# Game Theory and Oligopoly - DOC

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```					                               Game Theory and Oligopoly

Nash equilibrium: A best response to a best response
There may be another option with higher payoffs for both players, but it is not the
equilibrium

To Solve for a Nash Equilibrium
1       Take one of Firm 2’s actions as given (Price high)
2       Compare the payoffs for Firm 1’s alternatives (Enter or stay out)
(Read DOWN the column—compare 40 to 0)
3       Then take the other of Firm 2’s actions (Price low)
4       Compare the payoffs for Firm 1’s alternatives (Compare –30 to 0)
5       Reverse—Take Firm 1 actions as given and compare Firm 2’s payoffs
6       Two circles in a box is a NASH equilibrium—here Price High and Enter

Firm 2
Price High               Price low
Firm 1                 Enter                                  50                    -20
40                      -30
Stay out                               100                   40
0                       0

Types of firm interaction:
1.      Perfect competition
Many firms
Firm profits:  = P*q – C(q)
Solution is where P = MC as long as P>AVC in short run
In long run P = MC = AC and profits are zero

2.       Monopoly
One firm
Firm profits:  = P(Q)*Q – C(Q)
Solution is where MR = MC

3.       Cartel—cooperative oligopoly
Multiple firms split the monopolist’s profits
Firm 1’s profits:  = P(q1+q2)*q1 – C(q1)

4.       Cournot competitors
Two or more firms compete on basis of quantity
Firm Profits:  = P(q1+q2)*q1 – C(q1)
Profits are between Cartel and perfect competition

5.       Bertrand competitors
Two or more firms compete on the basis of price
Solution is competitive outcome
Solving for a Cournot Equilibrium

Steps:

1.    Express the demand function P(Q) as a function of q1+q2

2.    Write out profits for firm 1 : 1 = P(q1+q2)*q1 – C(q1)
The only variables in the profit function should be q1 and q2

3.    Maximize profits for firm 1
d/dq1 = 0

4.    Solve for q1 in terms of q2 This is firm 1’s REACTION FUNCTION

5.    Repeat steps 2-4 for firm 2 to get firm 2’s reaction function

6.    Plug firm 1’s reaction function into firm 2’s reaction function and solve for q2

7.    Plug q2 into firm 1’s reaction function to find q1

8.    Use both quantities to find the price

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