Oligopoly

Context

• Many real markets are neither perfectly competitive nor monopolies. Instead, they are oligopolies comprising a small number of firms that have large enough market shares and can influence prices.

• Nonetheless, firms' profits do not exclusively depend on their own choices. Their small numbers allow them to utilize a variety of cooperation and competition strategies.
  • How do firms strategically interact?
  • What means do they use to compete?
  • How do the welfare outcomes of oligopolies compare with those of monopolies and perfect competition?

Course Structure Overview

Lecture Structure and Learning Objectives

Structure

• Our Customers are our Enemies (Case Study)
• Basic Concepts
• The Cournot and Bertrand models
• Examples of the two models
• Welfare Comparisons
• Current Field Developments

Learning Objectives

• Explain how game theory models oligopolistic competition.
• Describe oligopolies with competition in quantities and their welfare output.
• Describe oligopolies with competition in prices and their welfare output.
• Illustrate the welfare differences between oligopolies, monopolies, and perfect competition.

Our Customers are Our Enemies

• Lysine is an amino acid that speeds the development of lean muscle tissue in humans and animals.
• It is essential for humans, but we cannot synthesize it.
• It has to be obtained from the diet.

The Lysine Industry

• At the end of the 1980s, the world lysine industry consisted of three significant sellers:
  • Ajinomoto,
  • Kyowa, and
  • Sewon.
• The three largest consumption regions were Japan, Europe, and North America.
• Most production took place in Japan, but it was based on imports of US dextrose.
• Ajinomoto had the largest share of the world market.

The ADM Entry

• In February 1991, Archer Daniel Midland Co. (ADM) entered the market and built by far the world’s largest lysine plant in the US.

• ADM hired biochemist Mark Whitacre, Ph.D., as head of the new division.
• ADM’s plant was three times the size of Ajinomoto’s largest plant.
• ADM gave Ajinomoto and Kyowa executives an unrestricted tour to show its production capacity.
• Companies engaged in a price war.
  • Three months before ADM’s entry, the average US lysine price was \($1.22\) per pound.
  • After an 18-month price war, the US price averaged \($0.68\) per pound.
  • ADM’s share of the US market reached \(80\%\).

The Lysine Association

• After the price war, ADM was willing to soften competition.

• In 1992, Mark Whitacre and his boss Terrance Wilson met with top Ajinomoto and Kyowa managers.
• Wilson proposed forming a world lysine association that would regularly meet.
• The new association would collect and distribute market information.
• Wilson also suggested that the new association could provide a convenient cover for illegal price-fixing discussions! (Connor 2001)
• After a year, the lysine association was founded, met quarterly, and performed the two functions that Wilson proposed.

Price Fixing

Connor (2000)

• There were \(25\) price fixing meetings in total.

• The first one took place in the Nikko Hotel in Mexico on June 23, 1992.
• The average Lysine price immediately jumped by more than \(12\%\).
• Consensus was not always easy to reach. The companies distrusted each other!
• There was a breakdown of the cartel during the spring and summer of 1993, and the lysine price plummeted.
• The crisis was resolved at a meeting in Irvine, California in October 1993 between ADM’s and Ajinomoto Executives. But…
• This meeting and many others were caught on video by the FBI.

Frenemies

WILSON: The only thing we need to talk here because we are gonna get manipulated by these God damn buyers, they’re sh, they can be smarter than us if we let them be smarter.

MIMOTO: (Laughs).

WILSON: Okay?

MIMOTO: (ui).

WILSON: They are not your friend. They are not my friend. And we gotta have ‘em. Thank God we gotta have ‘em, but they are not my friends. You’re my friend. I wanna be closer to you than I am to any customer. ‘Cause you can make us, I can make money, I can’t make money. At least in this kind of a market. And all I wanna is ta tell you again is let’s-let’s put the prices on the board.

