In this note we study a housing market where a heterogeneous housing stock is exogenously given and demands are derived on the basis of discrete choice models. We assume that prices are fixed by the government for policy reasons. The population is initially distributed over the housing stock in some exogenously given way and has to be redistributed in accordance with its own desires. Moreover, new households who do not yet occupy a dwelling may enter the market. Excess demand is assumed to occur for some or all housing types. The (re)distribution takes place by means of rationing. It has been shown that a rationed equilibrium can be reached by giving every household a probability of being able to realize the move to its desired dwelling type. It is observed in this paper that such equilibria are in general Pareto-inefficient, since voluntary exchange possibilities may persist. Furthermore, it is shown that in general Pareto-efficient rationed equilibria will exist as well. A proof is given by means of an algorithm of practical interest, and the procedure is illustrated by means of an example that refers to the Dutch housing market. In general, there will be a number of efficient equilibria, some of which can be regarded as more equitable than others.