The
Sonnenschein-Mantel-Debreu Theorem is a result in
General equilibriumGeneral equilibrium theory is a branch of theoretical neoclassical economics. It seeks to explain the behavior of supply, demand and prices in a whole economy with several or many markets, by seeking to prove that equilibrium prices for goods exist and that all prices are at equilibrium, hence...
economics. It states that the system of
excess demand functionsIn economics, Excess demand is when quantity demanded is more than quantity supplied.See Economic shortage....
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. Regarding microeconomics, its main implication is that with many interdependent markets the economic equilibrium may not be unique.
Formally, the theorem states that the Walrasian demand aggregate excess demand functions inherit only certain properties of individual excess demand functions:
- Continuity
In 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...
- Homogeneity of degree zero,
- 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 a
- boundary condition
In 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).
In turn, these inherited properties are not sufficient to guarantee that the aggregate excess demand functions obey the weak axiom of revealed preference which implies that it may have more than one
rootIn mathematics, a root of a real-, complex- or generally vector-valued function ƒ is a member x of the domain of ƒ such that ƒ vanishes at , that is,...
– more than one price vector at which excess demand is zero (the standard definition of equilibrium in this context).
The
Sonnenschein-Mantel-Debreu Theorem is a result in
General equilibriumGeneral equilibrium theory is a branch of theoretical neoclassical economics. It seeks to explain the behavior of supply, demand and prices in a whole economy with several or many markets, by seeking to prove that equilibrium prices for goods exist and that all prices are at equilibrium, hence...
economics. It states that the system of
excess demand functionsIn economics, Excess demand is when quantity demanded is more than quantity supplied.See Economic shortage....
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. Regarding microeconomics, its main implication is that with many interdependent markets the economic equilibrium may not be unique.
Formally, the theorem states that the Walrasian demand aggregate excess demand functions inherit only certain properties of individual excess demand functions:
- Continuity
In 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...
- Homogeneity of degree zero,
- 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 a
- boundary condition
In 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).
In turn, these inherited properties are not sufficient to guarantee that the aggregate excess demand functions obey the weak axiom of revealed preference which implies that it may have more than one
rootIn mathematics, a root of a real-, complex- or generally vector-valued function ƒ is a member x of the domain of ƒ such that ƒ vanishes at , that is,...
– more than one price vector at which excess demand is zero (the standard definition of equilibrium in this context). Occasionally the Sonnenschein-Mantel-Debreu Theorem is referred to as the “Anything Goes Theorem”.
The reason for the result is the presence of
wealth effectThe 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 effectThe wealth effect is an economic term, referring to an increase in spending that accompanies an increase in perceived wealth.-Effect on individuals:...
. 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' 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 ...
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 demand functions are well behaved, these assumptions do not guarantee that the aggregate demand is well behaved 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 functionIn 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 welfare of the 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 staticsIn 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 and the relevant equilibrium is stable.
Some critics have taken the theorem to mean that
General equilibriumGeneral equilibrium theory is a branch of theoretical neoclassical economics. It seeks to explain the behavior of supply, demand and prices in a whole economy with several or many markets, by seeking to prove that equilibrium prices for goods exist and that all prices are at equilibrium, hence...
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.