Silviase/jgov_v1.4_full_ann huggingface dataset:
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

$$ \frac{K l_{R}^{\gamma}}{\theta}=\frac{K^{*}l_{R}^{*}{}^{\gamma}}{\theta^{*}}. $$

With productivity growth equalized across countries, an examination of the time derivatives of K and $ K^{*} $ then reveals that $ \dot{\theta}/\theta = \dot{\theta}^{*}/\theta^{*} = \dot{K}/K = \dot{K}^{*} / K^{*} $ in equilibrium.

Next, we consider the equilibrium investment conditions. Combining $ \dot{\theta}/\theta = \dot{K}/K $ and $ \dot{\theta}^{*}/\theta^{*} = \dot{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:

$$ \rho=R(l_{R})\equiv\left[\gamma(\pmb{\sigma}-1)\left(1+\frac{l_{F}}{l_{R}}\right)-1\right]\frac{l_{R}^{\gamma}K}{\theta}, $$

$$ \rho=R^{*}(l_{R}^{*})\equiv\left[\gamma(\pmb{\sigma}-1)\left(1+\frac{l_{F}}{l_{R}^{*}}\right)-1\right]\frac{l_{R}^{*}\gamma K^{*}}{\theta^{*}}, $$

where $ R(l_{R}) $ and $ R^{*}(l_{R}^{*}) $ represent the internal rates of return on investment in process innovation. Invoking (18), it is clear from these conditions that $ l_{R}=l_{R}^{*} $ . Therefore, from the free entry conditions (14), we also have $ K/\theta=K^{*}/\theta^{*} $ and $ l_{X}/(1-\beta)=l_{X}^{*} $ 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/\theta = K^{*}/\theta^{*} $ to solve for $ s = s(\tilde{\theta}) $ . The second is found using $ l_{R} = l_{R}^{*} $ and the free-entry conditions (14) in the labor market conditions (13) to solve for $ s = s(L/L^{*}, \beta) $ . Setting these solutions equal yields the
following condition for the steady-state value of $ \tilde{\theta} $ :

$$ s=\frac{1-\delta\tilde{\theta}^{-1}}{1-\delta\tilde{\theta}+1-\delta\tilde{\theta}^{-1}}=\frac{L/L^{*}}{L/L^{*}+\left(1+\left(1-\beta\right)(\sigma-1)\right)/\sigma}. $$

Thus, as illustrated in Figure 1, relative productivity is determined implicitly by the relative labor endowment and the R&D subsidy rate: $ \tilde{\theta} = \tilde{\theta}(L/L^{*}, \beta) $ . Proportionate changes in the labor endowments have no effect on relative productivity. Moreover, setting $ L/L^{*} = 1 $ and $ \beta = 0 $ leads to a symmetric equilibrium with s = 1/2 and $ \tilde{\theta}(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^{*} $ ), and accordingly $ s > L/(L + L^{*}) $ for $ \beta > 0 $ . Note that with perfect knowledge dispersion ( $ \delta = 1 $ ), the $ s(\tilde{\theta}) $ curve becomes a vertical line and productivity is symmetric across countries ( $ \tilde{\theta} = 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 $ (\beta) $ raise the relative productivity of home firms.

Proof: Total differentiation of (21) gives

$$ \begin{align*}\frac{d\tilde{\boldsymbol{\theta}}}{d(L/L^{*})}&=\frac{(1+(1-\beta)(\boldsymbol{\sigma}-1))(\boldsymbol{\delta}-\tilde{\boldsymbol{\theta}})^{2}}{\boldsymbol{\sigma}\boldsymbol{\delta}(1-\boldsymbol{\delta}^{2}+(\tilde{\boldsymbol{\theta}}-\boldsymbol{\delta})^{2})(L/L^{*})^{2}}>0,\\\frac{d\tilde{\boldsymbol{\theta}}}{d\beta}&=\frac{(\boldsymbol{\sigma}-1)(\boldsymbol{\delta}-\tilde{\boldsymbol{\theta}})^{2}}{\boldsymbol{\sigma}\boldsymbol{\delta}(1-\boldsymbol{\delta}^{2}+(\tilde{\boldsymbol{\theta}}-\boldsymbol{\delta})^{2})(L/L^{*})}>0.\end{align*} $$

