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Definition 51 an allocation x is a δ equilibrium of

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Definition 5.1 An allocation ¯ x is a δ -equilibrium of an economy E , if there exist ε * IR + and p * Q such that the following conditions hold: ( i ) attainability: ¯ x i ¯ B i ( p, εδ i ) , i N, ( ii ) individual rationality: u i x ) = max x i ¯ B i ( p,εδ i ) u i x | x i ) , i N, ( iii ) market clearing: X i N ¯ x i = X i N w i . The specifics of δ -equilibria are twofold. First, the ratio of individual dividends is assumed to be given a priori and fixed, as in the case of Kajii’s equilib- ria with individual slacks, or, for instance, Mas-Colell’s (1992) equilibria with slack, where the uniform dividend scheme was applied. Second, dividends of all consumers have the same “order of smallness” ε. Therefore, income is re- distributed at most at one infinitesimality level, which may not generally be the case for non-standard dividend equilibria. Interpretation of the components δ i depends on the further specification of the model. For instance, they may represent initial stocks of coupons or paper money or express market shares of individuals. We continue with an example which illustrates that the system of dividends has to be specific for the number of non-standard equilibria to be finite. Let X 1 = X 2 = X 3 = { ( x 1 , x 2 ) : 0 x j 10 , j = 1 , 2 } , Q = { p IR l : k p k ≤ 2 } , w 1 = (2 , 1) , w 2 = w 3 = (2 , 0) , u 1 ( x ) = 5 - ( x 1 - 1) 2 - ( x 2 - 2) 2 , u 2 ( x ) = u 3 ( x ) = x 1 . Allocations ¯ x 1 = (1 , 1) , ¯ x 2 = (2 + λ, 0) , ¯ x 3 = (3 - λ, 0) , 0 λ 1 constitute a continuum of non-standard dividend equilibria for p = ( ε, 1) , ε 0 , ε > 0 , and dividends d = (0 , λε, (1 - λ ) ε ) . This example is robust against sufficiently small perturbations of utility functions. Observe that if variations of initial endowments are considered, then the number of non-standard equilibria is generically finite. Indeed, the survival assumption ( w i 0 , i = 1 , 2 , 3) is satisfied for almost all perturbations of initial endowments, in which case non- standard dividend equilibria coincide with ordinary Walrasian equilibria (see Proposition 2.4). 23
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6 - x 2 x 1 . m ‰… » &% ’$ . ¯ x 1 .w 1 - u 2 = u 3 ¡ ¡ x 2 , ¯ x 3 ) Figure 2: Continuum of non-standard dividend equilibria. Next, we will show that δ -equilibria exist even if the survival condition or any of its analogues is not satisfied. Note that the existence of non-standard dividend equilibria unlike that of hierarchic equilibria (see Florig (2001)) does not re- quire any conditions on consumption sets aside from convexity and compactness. Moreover, the compactness assumption can always be relaxed and substituted for closedness and boundedness from below. Theorem 5.2 Let Q = { p IR l : k p k ≤ 1 } . Assume that the set X i is convex and compact, and the utility function u i is continuous in x and strictly quasi-concave in x i for every i N. Then for each δ IR n ++ a δ -equilibrium exists.
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  • Spring '16
  • Equilibrium, Economic equilibrium, General equilibrium theory, Non-standard analysis, Florig

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