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# 3 total quantity is 2 c 3 and thus the equilibrium

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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 qi = . 2β 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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