Changes in $ L/L^{*} $ and $ \beta $ affect relative productivity through adjustments in the shares of firms locating in each country. For instance, returning to Figure 1, consider a rise in the R&D subsidy rate. The $ s = s(L/L^{*}, \beta) $ line shifts upwards resulting in an increase in the relative 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 share of firms locating in home. This rise in home’s share of firms coincides with an increase in relative productivity that brings knowledge spillovers back into balance, $ K/\theta = K^{*}/\theta^{*} $ , as the economy returns to steady-state equilibrium.
Figure 1: Firm Shares and Relative Productivity

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(\tilde{\theta}) $ from (21) into the expression for observable knowledge (9), and reorganizing the result, yields the equilibrium level of relative knowledge spillovers as follows:

$$ \frac{K}{\theta}=\frac{K^{*}}{\theta^{*}}=\frac{1-\delta^{2}}{2-\delta\tilde{\theta}-\delta\tilde{\theta}^{-1}}. $$

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 $ \tilde{\theta}=1 $ .

As shown in Figure 2, an increase in relative productivity lowers relative knowledge spillovers for $ \tilde{\theta}<1 $ and raises relative knowledge spillovers for $ \tilde{\theta}>1 $ . From (21), we can see that $ \tilde{\theta}=1/\delta $ when all firms locate in home (s=1), and that $ \tilde{\theta}=\delta $ 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 feature of 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 and the internal rate of return. Figure 3 provides an illustration of the no-arbitrage condition for
Figure 2: Relative Knowledge Spillovers

home. At every moment in time, firms choose optimal levels of in-house innovation with the aim of maximizing firm value. Thus, when the internal rate of return $ R(l_{R}) $ is greater than the risk free rate of return $ \rho(=r) $ , firms increase R&D employment, and when $ R(l_{R})<\rho $ , 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 of $ R(l_{R}) $ 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 the local dynamics of the economy around a symmetric long-run equilibrium for $ L/L^{*}=1 $ , and $ \beta=0 $ . We find that if $ 2>(\sigma-1)(1-\delta+2(1+\delta)\gamma) $ , then

$$ \frac{\partial R(l_{R})}{\partial l_{R}}=-\left[1-\gamma(\boldsymbol{\sigma}-1)+\frac{(1-\gamma)(\boldsymbol{\sigma}-1)l_{F}}{l_{R}}\right]\frac{\gamma l_{R}^{\gamma-1}K}{\theta}<0, $$

and the symmetric equilibrium is saddle-point stable. $ ^{4} $ Henceforth, we characterize the long-run equilibrium and investigate the determinants of productivity growth and product variety under the assumption that $ \partial R(l_{R})/\partial l_{R}<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 of relative productivity, it is now possible to examine the effects of changes in $ L/L^{*} $ and $ \beta $ on
Figure 3: In-house R&D Employment

firm-level employment in process innovation.

Lemma 3 Increases in the relative labor endowment $ (L/L^{*}) $ and the R&D subsidy rate $ (\beta) $ lower firm-level employment in innovation for $ \tilde{\theta}<1 $ and raise it for $ \tilde{\theta}>1 $ .

Proof: Substituting (22) into (19) and taking the derivative with respect to $ l_{R} $ and $ \tilde{\theta} $ yields:

$$ \frac{d l_{R}}{d\tilde{\theta}}=-\frac{1}{R^{\prime}(l_{R})}\frac{\rho\delta(1-\tilde{\theta}^{-2})(K/\theta)}{1-\delta^{2}}. $$

This derivative can be signed using (23) and Lemma 1.

Changes in $ L/L^{*} $ and $ \beta $ affect $ l_{R} $ indirectly through $ \tilde{\theta} $ and subsequently $ K/\theta $ . For example, consider the effect of an increase in the R&D subsidy rate. When home has a greater share of firms and $ \tilde{\theta} > 1 $ , the rise in $ \beta $ increases $ \tilde{\theta} $ , thereby raising relative knowledge spillovers. Returning to Figure 3, the internal rate of return to process innovation $ R(l_{R}) $ shifts upward and firms increase employment in process innovation until the internal rate of return falls back to the risk-free level $ \rho $ . On the other hand, when home has a smaller share of firms and $ \tilde{\theta} < 1 $ , the increase in $ \beta $ shifts the $ R(l_{R}) $ curve downwards and firms reduce innovation employment.
Figure 4: Product Variety and Productivity Growth

(a) Total number of Firms

(b) Rate of Productivity Growth

## 4 Product Variety and Productivity Growth

This 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 the steady-state condition $ l_{R} = l_{R}^{*} $ , we can solve for the equilibrium level of product variety as

$$ N=\frac{1}{l_{R}+l_{F}}\left(\frac{L}{1+(1-\beta)(\sigma-1)}+\frac{L^{*}}{\sigma}\right). $$

As illustrated in Figure 4a, equilibrium product variety and relative productivity have a concave relationship with a maximum occurring at $ \tilde{\theta}=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 $ \tilde{\theta} $ raises relative knowledge spillovers, and firm-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
obtain the following result.

