Firm Behavior in Oligopolistic Markets

Abstract

This chapter covers …

• how to apply techniques from game theory towards understanding firm behavior and equilibria in oligopolistic markets.

• the difference between the significance of price and the quantity competition on oligopolistic markets.

• how models of oligopolistic behavior can help one to better understand markets for oil, gas, etc.

• the logic of collusive behavior and the role of regulation in oligopolistic markets.

• how firms have to be organized that compete in such markets.

This chapter covers …

• how to apply techniques from game theory towards understanding firm behavior and equilibria in oligopolistic markets.

• the difference between the significance of price and quantity competition on oligopolistic markets.

• how models of oligopolistic behavior can help one to better understand markets for oil, gas, etc.

• the logic of collusive behavior and the role of regulation in oligopolistic markets.

• how firms have to be organized that compete in such markets.

1 Introduction

A horse never runs so fast as when he has other horses to catch up and outpace. (Ovid, 2002, Ars Amatoria)

Models of markets with perfect competition and monopolistic markets pinpoint extreme cases. They illustrate and help to understand how markets work. However, the stylized nature of these models makes it necessary for them to abstract from aspects of reality that may be relevant for understanding some markets. One such aspect—strategic interdependence of firm decisions —will be the topic of this chapter.

This chapter starts with a short summary of the central results from the theory of competitive and monopolistic markets:

• Perfect competition: There are many suppliers of an identical product and each seller assumes that she cannot influence the market price with her decisions. Under certain conditions, price-taking behavior leads to the “price-equals-marginal-costs” rule for the profit-maximizing choice of output and, at the same time, to the Pareto efficiency of this type of market, because all potential gains from trade are exploited. Two additional conditions, however, have to be met to make this rule rational. On the one hand, producer surplus has to exceed the relevant fixed costs . On the other hand, the production technology has to induce increasing or constant long-run marginal costs . If competitive markets work, then market entry and exit will drive profits to zero because positive (negative) profits encourage entry (force exits).

  The managerial implications of these findings point towards the crucial importance of having an effective accounting system: marginal and average costs of production have to be precisely reflected in the relevant indicators. In addition, given that profits are approximately zero with constant returns to scale or in the long run, the return on equity cannot exceed the return on debt, owners cannot expect larger profits from their investments than they would get in the capital market.

• Monopoly: Only one supplier of a product exists, which implies that customers see a relevant difference between this product and the closest substitute and that other firms cannot imitate it. Compared to competitive markets, a monopolist generates a higher producer surplus, such that it can sustain itself, even if fixed costs would drive competitive firms out of the market (if they are not too high). The efficiency of such a market depends on the monopolist’s ability to discriminate prices. The closer the monopolist gets to the ideal of perfect price discrimination, the more efficient the market becomes. However, there is a tension between the efficiency of monopolistic markets and the distribution of rents between the firm and the consumers, because in the efficient solution, the monopolist is able to transform all rents into producer surplus.

  In order to implement the optimal policy, the firm needs more information than under perfect competition. In addition to an accounting system, it needs a market-research department that estimates price elasticities and helps to segment demand into different groups that are targeted individually.

These findings give some mileage in understanding firm behavior and the functioning of markets, but the important topic of the strategic interdependence of firms’ decision-making has been left out of the picture. Strategic interdependence does not play any role in a monopolistic market, by definition, and it does not play a role in competitive markets, because each single firm is too small to influence aggregates. It becomes important, however, if there is more than one firm that is sufficiently large to influence the market price, such a decision made by one firm can influence the profit of another. This direct interdependency between firms’ objectives follows the same logic as the one analyzed in Chap. 6 and can therefore, in principle, be analyzed with the same toolbox of property rights and transaction costs. A direct interdependency can occur if several firms sell homogenous goods, but also if they sell differentiated goods that are closely linked (which happens, if cross-price elasticities between the goods are non-zero).

The latter situation is, to some extent, always present for a monopolist, but it is usually left out of the analysis to avoid additional complexities. This chapter will also neglect the analysis of several monopolists whose profits are interdependent, because they sell similar products. Instead, it focuses on oligopoly markets in which few sellers supply a homogenous good. The assumption that the goods are perfect substitutes, from the consumers’ point of view, simplifies the analysis and allows it to isolate the pure effect of strategic interdependence.

The central tool for understanding strategic interdependence is game theory and the definition of a game and a Nash equilibrium will be used to analyze the functioning of oligopolistic markets. Firms have, in principle, two instruments to maximize profits , if they are selling a given product. Both instruments are, however, not independent, because they are linked by the market demand function. This is why it is irrelevant, for the monopolist, whether he sets a price and lets quantities adjust passively or sets a quantity and lets the price adjust passively; both approaches lead to the same solution. This equivalence is lost in an oligopolistic market. As the following analysis will show, predictions for the functioning of an oligopoly market with price- and quantity-setting firms differ sharply. In order to understand the deeper reasons for this difference, one has to start by building models of price and quantity setting and then see what predictions they make.

The model of quantity setting is called the Cournot model and the model of price setting is called the Bertrand model , named after the French mathematicians Antoine Augustin Cournot, who developed his model as early as 1838, and Joseph Louis François Bertrand, who reworked the model by using prices in 1883. It is fascinating that Cournot’s analysis anticipated a lot of concepts from economics and game theory, like supply and demand as functions of prices, the use of graphs to analyze supply and demand, reaction functions, and the concept of a Nash equilibrium (limited to the oligopoly context).

