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Sonnenschein—Mantel—Debreu theorem Although generally assuming convexity an equilibrium will exist and will be efficient, the conditions under which it will be unique are much stronger. While the issues are fairly technical the basic intuition is that the presence of wealth effects which is the feature that most clearly delineates general equilibrium analysis from partial equilibrium generates the possibility of multiple equilibria.
When a price of a particular good changes there are two effects. First, the relative attractiveness of various commodities changes; and second, the wealth distribution of individual agents is altered. These two effects can offset or reinforce each other in ways that make it possible for more than one set of prices to constitute an equilibrium.
A result known as the Sonnenschein—Mantel—Debreu theorem states that the aggregate excess demand function inherits only certain properties of individual's demand functions, and that these ContinuityHomogeneity of degree zeroWalras' law and boundary behavior when prices are near zero are the only real restriction one can expect from an aggregate excess demand function: In particular uniqueness of equilibrium should not be expected.
There has been much research on conditions when the equilibrium will be unique, or which at least will limit the number of equilibria. One result states that under mild assumptions the number of equilibria will be finite see regular economy and odd see index theorem.
Furthermore, if an economy as a whole, as characterized by an aggregate excess demand function, has the revealed preference property which is a much stronger condition than revealed preferences for a single individual or the gross substitute property then likewise the equilibrium will be unique.
All methods of establishing uniqueness can be thought of as establishing that each equilibrium has the same positive local index, in which case by the index theorem there can be but one such equilibrium.
Determinacy[ edit ] Given that equilibria may not be unique, it is of some interest to ask whether any particular equilibrium is at least locally unique.
If so, then comparative statics can be applied as long as the shocks to the system are not too large. As stated above, in a regular economy equilibria will be finite, hence locally unique. One reassuring result, due to Debreu, is that "most" economies are regular. Work by Michael Mandler has challenged this claim.
Mandler accepts that, under either model of production, the initial endowments will not be consistent with a continuum of equilibria, except for a set of Lebesgue measure zero.
However, endowments change with time in the model and this evolution of endowments is determined by the decisions of agents e. Agents in the model have an interest in equilibria being indeterminate: Indeterminacy, moreover, is not just a technical nuisance; it undermines the price-taking assumption of competitive models.
Since arbitrary small manipulations of factor supplies can dramatically increase a factor's price, factor owners will not take prices to be parametric. The endowments where indeterminacy occurs systematically arise through time and therefore cannot be dismissed; the Arrow-Debreu-McKenzie model is thus fully subject to the dilemmas of factor price theory.
Stability[ edit ] In a typical general equilibrium model the prices that prevail "when the dust settles" are simply those that coordinate the demands of various consumers for various goods.
But this raises the question of how these prices and allocations have been arrived at, and whether any temporary shock to the economy will cause it to converge back to the same outcome that prevailed before the shock. This is the question of stability of the equilibrium, and it can be readily seen that it is related to the question of uniqueness.
If there are multiple equilibria, then some of them will be unstable. Then, if an equilibrium is unstable and there is a shock, the economy will wind up at a different set of allocations and prices once the convergence process terminates. However stability depends not only on the number of equilibria but also on the type of the process that guides price changes for a specific type of price adjustment process see Walrasian auction.
Consequently, some researchers have focused on plausible adjustment processes that guarantee system stability, i. When more than one stable equilibrium exists, where one ends up will depend on where one begins.
Unresolved problems in general equilibrium[ edit ] Research building on the Arrow—Debreu—McKenzie model has revealed some problems with the model. The Sonnenschein—Mantel—Debreu results show that, essentially, any restrictions on the shape of excess demand functions are stringent.
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In Example 4, the mode is 0, since 0 occurs most often in the set. Do not confuse a mode of 0 with no mode. Summary: The mode of a set of data is the value in the set that occurs most often.