R&D Subsidies, International Knowledge Dispersion, andFully Endogenous Productivity GrowthColin Davis
Doshisha University∗

Ken-ichi Hashimoto
Kobe University†

August, 2012

Abstract
This paper develops a two country model to investigate the effects of national R&D
subsidies on aggregate product variety and endogenous productivity growth withoutscale effects. In particular, monopolistically competitive firms invest in process innovation with the aim of lowering production costs. With imperfect knowledge dispersion, the larger of the two countries has a larger share of firms and a greater level ofproductivity. The higher concentration of relatively productive firms increases the sizeof knowledge flows between firms, leading to an increase in firm-level employment ininnovation. As a result, an economy with asymmetric countries produces a faster rateof growth than one with countries of similar size. The larger scale of firm-level innovation activity reduces market entry, however, and overall product variety falls. Using
this framework, we find that a national R&D subsidy has a positive effect on the industry share, relative productivity, and wage rate of the implementing country. Moreover,
if the smaller country introduces an R&D subsidy, overall product variety rises but therate of productivity growth falls. Alternatively, if the larger country introduces an R&D
subsidy, the rate of productivity growth rises, but overall product variety may rise or
fall. Finally, we briefly consider the effects of a national R&D subsidy on national andworld welfare levels.
Key Words: R&D Subsidy, Knowledge Dispersion, Productivity Growth, Scale EffectJEL Classifications: F43; O30; O40∗ The Institute for the Liberal Arts, Doshisha University, Karasuma Higashi, Imadegawa, Kamigyo, Kyoto,

Japan, 602-8580, cdavis@mail.doshisha.ac.jp.
† Graduate School of Economics, Kobe University, Rokkodai 2-1, Kobe, Japan, 657-8501,
hashimoto@econ.kobe-u.ac.jp.
Acknowledgements: We are grateful for helpful comments from T. Haruyama, T. Matsuoka, T. Ogawa, R.
Ohdoi, Y. Ono, and participants of seminars held at Kobe University, Tokyo Metropolitan University, Osaka
Prefecture University, and the Economic and Social Research Institute (ESRI) of the Cabinet Office of theGovernment of Japan. We also presented the paper at the Osaka University GCOE Conference and the Spring
Meeting of the Japanese Economic Association. We are responsible for all remaining errors.

1 Introduction
The discovery of novel ideas in the form of new products and production processes isbroadly recognized to be an essential driver of economic growth. As such, policy makers are perpetually looking for opportunities to accelerate growth by stimulating researchand development (R&D). In response, a diverse literature investigating the effects of economy policy designed to promote R&D activity has been developed with the objective ofproviding guidelines for policy makers. In this paper, we contribute to this literature witha study of the effects of national R&D subsidies on economic growth in the presence ofimperfect knowledge dispersion.
A key theme of the innovation-based endogenous growth literature has been the roleof technical knowledge dispersion amongst firms, and between regions and countries. Indeed, it is the non-rival nature of knowledge as an input in R&D that allows for perpetual
economic growth when the knowledge created by current innovation activity generates anintertemporal knowledge spillover that raises the labor productivity of all future R&D efforts (Romer, 1990). Given then the central role of knowledge dispersion in growth theory,a large body of empirical work has set out to verify the existence and measure the extent ofknowledge spillovers. While there is now convincing evidence that knowledge spillovers
do occur, their strength appears to diminish significantly with distance (Jaffe et al., 1993;
Mancusi, 2008; and Coe et al., 2009).1 It is this localized aspect of knowledge dispersion
that we focus on in our examination of the effects of national R&D subsidies.
Early comparative analysis of the policy implications of global versus local knowledge
spillovers can be found in Rivera-Batiz and Romer (1991) and Grossman and Helpman(1991, chp. 8). In these studies, an improvement in the flow of ideas between countries,
as a result of greater economic integration, raises labor productivity in R&D, thereby bolstering the rate of innovation. Within this context, we find that an exogenous reduction in1

See Keller (2004) for more detail on the various channels through which technical knowledge diffuses
between firms and across regions.

1

innovation costs induces a reallocation of resources from production to innovation activity, leading us to conclude that R&D subsidies affect economic growth positively, with thestrength of this effect increasing in the level of knowledge dispersion.
One potential concern with these results, however, is that they are inherently tied to ascale effect: a common feature of first-generation growth models whereby the growth rate is
positively related to population size, with an increase in the labor force raising the numberof researchers employed in innovation. In fact, there is a host of empirical evidence thatgenerally rejects the existence of scale effects (Jones, 1995a; Dinopoulus and Thompson,
1999; Barro and Sala-i-Martin, 2004; Lainez and Peretto, 2006), indicating that an appraisalof national R&D policy within a framework that corrects for scale effects and allows for imperfect knowledge dispersion would be a valuable exercise. Along these lines, Minniti andParello (2011) introduce diminishing technical opportunities into the intertemporal spilloverassociated with knowledge accumulation in a two country framework. This setting results
in a second-generation growth model (Jones, 1995b; Kortum, 1997; Segerstrom, 1998),
within which the rate of innovation depends solely on the rate of population growth andthe rate of diminishing technical opportunities, thus leaving no role for either imperfectknowledge dispersion or R&D policy in the promotion of economic growth. In contrast,
we develop a fully endogenous growth framework (Smulders and van de Klundert, 1991;
Peretto, 1996; Aghion and Howitt, 1998; Dinopoulus and Thompson, 1998; Young, 1998)
that accounts for scale effects while preserving a role for imperfect knowledge spillovers
and R&D policy. As there is now strong evidence supporting the validity of the fully endogenous growth approach over the semi-endogenous growth approach (Zachariadis 2003,
2004; Lainez and Peretto, 2006; Ha and Howitt, 2007; Madsen 2008, 2010; Madsen et al.
2010a; Madsen et al. 2010b), this paper contributes to the literature with a reexaminationof the implications for R&D policy when knowledge dispersion is incomplete, within an
empirically robust framework.
In particular, we develop a two country model in which monopolistically competitive2