Digression 15.1 (The Stackelberg Model and the Value of Commitment)

I can resist everything except temptation. (Oscar Wilde, 1892)

There is a third model of oligopolistic decision-making that goes back to Heinrich Freiherr von Stackelberg (1934). He returned to Cournot’s original analysis but assumed that two firms determine their quantities sequentially instead of simultaneously, as Cournot had assumed. This model will not be covered in this chapter, but I would like to focus attention on a figure of thought that emerged from this model and that proved to be of primary importance for economics and other social sciences : the idea of commitment.

It turns out that the firm that sets its quantity first (the “leader”) has an advantage over the other firm (the “follower”), in comparison to the Cournot model . However, if this were the case, both firms would like to be the leader and the factors that determine leadership are not obvious. Both firms would do whatever they could to be able to choose their strategy first. What is necessary is the existence of some mechanism or device to be available to one firm, but not to the other, which allows the firm to make its leadership position credible. Such a mechanism is called a commitment device .

The appreciation of the economic role and value of such a device offer an important new perspective on a number of social phenomena. One can interpret them as reactions to commitment problems . According to Dubner and Levitt (2005), a commitment device is “a means with which to lock yourself into a course of action that you might not otherwise choose but that produces a desired result.”

The ultimate commitment device can be found in Homer’s Odyssey, where Ulysses puts wax in his men’s ears so that they could not hear and had them tie him to the mast so that he could not jump into the sea, to make sure that he does not fall prey to the song of the sirens. (Franz Kafka, 1931, sees this as “[p]roof that inadequate, even childish measures, may serve to rescue one from peril.”)

Commitment problems exist on the individual as well as on the social level. Fitness goals are a good example of an individual commitment problem. Most people would like to exercise a little more, drink less alcohol, or eat healthier food. However, if it is time for a run, a friend asks if one is ready for a second glass of wine, or one has the choice between chocolate cake and broccoli, one can resist everything other than the temptation to give in. What would be needed in these situations is a device that forces one to stick to one’s resolutions. Some argue that emotions, like shame and embarrassment, can be interpreted as such a device: assume that one publicly announces a fitness goal (“I will run the Berlin Marathon next year”). If one makes such a public announcement and fails to stick to one’s goals, one’s friends will ridicule one and one will feel ashamed, which helps one’s future self persevere. These emotions make deviations from one’s plans costly (in this case, in a purely psychological sense), which is the most important property of a credible commitment device: if one wants to stick to a savings plan, sign a long-term contract that is costly to cancel; if one wants to prepare for an exam, lock oneself into a room without internet access and give the key to a friend, who will be away for the weekend; and so on.

The prisoner’s dilemma is the main example of a social commitment problem: both players would profit from a device that makes the cooperative strategy credible. If the dilemma is used as a metaphor for social interactions in general (a mainstream view since Thomas Hobbes claimed that life before organized, civil societies was solitary, poor, nasty, brutish, and short), then the state can be interpreted as one big attempt to make cooperation credible. This idea can refer to institutions like the rule of law, property rights, and their enforcement by means of material sanctions and punishments, but it can also refer to culture in general, where credibility stems from “softer” sanctions like feelings of guilt and shame.

Commitment problems have also been shown to be at the heart of phenomena like inflation and taxation . The phenomenon is also known as the time inconsistency of decision-making. For the case of monetary policy, politicians have an interest in promising low inflation for the future, in order to control the expectations of the people. However, once tomorrow comes, increasing inflation can have positive, short-run effects, like increasing employment. Hence, the announcement of a low-inflation goal may not be credible, if the government cannot commit to it, leading the economy into a high-inflation equilibrium. Independent central banks with high degrees of discretion in monetary policies are widely seen as a commitment device that can solve this problem. If the central banker’s objective is a zero-inflation policy, then taking away discretion from politicians can, in the end, help them in achieving their goals. The same is true for taxation. If a government wants to encourage investment, then it should announce very low rates of capital taxes but, once the investments are sunk and the factories are built, the corporations are locked in and it is rational for the politician to increase taxes again. If this incentive is anticipated, then firms will not invest in the first place. One of the reasons why Switzerland is considered an attractive place for investments is because it managed to establish a reputation for not falling prey to this incentive. A lack of such a reputation can be a serious impediment to economic development.

The above-mentioned firms are “locked in” with their investments. This lock-in effect is a widely used business practice that helps firms to make profits. Software standards are a good example. In order to be able to use software, one usually has to make large investments of time and effort. These investments lock one into a standard because, ex post, after one has made the investments, the opportunity costs of switching to another standard (called switching costs) are higher than ex ante, before one committed to it. This asymmetry in opportunity costs can be exploited by firms for setting higher prices, and so on.

Evolutionary biologists have used commitment problems to explain the evolution of moral sentiments, by arguing that the evolution of emotions that make cooperation rational (not in a material, but in a psychological sense) has a positive effect for the survival of groups.

A problem with any credible commitment device is that they reduce flexibility. If the future can be perfectly foreseen, then commitment incurs no additional costs but the more uncertain the future becomes, the more risky it is to constrain one’s choices. What would have happened to the epic poem “Odyssey” if Ulysses, tied to the mast, had drowned because of an unforeseen storm that hit his ship before he passed the island of the sirens? He would not be remembered for his brilliance and cunning intelligence, but for his ability to drown himself in an attempt to control his virility. That is not exactly the type of story that would be remembered forever.

