Question

5. Consider two firms selling differentiated varieties of a product, e.g., Coke and Pepsi. Each firm j chooses a price pj for its own variety. Since these varieties are close substitutes, the demand that each firm faces depends not only on its own price, but also the price of its competitor. Specifically, the demand for j’s variety is given by Dj (pj , p−j ) = max 0, 60 + p−j − 2pj

Suppose that both firms can produce any amount of their variety at no cost.

(a) Find firm j’s best response function.

(b) Assume that firms choose prices simultaneously and independently. Show that choosing pj = 18 is not rationalizable. [Hint: perform two rounds of iterated dominance]

(c) Assume that firms choose prices simultaneously and independently. Find the unique Nash equilibirum of the game. [Hint: symmetric games typically have a symmetric equilibrium]

(d) Assume that firm 1 chooses its price first, and firm 2 chooses its price second after seeing firms 1’s price. Find the Stackelberg equilibrium of the game. (e) Compare the equilibrium profits of each firm under part (c), withe their profits on part (d).

Answer 1

V. (a) consider two fioms selling differenticated product Coke and Pepsi The demand for j's variety is given by Dj ( Pj , Pj) = max {0, 60+ Pg - 2P;} Cost = 0 Firm j's best response function is given by the Clare Tj = Pj ( Dj (Pj , Pj) srop - 2a Crowe = Pi (60 + Pj - 2P) 23:23 paded out = 60 P; + Pi. Pr - 2P, muid anj ap = 60 + Pj. - 4P; - 0 3 festigio (3) x 60+ Page = 4 p d I neeft get to 293100 2200110 1130 nod i Pj = 60+ Paj cottile dmo golodd ord z rowolow- smo poled brorenggant ons dito isto firm is best response function.

Assume that firms chooses poices simultaneously and Independently. 11111 99999999999 9 . firm j's best response es Snodai Pj = 60 + P- j anturine o ziti 1900 Let's suppose firm ở choose Pj = file to i 18 = 60+ PhD 18x4 = 60 + lj I to fi = 72-60 = 12 D; (P; , P.y) = 60 + P-j - 2 Pojn = 60+12 - 2x 18 = 72-36 = 36 Ti = 36 X 18 = 648 Suppose, P; = 20 718, then Pj - 2084 -60 = 80-60 = 20 Dj = 60+20 - 2120) =80-40 = 40 Tg = 40(20) = 1800 > 648 ? Also let Pj = 19718, then Pj = 1984 - 60 = 176-60 = 16 Dj = 60 + 16 - 2119) = 76-38 = 38 Tj = 38(19) = 722 7 648

nultaneous y Mence, choosing Pi=18 is not a national decision C) Assume that firms choose brices simultaneously and independently. since, it is a symmetric game. od 0 s ja fiom 2 - Dj (Pj; Py) - 60+ Pj - 2- . -Jotion Tuj (P; , Pj) = 60 Paj & Pj. P-j - 2 P 23 am - 60+ P; - 4 PJ = 0 si ods. 4 Py = 60+ Pj Best response function . 60+ P; of fiom 1. Nash equilibrium is the one there Substitute firm 2's best response function in fiom 1's. Pj = 60+/60+ P-j) 4 4 Pj = 60+ 60 + Rs ५ ५ 4 Pj - P ļ = 60+15=15 os = 19 Pj = 75 dank Pj = 7544 - 300 eisen u

i P; = 60+ Pj - 60720 - 80 120) . Onique Nash Equilibrium is when both firms choose price equal to 20. (simultaneously and independently) ie Py = P.j = 1201 Dj (P; , - Pj) = D-j (P; , P-j) = 60+ Pg - 2 Pj tead out cari = 60-Paj =60-20 - MO Ti = 7-j = 40X20 – 800. Assume that from 1 chooses its boice first, and firm & chooses its prices second, after seeing from 's brice. We solve this by using backward induction, At stage 2, fionn 2 scelves its prices. Tj = Pj. Dj = 60 Pj + P; . Pj - 2 pt daj = 60+ P-j- 4 Pj = 0 ap; Pjo 60 + Pis At stage 1, firm I selve its prices Thij = 60 Pj + Pej. Pj - 2 Paj? Substitute the value of Pjes Thy = 60 Pj + Pj (60 + Pj) - 2 Pg² 4 2 Pige 60 Pj + 15 Pj + Pg = 756; -7 Pig

opy 75= 7 Pj os of love stod los P.j = 7542 Pj = 150 - 21.42 ~ 2105 7 ( Substitute the value of P-j into the best response function of firm j. brio di Lot B2B = 60 +150) Ps = 60+ lj L umont 22A D - 60 +50 3 200 od Votubi Jorow Good orlow pd aint gulos scu. 4 Pj = 60 + 2X8 ISO spate 36 4 P = 420 + 150 7 28 Py = 570 Pj = 570 = 120.352 20:50 galle? I mitt spate sa (from 1) Dij = 60 + Pj - 2P-j godt = 60+ 20:35 - 2(21,42) I = 80.35- 42.84 = 37.51 37.57 TH : 37.(2) - 922 - 803 96

for firm j Di Pj 20:35 Dj - 60+ 21:42 - 2(20•35) = 81.42 - 40.70 Di = 40.72 borba T = (40.72) (20:35) 70 Tj = 828.652 x X-3 l S 20 is 1. Stackelberg, equilibrium firma, Pji = 20:35 (fellower) Dj = 40.72 as = 828.652 SC ; tiom I (leader) P.; = 21:42 D-j = 37.51 -j = 803.46 (e) Equilibrium is hignes for from a when, when they play Stackelberg game. When tiom I is the leader and firm & choose after tiom ), But, for firm I profit is higher in part (c), when they play bestrand game and simultaneously choose prices. i. We can say that, in stackelberg botce competition, leader's profit is worse off while the follower's profit is better off than the case in bertrand borice competition