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# R&D Subsidies, International Knowledge Dispersion, and Fully Endogenous Productivity Growth

Colin Davis, Ken-ichi Hashimoto

August 2012

Economic and Social Research Institute

Cabinet Office

Tokyo, Japan

The views expressed in “ESRI Discussion Papers” are those of the authors and not those of the Economic and Social Research Institute, the Cabinet Office, or the Government of Japan.
# R&D Subsidies, International Knowledge Dispersion, and Fully Endogenous Productivity Growth

Colin Davis

Doshisha University $ ^{*} $

Ken-ichi Hashimoto

Kobe University $ ^{\dagger} $

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 without scale 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 of productivity. The higher concentration of relatively productive firms increases the size of knowledge flows between firms, leading to an increase in firm-level employment in innovation. As a result, an economy with asymmetric countries produces a faster rate of 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 the rate 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 and world welfare levels.

Key Words: R&D Subsidy, Knowledge Dispersion, Productivity Growth, Scale Effect

JEL Classifications: F43; O30; O40
## 1 Introduction

The discovery of novel ideas in the form of new products and production processes is broadly recognized to be an essential driver of economic growth. As such, policy makers are perpetually looking for opportunities to accelerate growth by stimulating research and 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 of providing guidelines for policy makers. In this paper, we contribute to this literature with a study of the effects of national R&D subsidies on economic growth in the presence of imperfect knowledge dispersion.

A key theme of the innovation-based endogenous growth literature has been the role of 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 an intertemporal 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 of knowledge 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 in
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 the strength of this effect increasing in the level of knowledge dispersion.

One potential concern with these results, however, is that they are inherently tied to a scale 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 number of researchers employed in innovation. In fact, there is a host of empirical evidence that generally rejects the existence of scale effects (Jones, 1995a; Dinopoulos and Thompson, 1999; Barro and Sala-i-Martin, 2004; Lainez and Peretto, 2006), indicating that an appraisal of 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 and Parello (2011) introduce diminishing technical opportunities into the intertemporal spillover associated 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 and the rate of diminishing technical opportunities, thus leaving no role for either imperfect knowledge 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; Dinopoulos 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 reexamination of 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 competitive
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 labor productivity 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 in both countries, the costs of innovation must equalize, inferring that a rise in the share of firms 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 technical knowledge between firms, thereby raising R&D productivity and inducing a greater level of 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 have a 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 of firms, 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 on
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 product variety 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 of product 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 prevent a general analysis of welfare issues, we are able to extract several conclusions from our framework 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 the other 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 fall depending on parameter values. In general, these results suggest that it may be possible to raise 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 of trade and endogenous productivity growth. Section 3 provides a characterization of long-run 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
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 horizontally differentiated 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 denote variables 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 expenditure-saving path in order to maximize lifetime utility over an infinite time horizon. The lifetime utility of a home household is

$$ U=\int_{0}^{\infty}e^{-\rho t}\ln C(t)d t, $$

where $ C(t) $ is household consumption at time t and $ \rho > 0 $ is the rate of time preference. Consumption takes the form of a constant elasticity of substitution quantity index:

$$ C(t)=\left(\int_{0}^{N(t)}c_{i}(t)^{\frac{\sigma-1}{\sigma}}d i\right)^{\frac{\sigma}{\sigma-1}}, $$

where N is the total number of product varieties, $ c_{i} $ is the consumption of a representative variety i, and $ \sigma > 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
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

$$ c_{m}(t)=\frac{p_{m}(t)^{-\sigma}E(t)}{\int_{0}^{n(t)}p_{i}(t)^{1-\sigma}d i+\int_{0}^{n^{*}(t)}p_{j}^{*}(t)^{1-\sigma}d j},\quad m\in N, $$

where $ p_{i} $ and $ p_{j}^{*} $ are the respective prices of home and foreign produced varieties. It then follows that the price index associated with aggregate consumption can be defined as

$$ P(t)=\left(\int_{0}^{n(t)}p_{i}(t)^{1-\sigma}d i+\int_{0}^{n^{*}(t)}p_{j}^{*}(t)^{1-\sigma}d j\right)^{\frac{1}{1-\sigma}}. $$

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:

$$ \int_{0}^{\infty}e^{-\int_{0}^{t}r(s)ds}E(t)dt\leq\int_{0}^{\infty}e^{-\int_{0}^{t}r(s)ds}\left(w(t)-T(t)\right)dt+B(0), $$

where r is the interest rate, w is the wage rate, T is a lump-sum tax, and B is asset holdings in the home country. $ ^{2} $

Substituting C = E/P into lifetime utility (1), and solving the households optimization problem, we derive the following Euler equation:

$$ \frac{\dot{E}(t)}{E(t)}=r(t)-\rho, $$

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: $ \dot{E}/E = \dot{E}^{*} / E^{*} = r - \rho $ .
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 ( $ l_{F} $ ) and the implementation ( $ l_{R} $ ) of innovation activity.

Home and foreign firms employ $ l_{X} $ and $ l_{X}^{*} $ units of labor with the following respective production technologies:

$$ x=\theta l_{X}, $$

$$ x^{*}=\theta^{*}l_{X}^{*}, $$

where x and $ x^{*} $ are firm-level outputs, and $ \theta $ and $ \theta^{*} $ 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 = c_{i}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 monopolistic competition, each firm maximizes operating profit on sales $ \pi (\equiv px - wl_{X}) $ by setting price equal to a constant markup over unit cost, that is,

$$ p=\frac{\sigma w}{(\sigma-1)\theta}, $$

$$ p^{*}=\frac{\sigma w^{*}}{(\sigma-1)\theta^{*}}, $$

for home and foreign firms, respectively. Accordingly, optimal operating profit on sales can
be obtained as $ \pi = wl_{X}/(\sigma - 1) $ . Substituting the pricing rules (6), the demand conditions (2), and the production functions (5) with aggregate world expenditure $ E^{W} \equiv EL + E^{*}L^{*} $ into $ x = c_{i}L + c_{i}^{*}L^{*} $ , and reorganizing the result, we obtain the following expressions for optimal firm levels of employment in production:

$$ l_{X}=\frac{(\sigma-1)p^{1-\sigma}E^{w}}{\sigma w P^{1-\sigma}},\quad l_{X}^{*}=\frac{(\sigma-1)p^{*1-\sigma}E^{w}}{\sigma w^{*}P^{1-\sigma}}, $$

where $ P = P^{*} $ . These conditions determine the optimal production scales of home and foreign firms.

