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homework3_sol - ECO 475 HW3 Solution Yu LIU 1 Properties of...

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Unformatted text preview: ECO 475 HW3 Solution Yu LIU September 22, 2010 1 Properties of Value Function T : C → C ( Tv )( x ) = max a ∈ Γ( x ) { u ( x,a ) + βv [ f ( x,a )] } C is the set of continuous and bounded real-valued functions on X . First of all, let us think about a set of su cient conditions that operator T is well de ned. I. u is continuous II. f is continuous III. Γ is a non-empty, compact-valued, continuous correspondence. Then, by Theorem of the Maximum (Theorem 3.6 in SLP), T maps a continuous function to a continuous function. Moreover, A ( x ) = arg max a ∈ Γ( x ) { u ( x,a ) + βv [ f ( x,a )] } is non-empty, compact-valued, and u.h.c. VI. u is bounded Condition VI combined with III implies that T maps a bounded function to a bounded function. Therefore, once conditions I-VI holds, T is well de ned. (a) Under following assumptions, T maps an increasing function into an increasing function. Assumptions : • u is increasing in x • f is increasing in x • Γ is monotone ( x ≤ x implies Γ( x ) ⊆ Γ( x ) ) • β ≥ . Proof : For any x ≥ x : Assume v is increasing. Then, v ( x ) ≥ v ( x ) . Let a ( x ) ∈ A ( x ) . Since x ≤ x , by monotonicity of Γ , a ( x ) ∈ Γ( x ) , that is, it is also feasible given x . Then, ( Tv )( x ) = max a ∈ Γ( x ) { u ( x ,a ) + βv [ f ( x ,a )] } ≥ u ( x ,a ( x )) + βv [ f ( x ,a ( x ))] ≥ u ( x,a ( x )) + βv [ f ( x,a ( x ))] = max a ∈ Γ( x ) { u ( x,a ) + βv [ f ( x,a )] } = ( Tv )( x ) 1 The second line is given by a ( x ) ∈ Γ( x ) ; the third line by the facts that u , f , and v are increasing and that β ≥ ; and the fourth line by a ( x ) ∈ A ( x ) . Hence, Tv is also increasing. T maps an increasing function into an increasing function. (b) Under following assumptions, T maps a strictly increasing function into a strictly increasing function. Assumptions : • the same as (a) • u strictly increasing in x . The inequality in the the third line of part (a) turns to be strict. With this condition, T maps an increasing function into a strictly increasing function, therefore a strictly increasing function into a strictly increasing function. For the next three questions to be well de ned, we need an additional assumption: X × Y and X × ˜ Y are convex, where X × Y is a domain of u and X × ˜ Y is a domain of f . (c) Under following assumptions, T maps a concave function into a concave function. Assumptions : • u is concave • f is linear • Γ is convex ( p ∈ Γ( x ) and q ∈ Γ( x ) implies that αp + (1- α ) q ∈ Γ( αx + (1- α ) x ) for any α ∈ [0 , 1] .) • β ≥ . Proof : For any x ∈ X, x ∈ X, α ∈ [0 , 1] : Assume v is concave. Then, by de nition, v ( x α ) ≥ αv ( x ) + (1- α ) v ( x ) where x α = αx + (1- α ) x ....
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This note was uploaded on 09/06/2011 for the course ECO 475 taught by Professor Hong during the Fall '07 term at Rochester.

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homework3_sol - ECO 475 HW3 Solution Yu LIU 1 Properties of...

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