Bertrand (Nash) equilibrium

 

Author Definition

 

Definition

Formally, a set of strategies is called a Nash equilibrium if, holding the strategies of all other firms constant, no firm can obtain a higher payoff by choosing a different strategy. In less formal terms, a Nash equilibrium holds when each firm is doing the best they can (i.e. earning the highest profit) given what all the other firms in the market are doing. A Bertrand Nash equilibrium describes the Nash equilibrium outcome in a Bertrand model of oligopoly. Bertrand models are widely used models of oligopoly in competition economics. The great attraction of Nash equilibria is that they are stable as no firm has an incentive unilaterally to change their behaviour. It is the most widely used equilibrium concept in industrial organisation economics and was introduced by John Nash in his seminal 1951 paper.

 

Commentary

The basic Bertrand model of competition involves firms setting the price for homogeneous products. It is assumed that each firm can supply as much product as is demanded at any given price. Under these assumptions, it can be shown that the Bertrand (Nash) equilibrium is that price is equal to marginal cost (i.e. the same outcome as under perfect competition). The logic for this result is simple. Suppose there are just two firms, A and B. Suppose Firm A prices above marginal cost. Then Firm B’s profit maximising price is to price just below Firm A but above marginal cost and capture all the market demand. But if Firm B does this, then Firm A can do better by pricing just below Firm B. This process of undercutting continues until price is at marginal cost. It would not make sense for either firm to price below marginal cost. The Bertrand (Nash) equilibrium is thus that price equals marginal cost.

This leads to the so-called Bertrand paradox: two firms are enough to generate the same outcome as under perfect competition. The “paradox” is that we normally assume that a duopoly will not be competitive and will price above marginal cost.

There are three ways to solve this apparent paradox. These involve relaxing one of the following assumptions:
- Products are homogeneous
- Each firm can supply the whole market
- Firms act in a short run non-cooperative fashion.

The most common way to change the model in order to get more intuitively plausible results is to assume that the firms sell differentiated products. Now firms do not lose all of their sales even if their competitors price slightly below them. This unravels the logic that leads to price equal to marginal cost in the standard (homogeneous products) Bertrand model. To see this, suppose that both Firm A and B did set their prices at marginal cost. Firm A would have an incentive to raise its price very slightly above marginal cost as it would not lose all its sales: those customers who prefer Firm A’s product to Firm B’s product would continue to buy it. This would increase Firm A’s profits as it would make a positive profit on those sales that it continues to make, rather than the zero profit when price is equal to marginal cost. But Firm B faces the same incentives. The result is that both will raise their prices above marginal cost until the point at which a further price rise is not profitable given the price set by the other firm. This is a Nash equilibrium. The extent to which prices are above costs is driven by the extent of differentiation between products i.e. prices increase more as products become more differentiated. To the extent that more firms in the market implies less differentiation between products, this means that prices will tend to be lower as more firms enter the market. This accords with our intuition for how prices should fall as the number of suppliers increases.

The Bertrand differentiated products model is a workhorse model for merger analysis. When two firms merge in a Bertrand differentiated products setting, the competition that previously existed between them is lost. The merged entity has an incentive to raise prices because sales that were previously lost by one of the firms but captured by the other firm are now recaptured within the merged entity. The stronger the competition between the two firms was pre-merger, the more the merged entity has an incentive to raise prices. A price increase by the merged entity incentivises the other firms in the market to also raise their prices, although these “second order effects” are typically much smaller than the merging parties’ price rise. This price rise is driven by the logic of the Nash equilibrium concept: if all the firms were previously doing the best they could given what all the other firms were doing, this will no longer be true if the merged entity raises its prices.

The Bertrand differentiated products model is applicable to markets where firms set prices for differentiated products and then customers choose whether or not to buy. It is therefore particularly relevant for retail markets. The UK Competition and Markets Authority review of the proposed Sainsbury’s/Asda merger in 2019 is a good example of this. Standard empirical measures of the effect of mergers on pricing incentives, such as upward pricing pressure indices and indicative price rise indices, are typically based on the Bertrand differentiated products model.

