Optimal patent licensing: from three to two part tariffs

We consider the licensing of a cost-reducing innovation in a Cournot oligopoly where an outside innovator uses three part tariffs that are combinations of upfront fees, per unit royalties and ad valorem royalties. The key insight of our analysis is per unit royalties have a location effect and ad valorem royalties have a scale effect on marginal costs. Using these two effects, we show that the same market outcome (price, quantities, operating profits) can be sustained by multiple combinations of per unit and ad valorem royalties. In the monopoly case, under three part tariffs it is optimal to set a pure upfront fee while the unique optimal two part royalty is a pure ad valorem royalty. In the case of a general oligopoly with linear demand, for relatively insignificant innovations, it is optimal to set a pure upfront fee; otherwise there is a continuum of optimal policies and there always exists an optimal policy consisting of a positive per unit royalty and upfront fee but no ad valorem royalty. For intermediate innovations, provided the demand intercept is relatively large, there exists an optimal policy that has both kinds of royalties but no fees. Finally in a Cournot duopoly it is illustrated that when the innovator is one of the incumbent firms rather than an outsider, market outcomes separately depend on two kinds of royalties and a pure ad valorem royalty is optimal among all three part tariffs.