2 Cournot Duopoly Model

In the Cournot duopoly model, it is assumed that two profit-maximizing firms, U ₁ and U ₂, simultaneously plan the quantities y ₁ and y ₂ of a homogenous good that they want to sell in a given period of time. Quantities are chosen from the set of positive real numbers (including zero). They produce with a technology that, for given factor prices, leads to convex cost functions , C ₁(y ₁) and C ₂(y ₂). Furthermore, there is an inverse demand function , P(y ₁ + y ₂), that gives the market price for market supply , y ₁ + y ₂ (customers see both goods as perfect substitutes ). In order to keep the analysis simple, assume that all firms are completely informed about all the cost functions and the inverse demand function, and that all of this is common knowledge .

If the profits of the firms are denoted by π ₁ and π ₂, they can be written as

$$\displaystyle \begin{aligned} \pi_{1}(y_{1},y_{2})=P(y_{1}+y_{2})\cdot y_{1}-C_{1}(y_{1}),\quad \pi_{2}(y_{1},y_{2})=P(y_{1}+y_{2})\cdot y_{2}-C_{2}(y_{2}). \end{aligned} $$

From the managers’ points of view, the problem is that profits depend not only on the firm’s own strategy , but also on the strategy chosen by the other firm, because the market price is a function of total quantity. In order to solve this problem, assume that manager 1 (2) expects that the other firm will supply a quantity $y_{2}^{e}$ ($y_{1}^{e}$). The managers determine the optimal quantities given these expectations. The first-order conditions for the profit-maximizing strategies are

for firm 1 and

for firm 2.

Both conditions have a simple economic interpretation: for an expected production level of the competitor, a firm chooses its quantity such that the marginal revenue of the last unit produced equals the unit’s marginal costs . This condition corresponds to the condition of a non-price-discriminating monopolist with the exception that marginal revenues depend on the expectations of the other firm’s production decision.

If one solves the first-order conditions for the respective decision variables, y ₁ and y ₂, one gets two functions $Y_{1}(y_{2}^{e})$ and $Y_{2}(y_{1}^{e})$, which determine the optimal quantity for one firm for a given expected supply of the other firm. These are the so-called reaction functions of the two firms.

Points on the reaction functions imply that firms behave optimally for any given expectation of the other firm’s strategy. However, plans do not have to be mutually consistent. There can be situations where both firms start with expectations, $y_{2}^{e}$ and $y_{1}^{e}$, choose their strategies optimally, but end up with quantities that deviate from the expectations of the other firm, $Y_{1}(y_{2}^{e})\neq y_{1}^{e}$ or $Y_{2}(y_{1}^{e})\neq y_{2}^{e}$. In order to guarantee consistency, one has to require that expectations and actual behavior coincide, $Y_{2}(Y_{1}(y_{2}^{e}))=y_{2}^{e}\wedge Y_{1}(Y_{2}(y_{1}^{e}))=y_{1}^{e}$: The best response of firm 2 to the best response of firm 1, at an expected quantity of $y_{2}^{e}$, is equal to the expected quantity $y_{2}^{e}$, and the best response of firm 1 to the best response of firm 2, at an expected quantity of $y_{1}^{e}$, is equal to the expected quantity $y_{1}^{e}$.

This is another way to say that one is looking for a Nash equilibrium in the game. Formally, a Nash equilibrium of a Cournot duopoly model is completely characterized by $Y_{2}(Y_{1}(y_{2}^{e}))=y_{2}^{e}\wedge Y_{1}(Y_{2}(y_{1}^{e}))=y_{1}^{e}$.

The general characterization of the Nash equilibrium does not contain anything interesting from an economic point of view, because it is just a formal way to say that firms follow their objectives rationally and that their behavior is mutually consistent. In order to gain more economic understanding, this chapter will proceed by assuming that the demand function is linear and that the cost functions are identical and linear, p(y ₁ + y ₂) = a − b ⋅ (y ₁ + y ₂), C ₁(y ₁) = c ⋅ y ₁, C ₂(y ₂) = c ⋅ y ₂ with a > c > 0 and b > 0. These functional specifications are called the linear model. A lot of the understanding that one can get from this model carry over to more general models with nonlinear functions for either demand, cost, or both. The model is illustrated in Fig. 15.1 . In the figure, y ₁ + y ₂ is plotted along the abscissa and demand, as well as the marginal-cost functions are plotted along the ordinate. The marginal-cost function intercepts the ordinate at c; the demand function interrupts at a and has a slope of −b.

Fig. 15.1

The linear model

From a mathematical point of view, there are a lot of different ways to determine the equilibrium . This subchapter will cover a long and rather complicated way for the purpose of exercise, by first computing the profit functions (in order to have a lean notation, one can skip the explicit mention of expected values):

$$\displaystyle \begin{aligned} \begin{array}{rcl} \pi_{1}(y_{1},y_{2})& =&\displaystyle (a-b\cdot(y_{1}+y_{2}))\cdot y_{1}-c\cdot y_{1},\\ \pi_{2}(y_{1},y_{2})& =&\displaystyle (a-b\cdot(y_{1}+y_{2}))\cdot y_{2}-c\cdot y_{2}. \end{array} \end{aligned} $$

They can be simplified to

$$\displaystyle \begin{aligned} \begin{array}{rcl} \pi_{1}(y_{1},y_{2})& =&\displaystyle (a-c-b\cdot y_{2})\cdot y_{1}-b\cdot y_{1}^{2},\\ \pi_{2}(y_{1},y_{2})& =&\displaystyle (a-c-b\cdot y_{1})\cdot y_{2}-b\cdot y_{2}^{2}. \end{array} \end{aligned} $$

