Stochastic control model for R&D race in a mixed duopoly with spillovers and knowledge stocks

Abstract

We consider the stochastic control model with finite time horizon for a mixed duopoly Research and Development (R&D) race between the profit-maximizing private firm and welfare-maximizing public firm. In our two-firm stochastic control R&D race model with input and output spillovers, the stochastic control variable is taken to be the private firm’s rate of R&D expenditure and the hazard rate of success of innovation has dependence on the R&D effort and knowledge stock. Given the fixed R&D effort of the public firm, the optimal control is determined so as to maximize the private firm’s value function subject to market uncertainty arising from the stochastic profit flow of the new innovative product. We conduct various sensitivity tests with varying model parameters to analyze the effects of input spillover, output spillover and knowledge stock on the optimal control policy and the value function of the profit-maximizing private firm. The R&D effort of the private firm is found to increase when the profit flow rate increases. Moreover, the optimal R&D effort level may decrease with increasing private firm’s knowledge stock and output spillover. The effects of input spillover on the optimal control policy and value function are seen to be relatively small. We examine the robustness of various observed phenomena of the two-firm R&D race with varying values of the fixed R&D effort of the public firm. With regard to public policy issue, we examine the level of the fixed public firm’s R&D effort so that social welfare is maximized.

Appendix: Derivation of \(C_{1} (\tau ; z_{1} )\) and \(C_{2} (\tau ; z_{1} )\), \(\tau =T-t\)

It is convenient to use $\tau =T-t$ as the temperal variable in $C_1$ and $C_2$. To determine $C_{1} (\tau ; z_{1} )$ and $C_{2} (\tau ; z_{1} )$, we substitute Eq. (2.6c) into the HJB formulation and obtain the following governing partial differential equations for $C_{1} (\tau ; z_{1} )$ and $C_{2} (\tau ; z_{1} )$:

$$\begin{aligned} \frac{\partial C_{1}}{\partial \tau }-(u_{1}+\beta _{1}u_{0})\frac{\partial C_{1}}{\partial z_{1}}&= -\left( r-\mu +h_{0}+h_{1}\right) C_{1} + \frac{h_{0}(1-\theta )\Pi }{r-\mu }\left[ 1-e^{-(r-\mu )\tau }\right] \\&+ ~\frac{h_{1}\theta \Pi }{r-\mu }\left[ 1-e^{-(r-\mu )\tau }\right] , \\ \frac{\partial C_{2}}{\partial \tau }-(u_{1}+\beta _{1}u_{0})\frac{\partial C_{2}}{\partial z_{1}}&= -(r +h_{0}+h_{1})C_{2} - c_{1}(u_{1}). \end{aligned}$$

The above equations share the following general form:

$$\begin{aligned} \frac{\partial C}{\partial \tau }-a \frac{\partial C}{\partial z_{1}} = h(\tau ; z_{1} )C + k(\tau ; z_{1} ), \end{aligned}$$

(6.1)

where $h(\tau ; z_{1} )$ and $k(\tau ; z_{1} )$ are functions of $\tau$ and $z_{1}$. Here, $a$ is constant ($u_0$ is constant and $u_1$ is also fixed by adopting ${\mathrm {sup}}~Q_1$). The general solution of Eq. (6.1) is given by

$$\begin{aligned} C(\tau ;u) = H(\tau ;u)\Big [\phi (u) + \int _{0}^{\tau } \frac{k(s;u-as)}{H(s;u)}~{\mathrm {d}}s\Big ], \end{aligned}$$

where

$$\begin{aligned} H(\tau ;u) = {\mathrm {exp}}\Big (\int h(s;u-as)~{\mathrm {d}}s\Big ), ~~~u=z_{1}+a \tau , \end{aligned}$$

and $\phi (u)$ is an arbitrary function to be determined from appropriate auxiliary conditions. Since $C_1 (0; z_{1} )=0$ and $C_2 (0; z_{1} )=0$, we obtain $\phi _{1} (u)=\phi _{2} (u)=0$. Therefore, the solution to Eq. (6.1) is found to be

$$\begin{aligned} C(\tau ;u) = H(\tau ;u)\int _{0}^{\tau } \frac{k(s;u-as)}{H(s;u)}~{\mathrm {d}}s. \end{aligned}$$