3. A homogeneous products duopoly faces a market demand function given by P = 500 − 10Q .
Both firms have a constant marginal cost of MC = 200.
1
a. What would the equilibrium price in this market be if it were perfectly competitive?
b. What are the firm’s outputs in a Nash equilibrium of Cournot’s model?
c. What is the market price in a Nash equilibrium of Cournot’s model?
d. What would the equilibrium price in this market be if the two firms colluded to set the
monopoly price?
was trying to do this question and im comepltly lost
Don't worry, I can help you step-by-step with this question.
a. In a perfectly competitive market, firms are price takers, meaning they take the market price as given. In this case, the equilibrium price can be found by setting the market demand equal to the marginal cost of production:
500 - 10Q = 200
Solving for Q gives us:
10Q = 300
Q = 30
Substituting Q back into the demand equation gives us:
P = 500 - 10(30)
P = 500 - 300
P = 200
Therefore, the equilibrium price in a perfectly competitive market would be $200.
b. In Cournot's model, firms simultaneously choose their outputs to maximize their profits, taking into account the reaction of their competitors. In this case, both firms have a constant marginal cost of 200. To find the firm's outputs in a Nash equilibrium, we need to solve for their best response functions.
Assuming both firms have the same cost structure, each firm's best response function can be derived by taking the derivative of their profit function and setting it equal to zero.
Each firm's profit is given by:
π = (P - MC)Q
= (500 - 10Q - 200)Q
= (300 - 10Q)Q
Taking the derivative with respect to Q and setting it equal to zero gives us:
dπ/dQ = 300 - 20Q = 0
Solving for Q gives us:
20Q = 300
Q = 15
Therefore, in a Nash equilibrium of Cournot's model, each firm would produce an output of 15 units.
c. To find the market price in a Nash equilibrium of Cournot's model, we substitute the firms' outputs into the market demand function:
Q = Q1 + Q2
= 15 + 15
= 30
P = 500 - 10(30)
P = 500 - 300
P = 200
Therefore, the market price in a Nash equilibrium of Cournot's model would also be $200.
d. If the two firms colluded to set the monopoly price, they would act as a single firm and maximize joint profits. In this case, they would choose the monopoly quantity that maximizes their combined profits.
To find the monopoly price, we need to find the quantity that maximizes their joint profit. Their joint profit function is:
π = (P - MC)Q
= (500 - 10Q - 200)Q
= (300 - 10Q)Q
To find the monopoly quantity, we take the derivative of the joint profit function and set it equal to zero:
dπ/dQ = 300 - 20Q = 0
Solving for Q gives us:
20Q = 300
Q = 15
Substituting Q back into the demand equation gives us:
P = 500 - 10(15)
P = 500 - 150
P = 350
Therefore, the equilibrium price in this market, if the two firms colluded to set the monopoly price, would be $350.
To solve this question, we need to understand the concepts of perfect competition and Cournot's model of duopoly. Let's take each part of the question step-by-step:
a. What would the equilibrium price in this market be if it were perfectly competitive?
In a perfectly competitive market, there are many sellers and buyers. Each seller has a negligible market share, and the products are identical. As a result, sellers have no market power and are price takers.
To find the equilibrium price, we need to set the total quantity demanded equal to the sum of the quantities supplied by all firms in the market. In this case, we have two firms, but assuming the market is perfectly competitive, we can treat them as one entity.
Given the demand function P = 500 - 10Q, we can find the equilibrium quantity by setting Qd (quantity demanded) equal to Qs (quantity supplied):
Qd = Qs
500 - 10Q = Q
500 = 11Q
Q = 500/11
Now, substitute this value of the equilibrium quantity Q back into the demand function to find the equilibrium price:
P = 500 - 10Q
P = 500 - 10(500/11)
P = 500 - (5000/11)
P = (55000/11) - (5000/11)
P = (50000/11)
P ≈ 454.54
Therefore, the equilibrium price in a perfectly competitive market would be approximately $454.54.
b. What are the firm’s outputs in a Nash equilibrium of Cournot’s model?
In Cournot's model, each firm determines its output quantity while assuming that the other firm's output will remain constant.
To find the Nash equilibrium, we need to set up each firm's reaction function. The reaction function specifies the output quantity that maximizes a firm's profit given the output of its competitor.
In this case, both firms have a constant marginal cost (MC) of 200. The profit function for each firm is given by:
π = (P - MC) * Q
To determine each firm's output in the Nash equilibrium, we need to solve the following simultaneous equations:
Firm 1's reaction function:
Q1 = (500 - 10Q2 - MC) / 20
Firm 2's reaction function:
Q2 = (500 - 10Q1 - MC) / 20
To solve these equations, we can use the method of iteration. Start by assuming an initial value for Q1 and Q2, then substitute these values into the reaction functions and solve them simultaneously. Repeat this process until the values of Q1 and Q2 converge to the same values.
c. What is the market price in a Nash equilibrium of Cournot’s model?
Once we have found the firm's outputs Q1 and Q2 from the previous step, we can find the market price by substituting the total quantity supplied (Q) into the demand function:
P = 500 - 10Q
Again, Q = Q1 + Q2, where Q1 and Q2 are the outputs of the firms obtained in the previous step.
d. What would the equilibrium price in this market be if the two firms colluded to set the monopoly price?
If the two firms collude to act as a monopoly, they would combine their profits and act as a single entity. In this case, they will maximize their total profit by jointly determining the monopoly quantity.
To determine the monopoly quantity, we can set up the total profit function as:
π = (P - MC) * Q
Where P is the price, MC is the marginal cost (given as 200), and Q is the total quantity produced by both firms.
To find the monopoly quantity, we need to maximize the total profit function by taking the derivative of π with respect to Q, setting it equal to zero, and solving for Q. Once we have the quantity, we can substitute it into the demand function (P = 500 - 10Q) to find the monopoly price.