Question

Exercise: Suppose in a Cournot oligopoly market with n firms, the inverse market demand is p='50 – 0.5Q. Each firm initially produces at a constant MC = 30. a) If a firm invests in R&D that reduces its MC to 20, find the firm's profit from the innovation as a function of number of firms in the market.

Answer 1

p=50-0.50 me=30=0 E1 Consider first 9 - market demand G = 9, +92 +93t ... In - {ai = 9 Q = 99 + ri where qi is sum - qi=9-9-1 and Eqi = (n-1)9 of quantity of all no firms, Before investing in R&D, 98 (a not i Roofit function of from i Theo Tri- Ti - b. i - mori Ti - poxa (p-mc)aitini, Take mC = C= 20 = (somasettore = ( 50-0.59-4°397 = (50-0.5 (99 19-0)- Ti = [50-0.591 -0.59-i - Jai Taking F.O.C. Ti - 50 - 90 -0.59 i-ci=0 - q;= 50 -0.59-i-ci (ii)

9° (9-)50 -0.594- C Ü equation C is Sertis pied for all n froms, By Summing The equation co obtain 39 - (50-0-59--69 - Son- 0.5 {qi - Ec CEL (=1 = 50h-0.5 (n-1)q-C where q = 5on-0.5 Cha-C V 78005- 0,5%A1OS UM 99. = R-6.997 (581815 Srl = 0451917 46.88-12 9 + 0.5 (27)9-500-c 9 (1+0.5 (n-1) )- 50n-c 9 (1+0.50-0.5) = 50-c q (osn +0.5) = 50 n-c q* = 5on-c qx = 0* = maxi. 0.5 Cht) quantity industry

9=9= 9; + 9 1 =) 9 i = 9-2i from (i) q = 50 - 0.5 q-i-ci - 50 -0.5 (9-4)-ci at equilibruin 93-94 9=9* qit = 50-0.5/9 * - qpt) - c 1 = 50-059*+0.592 28 - 0.5 g = 50-0.59-ci - 50 -0.5 (50n-c 10.5 Chri) - 56 - joci (hti) 0.50 = 50- (50n-cl_co (ht) 0.59 = 50 (n+1) - 500 + C - Chriki (ht) = 500+ 50-50h+ C-hep-ci (hti) 0.528* = 50-hop-Cai (ht)

qit - 50-hci +C-P 0.5 Cht) where i EG; from i 50-99 -0.59-8- ci=0 50 - 0.52p -0.591 - 0.5%-i-ci=0 50-0.591 -0.5 (9i+98) -ci = 0 50 - 05/21 - - -o-5% = 0 50 - 0.5 (91-ci = 0.528 equilibruim quantities, 50 -0.5 (2* J-CP = 0.590k Epiocih nasqot p-50-o.se pet-ci = 0.5 * - iv from (in) Т - Ср- mc ) MC=Go at equilibruim. * = (byt cilat

from us pt = 0.59pt . 99 * = 0.5 (at) 2. - 0.5 (50-ncit C-ig? (0.5 (tl) I (50 - hcit (-i) 2 0.5 (hti)? # = Case-I with ci= 30 and Cup is not given. Tok (50(30) 0.5 (hti) 2 Case-2 with ci= 30 and caj - (2-1) 30 To* = (50-n (30) + (1+)30)? 0.5 (n+1)2 = (50 - 300 + 301-30) ² 0.5 (hti) 2 (50-303² - 0.5 (N+1) 2 400 0.5 (n+1) 2

Case-3 with ci= 20 and (n-1) Co = (n-1)30 Ta 50-10) = (50 - h (20) + (n-1) 30) 2 0.5 (hti? (50 - 20n + 30n - 30) 0.5 (hti) 2 = (50- 30 tlong? 0.5 (h+Ij2 Tot = (20 + lon) ² 6.5 (h+122