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Multi-dimensional Virtual Values and Second-degree Price Discrimination
We consider a multi-dimensional screening problem of selling a product with
multiple quality levels and design virtual value functions to derive conditions
that imply optimality of only selling highest quality. A challenge of designing
virtual values for multi-dimensional agents is that a mechanism that pointwise
optimizes virtual values resulting from a general application of integration by
parts is not incentive compatible, and no general methodology is known for
selecting the right paths for integration by parts. We resolve this issue by
first uniquely solving for paths that satisfy certain necessary conditions that
the pointwise optimality of the mechanism imposes on virtual values, and then
identifying distributions that ensure the resulting virtual surplus is indeed
pointwise optimized by the mechanism. Our method of solving for virtual values
is general, and as a second application we use it to derive conditions of
optimality for selling only the grand bundle of items to an agent with additive
preferences
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