2 next we turn our attention to the characterization

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2 Next, we turn our attention to the characterization of non-standard dividend budget sets. For γ * IR ++ and p * IR l consider the set * B ( p, γ ) = { x * X : px γ } and denote by ¯ B ( p, γ ) its standardization: ¯ B ( p, γ ) = st * B ( p, γ ) . The following auxiliary lemma is useful. It says that small changes in prices and dividends do not alter a standardized dividend budget set. Lemma 3.5 Let X be a closed convex set, 0 X. Suppose that p, p 0 * IR l and that the non-standard numbers γ > 0 , γ 0 > 0 satisfy | p - p 0 | γ 0 and γ γ 0 1 , then ¯ B ( p, γ ) = ¯ B ( p 0 , γ 0 ) . 14
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Proof. Show first that ¯ B ( p, γ ) = ¯ B ( p, γ 0 ) . To this end assume γ 0 > γ and show that the inclusion ¯ B ( p, γ 0 ) ¯ B ( p, γ ) (10) holds. Let x ¯ B ( p, γ 0 ) . Then one can find ˜ x x, ˜ x * X such that p ˜ x γ 0 . Suppose that p ˜ x > γ (otherwise there is nothing to prove), and consider y = (1 - ε x where ε 0 satisfies γ 0 = (1 + ε ) γ. It is clear that y ˜ x x , y * X by convexity, and py = p ˜ x - εp ˜ x γ 0 - εγ = γ. Thus inclusion (10) follows. Next we shall establish ¯ B ( p, γ 0 ) = ¯ B ( p 0 , γ 0 ) . Let p 00 = p - p 0 . Since | p 00 | γ 0 0 , one can find ε 0, ε > 0 such that | p 00 | εγ 0 0 . Then for every near-standard y * X p 0 y - εγ 0 py p 0 y + εγ 0 . Therefore py γ 0 implies p 0 y γ 0 + εγ 0 and ¯ B ( p, γ 0 ) ¯ B ( p 0 , γ 0 + εγ 0 ) = ¯ B ( p 0 , γ 0 ) . Similarly, ¯ B ( p 0 , γ 0 ) ¯ B ( p, γ 0 ) . 2 Using representation (2) for p * IR l , assign to each non-standard γ > 0 its infinitesimality level , that is a number j = j ( p, γ ) ∈ { 1 , . . . , k + 1 } such that j ( p, γ ) = min { m | γ/λ m 6≈ 0 } , if γ/λ k 6≈ 0 , k + 1 otherwise. For j k, denote by μ = μ ( p, γ ) a standard part of the ratio γ/λ j : μ ( p, γ ) = ( γ/λ j ) . Thus μ is an element of IR ++ ∪ { + ∞} . Put μ = + if j = k + 1 . The next statement gives a complete characterization of a non-standard dividend budget set for a polyhedral set X. Proposition 3.6 Let X be polyhedral, p * IR l , γ * IR ++ . Assume that 0 X. Then one of the following alternatives is true: 15
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( i ) ¯ B ( p, γ ) = X ( q 1 , . . . , q j - 1 ) and μ = + ; ( ii ) ¯ B ( p, γ ) = { x X ( q 1 , . . . , q j - 1 ) | q j x μ } and μ < + ; ( iii ) ¯ B ( p, γ ) = B m ( p ) for some m < j and there exists y ¯ B ( p, γ ) such that q m y < 0 . Here q = ( q 1 , . . . , q k ) is a hierarchic price representing p, j = j ( p, γ ) is the infinitesimality level of γ. Proof. Consider an ( l +1)-dimensional set X * = X ×{ 1 } , and a “budget” set { x * X * | p * x 0 } , (11) where the price vector p * IR l +1 is given by p * = ( p 0 , - γ ) , if γ/λ j + , p 0 = p - t j λ t q t , ( p 0 , - λ j μ ) , if γ/λ j < + , p 0 = t j λ t q t . By construction | p - p 0 | 0 and λ j μ/γ 1 . Since 0 X, Lemma 3.5 implies that the projection of the standardization of the set defined in (11) onto the first l components of the set X * coincides with ¯ B ( p, γ ) . At the same time, p * = X t<j λ t q * t + λ * j q * j , where q * t = ( q t , 0) if t < j, q * j = (0 , - 1) , if γ/λ j + , ( q j , - μ ) , if γ/λ j < + , and λ * j = γ, if γ/λ j + , λ j , if γ/λ j < + .
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  • Spring '16
  • Equilibrium, Economic equilibrium, General equilibrium theory, Non-standard analysis, Florig

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