  We consider a multiproduct monopoly pricing model. We provide sufficient
conditions under which the optimal mechanism can be implemented via upgrade
pricing -- a menu of product bundles that are nested in the strong set order.
Our approach exploits duality methods to identify conditions on the
distribution of consumer types under which (a) each product is purchased by the
same set of buyers as under separate monopoly pricing (though the transfers can
be different), and (b) these sets are nested.
  We exhibit two distinct sets of sufficient conditions. The first set of
conditions is given by a weak version of monotonicity of types and virtual
values, while maintaining a regularity assumption, i.e., that the
product-by-product revenue curves are single-peaked. The second set of
conditions establishes the optimality of upgrade pricing for type spaces with
monotone marginal rates of substitution (MRS) -- the relative preference ratios
for any two products are monotone across types. The monotone MRS condition
allows us to relax the earlier regularity assumption.
  Under both sets of conditions, we fully characterize the product bundles and
prices that form the optimal upgrade pricing menu. Finally, we show that, if
the consumer's types are monotone, the seller can equivalently post a vector of
single-item prices: upgrade pricing and separate pricing are equivalent.
