Bertrand Duopoly Research Articles

On Bertrand duopoly game with differentiated goods

The paper investigates a dynamic Bertrand duopoly with differentiated goods in which boundedly rational firms apply a gradient adjustment mechanism to update their price in each period. The demand functions are derived from an underlying CES utility function. We investigate numerically the dynamical properties of the model. We consider two specific parameterizations for the CES function and study the Nash equilibrium and its local stability in the models. The general finding is that the Nash equilibrium becomes unstable as the speed of adjustment increases. The Nash equilibrium loses stability through a period-doubling bifurcation and the system eventually becomes chaotic either through a series of period-doubling bifurcations or after a Neimark–Sacker bifurcation.

Trade Costs, FDI Incentives, and the Intensity of Price Competition

Empirical evidence shows that an increase in trade liberalization causes an increase in foreign direct investments (FDIs). Here we propose an explanation to this apparent puzzle by exploiting the intensity of competition in a Bertrand duopoly with convex costs where the two firms enter in a new market. We adopt Dastidar's (1995) approach, delivering a continuum of Bertrand-Nash equilibria ranging above marginal cost pricing, to show that softening competition may indeed more than offset the standard effect generated by trade costs, thereby leading to a positive relationship between trade liberalization and FDIs.

Product Differentiation, the Volume of Trade and Profits under Cournot and Bertrand Duopoly

This article analyses how product differentiation affects the volume of trade under duopoly using Shubik-Levitan demand functions rather than the Bowley demand functions used by Bernhofen (2001). The drawback of Bowley demand functions is that an increase in product differentiation increases the size of the market so the increase in the volume of trade may be the result of the increase in the size of the market rather than the increase in product differentiation per se. The Shubik-Levitan demand functions have the advantage that an increase in product differentiation does not increase the size of the market, but consumers still have a ‘love of variety’. It is shown that the volume of trade in terms of quantities falls with increasing product differentiation when the trade cost is relatively low, but rises with increasing product differentiation when the trade cost is relatively high. Among the results, it is shown that the trade liberalisation is more likely to be profitable under Cournot duopoly than under Bertrand duopoly for differentiated products with a positive trade cost.

Complexity analysis of a Cournot–Bertrand duopoly game with different expectations

In this paper, we consider a Cournot–Bertrand mixed duopoly model with different expectations, where the market has linear demand and the firms have fixed marginal cost functions. Two firms choose output and price as decision variables, respectively, under the assumption that there is a certain degree of differentiation between the products offered by firms to avoid the whole market is occupied by the one that applies a lower price. The two players are considered to have bounded rational and static expectations. The existence and local stability condition of Nash equilibrium is investigated. We find the stability region of Cournot–Bertrand system is bigger than that of Cournot or Bertrand system under the same conditions. Furthermore, there are two different kinds of bifurcations when the parameters pass through the different boundary curves of the stability region, which is different from the Cournot or Bertrand model. Numerical simulation method is used to display the dynamic behaviors of the dynamical system, such as periodic cycles, bifurcation diagrams and strange attractors of the systems. The economic explanations of the complex dynamic behaviors are also given.

Are Cooperative R&D Agreements Good For The Society?

Technological progress is crucial for economic growth and enhancement of standard of living in any economy. But firms often have insufficient incentive for R&D, because in spite of patent protection, the benefits of R&D are not always limited to the firms that initially conduct the R&D activities. Consequently, governments around the world often undertake industrial policies to promote collaborative R&D efforts between firms in order to increase R&D. This paper examines the implications of cooperative R&D agreements for the societal well being. The R&D and price decisions are analyzed using a Bertrand Duopoly Model in presence of product differentiation in a two-stage game. It is shown that under cooperative R&D agreements R&D and output levels are larger and prices are lower than under non-cooperation. For complementary and independent goods, these results are valid for any degree of R&D spillover and for substitute goods they may hold even for sufficiently small R&D spillover. These results are more general than D'Aspremont and Jacquemin (1988) who have shown that cooperative R&D levels exceed those under non-cooperation only for large R&D spillover. As for the level of social welfare, this paper finds the cooperative as well as the non-cooperative R&D output and price levels to be socially inefficient. However, cooperative R&D agreements tend to dominate non-cooperative R&D ventures in terms of social welfare. This result also holds for any degree of R&D spillover for complementary and independent goods and even for sufficiently small spillover in the case of substitute goods.

Ex-post regret heuristics under private values (I): Fixed and random matching

In contexts in which players have no priors, we analyze a heuristic process based on ex-post regret as a guide to understand how to play games of incomplete information under private values. The conclusions depend on whether players interact within a fixed set (fixed matching) or they are randomly matched to play the game (random matching). The relevant long run predictions are minimal sets that are “closed under same or better reply” operations. Under additional assumptions in each case, the predictions boil down to Nash equilibria, ex-post equilibria or minimax regret equilibria. These three paradigms exhibit nice robustness properties in the sense that they are independent of beliefs about the exogenous uncertainty of type spaces. The results are illustrated in several applications, including second-price auctions, first-price auctions and Bertrand duopolies.

