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.