ECON
feweco20021216130138.pdf

# Definition 51 an allocation x is a δ equilibrium of

• 45

This preview shows pages 23–25. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

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.
This is the end of the preview. Sign up to access the rest of the document.
• Spring '16
• Equilibrium, Economic equilibrium, General equilibrium theory, Non-standard analysis, Florig

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern