Sonnenschein–Mantel–Debreu Theorem
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The Sonnenschein–Mantel–Debreu theorem is an important result in general equilibrium
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, proved by
Gérard Debreu Gérard Debreu (; 4 July 1921 – 31 December 2004) was a French-born economist and mathematician. Best known as a professor of economics at the University of California, Berkeley, where he began work in 1962, he won the 1983 Nobel Memorial Prize ...
, , and Hugo F. Sonnenschein in the 1970s. It states that the excess demand curve for an exchange economy populated with utility-maximizing
rational agents Rationality is the quality of being guided by or based on reason. In this regard, a person acts rationally if they have a good reason for what they do, or a belief is rational if it is based on strong evidence. This quality can apply to an a ...
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 curve A demand curve is a graph depicting the inverse demand function, a relationship between the price of a certain commodity (the ''y''-axis) and the quantity of that commodity that is demanded at that price (the ''x''-axis). Demand curves can be us ...
s, both for individual commodities and for the
aggregate demand In economics, aggregate demand (AD) or domestic final demand (DFD) is the total demand for final goods and services in an economy at a given time. It is often called effective demand, though at other times this term is distinguished. This is the ...
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 Neoclassical economics is an approach to economics in which the production, consumption, and valuation (pricing) of goods and services are observed as driven by the supply and demand model. According to this line of thought, the value of a go ...
.


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 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 Microfoundations are an effort to understand macroeconomic phenomena in terms of individual agents' economic behavior and interactions.Maarten Janssen (2008),Microfoundations, in ''The New Palgrave Dictionary of Economics'', 2nd ed. Research in mi ...
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 In economics, utility is a measure of a certain person's satisfaction from a certain state of the world. Over time, the term has been used with at least two meanings. * In a normative context, utility refers to a goal or objective that we wish ...
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 market
demand curve A demand curve is a graph depicting the inverse demand function, a relationship between the price of a certain commodity (the ''y''-axis) and the quantity of that commodity that is demanded at that price (the ''x''-axis). Demand curves can be us ...
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 Vector calculus or vector analysis is a branch of mathematics concerned with the differentiation and integration of vector fields, primarily in three-dimensional Euclidean space, \mathbb^3. The term ''vector calculus'' is sometimes used as a ...
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.


Significance

In the 1982 book ''Handbook of Mathematical Economics'', Hugo Sonnenschein explained some of the implications of his theorem for general equilibrium theory: In other words, it cannot be assumed that the demand curve for a single market, let alone 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 In mathematics and logic, a corollary ( , ) is a theorem of less importance which can be readily deduced from a previous, more notable statement. A corollary could, for instance, be a proposition which is incidentally proved while proving another ...
of Sonnenschein–Mantel–Debreu that a Walrasian auction will not always find a unique and stable equilibrium, even in ideal conditions:
Léon Walras Marie-Esprit-Léon Walras (; 16 December 1834 – 5 January 1910) was a French mathematical economics, mathematical economist and Georgist. He formulated the Marginalism, marginal theory of value (independently of William Stanley Jevons and Carl ...
' auction model requires that the price of a commodity will always rise in response to excess demand, and that it will always fall in response to an excess supply. But SMD shows that this will not always be the case, because the excess demand function need not be uniformly downward-sloping. The theorem has also raised concerns about the
falsifiability Falsifiability (or refutability) is a deductive standard of evaluation of scientific theories and hypotheses, introduced by the Philosophy of science, philosopher of science Karl Popper in his book ''The Logic of Scientific Discovery'' (1934). ...
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. For this reason, Andreu Mas-Colell referred to the theorem as the “Anything Goes Theorem” in his graduate-level
microeconomics Microeconomics is a branch of economics that studies the behavior of individuals and Theory of the firm, firms in making decisions regarding the allocation of scarcity, scarce resources and the interactions among these individuals and firms. M ...
textbook. Some economists have made attempts to address this problem, with Donald Brown and Rosa Matzkin deriving some
polynomial In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addit ...
restrictions on market variables by modeling the equilibrium state of a market as a topological
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a N ...
. 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.
Robert Solow Robert Merton Solow, GCIH (; August 23, 1924 – December 21, 2023) was an American economist who received the 1987 Nobel Memorial Prize in Economic Sciences, and whose work on the theory of economic growth culminated in the exogenous growth ...
interprets the theorem as showing that, for modelling macroeconomic growth, the
dynamic stochastic general equilibrium Dynamic stochastic general equilibrium modeling (abbreviated as DSGE, or DGE, or sometimes SDGE) is a macroeconomics, macroeconomic method which is often employed by monetary and fiscal authorities for policy analysis, explaining historical time-s ...
is no more microfounded than simpler models such as the Solow–Swan model. As long as a macroeconomic growth model assumes an excess demand function satisfying continuity, homogeneity, and Walras's law, it can be microfounded. The Sonnenschein–Mantel–Debreu results have led some economists, such as Werner Hildenbrand and Alan Kirman,Alan Kirman (1989) The intrinsic limits of modern economic theory: The emperor has no clothes. ''Economic Journal'' 99(395 Supplement: Conference Papers): 126-139. 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 microeconomics, the law of demand is a fundamental principle which states that there is an inverse relationship between price and quantity demanded. In other words, "conditional on ceteris paribus, all else being equal, as the price of a Goods, ...
in terms of the organization of society as a whole, and in particular the distribution of income.


Explanation

In mathematical terms, the number of equations that make up a market excess demand function is equal to the number of individual excess demand functions, which in turn equals the number of prices to be solved for. By Walras's law, if all but one of the excess demands is zero then the last one has to be zero as well. This means that there is one redundant equation and we can normalize one of the prices or a combination of all prices (in other words, only relative prices are determined; not the absolute price level). Having done this, the number of equations equals the number of unknowns and we have a determinate system. However, because the equations are non-linear there is no guarantee of a unique solution. Furthermore, even though reasonable assumptions can guarantee that the individual excess-demand functions have a unique root, these assumptions do not guarantee that the aggregate demand does as well. There are several things to be noted. First, even though there may be multiple equilibria, every equilibrium is still guaranteed, under standard assumptions, to be Pareto efficient. However, the different equilibria are likely to have different distributional implications and may be ranked differently by any given
social welfare function In welfare economics and social choice theory, a social welfare function—also called a social ordering, ranking, utility, or choice function—is a function that ranks a set of social states by their desirability. Each person's preferences ...
. Second, by the Hopf index theorem, in regular economies the number of equilibria will be finite and all of them will be locally unique. This means that comparative statics, or the analysis of how the equilibrium changes when there are shocks to the economy, can still be relevant as long as the shocks are not too large. But this leaves the question of the stability of the equilibrium unanswered, since a comparative statics perspective does not tell us what happens when the market moves away from an equilibrium.


Extension to incomplete markets

The extension to incomplete markets was first conjectured by Andreu Mas-Colell in 1986. 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. Hence, homogeneity is only saying that excess demand is the same if the budget sets are the same. This formulation extends to incomplete markets. So does Walras's law if seen as budget feasibility of excess-demand function. The first incomplete markets Sonnenschein–Mantel–Debreu type of result was obtained by Jean-Marc Bottazzi and Thorsten Hens. Other works expanded the type of assets beyond the popular real assets structures like Chiappori and Ekland. All such results are local. In 2003 Takeshi Momi extended the approach by Bottazzi and Hens as a global result.


Notes


References


Bibliography

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