Consider an economy in which a monopolistic firm serves two identical, but separate markets, called A and B.
The aggregate inverse demand in each market is given by 1000−q.
The cost function for the monopolist is given by (qA+qB)^2, where qA andqB denotes the amount sold in each market.
Suppose that each market is regulated by a separate government, and that the government of market A requires the monopolist to sell exactly 250 units on its market.
Suppose also that the monopolist is allowed to charge different prices on each market, but is not allowed to engage in more sophisticated forms of price discrimination.
QUESTION: Given these policies, what is the total amount produced by the monopolist in equilibrium?
To find the total amount produced by the monopolist in equilibrium, we need to analyze the monopolist's profit-maximizing decisions in each market.
Let's start with market A, where the government requires the monopolist to sell exactly 250 units. In this market, the monopolist has no control over the quantity sold but can determine the price.
To maximize profits, the monopolist will set the price in market A where marginal cost equals marginal revenue. The marginal cost is given by the derivative of the cost function with respect to qA, which is 2(qA + qB), and marginal revenue is equal to the inverse demand.
Setting marginal cost equal to marginal revenue:
2(qA + qB) = 1000 − qA
Simplifying the equation, we get:
3qA + 2qB = 1000
Now let's move on to market B, where the monopolist has full control over the quantity sold and the price. In this market, the monopolist will again maximize profits by setting marginal cost equal to marginal revenue.
The marginal cost is given by the derivative of the cost function with respect to qB, which is 2(qA + qB). The marginal revenue is equal to the inverse demand.
Setting marginal cost equal to marginal revenue:
2(qA + qB) = 1000 − qB
Simplifying the equation, we get:
2qA + 3qB = 1000
We now have a system of two equations:
3qA + 2qB = 1000
2qA + 3qB = 1000
Solving this system of equations will give us the values of qA and qB, which represent the quantity produced by the monopolist in each market.
By solving the system of equations, we find that qA = 400 and qB = 200.
Therefore, in equilibrium, the monopolist will produce a total quantity of 400 units in market A and 200 units in market B.
To find the total amount produced by the monopolist in equilibrium, we need to determine the profit-maximizing output levels in each market given the constraints imposed by the respective governments.
Let's start by considering market A, where the government requires the monopolist to sell exactly 250 units. We need to find the profit-maximizing output level in market B, taking into account the constraint in market A.
To maximize profit, a monopolist sets marginal revenue equal to marginal cost. Let's calculate the marginal revenue in market B:
The inverse demand in market B is given by 1000 - qB. Since the monopolist is a price setter, the marginal revenue (MRB) will be less than the inverse demand. We can find MRB by differentiating the inverse demand function with respect to qB:
MRB = d(1000 - qB) / dqB
MRB = -1
Now let's calculate the marginal cost (MC):
The cost function for the monopolist is given by (qA + qB)^2. We can find the marginal cost (MCB) by differentiating the cost function with respect to qB:
MCB = d((qA + qB)^2) / dqB
MCB = 2(qA + qB)
Since the monopolist wants to maximize profit, MRB should equal MCB:
-1 = 2(qA + qB)
Now we can solve for the value of qB:
-1 = 2(250 + qB)
-1 = 500 + 2qB
2qB = -501
qB = -250.5
It is important to note that negative quantities make no economic sense. Hence, we cannot produce a negative amount of goods. In this case, the value of qB is infeasible.
Therefore, there is no feasible profit-maximizing output level for market B. Hence, the monopolist will not produce any goods for market B.
As a result, the total amount produced in equilibrium by the monopolist is 250 units, which is the amount required by the government for market A.