Let's break down this problem step by step. ### **Step 1: Understanding the Problem** - There are 6 firms. - All firms charging **monopoly price**: Each firm's profit = $120,000 / 6 = $20,000 per month. - All firms charging **prevailing (competitive) price**: Each firm's profit = $60,000 / 6 = $10,000 per month. We are to find the **maximum monthly discount rate** (let's call it \( r \)), such that **collusion** (monopoly pricing by all) is still stable — i.e., each firm prefers to stick to the monopoly price rather than deviate and trigger a price war (reverting to the competitive price forever). ### **Step 2: Incentive Compatibility Condition** A firm will stick to the monopoly price if the present value of **cooperating** is at least as large as the present value of **deviating**. #### **Cooperation (Collusion) Payoff:** - Each month: $20,000 (monopoly profit) - Present value of infinite stream: \( V_C = \frac{20,000}{r} \) #### **Deviation Payoff:** - In the month of deviation: The deviator undercuts and gets a higher profit (not specified, but let's assume the classic "cheat" gets the whole market profit for one month, then gets the competitive profit forever after). - **One-period deviation gain:** In the deviation month, the deviator gets the monopoly profit for the whole market: $120,000 - **After deviation:** All firms revert to competitive price: $10,000 per month forever. So, present value for deviator: \[ V_D = \$120,000 + \frac{10,000}{r} \] ### **Step 3: Set up the Incentive Constraint** A firm will not deviate if: \[ V_C \geq V_D \] \[ \frac{20,000}{r} \geq 120,000 + \frac{10,000}{r} \] ### **Step 4: Solve the Inequality** Subtract \(\frac{10,000}{r}\) from both sides: \[ \frac{20,000}{r} - \frac{10,000}{r} \geq 120,000 \] \[ \frac{10,000}{r} \geq 120,000 \] Now, solve for \( r \): \[ \frac{10,000}{120,000} \geq r \] \[ r \leq \frac{1}{12} \] \[ r \leq 0.0833 \] ### **Step 5: Final Answer** **The maximum monthly discount rate is \(\boxed{0.0833}\) or \(\boxed{8.33\%}\) per month.** --- #### **Summary Table** | Scenario | Firm Profit (per month) | Present Value (infinite horizon, monthly discount rate \( r \)) | |-------------------|------------------------|---------------------------------------------------------------| | Collusion | $20,000 | \( \frac{20,000}{r} \) | | Deviation (cheat) | $120,000 (1 mo), then $10,000 | \( 120,000 + \frac{10,000}{r} \) | **Maximum discount rate for collusion to be stable:** \[ \boxed{r \leq 8.33\% \text{ per month}} \] --- **Explanation:** If the discount rate is higher than 8.33% per month, firms are too impatient — they'd rather grab the short-term gain from deviation than wait for the future collusive profits. If it's at or below 8.33%, collusion is stable.