Bertrand competition

Bertrand competition is a model of competition used in economics, named after Joseph Louis François Bertrand (1822-1900). Specifically, it is a model of price competition between duopoly firms which results in each charging the price that would be charged under perfect competition, known as marginal cost pricing.

The model has the following assumptions:

* There are at least two firms producing products;
* Firms do not cooperate;
* Firms have the same marginal cost (MC);
* Marginal cost is constant;
* Demand is linear;
* Firms compete in price, and choose their respective prices simultaneously;
* There is strategic behaviour by both firms;
* Both firms compete solely on price and then supply the quantity demanded;
* Consumers buy everything from the cheaper firm or half at each, if the price is equal.

Competing in price means that firms can easily change the quantity they supply, but once they have chosen a certain price, it is very hard, if not impossible, to change it. Some examples of firms that might operate in this way are bars, shops or other companies that publish non-negotiable prices.

Calculating the classic Bertrand model

*MC = Marginal cost
*p₁ = firm 1’s price level
*p₂ = firm 2’s price level
*p^M = monopoly price level

* Firm 1s optimum price depends on what it believes firm 2 will set prices at. Pricing just below the other firm will obtain full market demand (D), while maximizing profits. If firm 1 expects firm 2 to price below marginal cost, then its best strategy is to price higher, at marginal cost. In general terms, firm 1s best response function is p₁’’(p₂), this gives firm 1 optimal price for each price set by firm 2.

* Diagram 1 shows firm 1’s reaction function p₁’’(p₂), with each firms strategy on each axis. It shows that when P₂ is less than marginal cost (firm 2 pricing below MC) firm 1 prices at marginal cost, p₁=MC. When firm 2 prices above MC but below monopoly prices, then firm 1 prices just below firm 2. When firm 2 prices above monopoly prices (P^M) firm 1 prices at monopoly level, p₁=p^M.

*Because firm 2 has the same marginal cost as firm 1, its reaction function is symmetrical with respect to the 45 degree line. Diagram 2 shows both reaction functions.

*The result of the firms strategies is a Nash equilibrium, that is, a pair of strategies (prices in this case) where neither firm can increase profits by unilaterally changing price. This is given by the intersection of the reaction curves, Point N on the diagram. At this point p₁=p₁’’(p₂), and p₂=p₂’’(p₁). As you can see, point N on the diagram is where both firms are pricing at marginal cost.

Another way of thinking about it, a simpler way, is to imagine if both firms set equal prices above marginal cost, firms would get half the market at a higher than MC price. However, by lowering prices just slightly, a firm could gain the whole market, so both firms are tempted to lower prices as much as they can. It would be irrational to price below marginal cost, because the firm would make a loss. Therefore, both firms will lower prices until they reach the MC limit.

Implications

* There are two plausible outcomes: colluding to charge the monopoly price and supplying one half of the market each, or not colluding and charging marginal cost, which is the non-cooperative Nash equilibrium outcome.

* If one firm has lower average cost (a superior production technology), it will charge the highest price that is lower than the average cost of the other one (i.e. a price "just" below the lowest price the other firm can manage) and take all the business. This is known as "limit pricing"

Bertrand competition versus Cournot competition

* Although the models have similar assumptions, they have very different implications.
* Bertrand predicts a duopoly is enough to push prices down to marginal cost level, that duopoly will result in perfect competition.
* Neither model is necessarily "better". The accuracy of the predictions of each model will vary from industry to industry, depending on the closeness of each model to the industry situation.
* If capacity and output can be easily changed, Bertrand is generally a better model of duopoly competition. Or, if output and capacity are difficult to adjust, then Cournot is generally a better model.
* Under some conditions the Cournot model can be recast as a two stage model, where in the first stage firms choose capacities, and in the second they compete in Bertrand fashion.

Critical analysis of the Bertrand model

* The most critical flaw of the model is the assumption that firms compete in one period, the price being chosen and set forever. However, as it is unreasonable to expect the other firm to indefinitely keep higher prices and sell nothing, each firm must expect that lowering the price will almost immediately be met with the same move by the other firm, thus no firm can expect to get bigger market share by cutting price, and the preferred strategy is keeping prices at monopoly price level. The situation is analogous to the prisoner's dilemma, single-period version of which has completely opposite implications than the iterated version.

* Examining the assumptions reveals some inadequacies of the model: it assumes firms compete purely on price, ignoring non-price competition. Firms can differentiate their products and charge a higher price. For example, would someone travel twice as far to save 1% on the price of their vegetables?

* There are rarely just two firms in a market.

* If a firm does undercut a rival and get full market share, it now has to supply the whole market; many firms would not have the capacity to do this. In general, the greater the overall capacity constraints, the higher the price is than marginal cost.

See also

* Cournot competition
* Differentiated Bertrand competition
* Stackelberg competition
* Nash equilibrium
* Game theory
* Bertrand paradox (economics)

References

* Bertrand, J. (1883) "Book review of theorie mathematique de la richesse sociale and of recherches sur les principles mathematiques de la theorie des richesses", Journal de Savants 67: 499–508.