Sequential R&D and blocking patents in the dynamics of growth

Abstract

The incentives to conduct basic or applied research play a central role for economic growth. How does increasing early innovation appropriability affect basic research, applied research, innovation and growth? In a common law system an explicitly dynamic macroeconomic analysis is appropriate. This paper analyzes the macroeconomic effects of patent protection by incorporating a two-stage cumulative innovation structure into a quality-ladder growth model with endogenous skill acquisition. We focus on two issues: (a) the over-protection versus the under-protection of intellectual property rights in basic research; (b) the evolution of jurisprudence shaping the bargaining power of the upstream innovators. We show that the dynamic general equilibrium interactions may seriously mislead the empirical assessment of the growth effects of IPR policy: stronger protection of upstream innovation always looks bad in the short- and possibly medium-run. We also provide a simple “rule of thumb” indicator of the basic researcher bargaining power.

Appendix

Proof of Lemma 1

From Eqs. (22a) and (22c), we have

$$\begin{aligned} \frac{\beta }{1-\beta } = \frac{n_{B}^{a}}{\lambda _{0}}\left( \frac{r}{n_{A}}+ \frac{\lambda _{1}}{n_{A}^{a}}\right) \end{aligned}$$

(53)

which implies that $n_{B}$ is an increasing function of $n_{A}$ and $\beta$ , as is $\frac{n_{B}}{n_{A}}$.

Let us rewrite Eq. (11) as $\pi =\frac{1-\gamma }{1-\alpha }l$. Moreover, since $l=\theta _{0}$, Eq. (6) can be rewritten as $w_{H}=\frac{B}{l-\Gamma }$, where $B$ is a constant.

After solving for $v_{L}^{0}$ using Eqs. (22d) and (22e), and in light of the above-mentioned results, we can rewrite Eq. (22c) as:

$$\begin{aligned} \frac{B}{l-\Gamma }=\lambda _{1}n_{A}^{-a}\frac{1-\gamma }{1-\alpha }\frac{ l\left( 1-\beta \right) }{r+\lambda _{0}n_{B}^{1-a}}\left[ 1+\frac{\lambda _{0}n_{B}^{1-a}}{r+\lambda _{1}n_{A}^{1-a}}\right] . \end{aligned}$$

(54)

Equations (53) and (54) imply, by the implicit function theorem, that $l$ is a function of $l\left( n_{A},n_{B},\beta \right)$ increasing in all its three arguments.

From $l=\theta _{0}$, we can rewrite Eqs. (22a, 22b, 22c, 22d, 22e) as

$$\begin{aligned} h(l)=Q\left[ \left( 1-l^{2}\right) +2\Gamma (l-1)\right] , \end{aligned}$$

where $Q$ is a constant. Notice that $h^{\prime }(l)>0$ in the relevant range, because in equilibrium $l>\Gamma$. Equations (23) and (24) imply that:

$$\begin{aligned} \frac{\lambda _{1}+\lambda _{0}\frac{n_{B}}{n_{A}}}{\lambda _{0}+\lambda _{1}\left( \frac{n_{A}}{n_{B}}\right) ^{1-a}}n_{A}=h(l)-\frac{\alpha }{ 1-\alpha }\frac{l(l-\Gamma )}{B}. \end{aligned}$$

(55)

Notice that the right hand side of Eq. (55) is decreasing in $l$, and therefore decreasing in $n_{A}$, $n_{B}$, and $\beta$. Instead, the left hand side of Eq. (55) is increasing in $n_{A}$ and $\beta$.⁴⁴ Therefore, by the implicit function theorem, $n_{A}$ will be an increasing function of $\beta$. $\square$

Proof of Proposition 1

From Eqs. (68) and (25) follows that the steady state level of human capital per-capita is an increasing function of the skilled premium $w_{H}$, which we can write as $\overline{h}(w_{H})$.

