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answseven - Economics 206 Spring 2007(Prof G Loury Solution...

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Economics 206 Spring 2007 (Prof. G. Loury) Solution of Racing Problem in Assignment 7 [Racing Game] Let V be the value of the prize, and let C ( x ) be the cost to a player of taking a step toward the fi nish line of size x . A player gets to move every second period, so if δ is the per-period discount factor, then β = δ 2 is the discount factor that applies to any value which a player expects to receive on his next turn. Consider the single-player problem of optimally approaching the fi nish line from distance X : φ ( X ) = Max ( x n ,N ) { β N V N X n =0 β n C ( x n ) | N X n =0 x n X } , where N 0 is an integer, and n { 0 , 1 , ..., N } The expression above embodies a player’s choice of how many steps to take to reach the fi nish line ( N + 1 ), and of how big to make each step ( x n ). The value function φ ( X ) gives the net return to either player of having a “free run” to the fi nish line without competition from the other player. Now, consider the following recursion: C ( X 0 ) = V, and for k 0 : C ( X k +1 X k ) = βφ ( X k ) .
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