# 一般均衡理论

## 维基百科，自由的百科全书

• 每个消费者都能在给定价格下提供自己所拥有的生产要素，并在各自的预算限制下购买产品来达到自己的消费效用最大化
• 每个企业都会在给定的价格下决定其产量和对生产要素的需求，来达到其利润的极大化
• 每个市场（产品市场和要素市场）都会在这套价格体系下达到总供给与总需求的相等（均衡）

## 一般均衡的属性与特征

### 存在性

#### 大型经济体中的非凸性

Ross M. Starr （1969应用Shapley–Folkman–Starr theorem证明甚至无需凸性偏好仍存在一个近似均衡。当参与人数量超过商品维数时，Shapley–Folkman–Starr结论拉近了一个近似经济均衡与凸性经济均衡的距离。[4][5]:112。他还写道：

### 唯一性

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英语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

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英语comparative statics can be applied as long as the shocks to the system are not too large. As stated above, in a regular economy英语regular economy equilibria will be finite, hence locally unique. One reassuring result, due to Debreu, is that "most" economies are regular.

Recent work by Michael Mandler (1999) has challenged this claim. The Arrow-Debreu-McKenzie model 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 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.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 increase a factor's price, factor owners will not take prices to be parametric." (Mandler 1999, p. 17)

When technology 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 subject to the dilemmas of factor price theory." (Mandler 1999, p. 19)

Critics of the general equilibrium approach have questioned its practical applicability based on the possibility of non-uniqueness of equilibria. Supporters have pointed out that this aspect is in fact a reflection of the complexity of the real world and hence an attractive realistic feature of the model.

### Stability

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 Tatonnement). Consequently some researchers have focused on plausible adjustment processes that guarantee system stability, i.e., that guarantee convergence of prices and allocations to some equilibrium. When more than one stable equilibrium exists, where one ends up will depend on where one begins.

## 参考资料

1. ^ Walras, Elements of Pure Economics (trans Jaffe), Irwin, 1954
2. ^ Debreu, 1959
3. ^ Enrico Gallo Modena, 2013
4. ^ Starr, Ross M. Quasi-equilibria in markets with non-convex preferences. Econometrica. 1969, 37 (1): 25–38. JSTOR 1909201. doi:10.2307/1909201..称，Starr的论文之后，Shapley–Folkman–Starr结论“在理论文献中广为引用”
5. Guesnerie, Roger. First-best allocation of resources with nonconvexities in production. Bernard Cornet and Henry Tulkens (编). Contributions to Operations Research and Economics: The twentieth anniversary of CORE（Papers from the symposium held in Louvain-la-Neuve, January 1987）. Cambridge, MA: MIT Press. 1989: 99–143. ISBN 0-262-03149-3. MR 1104662.
6. ^ See pages 392-399 for the Shapley-Folkman-Starr results and see page 188 for applications in Arrow & Hahn: Arrow, Kenneth J.; Hahn, Frank H. Appendix B: Convex and related sets. General Competitive Analysis. Mathematical economics texts [Advanced textbooks in economics]. San Francisco, CA: Holden-Day, Inc. [North-Holland]. 1971: 375–401. ISBN 0-444-85497-5. MR 439057.
7. ^ Pages 52-55 with applications on pages 145-146, 152-153, and 274-275 in Mas-Colell, Andreu. 1.L Averages of sets. The Theory of General Economic Equilibrium: A Differentiable Approach. Econometric Society Monographs. Cambridge UP. 1985. ISBN 0-521-26514-2. MR 1113262.
8. ^ Hildenbrand, Werner. Core and Equilibria of a Large Economy. Princeton Studies in Mathematical Economics. Princeton, N.J.: Princeton University Press. 1974: viii+251. ISBN 978-0-691-04189-6. MR 389160.
9. ^ See section 7.2 Convexification by numbers in Salanié: Salanié, Bernard. 7 Nonconvexities. Microeconomics of market failures English translation of the (1998) French Microéconomie: Les défaillances du marché (Economica, Paris). Cambridge, MA: MIT Press. 2000: 107–125. ISBN 0-262-19443-0.
10. ^ An "informal" presentation appears in pages 63-65 of Laffont: Laffont, Jean-Jacques. 3 Nonconvexities. Fundamentals of Public Economics. MIT. 1988. ISBN 0-585-13445-6.