The Cournot-Bertrand model and the degree of product differentiation

We consider a duopoly model with a Cournot-type firm and a Bertrand-type firm. Both survive and the equilibrium is stable when there is sufficient product differentiation. With no differentiation, only the Cournot-type firm survives, and the outcome is perfectly competitive.     

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.     

Dynamic properties of a Cournot–Bertrand duopoly game with differentiated products

In this paper we consider a Cournot–Bertrand duopoly model with linear demand and cost functions and with product differentiation. We propose a dynamic framework for the study of the stability properties of this kind of mixed oligopoly game, a rather neglected topic in the existing literature despite its relevance. In particular, in this paper we highlight the role of best response dynamics and of an adaptive adjustment mechanism for the stability of the equilibrium.     

SPATIAL DISCRIMINATION,PRODUCT SUBSTITUTABILITY AND WELFARE

The paper examines a quantity–location duopoly game in a spatial discrimination model in which the delivered goods are assumed to be imperfect substitutes or complements. By extending the range of the unit transportation cost analysed in the existing literature, it is shown that a dispersed equilibrium arises in which the choice of the optimal locations is affected by the degree of product substitutability. The interaction between the latter and the size of the transportation cost is also discussed in order to verify its welfare implications. In particular, it is shown that in this spatial framework imperfect substitutability may increase welfare.     

A continuum of commitment

We examine a generic three-stage game for two players with alternating moves, where the first player can choose the level of adjustment cost to be paid in the last period to modify the action she announced in the first period. In the resulting continuum of commitment options, convexifying the choice between first-mover and second-mover advantage in pure strategies, we characterize when an intermediate adjustment-cost level is chosen in equilibrium. We show that the wastefulness of the adjustment cost may be in the players’ best interest, improving both of their individual net payoffs over making any fraction of the adjustment cost a side-payment from the first to the second player.     

R&D games in a Cournot duopoly with isoelastic demand functions: A comment

In a recent paper, Tramontana (Economic Modelling, 27; 350-357, 2010) investigates the stability properties of a Cournot Duopoly game when the demand function is isoelastic. In this note, we show that for some well known applications of two-stage Cournot games (D. Aspremont and Jacquemin, American Economic Review, 78, 1122-1137, 1988) an isoelastic demand function can guarantee both the existence and the uniqueness of a Nash Equilibrium even in cases where existence is not obtained with linear demands.     

Imitation and the evolution of Walrasian behavior: Theoretically fragile but behaviorally robust

A well-known result by Vega-Redondo (1997) [18] implies that in symmetric Cournot oligopolies, imitation leads to the Walrasian outcome. We show that this result is not robust to the slightest asymmetry in costs, since every outcome where agents choose identical actions will be played some fraction of the time in the long run. We then conduct experiments to check this fragility. We obtain that, contrary to the theoretical prediction, the Walrasian outcome is a good predictor of market outcomes. Finally, we suggest a new theory based on a mix of imitation and other learning processes that explains subjects' behavior fairly well.     

Imitation—theory and experimental evidence

We introduce a generalized theoretical approach to study imitation and subject it to rigorous experimental testing. In our theoretical analysis we find that the different predictions of previous imitation models are mainly explained by different informational assumptions, and to a lesser extent by different behavioral rules. In a laboratory experiment we test the different theories by systematically varying information conditions. We find significant effects of seemingly innocent changes in information. Moreover, the generalized imitation model predicts the differences between treatments well. The data provide support for imitation on the individual level, both in terms of choice and in terms of perception. Furthermore, individuals’ propensity to imitate more successful actions is increasing in payoff differences.     

A NOTE ON ENDOGENOUS TIMING WITH STRATEGIC DELEGATION: UNILATERAL EXTERNALITY CASE

We investigate the endogenous choice of roles by managerial firms in the presence of unilateral externality. The choice over timing can be taken either by managers or by owners. It is shown that: (i) the choice of the timing by managers entails the same profit that owners would have achieved by specifying the timing in the delegation contract; and (ii) firms move simultaneously if the degree of unilateral externality is small, while sequentially if the degree of unilateral externality is large, with the firm generating unilateral externality as a follower; the owner of the follower firm delegates to restrict output, while his/her counterpart does not delegate it.     

Elastic demand, sunk costs and the Kreps–Scheinkman extension of the Cournot model

Summary. The paper shows that, with any rationing mechanism between the efficient and proportional extremes, the Kreps-Scheinkman two-stage quantity-price game reduces to the Cournot model if demand is uniformly elastic and if all costs are sunk at the first stage, thus providing positive results to set against existing negative statements. Received: May 24, 1995; revised version: March 6, 1997     

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     

The folk theorem for repeated games with observation costs

The folk theorem literature has been relaxing the assumption on how much players know about each other's past action. Here we consider a general model where players can “buy” precise information. Every period, each player decides whether to pay a cost to accurately observe the actions chosen by other players in the previous period. When a player does not pay the cost, he obtains only imperfect private signals. Observational decisions are unobservable to others. Known strategies such as trigger strategies do not work since they fail to motivate players to pay for information. This paper shows that the folk theorem holds for any level of observation costs. Unlike existing folk theorems with private monitoring, ours imposes virtually no restriction on the nature of costless imperfect signals. The theorem does not use explicit or costless communication, thereby having implications on antitrust laws that rely on evidence of explicit communication. The main message is that accurate observation alone, however costly, enables efficient cooperation in general repeated games.     

