We consider a nonlinear pricing problem faced by a dominant firm competing with a minor firm. The dominant firm offers a general tariff first and then the minor firm responds with a per- unit price, followed by a buyer choosing her purchases. By developing a "mechanism design approach" to solve the subgame perfect equilibrium, we characterize the dominant firm's optimal nonlinear tariff, which exhibits convexity and yet can display quantity discounts. In equilibrium the dominant firm uses a continuum of unchosen offers to constrain its rival's potential deviations and extract more surplus from the buyer. Antitrust implications are also discussed.
We gratefully acknowledge helpful comments from two anonymous referees, Sandro Brusco, Giacomo Calzolari, Jimmy Chan, Yeon-Koo Che, Jiawei Chen, Ami Glazer, Fei Li, Giuseppe Lopomo, Preston McAfee, Alessandro Pavan, Patrick Rey, Mingjun Xiao, Huseyin Yildirim, Jidong Zhou, and seminar participants at Caltech, SHUFE, UC Davis, UCSB, HKU, USC, ZJU, UBC, University of Washington, UC Irvine, NCKU, CUHK, Nagoya University, UIUC, UNCC, UIBE, CityU, Wuhan U, 2017 Triangle Microeconomics Conference, 2017 Workshop on IOMS, 2018 NSF/NBER/CEME Decentralization Conference, 2018 CRESS-JUFE Conference, IO Workshop at University of St. Gallen, 17th IIOC, 2019 AMES, and Tsinghua BEAT 2019