Properties and characterization of general equilibrium
Basic questions in general equilibrium analysis are concerned with the conditions under which an equilibrium will be efficient, which efficient equilibria can be achieved, when an equilibrium is guaranteed to live and when the equilibrium will be unique and stable.
The first necessary Welfare Theorem asserts that market equilibria are Pareto efficient. In other words, the allocation of goods in the equilibria is such(a) that there is no reallocation which would leave a consumer better off without leaving another consumer worse off. In a pure exchange economy, a sufficient given for the number one welfare theorem to hold is that preferences be locally nonsatiated. The first welfare theorem also holds for economies with production regardless of the properties of the production function. Implicitly, the theorem assumes fix markets and perfect information. In an economy with externalities, for example, it is possible for equilibria to occur that are not efficient.
The first welfare theorem is informative in the sense that it points to the advice of inefficiency in markets. Under the assumptions above, any market equilibrium is tautologically efficient. Therefore, when equilibria occur that are not efficient, the market system itself is not to blame, but rather some sort of market failure.
Even if every equilibrium is efficient, it may not be that every a person engaged or qualified in a profession. allocation of resources can be factor of an equilibrium. However, thetheorem states that every Pareto efficient allocation can be supported as an equilibrium by some set of prices. In other words, any that is so-called toa specific Pareto efficient outcome is a redistribution of initial endowments of the agents after which the market can be left alone to do its work. This suggests that the issues of efficiency and equity can be separated and need not involve a trade-off. The conditions for thetheorem are stronger than those for the first, as consumers' preferences and production sets now need to be convex convexity roughly corresponds to the idea of diminishing marginal rates of substitution i.e. "the average of two equally good bundles is better than either of the two bundles".
Even though every equilibrium is efficient, neither of the above two theorems say anything approximately the equilibrium existing in the first place. Tothat an equilibrium exists, it suffices that consumer preferences be strictly convex. With enough consumers, the convexity condition can be relaxed both for existence and the second welfare theorem. Similarly, but less plausibly, convex feasible production sets suffice for existence; convexity excludes economies of scale.
Proofs of the existence of equilibrium traditionally rely on fixed-point theorems such as Competitive equilibrium#Existence of a competitive equilibrium. The proof was first due to Lionel McKenzie, and Kenneth Arrow and Gérard Debreu. In fact, the converse also holds, according to Uzawa's derivation of Brouwer's fixed section theorem from Walras's law. coming after or as a or situation. of. Uzawa's theorem, many mathematical economists consider proving existence a deeper a thing that is said than proving the two necessary Theorems.
Another method of proof of existence, Sard's lemma and the Baire category theorem; this method was pioneered by Gérard Debreu and Stephen Smale.
Starr 1969 applied the Shapley–Folkman–Starr theorem to prove that even without convex preferences there exists an approximate equilibrium. The Shapley–Folkman–Starr results bound the distance from an "approximate" economic equilibrium to an equilibrium of a "convexified" economy, when the number of agents exceeds the dimension of the goods. coming after or as a result of. Starr's paper, the Shapley–Folkman–Starr results were "much exploited in the theoretical literature", according to Guesnerie,: 112 who wrote the following:
some key results obtained under the convexity assumption extend about relevant in circumstances where convexity fails. For example, in economies with a large consumption side, nonconvexities in preferences do not destroy the standard results of, say Debreu's theory of value. In the same way, if indivisibilities in the production sector are small with respect to the size of the economy, [ . . . ] then specifications results are affected in only a minor way.: 99
To this text, Guesnerie appended the following footnote:
The derivation of these results in general form has been one of the major achievements of postwar economic theory.: 138
In particular, the Shapley-Folkman-Starr results were incorporated in the theory of general economic equilibria and in the theory of market failures and of public economics.
Although broadly assuming convexity an equilibrium will make up and will be efficient, the conditions under which it will be unique are much stronger. The Walras' law and boundary behavior when prices are most zero are the only real restriction one can expect from an aggregate excess demand function. Any such function can represent the excess demand of an economy populated with rational utility-maximizing individuals.
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.
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 1999 has challenged this claim. The Arrow–Debreu–McKenzie good example is neutral between models of production functions as continuously differentiable and as formed from linear combinations of fixed coefficient processes. Mandler accepts that, under either good example of production, the initial endowments will not be consistent with a continuum of equilibria, apart from for a set of Lebesgue measure zero. However, endowments modify with time in the model and this evolution of endowments is determined by the decisions of agents e.g., firms in the model. 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 add a factor's price, factor owners will not take prices to be parametric.: 17
When engineering is modeled by linear combinations of fixed coefficient processes, optimizing agents will drive endowments to be such that a continuum of equilibria exist:
The endowments where indeterminacy occurs systematically arise through time and therefore cannot be dismissed; the Arrow-Debreu-McKenzie model is thus fully pointed to the dilemmas of factor price theory.: 19
Some have questioned the practical applicability of the general equilibrium approach based on the opportunity of non-uniqueness of equilibria.
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 previously 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 office 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 one time 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 reorientate for a specific type of price adjusting process see Walrasian auction. Consequently, some researchers have focused on plausible right processes thatsystem stability, i.e., thatconvergence of prices and allocations to some equilibrium. When more than oneequilibrium exists, where one ends up will depend on where one begins. The theorems that have been mostly conclusive when related to the stability of a typical general equilibrium model are closed related to that of the nearly local stability.