On the comparison of price and quantity competition under endogenous timing

This paper compares Bertrand and Cournot equilibria in a differentiated duopoly with linear demand, and asymmetric constant marginal cost under endogenous timing. It shows that endogenous timing leads to two sequential play with both leader–follower configurations in Bertrand, but simultaneous play in Cournot. Moreover, every firm’s mark-up/output ratio and the two firms’ weighted ‘average’ price are all lower, but the two firms’ weighted ‘average’ output is higher in either of the two sequential Bertrand equilibria than in the simultaneous-move Cournot equilibrium.     

Bertrand versus Cournot revisited

Summary. Bertrand criticized Cournot's analysis of the competitive process, arguing that firms should be seen as playing a strategy of setting price below competitors' prices (henceforth, the Bertrand strategy) instead of a strategy of accepting the price needed to sell an optimal quantity (the Cournot strategy). We characterize Nash equilibria in a generalized model in which firms choose among Cournot and Bertrand strategies. Best responses always exist in this model. For the duopoly case, we show that iterated best responses converge under mild assumptions on initial states either to Cournot equilibrium or to an equilibrium in which only one firm plays the Bertrand strategy with price equal to marginal cost and that firm has zero sales. Received: December 11, 1995; revised version October 2, 1996     

Nature of Competition and New Technology Adoption

The incentives to adopt cost‐reducing technology by firms in a horizontally‐differentiated duopoly are analysed in the present paper. Cost reduction depends on the ‘quality’ of the ‘new technology’. A firm has to procure such technology in a ‘scoring auction’. When the quality offered by the suppliers of this technology lies in the interior of the feasible range, both firms adopt the new technology regardless of the nature of competition (Cournot or Bertrand). However, when there is a corner solution, there are equilibria where only one firm (or no firm) adopts the new technology. With a corner solution the nature of competition affects the equilibrium outcomes.     

Spatial price discrimination in the unidirectional Hotelling model with elastic demand

The unidirectional Hotelling model is extended to allow for elastic demand functions. A two-stage Bertrand-type model and a two-stage Cournot-type model are considered. If firms choose location and then set prices, agglomeration never arises; instead, if firms choose location and then set quantities, agglomeration arises at one endpoint of the segment when the transportation costs are low enough. Instead, when the transportation costs are high enough, a dispersed equilibrium arises in Cournot. The equilibrium distance between the firms is lower in Cournot than in Bertrand. When the transportation costs are high (low) the Bertrand equilibrium is welfare superior (inferior) to the Cournot equilibrium.     

Price and Quantity Competition in a Differentiated Duopoly With Network Compatibility Effects

We consider the efficiency of price and quantity competition in a network products market, where we observe product compatibility with network externalities (hereafter, network compatibility effects). In particular, if network compatibility effects between firms are sufficiently asymmetric, the Cournot equilibrium is more efficient than the Bertrand equilibrium in terms of larger consumer, producer and total surpluses. Then, we consider an endogenous choice of the strategic variables, price and quantity. If the degree of network compatibility effects of the rival firm is larger (smaller) than the degree of product substitutability, then choosing prices (quantities) is a dominant strategy for the firm. Thus, if the network compatibility effects of both firms are larger (smaller), the Bertrand (Cournot) equilibrium arises. Furthermore, if the network compatibility effects between the firms are sufficiently asymmetric, the firm with a larger (smaller) network compatibility effect than a certain level of product substitutability chooses quantities (prices). In this case, the Cournot–Bertrand equilibrium arises, which is less (more) efficient than the Cournot equilibrium in terms of consumer (producer) surplus.     

Collusion in Differentiated Duopolies with Quadratic Costs

The analysis of collusion in infinitely repeated duopoly games has generally assumed that marginal cost is constant, but this note uses quadratic costs (linear marginal costs) to compare the sustainability of collusion under Bertrand and Cournot duopoly with differentiated products. It is shown that when marginal costs are sufficiently increasing in output, then it is always easier to sustain collusion under Cournot duopoly than under Bertrand duopoly for any degree of product substitutability.     

Equilibrium prices in the Bertrand and Cournot oligopolies

The equilibrium prices for the Bertrand and Cournot oligopolies with product differentiation are compared. If all firms have linear demand and cost functions, and if, in addition, the Jacobian matrix of the demand functions has a dominant negative diagonal, the Cournot equilibrium prices are not lower than the Bertrand ones. The general condition for the comparison of the Bertrand and Cournot equilibrium prices can be derived even if the nonlinearity is involved in the cost and/or demand functions.     

Unionized Bertrand Duopoly and Strategic Export Policy

The paper reports that an export subsidy is optimal for a unionized Bertrand duopoly. Following results published by Brander and Spencer ( Journal of International Economics , 1988, pp. 217–34), this establishes the robustness of export subsidization to the mode of competition (Cournot or Bertrand), and contrasts with nonunion results in the literature. If both firms are unionized and both governments pursue active trade policies, a subsidy remains optimal except for a narrow range of extreme substitutability between products. Nations with a lower opportunity cost of labor employ more aggressive policies in equilibrium.     

Vertical differentiation in a Cournot industry: The Porter hypothesis and beyond

We modify the vertically differentiated duopoly model by André et al. (2009) replacing Bertrand with Cournot behaviour, and we characterise the region wherein a Porter-type result takes place. We show that the Porter hypothesis applies in an equilibrium taking always the form of a prisoner's dilemma. Moreover, whenever the asymmetry in the cost parameters between green and brown technology is not too high, a class of equilibria emerges wherein firms converge spontaneously onto the green standard as a result of dominant strategies, any environmental regulation being altogether absent.     

ENDOGENOUS TIMING AND STRATEGIC CHOICE: THE COURNOT‐BERTRAND MODEL

Cournot establishes a Nash equilibrium to a duopoly game under output competition; Bertrand finds a different Nash equilibrium under price competition. Both treat the strategic choice variable (output versus price) and the timing of play as exogenous. We investigate Cournot‐Bertrand models where one firm competes in output and the other competes in price in both static and dynamic settings. We also develop a general model where both the timing of play and the strategic choice variables are endogenous. Consistent with the conduct of Honda and Scion, we show that Cournot‐Bertrand behaviour can be a Nash equilibrium outcome.     

Dynamic efficiency of Cournot and Bertrand competition: input versus output spillovers

We consider the efficiency of Cournot and Bertrand equilibria in a duopoly with substitutable goods where firms invest in process R&D that generates input spillovers. Under Cournot competition firms always invest more in R&D than under Bertrand competition. More importantly, Cournot competition yields lower prices than Bertrand competition when the R&D production process is efficient, when spillovers are substantial, and when goods are not too differentiated. The range of cases for which total surplus under Cournot competition exceeds that under Bertrand competition is even larger as competition over quantities always yields the largest producers’ surplus.     

Comparing Cournot and Bertrand equilibria in an asymmetric duopoly with product R&D

We compare the Cournot and Bertrand equilibria in an asymmetric duopoly with product R&D competition. If a firm’s marginal cost is lower than that of its rival, then this firm (its rival) is referred to as the more (less) efficient firm. Under each mode of competition, there are three types of equilibria: blockaded-entry, deterred-entry, and accommodated-entry. Moreover, the presence of R&D investment makes it harder for the less efficient firm to survive. Cournot competition entails a unique equilibrium, whereas Bertrand competition may yield two equilibria. It is harder for the less efficient firm to survive under Bertrand competition than under Cournot competition. Versus Cournot competition, Bertrand competition yields higher industry output, and it shifts production from the less efficient firm to the more efficient firm. This result, together with the known size effect, explains the following three findings. First, the more efficient firm has a normal output ranking, whereas the less efficient firm may demonstrate an output reversal. Second, the more efficient firm may demonstrate a R&D reversal, whereas the less efficient firm has a normal R&D ranking (its Cournot R&D effort exceeds its Bertrand R&D effort). Third, Bertrand competition is more welfare-efficient than Cournot competition.     

The Cournot-Bertrand profit differential: A reversal result in a differentiated duopoly with wage bargaining

This paper compares Cournot and Bertrand equilibria in a downstream differentiated duopoly in which the input price (wage) paid by each downstream firm is the outcome of a strategic bargain with its upstream supplier (labor union). We show that the standard result that Cournot equilibrium profits exceed those under Bertrand competition - when the differentiated duopoly game is played in imperfect substitutes - is reversible. Whether equilibrium profits are higher under Cournot or Bertrand competition is shown to depend upon the nature of the upstream agents’ preferences and on the distribution of bargaining power over the input price. We find that the standard result holds unless unions are both powerful and place considerable weight on the wage argument in their utility function.     