firms invest in process innovation with the objective of increasing firm value through improvements in their production technologies that lower unit costs. We suppose that laborproductivity in process innovation reflects the average productivity of production technologies observable by the researchers employed at each firm. Then, with imperfect knowledge
dispersion, firm-level employment in innovation determines the relative production scales
and productivities of firms producing in each country. In order for innovation to occur inboth countries, the costs of innovation must equalize, inferring that a rise in the share offirms producing in a given country must be paralleled by an increase in these firms’ relative
productivities. We thus find that the country with the larger market, as measured by national
expenditure, attracts a larger share of firms, each with a greater level of productivity than
the firms producing in the smaller country.
The positive relationship between relative productivity and the share of firms producing
in a given country has important implications for overall product variety and the aggregate
rate of productivity growth. Specifically, the greater concentration of relatively productive
firms in the larger country generates a higher level of productivity in flows of technicalknowledge between firms, thereby raising R&D productivity and inducing a greater levelof employment in process innovation. With both labor productivity and labor employment
rising at the firm level, we find that an economy with asymmetric market sizes will havea greater rate of productivity growth than an economy with symmetric market sizes. As
the overall level of market entry is determined by optimal R&D employment, however,the lower cost of in-house process innovation associated with symmetric markets induces
greater market entry such that the symmetric equilibrium allows for the largest number offirms, and hence the greatest level of product variety.
Our framework has interesting implications for the effects of national R&D subsidies on
firm-level innovation activity. For example, consider the case where one country finances
an R&D subsidy to the process innovation of domestic firms by collecting a lump-sum income tax from domestic households. The national R&D subsidy has a positive effect on3

the industry share, relative productivity, and wage rate of the implementing country, causing adjustments in its terms of trade. The effects of the R&D subsidy on overall productvariety and aggregate productivity growth, however, depend on whether the implementing
country initially has the smaller or larger market. If the smaller of the two countries introduces an R&D subsidy, the larger country’s share of firms and relative productivity will
fall, pushing the economy towards the symmetric equilibrium, causing a rise in the level ofproduct variety, and inducing a fall in productivity growth. On the other hand, if the larger
country implements the R&D subsidy, its share of firms and relative productivity will rise,thus pushing the economy away from the symmetric equilibrium and raising productivity
growth. In this case, however, overall product variety may rise or fall.
While the large number of opposing effects associated with the R&D subsidy preventa general analysis of welfare issues, we are able to extract several conclusions from ourframework using simple numerical examples. First, when a national R&D subsidy is implemented by the larger country, its level will be too low from the perspective of the smaller
country. Second, when countries are symmetric, there is no incentive for either country
to promote domestic innovation activity, although both countries would prefer to have theother country provide an R&D subsidy. Third, if the smaller country implements an R&D
subsidy, its own welfare level will fall, but the welfare of the larger country may rise or falldepending on parameter values. In general, these results suggest that it may be possible toraise average world welfare through policy coordination, particularly if the larger country
agrees to provide an R&D subsidy that maximizes average world welfare. The results are,however, closely related to the level of knowledge spillovers with convergence in optimal
subsidy rates when there is perfect knowledge dispersion.
The remainder of this paper proceeds as follows. In Section 2 we introduce our model oftrade and endogenous productivity growth. Section 3 provides a characterization of longrun equilibrium, and Section 4 investigates the effects of R&D subsidies on the rate of
growth and the level of product variety. In Section 5, we discuss the implications of R&D

4subsidies for national and global welfare levels. Section 6 concludes.

2 Basic Model
This section develops a single sector model of trade in which firms produce horizontallydifferentiated product varieties and invest in in-house research with the aim of improving
their production technologies. There are two countries, home and foreign, which are symmetric with the exception of population sizes, L and L∗ , where an asterisk is used to denotevariables associated with the foreign country. Investment capital is perfectly mobile between countries, but there is no international labor migration. Home-based firms receive an
R&D subsidy, which is paid for with a lump-sum tax on home households. In what follows,
we focus on the home setup, but similar conditions can be derived for foreign.

2.1 Households
Each country is populated by dynastic households that chooses an optimal expendituresaving path in order to maximize lifetime utility over an infinite time horizon. The lifetime
utility of a home household is

U=

Z ∞

e−ρ t lnC(t)dt,(1)0where C(t) is household consumption at time t and ρ > 0 is the rate of time preference.
Consumption takes the form of a constant elasticity of substitution quantity index:

C(t) =

Z N(t)0ci (t)σ −1σdi

 σσ−1,

where N is the total number of product varieties, ci is the consumption of a representative
variety i, and σ > 1 is the elasticity of substitution between any two varieties.
At each moment in time, a home household allocates a given level of expenditure E

5across all available product varieties: N = n + n∗ , where n and n∗ are respectively the
numbers of product varieties produced in home and foreign. Thus, a home household’s
demand for a representative product variety m is

cm (t) = R n(t)0pm (t)−σ E(t)R n∗ (t) ∗ 1−σ ,p j (t)djm ∈ N,pi (t)1−σ di + 0(2)where pi and p∗j are the respective prices of home and foreign produced varieties. It thenfollows that the price index associated with aggregate consumption can be defined as

P(t) =

Z n(t)01−σpi (t)di +Z n∗ (t)0p∗j (t)1−σ d j 1−1σ.

(3)Thus, household expenditure is the product of the price and consumption indices: E = PC.
Households maximize lifetime utility (1) subject to the following intertemporal budget
constraint:Z ∞Rt

e− 0 r(s)ds E(t)dt ≤

0

Z ∞

Rt

e− 0 r(s)ds (w(t) − T (t))dt + B(0),0where r is the interest rate, w is the wage rate, T is a lump-sum tax, and B is asset holdingsin the home country.2
Substituting C = E/P into lifetime utility (1), and solving the households optimization
problem, we derive the following Euler equation:Ė(t)= r(t) − ρ ,
E(t)

(4)

where a dot over a variable denotes time differentiation. Perfect capital mobility ensures
a common evolution for home and foreign expenditures as the interest rate is equalized
between countries: Ė/E = Ė ∗ /E ∗ = r − ρ .
2 The value of asset holdings (B) will be driven to zero by free entry in the manufacturing sector.

Section 2.5 for more details.

6

SeeForeign households face a similar utility maximization problem and therefore have similar demand conditions. Time notation is suppressed where possible in order to economize
on notation.