Co-conspirator explains how end-of-year compensation scheme eliminates incentives to cheat on cartel

The Whistle-blower

• In 1992, Mark Whitacre became an FBI whistle-blower. He is the highest-level corporate executive to ever have done so.
• Whitacre’s wife pressured him into becoming a whistle-blower. She threatened to inform the FBI herself if Whitacre wouldn’t do it.
• Whitacre informed the FBI that he and other ADM executives performed illegal price-fixing operations.
• Over the next three years, Whitacre collected information and recorded conversations with ADM executives and competitors.
• The ADM settlement involved the greatest US federal charges at the time.
• Whitacre was convicted for embezzling and money laundering, which he performed while cooperating with the FBI in the price fixing case.

What can we Learn?

• Why did ADM initially engage in a price war?

• Why did it initiate the price fixing discussions afterward?
• Why was there so much distrust among companies?

Competition and Cooperation

• Oligopoly refers to market structures with a small number of interdependent firms.
  • Oligopolistic firms typically compete using non-cooperative strategies.
  • On some occasions, firms collude and use cooperative strategies.

Non-cooperative overview

• Oligopolies may compete using pricing strategies or by choosing quantities.
• Different means of competition strategies crucially affect the market outcome.
• The means of competition are decisive components of the market structure.

Cooperation and collusion

• Oligopolies have used explicitly collusive strategies in the past (e.g., cartels).
• Nowadays, collusion is usually illegal.
• Instances of tacit collusion have also been documented.
• Tacit collusion strategies do not explicitly require the exchange of information.

Competition in Quantities

• The Cournot model of oligopoly describes a market structure with two or more firms such that
  • the market does not suffer from any other market failure (imperfect information, externalities, etc.),
  • no other firms can enter the market,
  • firms sell a homogeneous product,
  • firms try to maximize their profits,
  • consumers are price takers; firms are simultaneously choosing the quantities that they produce, and
  • consumers try to maximize their utility.

A Cournot Competition Example

• Suppose that there are two firms \((i, j \in \{1,2\})\) in the market.
• They have the cost functions

\[c(q_{i}) = 4 q_{i}.\]

• Let the market inverse demand be

\[p(q_{i} + q_{j}) = 28 - 2 \left(q_{i} + q_{j}\right).\]

• Each firm maximizes its profit

\[\max_{q_{i}} \left\{ \left( 28 - 2 \left(q_{i} + q_{j}\right) \right) q_{i} - 4 q_{i} \right\}.\]

Best responses

• The necessary condition for each firm is

\[28 - 2 q_{j} - 4 q_{i} = 4\]

• Solving for \(q_{i}\) gives the best response of firm \(i\)

\[q_{i} = \frac{24 - 2 q_{j}}{4}.\]

Nash Equilibrium

• Combining the two best responses gives the Nash equilibrium

\[q_{i} = 4 = q_{j}.\]

• Profits are then

\[\pi_{i} = 32.\]

What happens if costs are not symmetric?

Competition in Prices

• The Bertrand model of oligopoly describes a market structure with two or more firms such that
  • the market does not suffer from any other market failure (imperfect information, externalities, etc.),
  • no other firms can enter the market,
  • firms sell a homogeneous product,
  • firms try to maximize their profits,
  • consumers are price takers, and firms are simultaneously choosing prices, and
  • consumers try to maximize their utility.

A Bertrand Competition Example

• Suppose that there are two firms \((i, j \in \{1,2\})\) in the market.

• Their marginal costs are equal to \(4\).

• Let the demand for firm \(i\) be

  \begin{align*} d_{i}(p_{i}, p_{j}) = \left\{\begin{aligned} &10 - \frac{1}{2}p_{i} & p_{i} < p_{j} \\ &5 - \frac{1}{4}p_{i} & p_{i}=p_{j} \\ &0 & p_{i} > p_{j} \end{aligned}\right.. \end{align*}

• The firm with the lowest price gets all the demand.

• If prices are equal, demand is equally split.

