Sonnenschein–Mantel–Debreu theorem

Empirical methods Prescriptive and policy The Sonnenschein–Mantel–Debreu theorem is an important result in general equilibrium economics, proved by Gérard Debreu, Rolf Mantel [es], and Hugo F. Sonnenschein in the 1970s.

[1][2][3][4] 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.

[7][8][9][10][note 1] 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.

is a continuous function that satisfies Walras's law, then there exists an economy with households indexed by

is a set-valued function with closed graph that satisfies Walras's law, then there exists an economy with households indexed by

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.

[14] 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").

In a 1973 paper, Hugo Sonnenschein posed the question of whether these were the only restrictions that could be placed on a market excess demand function.

These results were extended by Rolf Mantel,[3] and then by Gérard Debreu in 1974,[4] who proved that, as long as there are at least as many agents in the market as there are commodities, the market excess demand function inherits only the following properties of individual excess demand functions: These inherited properties are not sufficient to guarantee that the excess demand curve is downward-sloping, as is usually assumed.

There may be more than one price vector at which the excess demand function is zero, which is the standard definition of equilibrium in this context.

In a 1976 paper, Rolf Mantel showed that the theorem still holds even if the very strong assumption is added that all consumers have homothetic preferences.

Only in special cases can an economy be expected to act as an ‘idealized consumer.’ The utility hypothesis tells us nothing about market demand unless it is augmented by additional requirements.

[20] Frank Ackerman points out that it is a corollary of Sonnenschein–Mantel–Debreu that a Walrasian auction will not always find a unique and stable equilibrium, even in ideal conditions: In Walrasian general equilibrium, prices are adjusted through a tâtonnement ('groping') process: the rate of change for any commodity’s price is proportional to the excess demand for the commodity, and no trades take place until equilibrium prices have been reached.

But SMD shows that this will not always be the case, because the excess demand function need not be uniformly downward-sloping.

[14] The theorem has also raised concerns about the falsifiability of general equilibrium theory, because it seems to imply that almost any observed pattern of market price and quantity data could be interpreted as being the result of individual utility-maximizing behavior.

In other words, Sonnenschein–Mantel–Debreu raises questions about the degree to which general equilibrium theory can produce testable predictions about aggregate market variables.

[23] However, Abu Turab Rizvi comments that this result does not practically change the situation very much, because Brown and Matzkin's restrictions are formulated on the basis of individual-level observations about budget constraints and incomes, while general equilibrium models purport to explain changes in aggregate market-level data.

As long as a macroeconomic growth model assumes an excess demand function satisfying continuity, homogeneity, and Walras's law, it can be microfounded.

[25] The Sonnenschein–Mantel–Debreu results have led some economists, such as Werner Hildenbrand and Alan Kirman,[26] to abandon the project of explaining the characteristics of the market demand curve on the basis of individual rationality.

Instead, these authors attempt to explain the law of demand in terms of the organization of society as a whole, and in particular the distribution of income.

Second, by the Hopf index theorem, in regular economies the number of equilibria will be finite and all of them will be locally unique.

[29] To do this he remarks that Walras's law and homogeneity of degree zero can be understood as the fact that the excess demand only depends on the budget set itself.

The first incomplete markets Sonnenschein–Mantel–Debreu type of result was obtained by Jean-Marc Bottazzi and Thorsten Hens.