  The Hylland-Zeckhauser gave a classic pricing-based mechanism (HZ) for a
one-sided matching market; it yields allocations satisfying Pareto optimality
and envy-freeness (Hylland and Zeckhauser, 1979), and the mechanism is
incentive compatible in the large (He et al., 2018). They also studied the
exchange extension of HZ and gave an example showing that it may not even admit
an equilibrium. In this paper, we consider two models of two sided matching
markets: when utility functions are symmetric and when they are non-symmetric.
We ask if these models always admit allocations satisfying the two basic
properties of Pareto efficiency and envy freeness. Our results are negative. A
corollary of the former result is a negative result for non-bipartite matching
markets as well.
