CrystalWater Pty Ltd and ClearWater Pty Ltd are the only producers of spring wat

Question

CrystalWater Pty Ltd and ClearWater Pty Ltd are the only producers of spring water in the local market. The market demand for spring water is given by P = 70 – Q1 – Q2. CrystalWater and ClearWater compete by choosing quantities Q1 and Q2 respectively. Each firm has a marginal cost of $10 and no fixed cost.

a. Find Crystalwater’s and ClearWater’s reaction functions.
b. Suppose that the two firms choose their quantities supplied to the market simultaneously. What are the equilibrium price, quantities, and profits of the two firms in this market?
c. Now suppose that only CrystalWater has a chance to bribe the government and get the right to choose first the quantity it supplies. What is the maximum amount of money that CrystalWater is willing to pay as a bribe? Assuming Crystalwater does choose first the quantity its supplies, what are the equilibrium quantities and profits of CrsytalWater and ClearWater?
d. Assume that both firms are at the position described in part (b). Suppose that the firms decide to collude and share any profit equally. Both firms value their reputation and will conform to their agreement. What are the quantities supplied and profit achieved by each firm?
e. Continuing from part (d), both firms are now less tolerant of each other and willing to take advantage of the other. Foreseeing this change, they make a legally enforceable contract stating that if a firm does not produce the quantity agreed, it must pay some penalty to the other firm. What is the minimum amount of penalty that ensures each firm produces the quantity agreed in part (d)?

Explanation / Answer

a. This is a symmetric Cournot game. Let's call Y profit. Y1 will be the profit for firm 1 and Y2 will be the profit for firm 2. C is marginal cost. Y1 = (P-C)*Q1 Y1 = (70 - Q1 - Q2 - 10)*Q1 Y1 = (60 - Q1 - Q2)*Q1 Take the derivative to fine the marginal profit function. We'll call it MY1. It should equal zero at the maximum. MY1= 60 - 2Q1 - Q2 = 0 Now we can solve for firm 1's optimal Q1 in terms of Q2. Q1 = 30 - (1/2)Q2 This is firm 1's "reaction function." Since firm 2 has the same demand curve and marginal cost, firm 2's reaction function is: Q2 = 30 - (1/2)Q1 You can prove this by doing the same process as the one above for firm 2. This is a good exercise. b. If quantities are chosen simultaneously, then we can simply substitute one reaction function into another to find the equilibrium quantities. Q1 = 30 - (1/2)Q2 Q1 = 30 - (1/2)[30 - (1/2)Q1] Q1 = 30 - 15 + (1/4)Q1 Q1(1-(1/4))=15 Q1 = 15/(3/4) Q1 = 20 This equilibrium is "symmetric" because each firm has the same reaction function. This implies that the optimal quantity for firm 2 is equal to the optimal quantity for firm 1. So, Q2 = 20. You can prove this by following the same steps for firm 2 as we did above. This is a good exercise. Now, we simply substitute these quantities into the price function given to us. P = 70 – Q1 – Q2 P = 70 – 20 – 20 P = 30 Now, to find the profit, we substitute our values into the profit function Y1 = (P-C)*Q1 Y1 = (30-20)*20 Y1 = 200 Again, since this is a symmetric equilibrium, Y2 = 200. c. This is a Stackleburg game. This asks us to determine the Stackleburg equilibrium. The first firm will operate as a monopoly. Let's assume firm 1 enters first. P = 70 – Q1 Y1 = (P-C)*Q1 Y1 = (70 – Q1 - 20)*Q1 Y1 = (50 – Q1)*Q1 MY1 = 50 - 2Q1 = 0 Q1 = 25 The second firm will still have the same reaction function. Q2 = 30 - (1/2)Q1 Q2 = 30 - (1/2)25 Q2 = 17.5 Thus, we can calculate the Stackleburg duopoly price (say that three times fast.) P = 70 – Q1 – Q2 P = 70 – 25 – 17.5 P = 27.5 And we can calculate the Stackleburg profits, which are non-symmetric because the firms are different. One entered before the other. Y1 = (P-C)*Q1 Y1 = (27 - 20)*27.5 Y1 = 192.5 Y2 = (P-C)*Q2 Y2 = (27 - 20)*17.5 Y2 = 122.5 The maximum bribe "B" is the difference in profit between entering first and second. B = Y1 - Y2 B = 192.5 - 122.5 B = 70 Notice that both firms are better off in the simultaneous Cournot equilibrium. This indicates that, given time, each firm will adjust to the Cournot equilibrium in the long run. The Stackleburg result is only a subgame perfect Nash equilibrium. d. Here we assume that the two firms collude. This means that they, together, behave as a monopoly. We can denote the total quantity produced as Q and the total profit as Y. P = 70 – Q Y = (P-C)*Q Y = (70 – Q - 20)*Q Y = (50 – Q)*Q MY = 50 - 2Q = 0 Q = 25 This implies that Q1 = Q2 = 12.5 P = 70 – Q P = 70 – 25 P = 45 Y = (P-C)*Q Y = (45-20)*25 Y = 625 This implies Y1 = Y2 = 312.5 Notice that this is much greater than the profits that the firms would make in the Cournot equilibrium. e. If the firms collude, then each will make a profit of 312.5. But what if only one of them cheats? Let's say only firm 2 cheats. This means Q1 = 12.5 but firm 2 still has the reaction function: Q2 = 30 - (1/2)Q1 Q2 = 30 - (1/2)12.5 Q2 = 23.75 If firm 2 cheats, it will produce 23.75. Then, the price will be: P = 70 - Q1 - Q2 P = 70 - 12.5 - 23.75 P = 33.75 and firm 2's profits from cheating are: Y2 = (P-C)*Q2 Y2 = (33.75-20)*23.75 Y2 = 326.5625 So, the minimum penalty "F" that will prevent a firm from cheating is the difference in profits between cheating and colluding. F = Y2(cheat) - Y2(collude) F = 326.5625 - 312.5 F = 14.0625

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