Question

15.2 where a, b > 0 a. Suppose that firms' marginal and average costs are constant and equal to c and that inverse market demand is given by P = a - bQ. Calculate the profit-maximizing price-quantity combination for a monopolist. Also calculate the monopolist's profit. b. Calculate the Nash equilibrium quantities for Cournot duopolists, which choose quantities for their identical products simultaneously. Also compute market output, market price, and firm and industry profits. c. Calculate the Nash equilibrium prices for Bertrand duopolists, which choose prices for their identical products simultane- ously. Also compute firm and market output as well as firm and industry profits. d. Suppose now that there are n identical firms in a Cournot model. Compute the Nash equilibrium quantities as functions of n. Also compute market output, market price, and firm and industry profits. e. Show that the monopoly outcome from part (a) can be reproduced in part (d) by setting n = 1, that the Cournot duopoly outcome from part (b) can be reproduced in part (d) by setting n = 2 in part (d), and that letting n approach infinity yields the same market price, output, and industry profit as in part (c).

Answer 1

As per the HOMEWORKLIB POLICY, answering the first four parts

P- Q-bQ 2b po a- bo a. at equilibrium, under Monopoly MR = MC TR= px 0 = ao-ba² MR = 2 TR/aQ = a - 260 → a-2ba QM= arc An -"() pe 20-a+C- atc Ana 29 -4 ato Profit = (pM_ c) XQM- fate (0-c)? Ana 6. CUR NOT For from I: Profit, 1,= (-6)ą, = (a-bq-bę,-6)ą, » T,= (a-bezcla, - ba? Foci amilan, = 0 7 a-ban-c = 269, [orce -c][:] - 46

6R 0.5 은 - 0.541 - 0.5 0 05음 고 ⇒ 17591 = 0.25 = 0.5 - 0.5a-0.5c Solwingo &O - 0.5 4 [org -0.54-2.5g]-o54 2. 2.5 - 4.25g + 25, +0.75g - og 25 + a = eN - - 6.33 (ac) Ana . (ac) Mkk. prie, p= a- bf0.66 [c () -0.25 b Mt. Output, Q = 4,t92 는 . - a- 0.66 4 + o. 66c

P: 0.349 +0.660 Ago T infil, a, 3 T2 = (0.344+0.660-c) 0,033 (2-0) = 0.34 (a -c) 0.33 (a-c) 0.55(0-c)? industry profit = httle = 11 (a-c)² Am b b c. BERTRAND at equilibrium, p= MC So, P = P2 = vc Market output, a-6Q = vc Q: Q-C b Q, = 92 = a-C a e (60-03792) Profit, n= -0. d. psa-bo Ti - (P-C) = (2-69; -6Q.;-clai Q .qi+Q-1 FOC OTI = 0 (a-bQ_;-c) - 262i = 0 aqi Assuming Symmetry, we have 0 m Q* -i Inai -9 (n-022 So, ori/zi + a-b(n-174 - 24b-C=0 5a-c = 91 (6(n-1) + 2b] :-) a = Qoç b(n+1) Ano Total Output, a* = nak Coleh Aloo p= a-la- - Contra Pafit= (aune ht atne An *) () ni fla -c), (h+2

**if you liked the answer, then please upvote. Would be motivating for me. Thanks.