Non-Equilibrium Prices

• Suppose that firm \(j\) sets a price \(p_{j}\) that is greater than the marginal cost of firm \(i\) (i.e., \(4\)).
• Firm \(i\) can undercut by a small amount and grab all the market. For instance, set price \(p_{i} = \frac{p_{j} + 4}{2}\).
• Thus, firm \(i\) can only set a price equal to firm \(j\)’s marginal cost.
• Analogous arguments hold for firm \(j\)’s strategy.

Equilibrium

• The only possible equilibrium is to set a price equal to the (common) marginal cost.
• Firms do not have any incentive to deviate.
• Setting lower prices leads to losses.
• Setting higher prices leads to zero profit.
• Even with two firms, price competition leads to price setting similar to the competitive equilibrium case.

Current Field Developments

• There are two main types of extensions of the basic models (Cournot and Bertrand),
  • extensions incorporating dynamic decisions (e.g., Stackelberg)., and
  • extensions incorporating dynamics under uncertainty.
• Oligopoly models are primarily used in industrial organization (see Belleflamme and Peitz (2010) for an introduction) to examine
  • Market power
  • Pricing strategies,
  • Competition policies, and
  • R&D and innovation.
• Some recent micro-founded, general equilibrium macro models describe frictions with oligopolistic markets.

Concise Summary

• Competition is not always perfect.
• In reality, a few large firms have the lion’s share in many markets.
• Such markets are described by oligopoly models.
• Oligopolies can compete or collude. Explicit collusion is illegal in the US and EU.
• Depending on how firms compete (prices or quantities) and the number of firms, the oligopoly model gives predictions with welfare properties that range from perfect competition to monopoly.

Further Reading

• Varian (2010, chap. 28)
• Belleflamme and Peitz (2010, secs. 3.1.1, 3.2)
• Connor (2001)

Mathematical Details

Simultaneous quantity competition with two firms

Firms choose their strategies at the same time. Both firms choose their supplied quantities. This market structure is known as the Cournot model of competition.

The problem

Each firm solves \[\max_{q_{i}} \left\{ p(q_{1} + q_{2}) q_{i} - c(q_{i}) \right\}.\]

Best responses

The necessary condition for each firm is \[p’(q_{1} + q_{2}) q_{i} + p(q_{1} + q_{2}) = c’(q_{i}).\] From these conditions, the two best responses are obtained \[q_{i} = b_{i}(q_{j}) \quad\quad (i\neq j).\] Solving the system of these two equations gives the equilibrium point (if it exists).

An affine demand and symmetric, linear costs example

For inverse demand and costs given by

\begin{align*} p(q) &= p_{0} + p_{1} q \\ c(q) &= c_{1} q, \end{align*}

the best responses become \[q_{i} = \frac{c_{1} - p_{0} - p_{1} q_{j}}{2 p_{1}}.\] The equilibrium quantities are given by \[q_{i} = \frac{c_{1} - p_{0}}{3 p_{1}}.\] The profits are symmetric and can be calculated as \[\pi_{i} = -\frac{(c_{1} - p_{0})^{2}}{9 p_{1}}.\]

An affine demand and non-symmetric, linear costs example

What happens if costs are not symmetric in the affine example? Suppose demand is as before and costs are \[c_{i}(q) = c_{1, i} q, \quad\quad (i = \{1, 2\})\] The best responses become \[q_{i} = \frac{c_{1, i} - p_{0} - p_{1} q_{j}}{2 p_{1}}.\] Equilibrium ceases to be symmetric. The equilibrium quantities are given by \[q_{i} = \frac{2c_{1, i} - c_{1, j} - p_{0}}{3 p_{1}}.\] The symmetric equilibrium quantities are obtained as a special case from the last formula by setting \(c_{1, i} = c_{2, j}\).

