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.

Question

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.

Quizplus · Accepted Answer

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.

If the Inverse Demand for Bean Sprouts Were Given