Sonnenschein-Mantel-Debreu Theorem
Encyclopedia
The Sonnenschein–Mantel–Debreu theorem (named after Gérard Debreu
, Rolf Ricardo Mantel, and Hugo Freund Sonnenschein) is a result in general equilibrium
economics. It states that the excess demand function for an economy
is not restricted by the usual rationality restrictions on individual demands in the economy. Thus microeconomic rationality assumptions have no equivalent macroeconomic implications. Its main implications are that with many interdependent markets the economic equilibrium may neither be unique nor stable. As Rizvi has stated it, it is a "deeply negative result" for microeconomic research.
These inherited properties are not sufficient to guarantee that the aggregate excess demand functions obey the weak axiom of revealed preference
: this aggregate demand functions can have "any shape", which means that in perfect competition
model, it is impossible to deduce from a maximizing behavior of households and firms the shape of their supplies and demands. This has lots of consequences for the microeconomic field. Most notably, the uniqueness and the stability of the equilibrium is not guaranteed : it may have more than one root – more than one price vector at which excess demand is zero (the standard definition of equilibrium in this context) and no general process directing toward any equilibrium point (such as the famous Walrasian "tâtonnement" process
) can be deduced from the assumptions.
But, as Rizvy has explained, the range of implications are not here limited:
"There are problems with establishing general results on uniqueness (Ingrao and Israel 1990,chap. 11; Kehoe 1985, 1991; Mas-Colell 1991), stability (Sonnenschein 1973; Ingrao and Israel 1990, chap. 12; Rizvi 1990, 94–144), comparative statics (Kehoe 1985; Nachbar 2002, 2004), econometric identification (Stoker 1984a, 1984b), microfoundations of macroeconomics (Kirman 1992; Rizvi 1994b), and the foundations of imperfectly competitive general equilibrium (Roberts and Sonnenschein 1977; Grodal 1996). Subfields of economics that relied on well-behaved aggregate excess demand for much of their theoretical development, such as international economics, were also left in the lurch (Kemp and Shimomura 2002)."
Occasionally the Sonnenschein–Mantel–Debreu theorem is referred to as the “Anything Goes Theorem”.
s. A change in a price of a particular good has two consequences. First, the good in question is cheaper or more expensive relative to all other goods, which tends to increase or decrease the demand for that good, respectively – this is called the substitution effect. On the other hand the price change also affects the real wealth of consumers in society, making some richer and some poorer, which depending on their preferences will make some demand more of the good and some less – the wealth effect. The two phenomena can work in opposite or reinforcing directions, which means that more than one set of prices can clear all markets
simultaneously.
In mathematical terms the number of equations is equal to the number of individual excess demand functions which in turn equals the number of prices to be solved for. By Walras' 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
. 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 as a comparative statics point of view does not allow to know what happen when one moves from one equilibrium : it has no reason to move to a new one.
Some critics have taken the theorem to mean that General equilibrium analysis cannot be usefully applied to understand real life economies since it makes imprecise predictions (i.e. “Anything Goes”). Others have countered that there is no a priori reason why one should expect a real life economy to have a unique equilibrium and hence the possibility of multiple outcomes is in fact a realistic feature of the theory, with the saving grace that it is still possible to analyze local shocks in a 'comparative statics' view point.
was first conjectured by Andreu Mas Colell in 1986. To do this he remarks that Walras' law
and Homogeneity of degree zero can be understood as the fact that the excess demand only depends of 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 Law if seen as budget feasibility of excess demand function. The first incomplete markets Sonnenschein-Mantel-Debreu type of result was obtained Bottazzi and Hens (1996). Other works expanded the type of assets beyond the popular real assets structures like Chiappori and Ekland (1999). All such results are local.
Finally Momi (2010) extended Bottazzi and Hens’s approach as a global result.
Gerard Debreu
Gérard Debreu was a French economist and mathematician, who also came to have United States citizenship. 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 in Economics.-Biography:His father was the...
, Rolf Ricardo Mantel, and Hugo Freund Sonnenschein) is a result in general equilibrium
General equilibrium
General equilibrium theory is a branch of theoretical economics. It seeks to explain the behavior of supply, demand and prices in a whole economy with several or many interacting markets, by seeking to prove that a set of prices exists that will result in an overall equilibrium, hence general...
economics. It states that the excess demand function for an economy
Economy
An economy consists of the economic system of a country or other area; the labor, capital and land resources; and the manufacturing, trade, distribution, and consumption of goods and services of that area...
is not restricted by the usual rationality restrictions on individual demands in the economy. Thus microeconomic rationality assumptions have no equivalent macroeconomic implications. Its main implications are that with many interdependent markets the economic equilibrium may neither be unique nor stable. As Rizvi has stated it, it is a "deeply negative result" for microeconomic research.
Statement of the theorem
Formally, the theorem states that the Walrasian aggregate excess demand function inherits only certain properties of individual excess demand functions:- ContinuityContinuous functionIn mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...
- Homogeneity of degree zero,
- Walras' lawWalras' lawWalras’ Law is a principle in general equilibrium theory asserting that when considering any particular market, if all other markets in an economy are in equilibrium, then that specific market must also be in equilibrium. Walras’ Law hinges on the mathematical notion that excess market demands ...
