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

We consider a choice of options for an innovating firm in duopoly under vertical differentiation to enter the market with or without licensing its technology for producing a higher quality good 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 distribution function of consumers' taste parameter and cost function we will show that when the innovating firm licenses its technology to the incumbent firm without entry, the optimal royalty rate per output is zero with negative fixed fee, and when the innovating firm enters the market with a license to the incumbent firm, its optimal royalty rate is positive with positive or negative fixed fee. Also we show that when cost function is concave, the optimal royalty rate is one such that the incumbent firm drops out of the market; and when cost function is strictly convex, there is an internal solution of the optimal royalty rate under duopoly