The next step is to determine the first-order conditions:

$$\displaystyle \begin{aligned} \frac{\partial\pi_{1}(y_{1},y_{2})}{\partial y_{1}}=(a-c-b\cdot y_{2})-2\cdot b\cdot y_{1}=0 \end{aligned} $$

and

$$\displaystyle \begin{aligned} \frac{\partial\pi_{2}(y_{1},y_{2})}{\partial y_{2}}=(a-c-b\cdot y_{1})-2\cdot b\cdot y_{2}=0. \end{aligned} $$

Firm 1’s first-order condition is depicted in the left panel of Fig. 15.2 . Marginal costs are constant at c. Marginal revenues intersect the ordinate at a − b ⋅ y ₂ and have a slope of −2 ⋅b. They are falling with the supply of the other firm. A comparison with the monopoly case is illustrative: the marginal revenues of a non-price-discriminating monopolist have the same slope, −2 ⋅b, but they intersect the ordinate at a. One can, therefore, think of a Cournot duopolist i as a monopolist with a “curtailed” demand function $\tilde {a}=a-b\cdot y_{j}$. If one solves both conditions for the respective decision parameters, one ends up with the reaction functions:

$$\displaystyle \begin{aligned}\displaystyle Y_{1}(y_{2})&\displaystyle=\left\{\begin{array}[]{cc}(a-c-b\cdot y_{2})/(2\cdot b),&\text{if}\;\;y_{2}\leq(a-c)/b\\ 0&\text{if}\;\;y_{2}> (a-c)/b\end{array}\right.,\\ \displaystyle Y_{2}(y_{1})&\displaystyle=\left\{\begin{array}[]{cc}(a-c-b\cdot y_{1})/(2\cdot b),&\text{if}\;\;y_{1}\leq(a-c)/b\\ 0&\text{if}\;\;y_{1}> (a-c)/b\end{array}\right..\end{aligned} $$

They are illustrated in Fig. 15.2 where y ₁ is plotted along the abscissa and y ₂ along the ordinate. Given that firm 1’s reaction function has y ₂ and firm 2’s reaction function has y ₁ as explanatory variable, one has to look from the abscissa and ordinate simultaneously to understand the figure. The “flatter” graph is firm 2’s reaction function, which has the traditional orientation. The “steeper” graph is firm 1’s reaction function, which is symmetric to firm 2’s, but with the opposite orientation.

Fig. 15.2

Marginal revenues and marginal costs of the oligopolists (left) and reaction functions (right)

The figure reveals the following: (1) Y ₁(0) = Y ₂(0) = (a − c)∕(2 ⋅ b): thus, if the other firm produces nothing, the remaining duopolist behaves like a monopolist. (2) The profit-maximizing quantity of a firm falls with an increase in the quantity of the competitor. (3) The Nash equilibrium is given by the intersection of both reaction functions. Only at that point are the actual and expected supplies of the firms consistent with each other.

Formally, one can find the Nash equilibrium by inserting the reaction functions into one another, Y ₁(y ₂) = (a − c − b ⋅ y ₂)∕(2 ⋅ b) and Y ₂(y ₁) = (a − c − b ⋅ y ₁)∕(2 ⋅ b). A solution to these equations is given by

$$\displaystyle \begin{aligned} y_{1}^{*}=\frac{a-c}{3\cdot b},\quad y_{2}^{*}=\frac{a-c}{3\cdot b} \end{aligned} $$

for the individual equilibrium supplies and

$$\displaystyle \begin{aligned} y^{CN}=y_{1}^{*}+y_{2}^{*}\times\frac{2 \cdot (a-c)}{3\cdot b} \end{aligned} $$

for the equilibrium market supply.

Compare this solution to the monopoly solution, $y^{M}=\frac {a-c}{2\cdot b}$, and to the solution under perfect competition, $y^{PC}=\frac {a-c}{b}$. (One can obtain this number from the “price equals marginal costs” rule, which implies for the linear model that a − b ⋅ y = c. Solving for y gives the result.) It follows that y ^M < y ^CN < y ^PC, which reveals a lot about the effects of competition: comparing a monopoly with a duopoly and this, in turn, with perfect competition shows that competition reduces inefficiency. However, with only two firms, the competitive forces are not strong enough to enforce the solution under perfect competition. Accordingly, the equilibrium price in a duopolistic market lies between the price in a monopolistic market and the price under perfect competition, because the demand function is monotonically decreasing. The different prices can be determined as markups on marginal costs:

$$\displaystyle \begin{aligned} p^{PC}=c<p^{CN}=c+\frac{a-c}{3}<p^{M}=c+\frac{a-c}{2}. \end{aligned} $$

These markups play an important role as “rules of thumb” in the management literature, because they allow it to quickly assess the profitability of a market. They depend on the elasticity of demand, which is, in and of itself, a function based on the customers’ tastes and incomes (as reflected in a and b), as well as the competitiveness of the market, expressed by the number of firms. The markup under perfect competition is zero and it is smaller in a Cournot duopoly than in a monopoly.

