Simultaneous Competition

Context

• Many real markets are neither perfectly competitive nor monopolies. Instead, they are oligopolies comprised of 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 competition strategies.
  • How do firms strategically interact?
  • What means do they use to compete?
  • How do the welfare outcomes of oligopolies compare to those of monopolies and perfect competition?

Course Structure Overview

Lecture Structure and Learning Objectives

Structure

• Microsoft’s Pricing Strategies (Case Study)
• Basic Concepts
• The Cournot and Bertrand Models of Duopolies
• Extensions to Oligopolies with More Firms
• Spatial Competition
• Current Field Developments

Learning Objectives

• Describe oligopolies with competition in quantities and their welfare output.
• Describe oligopolies with competition in prices and their welfare output.
• Contrast the welfare outcomes with the perfectly competitive welfare.
• Illustrate the differences between the results of various modes of competition.

Microsoft’s Pricing Strategies

• In the early 1980s, several companies were competing in the operating system market of IBM-compatible PCs.
• In the 1990s and 2010s, Microsoft dominated the operating system market.
• In 2020s Microsoft’s dominance stopped, and its operating system is nowadays the second most used.

• How did Microsoft manage to dominate the operating system market?
• How did it lose its primacy?

MS-DOS

• In the early 1980s, the common practice of operating system companies was to charge hardware manufacturers for each operating system copy installed in a computer.

• Microsoft offered an alternative plan.
  • Charge computer manufacturers based on (the past number of) built computers.
  • The manufacturer was paying a general licensing fee and then could install the operating system in all the computers it produced.
  • Microsoft was offering low-priced licensing contracts making their operating system (MS-DOS) very attractive to manufacturers.

The Impact of Microsoft’s Early Pricing Strategy

• Effectively, manufacturers could purchase Microsoft’s operating system at much lower prices than the operating systems of other software companies.
  • A manufacturer had to pay \($50\ -\ $100 \) for installing an alternative operating system on an additional machine.
  • It cost nothing to install MS-DOS on an additional machine once a licensing contract with Microsoft has been signed.
  • MS-DOS ended up being the default operating system

Android

The Impact of Android on Microsoft’s Pricing Strategy

• Android is free. Anyone can install the operating system on her device after accepting the terms and conditions.
• For mobile device manufacturers, Android is a cheaper operating system alternative for their products.
• This leads to more competitive prices for consumers too.
  • Microsoft’s operating systems lost their primacy in the overall operating system market in \(2017\).
  • Microsoft’s operating systems are still dominant in less portable devices, such as desktop PCs and Laptops.
  • The rise of Android has also impacted Microsoft’s pricing strategies for its desktop operating systems.
  • Licensed users were able to upgrade to the last two versions of Microsoft’s operating system without paying for a new license.

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 is a decisive component of the market structure.

Cooperation and collusion

• Oligopolies have used explicitly written 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 require explicit contracts or communication.

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 market failure (imperfect information, externalities, etc.),
  • no other firms can enter the market,
  • firms sell a homogeneous product,
  • firms try to maximize their profits by simultaneously choosing the quantities they produce,
  • consumers are price takers, and
  • consumers try to maximize their utility.

Quantity Competition with two Firms

• Each firm solves

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

• 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).

The Linear Cost Case

• For inverse demand and costs given by

\begin{align*} p(q) &= p_{0} + p_{1} q \quad &&(p_{0}>0,\ p_{1}<0)\\ c(q) &= c_{1} q \quad &&(c_{1}>0). \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

\[\pi_{i} = -\frac{(c_{1} - p_{0})^{2}}{9 p_{1}}.\]

Quantity Competition with More than two Firms

• We extend the problem by allowing \(n>2\) firms that simultaneously choose their supplied quantities.
• 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 equilibria of this oligopoly model.

Best Responses

• Suppose inverse demand is given by

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

• All firms have (symmetric) costs

\begin{align*} c(q) &= c_{1} q . \end{align*}

• The best responses become

\[q_{i} = \frac{c_{1} - p_{0} - p_{1} \sum_{j\neq i} q_{j}}{2 p_{1}}.\]

Market Equilibrium

• The equilibrium quantities are given by

\[q_{i} = \frac{c_{1} - p_{0}}{(n + 1) p_{1}}.\]

• 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

\[p(q) = p_{0} + p_{1} n \frac{c_{1} - p_{0}}{(n + 1) p_{1}} = \frac{n c_{1} + p_{0}}{n + 1}.\]

• Profits are then

\[\pi_{i} = -\frac{(c_{1} - p_{0})^{2}}{(n + 1)^{2} p_{1}}.\]

• The case of two firms can be obtained by replacing \(n=2\) in the above results.

