Question

3. (35 points Suppose that there are K( 3) firms operate in a market with demand function given by P(Q) = 100-Q, where Q=91 +92 + ... +2K, and qi is the quantity produced by firm i. Each firm has a constant marginal cost of production, c = 10, and no fixed cost. The firms choose their quantities dynamically as follows: Firm 1, which is the industry leader, chooses qı € (0, 100). All other firms i = 2,..., K observe qi and then simultaneously choose 92, ..., 9k, respectively. (a) [5 points How many subgames does this extensive form game have? Explain briefly. (b) (5 points) What are the strategies of each firm in this extensive form game? (c) (5 points) For this part, assume that K = 2. Derive the unique subgame-perfect equilibrium and the equilibrium outcome. Be explicit about equilibrium strategies and equilibrium outcome. For parts (d) and (e), assume that K = 3. (d) (10 points) Find a subgame perfect equilibrium of this game. Argue briefly that it is unique. (e) (5 points) Find a Nash equilibrium that is not a subgame-perfect equilibrium. (f) (5 points) Now consider an arbitrary number of firms, K. Find the subgame perfect equi- librium of this game. Explain how firm l's quantity choice qu in the subgame perfect equilibrium depends on the number of firms.

Answer 1

All the assumptions under the Cournot model are applicable in Stakelberg's model except the assumption of simultaneous decision making. Under Stakelberg's model, we assume that firms enter the market sequentially.

It is assumed that the leader has information about the reaction curve of the follower(s). Therefore, the leader incorporates the reaction curve of the follower(s) in his profit function before maximizing it.

Extensive form game. This extensive form game have made two Subgames on (1) The game itself [(2) The second stage of the game.. • A subgame is a I begining with a including everything part of extensive jam decision node and that branches out to the alh Scanner

-1914. strategy is an action plan for playing the game Strategy & player 1 is to choose quantity between you and 100 that maximises its profits. 94 € 10,100) Bit Strategy set confines of quantities between 0 to intindy Utrategy of every followey_cie to choose_quantity I enot mond than 100) after observing that the quantity produced by player" ! and I considering the fact that other X-2 firms are also playing the game. .q t që € [0, 00). (cl K=L 1 Leader and -1 sellowex.. PCQ) = 100-Q - N UDE_ to p=9, +7 MC=10 Reaction cisne X firm 2:2 ADN To a TR, - TCD DON P. Qg1002 = 1.92 -1092 Up = 1100-9, - 4, 2. 9 - 109 = 100q, - qq - q -1092- Scanned with a Heartscanner 9042 -992-92

7 Kr. 90 q qiqr - 92 differentiatig with respect to for 212 = 90 - 91 - 291 30.. 90-9, = 2 - P9 - 1091_ Substituting 42 ur firm 1's profit function. - A - TR, - TC -- ___=(100-9-921.91 3.109– =_10097 = 92 - 9.9 - 1097 = 909, -92 -qiq putting q2 = 90-9 = 909, - q -9,190 -9.) = 904–914 - 24+ Scanned with 9 CamScanner

differentiating with respect to 9411 & dr. = 45. q. O n the loSubstitutina 9 =45 in the reaction curre of ..... 92 = 90 - 45 Subgame perfed- equilibrium (919x) es (45., 45.). These are the equilibrium setpet Scanned the CamScanner

(dill The game is of length two but each player has a very big action set consisting of L quantities of Couto let. Though quantities can be any positive number that is between 0 and unfinity but no um will prefer quantity greater than 100 ill give negative price. Subcame Perfect Equilibrium is obtained by backward induction Start with the Second stage this is just Cournot competition with K-1 funs. solve this for! Hash equilibrium by setting the first Il exdles condition fox sesim... men som k and solve these quantities by taking 9, as giren. This will give you quantities qz and q. q in terms of q plug these guardies into the profit Jundion of q, and maximise it. Scanned with CamScanner

POENO K = 3._ | P = 100 -8 q = 9, +q +93- Starting from stage 2 the nask equilibrium is solved by following two equatione : Ta' (91797790) -0. 72 19719819933. Thy - The = p.9 - 1097 = (P-10).925 = (100 = 2x=42-43 -10.92- = 60 - 91-92-94292 * 9041-9141-91- 9793 differentialing with respect to 9.2 dho = 90 -q - 2q - q₂ = 0 slianned diam. Scann.o le

PAGE NO.: 10-9, 9 - 92 Since I Player Cournot same. 2 game is Symmetric and player 3 are game. Cost of both Thin 1 - Thus in stage a playing like a Also, --- - q = 4₂ = fig.) 9 90 q = 92 2q + q = 90-9 SE 90-91 H _ 3 = 90-9 3 C Scanned with -Garn SeaTET

Substiluting the values of , and go in the profit function of player 1. A = P. q,- 10q, (P-10791 = (100-91-92=99 –10191 = 909, 9,2 9. 9. 9. 93 Subotititutie q2 = q = 90-9 in Iis = 90g, -9,2 - 9 i (90 = q) - 9, (90-91) 39091-92 - 3091-42 - 3041 - 42 differentiating wat ons dr = 90-29, - 30 + 2q - 30 + 29, = 0. 3 30-29, = 0 _30X3591 n CamScanner

PAGE NO.: Putting 9. = 45 in reaction curve of firm a and from 3 we get 42 = q = 90-45 = 42 33.. Pubgame perfect Nashe equilibriumi 119.92. gg) = ( 45 45 44 This equibrium is unique because Lat these quantities every firm makung maximum pouble only.. is " ( Silanned with amScanner