Exercises Solutions: Oligopoly

Exercises Solutions: Oligopoly Exercise - Quantity competition 1 Take firm 1 s perspective Total revenue is R(q 1 = (4 q 1 q q 1 and, hence, marginal revenue is MR 1 (q 1 = 4 q 1 q Marginal cost is MC 1 = 1 Apply MR 1 = MC 1 4 q 1 q = 1 We get that firm 1 s best response is q 1 = q Take firm s perspective Total revenue is R(q 1 = (4 q 1 q q and, hence, marginal revenue is MR (q = 4 q 1 q Marginal cost is MC = q Apply MR = MC 4 q 1 q = q We get that firm s best response is q = 4 q 1 Thus, the system of best response functions is: q 1 = q q = 4 q 1 By solving it for q 1 and q, we have the Cournot equilibrium is q = q 1 = 1 and p = Firm incorporates in its objective function the way in which firm 1 will react That is, firm anticipates firm 1 s reaction by plugging firm 1 s best response ( into the inverse demand: p = 4 q q = 5 q Thus, firm s total revenue is R(q = 5 q q and marginal revenue is MR(q = 5 q Apply MR = MC 5 q = q We obtain q = 5 4 > 1 Firm 1 chooses q 1 = 7 8 < 1 A cartel behaves as a monopolist Thus, we want to maximize the sum of profits of the two firms Note that since the two firms have different costs of production, we need to be a bit more careful We cannot simply apply MR = MC, We need to explicitly solve the joint maximization problem: The first-order conditions are: which gives the following optimal quantities: Π = (4 q q 1 (q 1 + q q 1 (q Π q 1 = (q 1 + q + (4 q q 1 1 = 0 Π q = (q 1 + q + (4 q q 1 q = 0 q 1 = q and q = 4 q 1 Solving the system yields q = 1, q 1 = 1 and p = 5 Therefore π 1 = 4 and π = One needs to show therefore that gives 5 8 to firm 1 1

Exercise - A merger Since the firms are symmetric, they produce the same quantity at the equilibrium demand is P = 0 q 1 q q n The marginal cost is MC = 6 The inverse 1 Total revenue for firm i is R i (q i = (0 q 1 q q n q i and marginal revenue is MR i (q i = 0 i j q j q i Applying MR i = MC 0 i j q j q i = 6, we obtain that the reaction (or best response function for firm i is q i = 4 i j q j As each firm produces the same quantity at the equilibrium, we have qi = q = 4 n+1 for all i Therefore, p = 0 4n n+1 = 6n+0 n+1 You can check that if the number of firms goes to infinity (the case of pure and perfect competition we have p = 6 = MC If there is only one firm, we find the monopoly price as a solution This is immediate For n =, we have that qi = 4 +1 = 6, p = 1, πi of the two firms is 7 = 6, and the sum of profits For n =, we have that q i = 4 +1 = 8, p = 14 and the profit is 64 < 7 It is therefore not optimal to merge 4 It is optimal to merge for all F > 0 that satisfy the following condition: 7 F < 64 F Or F > 8 5 The consumer surplus with three firms is CS = (0 1 18 = 16 Each firm has a profit of 6 and each pays the fixed costs The global surplus is therefore of 16 + 108 F = 70 F In the case of a merger, the surplus of consumers is now CS = (0 14 16 = 18 Each firm makes a profit of 64 and the fixed cost is paid twice The global surplus is then 18+18 F = 56 F If F > 14, it is efficient to merge Exercise 4 (final exam 005 1 The firms are symmetric, so they will produce the same quantity in equilibrium Total revenue for firm 1 is R 1 (q 1 = (100 q 1 q q 1 and marginal revenue is MR 1 (q 1 = 100 q 1 q Applying MR 1 = MC 100 q 1 q = 10 gives firm 1 s best response function q 1 (q = 45 q Firm s best response is symmetric In equilibrium, we