Under the condition the two cournot’s duopolistic firms have identical costs, assume that the market demand and cost functions are given as follows;
P=60-Q where Q=Q1+Q2
MC1=MC2=0
Where
Q1 is the output of firm 1
Q2 is the output of firm 2
P is the price of the market
MC1 is the marginal cost of firm 1
MC2 is the marginal cost of firm 2
By showing each of the formulas and steps required, calculate
A. The profit maximizing equilibrium out put of firm 1 and 2
B. The profit maximizing equilibrium price of firm 1 and 2
To calculate the profit maximizing equilibrium output and price for firms 1 and 2 in a Cournot duopoly, we need to follow a few steps.
Step 1: Find the total market demand equation
The given market demand equation is P = 60 - Q, where Q is the total output (Q = Q1 + Q2) and P is the price. This equation represents the demand curve for the whole market.
Step 2: Determine the reaction function of each firm
The reaction function shows how each firm's output depends on the output of the other firm. In a Cournot model, each firm assumes that the other firm's output is fixed and aims to maximize its own profit.
Firm 1's reaction function: Firm 1 will maximize its profit by setting its output level Q1, given Q2. The profit function (π1) for firm 1 is given by:
π1 = (P - MC1) * Q1
But we know P = 60 - Q, and MC1 = 0.
So, π1 = (60 - Q - 0) * Q1 = (60 - Q) * Q1
To find the reaction function, we differentiate π1 with respect to Q1 and set it equal to zero:
dπ1 / dQ1 = 60 - 2Q1 - Q = 0
Solving for Q1, we get Q1 = (60 - Q) / 2
Similarly, we can find the reaction function for firm 2:
Firm 2's reaction function: Q2 = (60 - Q) / 2
Step 3: Solve for the equilibrium output and price
To find the equilibrium output, we set the two reaction functions equal to each other:
Q1 = Q2
(60 - Q) / 2 = Q
Solving for Q, we get Q = 40.
Substituting this value back into the reaction functions, we find Q1 = Q2 = 20.
To find the equilibrium price, we substitute Q = 40 into the demand equation:
P = 60 - Q
P = 60 - 40
P = 20
Therefore, the answers to the questions are:
A. The profit maximizing equilibrium output for firm 1 and 2 is Q1 = Q2 = 20.
B. The profit maximizing equilibrium price for firm 1 and 2 is P = 20.