A DYNAMIC ENTRY AND PRICE GAME WITH CAPACITY INDIVISIBILITY

Strategic market interaction is here modelled as a two‐stage game in which potential entrants choose capacities and next active firms compete in prices. Due to capital indivisibility, the capacity choice is made from a finite grid and there are economies of scale. In the simplest version of the model with a single production technique, the equilibrium turns out to depend on the ratio between the level of total output at the long‐run competitive equilibrium and the firm's minimum efficient scale: if that ratio is sufficiently large (the market is sufficiently ‘large’), then the competitive price emerges at a subgame‐perfect equilibrium of the capacity and price game; if not, then the firms randomize in prices on the equilibrium path. The role of the market size for the competitive outcome is shown to be even more important if there are several available production techniques.     

Unilateral production restrictions in a dynamic duopoly

This paper examines a dynamic game of exploitation of a productive asset by a duopoly. A closed-loop Nash equilibrium of the game is constructed and used to analyze the effects of a unilateral production restriction. Surprisingly, such unilateral action may result in a decrease of the long-run asset's stock. We also exhibit production restrictions that can result simultaneously in an increase of the asset's stock and the long-run profits of the firm that is being imposed the production restriction. Moreover, a unilateral decrease of the production of one firm can induce its rival to also decrease its production.     

Entry, exit, and imperfect competition in the long run

An infinite-horizon, stochastic model of entry and exit with sunk costs and imperfect competition is constructed. A subgame perfect Nash equilibrium for the general dynamic stochastic game is shown to exist as a limit of finite-horizon equilibria. This equilibrium has a relatively simple structure characterized by two numbers per finite history. Under very general conditions, it tends to exhibit excessive entry and insufficient exit relative to a social optimum.     

The codetermined firm in a Cournot duopoly: A stability analysis

This paper aims to study the stability issue in a Cournot duopoly with codetermined firms. We show that when both firms codetermine employment together with decentralised employees' representatives, a rise in wages acts as an economic (de)stabiliser when the wage is fairly (high) low, while under profit maximisation a rise in wages always acts as a stabilising device because the parametric stability region monotonically increases with the wage in such a case. Moreover, a rise in the union's bargaining power has a de-stabilising effect, except when the wage is low and the firm power is already high. Therefore, under codetermination a change either in the wage or firm power in the Nash bargaining plays an ambiguous role on stability. We also show with numerical simulations that complex dynamics can also occur.     

The dynamics of a differentiated duopoly with quantity competition

We analyse the dynamics of a Cournot duopoly with heterogeneous players to investigate the effects of micro-founded differentiated products demand on stability. The present study, which indeed modifies and extends Zhang et al. (2007) (Zhang, J., Da, Q., Wang, Y., 2007. Analysis of nonlinear duopoly game with heterogeneous players. Economic Modelling 24, 138–148) and Tramontana, F., (2010) (Tramontana, F., 2010. Heterogeneous duopoly with isoelastic demand function. Economic Modelling 27, 350–357), reveals that a higher degree of product differentiation may destabilise the market equilibrium. Moreover, we show that a cascade of flip bifurcations may lead to periodic cycles and ultimately chaotic behaviours. Since a higher degree of product differentiation implies weaker competition, then a theoretical implication of our findings, that also constitute a policy warning, is that a fiercer (weaker) competition tends to stabilise (destabilise) the unique positive Cournot–Nash equilibrium.     

Product differentiation and location decisions under demand uncertainty

We investigate Hotelling's duopoly game of location-then-price choices with quadratic transportation costs and uniformly distributed consumers under the assumption that firms are uncertain about the exact location of demand. We characterize the unique equilibrium and the socially optimal locations. Contrary to the individual-level random utility models, location uncertainty is a differentiation force. In equilibrium, increases in the variance of the uncertainty lead to greater differentiation, higher expected equilibrium prices and profits, and a greater welfare loss.     

Uncertain spatial demand and price flexibility: A state space approach to duopoly

Sufficient conditions for a unique price equilibrium, in terms of the uncertainty distribution and the state contingent consumer distributions, are given for spatial duopoly. Also considered are efficiency and endogenous locations for the symmetric case and comparative statics on price flexibility.     

Intertemporal market division:: A case of alternating monopoly

In this paper, we report on an equilibrium with market dominance that exists in a simple two-firm model that features neither entry barriers nor sophisticated punishment strategies. This equilibrium induces an intertemporal market division in which the two firms alternate as monopolists - despite the fact that the model also sustains a Cournot duopoly. Even when initially both firms are active in the market, the alternating monopoly reveals itself rather quickly. Moreover, it Pareto dominates the Cournot equilibrium - as it is close to the cartel outcome. Several examples of what well may be such alternating monopolies are presented.