 + δ)(ω ∗ − ω)2
=−
< 0,
∂ϕ
αρ(1 − δ)(1 + ϕ)2 (b − b∗ )ω 2
(1 − ϕ)(ω ∗ − ω)2
∂Ω
=
> 0,
∂δ
αρ(1 − δ)2 (1 + ϕ)(b − b∗ )(1 + ωω ∗)ω 2nω ∗2(1 − αρ)(ωklF − ρ − λ)ω ∗
∂Φ
> 0,
=
+
∂ω
(ω + ω ∗ )ωα(ρ + λ)(1 + ωω ∗)2 lF∂Φnω
2(1 − αρ)(ωklF − ρ − λ)ω
=
+
> 0,
∂ω ∗
(ω + ω ∗ )ω ∗α(ρ + λ)(1 + ωω ∗)2 lF
∂Φ
n
=
> 0.
∂δ
1+δ
The slope of the share locus is (dω ∗/dω)|Ω=0 = −(∂Ω/∂ω)/(∂Ω/∂ω ∗ ), and is positive or negative depending on the sign of ∂Ω/∂ω. The slope of the investment locus
is (dω ∗/dω)|Φ=0 = −(∂Φ/∂ω)/(∂Φ/∂ω ∗ ) < 0. We study the stability of the steadystate described by nH = nL = n and Ω = Φ = 0 using a Taylor expansion of (21) and(22) to obtain the following determinant for the Jacobian matrix J1 of the linearized27

system:(1 + ωω ∗)(ρ + λ)lF
|J1 | =
2ω ∗



∂Φ
∂ω ∗




dω ∗
dω ∗
.
−
dω Φ=0dω Ω=0As the relative wage rate (ω) is a control variable and the level of market entry (n) isa state variable, we require one positive and one negative eigenvalue for saddle-pathstability. Accordingly, we consider long-run equilibria that satisfy |J1 | < 0, which isensured for ∂Φ/∂ω ∗ > 0 and (dω ∗/dω)|Ω=0 > (dω ∗/dω)|Φ=0, where ωklF > ρ + λ is asufficient condition for ∂Φ/∂ω > 0 and ∂Φ/∂ω ∗ > 0.
Appendix B: Proposition 1
The steady-state comparative statics associated with national labor allocations are
1 ∂Φ ∂Ω
dω=−,db|J2 | ∂ω ∗ ∂bdω
1 ∂Φ ∂Ω=−,
dϕ|J2 | ∂ω ∗ ∂ϕdω∂Φ ∂Ω ∂Φ ∂Ω
1
,
=−
−
dδ
|J2 | ∂ω ∗ ∂δ∂δ ∂ω ∗dω ∗1 ∂Φ ∂Ω=,
db|J2 | ∂ω ∂bdω ∗1 ∂Φ ∂Ω=,
dϕ|J2 | ∂ω ∂ϕ∂Φ ∂Ω ∂Φ ∂Ω
dω ∗
1
,
=
−
dδ
|J2 | ∂ω ∂δ∂δ ∂ωwhere |J2 | = (∂Ω/∂ω ∗ )(∂Φ/∂ω ∗ )((dω ∗/dω)|Φ=0 − (dω ∗/dω)|Ω=0 ) < 0 for |J1 | < 0.
These comparative statics are used to derive Proposition 1. First, we define ϕB as thethreshold value of the freeness of trade for which the denominator of ∂Ω/∂b equals
zero. Then, we have ∂Ω/∂b < 0, ω ∗ > ω, sX > 1/2, and sI > 1/2 for ϕ < ϕB , and∂Ω/∂b > 0, ω ∗ < ω, sX < 1/2, and sI < 1/2 for ϕ > ϕB .
Next, considering the effects of an increase in ϕ, since ∂Ω/∂ϕ < 0, we have dω/dϕ <0 and dω ∗ /dϕ > 0 for sX ∈ (0, 1). As such, when ϕ < ϕB , dsX /dϕ > 0 until sX = 1
at ϕ = ϕX , where ϕX is the threshold value of the freeness of trade at which theinvestment locus and the ω ∗ /ω = 1/δ line intersect in Figure 2. Similarly, dsI /dϕ > 0
until sI = 1 at ϕ = ϕI , where ϕI is the threshold value of the freeness of trade at28

