Abstract
This paper compares market profit and social welfare levels between differentiated Bertrand and Cournot duopoly. We start with a basic model in which a firm with a production technology can license its new technology to a potential rival who can use the technology to produce a differentiated product and compete with the incumbent firm. It is found that for any given technology level, Bertrand competition is necessarily more profitable but less socially desirable, due to its higher royalty rate. By contrast, if the licensee firm is an incumbent firm, the results hold if the technology level is high. Furthermore, if we assume the licensor firm can engage in product innovation and choose its optimal technology endogenously and the R&D efficiency is high (low), the welfare ranking is reversed (still holds).
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Appendix A
This appendix is to derive the result in Proposition 5. Firm 2 is now an incumbent, producing a differentiated product. The demand functions and the inverse demand functions are as follows.
By routine calculation, we can derive the equilibrium profits under Cournot and Bertrand competition with no licensing as follow:
We now move to derive the equilibria with licensing. Licensing will raise the price intercept of the demand for
Under Cournot competition, the objective function of firm 1 in the first stage can be written as follows:
where
Proceeding as before, we can derive the first-order condition for profit maximization as follows:
where
The objective function for firm 1 under Bertrand competition is specified as follows:
where
Proceeding as that in Cournot competition, we derive that the optimal royalty rate under Bertrand competition may have a corner or an interior solution:
where
By subtracting
By substituting the optimal royalty rates into the profits of firm 1 and firm 2, we can further compare market profits under Bertrand and Cournot competition as follows:
if
where
where
By comparing the social welfare levels under the two competition modes, we find that
where
Appendix B
This appendix is to compare the welfare levels in the long run under the two regimes as stated in Proposition 7. The social welfare function is expressed as follows:
By substituting eqs and into the social welfare function, we can derive the social welfare levels under the two regimes as follows:
From the above two equations, we can further derive that
It follows that Bertrand competition is socially more desirable than Cournot competition if the R&D efficiency is high.
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