Ex. 1 A market demand curve: P =120-Q Cost function: TCi(Q)=30Q for each firm For Cournot competition a. Response function? b. Optimum quantity and price profits? Assume output of firm 1 and firm 2 is Q1 and Q2 respectively → Qt=Q1+Q2 π1 =TR1-TC1 =P1.Q1 - TC1 =(120-Q1-Q2).Q1 - 30Q1 =120Q1 - Q1^2 - Q1Q2 - 30Q1 = -Q1^2 - Q1Q2 +90Q1 Π2 =TR2-TC2 =P2.Q2 - TC2 =(120-Q1-Q2).Q2 - 30Q2 =120Q2 - Q1Q2 - Q2^2 - 30Q2 = -Q2^2 - Q1Q2 + 90Q2 Firm 1 maximizes profit when: π1’=0 ⇔ -2Q1 - Q2 + 90 =0 (1) → Reaction function for firm 1: Q1= -0,5Q2+45 Firm 2 maximizes profit when: π2’=0 ⇔ -2Q2 - Q1 + 90 =0 (2) → Reaction function for firm 2: Q2= -0,5Q1+45 From (1), (2), the Cournot equilibrium is achieved: 2Q1+Q2=90 Q1+2Q2=90 → Q1=Q2=30 Market price: P=120-Q1-Q2=120-30-30=60 → π1=π2=900 Ex. 2 A market demand curve: P =120-Q Cost function: TC(Q)=30Q2 for each firm For Cournot competition a. Response function? b. Optimum quantity and price → profits? Assume output of firm 1 and firm 2 is Q1 and Q2 respectively → Qt=Q1+Q2 π1 =TR1-TC1 =P1.Q1 - TC1 =(120-Q1-Q2)Q1 - 30Q1^2 =120Q1 - Q1^2 - Q1Q2 - 30Q1^2 = -31Q1^2 - Q1Q2 + 120Q1 Π2 =TR2-TC2 =P2.Q2 - TC2 =(120-Q1-Q2).Q2 - 30Q2^2 =120Q2 - Q2^2 - Q1Q2 - 30Q2^2 = -31Q1^2 - Q1Q2 + 120Q1 Firm 1 maximizes profit when: π1’=0 ⇔ -62Q1 - Q2 + 120=0 (1) → Reaction function for firm 1: Q1= -Q2/62 + 60/31 Firm 2 maximizes profit when: π2’=0 ⇔ -62Q2 - Q1 + 120=0 (2) → Reaction function for firm 2: Q2= -Q1/62 + 60/31 From (1), (2), the Cournot equilibrium is achieved: 62Q1+Q2=120 Q1+62Q2=120 → Q1=Q2=40/21=1,9047 → Market price: P=120-Q1-Q2=116,1904 → π1=π2=112,47