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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 ( x ) { u ( x,a ) + v [ f ( x,a )] } C is the set of continuous and bounded realvalued 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. is a nonempty, compactvalued, 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 ( x ) { u ( x,a ) + v [ f ( x,a )] } is nonempty, compactvalued, 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 IVI 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 is monotone ( x x implies ( x ) ( x ) ) . 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 , a ( x ) ( x ) , that is, it is also feasible given x . Then, ( Tv )( x ) = max a ( x ) { u ( x ,a ) + v [ f ( x ,a )] } u ( x ,a ( x )) + v [ f ( x ,a ( x ))] u ( x,a ( x )) + v [ f ( x,a ( x ))] = max a ( x ) { u ( x,a ) + v [ f ( x,a )] } = ( Tv )( x ) 1 The second line is given by a ( x ) ( x ) ; the third line by the facts that u , f , and v are increasing and that ; 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 is convex ( p ( x ) and q ( x ) implies that p + (1 ) q ( x + (1 ) x ) for any [0 , 1] .) . Proof : For any x X, x X, [0 , 1] : Assume v is concave. Then, by de nition, v ( x ) v ( x ) + (1 ) v ( x ) where x = x + (1 ) x ....
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 Fall '07
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