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.     

AN ELEMENTARY PROOF OF ARROW'S IMPOSSIBILITY THEOREM*

This paper deals with a situation where a quantity-setting Coumot firm and a price- setting Bertrand firm coexist in the same industry. Under a set of regularity conditions
on demand and cost, we compute equilibrium prices on Cournot-Bertrand "mixed" duopolies, and compare them with those of "pure"Cournot and Bertrand duopolies.     

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.     

ON COURNOT-BERTRAND MIXED DUOPOLIES*

This paper deals;with a situation where a quality-setting Cournot firm and a price-seeting Bertrand firm coexist in the same industry. Under a set of regularity conditions on demand and coast, we compute equilibrium prices on Cournot-Bertrand "mixed"duopolies, and compare them with those of "pure" Cournot and Bertrand duopolies.     

Bertrand vs. Cournot equilibrium with risk averse firms and cost uncertainty

Summary. In an oligopoly game with cost uncertainty and risk averse firms, we show that Bertrand and Cournot equilibrium have different convergence properties when the market is replicated. The Cournot equilibrium price converges to the competitive price. Under very typical and somewhat general conditions, the highest Bertrand equilibrium price converges to one higher than the competitive equilibrium. We also give examples to show how to compute the limit of the highest Bertrand equilibrium prices and illustrate the ideas of the proof. We explore conditions under which the supply curve is upward sloping, a useful condition for our results. Received: April 20, 2000; revised version: May 10, 2001     

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     

Competition with Congestible Networks

We analyse competition between two network providers when the quality of each network depends negatively on the number of customers connected to that network. With respect to price competition we provide a sufficient condition for the existence of a unique pure strategy Nash equilibrium. Comparative statics show that as the congestion effect gets stronger quantities will decrease and prices increase, under both Bertrand and Cournot competition. In an example with endogenous capacities it turns out that equilibrium capacities are at first increasing and then decreasing in the strength of congestion. Furthermore, capacities are higher under Cournot competition. Welfare comparisons between Bertrand and Cournot competition are unambiguous for fixed capacities, but may turn around for endogenous capacities.     

Information sharing about a demand shock

This paper examines two questions in asymmetric Cournot and Bertrand oligopoly with a demand shock. Under which conditions is information sharing a subgame-perfect equilibrium? What is the welfare effect when firms are better off? Given these questions, the normal assumptions in the earlier literature can be relaxed in three ways: demand functions can be asymmetric; a demand shock can affect firms differently; distributions of the demand shock and information signals can be arbitrary. Under these general assumptions, the answer to the first question is: every firm's response to the demand shock is stronger when all firms have perfect information than when one firm does so alone; the answer to the second question is: social welfare increases in Cournot competition, and consumer surplus decreases in Bertrand competition.     

Coalition-proof supply function equilibria in oligopoly

In an industry where firms compete via supply functions, the set of equilibrium outcomes is large. If decreasing supply functions are ruled out, this set is reduced significantly, but remains large. Specifically, the set of prices that can be sustained by supply function equilibria is the interval between the competitive price and the Cournot price. In sharp contrast, when the number of firms is above a threshold we identify (e.g., three if demand is linear), only the Cournot outcome can be sustained by a coalition-proof supply function equilibrium.     

Strategic Cost Reduction and Cost Revelation

Strategic cost reduction requires cost transparency. When unilateral cost revelation is feasible, strategic cost reduction indeed arises as equilibrium. If it is not feasible, however, credible revelation has to be organized, possibly by a trade association. Then, firms face a prisoners' dilemma: in Cournot duopoly, cost revelation arises as an equilibrium, but hurts firms; in Bertrand duopoly, cost concealing is the equilibrium, while firms would be better off with cost revelation. Since cost revelation is socially desirable (undesirable) in Cournot (Bertrand) competition, it should be encouraged (discouraged).     

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.     

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.     

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.     

A note on merger in mixed duopoly: Bertrand versus Cournot

In this note we analyze the incentives to merge in a mixed duopoly if firms compete in prices or quantities. Our model framework mainly follows Barcena-Ruiz and Garzon (J Econ 80:27–42, 2003) who set up the model with quantity competition. We extend their analysis by analyzing the case of competition in prices. Further we compare the incentives to merge with Bertrand and Cournot competition. Comparing quantity with price competition we can show that a merger is more likely with Cournot competition than with Bertrand competition.     

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.     

Effect of credible quality investment with Bertrand and Cournot competition

We show how credible revelation and ability to commit to quality choice effect equilibrium qualities and welfare when product market is either Bertrand or Cournot competition. We show that results depend on the type of competition but not generally on the cost of quality function. We show that with Bertrand competition, the equilibrium qualities are lower with credible commitment. Competition is moderated and producer surplus is higher and consumer surplus lower. With Cournot competition, higher quality will be better but lower quality will be worse with credible commitment. Consumer surplus is always greater with credible commitment and if cost does not increase too quickly with quality, producer surplus will also increase. Thus credible commitment is a collusive device with Bertrand competition but it can improve social welfare with Cournot competition. Received: February 8, 2000; revised version: February 14, 2002 RID="*" ID="*" The idea of this paper originated in the weekly workshops of Mordecai Kurz at Stanford. I am forever in debted to Mordecai and fellow students – Luis Cabral, Peter DeMarzo, John Hillas, Michihiro Kandori, Steve Langois, Patrick McAllister, Steve Sharpe, Peter Streufert, Steve Turnbull and Gyu-Ho Wang – for their criticism and encouragement. I also benefited from comments from Yi-Heng Chen, Jin-Li Hu, Kala Krishna, Jinji Naoto, Thomas J. Prusa, and Shyh-Fang Ueng at various later stages of this work. Last but not least, I am grateful for the detailed comments of the referee.     

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.     

CARTEL STABILITY AND THE CURVATURE OF MARKET DEMAND*

The stability of collusion is analysed for a family of demand functions whose curvature is determined by a parameter varying between zero and infinity. When the number of firms is low, firms may prefer to act as quantity setters in order to increase cartel stability if demand is sufficiently convex. Otherwise, price-setting behaviour enhances their ability to collude. As the number of firms tends to infinity, Cournot behaviour is preferable to Bertrand behaviour in order to stabilize collusion, independently of the characteristics of market demand.     

Biased managers with network externalities

In network industry under Cournot and Bertrand competition, we examine a model when owners of firms hire biased managers who have incorrect market demand. Contrast to previous studies, we show that (i) regardless of the strength of network externalities when consumers form the responsive and passive expectations, owners realize strategic advantage by hiring biased managers to be more aggressive under Cournot and Bertrand competition, (ii) firms prefer facing passive expectations for the weak network externalities and vice versa for the strong network exteranlities under Bertrand and Cournot competition, (iii) if the network size is sufficiently large, then the prisoner's dilemma that firms hire aggressive managers no longer exists under both competition modes. As with no delegation case, we obtain the different rankings of firms' profit depending on both network externalities and forming of expectations under Cournot and Bertrand competition.