Duopoly Model with Asymmetric Buyer Distribution

We introduce asymmetric buyer distribution into a traditional duopoly model to study its impact on equilibria both in price-setting competition and quantity-setting competition. We show that non-negative output constraints become binding when products are close enough substitutes and buyer distribution is asymmetric enough. When these constraints are binding, a rival firm with less than 50% buyer distribution effectively exists the market and monopoly profit is accrued to the surviving firm. Since pricesetting competition is more intense than quantity-setting competition, the market exit occurs with less asymmetric buyer distribution in price-setting competition than in quantity-setting competition. Therefore, the reversal of usual profit ranking can arise in the sense that the surviving firm in price competition earns monopoly profit in price-setting competition while the same firm earns much less duopoly profit in quantity-setting competition. Moreover, three distinct limit equilibria are identified in price-setting competition with perfect substitutes and symmetric buyer distribution. These results highlight an important role played by buyer distribution.