mathematics

Sonnenschein–Mantel–Debreu theorem

Sonnenschein–Mantel–Debreu theorem is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Sonnenschein–Mantel–Debreu theorem rather than just read about it. In short: The Sonnenschein–Mantel–Debreu theorem is an important result in general equilibrium economics, proved by Gérard Debreu, Rolf Mantel, and Hugo F. Sonnenschein in the 1970s.

Key takeaways

Sonnenschein–Mantel–Debreu theorem belongs to mathematics; place it in that map before memorising details.
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Reference excerpt

The Sonnenschein–Mantel–Debreu theorem is an important result in general equilibrium economics, proved by Gérard Debreu, Rolf Mantel, 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 Z {\displaystyle Z} 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. 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. He conjectured that the answer was "yes," and made preliminary steps toward proving it. These results were extended by Rolf Mantel, and then by Gérard Debreu in 1974, 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:

Continuity Homogeneity of degree zero, and Walras's law These inherited properties are not sufficient to guarantee that the excess demand curve is downward-sloping, as is usually assumed. The uniqueness of the equilibrium point is also not guaranteed. 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.

Further developments In the wake of these initial publications, several scholars have extended the initial Sonnenschein–Mantel–Debreu results in a variety of ways. 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. This means that the utility that consumers assign to a commodity will always be exactly proportional to the amount of the commodity offered; for example, one million oranges would be valued exactly one million times more than one orange. Furthermore, Alan Kirman and Karl-Josef Koch proved in 1986 that the SMD theorem still holds even if all agents are assumed to have identical preferences, and the distribution of income is assumed to be fixed across time and independent of prices. 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 aggregate 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 the aggregate market demand curve may take on highly irregular shapes, even if all individual agents in the economy are perfectly rational.

Significance In the 1982 book Handbook of Mathematical Economics, Hugo Sonnenschein explained some of the implications of his theorem for general equilibrium theory:

…market demand functions need not satisfy in any way the classical restrictions which characterize consumer demand functions… The importance of the above results is clear: strong restrictions are needed in order to justify the hypothesis that a market demand function has the characteristics of a consumer demand function. 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. In other words, it cannot be assumed that the demand curve for an entire economy must be smoothly downward-sloping simply because the demand curves of individual consumers are downward-sloping. This is an instance of the more general aggregation problem, which deals with the theoretical difficulty of modeling the behavior of large groups of individuals in the same way that an individual is modeled. 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:

… excerpt ends here. Continue reading the full article.

Illustrations

Worked examples

Example 1 — a first encounter with Sonnenschein–Mantel–Debreu theorem

Start with the simplest possible case. Write down what Sonnenschein–Mantel–Debreu theorem claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Sonnenschein–Mantel–Debreu theorem before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Sonnenschein–Mantel–Debreu theorem ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Sonnenschein–Mantel–Debreu theorem

In research: Sonnenschein–Mantel–Debreu theorem appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Sonnenschein–Mantel–Debreu theorem in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Sonnenschein–Mantel–Debreu theorem is common in secondary-school and first-year university syllabi. It links to neighbouring topics Economics theorems, General equilibrium theory, Macroeconomic theories, so understanding it makes those chapters shorter.
In everyday life: Look for Sonnenschein–Mantel–Debreu theorem outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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Frequently asked questions

What is Sonnenschein–Mantel–Debreu theorem in simple terms?

The Sonnenschein–Mantel–Debreu theorem is an important result in general equilibrium economics, proved by Gérard Debreu, Rolf Mantel, and Hugo F. Sonnenschein in the 1970s.

Why does Sonnenschein–Mantel–Debreu theorem matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Sonnenschein–Mantel–Debreu theorem?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Sonnenschein–Mantel–Debreu theorem.