Question

2. Consider a Cournot dupoly, but assume that demand is p(q) = 12 – 9 (ignoring non-negativity) and and unit cost c= 3. Again, firms simultaneously pick quantities qı > 0 and q2 > 0 and the price is set to clear the market given the quantities chosen. 1. Write down the optimization problems that define the best responses, solve explicitly for the best response functions, and find all Nash equilibria. 2. Formulate the problem for a cartel that maximizes industry profits. Find all (21,92) that solves this problem. Also, let a symmetric cartel solution be all (91, 92) that maximize industry profits and have q1 = 42. Calculate the symmetric cartel solution, the associated profits and compare with the profits in the Nash equilibrium. 3. Instead consider the Stackelberg model with firm 1 being the leader and firm 2 being the follower. Carefully describe the strategy sets. 4. Find the subgame perfect equilibrium. 5. Construct a Nash equilibrium that is not subgame perfect.

Answer 1

pcq) = 12-01-2 Demand for Marginal Mefient = 3 Mc firm 2=3 Anss. fimm I will maximize the profit in which is given by Profit pq, -3a, (12-9, -2) 2 -39 129,-a2-9, 92-39, 2-222 qara (Applying First order de profit fim du Condition) 9-29, -2 = 0 a - Az = a, Profit firma a = a-an (1) This is the best response En of firma P4 - 34, (12-9, -a) 9-3an 1291 ke Ke-39 (Applying first order condition aq dc Profit firma) = 0 1/7

12-9-29 - B - q-al 2q2 a-g q (3) This is the best responue function of firm 2 Intersection of best responses of both from gives Nash equilibria. Susbstitite the value of a from equation (2) R900 in qi= 9-22 2 9,- 9-(9201) 29, = 9 -(4-2) 2a,- 18-ata 49,= ata, Bal 9. = 3 Substitue a=3 in eq (2) 92-9-91 9-3 & 6 = 3 the only ca, a? =(3,3) is a Wash 2 equilibria 2/7

Ansa. A cartel will maximize profit earte given by Profitartet Pxq, + Pxq - 39,-34 (12-4-2) a, +(12-9-90 % -39, -392 = 129,-9,7-99, +129992 2 -32,-39 ac Profiteartel da, = 12-29-2 4-3=0 = 9-29, 2q = 0 -(3) 9-22 a, 2 oc Profit cartell -9,+12-9, -2%-3=0 dar 9-297-292 = 0 Az = 9-29, 9-291-(4) = 2 Substitute a 9-a imega ype 9-7*(9-2017) 2 are 29,90(929) 29-9-9+29. So eq (3) and eq (4) barecally Same ire 29, +22=9 So all those ca, which satisfies 2214 229 maximize the industry (cartel) profits. 3/7

To find a symmetric cartel Solution that maximizes industry profit. So aq , + 2q = 9 Substitute q=2 for symmetric cartel Solution 291 +22=9 42,9 a=9/4 2 = 9/4 Total profits associated with Nash equilibria T. Profit = C12-9-9) a, +(12-22 2 -39,-392 = (12-3-3),3 +(12-3-3)*3-3x3-3x3 18+ 18 a-9 Nash 18 - に Spm. Carter Symmetric Total Profits associated with Cartel solution. To Profits (12-92 29, +(12-ar War 39,34 (12-4-44 12-9-94,-343, -504 2x(12-4-) - 2*309 2x (49-9-9) - 2 2 2 30 - 27 = 2 2 2 = 1 72 4/7

the 2 will maximize. aa-a, 9-92 Profits associated with Nash equilibria n's greater than hergmnhatrie cartel solution Ans 3. Stackelberg Model, with firm 1 being the leader and firma being follower Apply backward indiction. Assume that firm 1 has played and has chose a quantity. Now firm 2 (12-9-29 -32 1224, 222-392 d (Profit firma - 9-9-29 2 = 0 daa 9-9 = 292 (5) Now firms will maximize his profit keeping in mind the best response of firm 2. Profit firm, (12-2, 92, 9, -39, (12 substitute a=9-2 Profit fing = 2 =9-9 2 quan 5/7 01

234 nobt fim= 1290329일 990오른9월 - 12-4-(14) 94와 3일 4월 99일에 - 2 に 9 올외 외 d (Profet firm? 외 - 올 2 왜 2 substituting 온 9 2 = in eq(5) toget 9 오 다. Am 4) Hiances C4,월츠 (24) is subgame perfect equilibrium where firma produce a units and firma produces 9 units प 6/7

Solun dackell lberg model, Ans.5. Clearly in this game, the Nash equilibria found from Solung. Cournot duopoly model Cans 1) is different from Subgame perfect equilibrium found from So pca 2) = 12-9-92 $ each having marginal cost = 3 is a best exple where Nash equilibrium subgame perfect - Note? Every subgame perfect is a Nash equilibrium but not every Nash equilibrium is a subgame perfect équilibrion is not a 717