Lecture Notes 5: Monopoly
A firm has a monopoly when it is the only seller of a product that has no close substitutes. For example, your local water company is a monopoly. If you want running water at your house, the public utility is your only option. By contrast, we wouldn’t say that General Mills has a monopoly when it sells Cheerios. They may be the only seller of Cheerios, but other cereals are close substitutes.
Marginal Revenue and Price for a Price Maker
The first key to understanding monopoly pricing is that, unlike a competitive firm, a monopoly is a price-maker. It can choose its own price. This does not mean, though, that a monopoly is always going to set an extremely high price. Even if a firm is a monopoly, it still faces a downward-sloping demand curve. Consumers do not have infinite willingness to pay.
Now we see the tradeoff. A monopoly can choose the price it sets, but a higher price leads to a lower quantity sold. This is in contrast to a perfect competitor that faces a market price and its own output has no influence on the market price – it can sell whatever it wants at the market price.
Q P TR MR
0 12 0 --
1 11 11 11
2 10 20 9
3 9 27 7
4 8 32 5
5 7 35 3
6 6 36 1
7 5 35 -1
8 4 32 -3
The first unit sold makes sense. When the firm sets its price at $12, it sells nothing. But when it drops its price to $11, it sells one unit. Thus, its additional (marginal) revenue from selling the first unit is equal to $11.
But what about the second unit? The second unit sold is sold at a price of $10. But the marginal revenue from selling the second unit is only $9. Where did the extra dollar go? Well, in order to sell the second unit the firm has to set its price at $10. But this $10 price applies to both units sold. Thus, while the firm earns an extra $10 by selling the second unit, it loses $1 by cutting the price from the first unit to the second unit. All in all, the marginal revenue by expanding from one unit to two units sold is only $9 – not the full $10 for which the second unit is sold.
For the third unit, we can see the same thing. The third unit is sold at a price of $9. But selling 3 units instead of 2 units requires a price cut from $10 to $9 for all the units. Selling the third unit brings in revenue of $9, but the first two units are sold for $1 less. Thus, the marginal revenue from the sale of the third unit is only $7, as shown on the chart.
The important point is the following. In deciding to expand output, a price-maker always faces a tradeoff. Cutting price to reduce output produces more sales, but it cuts into revenue that could have been earned by selling fewer units but charging a higher price.
In fact, increasing sales can actually reduce revenue. Look at the table. Selling the seventh unit actually reduces sales revenues from $36 to $35. Marginal revenue can be negative. Increasing sales is not always a good thing. For a practical example, basketball teams frequently leave many seats empty at their games. Selling all the seats would require slashing ticket prices so much that the team is actually better off setting a high ticket price, but selling fewer seats.
Profit Maximization
Consider a monopoly facing a demand curve 𝑃𝑃 = 300 − 2𝑞𝑞 and that has a total cost function of
𝑇𝑇𝑇𝑇 = 100 + 𝑞𝑞2. What price and output should the monopoly choose to maximize profit?
Let’s set up the profit function.
Π = 𝑇𝑇𝑇𝑇 − 𝑇𝑇𝑇𝑇
= 𝑃𝑃𝑞𝑞 − (100 + 𝑞𝑞2)
= (300 − 2𝑞𝑞)𝑞𝑞 − (100 + 𝑞𝑞2)
= 300𝑞𝑞 − 2𝑞𝑞2− 100 − 𝑞𝑞2
The monopoly’s objective is to choose 𝑞𝑞 to maximize, which is done by setting the derivative of the profit function equal to zero.
𝑑𝑑Π
𝑑𝑑𝑞𝑞 = 300 − 4𝑞𝑞 − 2𝑞𝑞 = 0 ⇒ 𝑞𝑞 = 50
𝑃𝑃 = 300 − 2𝑞𝑞 = 300 − 2 ⋅ 50 = 200
This firm maximizes profit by charging a price of $200 and selling 50 units of output.
The firm’s total revenues are 𝑇𝑇𝑇𝑇 = 𝑃𝑃 ⋅ 𝑞𝑞 = $200 ⋅ 50 = $10,000. Its total costs can be found by plugging into the cost function 𝑇𝑇𝑇𝑇 = 100 + 𝑞𝑞2 = 100 + 502 = $2600. Thus, this monopoly earns a profit of $7400. There is no other level of output and price that would generate more profit.
