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Kaushik Basu
Spring 2008
Econ 367
Game Theoretic Methods
Problem Set 8
1.
Two firms, producing the same good, face the following inverse demand function:
)
(
10
2
1
x
x
p
+

=
where x
i
is the output produced by firm i and p is price in dollars.
Firm 2 can produce this good incurring zero cost.
Firm 1 can also produce this
good with no perunit cost but must incur an initial sunk cost of k dollars in order
to start production.
Firm 1 chooses x
1
in period 1 and then firm 2 (after seeing 1's
choice) chooses x
2
in period 2.
Describe the subgame perfect equilibrium for cases where (a) k = 0, (b) k = 15
and (c) k = 20.
Using backward induction, firm 2 in period 2 maximizes P*Q = [10(x
1
+x
2
)]x
2
which solves to 5/2 – x
1
/2.
Firm 1, now knowing that this is what firm two will choose,
maximizes P*Q = [10(x
1
+(5/2 – x
1
/2))]x
1
.
Solving, x
1
=5, x
2
=5/2.
This implies
that profit for firm 1 is 12.5.
So any sunk cost k greater than 12.5, firm 1 will not
produce.
So in both parts b & c, firm 1 does not produce and firm 2 produces the
monopoly amount, 5.
2.
Two firms face the following inverse demand function:
)
(
12
2
1
x
x
p
+

=
Each firm i's total cost of production is given by:
.
2
,
1
,
3
=
=
i
x
C
i
i
(a)
How much will each firm produce in a Cournot equilibrium?
(b)
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 Spring '08
 BASU

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