2.2 Production
There is a mass of firms locating in each country, with each firm producing a unique horizontally differentiated variety and competing according to Dixit and Stiglitz (1977) monopolistic competition. While no cost is incurred in market entry, in addition to production
costs, each period incumbent firms face fixed costs in both the management (lF ) and theimplementation (lR ) of innovation activity.
Home and foreign firms employ lX and lX∗ units of labor with the following respective
production technologies:x = θ lX ,x∗ = θ ∗ lX∗ ,(5)where x and x∗ are firm-level outputs, and θ and θ ∗ are firm-specific productivity coefficients. We suppose that firm-level productivity is symmetric within a country, but may
differ across countries.
Given its current production technology, each firm produces to meet the combined demands of both countries. For example, the output of a home firm is x = ci L + c∗i L∗ . The
large number of firms operating in each market eliminates strategic interaction between
firms as they set their optimal production levels. It thus follows that, under monopolisticcompetition, each firm maximizes operating profit on sales π (≡ px − wlX ) by setting priceequal to a constant markup over unit cost, that is,

p=

σw
,
(σ − 1)θ

p∗ =

σ w∗
,
(σ − 1)θ ∗(6)

for home and foreign firms, respectively. Accordingly, optimal operating profit on sales can
7be obtained as π = wlX /(σ − 1). Substituting the pricing rules (6), the demand conditions(2), and the production functions (5) with aggregate world expenditure E W ≡ EL + E ∗ L∗
into x = ci L + c∗i L∗ , and reorganizing the result, we obtain the following expressions for
optimal firm levels of employment in production:lX =(σ − 1)p1−σ E w,
σ wP1−σlX∗ =(σ − 1)p∗ 1−σ E w,
σ w∗ P1−σ(7)where P = P∗ . These conditions determine the optimal production scales of home andforeign firms.

2.3 Process Innovation
Each firm invests in process innovation with the objective of raising profit through technology improvements that raise the productivity (θ ) of production. In particular, the productivities of home and foreign firms evolve according toγθ̇ ∗ = K ∗ lR∗ γ ,θ̇ = KlR ,(8)where lR and lR∗ are firm-levels of employment in innovation, K and K ∗ are the labor productivities associated with innovation, and γ ∈ (0, 1) is a parameter that ensures diminishingmarginal products to R&D investment.
A key aspect of the innovation process is the application of production processes employed by other incumbent firms. Following the in-house process innovation literature(Smulders and van de Klundert, 1995; Peretto, 1996), we assume that technical knowledge regarding production processes accumulates within the firm and can be proxied forusing the productivity coefficients θ and θ ∗ . This allows us to model the labor productivities of home and foreign firms in in-house process innovation as a weighted average of the8

productivity of currently observable technical knowledge:
K = sθ + δ s∗ θ ∗ ,K ∗ = s∗ θ ∗ + δ sθ ,(9)where s ≡ n/N and s∗ = 1 − s ≡ n∗ /N. The parameter δ ∈ (0, 1) regulates the level ofinternational knowledge spillovers. As δ approaches unity perfect knowledge spillovers
arise between countries, and as δ approaches zero knowledge spillovers become completely
national in scope.3 Improvements in production technology raise the level of knowledge,
thereby lowering the future cost of process innovation, and generating perpetual growth.
A firm’s intertemporal optimization problem entails choosing the level of investment inprocess innovation that maximizes the future flow of profits. Total per-period profit equals
operating profit on sales minus the cost of investment in process innovation lR and the perperiod fixed cost lF :
Π = π − (1 − β )w(lR + lF ),Π∗ = π ∗ − w∗ (lR∗ + lF ),(10)

where β ∈ [0, 1) captures the R&D subsidy rate provided to firms in the home country. We
assume that per-period fixed costs are symmetric across countries (lF = lF∗ ).
Firms choose the optimal level of employment in innovation with the objective of maximizing firm value. For example, a home firm sets lR to maximize V = 0∞ Π(θ ,t)e− 0 r(s)ds dt
RtR

subject to the technology constraint (8). This intertemporal optimization problem can beγ

solved using the following current-value Hamiltonian function: H = Π + µ KlR , where µ is
the current shadow value of an improvement in the productivity of production. Supposingthat market shares are small enough that each firm takes the price index (3) and expenditurelevels as constant when considering the impact of changes in its price on its profit flow, thefirst order conditions are ∂ H/∂ lR = 0 and ∂ H/∂ θ = r µ − µ̇ . These can be used to solved
3This method of modeling imperfect knowledge dispersion is adapted from Baldwin and Forslid (2000).

9

for the following no-arbitrage conditions:
γ −1

r=

γ lX lR K (1 − γ )l˙R ẇ K̇
++ − ,(1 − β )θ
lR
w K

r∗ =

γ lX∗ lR∗ γ −1 K ∗ (1 − γ )l˙R∗ w˙∗ K˙∗++ ∗ − ∗.
θ∗
lR∗
w
K

(11)

These conditions determine the optimal levels of R&D employment for home and foreign
firms, respectively.

2.4 Government
As discussed above, the home government finances the R&D subsidy using a lump-sum tax
on households. Aggregating the subsidies provided to home firms, a balanced government
budget leads to a lump-sum tax of
T L = nβ w(lR + lF ),(12)

where T is tax per household, and the lefthand side thus denotes aggregate tax revenue. The
righthand side represents the total R&D subsidy payment. As we focus on the effects of
R&D subsidies provided to firms in home, the R&D subsidy rate is set to zero in foreign,that is, β ∗ = T ∗ = 0.
2.5 Market Equilibrium
With zero costs incurred in product development and free market entry, per-period profits
are driven to zero in both home and foreign. As a consequence, household expenditures
are determined solely as a function of wages earned. To see this, first note that full labor
employment requiresL∗ = n∗ (lX∗ + lR∗ + lF ).

L = n(lX + lR + lF ),

10

(13)

Then, setting per-period profits (10) equal to zero, the free entry conditions for home andforeign based firms arelX = (σ − 1)(1 − β )(lR + lF ),lX∗ = (σ − 1)(lR∗ + lF ).

(14)

Next, using (13) and (14), in the home country a balanced government budget leads to a
lump-sum tax on each home household of T = wβ /(1 + (σ − 1)(1 − β )). Once again,
R&D subsidies are not provided in foreign. Thus, home and foreign expenditures are
E = (1 − η (β ))w,

E ∗ = w∗ ,(15)

where η (β ) ≡ β /(1 + (1 − β )(σ − 1)) is the effective tax rate; η (0) = 0 and η ′ (β ) > 0.
Thus, from (4), we have ẇ/w = Ė/E = ẇ∗ /w∗ = Ė ∗ /E ∗ = r − ρ at all moments in time.

