If 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

In the Cournot duopoly model, each firm assumes that its competitor's output remains constant when deciding its own output level. Therefore, firm 1's profit is π1 = [P(Y) - TC(Y)]Y1, where Y1 is firm 1's output level. Similarly, firm 2's profit is π2 = [P(Y) - TC(Y)]Y2, where Y2 is firm 2's output level. 
To find the equilibrium output level, we need to find the Nash equilibrium, where neither firm has an incentive to change its output given the other firm's output level. This occurs when each firm maximizes its profit given the other firm's output level. 
Using the inverse demand and total cost functions given, we can find the industry profit function: 
Π = [P(Y) - TC(Y)](Y1 + Y2) 
Π = [300 - 3Y - 30(Y1 + Y2)](Y1 + Y2) 
Π = (300 - 33Y)(Y1 + Y2) - 30(Y1 + Y2)^2 
To find each firm's best response function, we take the partial derivative of its profit function with respect to its own output: 
∂π1/∂Y1 = 300 - 33Y - 60Y1 - 30Y2 
∂π2/∂Y2 = 300 - 33Y - 60Y2 - 30Y1 
Setting these equal to zero, we can solve for each firm's optimal output level as a function of the other firm's output level: 
Y1 = (210 - 30Y2)/60 
Y2 = (210 - 30Y1)/60 
Solving these two equations simultaneously, we get Y1 = Y2 = 30. 
Therefore, in equilibrium, each firm produces 30 units of bean sprouts.

If the Inverse Demand for Bean Sprouts Were Given by P(Y)=