Two firms (Firm 1 and Firm 2) compete in selling identical products. They choose their outputlevels simultaneously and face the inverse market demand curve given by P = 100 – Q, whereQ = Q1 + Q2 . Q is the total market quantity produced, while Q1 and Q2 represent the outputsproduced by the two firms, respectively. The total cost functions for firm 1 and firm 2 are givenby TC(Q1 ) = 40Q1 and TC(Q2 ) = 40Q2 , respectively.2.1 Determine the Cournot-Nash equilibrium in this market. [9]2.2 Calculate the profit (loss) for each firm.
Question
Two firms (Firm 1 and Firm 2) compete in selling identical products. They choose their outputlevels simultaneously and face the inverse market demand curve given by P = 100 – Q, whereQ = Q1 + Q2 . Q is the total market quantity produced, while Q1 and Q2 represent the outputsproduced by the two firms, respectively. The total cost functions for firm 1 and firm 2 are givenby TC(Q1 ) = 40Q1 and TC(Q2 ) = 40Q2 , respectively.2.1 Determine the Cournot-Nash equilibrium in this market. [9]2.2 Calculate the profit (loss) for each firm.
Solution
2.1 To determine the Cournot-Nash equilibrium, we first need to find the reaction functions of the two firms.
The profit function for each firm is given by:
π1 = P*Q1 - TC(Q1) = (100 - Q)Q1 - 40Q1 = 60Q1 - Q1^2 - Q1Q2
π2 = P*Q2 - TC(Q2) = (100 - Q)Q2 - 40Q2 = 60Q2 - Q2^2 - Q1Q2
Taking the derivative of the profit functions with respect to Q1 and Q2 and setting them equal to zero gives us the reaction functions:
dπ1/dQ1 = 60 - 2Q1 - Q2 = 0 => Q1 = 30 - 0.5Q2
dπ2/dQ2 = 60 - 2Q2 - Q1 = 0 => Q2 = 30 - 0.5Q1
Solving these two equations simultaneously gives us the Cournot-Nash equilibrium quantities:
Q1* = Q2* = 20
2.2 To calculate the profit for each firm, we substitute the equilibrium quantities into the profit functions:
π1* = (100 - 2*20)20 - 4020 = 800
π2* = (100 - 2*20)20 - 4020 = 800
So, each firm makes a profit of 800 in the Cournot-Nash equilibrium.
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