Innovators and imitators in a world economy

Abstract

R&D, an important determinant of productivity, is highly concentrated in rich economies because of the slow diffusion of knowledge to developing countries given the cumulative nature of technological progress. This study presents a dynamic general equilibrium model of international trade, innovation, and technology diffusion with a continuum of countries. A point of departure from traditional trade models is that comparative advantage based on innovative and imitative R&D productivities determines the identity of innovators and imitators. Through this mechanism, the division of the world economy into two groups, North and South, endogenously emerges as an equilibrium outcome. The model allows us to explore interesting issues related to economic policy and development, including the following questions. Does strengthening patent policy induce imitating economies to innovate? Does subsidizing innovation or imitation promote the industrialization of developing economies. If so, how does the inequality of the world economy change? We demonstrate that the answers to these important questions crucially depend on knowledge useful to imitative R&D.

Appendix

1.1 Appendix 1. The derivation of (25)

Using (7), we can rewrite (22) for a firm in country i as:

$$\begin{aligned} \frac{{\dot{E}}\left( t\right) }{E\left( t\right) }-\frac{\dot{w_{i}}\left( t\right) }{w_{i}\left( t\right) }+g(t)+h(t)+\rho =\dfrac{\left( 1-\alpha \right) p_{i}\left( t\right) ^{-\frac{\alpha }{1-\alpha }}E\left( t\right) }{N(t)V_{i}^{I}(t)}. \end{aligned}$$

(41)

Similarly, we derive a similar condition for Southern economies. For this, log-differentiate (23) to derive ${\dot{V}}_{S}(t)/V_{S}(t)=\dot{w_{S}}\left( t\right) /w_{S}\left( t\right) -\dot{K_{S}}\left( t\right) /K_{S}\left( t\right)$. Making use of this condition, (5), (10), (11) and (23), we can rewrite the Southern arbitrage equation given in (21) as:

$$\begin{aligned} \frac{{\dot{E}}\left( t\right) }{E\left( t\right) }-\frac{\dot{w_{S}}\left( t\right) }{w_{S}\left( t\right) }+\frac{\dot{K_{S}}\left( t\right) }{K_{S}\left( t\right) }+\rho =\dfrac{\left( 1-\theta \right) p_{S}\left( t\right) ^{-\frac{\alpha }{1-\alpha }}E\left( t\right) }{N(t)V_{S}(t)}. \end{aligned}$$

(42)

From the above two equations, we can obtain the relative value of innovative R&D to imitative R&D given in (25).

1.2 Appendix 2. A constant value of $E/w_{i}$ in the steady state equilibrium

The total supply of asset value V in this model is given by

$$\begin{aligned} V=\int _{0}^{m}n_{\iota }V_{\iota }d\iota +(M-m)n_{S}V_{S}. \end{aligned}$$

Define $\kappa _{i}\in [0,1]$ as the country $i'$s share of total asset wealth, then the asset holdings of the economy i is

$$\begin{aligned} A_{i}=\kappa _{i}V=\kappa _{i}n_{i}V_{i}\epsilon _{i},\qquad \epsilon _{i}\equiv \left[ \int _{0}^{m}\frac{n_{\iota }}{n_{i}}\frac{V_{\iota }}{V_{i}}d\iota +\frac{(M-m)n_{S}}{n_{i}}\frac{V_{S}}{V_{i}}\right] . \end{aligned}$$

(43)

Using (18), (22) and (23), we have the following relationships:

$$\begin{aligned} \epsilon _{i}&= \left[ \int _{0}^{m}\frac{n_{\iota }}{n_{i}}\frac{a_{i}}{a_{\iota }}\frac{w_{\iota }}{w_{i}}d\iota +(m-m)\frac{n_{S}}{n_{i}}\frac{a_{i}}{a_{S}}\frac{1}{f(\phi )(1-\phi )}\frac{1-s_{S}}{1-s_{N}}\frac{w_{S}}{w_{i}}\right] ,\;\;\;i\in {\mathcal {N}}\\ \epsilon _{i}&= \left[ \int _{0}^{m}\frac{a_{S}}{a_{\iota }}f(\phi )(1-\phi )\frac{1-s_{N}}{1-s_{S}}\frac{w_{\iota }}{w_{S}}d\iota +(M-m)\right] ,\;\;\;i\in {\mathcal {S}} \end{aligned}$$

Steady-state equilibrium is characterized by constant values of g, m, and $\phi$. In addition, $\kappa _{i}$ and $\epsilon _{i}$ are constant in steady state, because the relative values of $n_{i}/n_{\iota }$ and $w_{i}/w_{\iota }$ are constant. Thus, $\dot{A_{i}}/A_{i}=\dot{n_{i}}/n_{i}+\dot{V_{i}}/V_{i}=g+\dot{V_{i}}/V_{i}$ holds. Moreover, from $\dot{E_{i}}/E_{i}={\dot{E}}/E(=r-\rho ),$$E_{i}=\eta _{i}E$ holds, where $\eta _{i}$ is constant over time.