3 Steady-State Equilibrium
This section characterizes the long-run equilibrium of the economy in terms of the relative
productivity of home firms and the share of firms based in home. We use this characterization of equilibrium to examine the effects of changes in relative labor endowments and the
R&D subsidy rate on relative productivity and firm-level employment in process innovation.
We begin by defining the relative productivity of home firms as follows:θ̃ ≡ θ /θ ∗ .

(16)

The evolution of relative productivity is determined by the difference between the homeand foreign productivity growth rates, as seen from the time derivative of (16):γθ̃˙θ̇ θ̇ ∗ KlR K ∗ lR∗ γ,= − ∗=
−
θ θ
θ
θ∗
θ̃

11

(17)

where we have used the technology constraints (8).
As we are interested in a steady state with a constant allocation of labor across production and innovation activities, given a constant level of R&D employment, the rate of productivity growth is constant and common across countries in long-run equilibrium. Thus,
setting the evolution of relative productivity (17) equal to zero yields
γ

KlR K ∗ lR∗ γ
.
=
θ
θ∗

(18)

With productivity growth equalized across countries, an examination of the time derivativesof K and K ∗ then reveals that θ̇ /θ = θ˙∗ /θ ∗ = K̇/K = K˙∗ /K ∗ in equilibrium.
Next, we consider the equilibrium investment conditions. Combining θ̇ /θ = K̇/K andθ˙∗ /θ ∗ = K˙∗ /K ∗ with the technology constraints (8) and the free entry conditions (14) yields
the steady-state no-arbitrage conditions for home and foreign as follows:

 γ

l K
lF
−1 R ,
ρ = R(lR ) ≡ γ (σ − 1) 1 +
lR
θ



 ∗γ ∗lF
l K
ρ = R∗ (lR∗ ) ≡ γ (σ − 1) 1 + ∗ − 1 R ∗ ,
lR
θ


(19)
(20)

where R(lR ) and R∗ (lR∗ ) represent the internal rates of return on investment in process innovation. Invoking (18), it is clear from these conditions that lR = lR∗ . Therefore, from thefree entry conditions (14), we also have K/θ = K ∗ /θ ∗ and lX /(1 − β ) = lX∗ in long-run
equilibrium.
We are now ready to solve for relative productivity. The equilibrium conditions enable
the derivation of two expressions for the share of firms based in home (s). The first is
obtained directly using the technology constraints (9) in K/θ = K ∗ /θ ∗ to solve for s =s(θ̃ ). The second is found using lR = lR∗ and the free-entry conditions (14) in the labormarket conditions (13) to solve for s = s(L/L∗ , β ). Setting these solutions equal yields the12following condition for the steady-state value of θ̃ :s=1 − δ θ̃ −1
L/L∗.
=
1 − δ θ̃ + 1 − δ θ̃ −1 L/L∗ + (1 + (1 − β )(σ − 1)) /σ(21)

Thus, as illustrated in Figure 1, relative productivity is determined implicitly by the relativelabor endowment and the R&D subsidy rate: θ̃ = θ̃ (L/L∗ , β ). Proportionate changes in thelabor endowments have no effect on relative productivity. Moreover, setting L/L∗ = 1 and

β = 0 leads to a symmetric equilibrium with s = 1/2 and θ̃ (1, 0) = 1. A change in the R&D
subsidy rate has a similar effect to a change in the relative labor endowment (L/L∗ ), andaccordingly s > L/(L + L∗ ) for β > 0. Note that with perfect knowledge dispersion (δ =1), the s(θ̃ ) curve becomes a vertical line and productivity is symmetric across countries(θ̃ = 1). We summarize the determinants of relative productivity in the following lemma.
Lemma 1 Increases in the relative labor endowment (L/L∗ ) and the R&D subsidy rate (β )raise the relative productivity of home firms.
Proof: Total differentiation of (21) givesd θ̃
(1 + (1 − β )(σ − 1))(δ − θ̃ )2> 0,=d(L/L∗ ) σ δ (1 − δ 2 + (θ̃ − δ )2 )(L/L∗ )2d θ̃
(σ − 1)(δ − θ̃ )2
=
> 0.
dβ
σ δ (1 − δ 2 + (θ̃ − δ )2 )(L/L∗ )Changes in L/L∗ and β affect relative productivity through adjustments in the shares offirms locating in each country. For instance, returning to Figure 1, consider a rise in the
R&D subsidy rate. The s = s(L/L∗ , β ) line shifts upwards resulting in an increase in therelative productivity of home firms. Intuitively, the increase in the R&D subsidy lowers
per-period fixed costs for home firms leading to an increase in profits and raising the shareof firms locating in home. This rise in home’s share of firms coincides with an increase inrelative productivity that brings knowledge spillovers back into balance, K/θ = K ∗ /θ ∗ , as
the economy returns to steady-state equilibrium.
13

Figure 1: Firm Shares and Relative Productivityss = s(θ̃ )1s = s(L/L∗ , β )

0

δ

1/δ

θ̃

Next, we consider the relationship between relative productivity and knowledge spillovers,
as it will provide an important link in the determination of firm-level R&D employment.
Substituting s = s(θ̃ ) from (21) into the expression for observable knowledge (9), and reorganizing the result, yields the equilibrium level of relative knowledge spillovers as follows:
K K∗
1−δ2
= ∗ =
.
θ
θ
2 − δ θ̃ − δ θ̃ −1(22)This equilibrium condition provides the following result:Lemma 2 The steady-state level of relative knowledge spillovers is convex in relative productivity with a minimum at θ̃ = 1.
As shown in Figure 2, an increase in relative productivity lowers relative knowledge
spillovers for θ̃ < 1 and raises relative knowledge spillovers for θ̃ > 1. From (21), we cansee that θ̃ = 1/δ when all firms locate in home (s = 1), and that θ̃ = δ when all firms
locate in foreign (s = 0). Thus, it is clear that an increase in the concentration of industry
in one country raises labor productivity in R&D activity. This result is a common featureof new economic geography models that assume imperfect knowledge dispersion between
countries (Baldwin and Martin, 2004).
Finally, we investigate the relationship between firm-level innovation employment andthe internal rate of return. Figure 3 provides an illustration of the no-arbitrage condition for
14