Plugging Eq. (40) into the skilled labour market clearing condition (11) yields:

$$\begin{aligned} \left[ m(A_{0})+\left( 1-m(A_{0})\right) \left( \frac{\lambda _{1}}{\lambda _{0}}\frac{1-\beta }{\beta }\right) ^{\frac{1}{a}}\right] n_{B}=\overline{h} (w_{H})-x(w_{H})\equiv \Psi (w_{H}) \end{aligned}$$

(56)

with $\Psi ^{\prime } (w_{H})>0$. Inserting Eqs. (42) into (56) we obtain:

$$\begin{aligned} \frac{n_{B}}{\beta \left[ 1+\left( \frac{\lambda _{0}}{\lambda _{1}}\right) ^{\frac{1}{a}}\left( \frac{\beta }{1-\beta }\right) ^{\frac{1-a}{a}}\right] } =\overline{h}(w_{H})-x(w_{H})\equiv \Psi (w_{H}) \end{aligned}$$

(57)

Plugging (39) and Eq. (41) into Eqs. (14a) and (40) we obtain:

$$\begin{aligned} w_{H}&= \lambda _{0}n_{B}^{-a}\beta v_{L}^{0}=\lambda _{0}n_{B}^{-a}\beta \left[ \left( \frac{\lambda _{1}}{\lambda _{0}}\right) ^{\frac{1}{a}}\left( \frac{1-\beta }{\beta }\right) ^{\frac{1-a}{a}}+1\right] v_{L}^{1} \end{aligned}$$

(58a)

$$\begin{aligned} \pi&= \lambda _{1}n_{A}^{1-a}v_{L}^{1}=\lambda _{1}\left( \frac{\lambda _{1} }{\lambda _{0}}\frac{1-\beta }{\beta }\right) ^{\frac{1-a}{a} }n_{B}^{1-a}v_{L}^{1} \end{aligned}$$

(58b)

From the definition of profits and the steady state mass of unskilled labour, we know that $\pi =\pi (w_{H})$, with $\pi ^{\prime }(w_{H})<0$. Dividing the last two equations side by side implies:

$$\begin{aligned} n_{B}\frac{1}{\beta \left[ 1+\left( \frac{\lambda _{0}}{\lambda _{1}}\right) ^{\frac{1}{a}}\left( \frac{\beta }{1-\beta }\right) ^{\frac{1-a}{a}}\right] } =\frac{\pi (w_{H})}{w_{H}}. \end{aligned}$$

(59)

Plugging (59) into (57) gives:

$$\begin{aligned} 1=\Psi (w_{H})\frac{w_{H}}{\pi (w_{H})}\equiv \Phi (w_{H}) \end{aligned}$$

(60)

where $\Phi ^{\prime }(w_{H})>0$. Therefore there exists a unique steady state level of the skill premium obtained as the solution to Eq. (60). It is important to notice that, in this example, the steady state skill premium is independent of $\beta$.

The steady state innovation rate can be rewritten, after using (59), as:

$$\begin{aligned} \lambda _{0}n_{B}^{1-a}m(A_{0})&= \frac{\left[ \frac{\pi (w_{H})}{w_{H}} \right] ^{1-a}\beta ^{1-a}}{\left[ 1+\left( \frac{\lambda _{0}}{\lambda _{1}} \right) ^{\frac{1}{a}}\left( \frac{\beta }{1-\beta }\right) ^{\frac{1-a}{a}} \right] ^{a}} \end{aligned}$$

(61)

$$\begin{aligned}&= \frac{\left[ \frac{\pi (w_{H})}{w_{H}}\right] ^{1-a}}{\left[ \left( \frac{ 1}{\lambda _{0}}\right) ^{\frac{1}{a}}\left( \frac{1}{\beta }\right) ^{\frac{ 1-a}{a}}+\left( \frac{1}{\lambda _{1}}\right) ^{\frac{1}{a}}\left( \frac{1}{ 1-\beta }\right) ^{\frac{1-a}{a}}\right] ^{a}} \end{aligned}$$

(62)

The numerator does not change with $\beta$ as previously proved. The innovation rate is maximized when the denominator is minimized. Hence we need to find a value of $\beta$ such that $\left( \frac{1}{\lambda _{0}} \right) ^{\frac{1}{a}}\left( \frac{1}{\beta }\right) ^{\frac{1-a}{a}}+\left( \frac{1}{\lambda _{1}}\right) ^{\frac{1}{a}}\left( \frac{1}{1-\beta }\right) ^{\frac{1-a}{a}}$ is minimized, which implies expression (43 ).$\square$.