### 2.3 Process Innovation

Each firm invests in process innovation with the objective of raising profit through technology improvements that raise the productivity ( $ \theta $ ) of production. In particular, the productivities of home and foreign firms evolve according to

$$ \dot{\theta}=K l_{R}^{\gamma}, $$

$$ \dot{\theta}^{*}=K^{*}l_{R}^{*}{}^{\gamma}, $$

where $ l_{R} $ and $ l_{R}^{*} $ are firm-levels of employment in innovation, K and $ K^{*} $ are the labor productivities associated with innovation, and $ \gamma \in (0,1) $ is a parameter that ensures diminishing marginal 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 for using the productivity coefficients $ \theta $ and $ \theta^{*} $ . This allows us to model the labor productivities of home and foreign firms in in-house process innovation as a weighted average of the
productivity of currently observable technical knowledge:

$$ K=s\theta+\delta s^{*}\theta^{*},\quad K^{*}=s^{*}\theta^{*}+\delta s\theta, $$

where $ s \equiv n/N $ and $ s^{*} = 1 - s \equiv n^{*}/N $ . The parameter $ \delta \in (0,1) $ regulates the level of international knowledge spillovers. As $ \delta $ approaches unity perfect knowledge spillovers arise between countries, and as $ \delta $ 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 in process 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 $ l_{R} $ and the per-period fixed cost $ l_{F} $ :

$$ \Pi=\pi-(1-\beta)w(l_{R}+l_{F}),\quad\Pi^{*}=\pi^{*}-w^{*}(l_{R}^{*}+l_{F}), $$

where $ \beta\in[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 ( $ l_{F}=l_{F}^{*} $ ).

Firms choose the optimal level of employment in innovation with the objective of maximizing firm value. For example, a home firm sets $ l_{R} $ to maximize $ V = \int_{0}^{\infty} \Pi(\theta, t) e^{-\int_{0}^{t} r(s) ds} dt $ subject to the technology constraint (8). This intertemporal optimization problem can be solved using the following current-value Hamiltonian function: $ H = \Pi + \mu K l_{R}^{\gamma} $ , where $ \mu $ is the current shadow value of an improvement in the productivity of production. Supposing that market shares are small enough that each firm takes the price index (3) and expenditure levels as constant when considering the impact of changes in its price on its profit flow, the first order conditions are $ \partial H / \partial l_{R} = 0 $ and $ \partial H / \partial \theta = r \mu - \dot{\mu} $ . These can be used to solve
for the following no-arbitrage conditions:

$$ r=\frac{\gamma l_{X}l_{R}^{\gamma-1}K}{(1-\beta)\theta}+\frac{(1-\gamma)\dot{l_{R}}}{l_{R}}+\frac{\dot{w}}{w}-\frac{\dot{K}}{K},\quad r^{*}=\frac{\gamma l_{X}^{*}l_{R}^{*}\gamma^{-1}K^{*}}{\theta^{*}}+\frac{(1-\gamma)\dot{l_{R}^{*}}}{l_{R}^{*}}+\frac{\dot{w^{*}}}{w^{*}}-\frac{\dot{K^{*}}}{K^{*}}. $$

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

$$ TL=n\beta w(l_{R}+l_{F}), $$

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, $ \beta^{*} = 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 requires

$$ L=n(l_{X}+l_{R}+l_{F}), $$

$$ L^{*}=n^{*}(l_{X}^{*}+l_{R}^{*}+l_{F}). $$
Then, setting per-period profits (10) equal to zero, the free entry conditions for home and foreign based firms are

$$ l_{X}=(\boldsymbol{\sigma}-1)(1-\beta)(l_{R}+l_{F}),\quad l_{X}^{*}=(\boldsymbol{\sigma}-1)(l_{R}^{*}+l_{F}). $$

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 = w\beta / (1 + (\sigma - 1)(1 - \beta)) $ . Once again, R&D subsidies are not provided in foreign. Thus, home and foreign expenditures are

$$ E=(1-\eta(\beta))w, $$

$$ E^{*}=w^{*}, $$

where $ \eta(\beta)\equiv\beta/(1+(1-\beta)(\sigma-1)) $ is the effective tax rate; $ \eta(0)=0 $ and $ \eta'(\beta)>0 $ . Thus, from (4), we have $ \dot{w}/w=\dot{E}/E=\dot{w}^{*}/w^{*}=\dot{E}^{*}/E^{*}=r-\rho $ 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:

$$ \tilde{\theta}\equiv\theta/\theta^{*}. $$

The evolution of relative productivity is determined by the difference between the home and foreign productivity growth rates, as seen from the time derivative of (16):

$$ \frac{\dot{\tilde{\theta}}}{\tilde{\theta}}=\frac{\dot{\theta}}{\theta}-\frac{\dot{\theta}^{*}}{\theta^{*}}=\frac{K l_{R}^{\gamma}}{\theta}-\frac{K^{*}l_{R}^{*\gamma}}{\theta^{*}}, $$