  We study envy-free pricing mechanisms in matching markets with $m$ items and
$n$ budget constrained buyers. Each buyer is interested in a subset of the
items on sale, and she appraises at some single-value every item in her
preference-set. Moreover, each buyer has a budget that constraints the maximum
affordable payment, while she aims to obtain as many items as possible of her
preference-set. Our goal is to compute an envy-free pricing allocation that
maximizes the revenue, i.e., the total payment charged to the buyers. This
pricing problem is hard to approximate better than $\Omega({\rm min}
\{n,m\}^{1/2-\epsilon})$ for any $\epsilon>0$, unless $P=NP$. The hardness
result is due to the presence of the matching constraints given that the
simpler multi-unit case can be approximated up to a constant factor of $2$. The
goal of this paper is to circumvent the hardness result by restricting
ourselves to specific settings of valuations and budgets. Two particularly
significant scenarios are: each buyer has a budget that is greater than her
single-value valuation, and each buyer has a budget that is lower than her
single-value valuation. Surprisingly, in both scenarios we are able to achieve
a $1/4$-approximation to the optimal envy-free revenue. The algorithms utilize
a novel version of the Ausebel ascending price auction. These results may
suggest that, although it is difficult to approximate the optimal revenue in
general, ascending price auctions could achieve relatively good revenue in most
of the practical settings.