which the investment locus intersects the ω ∗ = 1 line. Alternatively, when ϕ > ϕB ,sX = 0 until ϕ ≥ ϕ∗X , after which dsX /dϕ > 0, where ϕ∗X is the threshold value ofthe freeness of trade at which the investment locus and the ω ∗ /ω = δ line intersect.
Likewise, sI = 0 until ϕ ≥ ϕ∗I , after which dsI /dϕ > 0, where ϕ∗I is the threshold levelof value of the freeness of trade at which the investment locus intersects the ω = 1
line.
The ranking of ϕX and ϕI depends on the degree of knowledge diffusion. Given
that dω/dδ|Φ=0; dω∗ =0 = −(∂Φ/∂δ)/(∂Φ/∂ω) < 0, an increase in δ shifts the investment
locus downwards. Hence, as a rise in δ rotates the ω ∗ /ω = 1/δ line clockwise aroundthe origin, we can define the threshold value δ = δ at which ϕX = ϕI . Thus, we have
ϕX > ϕI for δ < δ and ϕX ≤ ϕI for δ ≥ δ. It can be similarly shown that ϕ∗I > ϕ∗Xfor δ < δ and ϕ∗I ≤ ϕ∗X for δ ≥ δ.
Appendix C: Propositions 2 and 3
First, we find that b − sX ≥ 0 for ω ∗ /ω ≤ (b + δb∗ )/(b∗ + δb), where 1/δ > (b +δb∗ )/(b∗ + δb) > 1. In addition, evaluating (20) at ϕ = 0, we have
δ + ωω ∗ + (1 − δ)(b − b∗ ωω ∗)αρb + δb∗ω∗=<,ω Ω=0; ϕ=0 δ + ωω ∗ + (1 − δ)(b∗ − bωω ∗ )αρb∗ + δbgiven that 1 > αρ. As d(ω ∗/ω)|Ω=0/dϕ > 0, there is a threshold value ϕ = ϕXO ∈(0, ϕX ) at which b = sX . Thus, b < sX for ϕ ∈ (ϕXO , ϕB ) and b > sX for ϕ 6∈
(ϕXO , ϕB ). This proves Proposition 2.
Next, we set b = sI to obtain bω(1 − ω ∗2 ) = b∗ ω ∗(1 − ω 2 ). Given b > b∗ , forrelative wage combinations that satisfy this condition, we have ω ∗ > ω. In addition,
this condition is strictly concave with a positive slope:ω ∗ 2 (1 + ω 2 )b∗d2 ω ∗
2b∗ ω ∗ 2
dω ∗
= 2
>
0,
=
−
dω
ω (1 + ω ∗2 )bdω 2b(1 + ω ∗ 2 )ω 329



(1 − ω ∗2 )(1 + ω 2)2
1−(1 − ω 2)(1 + ω ∗ 2 )2



< 0,

where the second term in parentheses in d2 ω ∗ /dω 2 is less than one for ω ∗ > ω. Since
the investment locus has a negative slope, it intersects bω(1 − ω ∗2 ) = b∗ ω ∗ (1 − ω 2) onceat the threshold value ϕ = ϕIO ∈ (0, ϕI ). Thus, we have b < sI for ϕ ∈ (ϕIO , ϕB ),and b > sI for ϕ 6∈ (ϕIO , ϕB ). The existence of the range (ϕIO , ϕX ) depends on theposition of the ω ∗ /ω = 1/δ line in Figure 2. As increases in δ cause the ω ∗ /ω = 1/δ
line to rotate clockwise around the origin and the investment locus to shift downwards(dω/dδ|Φ=0; dω∗ =0 = −(∂Φ/∂δ)/(∂Φ/∂ω) < 0), there is a threshold value δ = δ atwhich ϕIO = ϕX . Therefore, we have ϕIO < ϕX for δ < δ and ϕIO ≥ ϕX for δ ≥ δ, asoutlined in Proposition 3.
Finally, as sX and sI are increasing functions of ω ∗ /ω, we consider the effects ofchanges in ϕ and δ on sX and sI using n
1 d(ω ∗/ω)= (1 + ωω ∗)
+
Θ
1 ,
Θ2 dϕ
ωω ∗

1 d(ω ∗/ω)(1 − ϕ)n2(1 − αρ)n(1 − ϕ)Θ1
=−
+
−
,
2
∗
∗
Θ2 dδ
(1 − δ) (1 + ϕ)ωω
(1 + δ)(1 + ωω ) (1 − δ)2 (1 + ϕ)(ρ + λ)where Θ1 = 4(1 − αρ)(ωklF − ρ − λ)/((ρ + λ)(1 + ωω ∗)2 αlF ) > 0, and Θ2 = −((ω ∗ −ω)2 ω ∗ )/(αρ(b − b∗ )(1 + ωω ∗)ω 3 |J2 |) > 0. Therefore, d(ω ∗/ω)/dϕ > 0, and we havedSX /dϕ < 0 and dSI /dϕ < 0. In addition, for ϕ < ϕB , we have d(ω ∗ /ω)/dδ < 0, anda sufficient increase in δ shifts the SX and SI curves upwards in Figure 3.
Appendix D: Propositions 4 and 5First, the effects of changes in ωk on market entry and productivity growth aren(ρ + λ)
dn
=−
,
d(ωk)
lF ωk − ρ − λdg(σ − 1)γlF
=
.
d(ωk)
σ(ν − ρ − λ)30Then, the following are used with |J2 | < 0 to obtain Propositions 4 and 5:d(ωk)
2(1 − αρ)(ωklF − ρ − λ)ωk(ω ∗ − ω)2
=
< 0,
db
α(ρ + λ)(b − b∗ )lF (1 + ωω ∗)2 ω 2 |J2 |2(1 + δ)(1 − αρ)(ωklF − ρ − λ)ωk(ω ∗ − ω)3
d(ωk)
= 2
,
dϕ
α ρ(ρ + λ)lF (1 − δ)(1 + ϕ)2 (b − b∗ )(ω + ω ∗)(1 + ωω ∗)2 ω 2|J2 |
 ∗

d(ωk)
2ω
2(1 − αρ)(ωklF − ρ − λ)ωk
(1 − δ)−2 (1 − ϕ)(ω ∗ − ω)3
.
=−
+
dδ
α(ρ + λ)(1 + ωω ∗)2 (ω + ω ∗ )lF |J2 |ωαρ(1 + ϕ)(b − b∗ )(1 + ωω ∗)ω 2

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