一般均衡理论

福利经济学第一定理强调市场均衡是帕累托有效的。在一纯交换经济中，福利经济学第一定理成立的一个充分条件是偏好的局部非满足性。生产经济中福利经济学第一定理也成立，无论生产函数的特性。该定理暗含了完全市场与完美信息的假定。比如，对于一个存在外部性的经济,产生的均衡可能是无效率的。

第一定理的启发意义在于其指出了市场无效率的来源。在上述假设之下，任何市场均衡均有效率。因此，当无效均衡率产生时并非市场体系须蒙谴责，而是发生了某种市场失灵。

纵然每个均衡都有效，显然也并非每个有效的资源配置均是均衡。然而，福利经济学第二定理指出每个有效的配置都有某个价格集支撑。换言之，达到某个特定结果所需的只是对参与人初始禀赋的再分配，此后的工作就可以交由市场完成。这意味着公平与效率的议题可以分割，其间不存在权衡取舍。第二定理成立的条件强于第一定理，即消费者的偏好需要为凸（凸性可粗略对应为边际替代率递减，或“平均优于极端”的偏好）。进一步，均衡分析第二定理引出了完美均衡分析[3]。

即使每个均衡都是有效率的，上述两个定理均未讨论均衡的存在性。为保证均衡的存在，消费者偏好满足凸性（虽然当消费者足够多时该假设对福利经济学第二定律与存在性定律均可放松）。与之类似但说服力稍欠，凸的可行生产集满足存在性条件，将规模经济排除在外。

传统的均衡存在性证明依赖于不定点定理，如函数的布劳威尔不动点定理（或更为一般的set-valued函数的角谷不动点定理）。实际上，根据Uzawa由瓦尔拉斯法则对布劳威尔不动点定理的推导反之亦成立。沿着Uzawa定理的思路，诸多数理经济学家在探索证明较两个福利经济学定理更为深入的结论。

另外还有证明存在性的全局分析方法，使用的是Sard's lemma与Baire category theorem，该方法的先驱是傑拉德·德布魯与Stephen Smale。

Ross M. Starr （1969）应用沙普利－福克曼－斯塔定理证明甚至无需凸性偏好仍存在一个近似均衡。当参与人数量超过財貨维数时，沙普利－福克曼－斯塔的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：

凸性假设之下得到的一些重要结论在凸性假设失效时仍（近似的）成立。例如，在消费方足够大的经济中，偏好的非凸性并不破坏Debreu的价值理论等标准结论。同样，如果生产领域的不可分割性较之经济规模较小，[ . . . ]，则标准结论所受影响是次要的。[5]^:99

对这段文字，Guesnerie增补了如下脚注：

在一般形式下推导出这些结论是战后经济理论的主要成就之一。[5]^:138

特别的，Shapley-Folkman-Starr结论被整合进一般均衡理论[6][7][8]与理论中的市场失灵[9]与公共经济学部分。[10]

尽管一般地（假定凸性）存在一个有效率的均衡，该均衡唯一性的条件要更强。虽然该问题相当技术性，其直觉在于财富效应的存在（此为一般均衡分析与局部均衡区别最为显著的特征）产生了多重均衡的可能性。一特定財貨价格的变化产生了两种效应。首先，不同財貨的相对吸引力变化；其次，参与人个体的财富分配变化。这两种效应可互相抵消或增强，使得一组以上的价格可能构成均衡。

名为Sonnenschein–Mantel–Debreu theorem的结论指出 宏观（过剩）需求函数只是继承了某些个体需求函数的性质， 并且这些（当价格都接近零的时候，连续性， 零度同质性，瓦尔拉斯定律和边界行为）是唯一真正的限制，它们可以从宏观过量函数中预期：任何这样的函数可以被理性化为这个经济体的过剩需求。 尤其是均衡点的独一性不应该被期望。

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.

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.

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.