The Baldonian shoe market is served by a monopoly firm. The demand for shoes in
ID: 1195937 • Letter: T
Question
The Baldonian shoe market is served by a monopoly firm. The demand for shoes in Baldonia is given by Q = 10 - P, where Q is millions of pairs of shoes (a right shoe and left shoe) per year, and P is the price of a pair of shoes. The marginal cost of making shoes is constant and equal to $2 per pair.
Baldonian authorities have concluded that the shoe sellers monopoly power is not a good thing. Inspired by the U.S. government’s attempt several years ago to break Microsoft into two pieces, Baldonia creates two firms: one that sells right shoes and the other that sells left shoes. Let P1 be the price charged by the right-shoe producer and P2 be the price charged by the left-shoe producer. Of course, consumers still want to buy a pair of shoes (a right one and a left one), so the demand for pairs of shoes continues to be 10 - P1 - P2. If you think about it, this means that the right-shoe producer sells 10 - P1 - P2 right shoes, while the left-shoe producer sells 10 - P1 - P2 left shoes. Since the marginal cost of a pair of shoes is $2 per pair, the marginal cost of the right-shoe producer is $1 per shoe, and the marginal cost of the left-shoe producer is $1 per shoe.
What is the Bertrand equilibrium price of shoes? How many pairs of shoes are purchased?
I have:
Q1 = 4.5 - 1/2P2
P1 = 5.5 - 1/2P2
and because of symmetry
Q2 = 4.5 - 1/2P1
P2 = 5.5 -1/2P1
I'm just not sure where to go from here to get the equilibrium? I feel like it's something very simple I'm missing so please show how you got the answer.
Thank you!
Explanation / Answer
First
Let’s begin by deriving Firm 1’s reaction function.
Firm 1’s demand curve can be written as: P1 = 10 – P2 – Q1
The corresponding MR curve is thus: MR1 = 10 – P2 – 2Q1
Set MR1 equal to MC and solve for Q1 in terms of P2
10 – P2 – 2Q1 = 1
Q1 = 4.5 – 0.5P2
Substitute back into the inverse demand curve to give us P1 in terms of P2.
P1 = [10 – P2] – [4.5 – 0.5P2] = 5.5 – 0.5P2
Firm 2’s reaction function is easy. The firms are symmetric so it takes the same form as firm 1’s reaction function:
P2 = 5.5 – 0.5P1
Solve for the equilibrium We have two equations in two unknowns.If we solve these reaction functions, we get
P1 = 5.5 – 0.5P2 = 5.5 – 0.5(5.5 – 0.5P1)
P1 = 3.67 = P2
Since P1 + P2 = the total price of a pair of shoes = 3.67 + 3.67 = 7.33
Therefore, Bertrand equilibrium price of shoes is $7.33
By putting the value of P in the demand function we get
Q = 10 -P = 10 - 7.33 = 2.67 ( if we round it. it becomes 3 pairs)
Related Questions
Navigate
Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.