Monopoly

Context

• The goal in the popular economic-themed board game “Monopoly” is to monopolize a fictional property market by driving other players to bankruptcy.

• In real markets, however, monopolies tend to have unpleasant connotations. In fact, the competition policies in both the US and EU aim (among others) to prevent the growth and abuse of monopoly power.
  • Why do regulators try to reduce monopoly power?
  • Are monopolies bad in terms of economic efficiency?
  • How do monopolies choose prices for their products?

Course Structure Overview

Lecture Structure and Learning Objectives

Structure

• The Baby Bells (Case Study)
• Basic Concepts
• An Example
• Welfare Analysis of Monopolies
• Current Field Developments

Learning Objectives

• Describe the monopoly market structure.
• Illustrate how monopolies make decisions to maximize their profits.
• Compare and contrast the welfare of competition and monopoly market structures.
• Explain the economic inefficiencies introduced by profit maximizing monopolies.
• Describe the conditions under which monopolies may arise.

The Baby Bells

• In 1974, the US Department of Justice filed an antitrust lawsuit against AT&T.
• AT&T was the sole provider of telephone services in the US.
• Its subsidiary, Western Electric, produced most telephonic equipment.
• Overall, AT&T had almost wholly controlled the communication technology sector in the US.

The AT&T breakup

wikipedia

• In 1982, a settlement between AT&T and the US Department of Justice was reached.

• The main provision in the settlement was the breakup of AT&T into seven independent companies.
• Two additional partial subsidiaries of AT&T were acquired by other companies after the breakup.
• These regional companies became known as Baby Bells.

AT&T Market Power

• The US vs. AT&T case (1982) is perceived as one of the most successful US antitrust cases.

• The US Department of Justice argued that AT&T’s market power led to higher prices and fewer services.
• The AT&T monopoly accounted for \(80-85\%\) of telephone lines in 1982.
• After the breakup, prices fell, and more services were provided.

Prices after the AT&T Breakup

Crandall (2007)

AT&T Price Setting

• The AT&T case does not seem to fit well the price taking behavior of firms in perfect competition.

• Was the influence of prices a special AT&T characteristic?
• What do price setting firms consider when they choose their prices?

What Firms Consider when Setting Prices?

Morris and de Vincent-Humphreys (2019)

Monopoly

• Perfect competition requires that a large number (formally an infinite number) of firms (sellers) exist in the market.

• What about the other extreme case of a single firm in a market?
• A market structure with exclusive possession of supply by a single seller is called a monopoly. The single firm (or seller) in a market is called a monopolist.
• Market demand and firm demand are identical in monopolistic markets.

Price Setting

• Is price taking behavior an appropriate assumption for monopolies?

• The justification for price taking is based on competition.
• Attempts to change prices do not work because other firms do not follow them.
• However, this is not a valid argument in a single seller market.
• A firm (or a consumer) is a price setter if it can influence the market price of the products it produces. Price setters consider market prices as (at least partially) endogenous.

An example

• Suppose that the monopolist’s cost function is \[c(q) = 2 q^{2} + 2.\]
• Let the inverse market demand be \[p(q) = 70 - 3 q \]
• The monopolist wants to maximize its profits \[\max_{q} \left\{ (70 - 3 q) q - (2 q^{2} + 2) \right\}.\]

The Profit Maximization Condition

• The monopolist wants to produce a quantity level for which its marginal cost equals its marginal revenue.
• If marginal cost is greater than marginal revenue, then the monopolist can increase its profit by reducing production.
• If marginal revenue is greater than marginal cost, then the monopolist can increase its profit by increasing production.
• Therefore, at the maximum, we should have \[70 - 6 q = 4 q\]

Monopoly vs. Competition

• The monopolist produces less than the competition quantity.
• The monopolistic price is greater than the competition price.

Monopolistic Profit

• The monopolist makes profit

  \begin{align*} \pi_{m} &= p_{m} q_{m} - c\left(q_{m}\right) \\ &= 343 - (98 + 2) \\ &= 243. \end{align*}

• In the competitive equilibrium, profit is

  \begin{align*} \pi_{c} &= p_{c} q_{c} - c\left(q_{c}\right) \\ &= 400 - (200 + 2) \\ &= 198. \end{align*}

The Welfare Effects of Monopolies

• We have shown that prices and quantities are different in competition and monopoly.