Firm \(i\) produces more than \(j\) if and only \[\frac{2c_{1, i} - c_{1, j} - p_{0}}{3 p_{1}} \ge \frac{2c_{1, j} - c_{1, i} - p_{0}}{3 p_{1}},\] which, because \(p_{1} < 0\), is equivalent to \(c_{1, i} \le c_{1, j}\). Thus, the lower-cost firm produces more.

The total market quantity is \[q = \frac{2c_{1, i} - c_{1, j} - p_{0}}{3 p_{1}} + \frac{2c_{1, j} - c_{1, i} - p_{0}}{3 p_{1}} = \frac{c_{1, i} + c_{1, j} - 2p_{0}}{3 p_{1}},\] and the market price \[p(q) = \frac{c_{1, i} + c_{1, j} + p_{0}}{3}.\] We can then calculate the profit of firm \(i\) as

\begin{align*} \pi_{i} &= \left(\frac{c_{1, i} + c_{1, j} + p_{0}}{3} - c_{1, i}\right) \frac{2c_{1, i} - c_{1, j} - p_{0}}{3 p_{1}} \\ & = -\frac{(2c_{1, i} - c_{1, j} - p_{0})^{2}}{9 p_{1}}. \end{align*}

The firm that produces more makes the greatest profit. The easiest way to get this result is to rewrite profits as \(\pi_{i} = -q_{i}^{2}p_{1}\). Since \(p_{1}<0\), we have \(\pi_{i}\ge \pi_{j}\) if and only if \(q_{i}\ge q_{j}\).

Simultaneous quantity competition with more than two firms

We extend the problem by allowing \(n>2\) firms that simultaneously choose their strategies. All firms choose their supplied quantities.

How is equilibrium affected when more than two firms are in the market? Each firm solves \[\max_{q_{i}} \left\{ p\left( \sum_{j=1}^{n} q_{j} \right) q_{i} - c(q_{i}) \right\}.\] Analogously to the two-firm case, we obtain \(n\) best response functions \[q_{i} = b_{i}\left((q_{j})_{j\neq i}\right) \quad\quad (i = 1,… , n).\] Solutions to the system of best responses (if any) are the Nash equilibrium of this oligopoly model.

An affine demand and symmetric, linear costs example

The best responses become \[q_{i} = \frac{c_{1} - p_{0} - p_{1} \sum_{j\neq i} q_{j}}{2 p_{1}}.\] The equilibrium quantities are given by \[q_{i} = \frac{c_{1} - p_{0}}{(n + 1) p_{1}}.\] Profits are then \[\pi_{i} = -\frac{(c_{1} - p_{0})^{2}}{(n + 1)^{2} p_{1}}.\] The total quantity is \[q = \sum_{i=1}^{n} q_{i} = \sum_{i=1}^{n} \frac{c_{1} - p_{0}}{(n + 1) p_{1}} = \frac{n}{n+1}\frac{c_{1} - p_{0}}{p_{1}},\] so the market price becomes \[p(q) = p_{0} + p_{1} n \frac{c_{1} - p_{0}}{(n + 1) p_{1}} = \frac{n c_{1} + p_{0}}{n + 1}.\] The case of two firms can be obtained by replacing \(n=2\) in the above results. We can also obtain the solution to the monopoly problem if we set \(n=1\).

Profits decrease as the number of firms in the market increases, with the limiting case being \[\pi_{i} \xrightarrow[n\to \infty]{} 0.\] In addition, we have \(p(q) \to c_{1}\), \(q_{i}\to 0\), and \(q\to\frac{c_{1}-p_{0}}{p_{1}}\) as \(n\to\infty\). The production of each firm becomes negligible, and the total market quantity and price approach those of perfect competition.

Exercises

Group A

1. Suppose that there are two firms in a market with an affine inverse demand function \(p(q) = p_{0} + p_{1}q\), where \(p_{0}>0\) and \(p_{1}<0\). The firms compete by simultaneously choosing quantities and have constant marginal costs equal to \(c_{1}\). Production has no fixed cost. Find the equilibrium outputs, price, and profits.