Let’s delve a little bit deeper into this decision. This monopolist’s total revenue is:
𝑇𝑇𝑇𝑇 = 𝑃𝑃 ⋅ 𝑞𝑞
= (300 − 2𝑞𝑞) ⋅ 𝑞𝑞 = 300𝑞𝑞 − 2𝑞𝑞2
Marginal revenue is the change in total revenue from selling one more unit.
𝑀𝑀𝑇𝑇 =𝑑𝑑𝑞𝑞𝑑𝑑 (𝑇𝑇𝑇𝑇) = 300 − 4𝑞𝑞
This is actually a general rule for linear demand. The marginal revenue curve has the same intercept as the demand curve, but twice the slope. Because our demand curve is 𝑃𝑃 = 300 − 2𝑞𝑞, the corresponding marginal revenue curve is 𝑀𝑀𝑇𝑇 = 300 − 4𝑞𝑞.
For out monopoly, 𝑀𝑀𝑇𝑇 = 300 − 4𝑞𝑞. The total cost is 𝑇𝑇𝑇𝑇 = 100 + 𝑞𝑞2. Remember that the
marginal cost is the first derivative: 𝑀𝑀𝑇𝑇 = 𝑑𝑑
𝑑𝑑𝑑𝑑(𝑇𝑇𝑇𝑇) = 2𝑞𝑞. Thus, profit for the firm is maximized where marginal revenue (MR) equals marginal cost (MC).
𝑀𝑀𝑇𝑇 = 𝑀𝑀𝑇𝑇 300 − 4𝑞𝑞 = 2𝑞𝑞
𝑞𝑞 = 50
Of course, this is the same solution that we obtained above from maximizing the profit function directly. It’s just that economists like to think in terms of marginal conditions sometimes.
Price Markup and the Lerner Index
Let’s derive an expression for the marginal revenue curve in general. Starting with total revenue.
𝑇𝑇𝑇𝑇 = 𝑃𝑃 ⋅ 𝑞𝑞
The key thing in calculating the derivative is to understand that the price 𝑃𝑃 is itself a function of the output 𝑞𝑞. The price that a firm charges is determined by the amount of output that it wants to sell. Thus, in differentiating this expression, we need to use the product rule.
𝑀𝑀𝑇𝑇 =𝑑𝑑𝑞𝑞𝑑𝑑 (𝑇𝑇𝑇𝑇)
= 𝑑𝑑𝑃𝑃𝑑𝑑𝑞𝑞 ⋅ 𝑞𝑞 + 1 ⋅ 𝑃𝑃
Let’s multiply the first term by 𝑃𝑃
𝑃𝑃 and then rearrange.
𝑀𝑀𝑇𝑇 =𝑑𝑑𝑃𝑃𝑑𝑑𝑞𝑞 ⋅ 𝑞𝑞 ⋅𝑃𝑃𝑃𝑃 + 𝑃𝑃
=𝑑𝑑𝑃𝑃𝑑𝑑𝑞𝑞 ⋅𝑃𝑃 ⋅ 𝑃𝑃 + 𝑃𝑃𝑞𝑞
= 𝑃𝑃 �𝑑𝑑𝑃𝑃𝑑𝑑𝑞𝑞 ⋅𝑞𝑞𝑃𝑃 + 1�
To finish, remember that the definition of price elasticity is 𝜀𝜀 =𝑑𝑑𝑑𝑑 𝑑𝑑𝑃𝑃⋅
𝑃𝑃
𝑀𝑀𝑇𝑇 = 𝑃𝑃 �1𝜀𝜀 + 1�
Notice that, for a perfectly competitive firm, the elasticity of demand is infinite, so the marginal revenue is equal to price, as we said in the last unit.
To characterize the profit-maximizing price, the monopoly maximizes profit by equating marginal revenue and marginal cost.
