Asked by Yadelis Carmona gonzalez on May 27, 2024
Verified
If the inverse demand for bean sprouts were given by P(Y) 640 3Y and the total cost of producing Y units for any firm were TC(Y) 10Y and if the industry consisted of two Cournot duopolists, then in equilibrium each firm's production would be
A) 52.50 units.
B) 70 units.
C) 105 units.
D) 35 units.
E) 53.33 units.
Inverse Demand
A function that expresses price as a function of quantity demanded, illustrating how the price of an item is determined by the quantity of it that consumers are willing to buy at different prices.
Cournot Duopolists
Firms in a Cournot duopoly model compete on the quantity of output they produce, assuming that each firm makes its output decision based on the output level of its competitor.
Total Cost
The complete cost of producing a specific amount of goods or services, including both fixed and variable expenses.
- Determine the steady state output for enterprises involved in a Cournot duopoly scenario.
- Activate the Cournot model amidst diverse market scenarios and demand outlines.
Verified Answer
MU
Muhammad UsmanMay 31, 2024
Final Answer :
B
Explanation :
In a Cournot duopoly, each firm assumes that its competitor's output will remain constant, and decides its own output level accordingly.
To find the equilibrium output level for each firm, we first need to find the market quantity, which is the sum of the quantities produced by each firm:
Q = q1 + q2
Next, we need to find the profit-maximizing output level for each firm, taking the other firm's quantity as given.
To do this, we need to find each firm's reaction function, which shows its optimal output level for any given quantity produced by the other firm.
To find firm 1's reaction function, we set its marginal revenue (MR) equal to its marginal cost (MC) and solve for q1:
MR1 = P(Y) + Q/2dY/dq1 * (1 - q1 - q2) - q1 * dP(Y)/dY = MC1
where Q = q1 + q2 is the total quantity produced by both firms.
Plugging in the given demand and cost functions, we get:
640 - 2/3Q - 2q1 = 10
Solving for q1, we get:
q1 = (315/2) - (1/3)q2
Similarly, firm 2's reaction function can be found by setting its MR equal to its MC:
MR2 = P(Y) + Q/2dY/dq2 * (1 - q1 - q2) - q2 * dP(Y)/dY = MC2
Plugging in the demand and cost functions, we get:
640 - 2/3Q - 2q2 = 10
Solving for q2, we get:
q2 = (315/2) - (1/3)q1
To find the equilibrium output levels, we need to solve the two reaction functions simultaneously. Substituting q2 from the second equation into the first equation, we get:
q1 = 70
Substituting q1 = 70 into the second equation, we get:
q2 = 70
Therefore, in equilibrium, each firm produces 70 units.
To find the equilibrium output level for each firm, we first need to find the market quantity, which is the sum of the quantities produced by each firm:
Q = q1 + q2
Next, we need to find the profit-maximizing output level for each firm, taking the other firm's quantity as given.
To do this, we need to find each firm's reaction function, which shows its optimal output level for any given quantity produced by the other firm.
To find firm 1's reaction function, we set its marginal revenue (MR) equal to its marginal cost (MC) and solve for q1:
MR1 = P(Y) + Q/2dY/dq1 * (1 - q1 - q2) - q1 * dP(Y)/dY = MC1
where Q = q1 + q2 is the total quantity produced by both firms.
Plugging in the given demand and cost functions, we get:
640 - 2/3Q - 2q1 = 10
Solving for q1, we get:
q1 = (315/2) - (1/3)q2
Similarly, firm 2's reaction function can be found by setting its MR equal to its MC:
MR2 = P(Y) + Q/2dY/dq2 * (1 - q1 - q2) - q2 * dP(Y)/dY = MC2
Plugging in the demand and cost functions, we get:
640 - 2/3Q - 2q2 = 10
Solving for q2, we get:
q2 = (315/2) - (1/3)q1
To find the equilibrium output levels, we need to solve the two reaction functions simultaneously. Substituting q2 from the second equation into the first equation, we get:
q1 = 70
Substituting q1 = 70 into the second equation, we get:
q2 = 70
Therefore, in equilibrium, each firm produces 70 units.
Learning Objectives
- Determine the steady state output for enterprises involved in a Cournot duopoly scenario.
- Activate the Cournot model amidst diverse market scenarios and demand outlines.