To find the marginal revenue functions and reaction functions for firm 1 and firm 2 in this Cournot duopoly, we need to follow these steps:
(a) Marginal Revenue Functions:
The marginal revenue (MR) function for a firm operating in a Cournot duopoly is calculated as follows:
MR = d(TR)/dQ
where TR is the total revenue and Q is the quantity produced by the respective firm.
Total revenue (TR) is given by:
TR = P * Q
where P is the price and Q is the quantity produced.
We know that the demand function is:
Q = 120 - 4P
Since there are two firms in the market, the total quantity produced, Q, is the sum of the quantities produced by both firms:
Q = Q1 + Q2
where Q1 is the quantity produced by firm 1, and Q2 is the quantity produced by firm 2.
Now, we can calculate the price P in terms of Q:
Q = 120 - 4P
4P = 120 - Q
P = 30 - 0.25Q
(b) Reaction Functions:
The reaction function of each firm represents the optimal quantity that a firm should produce, taking into account the output of the other firm. Each firm aims to maximize its profit based on the assumption that the other firm's output remains constant.
For firm 1:
The profit function (π1) for firm 1 is given by:
π1 = TR1 - TC1
π1 = P * Q1 - MC * Q1
π1 = (30 - 0.25Q1 - 0.25Q2) * Q1 - 3 * Q1
π1 = (30Q1 - 0.25Q1^2 - 0.25Q1Q2) - 3Q1
To find the reaction function for firm 1, we need to maximize its profit with respect to Q1:
∂π1/∂Q1 = 30 - 0.5Q1 - 0.25Q2 - 3 = 0
0.5Q1 = 27 - 0.25Q2
Q1 = 54 - 0.5Q2
For firm 2:
Following the same steps, the profit function (π2) for firm 2 is:
π2 = (30Q2 - 0.25Q1Q2 - 0.25Q2^2) - 3Q2
To find the reaction function for firm 2, we need to maximize its profit with respect to Q2:
∂π2/∂Q2 = 30 - 0.25Q1 - 0.5Q2 - 3 = 0
0.5Q2 = 27 - 0.25Q1
Q2 = 54 - 0.5Q1
These are the reaction functions for both firms. They represent the optimal quantity each firm should produce given the quantity produced by the other firm. To find the Cournot equilibrium, we need to solve these two reaction functions simultaneously.