A number of recent works have addressed the problem of describing the allocation of resources in an economy where prices are fixed at a value that may not achieve equilibrium of supply and demand in the classical sense. It seems that, by and large, through the works of Benassy (1975), Dreze (1975), Malinvaud-Younes (1977) and Younes (1975), agreement has been reached as to the properties that an allocation should possess when transactions take place at given arbitrary fixed prices. The purpose of this paper is to present a result in comparative statics which applies in this approach. In the framework of an exchange economy, a necessary and sufficient condition for the local uniqueness of the fixed price allocation in a neighbourhood of a competitive equilibrium is established. Under this condition, in a neighbourhood of the competitive allocation, the fixed price equilibrium is locally a continuous function of the price system, a prerequisite for most studies in comparative statics. This condition can be related to the stability of the quantity adjustment process associated with fixed price equilibria in a neighbourhood of the competitive equilibrium. As is suggested by Samuelson's principle of correspondence, stability implies local uniqueness, while the converse is not true. Finally, it is shown that the fixed price equilibria are both locally unique and stable when the commodities are substitutes for the rationed consumers. The paper is organized as follows. Section 2 presents the model and results, the main proofs are gathered in Section 3 while an appendix gives the properties of the derivatives of the consumer's demands. The mathematical tools involved are first a generalization of the inverse function theorem for Lipschitzian functions due to Clarke (1976) and second the identity of Sylvester for determinants (Gantmacher (1966, pp. 32-33)).