ok so i must have not payed attention in class when we talked about asymmetric costs in cournot duopoly's, so i am stuck on a homework problem. The problem reads as follows (*Note that my notation of c simply means Marginal Cost):
Consider a Cournot duopoly where inverse demand is P(Q) = a - Q but firms, 1 and 2, have asymmetric marginal costs, c1 and c2. What is the Nash equilibrium if 0 <ci < a/2 for each firm? What if c1 < c2 < a but 2c2 > a + c1?
Basically, I can get it down to each firm's individual reaction function's, but I am always used to assuming that the firms are identical (thus having the same marginal costs). I just wanted to maybe know the algebra behind this or simply the intution. If anyone could help that would be cool.
It has been 25 years since I needed to look at Cournot models. But, I remember that things got complicated in the case where the firms had different cost structures.
The intuition behind the Cournot model can be found in a Nash Equilibrium.
Sorry I cant be more helpful.
Sure, I can help you understand the concept of asymmetric costs in Cournot duopoly and guide you through solving the homework problem.
In Cournot duopoly, two firms compete in quantities rather than prices. Each firm chooses its quantity to maximize its profits, taking into account the quantities chosen by the other firm.
To find the Nash equilibrium, we need to determine the quantity that each firm will produce given their marginal costs and the inverse demand function.
In this case, the inverse demand function is given by P(Q) = a - Q, where Q is the total quantity produced by both firms and a is a constant representing the demand.
Now, let's focus on the two scenarios mentioned in the problem:
1. When 0 < ci < a/2 for each firm:
- To find the reaction functions, we need to determine how each firm's quantity choice depends on the other firm's quantity choice.
- Firm 1's reaction function can be derived by maximizing its profit with respect to its own quantity (Q1) while assuming Firm 2's quantity (Q2) remains constant.
- The profit function for Firm 1 is given by:
π1 = (P(Q) - c1) * Q1
Substitute P(Q) = a - Q and solve for Q1 to obtain Firm 1's reaction function.
- Similarly, derive Firm 2's reaction function assuming Firm 1's quantity (Q1) remains constant.
- Once you have the reaction functions for both firms, solve them simultaneously to find the Nash equilibrium, which corresponds to the intersection of the reaction functions.
2. When c1 < c2 < a, but 2c2 > a + c1:
- The steps to find the reaction functions remain the same as in the previous scenario.
- However, the conditions on the marginal costs imply that Firm 2 has a cost advantage over Firm 1, and both firms have costs below the equilibrium price level.
- In this case, you can expect Firm 2 to produce more than Firm 1, reflecting its cost advantage.
- Find the Nash equilibrium by solving the reaction functions as before.
To summarize, to solve these types of problems, you need to derive the reaction functions by maximizing each firm's profit function and then find their intersection to determine the Nash equilibrium. The conditions on the marginal costs will guide your intuition about the equilibrium outcomes.
I hope this explanation helps you understand how to approach and solve the problem.