Suggested Solutions to Problem Set 4
Ying Chen
ECN 416, Game Theory
1. Bertrand Model revisited
Suppose two
f
rms sell identical products and engage in price competition. The market demand
function is
= 100
−
and the consumers will buy from the
f
rm that o
f
ers a lower price. Unlike
what we assumed in class, let’s make the assumption here that if both
f
rmssetthesamepr
ice
,then
all consumers buy from
f
rm
1
. Moreover, assume that
f
rm
1
’s marginal cost of production is
10
whereas
f
rm
2
’s marginal cost of production is
15
. Neither
f
rm has
f
xed cost and both are pro
f
t
maximizers. Suppose the
f
rms choose prices simultaneously and they can charge any positive price.
(a) Write down the payo
f
functions of each
f
rm.
Firm
1
’s payo
f
function is
1
(
1
2
)=
½
(100
−
1
)(
1
−
10)
if
1
≤
2
0
if
1
2
Firm
2
’s payo
f
function is
2
(
1
2
)=
½
(100
−
2
)(
2
−
15)
if
1
2
0
if
1
≤
2
(b) Find pure strategy Nash Equilibrium of this game.
Pure strategy NE:
(
∗
1
∗
2
)
where
10
≤
∗
1
=
∗
2
≤
15
.
First, we can show that if a strategy pro
f
le
(
1
2
)
satis
f
es
1
6
=
2
,th
eni
ti
sn
o
taNE
.
(argument similar to what we made in class for the symmetric Bertrand model.)
Then, we can show that if
∗
1
=
∗
2
and the prices are above the marginal cost of
f
rm
1
and below
the marginal cost of
f
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 Spring '10
 Y.Chen
 Game Theory, player, pure strategy, mixed strategy

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