know that q 1 = q = q, thus q = 45 q gives q = q 1 = q = 0 The equilibrium price is p = 100 0 0 = 40 Demand faced by the monopoly is p = 100 Q, total revenue is R(Q = (100 QQ, and marginal revenue is MR(Q = 100 Q Then, MR = 100 Q = 10 = MC Q = 45 Thus, at the optimum, each firm produces q 1 = q = 5 and p = 55 Now, total revenue for firm 1 is R 1 (q 1 = (100 q 1 q q 1 + T q and total cost for firm 1 is C 1 (q 1 = 10q 1 + T q 1 Thus, marginal revenue is the same as before, MR 1 (q 1 = 100 q 1 q, whereas marginal cost is MC 1 = 10 + T Symmetrically for firm Apply MR i = MC i for both firms, we get that the reaction functions are q 1 = 45 05T 05q and q = 45 05T 05q 1 Solving this system of two equations and two unknowns, we get that the Cournot equilibrium is q 1 = q = 0 T We seek such a T that this equilibrium corresponds to q 1 = q = 5 Then T = 5

4 Firm s problem is the same as in Cournot So, its best response function is q (q 1 = 45 q 1 By moving first, Firm 1 can anticipate firm s reaction, therefore it will incorporate firm s best response into its maximization problem Thus, the effective demand faced by firm 1 is p = 100 q 1 45 + q 1 = 55 q 1 Total revenue is R 1 (q 1 = (55 q 1 q 1 implying that marginal revenue is MR 1 (q 1 = 55 q 1 MR = MC gives q1 = 45 Using the best response of firm, we get q =, 5, and using the inverse demand, p =, 5 5 Since firm 1 can revise its choice in period, any announcement made by firm 1 in period 1 is not credible, unless it is a best response in period Here, firm has commitment power (once it makes its choice in period, it cannot revise its choice, and, hence, firm is the Stackelberg leader! Firm has a first-mover advantage in this case At the end of period 1, it is likely that firm 1 will set q 1 = 45 (ie, it pretends to act as a Stackelberg leader In period, firm will ignore the choice made by firm 1 in period 1 (anticipating that firm 1 will want to revise its choice in period Hence, firm will choose q = 45 (acting as the real leader In period, the best firm 1 can do to maximize its own profit is to revise its choice and choose q 1 = 5 (ie, it acts as a Stackelberg follower Exercise 5 except points and 5 1 Cournot: The indirect demand curve is p (Q = 100 Q or p (Q = 100 q 1 q where Q = q 1 +q Take Firm 1 s perspective Total revenue is R(q 1 = (100 q 1 q q 1 and, hence, MR(q 1 = 100 q 1 q By applying MR = MC, we get 100 q 1 q = 0 So, firm 1 s best response to firm s production is q 1 (q = 50 1 q The firms are identical, so we have q (q 1 = 50 1 q 1 To find the Cournot-Nash Equilibrium, solve the system of best responses: q 1 = 50 1 q q = 50 1 q 1 and then we get q1 = q = 100 Then, the equilibrium market price is p = 100 ( 100 = 100 Stackelberg: suppose that firm moves second Apply Backward Induction Hence, we start from firm s problem Follower s problem: Set MR = MC to obtain that the best response of firm to firm 1 s production is q (q 1 = 50 1 q 1 (Notice that this is the same as in question 1 The follower s problem is the same as in Cournot Leader s problem: Firm 1 moves first and anticipates how firm is going to choose its output in response to its own output choice So, by backward induction, Firm 1 incorporates firm s best response in the total demand: p = 100 q 1 (50 1 q 1 = 50 1 q 1 Apply MR = MC 50 q 1 = 0, so q 1 = 50 Then q (q 1 = 5 Finally, the equilibrium market price is p = 100 (50 + 5 = 5 