𝑀𝑀𝑇𝑇 = 𝑀𝑀𝑇𝑇 𝑃𝑃 �1𝜀𝜀 + 1� = 𝑀𝑀𝑇𝑇
𝑃𝑃 �1 + 𝜀𝜀𝜀𝜀 � = 𝑀𝑀𝑇𝑇
𝑃𝑃 = �1 + 𝜀𝜀� 𝑀𝑀𝑇𝑇𝜀𝜀
There are two important things to understand about this expression. First, the price for a monopolist can be thought of as a mark-up over marginal cost. For example, if 𝜀𝜀 = −2, then the expression
gives 𝑃𝑃 = � −2
1+−2� 𝑀𝑀𝑇𝑇 = 2 ⋅ 𝑀𝑀𝑇𝑇. In other words, the monopolist sets a price of twice marginal cost. But if 𝜀𝜀 = −5, then 𝑃𝑃 = � −5
1+−5� 𝑀𝑀𝑇𝑇 = 1.25 ⋅ 𝑀𝑀𝑇𝑇. This makes good sense. As demand becomes more elastic, the firm is not able to mark up its price as much.
Second, notice that the expression only makes sense when demand is elastic, i.e. when 𝜀𝜀 < −1. An important lesson is that a monopolist always sets its price on the elastic part of its demand curve. Think about why. If demand were inelastic then total revenues rise if the monopolist raises its price. Furthermore, sales fall at least a little bit, so costs drop. With higher revenue and lower cost, profits rise for sure. Thus, a monopolist should never price on the inelastic part of its demand curve – if demand is inelastic, the monopolist could raise price to increase its profits.
There is a final way to express this condition that becomes very important later on.
𝑀𝑀𝑇𝑇 = 𝑀𝑀𝑇𝑇 𝑃𝑃 �1 +1𝜀𝜀� = 𝑀𝑀𝑇𝑇
𝑃𝑃 + 𝑃𝑃 �1𝜀𝜀� = 𝑀𝑀𝑇𝑇
𝑃𝑃 − 𝑀𝑀𝑇𝑇 = −𝑃𝑃 �1𝜀𝜀� 𝑃𝑃 − 𝑀𝑀𝑇𝑇
Now, since price elasticity of demand is negative, we can rewrite the last expression as:
𝑃𝑃 − 𝑀𝑀𝑇𝑇 𝑃𝑃 =
1 |𝜀𝜀|
This is a very famous expression. The left side of this equation 𝑃𝑃−𝑀𝑀𝑀𝑀
𝑃𝑃 is known as the Lerner Index or the price-cost markup. It represents the percentage of the total price that is a markup over marginal cost. Later in the course, the Lerner Index will be a key metric for examining market power and measuring the inefficiencies that result.
What the expression tells us is that that the markup in a market is crucially related to the elasticity of demand. As demand becomes more and more elastic and |𝜀𝜀| rises, the markup dwindles to zero. But markups can be substantial in markets where demand is more inelastic.
Deadweight Loss of Monopoly
Consider two markets with the same market demand and the same (constant) marginal cost of production. If the market is perfectly competitive, price will fall to marginal cost. The market operates efficiently, with all the surplus going to consumers. Firms earn no economic profits under this setup.
Because the price is marked up over marginal cost, the firm excludes some customers who are willing to pay more than the marginal cost of supplying the output. This represents lost surplus, and thus the outcome is allocatively inefficient because it does not generate maximum surplus for society. Another way to look at it is that the sum of the consumer and producer surplus under monopoly pricing is lower than the total surplus with competitive pricing. Society as a whole loses.
Other Sources of Inefficiency
An additional concern about monopolies is that, because they do not face the same pressures as competitive firms, they may not operate as efficiently as competitive firms do. In other words, because the monopoly earns substantial profits, it can afford some operational inefficiencies that increase costs while still remaining profitable. This is called x-inefficiency. Such inefficiencies are impossible in competitive markets since they earn no economic profits and would get driven out of business if other firms can produce at a lower cost.
A second source of inefficiency is rent-seeking behavior. That is, once a firm has a monopoly, it might be forced to expend resources to protect its monopoly power and preserve its economic profits. Economists see rent-seeking activities as a waste not only because they preserve monopoly power, but because they represent a waste of productive resources.
Benefits of Monopoly
While monopolies create efficiency losses for society, an important offsetting point is that monopoly profits can incentivize innovation and new product development. Think about a pharmaceutical firm spending hundreds of millions of dollars to develop a new drug. If, as soon as it invents the drug, the market becomes competitive, then our firm would earn no economic profit and would not even recoup its research and development costs. In other words, we might not like monopoly deadweight losses in a market, but without monopoly profits the market might not exist in the first place. This is exactly why we have patents, which confer monopoly protection. Overall, this is a thorny issue that we will treat extensively later in the course.
