Quantity Competition; An Example y2 Firm 1s reaction curve 60 y=R1(y2)=15-1 Firm 2's"reaction curve/ 53. y2=R2(y1) 45-y1 Cournot-Nash equilibrium (,y)=(3 13 48y1
Quantity Competition; An Example y2 y1 Firm 2’s “reaction curve” 48 60 Firm 1’s “reaction curve” y1 R1 y2 15 y2 1 4 = ( ) = − . 8 13 Cournot-Nash equilibrium (y1 y2) (13 8) * * , = , . y R y y 2 2 1 45 1 4 = = − ( )
Quantity Competition Generally, given firm 2s chosen output level y2, firm 1s profit function is I1(y;y2)=p(y1+y2y1-c(y1) and the profit-maximizing value of y, solves p(y1+y2) O,=p(y1+y2)+yi ay1 c1(y1)=0 The solution, y,=R,y2), is firm 1s Cournot- Nash reaction to y2
Quantity Competition 1 1 2 1 2 1 1 1 (y ;y ) = p(y + y )y − c (y ) 1 1 1 2 1 1 2 1 1 1 0 y p y y y p y y y = + + c y + − ( ) = ( ) ( ) . Generally, given firm 2’s chosen output level y2 , firm 1’s profit function is and the profit-maximizing value of y1 solves The solution, y1 = R1 (y2 ), is firm 1’s CournotNash reaction to y2
Quantity Competition Similarly, given firm 1's chosen output level y1, firm 2s profit function is I2(y2:y1)=p(y1+y2)y2-c2(y2) and the profit-maximizing value of y2 solves 0Ⅱ2=p(y+y2)+y2 ap(yi+y2) c2(y2)=0 ay2 ay2 The solution, y2=R2yi), is firm 2's Cournot- Nash reaction to y1
Quantity Competition 2 2 1 1 2 2 2 2 (y ;y ) = p(y + y )y − c (y ) 2 2 1 2 2 1 2 2 2 2 0 y p y y y p y y y = + + c y + − ( ) = ( ) ( ) . Similarly, given firm 1’s chosen output level y1 , firm 2’s profit function is and the profit-maximizing value of y2 solves The solution, y2 = R2 (y1 ), is firm 2’s CournotNash reaction to y1
Quantity Competition Firm1s“ reaction curve”y1=R1(y2) Firm 1's"reaction curve"y2=R2(y1) Cournot-Nash equilibrium y2? y1*= R1y2 *)and y2 " s r21*) y1 y1
Quantity Competition y2 y1 Firm 1’s “reaction curve” Firm 1’s “reaction curve” y1 = R1 y2 ( ). Cournot-Nash equilibrium y1 * = R1 (y2 *) and y2 * = R2 (y1 *) y2 * y2 = R2 y1 ( ). y1 *
Iso-Profit curves For firm 1, an iso-profit curve contains all the output pairs y1 y2) giving firm 1 the same profit level Il What do iso-profit curves look like?
Iso-Profit Curves For firm 1, an iso-profit curve contains all the output pairs (y1 ,y2 ) giving firm 1 the same profit level 1 . What do iso-profit curves look like?