Sonnenschein-Mantel-Debreu Theorem
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The Sonnenschein-Mantel-Debreu Theorem is a result in General equilibrium economics whose main implication is that with many interrelated markets the economic equilibrium may not be unique. Formally, the theorem states that the aggregate excess demand function inherits only certain properties of individual excess demand functions (Continuity, Homogeneity of degree zero, Walras' law, and a boundary condition assuring that as prices approach zero demand becomes large). In turn, these inherited properties are not sufficient to guarantee that the aggregate excess demand function obeys the weak axiom of revealed preference which implies that 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). Occasionally the Sonnenschein-Mantel-Debreu Theorem is referred to as the “Anything Goes Theorem”.
The reason for the result is the presence of wealth effects. 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 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 function. Second, by the Index theorem, in Regular economies the number of equilbria 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 and the relevant equilibrium is stable.
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.
