What is different about the two models Part 5 the Cournot and Bertrand models are different in that?

The Cournot (1838) and Bertrand (1883) models are cornerstones of the modern theory of oligopoly. In the former, firms' strategic variable is the quantity of output to produce while in the latter, firms choose price. Bertrand competition has traditionally been considered as more efficient in welfare terms than Cournot competition because it leads to lower prices and larger quantities (see for example Shubik, 1980, Vives, 1985, Singh and Vives, 1984). Indeed, if we assume that firms produce a homogeneous product at a common constant marginal cost, Bertrand competition will lead to a price equal to the marginal cost while Cournot competition will lead to a price which is intermediate between the competitive and the monopolistic price. If, to the contrary, we assume that firms produce differentiated products, then the Bertrand price will be above the marginal cost but it will be again lower than the corresponding Cournot price. Therefore, consumer surplus and total surplus are always higher in Bertrand competition than in Cournot competition. Furthermore, profits in Cournot competition are higher, equal or smaller than in Bertrand competition if the goods are substitutes, independent or complements.1

However, Singh and Vives (1984) state that the conclusion that Bertrand competition is more efficient than Cournot competition is not correct ‘if one considers supergame equilibria. Price-setting supergame equilibria may support higher prices than quantity-setting equilibria for either homogeneous or differentiated products. See Brock and Scheinkman (1981) and Deneckere (1983).’ That is, Singh and Vives restrict the validity of the conclusion to the class of static games only.2

Moreover, Vives (1984), analysing an incomplete information setting where firms receive signals about the uncertain demand, proves that the Bertrand Bayesian–Nash price (quantity) is, again, lower (higher) than the Cournot Bayesian–Nash one.3

Finally, Amir and Jin (2001) consider a model with linear demand and a mixture of substitute and complementary products and they find support for the conventional wisdom, though with some limitations. In particular, they prove that ‘Price competition is indeed more competitive according to the following criteria: lower mark-up/output ratios, larger average output, and lower average price.’

In this paper, I show that introducing incomplete information about rivals' costs of production brings interesting new insights to the debate and leads to results that are quite different from the traditional views. Indeed, I show that in a homogeneous oligopoly in which each firm knows the value of its own marginal cost and the distribution function of its rivals' ones, in equilibrium, the Bertrand price (quantity) might be higher (lower) than the Cournot price (quantity). This will be the case—rather surprisingly—when all firms are relatively efficient, that is, have sufficiently low costs. The intuition for this result is that when firms have low costs, they will all produce a relatively large quantity in the Cournot game so that the price will be relatively low. To the contrary, in the Bertrand game, only one firm will produce in equilibrium and will sell at a high price-cost margin as long as its cost is low. Moreover, individual firms' ex ante expected profits, i.e. before the game is actually played and the true costs of the rivals revealed, are often higher in the Bertrand game. This conclusion provides an interesting comparison with the results obtained by Vives (1984) in his model with uncertain demand. Vives shows that firms' expected profits are always higher in the Cournot game. Another interesting result is that while the ex post profit of the less efficient firms is generally positive in the Cournot game and always zero in the Bertrand game, the ex post profit of the most efficient firm is very likely to be higher in the latter. Finally, I show that, even though the Bertrand price can be higher than the Cournot price when the costs of production are sufficiently low, the Bertrand model may still be preferred to the Cournot model from a social welfare point of view. This is due to the fact that while in the Bertrand model the most efficient firm satisfies the whole market demand, in the Cournot model also the inefficient firms normally produce a positive quantity but at a higher cost of production; therefore, their price-cost margins as well as their profits will be lower than the corresponding values for the most efficient firm. This implies that even when the consumer surplus generated by the Bertrand model is lower than that of Cournot because the Bertrand price is higher than the Cournot price, the Bertrand model may still lead to a higher level of social welfare, because of a higher producers' surplus.4

The paper is organised as follows. In Section 2, I analyse the Bertrand and Cournot static games with incomplete information in an industry with n firms. In Section 3, I make comparisons between equilibrium prices, quantities, firms' ex ante and ex post profits as well as social welfare levels implied by the two models.

Cournot and Bertrand models with incomplete information

In both models I will use the following assumptions.

In the industry there are n firms producing a homogeneous product.

The demand function is a linear function of the price: i.e. Q=1−p where Q=∑i=1nqi is the aggregate quantity and p is the price.

The cost function for firm i is Ci(qi)=ciqi; i.e. there are no fixed costs and the marginal cost is constant.

The marginal cost ci is independently and uniformly distributed on [0, 1].

Each firm knows the value of its own cost, but only knows the

Comparison of the two models

We now have all the material necessary to compare the two models. In what follows, I will show that Bertrand competition can lead to a higher (lower) price (quantity) than Cournot competition. Moreover, in the Bertrand game, firm i's ex ante expected profit is often higher and also the ex post profit of the most efficient firm is likely to be higher than in the Cournot game. Finally, I will prove that, even when the Bertrand price is higher than the Cournot price, the Bertrand model may still

Acknowledgements

I am grateful to Claude d'Aspremont, Stephen Martin, Louis Phlips, Neri Salvadori as well as three anonymous referees for their helpful comments. All remaining errors are mine. The views expressed are those of the author and do not necessarily reflect the views of RBB Economics.

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