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MICROECONOMICS EV Prof. Davide Vannoni

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Exercise session 4 monopoly and deadweight loss 1.Exercise on monopoly and deadweight loss 2.Exercise on natural monopoly quantity choices 3.Exercise on oligopoly: simultaneous quantity choices (Cournot) and sequential (Stackelberg) collusion and price war 4.Oligopoly: collusion and price war price choices 5.Oligopoly: price choices simultaneous and sequential

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D. Vannoni e M. Piacenza Microeconomia C, A.A. 2007-2008 Esercitazione 4 3 Exercise n. 1 A monopoly maximizes the profit and bears a linear and increasing marginal cost. He produces 20 units, in correspondence of which the marginal cost (increasing) is equal to 40. The demand function (linear) crosses the marginal costs in correspondence with a price equal to 60 and with a quantity equal to 30 units. Determine: -The inverse demand function and the marginal cost function (both represented by lines); -The welfare deadweight loss.

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D. Vannoni e M. Piacenza Microeconomia C, A.A. 2007-2008 Esercitazione 4 4

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D. Vannoni e M. Piacenza Microeconomia C, A.A. 2007-2008 Esercitazione 4 5 02040 Demand MC MR 30 60 80100120 100 80 60 40 20 Deadweight loss:

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D. Vannoni e M. Piacenza Microeconomia C, A.A. 2007-2008 Esercitazione 4 6 Exercise n.2 In lake Wobegon, Minnesota, live 10 families; for each family the demand for electricity is Q i = 50 – P. The production cost of electricity of Lake Wobegon Electric’s (LWE) is TC = 500 + Q. a) If the regulatory authority of LWE wants to guarantee that in this market there is not a deadweight loss, which price would it impose to LWE? Which will be the production of electricity? Compute the consumer's surplus and the profit of LWE at such a price.

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D. Vannoni e M. Piacenza Microeconomia C, A.A. 2007-2008 Esercitazione 4 7 The market demand for electricity is the sum of the quantity demanded by the households. Graphically, we sum horizontally the demand for each family. Algebraically, the total demand is:

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D. Vannoni e M. Piacenza Microeconomia C, A.A. 2007-2008 Esercitazione 4 8 In order to avoid a deadweight loss, the regulatory authority will impose to LWE a price equal to the marginal cost. Since the total cost curve is TC = 500 + Q, the marginal cost is: MC = 1$ Equating price and MC and solving for quantity we get the quantity: P = 50 – 0.1Q = CM = 1 Q = 490 The profit of The profit of LWE in correspondence of such a price is: Π = TR – TC = 1∙490 – (500 + 490) = - 500$ (green area) The consumer's surplus is: CS = 0.5 ∙ (50 – 1) ∙ 490 = 12005 $ (equal to 1200.5 $ per household)

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D. Vannoni e M. Piacenza Microeconomia C, A.A. 2007-2008 Esercitazione 4 9

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D. Vannoni e M. Piacenza Microeconomia C, A.A. 2007-2008 Esercitazione 4 10 b) If the regulatory authority wants to avoid that LWE is not losing money (-500$), what will be the lowest price to be charged? Find the production level, the consumer's surplus and the profit at this price. Is there a deadweight loss? average cost of production The lowest price to be imposed without implying losses for LWE is equal to the average cost of production: AC = TC/Q = (500/Q) + 1 Imposing P = AC and solving for Q we get the production level: P = 50 – 0.1Q = AC = (500/Q) + 1

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D. Vannoni e M. Piacenza Microeconomia C, A.A. 2007-2008 Esercitazione 4 11 Solving for Q we get the quadratic equation: 0.1Q 2 – 49Q + 500 = 0 In our case:

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D. Vannoni e M. Piacenza Microeconomia C, A.A. 2007-2008 Esercitazione 4 12 Notice that: a) If Q = 10.4 MR >MC therefore the firm will increase profits if quantity increases; b) In correspondence of the second solution, since the quantity (Q = 479.6) is lower than the quantity for which P = MC (Q = 490), the price will be higher (2.04) and the consumer's surplus lower. The consumer's surplus is: CS = 0.5 ∙ (50 – 2.04) ∙ 479.6 = 11500 $ Finally, the deadweight loss (in violet color) is given by: P.S. = (2.04 - 1) ∙ (490 – 479.6) ∙ 0.5 = 5.4 $