Monopoly over Time
A related point is that monopolies might be willing to tolerate some losses in the short-run as long as it can make them up later. For a practical example, firms that laid down railroad tracks lost money for years, but the tracks last a long time and they eventually accumulated sufficient cash flows to offset their losses.
Where do monopolies come from?
Economists emphasize a few of the more important sources of monopoly.
• Knowledge advantage – Maybe only one firm knows how to produce a product or knows
how to produce at such a lower cost than any other firm that it faces no real competition.
• Government-created monopolies – Patents and government franchises can artificially
preserve monopoly power, even in a market that would feature competition otherwise. Examples are government liquor franchises or requiring a taxi medallion (although the latter is being quickly eroded away by Uber). Although this regulation has largely disappeared, for awhile the government would not allow the construction of new hospitals unless the firm could prove that the existing hospital had a shortage of beds. In other words, the government did not allow competitors to enter simply to chip away at monopoly power.
• Natural monopoly (economies of scale) – In some markets, a single firm may be able to
service the entire market at a lower total cost than a group of smaller firms. This is the case when large firms enjoy economies of scale and their average costs fall as they grow. It is often argued that this is the rationale for local monopolies in electricity and telecommunications services, although some economists dispute whether these industries actually feature economies of scale.
Dominant Firm Model
Some markets are characterized by multiple firms, but there is a single firm that is a clear, dominant leader. In the dominant firm model a dominant firm establishes a market price, and then there is a “competitive fringe” of smaller firms that take this price as given.
Let’s illustrate how this model works by means of an example. The market demand curve for a product is 𝑄𝑄 = 100 − 𝑃𝑃, where 𝑄𝑄 is total market output. The market consists of one dominant firm, which produces output of 𝑄𝑄𝑑𝑑𝑑𝑑𝑑𝑑. There are 10 fringe firms, each of which produce output level 𝑞𝑞. The output of all the fringe firms is 𝑄𝑄𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓. Thus, we can write total market output as
The dominant firm can produce output for $18 per unit. Each competitive fringe firm has a cost function 𝑇𝑇𝑇𝑇 = 10𝑞𝑞 + 20𝑞𝑞2.
To solve the model, we have to use backwards induction. The dominant firm knows that the competitive firms will respond accordingly to maximize profits, given the price that it sets.
Given the price 𝑃𝑃 set by the dominant firm, competitive firms will maximize profit.
Π𝑐𝑐𝑑𝑑𝑑𝑑𝑐𝑐𝑓𝑓𝑐𝑐𝑓𝑓𝑐𝑐𝑓𝑓𝑐𝑐𝑓𝑓 = 𝑇𝑇𝑇𝑇 − 𝑇𝑇𝑇𝑇
= 𝑃𝑃𝑞𝑞 − (10𝑞𝑞 + 20𝑞𝑞2)
= 𝑃𝑃𝑞𝑞 − 10𝑞𝑞 − 20𝑞𝑞2
Choosing output level to maximize profit:
𝑑𝑑Π
𝑑𝑑𝑞𝑞 = 𝑃𝑃 − 10 − 40𝑞𝑞 = 0 ⇒ 𝑞𝑞 = 1 40 𝑃𝑃 −
1 4
This is the output produced by each competitive fringe firm, given the price established by the dominant firm. As a result, total supply by the 10 fringe firms is:
𝑄𝑄𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 = 10𝑞𝑞 = 10 �40 𝑃𝑃 −1 14�
= 0.25𝑃𝑃 − 2.5
The dominant firm knows that, whatever price 𝑃𝑃 it sets, this will be the output by the competitive fringe. The output of the dominant firm is:
𝑄𝑄𝑑𝑑𝑑𝑑𝑑𝑑𝑓𝑓𝑓𝑓𝑑𝑑𝑓𝑓𝑐𝑐 = 𝑄𝑄 − 𝑄𝑄𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓
= (100 − 𝑃𝑃) − (0.25𝑃𝑃 − 2.5) = 100 − 𝑃𝑃 − 0.25𝑃𝑃 + 2.5 = 102.5 − 1.25𝑃𝑃
This last expression is sometimes called the residual demand curve. It shows how much demand is left over for the dominant firm to sell given the output that will be produced by the fringe firms.
