Suppose that the demand curve for a product x provided by a monopolist is given by p = 90−x and suppose further that the monopolist’s marginal cost curve is given by MC = x.
A: In this part, we will focus on a graphical analysis—which we ask you to revisit with some simple math in part B. (It is not essential that you have done Section B of the chapter in order to do (a) through (d) of part B of this question.)
(a) Draw a graph with the demand and marginal cost curves.
(b) Assuming that the monopolist can only charge a single per-unit price for x, where does the marginal revenue curve lie in your graph?
(c) Illustrate the monopolist’s profit maximizing" supply point”.
(d) In the absence of any recurring fixed costs, what area in your graph represents the monopolist’s profit. (There are actually two areas that can be used to represent profit— can you find both?) Answer: There are two ways you can represent profit in this case: First, we can simply use the usual way of indicating producer surplus as the area below the price down to the MC curve to get profit of (c + d + f). Second, we can also illustrate profit as the difference between MR and MC up to point A — i.e. area (a + c + f). This implies that (a) = (d) in the graph.
(e) Assuming that the demand curve is also the marginal willingness to pay curve, illustrate consumer surplus and deadweight loss.
(f) Suppose that the monopolist has recurring fixed costs of an amount that causes her actual profit to be zero. Where in your graph would the average cost curve lie? In particular, how does this average cost curve relate to the demand curve?
(g) In a new graph, illustrate again the demand, MR and MC curves. Then illustrate the monopolist’s average cost curve assuming the recurring fixed costs are half of what they were in part (f).
(h) In your graph, illustrate where profit lies. True or False: Recurring fixed costs only determine whether a monopolist produces—not how much she produces.
B: Consider again the demand curve and MC curve as specified at the beginning of this exercise.
(a) Derive the equation for the marginal revenue curve.
(b)What is the profit maximizing output level xM? What is the profit maximizing price pM (assuming that the monopolist can only charge a single per-unit price to all consumers)?
(c) In the absence of recurring fixed costs, what is the monopolist’s profit?
(d) What is consumer surplus and deadweight loss (assuming that demand is equal to marginal willingness to pay).
(e) What is the cost function if recurring fixed costs are sufficiently high to cause the monopolist’s profit to be zero?
(f) Use this cost function to set up the monopolist’s optimization problem and verify your answers to (b).
(g) Does the average cost curve relate to the demand curve as you concluded in part A(f )?