Consider an industry with two firms, each having marginal costs and total costs equal to zero. The industry demand is P = 100 − Q where Q = Q1 + Q2 is total output. 1. Find the cartel output and cartel profits assuming that the firms share the profit equally. Hint: In cartels, firms behave as if they are a monopoly. Hence, the cartel quantity is at the point where MR = MC. After finding the quantity, use the demand curve to find the cartel price. And then calculate Π = T R − T C. Divide the total profit by 2 to find each firm’s profit. 2. If each firm behaves as a Cournot competitor, what is firm 1’s optimal output given firm 2’s output? Hint: This part is asking the best response function of firm 1. Solve firm 1’s profit maximizatin problem by setting its MC = MR. Then, express Q1 as a function of Q2. 3. Calculate the Cournot equilibrium output and profit for each firm. Hint: You have already solved firm 1’s problem above. Now solve firm 2’s problem. Then, solve BR functions simultaneously to get Q1 and Q2. Use the demand function to find the equilibrium price and then calculate Π = T R − T C. 4. If the firms interact only once, is there a profitable deviation (cheating) from cartel for any firm? Find the profit of the firms in case of cheating. Hint: If a firm deviates from cartel (cheats) it will play its profit maximizing quantity, which is the quantity on its best response function. Now, use the best response function of firm 1 to find firm 1’s cheating quantity, when firm 2 follows the rules of the cartel. Use the demand curve to find the new price level and then calculate profits. 5. Design a 2x2 normal form payoff matrix with strategies cartel and cheat. Complete the payoffs using the profits you calculated in the previous parts. Cartel Cheat Cartel Cheat 6. Now, assume that players interact twice. Hence the game is a twice repeated game. Is it possible to have (cartel,cartel) as an outcome? Explain. 7. Now, assume that players interact infinitely. Hence the game is an infinitely repeated game. Is it possible to have cartel as an outcome using tit-for-tat? If yes, give the condition on the discount factor for which cartel is sustainable. Hint: Cartel is a sustainable outcome of the repeated game if while firm 2 plays grim-trigger, there exists no profitable one-shot deviation from tit-for-tat for firm 1. Sicne the firms are identical, you don’t need to check the profitable deviations of firm 2. They will be the same. Let’s start with completing the table below. Keep in mind that firm 2 is following tit-for-tat and firm 1 is deviating from tit-for-tat only in period 2. I have already completed the first 2 rows Period Action: Firm 1 Action: Firm 2 Payoff of Firm 1 without Deviation Payoff of Firm with One-Shot Deviation 1 Cartel Cartel 2 Cheat Cartel 3 4 5 ... ... 8. Now, using the discount factor δ calculate the total payoffs from the two cases. You will need to use formulas for infinite series. Now, find δ values which make the total payoff from no deviation bigger than the total payoff from one-shot deviation. For these values of the discount factor, firm 1 will not deviate from tit-for-tat. Since firms are identical, the same condition applies to firm 2 as well.
Consider an industry with two firms, each having marginal costs and total costs equal to zero.
The industry demand is P = 100 − Q where Q = Q1 + Q2 is total output.
1. Find the cartel output and cartel profits assuming that the firms share the profit equally.
Hint: In cartels, firms behave as if they are a
2. If each firm behaves as a Cournot competitor, what is firm 1’s optimal output given firm 2’s output?
Hint: This part is asking the best response function of firm 1. Solve firm 1’s profit maximizatin problem by setting its MC = MR. Then, express Q1 as a function of Q2.
3. Calculate the Cournot equilibrium output and profit for each firm.
Hint: You have already solved firm 1’s problem above. Now solve firm 2’s problem. Then, solve BR functions simultaneously to get Q1 and Q2. Use the demand function to find the
4. If the firms interact only once, is there a profitable deviation (cheating) from cartel for any firm? Find the profit of the firms in case of cheating.
Hint: If a firm deviates from cartel (cheats) it will play its profit maximizing quantity, which is the quantity on its best response function. Now, use the best response function of firm 1 to find firm 1’s cheating quantity, when firm 2 follows the rules of the cartel. Use the demand curve to find the new price level and then calculate profits.
5. Design a 2x2 normal form payoff matrix with strategies cartel and cheat. Complete the payoffs using the profits you calculated in the previous parts.
| Cartel | Cheat | |
| Cartel | ||
| Cheat |
6. Now, assume that players interact twice. Hence the game is a twice repeated game. Is it possible to have (cartel,cartel) as an outcome? Explain.
7. Now, assume that players interact infinitely. Hence the game is an infinitely repeated game. Is it possible to have cartel as an outcome using tit-for-tat? If yes, give the condition on the discount factor for which cartel is sustainable.
Hint: Cartel is a sustainable outcome of the repeated game if while firm 2 plays grim-trigger, there exists no profitable one-shot deviation from tit-for-tat for firm 1. Sicne the firms are identical, you don’t need to check the profitable deviations of firm 2. They will be the same. Let’s start with completing the table below. Keep in mind that firm 2 is following tit-for-tat and firm 1 is deviating from tit-for-tat only in period 2. I have already completed the first 2 rows
| Period | Action: Firm 1 | Action: Firm 2 | Payoff of Firm 1 without Deviation | Payoff of Firm with One-Shot Deviation |
| 1 | Cartel | Cartel | ||
| 2 | Cheat | Cartel | ||
| 3 | ||||
| 4 | ||||
| 5 | ||||
| ... | ||||
| ... |
8. Now, using the discount factor δ calculate the total payoffs from the two cases. You will need to use formulas for infinite series. Now, find δ values which make the total payoff from no deviation bigger than the total payoff from one-shot deviation. For these values of the discount factor, firm 1 will not deviate from tit-for-tat. Since firms are identical, the same condition applies to firm 2 as well.
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