National University of Singapore EC2101 Microeconomic Analysis I Department of Economics Semester 1 AY 2014/2015

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Practice Problems 10 – Solution Monopoly

Question 1 The marginal cost of preparing a large latte in a specialty coffee house is $1. The firm’s market research reveals that the elasticity of demand for its large lattes is constant, with a value of about -1.3. If the firm wants to maximize profit from the sale of large lattes, about what price should the firm charge? Using the inverse elasticity pricing rule, P −MCP

= −1ε⇒

P −1P

=11.3

⇒ P = 4.33

Thus the profit-maximizing price is about $4.33. Question 2 Suppose that Intel has a monopoly in the market for microprocessors in Brazil. During the year 2005, it faces a market demand curve given by P = 9 – Q, where Q is millions of microprocessors sold per year. Suppose you know nothing about Intel’s costs of production. Assuming that Intel acts as a profit-maximizing monopolist, would it ever sell 7 million microprocessors in Brazil in 2005? Marginal revenue is MR = 9 – 2Q. If Q=7, then MR=-5. To maximize profit, Intel should produce at where MR=MC. Unless marginal cost is negative, which is impossible, at Q=7, Intel is not maximizing profit because marginal revenue is not the same as marginal cost. In fact, since marginal cost is positive, Intel is producing at the region where MR<MC. To maximize profit, Intel should produce less. Alternatively, the midpoint of the demand curve is Q=4.5 and P=4.5. Based on the inverse elasticity pricing rule, we know that Intel should never produce at the inelastic region of the demand curve, which is the region where Q>4.5 since the demand curve is linear. So producing at Q=7 is not profit maximizing. Question 3 Suppose a monopolist faces the market demand function P = a - bQ. Its marginal cost is given by MC = c + eQ. Assume that a > c, b >0, and 2b + e > 0. a) Derive an expression for the monopolist’s optimal quantity and price in terms of a, b, c, and e. The monopolist will operate where MR MC= . With demand P a bQ= − , marginal revenue is given by 2MR a bQ= − . Setting this equal to marginal cost implies

 

2

2

a bQ c eQa cQb e

− = +

−=

+

 At this quantity price is

National University of Singapore EC2101 Microeconomic Analysis I Department of Economics Semester 1 AY 2014/2015

2

2

2

a cP a bb e

ab ae bcPb e

−⎛ ⎞= − ⎜ ⎟+⎝ ⎠+ +

=+

b) Show that an increase in c (which corresponds to an upward parallel shift in marginal cost) or a decrease in a (which corresponds to a leftward parallel shift in demand) must decrease the equilibrium quantity of output. Since

2a cQb e−

=+

It is easy to see that increasing c or decreasing a will reduce the numerator of the expression, reducing Q .

c) Show that when e ≥ 0 an increase in a must increase the equilibrium price. Since e≥ 0 and

2

ab ae bcPb e+ +

=+

Increasing a will increase the numerator for this expression. This will therefore increase the equilibrium price. Question 4 Two monopolists in different markets have identical, constant marginal cost functions. Suppose their linear demand curves have identical vertical intercepts but different slopes. Which monopolist will have a higher markup: the one with the flatter demand curve or the one with the steeper demand curve? Let the marginal cost function be MC(Q)=c. Let the inverse demand curves be P=a-bQ. The equation of marginal revenue is thus MR=a-2bQ. Equating it with the marginal cost, the profit-maximizing quantity is Q=(a-c)/2b and the profit-maximizing price is P=(a+c)/2. Interestingly, the profit-maximizing price does not depend on the slope of the demand curve. It only depends on the marginal cost and the vertical intercept of the inverse demand curve. Since the two monopolists have the same marginal cost and the same vertical intercept, the profit-maximizing price will be the same and the markup (P-MC) will also be the same. This finding is called the monopoly midpoint rule. If we have a linear demand curve and a constant marginal cost, the profit-maximizing price is halfway between the vertical intercept of the inverse demand curve, a, and the marginal cost, c. Question 5 The following diagram shows the average cost curve and the marginal revenue curve for a monopolist in a particular industry. What range of quantities could it be possible to observe this firm producing, assuming that the firm maximizes

National University of Singapore EC2101 Microeconomic Analysis I Department of Economics Semester 1 AY 2014/2015

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profit? You can read your answers off the graph, and therefore approximate values are permissible.

The minimum point on the AC appears to be at about 15 (or 16) units of output, and the point where the MR curve intersects the AC curve is at about 20 units. The monopolist’s profit maximizing output must fall between the minimum point on AC and the point where MR intersects AC. To see this, remember that the firm will produce where MR = MC. This cannot happen at any point less than 15 units because the AC curve is decreasing for Q < 15. Therefore the MC curve lies below the AC curve for Q < 15. From the graph, it is clear that MR > AC for Q < 15. Thus we have MR >AC > MC for Q < 15. Similarly, from the graph we can see that MR < AC for Q > 20. Since the MC curve must lie above the AC curve to the right of 15 units, we have MR < AC < MC for Q > 20.