Quantity Competition Suppose firm 1 takes firm 2s output level choice y2 as given. Then firm 1 sees its profit function as I1(y1iy2)=p(y1+y2y1-c1(y1) Given 2, what output level y1 maximizes firm 1s profit?
Quantity Competition Suppose firm 1 takes firm 2’s output level choice y2 as given. Then firm 1 sees its profit function as Given y2 , what output level y1 maximizes firm 1’s profit? 1 1 2 1 2 1 1 1 (y ;y ) = p(y + y )y − c (y )
Quantity Competition; An Example Suppose that the market inverse demand function is p(yT)=60-OT and that the firms' total cost functions are C1(y1=yi and C2(y2)=15y2+y2
Quantity Competition; An Example Suppose that the market inverse demand function is and that the firms’ total cost functions are p(yT) = 60 − yT c1 y1 y1 2 ( ) = c2 y2 y2 y2 2 and ( ) = 15 +
Quantity Competition; An Example Then, for given y2, firm 1's profit function is I(y1;y2)=(60-y1-y2y1-y1 So, given y2, firm 1s profit-maximizing output level solves =60-2y1-y2-2y1=0. dy1 I. e firm 1's best response to y2 Is y1=k1(y2)=15、1 y2
Quantity Competition; An Example (y ;y ) ( y y )y y . 1 2 1 2 1 1 2 = 60 − − − Then, for given y2 , firm 1’s profit function is So, given y2 , firm 1’s profit-maximizing output level solves y y y y 1 = 60 − 2 1 − 2 − 2 1 = 0. I.e. firm 1’s best response to y2 is y1 R1 y2 15 y2 1 4 = ( ) = −
Quantity Competition; An Example y2 Firm 1s reaction curve 60 y1=R1(y2)=15-y2 15 y1
Quantity Competition; An Example y2 y1 60 15 Firm 1’s “reaction curve” y1 R1 y2 15 y2 1 4 = ( ) = −
Quantity Competition; An Example Similarly, given 1, firm 2's profit function is I(y2:y1)=(60-y1-y2)y2-15y2-y2 So, given y1, firm 2s profit-maximizing output level solves =60-y1-2y2-15-2y2=0 dy2 I. e firm 1's best response to y2 Is y2=R2(y1)= 45-y
Quantity Competition; An Example (y ;y ) ( y y )y y y . 2 1 1 2 2 2 2 2 = 60 − − − 15 − Similarly, given y1 , firm 2’s profit function is So, given y1 , firm 2’s profit-maximizing output level solves y y y y 2 = 60 − 1 − 2 2 − 15 − 2 2 = 0. I.e. firm 1’s best response to y2 is y R y y 2 2 1 45 1 4 = = − ( )