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# class15_GEapplication_ - Econ 51 Winter 2011 Class#15 PS#5...

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1 Econ 51, Winter 2011 Class #15 • PS #5 due this Friday, PS #6 posted today. Today: Continue with applications of game theory in oligopolistic markets. Wednesday, February 23, 2011

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2 Game theory: Applications - Oligopoly We are done (for this course) with conceptual ideas of game theory, and we’ll now continue to see how we could apply them. Today we will continue analysis of oligopoly markets. Wednesday, February 23, 2011
3 Quantity competition (Cournot) Game theory Due to Cournot (1840’s). Suppose inverse demand in the market is given by p(Q)=100-Q, where Q is total quantity produced by two firms, i and j. Each firm simultaneously decides about its own quantity q i , and pays production costs c i (q i )=10q i . Therefore, we have that π i (q i ,q j )=(100-q i -q j )q i -10q i . What would be the Nash Equilibrium of this game? We need to calculate the best response functions. Wednesday, February 23, 2011

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4 Quantity competition (cont.) Game theory Firm i solves max q i (100-q i -q j )q i -10q i The first order condition: 100-q i -q j -q i - 10 = 0, so q i = 45 - q j /2------(1) This is the best response function , that is, a function representing i’s best response as a function of j’s strategy q j. Denote it by BR i (q j ), i.e., BR i (q j )= 45 - q j /2 Symmetrically, we can do the same for firm j , to get: q j = 45 - q i /2-------(2) (denote this by BR j (q i )) In a Nash equilibrium both firms should best respond to each other. That is, q i and q j satisfy equations (1) and (2). We can solve this to get q i =q j =30 . So, we have Q=60 and p=40 . Wednesday, February 23, 2011
5 Quantity competition (cont.) Game theory We can graph the best response functions for each firm: The graph shows decreasing best response functions: the more aggressive my opponent is, the less aggressive I am. This is an important feature of quantity competition, often called “strategic substitutes.” q i q j BR i (q j ) BR j (q i ) 90 90 45 45 The Nash equilibrium is the intersection of both firms’ best response functions. Wednesday, February 23, 2011

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6 Quantity competition: two remarks Game theory In Cournot competition, the outcome is not efficient for the firms: The firms could have done better by agreeing on the monopolistic quantity (which would maximize joint profits) and then splitting the revenues. In this example, they could agree on q i =q j =22.5 (half the monopoly quantity each). This will be better for both firms (verify it yourself!) But the problem is that both firms have incentives to produce more given that the other firm produces 22.5. This is similar to the recurrent theme of externality, and also to many examples we have seen such as Prisoner’s dilemma. On
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class15_GEapplication_ - Econ 51 Winter 2011 Class#15 PS#5...

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