Figure 2: Relative Knowledge Spillovers
K
θ

0

θ̃

1

home. At every moment in time, firms choose optimal levels of in-house innovation with theaim of maximizing firm value. Thus, when the internal rate of return R(lR ) is greater than
the risk free rate of return ρ (= r), firms increase R&D employment, and when R(lR ) < ρ ,they reduce R&D employment. This investment behavior implies that the internal rate of
return should be decreasing in R&D employment, as depicted by the negative slope ofR(lR ) in Figure 3, in order to ensure a positive and finite level of employment in process
innovation, thereby ensuring a balanced growth path. In the Appendix, we examine thelocal dynamics of the economy around a symmetric long-run equilibrium for L/L∗ = 1, and

β = 0. We find that if 2 > (σ − 1)(1 − δ + 2(1 + δ )γ ), then γ −1(1 − γ )(σ − 1)lF γ lR K
∂ R(lR )< 0,= − 1 − γ (σ − 1) +
∂ lR
lR
θ

(23)

and the symmetric equilibrium is saddle-point stable.4 Henceforth, we characterize thelong-run equilibrium and investigate the determinants of productivity growth and productvariety under the assumption that ∂ R(lR )/∂ lR < 0.
With relative productivity determined as a function of the relative labor endowment
and the R&D subsidy rate, and relative knowledge spillovers determined as a function ofrelative productivity, it is now possible to examine the effects of changes in L/L∗ and β on
4 See condition (A6) in the Appendix.

15

Figure 3: In-house R&D EmploymentrρR(lR )lR0firm-level employment in process innovation.
Lemma 3 Increases in the relative labor endowment (L/L∗ ) and the R&D subsidy rate (β )lower firm-level employment in innovation for θ̃ < 1 and raise it for θ̃ > 1.
Proof: Substituting (22) into (19) and taking the derivative with respect to lR and θ̃ yields:
1 ρδ (1 − θ̃ −2 )(K/θ )
dlR
.
=− ′
R (lR )1−δ2d θ̃This derivative can be signed using (23) and Lemma 1.
Changes in L/L∗ and β affect lR indirectly through θ̃ and subsequently K/θ . For example, consider the effect of an increase in the R&D subsidy rate. When home has a greater
share of firms and θ̃ > 1, the rise in β increases θ̃ , thereby raising relative knowledge
spillovers. Returning to Figure 3, the internal rate of return to process innovation R(lR )shifts upward and firms increase employment in process innovation until the internal rate of
return falls back to the risk free level ρ . On the other hand, when home has a smaller shareof firms and θ̃ < 1, the increase in β shifts the R(lR ) curve downwards and firms reduceinnovation employment.

16

Figure 4: Product Variety and Productivity GrowthgNg(θ̃ )

N(θ̃ )0θ̃1(a) Total number of Firms

01(b) Rate of Productivity Growthθ̃

4 Product Variety and Productivity GrowthThis section investigates the effects of changes in the home R&D subsidy on equilibrium
product variety and the rate of productivity growth.
Combining the labor market conditions (13) with the free-entry conditions (14) and thesteady-state condition lR = lR∗ , we can solve for the equilibrium level of product variety as
1N=
lR + lF




L
L∗
+
.
1 + (1 − β )(σ − 1) σ(24)

As illustrated in Figure 4a, equilibrium product variety and relative productivity have aconcave relationship with a maximum occurring at θ̃ = 1. The mechanics of this relationship can be understood by beginning from a symmetric equilibrium, in which productivity
is equalized across countries, and then considering the effects of a change in relative productivity. Either an increase or a decrease in θ̃ raises relative knowledge spillovers, andfirm-level innovation employment rises, leading to an increase in per-period fixed costs. As
a result, a portion of firms exit the market and equilibrium product variety falls.
Investigating the effects of the home R&D subsidy on equilibrium product variety, we

17obtain the following result.
Proposition 1 For θ̃ ≤ 1, an increase in the R&D subsidy rate (β ) raises product variety.
For θ̃ > 1, however, product variety may rise or fall.
Proof: Taking the total derivative of (24) and applying Lemma 3, we have
N dlR
(σ − 1)L
dN
−
=
.
2
dβ
(lR + lF )(1 + (1 − β )(σ − 1))lR + lF d β
The R&D subsidy has two effects on equilibrium product variety. The first is a positive
direct effect that shifts the N(θ̃ ) curve upward in Figure 4a. A rise in β directly lowers
per-period fixed costs causing a decrease in firm scale, through the free entry conditions(14), and allowing the market to support a larger number of firms for all levels of relative
productivity. The second, in contrast, is an indirect effect that moves the economy along
the N(θ̃ ) curve as firms adjust their optimal investment levels in response to the change
in relative knowledge spillovers which results from the increase in the R&D subsidy rate.
Invoking Lemma 3, through changes in lR , an increase in β will raise N for θ̃ < 1 andreduce N for θ̃ > 1. Thus, we conclude that when home has the larger number of firms andθ̃ < 1, an R&D subsidy can be used to raise product variety, but for θ̃ > 1 the overall effectwill be ambiguous.
Next, we calculate the steady-state rate of productivity growth. Substituting the noarbitrage conditions (19) into the innovation function (8) and manipulating the result yields
γ

l K
θ̇
ρ
.
g≡ = R =θθγ (σ − 1) − 1 + γ (σ − 1)lF /lR(25)

