Unformatted text preview: ure 2 Examples Examples: Cournot Competition
We now provide an explicit characterization of the Nash equilibrium
of Cournot for a speciﬁc demand function.
Suppose that both ﬁrms have marginal cost c and the inverse demand
function is given by P (Q ) = α − βQ , where Q = q1 + q2 , where
α > c . Then player i will maximize:
max π i (q1 , q2 ) = [P (Q ) − c ] qi
qi ≥ 0 = [α − β (q1 + q2 ) − c ] qi .
To ﬁnd the best response of ﬁrm i we just maximize this with respect
to qi , which gives ﬁrst-order condition [α − c − β (q1 + q2 )] − βqi = 0.
Therefore, the best response correspondence (function) of ﬁrm i can
be written as
α − c − β q−i
32 Game Theory: Lecture 2 Examples Cournot Competition (continued)
Now combining the two best response functions, we ﬁnd the unique
Cournot equilibrium as
q1 = q2 = α−c
3β Total quantity is 2 (α − c ) /3 β, and thus the equilibrium price is
P∗ = α + 2c
3 It can be veriﬁed that if the two ﬁrms colluded, then they could
increase joint proﬁts by reducing total quantity to (α − c ) /2 β and
increasing price to (α + c ) /2. 33 Game Theory: Lecture 2 Examples Examples: Bertrand Competition
An alternative to the Courn...
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- Spring '10
- Game Theory