Cooperative and Noncooperative R&D in Duopoly with Spillovers: Comment

Abstract

Claude d'Aspremont and Alexis Jacquemin (1988) employ a simple yet elegant symmetric duopoly model of R&D and spillovers to compare several equilibrium concepts. These concepts include (1) the two-stage noncooperative solution, (2) the two-stage mixed game,' (3) the two-stage fully cooperative solution,2 and (4) the social planner's optimum.3 For each of the cases stated above, they computed the equilibrium levels of output (Q=q1+q2) and R&D (xl=x2=x) and the required second-order conditions. They report (i) for large spillovers (i.e., /B > 0.5) x** > > x' > x* and Q**> Q> Q*> Q and (ii) for small spillovers (i.e., /3 x 2x*>x and Q**>Q*>Q>Q, where x denotes a firm's R&D level, Q denotes total industry output, ** denotes the social optimum, denotes the fully cooperative model, * the noncooperative two-stage case, and the mixed game. /3 is the spillover parameter. Here we show that comparing the pure cooperative and the pure noncooperative solutions as defined by d'Aspremont and Jacquemin is only meaningful when the noncooperative solution is stable, that is, when spillovers are not too small. We find that, for very small spillovers (in our example this occurs when 3 < 0.17), the d'Aspremont-Jacquemin observation holds because the noncooperative model is unstable. The importance of this result rests on the fact that even though the output reaction functions cross correctly when /3 < 0.17, the R&D reaction functions do not. When 0.17 < / < 0.41, stability obtains but R&D levels are higher in the noncooperative case than the fully cooperative one. For large spillovers the d'Aspremont-Jacquemin result is confirmed. Moreover, we find that the introduction of spillovers in the case of the noncooperative model tends to promote stability. In the case of the cooperative model, however, as the level of spillovers is increased, an equilibrium ceases to exist.

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