An examination of (25) shows that the rate of productivity growth is determined independently of the overall size of the labor force, and is therefore not biased by a scale effect,as θ̃ = θ̃ (L/L∗ , β ) is unaffected by proportionate changes in the labor endowments. Intuitively, an increase in the overall labor force is absorbed fully by a rise in the number ofincumbent firms (24). Then, although the aggregate level of employment associated with18

innovation activity (lR + lF )N = L/(1 + (1 − β )(σ − 1)) + L∗ /σ rises with proportionate
increases in L and L∗ , firm-level employment in process innovation remains unchanged. As
such, our model does not include a scale effect and is consistent with empirical evidence.
Figure 4b depicts the convex relationship between productivity growth and relative productivity, with the rate of productivity growth minimized at θ̃ = 1. Starting once again froma symmetric equilibrium in which productivity is equalized across countries, either a rise ora fall in relative productivity increases relative knowledge spillovers, leading to increases
in both labor employment and labor productivity in process innovation. As a consequence,
the rate of productivity growth rises. Intuitively, as investment in process innovation rises,
per-period fixed costs (lR + lF ) increase and firms are pushed out of the market. With thenumber of incumbent firms decreasing, the resources required to cover economy-wide perperiod fixed costs NlF fall, and the amount of labor available for employment in innovation
rises.
We examine the relationship between the home R&D subsidy and productivity growthand obtain the following proposition:
Proposition 2 An increase in the R&D subsidy (β ) lowers productivity growth for θ̃ < 1
and raises productivity growth for θ̃ > 1.
The R&D subsidy only effects productivity growth indirectly through changes in relative productivity. Returning once again to Lemma 1, d θ̃ /d β > 0 and an increase in the R&D
subsidy leads to a rightward movement along the g(θ̃ ) curve in Figure 4b. Accordingly, theR&D subsidy depresses productivity growth for θ̃ < 1 and accelerates it for θ̃ > 1. Notethat the R&D subsidy has no effect on productivity growth for a symmetric equilibrium
evaluated at β = 0.

19

5 Welfare Analysis
The welfare issues associated with our model are rather complex, given the opposing effects
of a national R&D subsidy on product variety, productivity growth, and the terms of trade.
In this section, we provide a simple discussion of welfare and use a numerical example
to show the optimal levels of the home R&D subsidy for the home, foreign, and world
economies.
Although the relative wage drops out of the model in the equilibrium analysis of thepreceding two sections, it assumes a role in the welfare effects associated with adjustmentsin the terms of trade. Substituting the pricing rules (6), the price index (3), and optimal
firm-level employment in production (7) into lX /(1 − β ) = lX∗ and simplifying yields therelative wage rate as w/w∗ = (1 − β )− σ θ̃ (L/L∗ , β )σ −1 , where w/w∗ = 1 in the symmetric1equilibrium for L/L∗ = 1 and β = 0. Hence, the terms of trade for the home country are1p= (1 − β )θ̃ (L/L∗ , β ) σ .
∗pThus, while the direct effect of an increase in the R&D subsidy rate is a deterioration ofhome’s terms of trade, a rise in relative productivity will lead to an improvement in p/p∗ .
Moving on to national welfare levels, the present values of utility flows to home andforeign households are found by substituting (3), (6), (15), (21), and the terms of trade into
lifetime utility (1):
− σσ−1
1gρ U0 = ln(1 − η (β ))A +ln s(β ) + (1 − s(β )) (1 − β )θ̃N+ ,(σ − 1)ρ σ −1
1gln s(β ) (1 − β )θ̃ σ + (1 − s(β )) N + ,ρ U0∗ = ln A∗ +(σ − 1)
ρ

(26)
(27)

where A = (σ − 1)θ (0)/σ and A∗ = (σ − 1)θ ∗ (0)/σ are constants, and s(β ) is defined in
the second expression of (21).
Focusing again on the home country, a change in the R&D subsidy affects welfare
20Figure 5: Optimal Home R&D Subsidies
U0 ,U0∗

U0

0

βo βW

βo∗

∗
U W U01βParameter values are L/L∗ = 2, σ = 1.75, δ = 0.5, γ = 0.5, lF = 0.01, and ρ = 0.02.

through four separate channels. First, η (β ) captures the negative income effect of the lumpsum tax placed on home households to pay for the R&D subsidy. Next, the second termin parenthesis describes the terms of trade effect that results from changes in production
shares (s), the relative wage rate (w/w∗ ), and relative productivity (θ̃ ). The third term isthe effect of the R&D subsidy on the level of product variety. Finally, the fourth term
describes the effect of the R&D subsidy on the productivity growth rate. The opposing
effects of the home R&D subsidy on each of these terms makes a general analytical analysis
of welfare intractable. As an alternative, we use a simple numerical example to discuss theimplications of changes in the R&D subsidy for specific cases.
Figure 5 plots home and foreign welfare against the home R&D subsidy rate for thecase where L/L∗ = 2, and the population of home is twice that of foreign. In this case,home hosts a greater share of firms, which are relatively more productive, that is, s > 1/2
and θ̃ > 1. The home R&D subsidy rates that maximize home and foreign welfare levels
are respectively βo and βo∗ , and accordingly, we can see that home will set the subsidy
rate at a lower level than is optimal for foreign. This suggests an opportunity for policy
coordination between home and foreign with the aim of maximizing the welfare level ofthe average household. Denoting welfare for the average household in the world economy
by U W ≡ (L/(L + L∗ ))U0 + (L∗ /(L + L∗ ))U0∗, the optimal home subsidy rate would then21

become βW , as depicted in Figure 5. Thus, as βW > βo , average world welfare can beincreased by raising the home R&D subsidy rate above the optimal level for home residents.
Generally, a positive level for βo only arises when the home country has a relatively
large population. For example, when country sizes are symmetric, that is L/L∗ = 1, we canevaluate the welfare effects of change in the R&D subsidy as follows:dU0(σ − 1)(1 − δ )=−< 0,d β β =0
4σ 2 δ ρ

dU0∗(σ − 1)(1 − δ )=> 0.
d β β =0
4σ 2 δ ρ

In this symmetric case, we find that the R&D subsidy lowers home welfare and raises
foreign welfare. Hence, while the optimal home R&D subsidy rate would be positive forforeign (βo∗ > 0), home would prefer to set the R&D subsidy to zero (βo = 0), or perhaps
even to implement an R&D tax (βo < 0). The optimal R&D subsidy rate for the averageworld household is zero in this case, as (dU W /d β )|βW = 0. Numerical analysis shows thatthese results continue to hold for L/L∗ < 1, although βo∗ > 0 falls to zero as L/L∗ approaches
zero. As a final point, we note that an increase in the level of knowledge dispersion leads
to policy convergence with the optimal subsidies βo and βo∗ equalized at δ = 1, given that
productivity is symmetric across countries, θ̃ = 1.