3 The Linear Cournot Model with n Firms

The above analysis suggests that the Cournot model builds a bridge between the model of non-price-discriminating monopolies and the model of perfect competition. In order to define this insight more precisely, it makes sense to analyze the equilibrium of an oligopolistic market with an arbitrary number of firms, to see how the number of competitors influences the outcome. The following paragraphs will determine the Nash equilibrium for the linear model with n firms in the market. For this purpose, some additional notations are needed. Denote the supply of any firm i by y _i and the supply sum of all firms, except for firm i, by y _−i. Then, firm i’s profit equation is

$$\displaystyle \begin{aligned} \pi_{i}(y_{i},y_{-i})=(a-b\cdot(y_{i}+y_{-i}))\cdot y_{i}-c\cdot y_{i},\quad i=1,\ldots,n. \end{aligned} $$

Given the quantity supplied by all other firms, firm i’s profit-maximizing supply can be determined by the first-order condition:

$$\displaystyle \begin{aligned} \frac{\partial\pi_{i}(y_{i},y_{-i})}{\partial y_{i}}=(a-c-b\cdot y_{-i})-2\cdot b\cdot y_{i}=0,\quad i=1,\ldots,n. \end{aligned} $$

In general, there are n first-order conditions and n unknown variables y ₁, …, y _n. If one assumes that identical firms behave identically in equilibrium, i.e., for any two firms i and j y _i = y _j, one can replace y _i with y and y _−extiti with (n − 1) ⋅ y. This substitution reduces the system of equations to one:

$$\displaystyle \begin{aligned} (a-c-b\cdot(n-1)\cdot y)-2\cdot b\cdot y=0. \end{aligned} $$

If one solves this equation for y, one obtains the Nash equilibrium quantity of a representative firm as y ^∗ = (a − c)∕((n + 1) ⋅ b). Market supply n ⋅ y ^∗ is then given by n∕(n + 1) ⋅ (a − c)∕b. To understand this result, compare it to the one under perfect competition, which was determined as (a − c)∕b. Then carry out the comparative-static analysis with respect to n by treating n as a continuous variable (which it is not, but the assumption facilitates the analysis):

$$\displaystyle \begin{aligned} \frac{\partial y^{*}}{\partial n}<0,\quad \frac{\partial(n\cdot y^{*})}{\partial n}> 0. \end{aligned} $$

Two implications follow: first, individual supply is falling due to the number of firms; the more competitors there are, the less each single firm produces. Second, market supply is increasing due to the number of competitors. Even though each single firm produces less, if more competitors are on the market, this effect is overcompensated by the sheer number of firms. Now let the number of firms become very large, such that one obtains lim_n→∞ n∕(n + 1) ⋅ (a − c)∕b = (a − c)∕b in the limit: the market tends towards the equilibrium under perfect competition, if the number of firms gets very large. The other extreme is a monopolistic market (n = 1). In this case, the optimal quantity is (a − c)∕(2 ⋅ b): the result from the monopoly model.

Alternatively, one can look at the markup the firms can charge. The equilibrium price is given by p ^∗ = a − b ⋅ n ⋅ y ^∗, which is equal to p ^∗ = c + (a − c)∕(n + 1). It follows that the markup is decreasing in the number of firms and converges to zero, if n becomes very large. Hence, the Cournot model provides a theoretical foundation for the idea that competition drives a market towards efficiency : the more competitors there are, the smaller the individual firm’s influence is on the outcome of the market. If the number of firms becomes arbitrarily large, then the influence of a firm completely disappears and it behaves as a price taker and sells according to the efficient “price-equals-marginal-costs” rule.

4 The Bertrand Duopoly Model

In order to see how price—instead of quantity setting—influences the behavior in such a market, assume that the duopolists choose the prices p ₁ and p ₂ instead of quantities. All other assumptions from the previous model persist and prices are assumed to be positive, real numbers (including zero). The only exception is that one directly assumes constant and identical marginal costs for both firms, C ₁(y ₁) = c ⋅ y ₁, C ₂(y ₂) = c ⋅ y ₂. Price competition with more general cost functions is very difficult to analyze, and the fundamental ideas of price competition are contained in the simplified model.

The firms’ profits are analogous to the previous model, but with the exception that, this time, prices are the strategic variables. Customers are confronted with two prices and they will choose their preferred firm and their optimal demand accordingly. Hence, x ₁(p ₁, p ₂) and x ₂(p ₁, p ₂) are the demand functions relevant for the two firms, for any given pair of prices p ₁ and p ₂. The profit functions become

$$\displaystyle \begin{aligned}\displaystyle\pi_{1}(p_{1},p_{2})&\displaystyle=p_{1}\cdot x_{1}(p_{1},p_{2})-c\cdot x_{1}(p_{1},p_{2}),\\ \displaystyle\pi_{2}(p_{1},p_{2})&\displaystyle=p_{2}\cdot x_{2}(p_{1},p_{2})-c\cdot x_{2}(p_{1},p_{2}).\end{aligned} $$

Both firms set prices simultaneously and independently. In order to be able to do so, they have to form expectations about the other firm’s price $p_{1}^{e},p_{2}^{e}$. A Bertrand–Nash equilibrium is a pair of prices, $p_{1}^{*}$ and $p_{2}^{*}$, such that both firms maximize their profits given price expectations for the other firm, and these expectations are correct, $p_{1}^{e}=p_{1}^{*},p_{2}^{e}=p_{2}^{*}$.

The maximization problems are non-standard, because the profit functions are not continuous in prices. Both goods are perfect substitutes from the point of view of the customers, so they will always buy the cheaper one. Assume that one firm charges a price that is a little bit higher than the price of the competitor. In that case, no one will buy from this firm. If the firm lowers the price just a little bit to undercut its competitor’s price, then all customers will change their minds and now buy from this firm instead. The firm can meet this demand, because it can produce with constant marginal costs and without any capacity constraint. Hence, demand is non-continuous at this point.

