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Economics 703
Advanced Microeconomics
Prof. Peter Cramton
Problem Set 3:
Suggested Answers
1.
Rubinstein's bargaining solution is (1 
δ
2
)/(1 
δ
1
δ
2
) where
d
i
i
=

e
t
∆
, t>0 is the time between offers
and
∆
i
is player i's interest rate.
We are interested in
lim
lim
(
)
,
(
)
(
)
t
t
t
t
t
t
e
e
e
e
B


+
B


+


=
+
=
+
0
0
2
1
2
2
1
2
1
1
2
1
2
2
1
2
∆
∆
∆
∆
∆
∆
∆
∆
∆
∆
∆
∆
where the first inequality follows from l'Hôpital's rule.
2.
Since
η
*
> 0, d(q
*
r)/da > 0, so the firms are farther from r during a boom (higher a).
The intuition is
that a high demand intercept means that prices are high, so the oneshot payoff from deviating from q
*
(by
dumping q>>q
*
on the market) is high.
Therefore, stern punishments are required to keep firms in line.
Alternatively, the cartel can relax the quantity restriction somewhat, as happens here.
3.
(a) There are two purestrategy equilibria:
player 1 always raises, so player 2 quits immediately, yielding a
payoff of ($9, 0); and player 2 always raises, so player 1 quits immediately, yielding a payoff of (0, $10).
(b) Past bids should be thought of as sunk costs, since they are paid regardless of whether you win or lose.
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 Fall '09
 ProfessorPeterCramton
 Microeconomics

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