Effects of Law-Enforcement Efficiency and Duration of Trials in an Oligopolistic Competition Among Fair and Unfair Firms

Abstract

Building on differential game theory involving asymmetric agents, an oligopoly game between two distinct groups of firms is analyzed and solved under open-loop information. One group develops Research & Development to reduce its marginal production costs and behaves fairly, whereas the other one violates intellectual property rights of the rival, using the stolen technology to reduce its own marginal costs. We investigate the effects of law enforcement in this setup, by discussing the appropriate fine to be determined and the profitability of unfair behavior. Finally, we assess how the duration of related trials can affect efficiency of enforcement policy.

Appendix

Proof of Proposition 3.2

First, we state necessary conditions for open-loop equilibrium before imposing symmetry. By Pontryagin’s maximum principle, the costate equations of the model are:

$$\begin{aligned} \dot{\lambda }_{ii}(t)= & {} \rho \lambda _{ii}(t)-\dfrac{\partial \widetilde{H}_i}{\partial c_i}\nonumber \\ \Longrightarrow \ \dot{\lambda }_{ii}(t)= & {} (\rho -\delta ) \lambda _{ii}(t)+\dfrac{-(N-h+1)\widetilde{c}(t)+(N-h)\overline{c}(t)+A}{N+1}, \ i=1, \ldots , h, \nonumber \\ \end{aligned}$$

(29)

$$\begin{aligned} \dot{\lambda }_{ij}(t)= & {} \rho \lambda _{ij}(t)-\dfrac{\partial \widetilde{H}_i}{\partial c_j}\nonumber \\ \Longrightarrow \ \dot{\lambda }_{ij}(t)= & {} (\rho -\delta ) \lambda _{ij}(t), \ i,j=1, \ldots , h, \ i \ne j, \end{aligned}$$

(30)

$$\begin{aligned} \dot{\lambda }_{ij}(t)= & {} \rho \lambda _{ij}(t)-\dfrac{\partial \widetilde{H}_i}{\partial c_j}\nonumber \\ \Longrightarrow \ \dot{\lambda }_{ij}(t)= & {} (\rho -\delta ) \lambda _{ij}(t), \ i=1, \ldots , h, \ j=h+1, \ldots , N, \end{aligned}$$

(31)

$$\begin{aligned} \dot{\mu }_{ii}(t)= & {} \rho \mu _{ii}(t)-\dfrac{\partial \overline{H}_i}{\partial c_i}\nonumber \\ \Longrightarrow \ \dot{\mu }_{ii}(t)= & {} (\rho -\delta ) \mu _{ii}(t)+\dfrac{h \widetilde{c}(t)-(h+1)\overline{c}(t)+A}{N+1}, \ i=h+1, \ldots , N, \end{aligned}$$

(32)

$$\begin{aligned} \dot{\mu }_{ij}(t)= & {} \rho \mu _{ij}(t)-\dfrac{\partial \overline{H}_i}{\partial c_j}\nonumber \\ \Longrightarrow \ \dot{\mu }_{ij}(t)= & {} (\rho -\delta ) \mu _{ij}(t), \ i,j=h+1, \ldots , N, \ i \ne j, \end{aligned}$$

(33)

$$\begin{aligned} \dot{\mu }_{ij}(t)= & {} \rho \mu _{ij}(t)-\dfrac{\partial \overline{H}_i}{\partial c_j}\nonumber \\ \Longrightarrow \ \dot{\mu }_{ij}(t)= & {} (\rho -\delta ) \mu _{ij}(t), \ i=h+1, \ldots , N, \ j=1, \ldots , h. \end{aligned}$$

(34)

The associated transversality conditions are:

$$\begin{aligned}&\lim \limits _{t \rightarrow T_2} e^{-\rho t}\lambda _{ii}(t)c_i(t)=0, \end{aligned}$$

(35)

$$\begin{aligned}&\lim \limits _{t \rightarrow T_2} e^{-\rho t}\lambda _{ij}(t)c_j(t)=0, \end{aligned}$$

(36)

for $i=1, \ldots , h$.

$$\begin{aligned}&\lim \limits _{t \rightarrow T_1} e^{-\rho t}\mu _{ii}(t)c_i(t)=0, \end{aligned}$$

(37)

$$\begin{aligned}&\lim \limits _{t \rightarrow T_1} e^{-\rho t}\mu _{ij}(t)c_j(t)=0, \end{aligned}$$

(38)

for $i=h+1, \ldots , N$.

Note that the concavity of all Hamiltonian functions, with respect to the control variables, ensures that all the second-order conditions hold.

The costate equations deserve further investigation: Eqs. (30) and (33) are not actually relevant for the model dynamics, because the related costate variables do not appear in (11) and (13). On the other hand, (31) and (34) trivially admit the nil solutions $\lambda _{ij}=\mu _{ij} \equiv 0$, but if such solutions were plugged into (11) and (13), then the spillover effects measured by $\gamma$ would vanish in the strategic equations.

After inserting the optimal controls obtained in (11) and (13) into (29) and (32), we achieve the dynamic equations for the open-loop strategies of controls $\widetilde{k}(t)$ and $\overline{k}(t)$, thus entailing the state-control dynamic system (17), having initial conditions on marginal costs, i.e., $\widetilde{c}(0)=\widetilde{c}_0$, $\overline{c}(0)=\overline{c}_0$, and final conditions on Research & Development levels. This choice of final conditions is due to the transversality conditions: (36) and (38) trivially hold, because the related costate variables vanish at all t. On the other hand, (35) and (37) provide us with $\widetilde{k}(T_1)=\overline{k}(T_1)=0$, because $\lambda _{ii}$ and $\mu _{ii}$ cannot be identically zero; otherwise, the model would collapse to a static game.