A Comparison of Bertrand and Cournot Profits in Oligopolies with Differentiated Products

We compare equilibrium profits of Bertrand (price) and Cournot (quantity) competition in oligopolies with an arbitrary number of nonsymmetric firms offering differentiated substitutable products under an affine demand function. We provide a precise characterization of the profit relationship in terms of (1) the number of firms, (2) their relative quality and cost differences, and (3) the competition intensity, defined as the maxiumum absolute value of total change in competitors' demand over change in own demand due to a unit change in own price. We first examine the case where firms have the same demand sensitivity to own price and the same demand sensitivity to competitor prices but different cost and quality parameters. For this case, we prove that the total profit of the industry under Cournot competition is at least as high as the total profit under Bertrand competition if the number of firms is less than 28 or if the competition intensity is less than 0.909 or if the differences in quality and cost competitiveness between firms are small. We also prove that for each firm, the profit achieved under Cournot competition is at least as high as the profit achieved under Bertrand competition if the number of firms is less than eight or if the competition intensity is less than 0.739. We then provide numerical and analytical results that qualitatively support the same conclusions for general affine demand functions with variable demand sensitivities to prices.