Design of the contingent royalty rate as related to the type of investment

This study investigates the design of the royalty rate in a first-price auction across three types of investments: incremental and lumpy with or without an exogenously given intensity. A bidder’s investment cost comprises private information. This, together with the stochastic evolution of the price of the output generated from the auctioned project, precludes the seller from setting the exact dates of investment with the winner. However, the seller can set the royalty rate to equate the winner’s royalty payment with the winner’s information rent so that the winner acts as if to maximize the seller’s revenue. We derive two main conclusions. First, compared with the case in which investment is lumpy with an exogenously given intensity, the seller can set a lower royalty rate on incremental investment because she can collect additional royalty payments from the winner, who has the option to later expand capacity. Second, the impact of output price uncertainty on the optimal royalty rate for the three types of investments exhibits two different patterns. When investment is either incremental or lumpy with an exogenously given intensity, greater output price uncertainty reduces the royalty rate. When investment is lumpy with variable intensity, greater output uncertainty raises the royalty rate. Our results imply that auctioneers may charge differential royalty rates for different types of investments.