  The problem of designing a profit-maximizing, Bayesian incentive compatible
and individually rational mechanism with flexible consumers and costly
heterogeneous supply is considered. In our setup, each consumer is associated
with a flexibility set that describes the subset of goods the consumer is
equally interested in. Each consumer wants to consume one good from its
flexibility set. The flexibility set of a consumer and the utility it gets from
consuming a good from its flexibility set are its private information. We adopt
the flexibility model of [1] and focus on the case of nested flexibility sets
-- each consumer's flexibility set can be one of k nested sets. Examples of
settings with this inherent nested structure are provided. On the supply side,
we assume that the seller has an initial stock of free supply but it can
purchase more goods for each of the nested sets at fixed exogenous prices. We
characterize the allocation and purchase rules for a profit-maximizing,
Bayesian incentive compatible and individually rational mechanism as the
solution to an integer program. The optimal payment function is pinned down by
the optimal allocation rule in the form of an integral equation. We show that
the nestedness of flexibility sets can be exploited to obtain a simple
description of the optimal allocations, purchases and payments in terms of
thresholds that can be computed through a straightforward iterative procedure.
