A Dynamic Oligopoly Game
Consider the following Stackelberg game with two
fi
rms and two states of
nature. In the high state, the inverse demand function is given by
p
H
= 120
−
q
1
−
q
2
where
p
H
is the price and
q
i
is the quantity supplied by
fi
rm
i
(
i
= 1
,
2
). In the
low state, the inverse demand function is given by
p
L
= 80
−
q
1
−
q
2
The probability of each state is
1
2
.
Assume that production costs are zero.
The timing of moves is the following.
Firm 1 moves
fi
rst and selects its
quantity.
Then
fi
rm 2 chooses its quantity after observing
fi
rm 1’s quantity.
That is,
q
2
can be a function of
q
1
.
(a)
Suppose
fi
rm 2 can observe the true state of nature before choosing
its quantity, but
fi
rm 1 cannot. Calculate the unique subgame perfect Bayesian
Nash equilibrium. Be careful to fully specify
fi
rm 2’s strategy.
(b)
Suppose
fi
rm 1 can observe the true state of nature before choosing
its quantity, but
fi
rm 2 cannot. Calculate one weak perfect Bayesian equilibrium
(WPBE) of this game. Remember to specify beliefs as well as strategies.
(c)
Without doing any calculations, carefully explain the intuition for
why there are many WPBE of the game in part (b).
Answers
(a) Since
fi
rm 1 does not know the state, we have a subgame starting when
fi
rm 2 observes the state and
fi
rm 1’s quantity.
Given
q
1
,
fi
rm 2’s pro
fi
t in the
high state is given by
π
H
2
= (120
−
q
1
−
q
H
2
)
q
H
2
.
(1)
Di
ff
erentiating (1) with respect to
q
H
2
, and setting the expression equal to zero,
we have
fi
rm 2’s optimal strategy for the subgame,
q
H
2
= 60
−
q
1
2
.
(2)
A similar calculation for the low state yields
q
L
2
= 40
−
q
1
2
.
(3)
Thus, (2) and (3) constitute
fi
rm 2’s subgame perfect equilibrium strategy.
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 Spring '08
 PECK
 Oligopoly, Firm, 2 L, H L

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