Transcript n Bertrand`s model of oligopoly

Game Theory Lecture Jan 18
In Bertrand’s model of oligopoly
A) Each firm chooses its quantity as the best
response to the quantity chosen by the other(s).
B) Each firm chooses its price as the best response
to the price chosen by the other(s).
C) The firms set quantities sequentially. The
second firm’s quantity is the best response to
the first firm’s quantity.
D) The first firm sets a quantity. The second firm
follows by setting a price.
E) The firms jointly set the price that maximizes
industry profits.
Ex 42.2 (a joint project)
Two players, choose effort levels x1 and x2
between 0 and 1.
Cost of effort to player i is xi2
Part a) Total output is 3x1x2. They divide output
equally.
Payoff to Player 1 is:
Find Player 1’s best response
• Maximize
Player 1’s reaction function
The two reaction functions
Reaction Function Graph
Part b
• Total output is 4x1x2 and is split between two
players.
• Cost of effort xi is c(xi)=xi
• Payoff to player 1 is:
Find Player 1’s best response function
• Take derivative: What does it tell us?
Ex 58.2 (Cournot)
• Two firms, linear inverse demand function
P=a-Q=a-q1-q2. Firms have constant marginal
costs, c1 and c2
• Profit function for firm 1 is:
Reaction Functions
Graph reaction functions
Algebra
Ex 59.1 (Cournot 2)
• Linear inverse demand P=a-Q
• Quadratic Cost function: C(qi)=qi2
• What is profit function for firm 1?
Bertrand model
• Each player does best response to other’s
price.
• Constant marginal cost
• Buyers will purchase only from seller with
lowest price. If prices are equal, demands are
split.
• What can be an equilibrium?
Ex 69.1 Bertrand with fixed costs
Mixed Strategies
Matching pennies
Player 2’s best response
• If Player 1 plays heads with probability p>1/2,
what is Player 2’s best response?
• What if p<1/2?
• What if p=1/2
Both players’ best responses
Hide and seek game