  Walrasian equilibrium is a prominent market equilibrium notion, but rarely
exists in markets with indivisible items. We introduce a new market equilibrium
notion, called two-price equilibrium (2PE). A 2PE is a relaxation of Walrasian
equilibrium, where instead of a single price per item, every item has two
prices: one for the item's owner and a (possibly) higher one for all other
buyers. Thus, a 2PE is given by a tuple
$(\textbf{S},\mathbf{\hat{p}},\mathbf{\check{p}})$ of an allocation
$\textbf{S}$ and two price vectors $\mathbf{\hat{p}},\mathbf{\check{p}}$, where
every buyer $i$ is maximally happy with her bundle $S_i$, given prices
$\mathbf{\check{p}}$ for items in $S_i$ and prices $\mathbf{\hat{p}}$ for all
other items. 2PE generalizes previous market equilibrium notions, such as
conditional equilibrium, and is related to relaxed equilibrium notions like
endowment equilibrium. We define the {\em discrepancy} of a 2PE -- a measure of
distance from Walrasian equilibrium -- as the sum of differences
$\hat{p}_j-\check{p}_j$ over all items (normalized by social welfare). We show
that the social welfare degrades gracefully with the discrepancy; namely, the
social welfare of a 2PE with discrepancy $d$ is at least a fraction
$\frac{1}{d+1}$ of the optimal welfare. We use this to establish welfare
guarantees for markets with subadditive valuations over identical items. In
particular, we show that every such market admits a 2PE with at least $1/7$ of
the optimal welfare. This is in contrast to Walrasian equilibrium or
conditional equilibrium which may not even exist. Our techniques provide new
insights regarding valuation functions over identical items, which we also use
to characterize instances that admit a WE.