Proposition 1 For $ \tilde{\theta} \leq 1 $ , an increase in the R&D subsidy rate ( $ \beta $ ) raises product variety. For $ \tilde{\theta} > 1 $ , however, product variety may rise or fall.

Proof: Taking the total derivative of (24) and applying Lemma 3, we have

$$ \frac{d N}{d\beta}=\frac{(\boldsymbol{\sigma}-1)L}{(l_{R}+l_{F})(1+(1-\beta)(\boldsymbol{\sigma}-1))^{2}}-\frac{N}{l_{R}+l_{F}}\frac{d l_{R}}{d\beta}. $$

The R&D subsidy has two effects on equilibrium product variety. The first is a positive direct effect that shifts the $ N(\tilde{\theta}) $ curve upward in Figure 4a. A rise in $ \beta $ 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(\tilde{\theta}) $ 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 $ l_{R} $ , an increase in $ \beta $ will raise N for $ \tilde{\theta} < 1 $ and reduce N for $ \tilde{\theta} > 1 $ . Thus, we conclude that when home has the larger number of firms and $ \tilde{\theta} < 1 $ , an R&D subsidy can be used to raise product variety, but for $ \tilde{\theta} > 1 $ the overall effect will be ambiguous.

Next, we calculate the steady-state rate of productivity growth. Substituting the no-arbitrage conditions (19) into the innovation function (8) and manipulating the result yields

$$ g\equiv\frac{\dot{\theta}}{\theta}=\frac{l_{R}^{\gamma}K}{\theta}=\frac{\rho}{\gamma(\sigma-1)-1+\gamma(\sigma-1)l_{F}/l_{R}}. $$

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 $ \tilde{\theta} = \tilde{\theta}(L/L^{*}, \beta) $ 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 of incumbent firms (24). Then, although the aggregate level of employment associated with
innovation activity $ (l_{R}+l_{F})N=L/(1+(1-\beta)(\sigma-1))+L^{*}/\sigma $ 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 $ \tilde{\theta}=1 $ . Starting once again from a symmetric equilibrium in which productivity is equalized across countries, either a rise or a 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 $ (l_{R}+l_{F}) $ increase and firms are pushed out of the market. With the number of incumbent firms decreasing, the resources required to cover economy-wide per-period fixed costs $ N l_{F} $ fall, and the amount of labor available for employment in innovation rises.

We examine the relationship between the home R&D subsidy and productivity growth and obtain the following proposition:

Proposition 2 An increase in the R&D subsidy ( $ \beta $ ) lowers productivity growth for $ \tilde{\theta} < 1 $ and raises productivity growth for $ \tilde{\theta} > 1 $ .

The R&D subsidy only effects productivity growth indirectly through changes in relative productivity. Returning once again to Lemma 1, $ d\tilde{\theta}/d\beta > 0 $ and an increase in the R&D subsidy leads to a rightward movement along the $ g(\tilde{\theta}) $ curve in Figure 4b. Accordingly, the R&D subsidy depresses productivity growth for $ \tilde{\theta} < 1 $ and accelerates it for $ \tilde{\theta} > 1 $ . Note that the R&D subsidy has no effect on productivity growth for a symmetric equilibrium evaluated at $ \beta = 0 $ .
## 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 the preceding two sections, it assumes a role in the welfare effects associated with adjustments in the terms of trade. Substituting the pricing rules (6), the price index (3), and optimal firm-level employment in production (7) into $ l_{X}/(1-\beta) = l_{X}^{*} $ and simplifying yields the relative wage rate as $ w/w^{*} = (1-\beta)^{-\frac{1}{\sigma}} \tilde{\theta}(L/L^{*},\beta)^{\sigma-1} $ , where $ w/w^{*} = 1 $ in the symmetric equilibrium for $ L/L^{*} = 1 $ and $ \beta = 0 $ . Hence, the terms of trade for the home country are

$$ \frac{p}{p^{*}}=\left[(1-\beta)\tilde{\theta}(L/L^{*},\beta)\right]^{\frac{1}{\sigma}}. $$