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D. Vannoni e M. Piacenza Microeconomia C, A.A. 2007-2008 Esercitazione 4 13

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D. Vannoni e M. Piacenza Microeconomia C, A.A. 2007-2008 Esercitazione 4 14 Exercise n. 3 Two firms manufacture the same product. Firm 1 has a marginal cost of 1 $ and firm 2 has a marginal cost of 2 $. Fixed costs are zero for both firms. The inverse demand is P = 6 - 0.01Q (Q = q 1 + q 2 ) Cournot equilibrium a)Compute the reaction functions and find the Cournot equilibrium 2Stackelberg b) Assume that firm 2 is a Stackelberg leader and find the new equilibrium

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D. Vannoni e M. Piacenza Microeconomia C, A.A. 2007-2008 Esercitazione 4 15 a)Profits of firm 1 and 2 are: 1 = (6 - 0.01·q 1 - 0.01·q 2 )·q 1 - q 1 2 = (6 - 0.01·q 2 - 0.01·q 1 )·q 2 - 2·q 2 Maximizing profits: d /dq 1 = ’ 1 = 6 - 0.02·q 1 - 0.01·q 2 - 1 = 0 q 1 = 250 - ½ q 2 from which q 1 = 250 - ½ q 2 (inverting: q 2 = 500 - 2q 1 ) d /dq 2 = ’ 2 = 6 - 0.02·q 2 - 0.01·q 1 - 2 = 0 q 2 = 200 - ½ q 1 from which q 2 = 200 - ½ q 1

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D. Vannoni e M. Piacenza Microeconomia C, A.A. 2007-2008 Esercitazione 4 16 The equilibrium point is where the reaction functions intersect: 500 - 2q 1 = 200 - ½ q 1 q 1 = 300/1.5 = 200; q 2 = 500 - 2·200 = 100 p = 6 - 2 -1 = 3 1 = 600 - 200 = 400 and 2 = 300 – 200 = 100

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D. Vannoni e M. Piacenza Microeconomia C, A.A. 2007-2008 Esercitazione 4 17 Q2Q2 Q1Q1 Reaction function firm 1 500 250 Reaction function firm 2 200 400 100 200 Cournot Equilibrium

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D. Vannoni e M. Piacenza Microeconomia C, A.A. 2007-2008 Esercitazione 4 18 Stackelberg model, firm 2 (leader) b) In the Stackelberg model, firm 2 (leader) includes in its profit function the reaction function of firm 1: q 1 2 = (6 - 0.01·q 2 - 0.01·q 1 )·q 2 - 2·q 2 = 250 - ½ q 2 = [6 - 0.01·q 2 - 0.01·(250 - ½ q 2 )]·q 2 - 2·q 2 ’ 2 = 6 - 0.01·q 2 - 2.5 - 2 = 0 from which q 2 = 1.5/0.01 = 150 and q 1 = 250 – 75 = 175 p = 6 - 1.5 - 1.75 = 2.75 1 = (2.75 - 1)·175 = 306 and 2 = (2.75 - 2)·150 = 112.5 It exists a so called first mover advantage

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D. Vannoni e M. Piacenza Microeconomia C, A.A. 2007-2008 Esercitazione 4 19 Q2Q2 Q1Q1 Reaction function Firm 1 500 250 Reaction function Firm 2 200 400 100 200 Cournot Equilibrium Stackelberg Equilibrium 175 150

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D. Vannoni e M. Piacenza Microeconomia C, A.A. 2007-2008 Esercitazione 4 20 Exercise n. 4 In an oligopoly there are 4 firms manufacturing an homogenous good with the following marginal costs c1=10, c2=10, c3=12, c4=11. The firms start to collude and, after a recession, adopt price strategies à la Bertrand. a) Which will be the equilibrium prevailing after the price war? b) Is it possible to find equilibrium quantities and prices? c) If firm 1 and firm 2 are colluding, which will be the market price?

