In this paper we consider a market where a heterogeneous population of individual actors demands units of various types of a heterogeneous good (for example, housing). It is assumed that allocation of the market does not take place completely by means of the price mechanism, either because prices are fixed, or because they can only vary within some limited range. Rationing is assumed to take place by preventing some actors from realising the alternative they have chosen. We will prove three existence theorems. First of all, we demonstrate the existence of a rationed equilibrium when all prices are completely fixed. Second, we show the existence of a mixed equilibrium, that is, an equilibrium where demand and supply are matched partly by means of price adjustments and partly by means of quantity rationing. Third, we prove the existence of a mixed equilibrium in a situation where the rationing is allowed to vary over the different classes of consumers and over their present situation. This introduces the possibility of giving a preferential treatment to some groups of consumers. The second theorem is a generalisation of the first and the third a generalization of the second.