Suppose that firm 1 is the leader and firms and are the followers To solve this sequential game we first need to determine the quantities chosen by the followers and simultaneously at stage Take the perspective of one of the followers, say firm Then, firm s inverse (residual demand is p = 100 q 1 q q Total revenue is R(q = (100 q 1 q q q, so that marginal revenue is MR(q = 100 q 1 q q Apply MR = MC 100 q 1 q q = 0 which gives firm s best response as a function of the quantities chosen by firm 1 and : q (q 1, q = 50 1 q 1 1 q By symmetry, the best response of firm (as it faces same demand and same cost structure as firm is q (q 1, q = 50 1 q 1 1 q

To find the Nash Equilibrium, solve the system of best responses: q = 50 1 q 1 1 q q = 50 1 q 1 1 q which gives q (q 1 = q (q 1 = 100 1 q 1 Now, Firm 1 moves first and recognizes how firms and will respond to its choice of output So, firm 1 takes into account firms and s best responses into its own problem: The demand is p = 100 q 1 q q = 100 q 1 ( 100 1 q 1 = 100 q 1 Total revenue for firm 1 is R(q 1 = ( 100 q 1 q 1 and marginal revenue for firm 1 is MR(q 1 = 100 q 1 Apply MR = MC 100 q 1 = 0 So, q1 = 50 Then, q = q = 50 and p = 100 ( 50 + 50 = 50 4 Suppose that firm 1 moves first, then and then We apply backward induction and start from the decisions taken by firm Firm s problem: This is the same type of problem that the firm would solve if it were playing à la Cournot Total revenue for firm is R(q = (100 q 1 q q q and MR(q = 100 q 1 q q Apply MR = MC 100 q 1 q q = 0 Then, firm s best response is q = 50 1 q 1 1 q Firm s problem: Firm recognizes that firm will react to its output choice and then takes into account firm s best response into its own problem So, the inverse demand for firm looks like p = 100 q 1 q q = 100 q 1 q (50 1 q 1 1 q p = 50 1 q 1 1 q Now, total revenue for firm is R(q = (50 1 q 1 1 q q and MR(q = 50 1 q 1 q Apply MR = MC 50 1 q 1 q = 0 Hence, firm s best response to firm 1 s choice is q = 50 1 q 1 Moreover, we can plug q into firm s best response and obtain firm s best response a s a function of firm 1 s choice: q = 50 1 q 1 1 ( 50 1 q 1 = 5 1 4 q 1 Firm 1 s problem: Firm 1 applies, again, backward induction and incorporates into its problem both firm and firm s best responses p = 100 q 1 q q = 100 q 1 (50 1 q 1 (5 1 4 q 1 p = 5 1 4 q 1 Therefore, total revenue for firm 1 is R(q 1 = (5 1 4 q 1q 1 and marginal revenue MR(q 1 = 5 1 q 1 Apply MR = MC 5 1 q 1 = 0 to get q1 = 50 Then, the optimal quantities for firms and are q = 50 1 50 = 5 and q = 15 Finally, the equilibrium market price is p = 100 (50 + 5 + 15 = 15 Exercise 6 1 For firm 1, the residual demand is q 1 = 10 p q Then profit is Π = (10 q 1 q q 1 4q 1, marginal revenue is MR 1 (q 1 = 10 q 1 q, and marginal cost is MC 1 = 4 Hence, MR 1 = MC 1 gives the best response function for firm 1: q 1 (q = q Similarly, we can get that for firm q (q 1 = q 1 (Note that the firms are symmetric For the Cournot-Nash equilibrium, we have to solve the system q 1 (q = q q (q 1 = q 1 which gives q 1 = q = Then p = 6 Π 1 = Π = 6 4 = 4 Marginal revenue is the same, MR 1 (q 1 = 10 q 1 q, but now MC 1 = 1 Applying MR 1 = MC 1 gives the best response function for firm 1: q1 (q = 9 q Since firms costs did not change, we have that firm s best response is unchanged: q (q 1 = q 1 To find the Cournot-Nash 4

