ECON 468: Industrial Organization
Problem Set 3
April 24, 2008
1. Consider linearcity market in which 2 firms are located at the opposite end of the line:
Figure 1: Linear city with two stores
G
1
and
G
2
G
1
= 0
x
G
2
= 1
We know that in this market the demand functions at each store are given by:
D
1
(
p
1
, p
2
) =
p
2

p
1
2
t
+
t
2
D
2
(
p
1
, p
2
) =
p
1

p
2
2
t
+
t
2
Consider the following twoperiod game:
•
In period 1, Firm 1 can reduce its marginal cost by investing in research and development.
•
In period 2, both firms compete in prices (i.e. Bertrand game with product differentita
tion).
The cost function of the two firms is given by:
C
1
(
q
) = (
c

x
)
q
C
2
(
q
) =
cq
where
x
is the result of the investment made in period one. The cost of reducing marginal
cost by
x
is given by:
g
(
x
) =
1
2
x
.
(a) Given an arbibtrary value for
x
, what are the equilibrium profits of both firms in the
second stage of the game?
1
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The two profit functions are given by:
π
1
(
x
) =
(3
t
2
+
c

(
c

x
))
2
18
t
π
2
(
x
) =
(3
t
2
+ (
c

x
)

c
)
2
18
t
(b) Draw the reaction functions of both firms for two values of
x
.
In which direction are
equilibrium prices changing when we increase the amount of investment
x
? What about
profits?
The reaction functions are given by:
R
1
(
p
2
) =
t
2
+
c

x
2
+
p
2
2
R
2
(
p
1
) =
t
2
+
c
2
+
p
1
2
If we evaluate
R
1
(
p
2
)
at
x
0
< x
, the bestresponse function shift down as we go from
x
0
to
x
. Therefore the equilibrium price is decreasing in the level of investment. Similarly,
the profits of firm
1
are increasing in
x
, while firm
2
’s are decreasing in
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 Spring '08
 HOUDE
 Economics, Game Theory, Perfect Competition, Supply And Demand, k2, production level K1, total production K1

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