Market Power

• We can also obtain the solution of the monopoly problem if we set \(n=1\).
• With \(n=1\), we get the monopolistic profit

\[\pi_{1} = -\frac{(c_{1} - p_{0})^{2}}{4 p_{1}}.\]

• Profits decrease as the number of firms in the market increases.
• The limiting case is

\[\pi_{i} \xrightarrow[n\to \infty]{} 0.\]

• In addition, \(p(q) \to c_{1}\), \(q_{i}\to 0\), 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.

A Cournot Competition Exercise

• Suppose that both firms \((i, j \in \{1,2\})\) have the cost function

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

• Let market inverse demand be

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

• Each firm maximizes its profits

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

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 market failure (imperfect information, externalities, etc.),
  • no other firms can enter the market,
  • firms sell a homogeneous product,
  • firms try to maximize their profits by simultaneously choosing prices,
  • consumers are price takers, and
  • consumers try to maximize their utility.

A Bertrand Competition Exercise

• Suppose that both firms \((i, j \in \{1,2\})\) produce at a marginal cost 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 \(j\) can only set a price equal to firm \(i\)’s marginal cost.
• Analogous arguments hold for firm \(i\)’s pricing strategy.

Equilibrium

• The only possible equilibrium is to set prices 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 profits.
• Even with two firms, price competition leads to prices similar to perfect competition.

Spatial Competition

• There are two firms on a street.
• Points on the street are given by \([0, 1]\).
• Each firm chooses a point.
• Firms have the same cost and charge the same price.
• Customers on the street prefer the firm that is the closest.

An illustration of the game

Non equilibrium placements

• If firm \(2\) chooses \(x_{2} > \frac{1}{2}\), firm \(1\) would like to undercut by a small amount and set \(x_{1} = x_{2} - \varepsilon > \frac{1}{2}\).
• Then firm \(2\) has a profitable deviation by changing to \(x_{2} = \frac{1}{2}\).
• Thus any \(x_{2} > \frac{1}{2}\) cannot be an equilibrium.
• Similarly, any \(x_{2} < \frac{1}{2}\) cannot be an equilibrium.
• Analogous arguments hold for firm \(1\) because of symmetry.

Equilibrium placements

• Therefore, the only possible equilibrium is \(x_{1} = \frac{1}{2} = x_{2}\).
• Firms split the market and make equal profits.
• Any deviation leads to fewer profits for the firm that moved.

Current Field Developments

• There are two main types of extensions of the basic models (Cournot and Bertrand).
  • Extensions incorporating dynamic decisions (see Dynamic Competition topic) and
  • Extensions incorporating decisions under uncertainty (see Competition with Incomplete Information topic)
• Oligopoly models are primarily used in industrial organization to examine
  • market power,
  • pricing strategies,
  • competition policies, and
  • R&D and innovation.
• Some recent micro-founded, general equilibrium macro models use oligopoly models to describe markets with frictions.

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.
• Depending on how firms compete (prices or quantities) and the number of firms, oligopoly models give predictions with welfare properties that range from perfect competition to monopoly.

Further Reading

• Watson (2008, chaps. 10, 11)
• Belleflamme and Peitz (2010, secs. 3.1.1, 3.2)
• Varian (2010, chap. 28)

Mathematical Details

An Example of Competition with More than two Firms

What happens to welfare, prices, and production when we have oligopolies with more than two firms?

Price Competition with More than two Firms

In price competition, the Nash equilibrium and the outcomes do not change when more firms are included in the market. With two firms, the competitive outcome is already achieved; prices are set equal to the (common) marginal cost, profits are zero, and the welfare outcome is Pareto efficient.

