Selling two units of a customizable good

A seller can produce two units of a good. Each unit can be customized into either version a or version b, and buyers privately learn their valuations for each version. First I consider the case when the second unit is more costly to produce than the …rst. Under certain distributional conditions the search for the optimal mechanism can be restricted to a class where there is no uncertainty about the number of units the buyer will receive, i.e. the buyer chooses whether to get 0, 1, or 2 units. However, there still may be uncertainty over which version the buyer will get. Buyers whose valuation for their favorite version is high, both in absolute terms and relative to the other version, purchase two units of their favorite version with certainty. Buyers with low values are excluded from purchasing. Buyers with values in the intermediate range typically get a lottery over di¤erent versions. I compare the fully optimal mechanism with the mechanism that optimally sells each unit separately and show that the solutions coincide when the fully optimal mechanism is deterministic but they may di¤er otherwise. In the case when the cost of the second unit is not higher than the cost of the …rst the I thank Maria Goltsman, Linjie Hao, and seminar participants at University of Waterloo and Conference on Mechanism and Institution Design (virtual, 2020) for helpful comments. Financial support from Social Sciences and Humanities Research Council of Canada is gratefully acknowledged (Grant #410-2013-001052). All remaining errors are mine. yDepartment of Economics, University of Western Ontario, Social Science Centre, London, Ontario N6A 5C2, Canada, gpavlov@uwo.ca 1 optimal mechanisms only contain contracts such that the buyer chooses whether to get 0 or 2 units. Thus the solution in this case can be shown to be isomorphic to a single unit case.