Turning to the budget constraint of country i’s household $\dot{A_{i}}=rA_{i}+w_{i}L-E_{i}-T_{i}$, we can obtain the following relationships

$$\begin{aligned} \eta _{i}\frac{E}{w_{i}}&= L-s_{N}R_{i}-\frac{\kappa _{i}\epsilon _{i}(1-s_{N})}{a_{i}}\frac{n_{i}}{N}g+\kappa _{i}\epsilon _{i}\frac{1-\alpha }{\alpha }(L-R_{i}),\;\;\;i\in {\mathcal {N}}\\ \eta _{i}\frac{E}{w_{i}}&= L-s_{S}R_{i}-\frac{\kappa _{i}\epsilon _{i}(1-s_{S})}{a_{S}}\frac{n_{S}}{N_{S}f(\phi )}g+\kappa _{i}\epsilon _{i}(1-\theta )(L-R_{i}),\;\;\;i\in S \end{aligned}$$

where we used government budget constraint $T_{i}=s_{N}w_{i}R_{i}$ for $i\in {\mathcal {N}}$, $T_{i}=s_{S}w_{i}R_{i}$ for $i\in {\mathcal {S}}$, $\dot{A_{i}}/A_{i}=g+\dot{V_{i}}/V_{i}$, $r=\dot{V_{i}}/V_{i}+\pi _{i}/V_{i}$, (22), (23), (43), $n_{i}\pi _{i}=(1-\alpha )p_{i}n_{i}x_{i}=\frac{1-\alpha }{\alpha }w_{i}\left( L-R_{i}\right)$ for $i\in {\mathcal {N}}$, and $n_{i}\pi _{i}=(1-\theta )p_{i}n_{i}x_{i}=(1-\theta )w_{S}\left( L-R_{i}\right)$ for $i\in {\mathcal {S}}$. Therefore, $E/w_{i}$ is constant, because $n_{i}/N$, $n_{S}/N_{S}$ and $R_{i}$ are constant in the steady state equilibrium.

1.3 Appendix 3. Derivation of $w_{m}/w_{S}=1$

From (41) and (42) with ${\dot{E}}/E={\dot{w}}_{i}/w_{i}$, (22), (23) and (18), we obtain

$$\begin{aligned} \frac{E}{w_{i}}&= (g+h+\rho )\frac{(1-s_{N})}{(1-\alpha )a_{i}}p_{i}^{\frac{\alpha }{1-\alpha }},\,\,i\in {\mathcal {N}} \end{aligned}$$

(44)

$$\begin{aligned} \frac{E}{w_{i}}&= (g+\rho )\frac{(1-s_{S})}{(1-\theta )a_{S}}\frac{N}{f(\phi )N_{S}}p_{S}^{\frac{\alpha }{1-\alpha }},\,\,i\in {\mathcal {S}} \end{aligned}$$

(45)

From (44) and (45) with (6), (9) and $N_{S}/N=1-\phi$, we obtain the following equations:

$$\begin{aligned} g+h+\rho&= \dfrac{\left( 1-\alpha \right) \alpha ^{\frac{\alpha }{1-\alpha }}a_{i}E}{(1-s_{N})}w_{i}^{\frac{-1}{1-\alpha }}, \end{aligned}$$

(46)

$$\begin{aligned} g+\rho&= \dfrac{\left( 1-\theta \right) a_{S}E}{(1-s_{S})}f(\phi )(1-\phi )w_{S}^{\frac{-1}{1-\alpha }}. \end{aligned}$$

(47)

After using (46) and (47) to eliminate E, we can rewrite the resulting equation as

$$\begin{aligned} \left( \frac{w_{i}}{w_{S}}\right) ^{\frac{1}{1-\alpha }}=\dfrac{\left( 1-\alpha \right) \alpha ^{\alpha /(1-\alpha )}(1-s_{S})}{\left( 1-\theta \right) (1-s_{N})}\frac{g+\rho }{g+h+\rho }\frac{a_{i}}{a_{S}}\frac{1}{f(\phi )(1-\phi )}. \end{aligned}$$

(48)

The following two equations can be derived from (22), (23), (46), and (47):

$$\begin{aligned} V_{S}=\dfrac{\left( 1-\theta \right) w_{S}^{-\alpha /(1-\alpha )}f(\phi )(1-\phi )E}{(g+\rho )K_{S}},\quad V_{i}=\dfrac{\left( 1-\alpha \right) \alpha ^{\alpha /(1-\alpha )}w_{i}^{-\alpha /(1-\alpha )}E}{(g+h+\rho )N}. \end{aligned}$$

(49)