Perpetual Leapfrogging in Bertrand Duopoly

We consider different patterns of infinite technological adoption choices by firms in a Bertrand duopoly. Every period technological progress provides a sequence of cost reducing innovations. The equilibrium concept is Markov perfect equilibrium. We analyze conditions for which equilibrium adoption leads to persistent leadership and those where firms alternate in adoption inducing leapfrogging. Only leapfrogging leads to technological improvement in the long run. Demand conditions play a crucial role in determining whether leapfrogging can be perpetual in Bertrand duopoly.     

OPTIMUM-WELFARE AND MAXIMUM-REVENUE TARIFFS UNDER BERTRAND DUOPOLY

This article derives the maximum‐revenue tariff and the optimum‐welfare tariff under Bertrand duopoly with differentiated products. It is shown that both tariffs are lower under Bertrand duopoly than under Cournot duopoly. Also, the optimum‐welfare tariff may exceed the maximum‐revenue tariff under both Bertrand duopoly and Cournot duopoly. This result is more likely the lower the costs of the home firm relative to the costs of the foreign firm, and the greater the degree of product substitutability. Also, it is shown that the optimum‐welfare tariff is less likely to exceed the maximum‐revenue tariff under Bertrand duopoly than under Cournot duopoly.     

Licensing under vertical product differentiation: Price vs. quantity competition

This paper develops a duopoly model of vertical product differentiation where two domestic firms incur variable costs of quality development. These domestic firms can purchase a superior foreign technology through licensing. Outcomes between Bertrand and Cournot competition are compared. We find that licensing raises domestic welfare, and domestic welfare is higher in Bertrand than in Cournot competition regardless of whether or not domestic firms engage in licensing. Non-exclusive licensing is also found to benefit the domestic country more than exclusive licensing.     

PRICE AND QUANTITY COMPETITION WITH ASYMMETRIC COSTS IN A MIXED DUOPOLY: A TECHNICAL NOTE

We consider a mixed duopoly in which private and public firms can choose to strategically set prices or quantities when the public firm is less efficient than the private firm. Thus, even with cost asymmetry, we obtain exactly the same result (i.e., Bertrand competition) of Matsumura and Ogawa (2012) if Singh and Vives’ (1984) assumption of positive primary outputs holds. However, compared to endogenous determination of the type of contract without cost asymmetry, our main finding is that in the wider range of cost asymmetry, different type(s) of equilibrium related to or not related to the limit‐pricing strategy of the private firm can be sustained. Thus, when considering an implication on privatization, we may overestimate the welfare gain of privatization because Cournot competition takes place after privatization even though cost asymmetry exists between firms. While the result of Matsumura and Ogawa (2012) holds true if the goods are complements, we find the novel results in the case of substitutes.     

PRICE AND QUANTITY COMPETITION IN A UNIONISED MIXED DUOPOLY: THE CASES OF SUBSTITUTES AND COMPLEMENTS*

We investigate a differentiated mixed duopoly in which private and public firms can choose to strategically set prices or quantities when unions are present. For the case of a unionised mixed duopoly, there exists a dominant strategy only for the public firm that chooses Bertrand competition irrespective of whether the goods are substitutes or complements; there is no dominant strategy for a private firm. Thus, we show that regardless of the nature of goods, social welfare under Bertrand competition is always determined in equilibrium, wherein Bertrand competition entails higher social welfare than Cournot competition. Moreover, our main results hold irrespective of the nature of goods, with the exception that when a sufficiently large parameter of complements is employed, the ranking of a private firm's profit is not reversed, which is in contrast to the standard findings.     

Technology adoption in a differentiated duopoly: Cournot versus Bertrand

This paper shows that the cost as well as the effectiveness of technology has a differential impact on technology adoption under two alternative modes of competition. If the cost of the technology is high, Bertrand competition provides a stronger incentive to adopt technology than Cournot competition unless the effectiveness of the technology is very low. On the contrary, if the cost of the technology is low, Cournot competition fares better than Bertrand competition in terms of technology adoption by firms. This demonstrates that the commonly subscribed assumption of ‘positive primary outputs’ restricts (inflates) the scope of higher degree of technology adoption under Bertrand (Cournot) competition. Moreover, in contrast to standard welfare ranking, it shows that Cournot competition leads to higher social welfare than Bertrand competition under certain situations.