Question

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Exercise 59.1 (Cournot's duopoly game with linear inverse demand and a quadratic cost function) Find the Nash equilibrium of Cournot's game when there are two firms, the inverse demand function is given by P(Q) = a-Q if Q<a, = o if Q> a, and the cost function of each firm i is C;(q;) = qť.

Answer 1

920 T = (2-9, 9,)9,-9, 2 Anso- Nash equilibrium of cousnot's duo duopoly Given inverse demand function P(Q)=x-a if o it god and Given cost function of each firm ci (ai) = 9; 2 say that quantity of first firm be q, and quantity of second firm be us, Then cost function of format 9 (9,)=9, cost function of firm-2 SCQ) = 9 Then Profit functions are [q=9, +9) Cgenerally profit = TR-7) Lets 2 2 T2 = (2-9, 2) q = 2 2 ²

The Nash equilibrium by folving can be found ati Jai =2-297 & ₂ 29,=0 and az du 2-9, 29-29=0 2 from ① and @ 2=49, +9 = 49,+q, » 44, tq = 4q+q, 39 = 3% . q if we alsume q = q = q then total aty Q=q, t = 29. Price P(q) = x-29 if aq sa if qq>a These are solutions of wash-carnout duopoly equilibrium.