License and entry strategies for an outside innovator under duopoly with combination of royalty and fixed fee

We consider a choice of options for an innovating firm to enter the market with or without licensing its new cost-reducing technology to the incumbent firm using a combination of a royalty per output and a fixed license fee, or to license its technology without entry. With general demand and cost functions we show the following results. When the innovating firm licenses its technology to the incumbent firm without entry, the optimal royalty rate per output for the innovating firm is zero with negative fixed fee, and when the innovating firm enters the market and at the same time licenses its technology to the incumbent firm, the optimal royalty rate is positive with positive or negative fixed fee. Also we show that when cost functions are concave, the optimal royalty rate is one such that the incumbent firm drops out of the market and license without entry strategy and entry with license strategy are optimal for the innovator; and when cost functions are strictly convex, there is an internal solution of the optimal royalty rate under duopoly and entry with license strategy is optimal for the innovator.