• How can we measure which situation is better?
• Which situation is in favor of the firm?
• Which situation is in favor of the consumers?
• Which situation is overall better?

Total Welfare

• The consumers' surplus is the consumer benefit from buying a commodity or service at a price lower than her reservation value.
• The producers' surplus is the producer’s benefit stemming from selling a commodity or service at a price greater than her reservation value.
• The total welfare (or economic surplus) is the sum of consumers' and producers' surplus.

Deadweight Loss

• We can compare the total welfare of the two market structures to examine which one is economically more efficient.
• The deadweight loss (or excess burden) is a measure of lost economic efficiency compared to when the socially optimal quantity of a commodity or a service is produced.
• The deadweight loss of a monopoly is the difference between the competition and the monopoly’s total welfare.

A Deadweight Loss Example

Markup Pricing

• The monopolist sets the price of the commodity it produces using a markup.
• The monopolist pricing rule can be calculated as \[p_{m} = \frac{\text{Marginal Cost}}{\frac{1}{\text{Demand Elasticity}} + 1}\]

How do Marginal Cost Changes Affect Profit?

How do Elasticity Changes Affect Profit?

Natural Monopolies

• A natural monopoly is a type of monopoly that manifests because of high start-up costs, technological barriers, or increasing returns of scale.
• Examples of entry barriers that can lead to natural monopolies are:
  • Infrastructure costs such as those of the electricity or water supply networks.
  • High-end technologies such as those needed in the production of semiconductor chips.
  • Exclusive access to a needed production factor.

When are Monopolies Economically more Likely to Appear?

Minimum Efficient Scale

• The minimum efficient scale is the lowest production scale where a firm minimizes its average cost.

Current Field Developments

• Antitrust laws and competition policies are well integrated into the US and EU economic systems.
• For instance, the EU has investigated Google on a variety of occasions for breaches of the EU competition laws since 2010.
• Extensions of the monopoly model that we have seen are actively used in economic research.
• Dynamic extensions of the monopoly model are used to study the impact of firm size on R&D.

Concise Summary

• The monopoly represents the opposite market structure of perfect competition in terms of the number of firms in the market.
• Instead of having many small, price taking firms, a monopoly consists of a single, large, price setting firm.
• Monopolies introduce economic inefficiencies due to their market power.
• They result in deadweight losses for the economy as a whole.
• Entry barriers can foster the creation of (natural) monopolies.

Further Reading

• Varian (2010, chap. 25)
• CORE Team (2017, secs. 7.5-8)
• Morris and de Vincent-Humphreys (2019)
• Crandall (2007)

Mathematical Details

Another extreme market structure case is to examine markets that consist of exactly one firm. The single firm is called a monopolist. Price taking behavior is less of a realistic assumption one can make for monopolistic markets. Market demand and firm demand are identical in monopolistic markets.

A profit maximizing monopolist

Using the cost function, the profit maximization problem of a monopolist is given by \[\max_{p} \left\{ p d(p) - c\left(d(p)\right) \right\}.\] The cost function incorporates the technological constraints faced by the firm. The profit can be decomposed into two parts, namely

• revenue \(p d(p)\) and
• cost \(c\left(d(p)\right)\).

The difference with the competition case stems from the dependence of prices on the supplied quantity via the demand function \(d(p)\).

As long as demand is a one-to-one function (i.e., there is a single price for each demanded quantity and vice versa), maximizing over quantities or prices does not affect the optimization outcome. This condition will be satisfied in most healthy markets. The problem is then written using quantities as an optimization control as \[\max_{q} \left\{ p(q) q - c\left(q\right) \right\}.\]

For non-boundary solutions, the marginal revenue and marginal cost are equalized at the maximum (why?). With price as the control, this means \[d(p) + p d'(p) = c'\left(d'(p)\right) d'(p).\] With quantity as the control, the variational condition becomes \[p_{d}'(q) q + p_{d}(q) = c'\left(q\right),\] where \(p_{d}\) is the inverse demand. The variational condition is necessary. It gives a maximum if \[p_{d}''(q) q + 2 p_{d}'(q) - c''\left(q\right) < 0.\]

Markup Pricing

The optimal price is set by \[p = \frac{c'\left(d(p)\right)}{\frac{1}{e_{d}(p)} + 1}.\]

Market Power

The market power of the monopolist depends on the elasticity of demand. From the variational condition with price as the control, we get

\begin{align*} c'\left(d(p)\right) &= \frac{d(p)}{d'(p)} + p \\ &= p\frac{d(p)}{p}\frac{1}{d'(p)} + p \\ &= p \left(\frac{1}{e_{d}(p)} + 1 \right) \\ \end{align*}

The greater is \(e_{d}\) (i.e., the smaller is the absolute value of the elasticity), demand becomes more inelastic, and the monopolist’s profit increases.

