4. Consider a market with inverse demand function
p
= 14
°
Q
. Firms have constant marginal cost 2 and
°xed cost 2. Firms compete by simultaneously choosing quantities.
(a) Suppose there are
n
°rms in this market. Derive the Nash equilibrium prices, quantities and pro°ts.
Firm
i
maximizes
°
i
=
p
(
Q
)
q
i
°
2
q
i
°
2
:
The FOC is
²
14
°
X
j
6
=
i
q
j
³
°
2
q
i
°
2 = 0
:
We are looking for a symmetric equilibrium, and so this implies
q
j
=
12
n
+ 1
for all
j:
2

We also have
p
= 14
°
Q
= 14
°
12
n
n
+ 1
=
2
n
+ 14
n
+ 1
and
°
i
=
2
n
+ 14
(
n
+ 1)
12
(
n
+ 1)
°
2
12
(
n
+ 1)
°
2
=
°
12
n
+ 1
±
2
°
2
:
(b) Treating
n
as a continuous variable (i.e. ignoring the integer constraint), solve for the equilibrium
number of °rms when there is free entry.
Entry occurs until the pro°t equals to zero:
°
i
=
°
12
n
+ 1
±
2
°
2 = 0
,
n
fe
=
6
p
2
°
1
²
7
:
49
:
(c) What is the e¢ cient number of °rms? How does this compare to the equilibrium number of °rms?
Provide some intuition for this discrepancy.
For given
n
, the consumer surplus is
CS
(
n
)
=
1
2
(14
°
p
)
Q
=
1
2
°
12
n
n
+ 1
±
2
:
The total surplus is
TS
(
n
)
=
CS
(
n
) +
n
X
i
=1
°
i
=
1
2
°
12
n
n
+ 1
±
2
+
°
12
n
+ 1
±
2
n
°
2
n:
Di/erentiate this with respect to
n
to get
°
12
n
n
+ 1
±
12 (
n
+ 1)
°
12
n
(
n
+ 1)
2
+
144 (
n
+ 1)
2
°
288
n
(
n
+ 1)
(
n
+ 1)
4
°
2 = 0
,
n
°
=
72
1
3
°
1
²
3
:
16
:
When making an entry decision, °rms do not take into account the e/ect of entry on other °rms±
pro°t.
Without °xed costs, entry leads to a higher total surplus, since an increase in consumer
surplus associated with °rm±s entry is always larger than the decrease in °rm±s pro°t.
However,
with the °xed cost, the increase in consumer surplus gets smaller as n gets bigger, but the °xed cost
incurred is the same for any additional entry.

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- Game Theory, Supply And Demand, best response