feweco20021216130138.pdf

# This is the meaning of fuzzy rejection denote the set

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This is the meaning of fuzzy rejection. Denote the set of non-standard dividend equilibria of an economy E by W ns ( E ) . Theorem 7.1 Assume that all consumption sets X i are polyhedral, and that the sets P i ( x i ) are open, convex and do not contain x i for all x i X i , i N. Then the fuzzy rejective core of an economy E coincides with the set of non-standard dividend equilibria: C fr ( E ) = W ns ( E ) . Proof. First, we show that C fr ( E ) W ns ( E ) . Let ¯ x = (¯ x i ) i N C fr ( E ) and consider the sets G i x i ) = { y i - ¯ x i | y i ∈ P i x i ) } , i N, 4 Strictly speaking, we need ¯ x i to be individually rational and preference order transitive for a trade to be beneficial for an agent i who kept his initial endowments. 40

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and G i ( w i ) = { y i - w i | y i ∈ P i x i ) } , i N. Denote the convex hull of the union of these sets by G G = conv [ i N ( G i x i ) ∪ G i ( w i )) and show that G does not contain zero. Suppose it does. Then, by convexity of G i x i ) and G i ( w i ) , there exist t = ( t i ) i N , s = ( s i ) i N , y, z X such that X i N t i + X i N s i = 1 , t i , s i 0 , i N, X i N t i y i + X i N s i z i = X i N t i ¯ x i + X i N s i w i , (33) where y i ∈ P i x i ) if i supp t, and z i ∈ P i x i ) if i supp s. This implies that ¯ x is rejected by a fuzzy coalition t + s, a contradiction. Therefore, a zero point does not belong to the convex set G. By the non-standard separating hyperplane theorem (see Konovalov (2001), Theorem 2.9.3), there exists p * IR l such that for every y i ∈ P i x i ) , i N, the following conditions hold simultaneously py i > pw i , (34) and py i > p ¯ x i . (35) Define the components of the vector d * IR n + by d i = max { 0 , p ¯ x i - pw i } , i N. (36) Then p ¯ x i pw i + d i , i N, which implies attainability of ¯ x. To prove the validity of the required inclusion we have to show that ¯ x satisfies the equilibrium property of individual rationality: ¯ B i ( p, d i ) ∩ P i x i ) = . (37) What we do have so far is { x * X i : px pw i + d i } ∩ P i x i ) = , i N. But then (37) is a consequence of Proposition 3 . 9 and openness of the sets P i x i ) , i N. To prove the converse inclusion W ns ( E ) C fr ( E ) , let ¯ x W ns ( E ) and assume that p * IR l and d * IR n + are corresponding non-standard equilibrium prices and dividends. Suppose that there exists a fuzzy coalition ξ = ( ξ i ) i N that blocks ¯ x. Then, there exist y i ∈ P i x i ) , t i , s i IR + , i supp ξ such that X i N t i y i + X i N s i y i = X i N t i ¯ x i + X i N s i w i , (38) 41
where t i + s i = ξ i . By individual rationality, y i 6∈ ¯ B i ( p, d i ) , i supp ξ, (39) which implies that py i > pw i , i supp ξ. Fix i supp ξ and consider a hierarchic representation ( q 1 , . . . , q k ) of prices p. For every y i find a number h = h ( i ) ∈ { 1 , . . . , k } such that q h y i > q h w i , q r y i = q r w i , r < h.

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• Spring '16
• Equilibrium, Economic equilibrium, General equilibrium theory, Non-standard analysis, Florig

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