Welfare Analysis

Market power usually introduces inefficiencies in the market structure. The monopolist produces less than the competitive quantity and sells at a price higher than the competitive price. How can we measure the economic loss? Let \(p_{m}\) and \(p_{c}\) correspondingly denote the monopoly and competition prices.

The total welfare in the competition case can be calculated by

\begin{align*} W(q_{c}) &= \Pi_{c}(q_{c}) + \Pi_{s}(q_{c}) \\ &= \int_{0}^{q_{c}} \left( p(q) - c'(q) \right) \mathrm{d}q \\ &= \int_{0}^{q_{c}} p(q) \mathrm{d}q - \mu(q_{c}) . \end{align*}

Similarly, the total welfare in the monopoly case is given by \[W(q_{m}) = \int_{0}^{q_{m}} p(q) \mathrm{d}q - \mu(q_{m}).\] The difference is called deadweight loss

\begin{align*} D(q_{c}, q_{m}) &= W(q_{c}) - W(q_{m}) \\ &= \int_{q_{m}}^{q_{c}} p(q) \mathrm{d}q - \left[ \mu(q_{c}) - \mu(q_{m}) \right] . \end{align*}

Minimum efficient scale

The minimum efficient scale is the level of output that minimizes average cost, namely \[\mathrm{MES} = \mathrm{arg\,min}_{q} \bar{c}(q).\] The greater is the ratio of the minimum efficient scale to the demanded quantity corresponding to the price being equal to the minimum average cost, i.e., \[\frac{\mathrm{MES}}{d(\bar{c}(\mathrm{MES}))},\] the more possible become monopolistic structures.

An affine demand and cost example

For this example let the cost function be given by \[c(q) = c_{1} q + c_{0} \quad\quad (c_{1},c_{0} > 0).\] Let also the inverse demand function be \[p(q) = p_{0} + p_{1} q \quad\quad (p_{0} > 0, p_{1} < 0).\]

Monopoly and competition quantities

For the monopoly, some simple calculations give \[q_{m} = \frac{c_{1}-p_{0}}{2p_{1}}.\] For the competition case, we have \[q_{c} = \frac{c_{1}-p_{0}}{p_{1}}.\] Solutions are valid for \(p_{0} > c_{1}\) because otherwise, the quantities become negative.

How do marginal cost changes affect profit?

The monopolist’s profit is

\begin{align*} \pi_{m} &= p(q_{m})q_{m} - c(q_{m}) \\ &= -\frac{(c_{1}-p_{0})^{2}}{4 p_{1}} - c_{0} . \end{align*}

The profit decreases as marginal cost (here \(c_{1}\)) increases \[\frac{\mathrm{d}\pi_{m}}{\mathrm{d} c_{1}} = -\frac{c_{1}-p_{0}}{2 p_{1}} < 0.\]

How do elasticity changes affect profit?

The elasticity of demand at the monopolistic quantity is \[e_{d} = \frac{1}{p_{1}} \frac{p_{1}q_{m} + p_{0}}{q_{m}} = \frac{c_{1} + p_{0}}{c_{1} - p_{0}}.\] Keeping the marginal cost constant, elasticity changes with \(p_{0}\). That is \[\frac{\partial e_{d}}{\partial p_{0}} = \frac{2 c_{1}}{(c_{1} - p_{0})^{2}} > 0.\] Therefore, profit increases as the elasticity \(e_{d}\) increases (\(\left|e_{d}\right|\) decreases and demand becomes more inelastic), i.e.,

\begin{align*} \frac{\mathrm{d}\ \pi_{m}}{\mathrm{d}\ {e_{d}}} &= \frac{\mathrm{d}\ \pi_{m}}{\mathrm{d}\ p_{0}} \frac{1}{\frac{\partial e_{d}}{\partial p_{0}}} \\ &= \frac{c_{1}-p_{0}}{2 p_{1}} \frac{(c_{1} - p_{0})^{2}}{2 c_{1}} > 0 . \end{align*}