equilibrium, solve the system q 1 (q = 9 q q (q 1 = q 1 We get q 1 = 4 and q = 1 Then p = 5 and Π 1 = 5 4 1 4 = 16 and Π = 5 1 4 1 = 1 Then the best response functions for both firms are going to be q1 (q = 9 q and q (q 1 = 9 q 1 The solution for an equilibrium gives now q1 = q = and p = 4 and Π 1 = Π = 4 1 = 9 4 The outcome (Not Invest, Not Invest was calculated in question 1 The situation when one of the firms invests and the other does not is described in question In question we found the profits (ignoring the fixed cost F of firms when they both improve their technology 5 When is not investing the dominant strategy? The values are 9 F 1 and 16 F 4 which means, taking the most restrictive of the conditions, F 1 If F > 1, not investing is a strict dominant strategy 6 This amount to ask: When is (Invest, Invest a Nash equilibrium? When neither of the firms would wish to unilaterally deviate, ie, whenever 9 F 1 or F 8 7 The payoff matrix with F = 10 is Firm 1 \ Firm Invest Not Invest Invest 1, 1 6, 1 Not Invest 1, 6 4, 4 The Nash equilibria are (Not Invest, Invest and (Invest, Not Invest 8 If firm 1 Invests, Firm prefers not to If firm 1 does not invest, Firm will indeed invest Hence, knowing this, Firm 1 prefers to Invest as then it will get 6 instead of 1 There is a first-mover advantage here 9 The last question of the exercise is modified as follows: What if the firms would compete as in a Stackelberg competition when MC = 4 and Firm 1 moves first? We solve the game by backward induction Then we know Firm reaction function remains q (q 1 = q 1 Firm 1 knows exactly how Firm will react, so it s profit is Π 1 = (10 q 1 q (q 1 q 1 4q 1 = ( 10 q 1 + q 1 q1 4q 1 = ( 7 q 1 q1 4q 1 Profit maximization of Firm 1 leads to a condition 7 q 1 = 4 or q 1 = Then we can find q (q 1 = = p = 65 Exercise 7 except q 4, 5, 6, and 9 - Pepsi and Coke (see optional slides - difficult 1 The best response curves are: P 1 = 105 + 05P and P = 7 + 01P 1 Bertrand equilibrium price: 1, 56 for Coke and 8, 6 for Pepsi Corresponding equilibrium quantities: 08 per quarter for Coke and 16 for Pepsi (Done in slides Profits: 891 for Coke and 906 for Pepsi To find the new equilibrium, we start by finding the new reaction function for Coke: Pepsi s reaction function stays the same: P = 7 + 01P 1 Coke s new reaction function: Step 1: Start with Coke s new demand curve: Q 1 = 100 4P 1 + P ; Step : Invert: P 1 = [5 + 05P ] 05Q 1 ; Step : Find MR 1 : (We just double the slope on Q 1 MR 1 = 5 + 05P 050Q 1 ; Step 4: Equate MR to MC, solve for Q 1 in terms of P : 5 + 05P 050Q 1 = 5 Q 1 = 40 + P Step 5: Plug result of step 4 back into demand inverse demand curve to solve for P 1 in terms of 5

P : P 1 = 5 + 05P 05[40 + P ] 1 = 15 + 05P To find the new equilibrium, we solve the reaction functions simultaneously for P 1 and P 01P 1 + P = 7P 1 05P = 15 Solving this system gives us: P 1 = 1718, P = 87 Is Pepsi better off? Pepsi s profit initially: (86 4(16 = 906 Pepsi s profit at the new equilibrium: (87 4(50 5(87 + 1(1718 = 111 Pepsi is better off after the increase in Coke s demand! As the demand for Coke shifts out, and Coke wishes to post higher prices no matter what price Pepsi will have, Pepsi would enjoy an increase in profit even if it kept prices constant, as it would be getting some customers turned away from Coke s higher prices Moreover, as Coke raises these prices and Pepsi has a higher residual demand, it wishes