An example is two neighboring bakeries that are on the way to work for a number of people. If one bakery sets a higher price for a croissant than the other bakery, then no one will buy there (one abstains from queuing or transaction costs of queuing). Hence, demand as a function of both prices can be written as follows. Let X(p _i), i = 1, 2 be the market demand function :

$$\displaystyle \begin{aligned}\displaystyle x_{1}(p_{1},p_{2})&\displaystyle=\left\{\begin{array}[]{cc}X(p_{1}),&p_{1}<p_{2}\\ 0.5\cdot X(p_{1}),&p_{1}=p_{2}\\ 0&p_{1}>p_{2}\end{array}\right.,\\ \displaystyle x_{2}(p_{1},p_{2})&\displaystyle=\left\{\begin{array}[]{cc}X(p_{2}),&p_{1}> p_{2}\\ 0.5\cdot X(p_{2}),&p_{1}=p_{2}\\ 0&p_{1}<p_{2}\end{array}\right.,\end{aligned} $$

using the convention that consumers will be split up equally between the two firms, if prices are identical.

The non-continuity of the profit functions implies that one cannot characterize the best-response functions using partial derivatives of the profit functions. The non-continuity occurs at p ₁ = p ₂ because, at this point, demand switches from one firm to the other. To characterize best responses, the following paragraphs will focus on firm 1. A similar argument holds for firm 2 because of the symmetry of the problem.

If the purpose is to characterize just one equilibrium, then the task is simple: start with the conjecture that both firms offer a price that equals marginal costs, p ₁ = p ₂ = c. Market demand splits equally between the firms for this pair of strategies and both firms make zero profits. If a firm sets a higher price, it loses the demand and still makes zero profits. If it sets a lower price, it wins over all the customers, but sells at a price that is lower than its marginal costs, so it incurs losses. In other words, it cannot improve its profits by deviating to another price, which is the definition of a Nash equilibrium. Therefore, $p_{1}^{*}=p_{2}^{*}=c$ is a Bertrand–Nash equilibrium.

It is slightly more complex to prove that the equilibrium is unique. In order to show uniqueness, start with the scenario in which at least one firm sets a price below marginal costs. This price leads to losses for at least one firm (the one with the lower price). This firm can avoid these losses by increasing its price above that of its competitor. (If both firms set equal prices, the same logic applies.) Now, assume that at least one firm sets a price that is strictly larger than its marginal costs. If the other firm sets a price below marginal costs, then one is back at the case analyzed above. Thus, assume that the other firm sets a price above or equal to its marginal costs. If they are equal to marginal costs, both firms make a profit of zero, because one of them has no customers and the other is selling at marginal costs. The firm that is selling at marginal costs can increase its profits by increasing its price a little bit, making sure that it is above marginal costs, but below the price of the competitor. If the price is larger than marginal costs, but smaller than the competitor’s price, it wins the whole market and also makes a profit. However, it is not rational for the competitor to stick to the higher price. She can increase his profits by slightly undercutting the other price, making sure that it is still above marginal costs. In this case, he wins over the market, which increases profits from zero to something strictly positive. Last, but not least, one has to focus on situations in which both firms set equal prices above marginal costs. In this case, they share the market equally, making positive profits. Denote the prices by p > c. Formally, this leads to π ₁(p, p) = 0.5 ⋅ X(p) ⋅ (p − c) > 0. What happens if firm 1 deviates to a price p ₁ = p − 𝜖, where 𝜖 is a small positive number, 𝜖 > 0, 𝜖 → 0? Given that all customers buy from firm 1 now, profits become π ₁(p − 𝜖, p) = X(p − 𝜖) ⋅ (p − 𝜖 − c). Given that the firm wins half of the market by this change, there exists an 𝜖 that is small enough such that profits go up.

To summarize, the above line of reasoning has shown that the equilibrium is, in fact, unique. The model of Bertrand price competition has a stark implication: price competition drives prices all the way down to marginal costs. This result is remarkable: even with only two firms, the market behaves as if it were perfectly competitive. This result has an important implication for competition policy: the number of firms in a market is, in general, a poor indicator for the functioning of the market. No conclusive evidence about the intensity of competition can be drawn from the number of firms alone. Further information about the type of competition is necessary.

This result has been derived under very specific assumptions, especially regarding the absence of capacity constraints and identical marginal costs. In order to figure out how robust the results are, one must start with an analysis of the consequences of different marginal costs, c ₁ < c ₂. In this case, setting prices equal to marginal costs leads to different prices and only the low-cost firm 1 is able to sell its products. However, it no longer has an incentive to stick to a price that equals marginal costs, because it can still serve the whole market at higher prices, as long as it sets a price below firm 2’s marginal costs (which define the lower limit for the price of this other firm). The exact strategy of firm 1 depends on the difference between both firms’ marginal costs. Let $p_{1}^{M}$ be the price that firm 1 would set, if it had a monopoly.

• If $c_{2}> p_{1}^{M}$, then firm 1 is able to set the monopoly price without being threatened by firm 2. Due to a sufficiently large cost differential, firm 1 has a de facto monopoly, even though another firm exists that could enter the market. Firm 1 is protected against market entries, due to its cost leadership.

• If $c_{2}<p_{1}^{M}$, then firm 1 cannot enforce the monopoly price, because it would encourage market entry by firm 2. This case is not only interesting because of its economic implications, but also because it shows a tension between economic intuition and mathematical modeling, where one has to ask which source is more trustworthy: one’s intuition or the results from the theoretical model. Here is the problem: intuitively one would expect that the low-cost firm would set the highest price it can that is still lower than the marginal costs of the competitor, i.e., p ₁ = c ₂ − 𝜖 with 𝜖 > 0, 𝜖 → 0. Such a price keeps firm 2 out of the market and is, at the same time, as close to the monopoly price as possible. Such a strategy does not exist from a mathematical point of view, however, because the set p ₁ < c ₂ is an open set (the boundary p ₁ = c ₂ does not belong to it). Hence, for each price, p ₁ = c ₂ − 𝜖, there exists a larger price, $\tilde {p}_{1}<c_{2}-0.5\epsilon$, that leads to higher profits, which follows from the denseness of real numbers. The implication of the denseness of real numbers is that firm 1 has no optimal strategy, which in turn implies that there is no Nash equilibrium. This result is highly unsatisfactory, because intuition tells one that this is highly unlikely; that this problem is merely an artifact of an abstract property of real numbers.