Deadweight loss

The deadweight loss is

\begin{align*} D(q_{c}, q_{m}) &= \int_{q_{m}}^{q_{c}} \left[ p(q) - c'(q) \right] \mathrm{d}q \\ &= \int_{q_{m}}^{q_{c}} \left[ p_{0} + p_{1}q - c_{1} \right] \mathrm{d}q \\ &= \left[ \left( p_{0} - c_{1} \right) q + \frac{p_{1}}{2} q^{2} \right]_{q_{m}}^{q_{c}} \\ &= -\frac{(c_{1} - p_{0})^{2}}{8 p_{1}} . \end{align*}

An affine demand, cubic cost example

The calculations of this example are used for producing the figures of sections How do Marginal Cost Changes Affect Profit? and How do Elasticity Changes Affect Profit?. Let the cost function be given by \[c(q) = q^{3} - 2 q_{2} q^{2} + (q_{2}^{2} + q_{1}) q + q_{0} \quad\quad (q_{2},q_{1},q_{0} > 0).\] Let also the inverse demand function be \[p(q) = p_{1} q + p_{0} \quad\quad (p_{0} > 0, p_{1} < 0).\]

Monopoly and competition quantities

For the monopolistic market structure, some tedious calculations give

\begin{align*} p'(q) q + p(q) &= c'(q) \implies \\ q_{m} &= \frac{1}{3} \left( 2 q_{2} + p_{1} + \sqrt{q_{2}^{2} - 3q_{1} + p_{1}^{2} + 4q_{2}p_{1} + 3p_{0}} \right) . \end{align*}

For the competition case, we have

\begin{align*} p(q) &= c'(q) \implies \\ q_{c} &= \frac{1}{6} \left( 4 q_{2} + p_{1} + \sqrt{4q_{2}^{2} - 12q_{1} + p_{1}^{2} + 8q_{2}p_{1} + 12p_{0}} \right) . \end{align*}

Exercises

Group A

1. Consider a monopolist with cost function \(c(q) = 2 q\). Suppose that market demand is given by \(d(p) = 100 - 2p\). Calculate the monopolist’s price, quantity, and profit. In addition, calculate the deadweight loss.

   The monopolist solves \[ \max_{p}\left\{ p d(p) - c\left(d(p)\right) \right\}. \] The first order condition of the problem is \[ d(p) + p d'(p) - c'\left(d(p)\right) d'(p) = 0. \] Substituting the exercise’s functional forms and calculating gives \(p_{m} = 26\). Substituting back to demand gives \(q_{m} = d(26) = 48\). The monopolist achieves profit \[ \pi(26) = 2 d(26) - c\left(d(26)\right) = 26 \cdot 48 - 2 \cdot 48 = 1152. \] When demand is affine, the deadweight loss can be calculated using a geometric argument and the formula for the area of a triangle. For this, we need the perfect competition price and quantity. The perfect competition price is given by \(p_{c} = 2\) (why?) and, the perfect competition quantity is \(q_{c} = 96\). One then calculates \[ D\left(q_{c}, q_{m}\right) = \frac{1}{2} \left(q_{c} - q_{m}\right) \left(p_{m} - p_{c}\right) = \frac{1}{2} (96 - 48)(26 - 2) = 576 \]

2. Consider a monopolist with cost function \(c(q) = q^{2}\). Suppose that market demand is given by \(d(p) = 100 / p\). Calculate the quantity that the monopolist would like to produce.

