Consider an infinitely repeated Cournot duopoly with discount factor delta <1, unit costs of c>0, and inverse demand
functions p(Q)=a-bQ, with a>c and b>0. Find the condition on the discount factor,delta , for which the two firms could
successfully collude over the monopoly output and hence share the monopoly profit using trigger strategies.
To find the condition on the discount factor delta for successful collusion in the infinitely repeated Cournot duopoly, we need to analyze the trigger strategy.
In a trigger strategy, firms commit to certain actions based on the history of previous actions taken by their competitors. This helps sustain collusion by deterring any deviations from the collusive outcome.
Here's how we can proceed to find the condition while explaining the steps:
Step 1: Set up the problem
- Let's assume Firm 1 and Firm 2 are the two duopolists.
- The inverse demand function is given as p(Q) = a - bQ, where Q is the total quantity produced.
- Both firms have the same unit costs, c > 0.
- The discount factor is represented as delta, where 0 < delta < 1.
Step 2: Determine the monopoly outcome
- In a collusion scenario, the duopoly would act as a monopolist and maximize joint profits.
- The monopoly output is found by setting the total quantity produced (Q) equal to the output that maximizes joint profits.
- To find this output, we need to take the derivative of the joint profit function with respect to Q and set it equal to zero.
Step 3: Calculate monopoly profit
- Once we find the monopoly output, we can substitute it back into the inverse demand function to get the corresponding price (p).
- The monopoly profit is calculated by multiplying the price (p) by the monopoly output (Q).
Step 4: Analyze deviation cases
- Any deviation from the collusion strategy can adversely impact the collusive outcome.
- Analyze the possible situations where a firm chooses to deviate and produces more or less than the collusion output.
- Calculate the resulting profits for both firms in each deviation case.
Step 5: Compare payoffs and set trigger strategies
- Compare the profits in each deviation case to the collusive profit.
- If the collusive profit is higher than the profits from any deviation case, the trigger strategies can be sustained successfully.
- Find the condition on the discount factor (delta) that ensures the collusive profit is higher.
By following these steps, you can calculate the condition on the discount factor delta for successful collusion in the infinitely repeated Cournot duopoly using trigger strategies.