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PS3 - Problemset 3 Due Tuesday November 8th before 5pm 1...

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Problemset3 Due Tuesday November 8th before 5pm 1. Consider the infinitely-repeated price-competition game where two firms make simultaneous choices of their monthly prices. Here the stage-game is the standard Bertrand-model of price competition. Each of two firms chooses a price for the month of p t i 0, and the profits are given by: π i ( p t 1 , p 2 ) = ( p t i - c ) D i ( p t 1 , p t 2 ) where the demand for firms i ’s products in a particular month is given as a function of their own price p i and the competitor’s price p j is: D i ( p i , p j ) = a - p i if p i < p j and p i < a , 1 2 ( a - p i ) if p i = p j < a , 0 if p i > p j or min { p i , p j } > a . Both firms care about the streams of revenue with discount factor 1 1+ r , where r > 0 is the interest rate. So given an infinite price history for both firms { ( p t 1 , p t 2 ) } t =1 , each firm ranks the history via: Π i ( braceleftbig ( p t 1 , p t 2 ) bracerightbig t =1 ) = summationdisplay t =1 parenleftbigg 1 1 + r parenrightbigg t - 1 π i ( p t 1 , p t 2 ) (a) Provide an expression for the average-discounted payoff.
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