Suppose there are two firms (called A and B) that compete in a Cournot market. Market demand is given by the equation P = 120 - ¼(QA + QB). Let the marginal costs for both firms be MCA = 30 and MCB = 30, which means that costs equal 30Q.
If firm B sets its output to QB = 0, we need to determine firm A's marginal revenue, optimal output, market prices, and monopoly profits.
To find firm A's marginal revenue, we need to differentiate the market demand equation with respect to QA:
MR = d(P)/d(QA) = 120 - ¼(QA + QB) - ¼(dQA/dQA) = 120 - ¼(QA + QB) - ¼(1) = 120 - ¼(QA + QB) - ¼ = 120 - ¼(QA + QB) - ¼ = 120 - ¼(QA + 0) - ¼ = 120 - ¼QA - 0 = 120 - ¼QA
The optimal output for firm A can be found by setting marginal revenue equal to marginal cost:
MR = MC
120 - ¼QA = 30
- ¼QA = 30 - 120
- ¼QA = -90
QA = (-90)/(- ¼)
QA = 360
The market price can be found by substituting the optimal output for firm A into the market demand equation:
P = 120 - ¼(QA + QB)
P = 120 - ¼(360 + 0)
P = 120 - ¼(360)
P = 120 - 90
P = 30
To calculate firm A's monopoly profits, we need to find the difference between total revenue and total costs:
Total revenue = P * QA
Total revenue = 30 * 360
Total revenue = 10,800
Total costs = MC * QA
Total costs = 30 * 360
Total costs = 10,800
Monopoly profits = Total revenue - Total costs
Monopoly profits = 10,800 - 10,800
Monopoly profits = 0
If firm B sets its output to QB = 360, we need to determine the optimal QA.
Using firm A's reaction function, QA = 180 - ½QB:
QA = 180 - ½(360)
QA = 180 - 180
QA = 0
Firm A's optimal output is QA = 0.
To sketch both firm's reaction functions, we plot QA on the y-axis and QB on the x-axis. Firm A's reaction function is QA = 180 - ½QB, and firm B's reaction function is QB = 180 - ½QA. The resulting graph will show symmetric reaction functions due to identical costs for both firms.
To find the Cournot Nash Equilibrium outputs QA and QB and the market price (Pcournot), we need to solve the reaction functions simultaneously:
QA = 180 - ½QB
QB = 180 - ½QA
Substituting the second equation into the first equation:
QA = 180 - ½(180 - ½QA)
QA = 180 - 90 + ¼QA
¾QA = 90
QA = (90)/(¾)
QA = 120
Substituting the value of QA into the second equation:
QB = 180 - ½(120)
QB = 180 - 60
QB = 120
The Cournot Nash Equilibrium outputs are QA = 120 and QB = 120. To find the market price, we substitute these values into the market demand equation:
Pcournot = 120 - ¼(QA + QB)
Pcournot = 120 - ¼(120 + 120)
Pcournot = 120 - ¼(240)
Pcournot = 120 - 60
Pcournot = 60
To calculate the profits for each firm (πA and πB), we need to find the difference between total revenue and total costs for each firm:
Total revenue for firm A = Pcournot * QA
Total revenue for firm A = 60 * 120
Total revenue for firm A = 7,200
Total costs for firm A = MC * QA
Total costs for firm A = 30 * 120
Total costs for firm A = 3,600
Ï€A = Total revenue for firm A - Total costs for firm A
Ï€A = 7,200 - 3,600
Ï€A = 3,600
Total revenue for firm B = Pcournot * QB
Total revenue for firm B = 60 * 120
Total revenue for firm B = 7,200
Total costs for firm B = MC * QB
Total costs for firm B = 30 * 120
Total costs for firm B = 3,600
Ï€B = Total revenue for firm B - Total costs for firm B
Ï€B = 7,200 - 3,600
Ï€B = 3,600
The total profits (πA + πB) in the Cournot equilibrium are 3,600 + 3,600 = 7,200. This is less than if the firms acted as a monopoly because in a monopoly, there would be no competition, allowing the monopolistic firm to charge a higher price and earn higher profits.