Question

0. Let us consider a market where 3 firms I = {1, 2, 3} compete `a la Cournot (quantity-setting competition). The inverse demand function is given by p(Q) = 300 − 5Q, where Q = q1 + q2 + q3. The cost function is homogeneous and it is C1(q) = C2(q) = C3(q) = 30q.

   • Write explicitly the profit functions of each i ∈ I.

   • Derive best reply functions and the Nash equilibrium of the game.

Answer 1

P= 300 - 50 P = 300 - 5121+22+ 23) From = = PEI- 3091 [ 300 - 5(21+92 +93)] 2, - 309, 30001 - 5121+22+23) &, - 302, = 2702, - 59? - 52,92 - 52,23 = 270 - 102 - 582-593 put oli = 0 221 OT 02, 270 - 102,- 522 - 523 = 0 270 = 1021+592 + 523 54 = 22 +22+ 23 54-92-23 = 28, 2= 54 - 22 - 23 W This is best response fu of firma Firma [T2 = = = OIT2 P22 - 3022 (300- 5121 + 92 +23)] 92 - 3022 30092 - 52192 - 542 - 52322 - 3022 - 300 - 521 - 1092 - 593 - 30 022 = 270 - 52, - 1022 - 523 put on so Pa 022 270 59,- 1022 593 0 270-521- 543 = 1022 5u - q - q3 = 222

54 - 2,- qz = 222 q2 = 54-91-23 firmiz's best response ph q2 = 54-91-93 in Similarly firm 3. 53= P23 - 3093 = [ 300 - 51917 92+232] 23 - 3093 = 30093 - 59,93 - 52223 -54²3 - 3083 = 270 q3 52123 - 59223 - 59 2 013 = 270 - 52, 592 - 1083 DE3 put 0113 =0 023. 270 - 521-522 - 1023=0 54 - 21-q2 - 223 =0 54- 2,92 = 223 23 live = 54 - 21-22 2 response tu is . firm 3's best 23 = 54 - {1-22 2 from (i); (i) and ciii)

&, = 54 - 22-93 2 29,= 54-92-93 29, +q2t 93 = 54 92 = 54 - 21-93 2 292 = 54-2,-23 21+2q2 +93 = 54 23= 54-21-92 283= 54-{1-92 21+q2+293 = 54 Now 221+ 22 +23 = 54 2ut 222 +23=54 2,+q2 + 293 = 54 we can solue there equations by any method, but Iam uring cramme sue 2 ila / 54 12 | 54 i 2 23 54 92 JA) = 2 ' 2 1 12 = 2(4-1) -1 (2-1) +147-27 6-1-1 = = 4

TAL = 54 54 2 54 1 2 = 54(4-1) - 1(108 - 54) +1 (54-108) = 162 - 54-54 = 54 A2 = 2 54 1 54 ) = 1 54 2 2.0108 - 54) - 54 (2-1) +1(54 - 54 ) 108 - 54 to = 54 - JA 3) = 2 , 54 2 54 i 54 2(108- 54) - 1(54-54) + 54 (1-2) = 108 -o- 54 = 54 = q, = All = 59 = 13.5 23 = 183! = 5 = 13.5 TAH