Asked by Blake Ellington on Mar 10, 2024
Verified
A duopoly faces the inverse demand curve p 160 2q.Both firms in the industry have constant costs of $10 per unit of output.In a Cournot equilibrium how much output will each duopolist sell?
A) 75
B) 54
C) 25
D) 35
E) 48
Cournot Equilibrium
A concept in economic theory where firms reach a state in which each firm chooses the optimal output level assuming the output levels of the other firms.
Duopoly
A market structure characterized by two producers/sellers in competition with each other.
Constant Costs
Costs that remain unchanged regardless of the level of output or activity within a certain range.
- Calculate equilibria in various oligopolistic market structures, including Cournot and Stackelberg competition.
- Analyze the effect of cost structures on firm behavior and market outcomes.
Verified Answer
MC
Michael CharronMar 10, 2024
Final Answer :
C
Explanation :
To find the Cournot equilibrium, we need to find the quantity that each firm will produce, given the quantity produced by the other firm. We can do this by using the reaction functions:
q1 = (F11 - 0.5F21 - 80)/2
q2 = (F21 - 0.5F11 - 80)/2
where q1 and q2 are the quantities produced by firms 1 and 2, respectively, and F11 and F21 are the quantities produced by firms 1 and 2 when they act simultaneously.
To find F11 and F21, we can use the total demand equation:
Q = 80 - 0.5F11 - 0.5F21
Substituting the constant cost of $10 per unit, we get:
profit1 = p*q1 - 10*q1 = (160 - q1 - 0.5q2)*q1 - 10*q1
profit2 = p*q2 - 10*q2 = (160 - q2 - 0.5q1)*q2 - 10*q2
Taking the first derivative of each firm's profit function with respect to its quantity produced and setting them equal to zero, we can solve for the optimum quantities:
q1 = 25
q2 = 25
Therefore, the Cournot equilibrium is for each duopolist to sell 25 units of output.
q1 = (F11 - 0.5F21 - 80)/2
q2 = (F21 - 0.5F11 - 80)/2
where q1 and q2 are the quantities produced by firms 1 and 2, respectively, and F11 and F21 are the quantities produced by firms 1 and 2 when they act simultaneously.
To find F11 and F21, we can use the total demand equation:
Q = 80 - 0.5F11 - 0.5F21
Substituting the constant cost of $10 per unit, we get:
profit1 = p*q1 - 10*q1 = (160 - q1 - 0.5q2)*q1 - 10*q1
profit2 = p*q2 - 10*q2 = (160 - q2 - 0.5q1)*q2 - 10*q2
Taking the first derivative of each firm's profit function with respect to its quantity produced and setting them equal to zero, we can solve for the optimum quantities:
q1 = 25
q2 = 25
Therefore, the Cournot equilibrium is for each duopolist to sell 25 units of output.
Learning Objectives
- Calculate equilibria in various oligopolistic market structures, including Cournot and Stackelberg competition.
- Analyze the effect of cost structures on firm behavior and market outcomes.