to raise its own prices as well, furthermore increasing its profit Let s return to the case in which Coke s demand curve is Q 1 = 64 4P 1 + P, but now suppose that Coke s marginal cost decreases to 4 per unit (a If prices and quantities do not change, Coke s profit goes up from (156 5 06 = 88 to (156 4 06 = 590 (b To find the new Bertrand equilibrium when Coke has a marginal cost of 4 per unit, we start by finding the new reaction function for Coke As in Task 1, Pepsi s reaction function stays the same, P = 7 + 01P 1 Coke s new reaction function: Step 1: Start with Coke s demand curve: Q 1 = 64 4P 1 + P Step : Invert: P 1 = [16 + 05P ] 05Q 1 Step : Find MR 1 : (We just double the slope MR 1 = 16 + 05P 050Q 1 Step 4: Equate MR to MC, solve for Q 1 in terms of P : 16+05P 050Q 1 = 4 Q 1 = 4+P Step 5: Plug result of step 4 back into demand inverse demand curve to solve for P 1 in terms of P : P 1 = 16 + 05P 05[4 + P ] P 1 = 10 + 05P Solving the reaction functions simultaneously gives us the new equilibrium prices: P 1 = 105, P = 81 This new Bertrand equilibrium is shown in the picture below Note that the drop in Coke s MC leads to a lower price for both Coke and for Pepsi 7 If firms simultaneously choose quantities rather than prices, we must first invert the demand curves, to get prices as a function of quantities Note that the products here are not homogenous, there is no unique price! Solving for P 1 and P, we get: Therefore, P 1 = (40 5Q 1 Q 18 and P = (64 Q 1 4Q 18 MR 1 (Q 1 = (40 10Q 1 Q 18 and MR (Q = (64 Q 1 8Q 18 Solving MR 1 (Q 1 = MC 1, and MR (Q = MC, we obtain the best-response functions: Q 1 = Q /5, and Q = 4 Q 1 /8 The Cournot-Nash equilibrium is found by solving for the intersection: we get Q 1 = 76/1 89 and Q = 65/1 08 Prices are respectively P 1 = 155/117 1 and P = 998/117 85; profits are 5440/151 7 and 140450/151 94 Comparing with simultaneous price competition, we see that prices and profits are higher under simultaneous quantity competition; this is not surprising: when we take quantities as given, by increasing one s quantity, the price of the rival automatically decreases a little bit (so that his 6

quantity remains fixed Having his rival automatically decreasing his prices makes such an output increase less attractive, and therefore, under quantity competition, firms settle on lower quantities and higher prices 8 Since firm moves second, we use its best-response function Q = 4 Q 1 /8 and substitute into firm 1 s demand curve: P 1 = (40 5Q 1 Q 18 = 6/ 19Q 1 /7 The marginal revenue is MR 1 (Q 1 = 6/ 19Q 1 /6 Solving for MR 1 (Q 1 = MC 1, we obtain Q 1 = 564/19 968 and Q = 771/8 09 Profits are therefore 4418/19 5 and 66049/7 9148: there is a first-order mover advantage, as the first mover gains more than the second mover from the sequential game: in fact, firm 1 gains something while firm loses (relative to the profit under simultaneous quantity setting Exercise 8 - Price competition 1 Rewrite the demand functions expressing the price of each firm as a function of its produced quantities and the price of competitors Therefore Using now MR = MC: the best response functions are p 1 = 5 + p q 1 p = 6 + p 1 q q 1 = (5 c p q = 1 (6 + p 1 p 1 = 5 + p 4 + c p = 9 + 8 p 1 Therefore p 1 = 4 + 16 9 c p = 6 + 6 9 c The profits are easy to calculate Exercise 9 - Homogeneous products Let s refer to the two bakeries as firm 1 and firm Market Demand P = 100 Q where Q = q 1 + q Fixed cost: F = 500 Variable cost: V C 1 (q 1 = q 1 and V C (q = q Hence, C(q 1 = q 1 + 500, C(q = q + 500, and MC 1 = MC = 7

