Question

Can someone help with the problem below?

Suppose two oligopolistic firms face a market (inverse) demand curve P(Y + Y2) = 20 - (Y1 + Y). Both firms produce at constant marginal cost, but they are not symmetric: firm 1 has marginal cost 2 and firm 2 has marginal cost 4. For each of the following competitive situations below, compute: • The equilibrium price. • The equilibrium quantities produced by each firm. • The profits received by each firm. (a) Stackelberg with firm 1 leading. (b) Cournot (simultaneous quantity-setting)

Answer 1

Innerse Denrand come i Ply 1+ Y₂) = 20-("1+y) Marginal cost of firm 1 MG = 2 . Marginal cost of Firni 2. men = 4 . Let y = Y( + Y2 . . (a) stackelberg with firmil reading ... Here we have to soline for the equilibrium by the method of backward induction and hence, at first we have to solue for firm 2's Reaction Funition Firm 2's profit function, la is given by t he The = P Yg – (MCs) x Y> ; = [20- (Y+Y2)742 - 442 ; ; - 20 Y2 - 4,42 - 42-442 = 16 Y2 - Y, Yg - Yg2.. Partially differentiatina Tle with respect to Y and applying the first order condition, we get 012 - 16-4, -242 - 0 : 342 18 » Y= 16-Y, Ry (1) - 0 kg (9) is the reaction fonction of Firm 2 wit

This reaction by firma is known to firm 1 wid he would choose a quantity y after accounting for Rp (1) in his/her profit fonction. from 1's profit function, II, is given by 111 = P4,- (mei) Y; . -720 - (91+Y9) ] Y. - 27, Now puting the value of Ye from 11 = 720-4, - 16,4 we get : - 24. .. = 5.40 - 2416 + 9'] Yi - BY = | 24 - 4; -74 – 24, - 24 - 24Y. - * * - 241., = 104, - Y, 22 Pentiatlet Bifferentiating with with respect to 4, and : applying the first order condition, we get to The dti - 10 - 4, 5,0 5, 041 => Y =10 4 we get : E putting this value of 4, in Y = 16-41 Y = 0 >> Y = 16-1.0.3. >> Yg = 3 .

.. 1: + + + 10 + 2 = 13 . => Y=13: M. Po 20 - y = 20-13='7 ." =) p=7. :. Equilibrium Price : P = 7 units Y = 10 units ; # Y = 3 units Profit of firm 1, 71, 11. = P4, - MG x Y1 : = [7 x 10) - (2x Yi) 2.70-20.-50.. profit of Firm 2,112! T2 = P49,- MC2 xYg . , - (783) - (4 x3). :- 21-12=9 8. Tis = 50 units; T = 9 units .. (6) Con mot Equilibrium i i 1. In this site model, both the firms set their quantities together. i o firmi's profit finition, Th, in this case will be ginen by '. .!? Ti = py, - MC XAYI = [20-4+42]]4.-2413 = 2041 – 4,2 - Y:42 -24, =184, - Y,2 – 4,42.11.

Partially differentiating HT ustth respect to Yo, and esplying the first order condition, we get: 3 The = 18 - 24,-49 50 =) Y, = 18-42- DY, - R, (Y) = 4, = .18-42 "?). "I 2 The above equation is the reaction function of firmi: firm a's reaction finition would remain. The same as in 0.1 .1.e, =4, - R2 (4Y2 - 16-Y 16 mm 2 n Putting the value of Ye from ④ in ① ; we get 1. 18-Y2 18-16mm Y, E ī 17:2: Y = 36 - 16+4," >> 44, -4, =20 : 3.4, = 20 = 6667. => Y, -1 66.67 , we get : Putting this value of Tiin Y2 = - 16-6.667 = 4.667. 2

oYE Y, + y2 = '63667 44*667 > y = 11. 334 : os P = 20 - 11.334 = 8.666. => P: 8-666..!! ; . Equilibrium Price : p= 8.666 units. 1. y q = 6:667 units. You = 4.667 units. Profit of firm 1, here, Ni, is given by : - T = P4 - MG XY, • 8.666 $6.667) - 12 x6:667). = 44.442 Profit of from 2, M2; here, is given by : 112 - PYg - MC, x42 =(8.666 x 4.667) -(4x4-66 7)'. = 21.776 i. TiC = 441442 ; 794 = 21.776 (C7 Collusion since the marginal cost of the forms are not symmetric, the firms won't agree to collude. firm thas a cost advantage over firm 2 and If the profit is splitted evenly, then it will be unfair on part of firms word Hence, collurion won't take place, but Firm I can become monopoly

. Therefore, under Monopoly, we know that me equilibrium is established at a point where, . Marginal Revenue - Marginal lost In this case, 1 Y=Y, in p= 20 - 4, (sime, 42 = 0) . Further, me=me -) MC = 2. At equilibrium : dipyn o me esime, Total Re DY - IMC since, Total Revenue : TR = P41 - and hence, Marginal Revenue MR=dTk d(04), => S [204. – Y,?] =2. 5) 20-24, = 2 =) Y, = 20-2 -9. => 4, = 9. in p=20-9 =11. =) p = 11: Profit : 1 = P4,- (MC)Y, -*- (1149) - (2x9). 11 = 81. :. Equilibriom Price : pall units - Equilibrium quantity: Y, = 9 units (EY) Equilibriom Profit lof firm ): 71 = 81 units :