2/5/2009
1
A Note About Equilibria
Some games –
Get feel for what an equilibrium might be
Show that if everyone does it, nobody has an
incentive to change
Show everything else isn’t equilibrium: for
each other possible strategy profile, somebody
has an incentive to change
Others games – when you’re clueless
Calculate best responses
Equilibria occur where best responses cross
Imperfect Competition: Bertrand
Two firms set prices
in a market with
complete information, especially of prices
Consumers
only
buy from the cheaper firm
If two firms set the same price, each firm gets
2
,
1
p
p
If two firms set the same price, each firm gets
half the market demand at that price
Again market demand is given by
Each firm’s cost is given by
Firm 1’s profit:
P
P
D
Q
−
=
=
302
)
(
q
q
C
2
)
(
=
1
2
1
1
1
q
q
p
−
×
=
Π
Imperfect Competition: Bertrand
Firm 1’s demand looks like…
1
2
1
p
p
<
1
302
1
p
q
−
=
if
Recall the monopolists price:
0
1
=
q
2
1
p
p
>
2
1
p
p
=
)
1
302
(
2
1
p
q
−
=
if
if
152
=
M
P
2
p
302
M
P
Imperfect Competition: Bertrand
1
p
2
2
302
M
P
Imperfect Competition: Bertrand
Finding equilibrium through systematic
checking:
0
2
Possible locations for p2
p2 < 2 = MC
p2 > 2 = MC
p2 = 2 = MC
Check for equilibrium in each range
Imperfect Competition: Bertrand
p2 < 2 = MC
p1 < p2
p1 = p2
p1 > p2
Firm 1 wants to raise price
Firm 1 wants to raise price
Firm 2 wants to raise price
p2 > 2 = MC
p1
≤
2 < p2
2 < p1
≤
p2
p1
≥
p2
Firm 1 wants to raise price
Firm 2 wants to lower price
Firm 1 wants to lower price

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