PartIIISec3_5

PartIIISec3_5 - Game Theory Solutions to Exercises: Models...

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Game Theory Solutions to Exercises: Models of Duopoly Jan-Jaap Oosterwijk Fall 2007
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3 Models of Duopoly 3.5.1 (a) Suppose in the Cournot model that the firms have different production costs. Let c 1 and c 2 be the costs of production per unit for firms 1 and 2 respectively, where both c 1 and c 2 are assumed less than a/ 2 . Find the Cournot equilibrium. Solution: The price function stays the same. The only thing that changes is the payoff function for each of the two players: ± u 1 ( q 1 ,q 2 ) = q 1 P ( q 1 + q 2 ) - c 1 q 1 = q 1 ( a - q 1 - q 2 ) + - c 1 q 1 u 2 ( q 1 ,q 2 ) = q 2 P ( q 1 + q 2 ) - c 2 q 2 = q 2 ( a - q 1 - q 2 ) + - c 2 q 2 . Setting the partial derivates to zero, ( ∂q 1 u 1 ( q 1 ,q 2 ) = a - 2 q 1 - q 2 - c 1 = 0 ∂q 2 u 2 ( q 1 ,q 2 ) = a - q 1 - 2 q 2 - c 2 = 0 , gives ± q * 2 = a - 2 q * 1 - c 1 q * 1 = a - 2 q * 2 - c 2 , i.e. q * 1 = a - 2( a - 2 q * 1 - c 1 ) - c 2 = - a + 4 q * 1 + 2 c 1 - c 2 . Therefore, q * 1 = a + c 2 - 2 c 1 3 = ( a - c 1 ) + ( c 2 - c 1 ) 3 , and because of symmetry, q * 2 = a + c 1 - 2 c 2 3 = ( a - c 2 ) + ( c 1 - c 2 ) 3 . The payoff that Player I receives from this Cournot equilibrium is u 1 ( q * 1 ,q * 2 ) = q * 1 ( a - q * 1 - q * 2 ) + - c 1 q
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PartIIISec3_5 - Game Theory Solutions to Exercises: Models...

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