, and a - boundary conditionBoundary value problemIn mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional restraints, called the boundary conditions...
assuring that as prices approach zero demand becomes large.
These inherited properties are not sufficient to guarantee that the aggregate excess demand functions obey the weak axiom of revealed preference
Revealed preference
Revealed preference theory, pioneered by American economist Paul Samuelson, is a method by which it is possible to discern the best possible option on the basis of consumer behavior. Essentially, this means that the preferences of consumers can be revealed by their purchasing habits...
: this aggregate demand functions can have "any shape", which means that in perfect competition
Perfect competition
In economic theory, perfect competition describes markets such that no participants are large enough to have the market power to set the price of a homogeneous product. Because the conditions for perfect competition are strict, there are few if any perfectly competitive markets...
model, it is impossible to deduce from a maximizing behavior of households and firms the shape of their supplies and demands. This has lots of consequences for the microeconomic field. Most notably, the uniqueness and the stability of the equilibrium is not guaranteed : it may have more than one root – more than one price vector at which excess demand is zero (the standard definition of equilibrium in this context) and no general process directing toward any equilibrium point (such as the famous Walrasian "tâtonnement" process
Walrasian auction
A Walrasian auction, introduced by Leon Walras, is a type of simultaneous auction where each agent calculates its demand for the good at every possible price and submits this to an auctioneer. The price is then set so that the total demand across all agents equals the total amount of the good...
) can be deduced from the assumptions.
But, as Rizvy has explained, the range of implications are not here limited:
"There are problems with establishing general results on uniqueness (Ingrao and Israel 1990,chap. 11; Kehoe 1985, 1991; Mas-Colell 1991), stability (Sonnenschein 1973; Ingrao and Israel 1990, chap. 12; Rizvi 1990, 94–144), comparative statics (Kehoe 1985; Nachbar 2002, 2004), econometric identification (Stoker 1984a, 1984b), microfoundations of macroeconomics (Kirman 1992; Rizvi 1994b), and the foundations of imperfectly competitive general equilibrium (Roberts and Sonnenschein 1977; Grodal 1996). Subfields of economics that relied on well-behaved aggregate excess demand for much of their theoretical development, such as international economics, were also left in the lurch (Kemp and Shimomura 2002)."
Occasionally the Sonnenschein–Mantel–Debreu theorem is referred to as the “Anything Goes Theorem”.
Explanation
The reason for the result is the presence of wealth effectWealth effect
The wealth effect is an economic term, referring to an increase in spending that accompanies an increase in perceived wealth.-Effect on individuals:...
s. A change in a price of a particular good has two consequences. First, the good in question is cheaper or more expensive relative to all other goods, which tends to increase or decrease the demand for that good, respectively – this is called the substitution effect. On the other hand the price change also affects the real wealth of consumers in society, making some richer and some poorer, which depending on their preferences will make some demand more of the good and some less – the wealth effect. The two phenomena can work in opposite or reinforcing directions, which means that more than one set of prices can clear all markets
Market clearing
In economics, market clearing refers to either# a simplifying assumption made by the new classical school that markets always go to where the quantity supplied equals the quantity demanded; or# the process of getting there via price adjustment....
simultaneously.
In mathematical terms the number of equations is equal to the number of individual excess demand functions which in turn equals the number of prices to be solved for. By Walras' 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
Social welfare function
In economics, a social welfare function is a real-valued function that ranks conceivable social states from lowest to highest. Inputs of the function include any variables considered to affect the economic welfare of a society...
. 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
Comparative statics
In economics, comparative statics is the comparison of two different economic outcomes, before and after a change in some underlying exogenous parameter....
, 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 as a comparative statics point of view does not allow to know what happen when one moves from one equilibrium : it has no reason to move to a new one.
Some critics have taken the theorem to mean that General equilibrium analysis cannot be usefully applied to understand real life economies since it makes imprecise predictions (i.e. “Anything Goes”). Others have countered that there is no a priori reason why one should expect a real life economy to have a unique equilibrium and hence the possibility of multiple outcomes is in fact a realistic feature of the theory, with the saving grace that it is still possible to analyze local shocks in a 'comparative statics' view point.
Extension to incomplete markets
The extension to incomplete marketsIncomplete markets
In economics, incomplete markets refers to markets in which the number of Arrow–Debreu securities is less than the number of states of nature...
was first conjectured by Andreu Mas Colell in 1986. To do this he remarks that Walras' law
Walras' law
Walras’ Law is a principle in general equilibrium theory asserting that when considering any particular market, if all other markets in an economy are in equilibrium, then that specific market must also be in equilibrium. Walras’ Law hinges on the mathematical notion that excess market demands ...
and Homogeneity of degree zero can be understood as the fact that the excess demand only depends of 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 Law if seen as budget feasibility of excess demand function. The first incomplete markets Sonnenschein-Mantel-Debreu type of result was obtained Bottazzi and Hens (1996). Other works expanded the type of assets beyond the popular real assets structures like Chiappori and Ekland (1999). All such results are local.
Finally Momi (2010) extended Bottazzi and Hens’s approach as a global result.