  One way to bring intuition in line with the mathematical model is to impose a certain “granularity” on the set of admissible prices. If one assumes that prices are elements of a finely structured set of possible prices (the smallest change in prices could, for example, be 1/10 of a Rappen), then an equilibrium exists where firm 1 chooses the highest price lower than the marginal costs of the second firm (provided that it is higher than its own marginal costs).

  If the granularity of prices solves the problem, one may ask why this assumption was not used right from the beginning. The reason is twofold. First of all, the necessary notation would be more complex. Second of all, discrete price changes have unintended side effects of their own. For example, in the case of identical marginal costs, one would get the potential for multiple equilibria or positive profits in the equilibrium. These problems illustrate the role mathematics plays in economics: there is no deeper truth behind the mathematical formalism used in most theories. Mathematics helps one to understand the logical structure of arguments: it does no more nor less.

5 Conclusion and Extensions

The Cournot and Bertrand models lead to radically different predictions about the functioning of oligopolistic markets. The natural question then becomes which model is more adequate to describe oligopolistic behavior. Unfortunately, the answer to this question is not that simple. The Cournot and Bertrand models are only the tip of the iceberg of models of oligopolistic behavior that have been developed over the years and that focus on different aspects of firm strategies in such a market environment. Firms can, for example, also compete in the positioning of their products, technological innovations, marketing, or reputation. It depends on the specific industry, maybe on the exact period of time, as well as on other factors that are hard to predict whether a market is more adequately described by quantity or by price competition. While both models are useful, a metatheory that explains and clarifies the conditions under which each model is more adequate is still missing.

In a nutshell, it can be argued that the Bertrand model is useful for the analysis of price wars . It shows that the results of the model of perfect competition may also hold in markets with few firms. This has important methodological consequences, because it implies that the much easier model of perfect competition can also be used to analyze industries with few competitors, as long as there is evidence that they engage in price competition.

The Cournot model is useful for the analysis of firms’ behavior in less competitive situations. It builds a bridge between the monopoly model and the model of perfect competition, because it predicts a continuous adjustment from the monopolistic to the perfectly competitive equilibrium as the number of firms increases.

Economists have tried to develop a “unified” approach to the Cournot–Bertrand problem . An interesting one is to disentangle the problem of an oligopolist into two stages. The idea is to assume that a firm’s production capacity has to be planned at a relatively early stage (stage 1) when there is still uncertainty regarding demand and that the firm is then committed to produce within the chosen capacity constraint. The production decision (stage 2) takes place under conditions of price competition. Interestingly, such a two-stage game is able to predict Bertrand-type price competition in periods of low demand and overcapacity (capacity constraints are not binding). At the same time, the market transforms into Cournot competition if capacity constraints are binding. Given that firms try to avoid overcapacities (they are costly), Cournot competition can therefore be regarded as the normal case if demand is relatively predictable. However, if demand fluctuates widely over time, there will be periods of Bertrand competition again and again.

Independently of whether one is confronted with price or quantity competition, firms have a strong incentive for coordinated or collusive behavior . The reason is that the joint industry profits are maximal, if the firms coordinate on the monopolistic solution and share the profits equally. To see this assume, on the contrary, that industry profits would be maximized in the oligopolistic equilibrium. If this were the case, the monopolist could imitate the oligopolists and choose the Cournot or Bertrand solution instead. The fact that a profit-maximizing monopolist prefers another solution shows that he must be better off. Thus, it is in the interest of the oligopolists to collude and constrain their production in an attempt to move closer to the monopolistic outcome, which creates a tension between profits and efficiency of the market. Different strategies are possible to achieve this goal:

• Firms can try to make explicit price-fixing agreements. However, this is illegal in most countries, exactly because it would make the market less efficient. Hence, firms have developed more subtle strategies to coordinate their outputs.

• One way of reducing competition is through a merger or an acquisition (M&A). These measures usually have to be approved by the national or supranational competition authorities. However, even if M&As are not an option, in practice it is sometimes possible to gain control over some other firm’s strategies by complicated cross-ownership or holding structures.

• It is also possible to reach implicit agreements on prices or quantities that fly below the radar of the competition authorities. These agreements are relatively easy to achieve, because of the limited number of firms that all operate in the same industry but, at the same time, difficult to enforce. However, enforcement is crucial, because every single firm has an incentive to break the agreement and sell a little more at a lower price. The reason is that the monopoly solution is not a Nash equilibrium, so every single firm can profit from unilaterally deviating from a non-equilibrium strategy. Coordination in an oligopolistic market has the structure of a prisoner’s dilemma. A way out of this dilemma opens, if firms compete repeatedly. If firms compete not only today, but also in the future, then trust can build and they can, in principle, punish deviations from cooperative behavior over time. The exact conditions under which cooperation can be stabilized, by repeated interactions, are complicated to characterize, but an important factor is how forward-looking firms are. If they focus heavily on the present, then future gains and losses are of only secondary importance, which makes the enforcement of cooperative behavior difficult.