   There are two differences compared to exercise 1. Firstly, demand is hyperbolic instead of affine, and secondly, the cost function is quadratic instead of linear. In this case, the maximization problem of the monopolist is \[ \max_{p} \left\{ p \frac{100}{p} - \frac{100^{2}}{p^{2}} \right\} = \max_{p} \left\{ 100 - \frac{100^{2}}{p^{2}} \right\}. \] The derivative of the objective is \(2 \cdot 10^{4} / p^{3}\), which is positive for all positive prices. This implies that the objective is increasing and, therefore, the maximization problem does not have a solution. The monopolist would like to let prices go to infinity. Taking this limit in the demand equation, we see that the monopolist would like to abstain from producing, i.e., \[ \lim_{p\to\infty} d(p) = \lim_{p\to\infty} \frac{100}{p} = 0. \]

3. Consider a monopolist operating in a market where the commodity’s demand depends on a qualitative attribute of the product besides its price. Specifically, demand is given by \(d(s, p) = 3 \sqrt{s} / p^{3}\), where \(s\) is the qualitative attribute of the market’s commodity. The cost function of the monopolist also directly depends on the qualitative attribute, i.e., \(c(s,q) = s/2 + 2 q\). The monopolist’s goal is to maximize profit by simultaneously choosing both the qualitative attribute level and the commodity’s price. Calculate the monopolist price, the optimal level of the qualitative attribute, the output level, and the profit.

   The problem of the monopolist is \[ \max_{s, p} \left\{ p d(s, p) - \frac{s}{2} - 2 d(s, p) \right\} = \max_{s, p} \left\{ 3 \sqrt{s} \left( \frac{1}{p^{2}} - 2 \frac{1}{p^{3}} \right) - \frac{s}{2} \right\}. \] The first order conditions of the problem are

   \begin{align*} -2 \frac{1}{p^{3}} + 6 \frac{1}{p^{4}} &= 0, \\ \frac{3}{2} \frac{1}{\sqrt{s}} \left( \frac{1}{p^{2}} - 2 \frac{1}{p^{3}} \right) - \frac{1}{2} &= 0. \end{align*}

   The first condition involves only \(p\), and the monopolistic price can be calculated from it. Solving for prices gives \(p=3\). By substituting \(p\) in the second condition, one can then solve for the qualitative attribute to get \(s=1/81\). Using the demand function, the output quantity level that the monopolist produces is \(q = 1/81\). Finally, the profit is \[ \pi = 3 \sqrt{\frac{1}{81}} \left( \frac{1}{9} - 2 \frac{1}{27} \right) - \frac{1}{2}\frac{1}{81} = \frac{1}{162} . \]

4. Can the monopolist select a price level for which demand is inelastic (i.e., \(0 > e_{d}>-1\))?

   No. By the monopolistic pricing condition, one can see that the monopolist would like to set a negative price in such a case. Specifically,

   \begin{align*} p = \frac{c'\left(d(p)\right) e_{d}}{e_{d} + 1} < 0. \end{align*}

5. Suppose that the elasticity of demand \(e_{d}<-1\) and the marginal production cost \(c_{1}>0\) are constant. Calculate the marginal effect on prices of an improvement in the production technology reducing the marginal cost in a monopolistic market.

   The monopolistic pricing condition is

   \begin{align*} p = \frac{c_{1} e_{d}}{e_{d} + 1}, \end{align*}

   where elasticity does not depend on prices by the assumptions of the exercise. We can calculate

   \begin{align*} \frac{\mathrm{d}\, p}{\mathrm{d}\, c_{1}} &= \frac{e_{d}}{e_{d} + 1}. \end{align*}

   Since \(e_{d} < -1\), then \(\mathrm{d}\, p / \mathrm{d}\, c_{1} > 0\). This implies that whenever a technological improvement reduces marginal cost, the monopolistic price in the market falls.

Group B

1. Consider a monopolist with cost function \(c(q) = c_{1} q\). Suppose that market demand is given by \(d(p) = a / p^{b}\) for \(a>0\) and \(b \ge 1\). Calculate the monopolist’s price, quantity, and profit. In addition, calculate the deadweight loss for \(c_{1}=2\), \(a=16\), and \(b=2\).

   The difference with exercise 1 is that demand, in this case, is hyperbolic instead of affine. The first order condition of this problem yields \[ \frac{a}{p^{b}} + p \frac{-a b}{p^{b+1}} - c_{1} \frac{-a b}{p^{b+1}} = 0, \] which gives \(p_{m}= c_{1}b / (b-1)\) and \(q_{m} = a (b-1)^{b} / (c_{1}b)^{b}\). The monopolist makes profit \[ \pi_{m} = \frac{c_{1}b}{b-1} \frac{a (b-1)^{b}}{(c_{1}b)^{b}} - c_{1} \frac{a (b-1)^{b}}{(c_{1}b)^{b}} = \frac{c_{1} a (b-1)^{b-1}}{(c_{1}b)^{b}}. \] The competition price is \(p_{c}=c_{1}\) and the competition quantity is \(q_{c} = a / c_{1}^{b}\). The deadweight loss can be calculated by \[ D\left(q_{c}, q_{m}\right) = \int_{q_{m}}^{q_{c}} \left(p(z) - c'(z)\right) \mathrm{d} z, \] where \(p(q) = (a / q)^{1/b}\) is the inverse demand function. For the given parameter values, we get \(q_{c} = 4\), \(q_{m} = 1\), and

