If in Problem 4, the inverse demand for bean sprouts were given by P(Y) = 300 - 3Y and the total cost of producing Y units for any firm were TC(Y) = 30Y and if the industry consisted of two Cournot duopolists, then in equilibrium each firm's production would be
A) 45 units.
B) 22.50 units.
C) 30 units.
D) 15 units.
E) 25 units.

Question

Quizplus · Accepted Answer

To find the equilibrium level of output for each firm in a Cournot duopoly, we need to solve for the Nash equilibrium where each firm maximizes their profits given the output of the other firm.

First, we can find the market demand function by adding up the individual demands: 
Q = q1 + q2 = (300 - 3q1) + (300 - 3q2) 
Q = 600 - 3q1 - 3q2

Next, we can find the inverse marginal cost function (MC^-1) by taking the derivative of the total cost function: 
MC(Y) = TC'(Y) = 30

So, MC^-1 = 1/30

To find the best response for Firm 1, we set up their profit function: 
π1 = (300 - 3q1 - 3q2)q1 - 30q1

We take the derivative of this function with respect to q1 and set it equal to zero to find the profit-maximizing output for Firm 1: 
∂π1/∂q1 = 297 - 6q1 - 3q2 = 0 
6q1 = 297 - 3q2 
q1 = 49.5 - 0.5q2

Similarly, for Firm 2: 
∂π2/∂q2 = 297 - 6q2 - 3q1 = 0 
6q2 = 297 - 3q1 
q2 = 49.5 - 0.5q1

Now we can substitute Firm 2's best response function into Firm 1's best response function: 
q1 = 49.5 - 0.5(49.5 - 0.5q1) 
q1 = 30

So, each firm would produce 30 units in equilibrium.

If in Problem 4, the Inverse Demand for Bean Sprouts