   Let \(i\neq j\) for \(i,j=1,2\). Firm \(i\) chooses its strategy by solving \[ \max_{q_{i}} \left\{ p(q_{i} + q_{j}) q_{i} - c_{1} q_{i} \right\}. \] For interior solutions, the firm sets

   \begin{align*} p_{0} + 2p_{1}q_{i} + p_{1}q_{j} - c_{1} = 0, \end{align*}

   which gives the best response function

   \begin{align*} b_{i}(q_{j}) = \frac{c_{1} - p_{0} - p_{1}q_{j}}{2p_{1}} . \end{align*}

   Since the problem is symmetric, we can set \(q_{i}=q_{j}\) in the first order condition to easily obtain the equilibrium output

   \begin{align*} q_{i} = \frac{c_{1} - p_{0}}{3p_{1}} . \end{align*}

   Thus, the total market quantity is

   \begin{align*} q = q_{1} + q_{2} = 2\frac{c_{1} - p_{0}}{3p_{1}}, \end{align*}

   which implies that the equilibrium price is

   \begin{align*} p(q) = \frac{p_{0} + 2 c_{1}}{3}. \end{align*}

   Finally, each firm \(i\) makes profit \[ \pi_{i}= (p(q) - c_{1}) q_{i} = - \frac{\left(c_{1} - p_{0}\right)^{2}}{9p_{1}} . \]

2. Can an oligopolistic market structure result in an efficient level of output?

   Yes. The simultaneous price competition with homogeneous products leads firms to choose the competitive price, which, in turn, results in the production of the efficient output level. In contrast, competition in quantities results in deadweight losses.

3. Consider a market with inverse demand function \(p(q) = 100 - 2q\). The total cost function for any firm in the market is given by \(c(q) = 4q\).

   1. What is the marginal cost for any firm in the market?
   2. Calculate the perfect competition market quantity and price.
   3. Suppose that the market consists of two firms that compete by simultaneously choosing their supplied quantities. Find the best responses and the equilibrium market output and price?
   4. Draw the two best responses in a single graph to indicate the equilibrium point.
   5. Suppose that the two firms collude by maximizing and splitting their joint profit. Find market output and the market price. Compare them with the non-collusive market output and price.
   6. Suppose that a firm decides to deviate from the collusive output, while the other one keeps its collusive strategy unchanged. What are the firms’ output levels and profits?
   7. Is the collusion strategy sustainable in this game?

   1. \(c'(q) = 4\).

   2. The perfect competition price is equal to the marginal production cost, i.e., \(p_{c}=4\). By inverting the inverse demand and substituting the competitive price, we obtain

      \begin{align*} q_{c} = \frac{100 - p_{c}}{2} = 48. \end{align*}

   3. Each firm solves

      \begin{align*} \max_{q_{i}} \left\{ p\left( q_{i} + q_{j} \right) q_{i} - c(q_{i}) \right\}, \end{align*}

      with first order condition

      \begin{align*} 100 - 4q_{i} - 2 q_{j} -4 = 0. \end{align*}

      Therefore, the best response of each firm is

      \begin{align*} b_{i}(q_{j}) = \frac{48 - q_{j}}{2} \end{align*}

      and the firm’s optimal quantity is \(q_{i} = 16\). The market quantity is \(q_{s} = 32\), and the market price is \(p_{s} = 36\).

   4. The intersection of the best responses indicates the equilibrium quantities.

   5. In this case, the two firms solve the monopolist problem and split the profit. The monopolistic quantity is given by

      \begin{align*} q_{m} = \frac{100 - 4}{4} = 24, \end{align*}

      in which case, the market price is \(p_{m}=52\). With collusion, the market power of the firms increases, the produced output level decreases (\(q_{m} = 24 < 32 =q_{s}\)), while the market price increases (\(p_{m} = 52 > 36 =p_{s}\)).

