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# 19 theorem 41 suppose that x i is a polyhedral set

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Theorem 4.1 Suppose that X i is a polyhedral set for every i N. Then there exists a set ¯ Θ Θ such that (i) for each non-standard dividend equilibrium ¯ x there exists q ¯ Θ such that q is an equilibrium hierarchic price for ¯ x (or, which is equivalent, such that q represents non-standard equilibrium prices p corresponding to the equilibrium ¯ x ). (ii) ¯ Θ is a union of manifolds of dimension less than or equal to l - 1 . Proof. We continue to use the convention w i = 0 for every i N. Suppose that ¯ x is a non-standard dividend equilibrium and p * IR l and d * IR n + are corresponding prices and dividends. It is possible to classify consumers according to the structure of their budget sets ¯ B ( p, d i ) . Proposition 3.6 implies that each consumer i faces (for some m = m ( i ) ∈ { 1 , . . . , k + 1 } ) m - 1 budget constraints in the form of the equalities: q t x = 0 , t ∈ { 1 , . . . , m - 1 } , and (possibly) one budget constraint in the form of the inequality: q m x 0 or q m x μ j , where μ i = ( d i j ( p,d i ) ). Let N m N be the set of all agents for whom the last budget restriction involves the vector q m . For all such agents we have ¯ B i ( p - X t>m λ t q t , d i ) = ¯ B i ( p, d i ) . Put i N 1 if agent i faces no restrictions at all, that is if ¯ B i ( p, d i ) = X i . Thus, given non-standard prices p and dividends d i we partition the set of agents N into k subsets N 1 , . . . , N k . Consider an arbitrary agent i ; i N m for some m. His budget set is a subset of the set X i ( q 1 , . . . , q m - 1 ) = { x X i | q 1 x = 0 , . . . , q m - 1 x = 0 } . Each vector q t , t = 1 , . . . , m - 1 , supports X i ( q 1 , . . . , q t - 1 ) , (though it does not have to support X i ), which implies that the sets X i X i ( q 1 ) . . . X i ( q 1 , . . . , q m - 1 ) form a finite sequence of faces of X i contained in each other. Denote the face X i ( q 1 , . . . , q t - 1 ) by F t i ( p ) (note that the superscript t specifies that there are t - 1 equalities). Let us construct a set Θ p which contains a hierarchic price q p = ( q 1 , . . . , q k ) representing p. It will be done in k steps, and Θ p will finally be obtained as a product of k spheres in some linear space. 20
Consider the set N k . Without loss of generality it is not empty (otherwise one can throw away the last component of q p and consider a new equilibrium price p 0 = p - λ k q k ). Let L k be the linear hull of faces F k i , i N k , L k = span ˆ [ i N k F k i ( p ) ! . It is clear that the vector q k must belong to this subspace (if necessary, q k can be replaced by its projection on L k ). Take the unit sphere S k in L k as the last component of Θ p , S k = { x L k | k x k = 1 } . Secondly, consider F k - 1 i ( p ) — superfaces of F k i ( p ) for i N k — that is the sets { x X i | q t x = 0 , t k - 2 } , and faces F k - 1 i ( p ) for i N k - 1 . Taking a linear hull of the union of the sets F k - 1 i ( p ) for all i from N k and N k - 1 , we obtain a linear space M k - 1 that contains L k . Denote by L k - 1 the orthogonal complement to L k in the space M k - 1 , L k - 1 = M k - 1 ( L k ) , and take the unit sphere in L k - 1 as the next component of Θ p , S k - 1 = { x L k - 1 | k x k = 1 } .

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

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