1.1 Labour supply and education dynamics

1.1.1 Unskilled labor supply

As previously shown, individuals born at $t$ with ability $\theta (t)\in [0,\theta _{0}(t)]$ optimally choose not to educate themselves, thereby immediately joining the unskilled labour force. Hence a fraction $\theta _{0}(t)$ of cohort $t$ remains unskilled their whole life. Summing up over all the older unskilled who are still alive—hence born in the time interval $[t-D,t]$—we obtain the total stock of unskilled labour as of time $t$:

$$\begin{aligned} L(t)=\int _{t-D}^{t}bN(s)\theta _{0}(s)ds=b\int _{t-D}^{t}e^{gs}\theta _{0}(s)ds \end{aligned}$$

where $b$ is the birth rate, $N(s)$ is the population at time $s$.

To stationarize variables, we divide by current (time $t$) population $e^{gs}$, obtaining:

$$\begin{aligned} l(t)\equiv \frac{L(t)}{N(t)}=b\int _{t-D}^{t}e^{g(s-t)}\theta _{0}(s)ds. \end{aligned}$$

Its steady state level is:

$$\begin{aligned} l=b\frac{1-e^{g(-D)}}{g}\theta _{0}=\theta _{0}. \end{aligned}$$

The change in the stock of the population-adjusted stock of unskilled labour is obtained by derivating $l(t)$ with respect to time:

$$\begin{aligned} \dot{l}(t)=b\theta _{0}(t)-be^{-gD}\theta _{0}(t-D)-gl(t) \end{aligned}$$

(63)

As in Boucekkine et al. (2002) and Boucekkine et al. (2007) we obtain a crucial role for delayed differential equations.

1.1.2 College population

The individuals born in $t$ with ability $\theta (t)\in [\theta _{0}(t),1]$ optimally choose to educate themselves, thereby becoming college students for a training period of duration $Tr$. Hence summing up over all the previous cohorts who are still in college—hence born in the time interval $[t-Tr,t]$—we obtain the total stock of college population as of time $t$:

$$\begin{aligned} \widetilde{C}(t)=b\int _{t-Tr}^{t}N(s)(1-\theta _{0}(s))ds=b\int _{t-Tr}^{t}e^{gs}(1-\theta _{0}(s))ds. \end{aligned}$$

In per-capita terms:

$$\begin{aligned} \widetilde{c}(t)\equiv \frac{\widetilde{C}(t)}{N(t)}=b\int _{t-Tr}^{t}\frac{ N(s)}{N(t)}(1-\theta _{0}(s))ds=b\int _{t-Tr}^{t}e^{g(s-t)}(1-\theta _{0}(s))ds. \end{aligned}$$

(64)

In a steady state:

$$\begin{aligned} \widetilde{c}=b\frac{1-e^{g(-Tr)}}{g}(1-\theta _{0}). \end{aligned}$$

(65)

Taking the derivative of Eq. (64) with respect to time we obtain:

$$\begin{aligned} \overset{.}{\widetilde{c}}(t)=b\left( 1-\theta _{0}(t)\right) -be^{-gTr}\left( 1-\theta _{0}(t-Tr)\right) -g\widetilde{c}(t). \end{aligned}$$

(66)

1.1.3 Human capital

The stock of skilled workers will coincide with those students who have completed their education and are still alive, born in $[t-D,t-Tr]$:

$$\begin{aligned} \widetilde{H}(t)=b\int _{t-D}^{t-Tr}N(s)(1-\theta _{0}(s))ds=bN(t)\int _{t-D}^{t-Tr}e^{g(s-t)}(1-\theta _{0}(s))ds \end{aligned}$$

The total workforce (including students) in equilibrium equals total population, hence:

$$\begin{aligned} L(t)+\widetilde{H}(t)+\tilde{C}(t)=e^{gt}. \end{aligned}$$

Due to heterogeneous learning abilities, in order to obtain the aggregate skilled labour supply, we need to multiply each skilled worker by the average amount of human capital that she can supply, given by the average skill of her cohort net of dispersion parameter $\Gamma$:

$$\begin{aligned} \int _{\theta _{0}(t)}^{1}(\theta -\Gamma )\frac{1}{1-\theta _{0}(t)}d\theta = \frac{1+\theta _{0}(t)-2\Gamma }{2}. \end{aligned}$$

Therefore the aggregate amount of skilled labour in efficiency units (skilled labor supply) is:

$$\begin{aligned} H(t)=bN(t)\int _{t-D}^{t-Tr}\frac{e^{g(s-t)}(1-\theta _{0}(s))\left( 1+\theta _{0}(s)-2\Gamma \right) }{2}ds \end{aligned}$$

