Robustness of subsidy in licensing under vertical differentiation: General distribution and cost functions

We extend the analysis of a possibility of negative royalty in licensing under oligopoly with an outside or an incumbent innovator by Liao and Sen (2005) to a case of oligopoly with vertical product differentiation under general distribution function of consumer' taste parameter and general cost functions. We consider both outside innovator case and incumbent innovator case. When the non-licensee does not drop out of the market; in the outside innovator case, if the goods of the firms are strategic substitutes (or complements), the optimal royalty rate is negative (or may be negative or positive); in the incumbent innovator case, if the goods are strategic substitutes (or complements), the optimal royalty rate may be negative or positive (is positive). When the non-licensee drops out of the market with negative royalty; in both cases, 1) If the goods are strategic substitutes, the optimal royalty rate is negative, 2) If the goods are strategic complements, the optimal royalty rate is positive