   6. Suppose that firm \(i\) deviates from collusion, while firm \(j\) keeps producing half of the monopolistic quantity. The best deviation of firm \(i\) can be calculated from its best response function, namely

      \begin{align*} q_{b} = b_{i}\left(\frac{q_{m}}{2}\right) = \frac{48 - 12}{2} = 18. \end{align*}

      The best deviation for the firm is to produce at an output level greater than the level of the simultaneous quantity competition (i.e., \(q_{b} = 18 > 16 = q_{s}/2\)). Firm \(i\) takes advantage of the fact that firm \(j\) is not producing at its best response output level and produces more to grab a greater share of the market. The market quantity is

      \begin{align*} q_{w} = q_{b} + \frac{q_{m}}{2} = 18 + 12 = 30. \end{align*}

      Then, the price is \(p_{w} = 40\), and the firms' profits are

      \begin{align*} \pi_{i} &= (p_{w}-c)q_{b} = 36 \cdot 18 = 648 \\ &> 576 = 48 \cdot 12 = (p_{m}-c)\frac{q_{m}}{2} = \frac{\pi_{m}}{2} \end{align*}

      and

      \begin{align*} \pi_{j} &= (p_{w}-c)\frac{q_{m}}{2} = 36 \cdot 12 = 432 < 576 = \frac{\pi_{m}}{2} . \end{align*}

   7. No. Each firm has a profitable deviation from producing half of the monopolistic output in this static game. More than a single date is needed for firms to be able to collude successfully.

Group B

1. Give an example in which there is no equilibrium in a duopoly market in which firms simultaneously compete using quantities. Draw the best responses of the firms in this market.

   Suppose that the inverse demand function of the market is given by \(p(q) = p_{0} + p_{1} q = 25 - 4 q\). Let the marginal cost of firm 1 be \(c_{1}=15\) and that of firm 2 to be \(c_{2}=1\). For these values, if there was an equilibrium in this market, according to the best response functions, firm 1 should produce

   \begin{align*} q_{1} = \frac{2c_{1} - c_{2} - p_{0}}{3 p_{1}} = -\frac{30 - 1 - 25}{12} < 0. \end{align*}

   Therefore, this market does not have an equilibrium. Visually, this results in two best response functions that do not cross each other.

2. Suppose there are \(n\) identical firms in a market competing by simultaneously choosing quantities. Show that the elasticity of the market demand must be less than \(-1/n\) for an equilibrium to exist.

   This result is the analog of the upper bound for the elasticity of monopolistic markets discussed in exercise 4. In particular, the exact result of the monopolistic case is obtained for \(n=1\). To simplify notation, let \(q = \sum_{j=1}^{n} q_{j}\). Each firm solves

   \begin{align*} \max_{q_{i}} \left\{ p\left( q \right) q_{i} - c(q_{i}) \right\}, \end{align*}

   with the first order condition of firm \(i\) being

   \begin{align*} p'\left(q\right) q_{i} + p\left(q\right) - c'(q_{i}) = 0. \end{align*}

   By the inverse function theorem, we can rewrite the above condition as

   \begin{align*} 0 &= \frac{1}{d'\left(p(q)\right)} q_{i} + p\left(q\right) - c'(q_{i}), \end{align*}

   where \(d\) is the demand function, and, then, express it in terms of the demand elasticity as

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

   Since all firms are identical, the equilibrium, if it exists, is symmetric. This means that \(q_{i} = q / n\). Substituting the symmetry condition in the first order condition gives

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

   Since the marginal cost is positive, positive prices require that

   \begin{align*} \frac{1}{e_{d}\left(p(q)\right)} \frac{1}{n} + 1 > 0, \end{align*}

   which is equivalent to \(e_{d}\left(p(q)\right) < - 1 / n\).

References

References

Topic's Concepts

• bertrand
• cournot
• oligopoly