Question

The market for widgets consists of two firms that produce identical products. Competition in the market is such that each of the firms independently produces a quantity of output, and these quantities are then sold in the market at a price that is determined by the total amount produced by the two firms. Firm 2 is known to have a cost advantage over firm 1. A recent study found that the (inverse) market demand curve faced by the two firms is P=580-2(q1 and q2), and costs are C1(q1) = 3q1, and C2(q2)= 2q2.

a) Determine the marginal revenue curve for each firm

b) Determine the reaction function for each firm

c) How much output will each firm produce in equilibrium?

d) What are the equilibrium profits for each firm?

e) A third firm is considering to enter the market with C3(q3)= 2q3, at what maximum investment cost would this firm enter the market?

Answer 1

we have see Pa 580–2001+92). Glad = 39, Bilik - (a) - 24 a) Total Revenue for from: 1, Try as given by a n TRY = pq, = 1580 –2 (0i+92)] : ? = $800,- 24,3 - 20122 do Marginal Revenue, MR, is given by a MR, 3 D.Trueta d',($800,–29,224W). dai. Tai da te

= 580 -40..-29214 Thus, Marginal Revenue wree for Firm is given by: MR, = 580 – 49,-292: Total Revenue for firm 2 is given by a - The = P.q2 = [5.80 - 2 (2, +92)] Qiz : .:: 58092 – 24; 92 - +243 e Marginal Revenue, MR2 is given by MR s ditRe d (58092 - 20122=262) MRizdas tidão Hi=580 – 29-4qiz. Thus; marginal revenue cuane for from 2 is given by MR2 = 580 - 29,- 492 : by Profit function for firm 1, I, is given by : Ti = pq, -4 = (580-2 (ai+.92)]q , .-39. Partially differentiating Ili with respect to q and applying the first order condition, we get

>> Q..?!:45 A 2= 580 – 49. – 292 – 350 =) 577 - 49, -222 = 0 .' 577-292 2 =R1 (29):- - Re (ave) in. equation @ gives us the reaction fonction of firm 1 Profit Fonetion for firm 2 is ginen toy * T2 = PQ2-C = [580-2(a +22)] 42 - 242 Partially differentlating 12 with respect to qe me get and applying the first Order condition, we get? 1996= 580-247- 492 -2 -0. a = 598 -29 R. (a) - . Rq (a) in equationi 6 giver is the realtimi funition of firm 2 ..... Thus, we havie : 144.. . Reaitin Feriction of Fiom tar R (9) - 577 - 292 reaction Function of Firu.2.& Ry ( 578 - 2915

@ in @ o let us put the value of an from Therefore, we get: q, = 577 578-291 . 577-2 4 - 577- 578-290 1154 - 578 +291,1.. => 89, = 1154-578+291.... 576 >) ar's:- =96: => Q1 = 96512 hutting an=96 in 6, me get : q2 = 5:48 - 221 = 578 - (2x96) = 96.5 CM = 92 = 96.5 In equilium, quantity produced by firm 1, q, = 96 and quantity produced by Fiom 2, 92 = 965

do we have a = 96 and q2 = 465 t Therefore, p = 2,+92 - 96+96.53 192-52 Thm, P = 580 - 2(192.5) - 195. -) P=195. I f Now, G. = 39, = 3x96 = 288 (n = 292 = 2x96.5 = 193 :.. "TI, = pq, -= 1195x96) - 288 - = 18482 -) TI = 184321%!! !! Again, Ty = pq 2 - C2 = (195x96.5) - 193 =18 =18624.SI. => 1 =18624.5. Home, Profit function of firm I, at equilibrium a T, '= 18432 Profit of firm 2 at equilibrium : T2 = 1862405