Quantity Competition with More than two Firms

Suppose that we have \(n\ge2\) firms in the market, inverse demand is given by \[ p(q) = 10 - q. \] and firms have symmetric costs \[ c(q_{i}) = 2q_{i}. \] The market quantity with \(n\) firms is the sum of the individual quantities produced by each firm, namely \[ q = \sum_{k=1}^{n}q_{k}. \]

Firm \(i\) maximizes its profits by choosing the quantity \(q_{i}\) it produces, i.e., \[ \max_{q_{i}} \left\{ p(q) q_{i} - c(q_{i}) \right\} = \max_{q_{i}} \left\{ p\left(\sum_{k=1}^{n}q_{k}\right) q_{i} - c(q_{i}) \right\} . \] The optimal choice is set when the firm equalizes its marginal revenue with its marginal cost (why?), which implies \[ -q_{i} + 10 - \sum_{k=1}^{n}q_{k} = 2. \] Solving for \(q_{i}\) gives the best response (or reaction function) \[ q_{i} = 4 - \frac{1}{2} \sum_{k \neq i} q_{k}. \]

Since the game is symmetric, the strategies of all firms should be equal in Nash equilibrium (i.e., q_i=q_k for all \(k=1,\dots,n\)), which simplifies the problem of firm \(i\) to \[ -q_{i} + 10 - n q_{i} = 2. \] Therefore, firm \(i\) (and any other firm \(k\)) produces \(q_{i}^{\star} = 8/(n+1)\) in equilibrium. Then, the market quantity is \[q^{\star} = \sum_{k=1}^{n} q_{k}^{\star} = n q_{i}^{\star} = \frac{8n}{n+1} , \] and the market price is \[p^{\star} = p(q^{\star}) = 10 - q^{\star} = \frac{2n + 10}{n+1}. \]

Welfare and Deadweight Loss

The profit of firm \(i\) is \[ \pi_{i}^{\star} = p^{\star} q_{i}^{\star} - 2 q_{i}^{\star} = \frac{64}{(n+1)^{2}}. \] The producers' surplus is obtained by summing up the profits of all firms, which gives \[ W_{p}^{\star} = \sum_{k=1}^{n} \pi_{k}^{\star} = n\pi_{i}^{\star} = \frac{64n}{(n+1)^{2}}. \] Using a geometric argument, the consumers' surplus is \[ W_{c}^{\star} = \frac{1}{2}\left(10 - p^{\star}\right)q^{\star} = \frac{32n^{2}}{(n+1)^{2}}. \] Hence, the total welfare of the Nash equilibrium in the Cournot competition is \[ W^{\star} = W_{p}^{\star} + W_{c}^{\star} = \frac{32n^{2} + 64n}{(n+1)^{2}}. \]

To examine if the welfare outcome is Pareto efficient, we can compare it with that of the competitive equilibrium. In the competitive equilibrium, we have \(p(q^{c})=p^{c}=c'(q^{c})\), which is equivalent to \(q^{c}=8\) and \(p^{c}=2\). Profits are zero for all firms, so the producers' surplus is \(W_{p}^{c}=0\). Using another geometric argument, we can calculate the consumers' surplus as \[ W_{c}^{c} = \frac{1}{2}\left(10 - p^{c}\right)q^{c} = 32. \] Thus, the total welfare in competition is \(W^{c} = W_{c}^{c} = 32\) and, therefore, the deadweight loss of the Cournot oligopoly is \[ DW^{\star} = W^{c} - W^{\star} = 32 - \frac{32n^{2} + 64n}{(n+1)^{2}}. \]

Asymptotics and Competition

As the number of firms in the market increases, competition intensifies, profit margins decrease, the deadweight loss decreases, and the total market welfare increases.

Asymptotically (as the number of firms goes to infinity), each firm’s production becomes negligible \[ q_{i}^{\star} = \frac{8}{n+1} \xrightarrow[n\to\infty]{} 0. \] The total market quantity increases approaching the competition quantity \[ q^{\star} = \frac{8n}{n+1} \xrightarrow[n\to\infty]{} 8 = q^{c}. \] The market price decreases approaching the competition price \[ p^{\star} = \frac{2n + 10}{n+1} \xrightarrow[n\to\infty]{} 2 = p^{c}. \] The profit of firm \(i\) decreases to zero (as in competition) \[ \pi_{i}^{\star} = \frac{64}{(n+1)^{2}} \xrightarrow[n\to\infty]{} 0. \] Similarly, the producers' surplus decreases to zero \[ W_{p}^{\star} = \frac{64n}{(n+1)^{2}} \xrightarrow[n\to\infty]{} 0 = W^{c}_{p}. \] The consumers' surplus increases to that of the competition \[ W_{c}^{\star} = \frac{32n^{2}}{(n+1)^{2}} \xrightarrow[n\to\infty]{} 32 = W^{c}_{c}. \] The total welfare grows toward becoming Pareto efficient \[ W^{\star} = \frac{32n^{2} + 64n}{(n+1)^{2}} \xrightarrow[n\to\infty]{} 32 = W^{c}. \] Finally, the deadweight loss vanishes \[ DW^{\star} = 32 - \frac{32n^{2} + 64n}{(n+1)^{2}} \xrightarrow[n\to\infty]{} 0. \]

