Question

Consider an asymmetric Cournot duopoly game, where the two firms have different costs of production. Firm 1 selects quantity q1 and pays the production cost of 2q1 . Firm 2 selects quantity q2 and pays the production cost 4q2 . The market price is given by p = 12 − q1 − q2 . Thus, the payoff functions are u1 (q1,q2) = (12 − q1 − q2 ) q1 − 2q1 and u2 ( q1 , q2 ) = (12 − q1 − q2) q2 − 4q2 .

a.) Calculate the firms’ best-response functions BR1( q2 ) and BR2( q1 )

b.) Find the Nash equilibrium of this game.

Answer 1

a) Pay-off funtion for firm -1,

For maximiing payoff, the partial differential of payoff with respect to q₁ has to be 0.

This is the best response function for Firm-1

Payoff for firm-2,

For maximiing payoff, the partial differential of payoff with respect to q₂ has to be 0.

This is the best response function for Firm-2

b) For obtaining the Nash Equlibrium, we need to solve the best response functions for both firms simultaneously

That is the Nash equlibrium is (q₁,q₂) = (4,2)