   \begin{align*} D\left(q_{c}, q_{m}\right) &= \int_{q_{m}}^{q_{c}} \left(a^{1/b} z^{-1/b} - c_{1} \right) \mathrm{d} z \\ &= \int_{1}^{4} \left(4 z^{-1/2} - 2 \right) \mathrm{d} z \\ &= \left. 8 z^{1/2} - 2 z \right|_{1}^{4} \\ &= 16 - 8 - (8 - 2) \\ &= 2 \end{align*}

2. Show that if demand is affine and the marginal cost is constant, the rate of change of prices with respect to marginal cost in a monopolistic market is \(1/2\).

   Suppose that marginal cost is equal to \(c\). The monopolistic pricing condition is

   \begin{align*} p = \frac{c e_{d}(p)}{e_{d}(p) + 1}. \end{align*}

   Therefore, we can calculate

   \begin{align*} \frac{\mathrm{d}\, p}{\mathrm{d}\, c} &= \frac{e_{d}(p)}{e_{d}(p) + 1} + \frac{c}{\left(e_{d}(p) + 1\right)^{2}} \left(e_{d}'(p) \frac{\mathrm{d}\, p}{\mathrm{d}\, c} \left(e_{d}(p) + 1\right) - e_{d}(p) e_{d}'(p) \frac{\mathrm{d}\, p}{\mathrm{d}\, c}\right) \\ &= \frac{e_{d}(p)}{e_{d}(p) + 1} + \frac{c}{\left(e_{d}(p) + 1\right)^{2}} e_{d}'(p) \frac{\mathrm{d}\, p}{\mathrm{d}\, c}, \end{align*}

   from which we get

   \begin{align*} \frac{\mathrm{d}\, p}{\mathrm{d}\, c} &= \frac{e_{d}(p) \left(e_{d}(p) + 1\right)}{\left(e_{d}(p) + 1\right)^{2} - c e_{d}'(p)}. \end{align*}

   Thus, we also need to calculate

   \begin{align*} e_{d}'(p) &= \frac{\mathrm{d}\, \left(d'(p)\frac{p}{d(p)}\right)}{\mathrm{d}\, p} \\ &= d''(p)\frac{p}{d(p)} + d'(p)\frac{1}{\left(d(p)\right)^{2}} \left(d(p) - p d'(p) \right) \\ &= d''(p)\frac{p}{d(p)} + \frac{1}{p} d'(p)\frac{p}{d(p)} \left(1 - d'(p)\frac{p}{d(p)} \right) \\ &= d''(p)\frac{p}{d(p)} + \frac{1}{p} e_{d}(p) \left(1 - e_{d}(p) \right) . \end{align*}

   Firstly, since demand is affine, we have \(d'' = 0\) and, secondly, by the monopolist’s pricing condition, we can rewrite the last expression as

   \begin{align*} e_{d}'(p) &= \frac{e_{d}(p) + 1}{c} \left(1 - e_{d}(p) \right) = \frac{1 - \left(e_{d}(p)\right)^{2}}{c} . \end{align*}

   Substituting this result in the expression for \(\mathrm{d}\, p / \mathrm{d}\, c\), we obtain

   \begin{align*} \frac{\mathrm{d}\, p}{\mathrm{d}\, c} &= \frac{e_{d}(p) \left(e_{d}(p) + 1\right)}{\left(e_{d}(p) + 1\right)^{2} - 1 + \left(e_{d}(p)\right)^{2}} = \frac{1}{2}. \end{align*}

References

References

Topic's Concepts

• minimum efficient scale
• natural monopoly
• deadweight loss of a monopoly
• deadweight loss
• total welfare
• producers surplus
• consumers surplus
• price setter
• monopolist
• monopoly