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D. Vannoni e M. Piacenza Microeconomia C, A.A. 2007-2008 Esercitazione 4 21 a)At the end of the price war, only firm 1 and 2 will remain (lowest marginal costs); a)The equilibrium price will be equal to 10, while the quantity in undetermined. A usual assumption is to divide it equally among firms. a)A collusive agreement will allow firm 1 and 2 to fix a price equal (or slightly lower than) 11, i.e. firm 4's marginal cost.

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D. Vannoni e M. Piacenza Microeconomia C, A.A. 2007-2008 Esercitazione 4 22 Exercise n. 5 Two firms compete using price as strategic variable. The demand functions are: Q 1 = 20 – P 1 + P 2 Q 2 = 20 + P 1 – P 2 Marginal costs are equal to zero.

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D. Vannoni e M. Piacenza Microeconomia C, A.A. 2007-2008 Esercitazione 4 23 a) Suppose that the two firms are simultaneously chosing the price. Find the Nash equilibrium. Which will be the price, quantities and profits? In order to find the Nash equilibrium we compute for each firm the reaction function with respect to the price of the other firm, then we solve the system of two equations with two unknown variables. With marginal costs equal to zero, the profits are: 1 = P 1 Q 1 = P 1 (20 - P 1 + P 2 ) = 20P 1 - P 1 2 + P 2 P 1 ’ 1 = MR 1 = 20 - 2P 1 + P 2

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D. Vannoni e M. Piacenza Microeconomia C, A.A. 2007-2008 Esercitazione 4 24 Al the profit maximizing price MR 1 = 0, from which we get the reaction function of firm 1: P 1 = (20 + P 2 )/2 2: Symmetrically for firm 2: P 2 = (20 + P 1 )/2. Substituting one reaction function into the other: P 1 = [20 + (20 + P 1 )/2]/2 = 15 + P 1 /4P 1 = 20 $ Due to symmetry, P 2 = 20 $ Q 1 = 20, Q 2 = 20 and 1 = 2 = 400

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D. Vannoni e M. Piacenza Microeconomia C, A.A. 2007-2008 Esercitazione 4 25

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D. Vannoni e M. Piacenza Microeconomia C, A.A. 2007-2008 Esercitazione 4 26 If firm 1 chooses the price before firm 2, it can take into account the reaction function of firm 2. The profit of firm 1 is: 1 = P 1 Q 1 = P 1 [20 - P 1 + (20 + P 1 )/2] To determine the price which maximizes the profit: 1 ’ = 20 - 2P 1 + 10 + P 1 Imposing 1 ’= 0 P 1 = 30 $ b) Suppose now that firm 1 chooses as first the price, followed by firm 2. Which will the the equilibrium prices, quantities and profits?

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D. Vannoni e M. Piacenza Microeconomia C, A.A. 2007-2008 Esercitazione 4 27 Substituting in the reaction function of firm 2, we get P 2 = (20 + 30)/2 = 25 $. In correspondence of such prices we get: Q 1 = 20 – 30 + 25 = 15 Q 2 = 20 + 30 - 25 = 25 Profits are therefore: 1 = 30·15 = 450 $ 2 = 25·25 = 625 $ If firm 1 fixes its price acting as the first mover, firm 2 can exploit this and gain more profits (second mover advantage)

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D. Vannoni e M. Piacenza Microeconomia C, A.A. 2007-2008 Esercitazione 4 28

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D. Vannoni e M. Piacenza Microeconomia C, A.A. 2007-2008 Esercitazione 4 29 c) In this example of price competition with differentiated products, which situation will you prefer? (i) two firms choosing simultaneously the price; (ii) you are the first choosing the price; (iii) your competitor is the first mover choosing the price Looking at the profits attainable in the different cases, the ranking should be (iii), followed by (ii) and (i).

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