The model is not relevant where prices are set separately for each buyer (i.e. markets involving substantial price discrimination) or where firms choose quantities rather than prices (i.e. Cournot competition).

Another way to escape the Bertrand paradox is to remove the assumption that each firm can supply the whole market. Where there are several firms in a market, this is unlikely to be a plausible assumption, at least not in the short run. The Bertrand-Edgeworth model assumes that no firm can supply the whole market. This allows prices to rise above marginal cost as the logic of the standard model (undercutting to win the whole market) no longer holds. This model illustrates another important aspect of the Nash equilibrium concept: it relates to strategies, rather than just to prices. In the Bertrand-Edgeworth model firms do not set a single price, but instead set different prices for each period based on a mixed strategy equilibrium (e.g. set a price of 10 with probably 0.2; a price of 11 with probability 0.18; etc.). This is because there is no single Nash equilibrium price in the Bertrand-Edgeworth model, but there is a Nash equilibrium set of mixed strategy equilibria. The European Commission’s decision on the Inoxum/Outokumpu merger is a good example of the application of the Bertrand-Edgeworth model to a case.

The third way to escape the Bertrand paradox is to drop the assumption that firms think only about the outcome of period. The Bertrand paradox arises because the model is a one-period model. But suppose firms compete over many periods. Is it plausible that in each period they will set price equal to marginal cost and earn zero profits? Maybe. But where there are only a few firms it is also plausible that they will “soften” their pricing in the hope that other firms will as well.

Economists have traditionally assumed that collusion between firms would tend to be unstable as collusion was not a Nash equilibrium. The argument was that each firm has an incentive to cheat on the collusion (i.e. lower prices to win more demand) because, given what the other firms were doing, cheating is profit maximising. However, this argument is hard to square with the observed facts that cartels do exist and do manage to raise prices significantly. The problem with the standard argument is that it is very short termist. When firms compete over time, they aim to maximise profits over time, not just in one period. Fudenberg and Maskin (1986) showed that any collusive output can be sustained as a Nash equilibrium of a pricing game when discount rates are high enough and when punishment mechanisms are credible.

 

Bibliography

Carlton, D. and Perloff, J. 2015. Modern Industrial Organization Fourth edition (Pearson)

Fudenberg, D. and Tirole, J. 1989. “Noncooperative game theory for industrial organization: an introduction and overview” in Handbook of Industrial Organization ed. Schmalensee, R. and Willig, R. (North-Holland)

Nash, J. 1951. “Non-cooperative games” Annals of Mathematics (vol. 54, no 2. Sept 1951)

Fudenberg, D. and Maskin, E. 1986. “The Folk Theorem in repeated games with discounting and with incomplete information” Econometrica (vol. 54)

Edgeworth, F. 1989. “The pure theory of monopoly” reprinted in Collected Papers relating to Political Economy Macmillan (1925)

This article is being reviewed by the Editors of the Dictionary.

Author

  • United Kingdom’s Competition Authority - CMA (London)

Quotation

Mike Walker, Bertrand (Nash) equilibrium, Global Dictionary of Competition Law, Concurrences, Art. N° 85397

Visites 288

Publisher Concurrences

Date 1 January 1900

Number of pages 500

 

Institution Definition

In a Bertrand model of oligopoly, firms independently choose prices (not quantities) in order to maximize profits. This is accomplished by assuming that rivals’ prices are taken as given. The resulting equilibrium is a Nash equilibrium in prices, referred to as a Bertrand (Nash) equilibrium.

When the industry is symmetric, i.e., comprising firms of equal size and identical costs, and the costs are constant and the product homogenous, the Bertrand equilibrium is such that each firm sets price equal to marginal cost, and the outcome is Pareto efficient. This result holds regardless of the number of firms and stands in contrast to the Cournot equilibrium where the deviation from Pareto efficiency increases as the number of firms decreases.

However, when products are differentiated even the Bertrand model results in prices which exceed marginal cost, and the difference increases as products become more differentiated. (...)

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