6 Conclusion
In this paper we have developed a two country model of fully endogenous growth in order
to investigate the effects of national R&D subsidies on aggregate product variety and economic growth. Monopolistically competitive firms assume a central role in the model as
they invest in process innovation with the aim of improving production technologies andlowering unit costs. In a world of perfect capital mobility, but imperfect knowledge dispersion, the country with the larger market maintains a larger share of manufacturing firms, agreater level of productivity, and a higher wage rate. The greater concentration of relatively
productive firms in the larger country generates higher average productivity in flows of tech22

nical knowledge between firms, thereby raising R&D productivity and inducing a greaterlevel of employment in in-house process innovation. As a result, we find that equilibria withasymmetric country sizes produce faster rates of growth than a symmetric equilibrium. As
the overall level of market entry is determined by optimal R&D employment, however, thelower cost of process innovation associated with symmetric markets generates greater market entry so that the symmetric equilibrium produces the largest number of firms, and thegreatest level of product variety.
We use this framework to investigate how national R&D subsidies affect firm-level innovation activity. In particular, we consider the case where one country finances an R&D
subsidy to the process innovation of domestic firms through a lump-sum income tax collected from domestic households. The R&D subsidy has positive effects on industry share,
relative productivity, and the wage rate in the implementing country. The effects of the
R&D subsidy on overall product variety and aggregate productivity growth, however, depend on whether the implementing country initially has the smaller or larger market. If thesmaller of the two countries introduces an R&D subsidy, the larger country’s share of firms
and relative productivity will fall, pushing the economy towards the symmetric equilibrium,
causing a rise in product variety, and inducing a fall in productivity growth. In contrast, if
the larger country implements the R&D subsidy, its share of firms and relative productivity will rise, thus pushing the economy away from the symmetric equilibrium and raisingproductivity growth. In this case, however, overall product variety may rise or fall.
Although the large number of opposing effects generated by national R&D subsidies
makes a direct study of welfare issues intractable, simple numerical examples lead us toconclude that the optimal level for a national R&D subsidy that is implemented by thelarger country will be too low from the perspective of the smaller country. As such, there
may be opportunities for policy coordination between the two countries with the aim ofraising average world welfare by increasing the R&D subsidy.
Given these preliminary results, an interesting extension of our framework might be an23

analysis of strategic R&D policy. For example, Kondo (2012) investigates the relationshipbetween R&D subsidies and economic growth in a variety expansion model that allows
for strategic policy interaction in the present of imperfect knowledge dispersion, but that is
subject to a scale effect. An simplified version of the model presented in this paper mightallow for an investigation of the relationship between strategic R&D policy and endogenous
growth that is not biased by a scale effect. We leave this issue as a topic for future work.

24Appendix: Stability of Symmetric Equilibrium
This appendix derives a sufficient condition for the stability of the symmetric equilibrium.
In order to examine the stability of the model, we first require a condition for home’s shareof firms outside the steady-state equilibrium. As both countries have the same rate of time
preference and there are no flows of investment income, trade flows balance at every moment in time. Therefore, using the product demand conditions (2), the pricing rules (6), andhousehold expenditures (15) into the trade balance condition npc∗i L∗ = n∗ p∗ c j L, we obtain
the home share of firms on the dynamic path as follows:s=(1 − η (β ))L/L∗.
(1 − η (β ))L/L∗ + (w/w∗ )−σ θ̃ σ −1

(A1)

Next, conditions (11) and (17) can be combined with K/θ = δ θ̃ −1 + s(1 − δ θ̃ −1 ), andK ∗ /θ ∗ = 1 + s(δ θ̃ − 1) to obtain the following dynamic system:
 γlR K lR∗ γ K ∗˙,θ̃ = θ̃
−
θ
θ∗



γ 
K
γ
(
σ
−
1)l
l
l
K̇
R
F
R
l˙R =
1+
ρ−
+
,
1−γ
θ
lR
K

∗γ ∗ 
∗
∗ 
K
γ
(
σ
−
1)l
K̇
l
l
F
R
R
∗
ρ−
l˙R =
1+ ∗ + ∗ ,
1−γ
θ∗
lR
K

(A2)
(A3)
(A4)

where
 ∗γ
(1
−
s)
1
+
(
δ
θ̃
−
1)s
δ lR
s(1 − s)(σ − 1)(δ − θ̃ ) θ̃˙K̇γ= slR ++,
K
δ + (θ̃ − δ )s
δ + (θ̃ − δ )sθ̃s δ + (θ̃ − δ )s δ lR γ s(1 − s)(σ − 1)(1 − δ θ̃ ) θ̃˙
K̇ ∗
∗γ
 +

= (1 − s)lR +.
K∗
1 + (δ θ̃ − 1)s
1 + (δ θ̃ − 1)sθ̃In our study of this system, we set the relative wage equal to its steady-state level, w/w∗ =˙ ∗ ) = (1 − η (β ))−1 (E/E˙ ∗ ) = 0. Then, (A1), (A2),(1 − β )−1/σ θ̃ (L/L∗ , β )σ −1 , since (w/w
(A3), and (A4) provide an autonomous system in θ̃ , lR , and lR∗ . We investigate the local

25

dynamics around the symmetric equilibrium that arises for L/L∗ = 1 and β = 0:θ̃ = 1,w= 1,w∗

lR = lR∗ = l¯R ,

1
s= ,
2

K K∗ 1 + δ
= ∗ =
.
θ
θ
2

(A5)

Taking a linear expansion of (A2), (A3), and (A4) and evaluating it around (A5) yields thefollowing Jacobian matrix:γ(2δ +(1−δ )(σ −1))lR
2
 −

∂ l˙R
J=

∂ θ̃

∂ l˙R∗∂ θ̃

γ −1

γ (1+δ )lR
2
∂ l˙R
∂ lR
γ
(2δ +(1−δ )(σ −1))γ lR4(1−γ )γ −1

γ (1+δ )lR2γ(2δ +(1−δ )(σ −1))γ lR4(1−γ )
∂ l˙R∗
∂ lR∗





,



where
γ
∂ l˙∗ [2δ + (1 − δ )(σ − 1)][(1 − δ )(σ − 2)lR + γ (σ − 1)(1 + δ )(lR + lF )] lR∂ l˙R,=− R =
4(1 − γ )(1 + δ )
∂ θ̃
∂ θ̃
γ −1
∂ l˙R ∂ l˙R∗ [2(1 − γ )(1 + δ )(σ − 1)lF − (2(σ − 2) + (1 − δ )(σ − 1)) lR ] γ lR
= ∗ =.
∂ lR ∂ lR
4(1 − γ )As the system consists of two jump variables (lR and lR∗ ) and one state variable (θ̃ ),we require two eigenvalues with positive real parts and one eigenvalue with a negative realpart for saddle-point stability. While we cannot solve for the eigenvalues directly, saddlepoint stability is established for tr(J) > 0 > |J|, where tr(J) and |J| denote the trace anddeterminant of J, respectively. Calculating tr(J) and |J|, we obtainγ(2δ + (1 − δ )(σ − 1))lR
2lR ∂ R(lR ),−tr(J) = −2(1 − γ )(1 − γ ) ∂ lR
2γ

