Budget et systèmes mixtes de demande

P.A. Samuelson (1960) has pointed out that, in equilibrium systems, n (primal) variables (x1, ..., xn) and n (dual) variables (y1, ..., yn) are pairwise conjugate. If, with each pair of conjugate variables, we select one and only one of the elements of the pair, we get 2n equivalent equilibrium systems. This approach to "mixed equilibrium systems" does not extend mutatis mutandis to consumer theory, where mixed demand is derived through maximization of a mixed utility function (cf. Samuelson, 1965, or Chavas, 1984) or by partially inverting the local demand system (cf. Salvas, Bronsard et Leblanc, 1977). In this paper, we study the existence problem of such mixed demand systems, defined intrinsically, with particular focus on the, yet unexplored, potential conjugacy relation between numeraire and budget constraint. In particular our proposition 3 states that, if preferences are defined and twice continuously differentiable over the strictly positive orthant Rn+, then there exists an open unbounded subset of Rn+, on which the numeraire can be substituted to the budget as exogenous variable of a mixed demand system. We also provide counterexamples, which show this is generally not true over the whole set on which preferences are defined (or even on an open dense subset of it). However, since our counterexamples deal with preferences which either exhibit satiation or are not defined over the whole of Rn+, the question to replace "open unbounded" in proposition 3 by "open dense" (so that the complementary set would be a null set) remains open. Dans le cadre de l’étude des systèmes mixtes introduits par Samuelson (1960), on est amené à établir une relation de conjugaison entre chaque variable primale et une variable duale. On se demande alors si, par exemple dans le cas des systèmes de demande, il existe un système mixte où les k premières variables primales et les n-k dernières variables duales du système peuvent être exprimées en fonction des variables complémentaires. Cependant, pour la contrainte de budget, considérée comme variable duale, la seule variable primale candidate au rôle de variable primale conjuguée est le numéraire.