1 Bertrand Competition: The strategic variable is price There will be a Bertrand price war and p 1 = p = MC 1 = MC = Then, the total quantity is Q = 98 The profits of the bakeries are Π 1 = Π = F = 500 To see that the outcome (p 1, p = (, is a Nash equilibrium, we can check that neither of the firms has a unilateral profitable deviations Take firm 1 s perspective and argue that, given that firm chooses p =, firm 1 has no incentives to set a price different from Suppose that firm 1 chooses a price p 1 >, given that p = Then, nobody wants to buy from firm 1 as firm still quotes a price of Hence, firm 1 s sales are zero and profit is Π 1 = F = 500 Therefore, the firm is not better off and p 1 > is not a profitable deviation Suppose now that firm 1 chooses a price p 1 < Then, it captures the entire market demand since it is selling below its competitors price of So D 1 ( p 1, p = = 100 p 1 But now, firm 1 is selling below cost and is making a loss on every unit it sells Therefore, Π 1 = ( p 1 (100 p 1 F < F = Π 1 so p 1 < is not a profitable deviation We conclude that firm 1 cannot improve its payoff by playing something else than p 1 = when p =, which means that p 1 = is the best response to p = The same reasoning will hold for firm as they are symmetric - identical in terms of costs This completes our proof that (p 1, p = (, is a Nash equilibrium Cournot Competition: The strategic variable is now quantity Step 1: Take firm 1 s perspective The total revenue is T R 1 (q 1 = (100 q 1 q q 1 Hence, the marginal revenue of firm 1 is MR 1 (q 1 = 100 q 1 q By applying MR 1 (q 1 = MC 1 (q 1 we get 100 q 1 q = and the best response curve is q 1 (q = 49 q Step : Since firms have the same cost, the problem is symmetric, so we also have q (q 1 = 49 q 1 Step : In a Nash equilibrium it must be true that both bakeries are going to be best-responding to each others actions, so we have to solve the system of best response functions: q 1(q = 49 q q (q 1 = 49 q 1 which yields q1 = q = 49 The equilibrium market price is P = 100 4 profits Π 1 = Π = 104 49 49 500 49 = 104 The In a cartel, they act as if they were a monopolist Then, total revenue is T R(Q = (100 QQ Marginal revenue is MR (Q = 100 Q and MR = MC leads to Q = 49 and P = 51 If the bakeries share the profits equally, their profits are now Π 1 = Π = 51 1 49 ( 1 49 500 The quantity produced by each bakery is Q = 45 Exercise 0 1 Inverse demand is p = 10 q 1 q Take firm 1 s perspective Then, marginal revenue is MR 1 (q 1 = 10 q 1 q Apply MR = MC q = 6 Firm 1 s best response is q 1 = q By symmetry, Firm s best response is q = q 1 Solve the system of best responses to obtain the equilibrium quantities and price: q1 = q = 4 and p = Suppose MC 1 = and MC = 6 Firm s best response stays the same: q = q 1 Find the new best response for firm 1 by setting MR 1 = 10 q 1 q = = MC q 1 = 4 q Find the CNE by solving the system of best responses We obtain that q 1 = 4, q = 0, and p = 6 Firm 1 would price firm out of the market Firm 1 would set a price slightly lower than the marginal cost of firm, so that firm would leave the market 8

