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Some applications of the Cartan-Kähler theorem to economic theory

The purpose of this short paper is to show a fundamental problem of economic theory gives rise to a system of nonlinear PDEs of the first order which up to now can only be solved by applying the celebrated Cartan-Kahler theorem on integral manifolds of exterior differential systems (see [3], [2]). Following standard definitions in microeconomics (see [11] for an exhaustive review of the theory), an economy is described by N agents and K goods. Agents trade and consume, not individual goods, but bundles, each bundle being described by a point x = i x, ..., x ¢ ∈ R , where xk denotes the quantity of good k ; for instance, we do not eat of buy bread and butter separately, but sandwiches, which are certain bundles of bread and butter. There will also be a set of prices p = (p1, ..., pK) ∈ R . It the prevailing price system is p, the cost of bundle x is p0x = P pkx . Agent n is fully described by his utility function U : R → R and by his initial endowment ; the latter can be given,