Now, the objective of the dominant firm is to set the price that maximizes its profit.
Π𝑑𝑑𝑑𝑑𝑑𝑑= 𝑇𝑇𝑇𝑇 − 𝑇𝑇𝑇𝑇
Substituting in the residual demand function faced by the dominant firm:
Π𝑑𝑑𝑑𝑑𝑑𝑑 = 𝑃𝑃 ⋅ (102.5 − 1.25𝑃𝑃) − 18 ⋅ (102.5 − 1.25𝑃𝑃)
= 102.5𝑃𝑃 − 1.25𝑃𝑃2− 1845 + 22.5𝑃𝑃
= 125𝑃𝑃 − 1.25𝑃𝑃2 − 1845
The dominant firm will choose the price that maximizes its profit:
𝑑𝑑Π𝑑𝑑𝑑𝑑𝑑𝑑
𝑑𝑑𝑃𝑃 = 125 − 2.5𝑃𝑃 = 0 ⇒ 𝑃𝑃 = 50
This implies that the dominant firm’s sales are 𝑄𝑄𝑑𝑑𝑑𝑑𝑑𝑑𝑓𝑓𝑓𝑓𝑑𝑑𝑓𝑓𝑐𝑐 = 102.5 − 1.25𝑃𝑃 = 40. The dominant firm earns a profit of Π = 𝑇𝑇𝑇𝑇 − 𝑇𝑇𝑇𝑇 = $50 ⋅ 40 − $18 ⋅ 40 = $1280
Each competitive fringe firm produces 𝑞𝑞 = 1 40𝑃𝑃 −
1
4 = 1 unit of output. Thus, the total output of the competitive fringe is 𝑄𝑄𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓= 10. Each competitive firm earns revenue of $50 and incurs costs of 𝑇𝑇𝑇𝑇 = 10𝑞𝑞 + 20𝑞𝑞2 = 30, so each fringe firm earns a profit of $20.
A few points to make here.
• The dominant firm will not try to drive the competitive firms out of business. The shut-down price for the competitive firms is $10, so in order to drive them out of business the dominant firm would have to reduce its price so much that it would be losing money. One important lesson behind this model is that it’s not necessarily the case that dominant firms will always seek to drive smaller firms out of business. It might be too costly to do so.
• This outcome is better for consumers than monopoly without a competitive fringe. Without a competitive fringe, the monopoly’s profit-maximizing price is $59, compared to the $50 market price with the competitive fringe. A second important lesson behind this model is that, even if there is a dominant firm with pricing power, it’s still better for consumers for there to be at least some competition.
Exercises
Problem 1
You are the manager of a monopoly firm facing demand curve 𝑃𝑃 = 100 − 2𝑞𝑞 and total cost function 𝑇𝑇𝑇𝑇 = 2000 + 3𝑞𝑞2.
a. How much output should the firm sell to maximize profit? b. What price should the firm set to maximize profit?
c. Calculate the profit earned by the firm.
d. Calculate the elasticity of demand at the profit-maximizing price. (Hint: Solve the demand curve for Q). Is demand elastic or inelastic at this price?
Problem 2
A firm produces a new drug and holds a patent on the drug. It estimates that the elasticity of demand it faces for the drug is 𝜀𝜀 = −2.25. The firm incurred an R&D cost of $12 million to develop the drug, although its marginal cost to produce each pill is only $0.25. What price should the firm set in order to maximize its profits?
Problem 3
Consider a market where the market demand curve is 𝑄𝑄 = 200 − 4𝑃𝑃, and the marginal cost of production is constant at $10 per unit.
a. Graph the demand curve and the marginal cost function.
b. Suppose first that the market is perfectly competitive. Calculate the consumer surplus and the producer surplus.
c. Now suppose that the market is a monopoly. Calculate the consumer surplus, the producer surplus and the deadweight loss.
Problem 4
A firm that faces a marginal cost of $35 has a Lerner index of 0.2.
a. What elasticity of demand does the firm face? b. What price is the firm charging?
Problem 5
