We consider a multi-dimensional screening problem of selling a product withmultiple quality levels and design virtual value functions to derive conditionsthat imply optimality of only selling highest quality. A challenge of designingvirtual values for multi-dimensional agents is that a mechanism that pointwiseoptimizes virtual values resulting from a general application of integration byparts is not incentive compatible, and no general methodology is known forselecting the right paths for integration by parts. We resolve this issue byfirst uniquely solving for paths that satisfy certain necessary conditions thatthe pointwise optimality of the mechanism imposes on virtual values, and thenidentifying distributions that ensure the resulting virtual surplus is indeedpointwise optimized by the mechanism. Our method of solving for virtual valuesis general, and as a second application we use it to derive conditions ofoptimality for selling only the grand bundle of items to an agent with additivepreferences.
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