Dynamical analysis of the R&D-based growth model with a regime switch

Abstract

Several empirical studies suggest that advanced economies experience a growth regime switch from factor accumulation to knowledge accumulation. To investigate the mechanism of such a regime switch, this study develops a concise and flexible dynamic model based on Romer (J Polit Econ 98:S71–S102, 1990) by introducing two types of endogenously supplied R&D input capital. The model replicates the growth patterns of developed and underdeveloped nations, clarifies the important role that capital plays in the difference between them, and presents several implications for interest-rate subsidies and official development assistance. Further, it shows that if a country enjoying long-run growth has little initial capital, its initial economic development will be based on capital accumulation. When the capital stock becomes sufficient for supporting R&D, the economy will achieve long-run growth through R&D.

Appendix A

1.1 A1. Positive initial endowment of human capital

In this appendix, we consider the case of $H(0)=H_0>0$. If the profit rate of human capital is equated with physical capital, it is simply the Romer regime, and as such, in this section, we consider the case wherein the profit rate of human capital is smaller than that of physical capital. In this case, $\dot{H}=0$ and $r>r_H$ hold, and therefore, human capital endowment is fixed at the initial endowment $H_0$, and R&D is executed to hold the no-arbitrage condition on capital profit rate $r$. Then $r_H$ is determined at the level that satisfies the zero-profit condition on R&D.

Therefore, the equilibrium condition on this regime is given by

$$\begin{aligned} \frac{\partial \Pi }{\partial K_A}=(1-\gamma )\delta \frac{\xi _A^{-\gamma }}{N(t)}\bar{V}-r=0\nonumber , \quad \frac{\partial \Pi }{\partial H}=\gamma \, \delta \frac{\xi _A^{1-\gamma }}{N(t)}\bar{V}-r_H=0, \end{aligned}$$

where $\xi _A(t)\equiv \frac{K_A}{H}$. From these two equations, we have

$$\begin{aligned} \tilde{v}=\frac{\xi _A^\gamma }{(1-\gamma )\delta }r,\quad \text{ and}\quad r_H=\frac{\gamma \xi _A}{1-\gamma }r. \end{aligned}$$

Thus, for the steady state, $\xi _A=(const)$ is also necessary in this case. Because $H_0$ is constant, $K_A$ also is constant, and we denote it as $\bar{K}_A$. However, the growth rate under $\bar{K}_A$ is given as

$$\begin{aligned} g_A(t)=\frac{\bar{K}_A}{A(t)N(t)}\xi _A^\gamma . \end{aligned}$$

Therefore, after the long run, the growth rate converges to $\lim _{t \rightarrow \infty }g_A(t)=0;$ namely, the system asymptotically deceases the growth rate and converges to the Solow regime.

1.2 A2. Command economy and optimal conditions

To obtain the welfare properties of the decentralized solution, we consider the social planner formulation of this growth model. A benevolent government is assumed to maximize the representative household’s utility function Eq. (8). Therefore, the Hamiltonian of this government can be written as

$$\begin{aligned} \mathcal{H}(t)&= \frac{c(t)^{1-\sigma }-1}{1-\sigma }+\lambda (t) (\underbrace{\eta ^{-\alpha }K_Y(t)^\alpha N(t)^{1-\alpha }A(t)^{1-\alpha }}_{Y}-c(t)N(t)-I_H(t))\\&+\mu (t)\delta \frac{(K(t)-K_Y(t))^{1-\gamma } H(t)^{\gamma }}{N(t)}+\lambda _H(t)I_H(t), \end{aligned}$$

where $\lambda$, $\mu$, and $\lambda _H$ are the shadow prices of per capita capital stock, knowledge, and human capital, respectively. The optimal conditions are obtained as follows:

$$\begin{aligned} \lambda (t)N(t)&= c(t)^{-\sigma }, \qquad \end{aligned}$$

(31)

$$\begin{aligned} \lambda (t)&= \lambda _H(t), \end{aligned}$$

(32)

$$\begin{aligned} \lambda (t) \alpha \eta ^{-\alpha }K_Y(t)^{\alpha -1}\bigl \{A(t)N(t)\bigr \}^{1-\alpha }&= (1-\gamma )\mu (t) \delta \frac{(K(t)-K_Y(t))^{-\gamma }H(t)^{\gamma }}{N(t)}, \qquad \nonumber \\ \end{aligned}$$

