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The Sonnenschein–Mantel–Debreu theorem is an important result in general equilibrium economics, proved by Gérard Debreu, , and Hugo F. Sonnenschein in the 1970s. It states that the excess demand curve for a market populated with utility-maximizing rational agents can take the shape of any function that is continuous, has homogeneity degree zero, and is in accordance with Walras's law. This implies that market processes will not necessarily reach a unique and stable equilibrium point. More recently, Jordi Andreu, Pierre-André Chiappori, and Ivar Ekeland extended this result to market demand curves, both for individual commodities and for the aggregate demand of an economy as a whole.
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The only income distribution that is not permissible is a uniform one where all individuals have the same income and therefore, since they have the same preferences, they are all identical. For a while it was unclear whether SMD-style results also applied to the market demand curve itself, and not just the excess demand curve. But in 1982 Jordi Andreu established an important preliminary result suggesting that this was the case, and in 1999 Pierre-André Chiappori and Ivar Ekeland used vector calculus to prove that the Sonnenschein–Mantel–Debreu results do indeed apply to the market demand curve. This means that market demand curves may take on highly irregular shapes, quite unlike textbook models, even if all individual agents in the market are perfectly rational.
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