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 an exchange economy 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 the excess demand function does not take a well-behaved form even if each agent has a well-behaved utility function. 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. This means that demand curves may take on highly irregular shapes, even if all individual agents in the market are perfectly rational. In contrast with usual assumptions, the quantity demanded of a commodity may not decrease when the price increases. Frank Hahn regarded the theorem as a dangerous critique of mainstream neoclassical economics. Formal statement There are several possible versions of the theorem that differ in detailed bounds and assumptions. The following version is formulated in the Arrow–Debreu model of economy. For the notation, see the Arrow–Debreu model page.Similarly, changing to a set-valued, closed graph function, we obtain another History of the proof The concept of an excess demand function is important in general equilibrium theories, because it acts as a signal for the market to adjust prices. If the value of the excess demand function is positive, then more units of a commodity are being demanded than can be supplied; there is a shortage. If excess demand is negative, then more units are being supplied than are demanded; there is a glut. The assumption is that the rate of change of prices will be proportional to excess demand, so that the adjustment of prices will eventually lead to an equilibrium state in which excess demand for all commodities is zero. In the 1970s, mathematical economists worked to establish rigorous microfoundations for widely used equilibrium models, on the basis of the assumption that individuals are utility-maximizing rational agents (the "utility hypothesis"). It was already known that this assumption put certain loose restrictions on the excess demand functions for individuals (continuity and Walras's law), and that these restrictions were "inherited" by the market excess demand function.