Two firms produce the same good and compete against each other in a Cournot market. The market demand for their product is P = 204 - 4Q, and each firm has a constant marginal cost of $12 per unit. MR1 = 204 - 8q1 - 4q2. Let q1 be the output produced by firm i, where i = 1,2.
1. Firm 1's reaction function is:
2. In the Cournot equilibrium for this market, each firm will produce ____ unites of output, and the market price will be___.
3. Each firm will earn a profit of ___.
"Two firms produce the same good and compete against each other in a Cournot market. The market demand for their product is P = 204 - 4Q, and each firm has a constant marginal cost of $12 per unit. MR1"
1. To find Firm 1's reaction function, we need to maximize its profit by choosing the optimal output level, q1. Firm 1's profit can be calculated as follows:
Profit1 = TR1 - TC1
Total revenue (TR1) is calculated by multiplying the firm's output (q1) by the market price (P):
TR1 = q1 * P
= q1 * (204 - 4Q)
= q1 * (204 - 4(q1 + q2))
= 204q1 - 4q1^2 - 4q1q2
Total cost (TC1) is calculated by multiplying the firm's output (q1) by the marginal cost (MC):
TC1 = q1 * MC
= q1 * 12
= 12q1
Profit1 = TR1 - TC1
= 204q1 - 4q1^2 - 4q1q2 - 12q1
= 192q1 - 4q1^2 - 4q1q2
To find Firm 1's reaction function, we differentiate this profit function with respect to q1 and set it equal to zero:
d(Profit1)/dq1 = 192 - 8q1 - 4q2 = 0
Solving for q1, we get Firm 1's reaction function:
Firm 1's reaction function: q1 = (192 - 4q2)/8
q1 = 24 - 0.5q2
2. In the Cournot equilibrium, each firm assumes that its competitors’ output is fixed. Thus, each firm maximizes its profit by setting its output level (q1 and q2) simultaneously, knowing the other firm's reaction function. In the Cournot equilibrium, the output levels chosen by both firms will be such that neither firm has an incentive to unilaterally change its output quantity.
To find the Cournot equilibrium quantity, we substitute Firm 1's reaction function into Firm 2's reaction function and vice versa:
q1 = 24 - 0.5q2
q2 = 24 - 0.5q1
Solving these two equations simultaneously, we find the equilibrium quantities:
q1 = 16
q2 = 16
To find the market price, we substitute the equilibrium quantities back into the market demand function:
P = 204 - 4Q
= 204 - 4(q1 + q2)
= 204 - 4(16 + 16)
= 204 - 128
= $76
Therefore, in the Cournot equilibrium for this market, each firm will produce 16 units of output, and the market price will be $76.
3. To calculate the profit of each firm, we substitute the equilibrium quantities into the profit function:
Profit1 = 192q1 - 4q1^2 - 4q1q2 - 12q1
= 192(16) - 4(16)^2 - 4(16)(16) - 12(16)
= $1,536
Profit2 = 192q2 - 4q2^2 - 4q1q2 - 12q2
= 192(16) - 4(16)^2 - 4(16)(16) - 12(16)
= $1,536
Thus, each firm will earn a profit of $1,536 in the Cournot equilibrium.
1. To find Firm 1's reaction function, we need to determine how Firm 1 will respond to changes in the quantity produced by Firm 2, while maximizing its own profit.
In a Cournot market, each firm assumes that the output of its competitor remains constant. Therefore, Firm 1 maximizes its profit by choosing the quantity that maximizes its own profit, given Firm 2's output.
To find Firm 1's reaction function, we can set the derivative of Firm 1's profit with respect to its own output, q1, equal to zero. The profit function for Firm 1 can be calculated as follows:
Profit1 = (P - MC1) * q1
= (204 - 4Q - 12) * q1
= (192 - 4q1 - 4q2) * q1
= 192q1 - 4q1^2 - 4q1q2
Now, let's find the derivative of the profit function with respect to q1:
d(Profit1)/dq1 = 192 - 8q1 - 4q2
Setting this derivative equal to zero gives us Firm 1's reaction function:
192 - 8q1 - 4q2 = 0
8q1 = 192 - 4q2
q1 = (192 - 4q2)/8
q1 = 24 - 0.5q2
Therefore, Firm 1's reaction function is q1 = 24 - 0.5q2.
2. In the Cournot equilibrium, each firm chooses the quantity that maximizes its own profit, taking into account the output of its competitor. To find the Cournot equilibrium, we need to find the quantity that satisfies both reaction functions, where the output of each firm is simultaneously maximized.
Setting Firm 1's reaction function equal to Firm 2's reaction function, we have:
24 - 0.5q2 = q2
24 = 1.5q2
q2 = 16
Substituting the value of q2 into Firm 1's reaction function, we can find the quantity produced by Firm 1:
q1 = 24 - 0.5(16)
q1 = 24 - 8
q1 = 16
Therefore, in the Cournot equilibrium, each firm will produce 16 units of output.
To find the market price, we substitute the quantity produced by both firms into the market demand equation:
Q = q1 + q2
Q = 16 + 16
Q = 32
P = 204 - 4Q
P = 204 - 4(32)
P = 204 - 128
P = 76
Therefore, the market price in the Cournot equilibrium will be $76.
3. To calculate each firm's profit, we need to multiply their respective quantities by the market price and subtract their total costs.
For Firm 1:
Profit1 = (P - MC1) * q1
= (76 - 12) * 16
= $960
For Firm 2:
Profit2 = (P - MC2) * q2
= (76 - 12) * 16
= $960
Therefore, each firm will earn a profit of $960 in the Cournot equilibrium.