γ (1 + δ )(σ − 1)(2δ + (1 − δ )(σ − 1))lR lF ∂ R(lR )|J| =,4(1 − γ )2
∂ lR
γ −1

where ∂ R(lR )/∂ lR = − [1 − γ (σ − 1) + (1 − γ )(σ − 1)lF /lR ] γ (1 + δ )lR /2. Thus, saddlepoint stability is achieved if ∂ R(lR )/∂ lR is sufficiently negative. Returning to the steadystate no-arbitrage condition (19), we can see that lF /lR > (1 − γ (σ − 1))/(γ (σ − 1)) is re26

quired for active process innovation. Then, substituting lF /lR = (1 − γ (σ − 1))/(γ (σ − 1))with ∂ R(lR )/∂ lR into tr(J), and manipulating the result, we obtain(1 + δ ) (1 − γ (σ − 1)) >2δ + (1 − δ )(σ − 1)
> 0.
2

(A6)

This inequality can be reduced to 2 > (σ − 1)(1 − δ + 2(1 + δ )γ ), which represents a sufficient condition for ∂ R(lR )/∂ lR < 0, tr(J) > 0, |J| < 0, and the saddle-point stability of the
dynamic path around the symmetric equilibrium.

27

References[1] Aghion, P., Howitt, P., 1998. Endogenous growth. Cambridge: MIT Press.
[2] Baldwin, R., Forslid R., 2000. The core-periphery model and endogenous growth:
Stabilizing and destabilizing integration. Economica 67, 307-324.
[3] Baldwin, R., Martin, P., 2004. Agglomeration and regional growth. Handbook ofRegional and Urban Economics 4, 2671-2711.
[4] Barro, R., Sala-i-Martin, X., 2004. Economic growth. Cambridge: MIT Press.
[5] Coe, D., Helpman, E., Hoffmaister, A., 2009. International R&D spillovers andinstitutions. European Economic Review 53, 723-741.
[6] Dinopoulos, E., Thompson, P., 1998. Schumpeterian growth without scale effects.
Journal of Economic Growth 3, 313-335.
[7] Dinopoulos, E., Thompson, P., 1999. Scale effects in Schumpeterian models of
economic growth. Journal of Evolutionary Economics 9, 157-185.
[8] Dixit, A., Stiglitz, J. E., 1977. Monopolistic competition and optimum productdiversity. American Economic Review 67, 297-308.
[9] Grossman, G., Helpman, E., 1991. Innovation and growth in the global economy.
Cambridge: MIT Press.
[10] Ha, J., Howitt, P., 2007. Accounting for trends in productivity and R&D: A Schumpeterian critique of semi-endogenous growth theory. Journal of Money, Credit andBanking 39, 733-774.
[11] Jaffe, A., Trajtenberg, M., Henderson, R., 1993. Geographical localization of knowledge spillovers as evidenced by patent citations. The Quarterly Journal of Economics
108, 577-598.
[12] Jones, C., 1995a. Times series tests of endogenous growth models. Quarterly Journalof Economics, 110, 495-525.
[13] Jones, C., 1995b. R&D-based models of endogenous growth. Journal of Political
Economy, 103, 759-784.
[14] Keller, W., 2004. International technology diffusion. Journal of Economic Literature
42, 752-782.
[15] Kondo, H., 2012. International R&D subsidy competition, industrial agglomeration,and growth. Journal of International Economics, doi: 10.1016/j.jinteco.2012.04.004.
[16] Kortum, S., 1997. Research, patenting and technological change. Econometrica 65,
1389-1419.
[17] Lainez, C., Peretto, P., 2006. Scale effects in endogenous growth theory: An error of
aggregation not specification. Journal of Economic Growth 11, 263-288.
[18] Madsen, J., 2008. Semi-endogenous versus Schumpeterian growth models: testing
the knowledge production function using international data. Journal of Economic
Growth 13, 1-26.

28

[19] Madsen, J., 2010. The anatomy of growth in the OECD since 1870. Journal of
Monetary Economics 57, 753-767.
[20] Madsen, J., Saxena, S., Ang, J., 2010a. The Indian growth miracle and endogenous
growth. Journal of Development Economics 93, 37-48.
[21] Madsen, J., Ang, J., Banerjee, R., 2010b. Four centuries of British economic growth:
the roles of technology and population. Journal of Economic Growth 15, 263-290.
[22] Mancusi, M., 2008. International spillovers and absorptive capacity: A cross-country
cross-sector analysis based on patents and citations. Journal of International Economics 76, 155-165.
[23] Minniti, A., Parello, C., 2011. Trade integration and regional disparity in a model ofscale-invariant growth. Regional Science and Urban Economics 41, 20-31.
[24] Peretto, P., 1996. Sunk costs, market structure, and growth. International Economic
Review 37, 895-923.
[25] Rivera-Batiz, L. Romer, P., 1991. Economic integration and endogenous growth. The
Quarterly Journal of Economics 106, 531-555.
[26] Romer, P., 1990. Endogenous technological change. Journal of Political Economy 98,
S71-S102.
[27] Segerstrom, P., 1998. Endogenous growth without scale effects. American Economic
Review 88, 1290-1310.
[28] Smulders, S., van de Klundert, T., 1995. Imperfect competition, concentration, andgrowth with firm-specific research. European Economic Review 39, 139-160.
[29] Young, A., 1998. Growth without scale effects. Journal of Political Economy 106,
41-63.
[30] Zachariadis, M., 2003. R&D, innovation, and technological progress: a test of the
Shumpeterian framework without scale effects. Canadian Journal of Economics 36,
566-586.
[31] Zachariadis, M., 2004. R&D-induced growth in the OECD? Review of Development
Economics 8, 423-439.

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