Exercises

Group A

1. Consider a Cournot duopoly with inverse demand \(p(q) = p_{0} + p_{1} q\) and costs \(c_{i}(q) = c_{1, i} q\), for \(i = \{1, 2\}\), where \(p_{0},c_{1}>0\) and \(p_{1}<0\). What happens if costs are not symmetric? Which firm produces more? Which firm makes more profit?

   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 firm with the lower cost 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 \(q_{i}\ge q_{j}\).

Group B

1. Consider a differentiated product duopoly with firms competing in prices. The market consists of a homogeneous unit mass of consumers. Each consumer \(h\in[0,1]\) has unit demand for commodity \(i=1,2\). The (gross) valuation of each consumer for commodity \(i\) is \(v_{i}\) with \(v_{1} > v_{2}\). If the net (gross minus price) valuations of the two commodities are equal, commodity \(1\) gets all the demand. The marginal production costs of both firms are zero.

   1. Suppose that each consumer can only buy one of the two products. Find the Nash equilibria.
   2. Suppose that consumers can also buy both products in a bundle. The bundle has valuation \(v_{b}\) with \(v_{1} + v_{2} > v_{b} > v_{1}\). Find the Nash equilibrium.
   3. Calculate the consumer surpluses of the two cases and compare them.

   1. The Nash equilibrium is characterized by \(p_{1}^{\ast} = v_{1}-v_{2}>0\) and \(p_{2}^{\ast}=0\). We work our way to this result by showing that there is no profitable unilateral deviation. In the candidate equilibrium, the profits are \(\pi_{1}^{\ast} = v_{1}-v_{2}\) and \(\pi_{2}^{\ast} = 0\).

      Suppose that firm \(2\) deviates to \(p_{2}>0\). We then have \(p_{1} < p_{2} + v_{1} - v_{2}\). This implies that \(v_{2} - p_{2} < v_{1} - p_{1}\), so the consumers prefer commodity \(1\). As a result, \(\pi_{2} = 0 = \pi_{2}^{\ast}\). Thus, firm \(2\) has no profitable deviation.

      Consider the deviations of firm \(1\). There are two potential deviations. Firstly, suppose that firm deviates to \(p_{1} > v_{1} - v_{2} = p_{2} + v_{1} - v_{2}\). Then, commodity \(2\) is preferred, and firm \(1\) makes zero profit, which is less than \(\pi_{1}^{\ast}\). Secondly, consider a deviation \(p_{1} < v_{1} - v_{2} = p_{2} + v_{1} - v_{2}\). Then, commodity \(1\) is preferred, and firm \(1\) earns \(\pi_{1} = p_{1} < v_{1} - v_{2} = \pi_{1}^{\ast}\).

   2. In equilibrium, it holds \[v_{1}-p_{1}^{\ast\ast} = v_{2}-p_{2}^{\ast\ast} = v_{b}-p_{1}^{\ast\ast}-p_{2}^{\ast\ast}\] because at least one of the two firms has a profitable deviation otherwise. For example, suppose that \(v_{1}-p_{1} < v_{2}-p_{2}^{\ast\ast}\). Then, only commodity \(2\) is consumed, and the profit of firm \(1\) is zero. Firm \(1\) can deviate to \(p_{1}^{\ast\ast}\) and earn profit \(p_{1}^{\ast\ast} = v_{b} - v_{2} > 0\). Similarly, other cases can be excluded.

   3. The consumer surplus of the market in part one is \[C^{\ast} = v_{1} - p_{1}^{\ast} = v_{1} - v_{1} + v_{2} = v_{2}.\] The consumer surplus of the market in part two is \[C^{\ast\ast} = v_{1} - p_{1}^{\ast\ast} = \underbrace{v_{1} - v_{b}}_{<0} + v_{2} > 0.\] Therefore, \(C^{\ast}>C^{\ast\ast}\) and consumers are better off in the first case.

References

References

Topic's Concepts

• bertrand
• cournot
• oligopoly