For $i=m$, using $V_{S}=V_{m}$ and $K_{S}/N=(K_{S}/N_{S})(N_{S}/N)=f(\phi )(1-\phi )$ and substituting (49) in $i=m$ into (26) yields

$$\begin{aligned} \left( \frac{w_{m}}{w_{S}}\right) ^{\frac{\alpha }{1-\alpha }}=\dfrac{\left( 1-\alpha \right) \alpha ^{\alpha /(1-\alpha )}(1-s_{S})}{\left( 1-\theta \right) (1-s_{N})}\frac{g+\rho }{g+h+\rho }\frac{a_{m}}{a_{S}}\frac{1}{(1-\phi )f(\phi )}. \end{aligned}$$

(50)

Equating (48) in $i=m$ and (50) gives $w_{m}/w_{S}=1$.

1.4 Appendix 4. Derivation of the partial derivatives of $\phi =\phi (g,m;\alpha ,\theta ,s_{N},s_{S},a_{S},\delta )$.

We can rewrite (32) as

$$\begin{aligned} a_{m}=\varLambda \dfrac{g/\phi +\rho }{g+\rho }[1-\phi (1-\delta )],\quad \varLambda \equiv \dfrac{\left( 1-\theta \right) (1-s_{N})a_{S}}{\left( 1-\alpha \right) \alpha ^{\alpha /(1-\alpha )}(1-s_{S})}. \end{aligned}$$

The total differentiation of $\phi =\phi (g,m;\alpha ,\theta ,s_{N},s_{S},a_{S},\delta )$ yields

$$\begin{aligned}{}[\Phi ]d\phi =\left[ \dfrac{\rho (1-\phi )}{(g+\rho \phi )(g+\rho )}\right] dg+\dfrac{1}{a_{m}}\left[ -\dfrac{\partial a_{m}}{\partial m}\right] dm+\left[ \dfrac{\phi }{1-\phi (1-\delta )}\right] d\delta +\frac{d\varLambda }{\varLambda }, \end{aligned}$$

where

$$\begin{aligned}&\Phi \equiv \left[ \frac{g/\phi }{g+\rho \phi }+\frac{1-\delta }{1-\phi (1-\delta )}\right] >0,\\&\quad \dfrac{d\Lambda }{\Lambda }=-\frac{1}{1-\theta }d\theta -\frac{1}{1-s_{N}}ds_{N}+\frac{1}{1-s_{S}}ds_{S}+\frac{1}{a_{S}}da_{S}+\frac{-\ln \alpha }{(1-\alpha )^{2}}d\alpha . \end{aligned}$$

From the above equation, we can obtain the partial derivatives of $\phi$ with respect to $g,m,\alpha ,\theta ,a_{S},s_{N},s_{S}$ and $\delta$

$$\begin{aligned} \phi _{g}&= \dfrac{\rho (1-\phi )}{(g+\rho \phi )(g+\rho )}\frac{1}{\Phi }>0,\quad \phi _{m}=\dfrac{1}{a_{m}}\left[ -\dfrac{\partial a_{m}}{\partial m}\right] \frac{1}{\Phi }>0,\\ \phi _{\alpha }&= \frac{-\ln \alpha }{(1-\alpha )^{2}}\frac{1}{\Phi }>0,\quad \phi _{\theta }=-\frac{1}{1-\theta }\frac{1}{\Phi }<0,\quad \phi _{a_{S}}=\frac{1}{a_{S}}\frac{1}{\Phi }>0,\\ \phi _{s_{N}}&= -\frac{1}{1-s_{N}}\frac{1}{\Phi }<0,\quad \phi _{s_{S}}=\frac{1}{1-s_{S}}\frac{1}{\Phi }>0,\quad \phi _{\delta }=\dfrac{\phi }{1-\phi (1-\delta )}\frac{1}{\Phi }>0. \end{aligned}$$

1.5 Appendix 5. $d(\mathrm {Gini})/dm<0$

From the definition of the Gini Index, by differentiating it with respect to m, we obtain the following result:

$$\begin{aligned} \frac{d(\mathrm {Gini})}{dm}&= -2\frac{d\left( \int _{0}^{1}L\left( x\right) dx\right) }{dm}\\&= -2\frac{\int _{\frac{M-m}{M}}^{1}z\frac{\partial q}{\partial m}dz}{\int _{0}^{\frac{M-m}{M}}zdz+\int _{\frac{M-m}{M}}^{1}z\left( \frac{a_{M(1-z)}}{a_{m}}\right) ^{1-\alpha }dz}\left[ 1-\int _{0}^{1}L\left( x\right) dx\right] <0, \end{aligned}$$

where

$$\begin{aligned} \frac{\partial q}{\partial m}=q(1-\alpha )\frac{-(\partial a_{m}/\partial m)}{a_{m}}>0. \end{aligned}$$