(33)

$$\begin{aligned} \rho \lambda (t)-\dot{\lambda }(t)&= (1-\gamma )\mu (t)\delta \frac{(K(t)-K_Y(t))^{-\gamma }H(t)^{\gamma }}{N(t)}, \qquad \nonumber \\ \end{aligned}$$

(34)

$$\begin{aligned} \rho \lambda _H(t)-\dot{\lambda }_H(t)&= \gamma \mu (t)\delta \frac{(K(t)-K_Y(t))^{1-\gamma }H(t)^{\gamma -1}}{N(t)}, \qquad \end{aligned}$$

(35)

$$\begin{aligned} \rho \mu (t)-\dot{\mu }(t)&= \lambda (t)(1-\alpha ) \eta ^{-\alpha } K_Y(t)^{\alpha }N(t)^{1-\alpha }A(t)^{-\alpha }. \nonumber \\ \end{aligned}$$

(36)

Using Eqs. (31), (34), and (36) and the notation $\tilde{k}_Y$, we derive the following equations:

$$\begin{aligned} \rho -g_{\lambda }(t)&= \alpha \eta ^{-\alpha }\tilde{k}_Y(t)^{\alpha -1}, \end{aligned}$$

(37)

$$\begin{aligned} \rho -g_{\mu }(t)&= \frac{1-\alpha }{\alpha }\delta \Gamma (\gamma )\tilde{k}_Y(t). \end{aligned}$$

(38)

(34) and (35) yield the optimal rate between physical capital $K_Y$ and human capital $H$ as $H=\frac{\gamma }{1-\gamma }K_Y$; namely, we have the rate between physical capital and human capital, $\xi$, in both the market and command economies in this study. Using $\xi$, the growth rate can also be written as $g_A=\delta (\tilde{k}-\tilde{k}_Y)\xi ^\gamma$.

From Eqs. (31) and (37) and the definitions of $\tilde{c}$ and $\tilde{k}$, the following Euler equation is obtained:

$$\begin{aligned} g_{\tilde{c}}(t)=\frac{1}{\sigma }\left\{ \alpha \eta ^{-\alpha }\tilde{k}_Y(t)^{\alpha -1}-\rho -n-\sigma \delta \bigl (\tilde{k}(t)-\tilde{k}_Y(t)\bigr )\xi ^\gamma \right\} . \end{aligned}$$

(39)

From Eqs. (33), (37), and (38), eliminating $\lambda$ and $\mu$ yields

$$\begin{aligned} g_{\tilde{k}_Y}(t)=\frac{1}{1-\alpha }\left[\frac{\delta (1-\alpha )\Gamma (\gamma )}{\alpha }\tilde{k}_Y(t)+n-\alpha \eta ^{-\alpha }\tilde{k}_Y(t)^{\alpha -1}\right]. \end{aligned}$$

(40)

The system consists of the following dynamic equations: (2), (39), and (40). These equations imply that $g_{\tilde{k}_Y}=g_{\tilde{k}}$ and $g_A=g_k=g_c=\delta (\tilde{k}-\tilde{k}_Y)$ must hold in a steady state. Substituting these conditions into Eqs. (2), (39), and (40), we obtain the following equations denoting the steady state of the command economy:

$$\begin{aligned}&\displaystyle \rho +\sigma \delta (\tilde{k}^{*op}-\tilde{k}_Y^{*op})\xi ^\gamma =\alpha \eta ^{-\alpha }\tilde{k}_Y^{*op\, \alpha -1}-n,\end{aligned}$$

(41)

$$\begin{aligned}&\displaystyle \eta ^{-\alpha }\tilde{k}_Y^{*op \, \alpha }-\tilde{c}^{*op}-\bigl \{n+\delta (\tilde{k}^{*op}-\tilde{k}_Y^{*op})\xi ^\gamma \bigr \}\tilde{k}^{*op}=0\end{aligned}$$

(42)

$$\begin{aligned}&\displaystyle \alpha \eta ^{-\alpha }\tilde{k}_Y^{*op \alpha -1}-n=\frac{(1-\alpha )\Gamma (\gamma )}{\alpha }\delta \tilde{k}_Y^{*op}. \end{aligned}$$

(43)

By eliminating $\alpha \eta ^{-\alpha }\tilde{k}_Y^{*op\, \alpha -1}-n$ from (41) and (43), we obtain the result that optimal capital allocation in the steady state in a command economy is the same as that in a market economy given by (12). Furthermore, the growth rate for given R&D-difficulty-adjusted capital stock in a command economy is the same as that in a market economy given by (13).