Thus, while the direct effect of an increase in the R&D subsidy rate is a deterioration of home’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 and foreign households are found by substituting (3), (6), (15), (21), and the terms of trade into lifetime utility (1):

$$ \rho U_{0}=\ln(1-\eta(\beta))A+\frac{1}{(\sigma-1)}\ln\left[s(\beta)+(1-s(\beta))\left[(1-\beta)\tilde{\theta}\right]^{-\frac{\sigma-1}{\sigma}}\right]N+\frac{g}{\rho}, $$

$$ \rho U_{0}^{*}=\ln A^{*}+\frac{1}{(\sigma-1)}\ln\left[s(\beta)\left[(1-\beta)\tilde{\theta}\right]^{\frac{\sigma-1}{\sigma}}+(1-s(\beta))\right]N+\frac{g}{\rho}, $$

where $ A = (\sigma - 1)\theta(0)/\sigma $ and $ A^{*} = (\sigma - 1)\theta^{*}(0)/\sigma $ are constants, and $ s(\beta) $ is defined in the second expression of (21).

Focusing again on the home country, a change in the R&D subsidy affects welfare
Figure 5: Optimal Home R&D Subsidies

Parameter 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, $ \eta(\beta) $ captures the negative income effect of the lump-sum tax placed on home households to pay for the R&D subsidy. Next, the second term in parenthesis describes the terms of trade effect that results from changes in production shares $ (s) $ , the relative wage rate $ (w/w^{*}) $ , and relative productivity $ (\tilde{\theta}) $ . The third term is the 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 the implications 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 the case 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 $ \tilde{\theta}>1 $ . The home R&D subsidy rates that maximize home and foreign welfare levels are respectively $ \beta_{o} $ and $ \beta_{o}^{*} $ , 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 of the average household. Denoting welfare for the average household in the world economy by $ U^{W}\equiv(L/(L+L^{*}))U_{0}+(L^{*}/(L+L^{*}))U_{0}^{*} $ , the optimal home subsidy rate would then
become $ \beta_{W} $ , as depicted in Figure 5. Thus, as $ \beta_{W} > \beta_{b} $ , average world welfare can be increased by raising the home R&D subsidy rate above the optimal level for home residents.

Generally, a positive level for $ \beta_{b} $ only arises when the home country has a relatively large population. For example, when country sizes are symmetric, that is $ L/L^{*}=1 $ , we can evaluate the welfare effects of change in the R&D subsidy as follows:

$$ \frac{d U_{0}}{d\beta}\bigg|_{\beta=0}=-\frac{(\sigma-1)(1-\delta)}{4\sigma^{2}\delta\rho}<0,\qquad\quad\frac{d U_{0}^{*}}{d\beta}\bigg|_{\beta=0}=\frac{(\sigma-1)(1-\delta)}{4\sigma^{2}\delta\rho}>0. $$

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 for foreign $ (\beta_{o}^{*}>0) $ , home would prefer to set the R&D subsidy to zero $ (\beta_{o}=0) $ , or perhaps even to implement an R&D tax $ (\beta_{o}<0) $ . The optimal R&D subsidy rate for the average world household is zero in this case, as $ (dU^{W}/d\beta)|_{\beta_{W}}=0 $ . Numerical analysis shows that these results continue to hold for $ L/L^{*}<1 $ , although $ \beta_{o}^{*}>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 $ \beta_{o} $ and $ \beta_{o}^{*} $ equalized at $ \delta=1 $ , given that productivity is symmetric across countries, $ \tilde{\theta}=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 and lowering 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, a greater 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 tech-
The critical knowledge between firms, thereby raising R&D productivity and inducing a greater level of employment in in-house process innovation. As a result, we find that equilibria with asymmetric 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, the lower cost of process innovation associated with symmetric markets generates greater market entry so that the symmetric equilibrium produces the largest number of firms, and the greatest 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 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 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 raising productivity 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 to conclude that the optimal level for a national R&D subsidy that is implemented by the larger 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 of raising average world welfare by increasing the R&D subsidy.

Given these preliminary results, an interesting extension of our framework might be an
analysis of strategic R&D policy. For example, Kondo (2012) investigates the relationship between R&D subsidies and economic growth in a variety of expansion models 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 might allow 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.