Dividing by time $t$ population, we can express per-capita human capital as:

$$\begin{aligned} h(t)\equiv \frac{H(t)}{N(t)}=\frac{b}{2}\int _{t-D}^{t-Tr}e^{g(s-t)}(1-\theta _{0}(s))\left( 1+\theta _{0}(s)-2\Gamma \right) ds. \end{aligned}$$

(67)

The steady state value is:

$$\begin{aligned} h=b\frac{\left[ e^{g(-Tr)}-e^{g(-D)}\right] (1-\theta _{0})\left( 1+\theta _{0}-2\Gamma \right) }{2g} \end{aligned}$$

(68)

The dynamics of human capital can be studied by derivating both sides of Eq. (67) with respect to time:

$$\begin{aligned} \overset{.}{h}(t)&= -gh(t)+\frac{b}{2}e^{-gTr}(1-\theta _{0}(t-Tr))\left( 1+\theta _{0}(t-Tr)-2\Gamma \right) - \nonumber \\&+\frac{b}{2}e^{-gD}(1-\theta _{0}(t-D))\left( 1+\theta _{0}(t-D)-2\Gamma \right) . \end{aligned}$$

(69)

1.2 Transitional properties of educational choice

The study of the transition dynamics of this model is complicated by the skilled/unskilled labour dynamics and by the endogenous education choice under perfect foresight. Key to the solution is the transformation of the integral equation for the ability threshold level for education into a set of differential equations.

Defining the present value of the unskilled wage incomes as $W_{U}(t)=\int _{t}^{t+D}e^{-\int _{t}^{s}i(\tau )d\tau }ds$ and the present value of the skilled wage income as $W_{S}(t)=\int _{t+Tr}^{t+D}e^{- \int _{t}^{s}i(\tau )d\tau }w_{H}(s)ds$, we know from (24) that

$$\begin{aligned} \theta _{0}(t)=\Gamma +\frac{W_{U}(t)}{W_{S}(t)}. \end{aligned}$$

(70)

Defining

$$\begin{aligned} R_{1}(t)&= e^{-\int _{t}^{t+D}i(\tau )d\tau },\;\text {and} \end{aligned}$$

(71)

$$\begin{aligned} R_{2}(t)&= e^{-\int _{t}^{t+Tr}i(\tau )d\tau } \end{aligned}$$

(72)

we can write:

$$\begin{aligned} \dot{W}_{U}(t)&= R_{1}(t)-1+i(t)W_{U}(t) \end{aligned}$$

(73)

$$\begin{aligned} \dot{W}_{S}(t)&= R_{1}(t)w_{H}(t+D)-R_{2}(t)w_{H}(t+Tr)+i(t)W_{S}(t). \end{aligned}$$

(74)

Differentiating Eqs. (71)–(72) with respect to time we obtain:

$$\begin{aligned} \dot{R}_{1}(t)&= R_{1}(t)(i(t)-i(t+D))\text {, and} \end{aligned}$$

(75)

$$\begin{aligned} \dot{R}_{2}(t)&= R_{2}(t)(i(t)-i(t+Tr)). \end{aligned}$$

(76)

These equations allow us to cast our model in a framework that can be studied in terms of delayed differential equations.

1.3 Expenditure and manufacturing dynamics

From Eq. (10) follows:

$$\begin{aligned} \frac{\gamma -1}{\gamma } e(t)=\left( \gamma -1\right) \frac{1}{1-\alpha } l(t) . \end{aligned}$$

(77)

Log-differentiating with respect to time, using Euler equation (3) and the unskilled law of motion (63) yield:

$$\begin{aligned} i(t)-\left( \rho +g\right) =\frac{\dot{e}(t)}{e(t)}=\frac{\dot{l}(t)}{l(t)}= \frac{b\theta _{0}(t)-be^{-gD}\theta _{0}(t-D)}{l(t)}-g \end{aligned}$$

(78)

that—since $r(t)=i(t)-g$—can be rewritten as

$$\begin{aligned} r(t)-\rho =\frac{b\theta _{0}(t)-be^{-gD}\theta _{0}(t-D)}{l(t)}-g, \end{aligned}$$

(79)

In the steady state: $r(t)=\rho$.