4 Apply backward induction Freel moves second and its best response function is the same as in Cournot: q 1 = 4 q LFree anticipates this and incorporates Freel s best response into its own problem Thus, the inverse demand faced by Lfree is p = 10 4 + q q = 6 q Then, Lfree s marginal revenue is MR (q = 6 q Apply MR = MC q = 0 Then, q 1 = 4 There is no difference with the Cournot competition 5 Start with the problem of the follower (firm 1 or Freel Marginal revenue is MR 1 = 1 q 1 q and the marginal cost is MC 1 = By setting MR 1 = MC 1, we obtain firm 1 s best response, q 1 = 5 q Now, apply backward induction and analyze firm s leader problem Its demand is p = 1 5 + q q = 7 q Then, apply MR = MC 7 q = 6 Hence, we now have q = 1, q 1 = 45 and p = 65 Exercise 1 1 Take firm 1 s perspective Total revenue is R 1 (q 1 = (80 q 1 q q 1 and marginal revenue is MR 1 (q 1 = 80 q 1 q Apply MR 1 = MC to obtain firm 1 s best response function: q 1 = 0 q By symmetry, firm s best response function is q = 0 q 1 By solving the system of best responses, we obtain that the Cournot-Nash equilibrium is q1 = q = 0 and the equilibrium price is p = 80 0 0 = 40 Profit is π 1 = π = 0 (40 0 = 400 For n =, inverse demand is p = 80 q 1 q q Take firm 1 s perspective Then, total revenue is R 1 (q 1 = (80 q 1 q q q 1 and marginal revenue is MR 1 (q 1 = 80 q 1 q q Apply MR 1 = MC to obtain firm 1 s best response function: q 1 (q, q = 0 ( q +q By symmetry, we have that firms and s best response functions are, respectively, q (q 1, q = 0 ( q 1 +q and q (q 1, q = 0 ( q 1 +q By symmetry, we know that in equilibrium we will have q 1 = q = q = q Hence, by using any of the above best responses (indeed, it is sufficient to compute the best response ( of only one firm, we have that at the equilibrium quantity q, the equation q = 0 q +q must hold It follows that, q = 15, p = 80 15 = 5, and π = 15 (5 0 = 5 The demand is p = 80 Q Total revenue for the cartel is R(Q = (80 QQ and marginal revenue is MR(Q = 80 Q Then, set MR = MC to get that the equilibrium is Q = 0, P = 50, and π = 900 Each firm makes 00 in profit 4 This is the same situation as in question 1, with the only difference that the merged firms share their profit So, we have that q1 = q + = 0, p = 40 The profits of firm 1 and the merged firm + are π1 = π + = 400 Firm and make each 00 in profit, whereas in question each firm makes 5 Thus the merger is not profitable 5 Firm 1 is the Stackelberg leader; firms and are the Stackelberg followers and play a Cournot game between themselves Apply backward induction and solve the Cournot game between firms and From question, we directly know that the best response functions are ( ( q1 + q q1 + q q = 0 and q = 0 Solve( the system of best responses: by symmetry, we know that q = q = q Thus, q = 0 q1 +q q = 0 q 1 By backward induction, plug q = q = q into the leader s problem: the inverse demand is then p = 80 q 1 ( 0 q 1 = 40 q 1 Firm 1 s total revenue is R 1 (q 1 = (40 q 1 q 1 and marginal revenue is MR 1 (q 1 = 40 q 1 Apply MR 1 = MC 40 q 1 = 0 to obtain that q1 = 0 Finally, q = q = q = 0 0 = 10 The equilibrium price is p = 80 0 10 = 40 Profits are π 1 = 600 and π = π = 00 9

6 Apply backward induction From question 4, we know that the best response function of the merged firm + is q + = 0 q 1 Plug this into firm 1 s problem Thus, p = 80 q 1 ( 0 q 1 = 50 q 1 Marginal revenue is MR 1 = 50 q 1 By MR = MC, we have q 1 = 0, q + = 15, and p = 5 The profits are π 1 = 450 and π + = π 5 < 400 as found in question 5 10