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Problem Set 4 - Solutions

# Problem Set 4 - Solutions - Suggested Solutions to Problem...

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Suggested Solutions to Problem Set 4 Ying Chen ECN 416, Game Theory 1. Bertrand Model revisited Suppose two f rms sell identical products and engage in price competition. The market demand function is = 100 and the consumers will buy from the f rm that o f ers a lower price. Unlike what we assumed in class, let’s make the assumption here that if both f rmssetthesamepr ice ,then all consumers buy from f rm 1 . Moreover, assume that f rm 1 ’s marginal cost of production is 10 whereas f rm 2 ’s marginal cost of production is 15 . Neither f rm has f xed cost and both are pro f t maximizers. Suppose the f rms choose prices simultaneously and they can charge any positive price. (a) Write down the payo f functions of each f rm. Firm 1 ’s payo f function is 1 ( 1  2 )= ½ (100 1 )( 1 10) if 1 2 0 if 1  2 Firm 2 ’s payo f function is 2 ( 1  2 )= ½ (100 2 )( 2 15) if 1  2 0 if 1 2 (b) Find pure strategy Nash Equilibrium of this game. Pure strategy NE: ( 1  2 ) where 10 1 = 2 15 . First, we can show that if a strategy pro f le ( 1  2 ) satis f es 1 6 = 2 ,th eni ti sn o taNE . (argument similar to what we made in class for the symmetric Bertrand model.) Then, we can show that if 1 = 2 and the prices are above the marginal cost of f rm 1 and below the marginal cost of f

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Problem Set 4 - Solutions - Suggested Solutions to Problem...

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