Digression 15.2 (The Prisoner’s Dilemma and Frames of Reference)

From the point of view of the competing firms, Cournot and Bertrand equilibria have the character of the prisoner’s dilemma: both firms could be better off by coordinating on the monopoly solution, but individual rationality leads them to a different outcome.

At this point, one could argue that, as with the prisoners in their interrogation rooms, this solution is no dilemma at all, because the general public profits from the inability of the firms or prisoners to cooperate. The prisoners are guilty and end up in jail and the outcome of oligopolistic competition is closer to the Pareto optimum than the monopolistic one is.

What this discussion shows is that the perception of a problem depends on the frame of reference. Oligopolistic competition is a cooperation problem from the point of view of the firms, but not from the point of view of society. On the contrary, society can make use of the dilemma structure between the firms to make markets more efficient.

Thus, the existence of a cooperation problem does not automatically imply that society should do something about it. It depends on the frame of reference (the most adequate one from a normative perspective), whether a cooperation problem is perceived as a vice or as a virtue.

Empirical industry studies usually identify many factors that influence market behavior, but that change rather frequently, which makes it very difficult to empirically identify and control, ceteris paribus , experiments to test the theory. One way out of this dilemma is to test the theories in the lab by means of market experiments. The advantage of this approach is that the researcher can control a lot of the relevant factors by the design of the experiment. However, the validity of experiments is limited, because participants are aware that they are not in real markets, but in the lab. There is an extensive debate about the so-called external validity of experiments that this chapter will not cover. Instead, this subchapter will briefly summarize the main findings from the literature on experimental oligopoly theory.

In experiments about Cournot quantity competition studies find a lot of support for the predictions of the model, if the experiment is run for a single round and subjects are anonymous and cannot communicate with each other. Repeated interaction and the possibility to communicate reduce the intensity of competition and collusive behavior becomes more likely. However, collusion is fragile and depends on the number of firms (players) in the experiment. In a duopoly, collusive behavior can be frequently observed, but it breaks down quickly, if the number of players increases. With four firms (players), the intensity of competition is generally higher in experiments than is predicted by the theory and the solution converges very quickly to the competitive equilibrium. The Bertrand model has also been experimentally tested, and the experimental findings are in line with the theoretical predictions.

Digression 15.3 (The Three Cs of Economics)

Chapter 9 concluded with the conjecture that games can be interpreted as structural metaphors that allow one to gain insight into the logic of individual decision-making and collective outcomes. It asserts that society has to overcome two types of challenges, if it wants to alleviate scarcity , cooperation problems , and coordination problems . At the beginning of this chapter, the argument is brought that a third type of problem exists: commitment. Commitment problems lie at the heart of the solution to cooperation, as to coordination problems. To see why, take a prisoner’s dilemma as an example. In this cooperation problem, players would like to mutually coordinate on the cooperative strategy, but individual rationality makes cooperation not credible. Hence, what is missing is a commitment device that allows them to overcome the credibility problem. Coordination problems have a different logic, but commitment mechanisms play a crucial role as well. If all players could publicly commit to a specific strategy, the equilibrium-selection problem would be solved.

Hence, coordination, cooperation, and commitment problems define the structural landscape of economics . This is why they can be called the three Cs of economics.

Such a structural approach to economics has two main advantages:

• First, the simplicity of the three Cs approach gives one a frame of reference for the interpretation and understanding of societal problems. Is it a coordination problem or a cooperation problem? What kind of commitment device might help to overcome it? Additionally, if there is no problem, then what kind of commitment mechanism is in the background that helps in stabilizing the efficient outcome?

  Here are two examples that illustrate this approach: Chap. 14 demonstrated that externalities can be interpreted as unresolved cooperation problems. Hence, the next step is to think about commitment mechanisms that help in internalizing them. On the other hand, previous chapters have argued that a complete set of competitive markets leads to efficiency under certain assumptions. The commitment device in the background is a system of perfectly enforced property rights. But is this the end of the story? Who enforces the property rights and is it in the interest of this person to do so? Does one have to dig deeper to identify commitment mechanisms for law enforcers? etc.

• Second, the three Cs are a tool for future studies. When one begins to study more elaborate and advanced economic theories, it is easy to lose track of the basic story underlying the theory. Yet most, if not all, theories are variations of coordination or cooperation problems, plus some more or less elaborate ideas on commitment. Approaching these theories with a three-Cs perspective helps one to make sense of them. It also helps one to scrutinize the basic ideas of these theories. Is the problem at hand adequately described as a coordination or cooperation problem? Are the institutions the theory focuses upon convincing, in the sense that they are credible commitment devices and, if not, why?

Part III of this book gave an introduction into the functioning of different prototypical markets. The following table summarizes the main findings from those chapters.

Overview of market structures (long run)	Sellers	Buyers	Price	Profits	Efficiency
Perfect competition	Many (homogeneous goods)	Many	p = MC, in the long run $p=\min AC$	π = 0	Efficient
Bertrand oligopoly	Few (same cost structure)	Many	p B = MC	π B = 0	Efficient
Cournot oligopoly	Few	Many	p C > MC	π C > 0	Inefficient
Monopoly (no price discrimination)	One	Many	p M > p C > MC	π M > π C > 0	Inefficient
Monopoly (1st degree price discrimination)	One	Many	$p^{j}_{M}=$ individual j’s willingness to pay	π M =  maximum sum of CS and PS > π M > π C > 0	Efficient
Monopolistic competition	Many (heterogeneous goods)	Many	p = MC+μ = AC, μ =  markup	π = 0	Inefficient