  We consider markets consisting of a set of indivisible items, and buyers that
have {\em sharp} multi-unit demand. This means that each buyer $i$ wants a
specific number $d_i$ of items; a bundle of size less than $d_i$ has no value,
while a bundle of size greater than $d_i$ is worth no more than the most valued
$d_i$ items (valuations being additive). We consider the objective of setting
prices and allocations in order to maximize the total revenue of the market
maker. The pricing problem with sharp multi-unit demand buyers has a number of
properties that the unit-demand model does not possess, and is an important
question in algorithmic pricing. We consider the problem of computing a revenue
maximizing solution for two solution concepts: competitive equilibrium and
envy-free pricing.
  For unrestricted valuations, these problems are NP-complete; we focus on a
realistic special case of "correlated values" where each buyer $i$ has a
valuation $v_i\qual_j$ for item $j$, where $v_i$ and $\qual_j$ are positive
quantities associated with buyer $i$ and item $j$ respectively. We present a
polynomial time algorithm to solve the revenue-maximizing competitive
equilibrium problem. For envy-free pricing, if the demand of each buyer is
bounded by a constant, a revenue maximizing solution can be found efficiently;
the general demand case is shown to be NP-hard.
