Review Questions 1
Yuya Takahashi
7
=
26
=
2018
1. Consider an industry where there is only one °rm. Demand is given by
Q
= 10
°
p:
Assume the °rm±s
cost function is
c
(
Q
) = 2
Q:
Find a competitive equilibrium.
Since we are °nding a competitive equilibrium, the °rm maximizes its pro°t taking the price as given.
Thus,
p
=
MC
gives
p
= 2
:
From the demand function, we know
Q
= 8
.
2. Consider a market of homogenous products with two active °rms (the number of °rms is °xed). Demand
is given by
Q
(
p
) = 40
°
p:
Two °rms are identical and the total cost function is given by
TC
(
q
) = 10
q:
(a) Find a competitive equilibrium (i.e., °nd
p
°
and
Q
°
).
p
=
MC
implies
p
°
= 10
:
Using the demand function, we have
Q
°
= 30
:
(b) Find a Cournot-Nash equilibrium (i.e., °nd
p
C
; q
C
1
;
and
q
C
2
).
Firm 1 maximizes
°
1
=
pq
1
°
10
q
1
= (30
°
q
1
°
q
2
)
q
1
:
The °rst order condition is
30
°
q
2
°
2
q
1
= 0
:
In the same way, we also obtain the best response function of °rm 2:
30
°
q
1
°
2
q
2
= 0
:
Solving
this system gives
q
C
1
= 10
; q
C
2
= 10
;
and
p
C
= 40
°
q
C
1
°
q
C
2
= 20
:
(c) Find an equilibrium for a sequential game where °rm 1 moves °rst and °rm 2 moves second. (i.e.,
°nd
p
S
; q
S
1
;
and
q
S
2
).
We solve this problem backward. Firm 2±s best response function for any given
q
1
is
q
2
(
q
1
) =
30
±
q
1
2
:
This is the same best response function for °rm 2 as in question (b). Then, °rm 1 maximizes
°
1
=
pq
1
°
10
q
1
=
(30
°
q
1
°
q
2
(
q
1
))
q
1
=
°
30
°
q
1
2
±
q
1
:
The °rst order condition implies
q
S
1
= 15
:
Using
q
2
(
q
1
)
;
we have
q
S
2
=
15
2
:
Finally,
p
S
=
35
2
:
(d) Find a Bertrand-Nash equilibrium (i.e., °nd
p
B
; q
B
1
;
and
q
B
2
).

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- Spring '08
- Staff
- Game Theory, Supply And Demand, best response