Toward Environmentally Sustainable Mobile Computing Through an Economic Framework

Amid the plethora of initiatives and research endeavors targeting the minimization of power and energy consumption of information and communication technologies, what has been largely missing is an effort to reduce the energy consumption and electronic waste generated by the rapidly growing segment of mobile computing and communication devices. Prior work with energy efficiency in mobile devices has primarily focused on the goal of maximizing battery life of these devices and not on the broader concept of environmentally sustainable mobile computing. This paper provides an overview of the concept of environmentally sustainable mobile computing and identifies reduction in manufacturing energy costs and electronic waste generated as two important outcomes that can be achieved by increasing device lifespan. Increased device lifespans, however, are possible only if the underlying market forces support such a paradigm shift. This paper develops an analytical economic framework as it applies to mobile phones by analyzing a market scenario of two firms competing under a differentiated Bertrand duopoly model. The framework and its analysis helps verify intuitions about the reasons that affect a firm's decision to offer an environmentally sustainable choice for consumers and considers the feasibility, possible benefits, and challenges in increasing device lifespan, including technical challenges. The results of this paper also provide guidance on the relative impacts of various factors involved on device lifespan such as user-experience, subsidies, and differences in underlying costs to providers.

Intra-Industry Trade with Substitutes and Complements under Bertrand Duopoly

Intra-Industry Trade with Substitutes and Complements under Bertrand Duopoly

Patent Licensing with Bertrand Competitors

This paper analyses an optimal two‐part licensing scheme based on ad valorem royalties within a differentiated Bertrand duopoly where the innovator is also the downstream producer, and compares it with the optimal two‐part per‐unit royalty mechanism. After showing that the optimal two‐part ad valorem licensing scheme reduces to a pure ad valorem royalty scheme, we show that per‐unit contracts are typically preferred to ad valorem contracts by the patentee, as, under price competition, the per‐unit royalty has a stronger strategic effect than the ad valorem royalty. In contrast, welfare is higher under the ad valorem contract than under the per‐unit mechanism.

Analysis of Price Stackelberg Duopoly Game with Bounded Rationality

The classical Stackelberg game is extended to boundedly rational price Stackelberg game, and the dynamic duopoly game model is described in detail. By using the theory of bifurcation of dynamical systems, the existence and stability of the equilibrium points of this model are studied. And some comparisons with Bertrand game with bounded rationality are also performed. Stable region, bifurcation diagram, The Largest Lyapunov exponent, strange attractor, and sensitive dependence on initial conditions are used to show complex dynamic behavior. The results of theoretical and numerical analysis show that the stability of the price Stackelberg duopoly game with boundedly rational players is only relevant to the speed of price adjustment of the leader and not relevant to the follower’s. This is different from the classical Cournot and Bertrand duopoly game with bounded rationality. And the speed of price adjustment of the boundedly rational leader has a destabilizing effect on this model.

Price discrimination or uniform pricing: Which colludes more?

Conventional wisdom attributes different economic outcomes of uniform pricing and price discrimination to the heterogeneity in market conditions or market participants, such as differences in demand elasticity or production costs. We offer a new explanation for the observed differences that relates to behavioral aspects rather than demand- or supply-side effects. In particular, in a symmetric Bertrand duopoly laboratory experiment, for which theory predicts no differences between the two pricing regimes, we find that tacit price collusion is systematically higher under price discrimination than under uniform pricing.

The dynamics of a Bertrand duopoly with differentiated products: Synchronization, intermittency and global dynamics

We study the dynamics of a duopoly game à la Bertrand with horizontal product differentiation as proposed by Zhang et al. (2009) [35] by introducing opportune microeconomic foundations. The final model is described by a two-dimensional non-invertible discrete time dynamic system T. We show that synchronized dynamics occurs along the invariant diagonal being T symmetric; furthermore, we show that when considering the transverse stability, intermittency phenomena are exhibited. In addition, we discuss the transition from simple dynamics to complex dynamics and describe the structure of the attractor by using the critical lines technique. We also explain the global bifurcations causing a fractalization in the basin of attraction. Our results aim at demonstrating that an increase in either the degree of substitutability or complementarity between products of different varieties is a source of complexity in a duopoly with price competition.

Progressive managerial bonuses in a spatial Bertrand duopoly

The relationship of managerial bonuses and profit maximization is interesting both from an economic and a managerial viewpoint. Our contribution to this literature is showing that progressive managerial bonuses can increase profits in a spatial Bertrand competition, and furthermore they can help collusion.

The Classical Notion of Competition Revisited

This article seeks to fill a lacuna within classical economics concerning the process of market price determination in situations of market disequilibrium. To this aim, first we distinguish the classical notion of free competition from the Walrasian notion of perfect competition and we argue that the latter is beset with some theoretical difficulties alien to the former. Second, we reconstruct in some detail Smith’s and Marx’s views concerning market price determination and show that Marx’s extensive use of metaphors and numerical examples foreshadows the modern taxonomy of buyers’ market, sellers’ market, and mixed strategy equilibrium in the capacity space of a standard Bertrand duopoly model. Finally, we highlight similarities and differences between the classical notion of competition and contemporary Bertrand competition models.

The research on Cournot–Bertrand duopoly model with heterogeneous goods and its complex characteristics

This paper details the research of the Cournot–Bertrand duopoly model with the application of nonlinear dynamics theory. We analyze the stability of the fixed points by numerical simulation; from the result we found that there exists only one Nash equilibrium point. To recognize the chaotic behavior of the system, we give the bifurcation diagram and Lyapunov exponent spectrum along with the corresponding chaotic attractor. Our study finds that either the change of output modification speed or the change of price modification speed will cause the market to the chaotic state which is disadvantageous for both of the firms. The introduction of chaos control strategies can bring the market back to orderly competition. We exert control on the system with the application of the state feedback method and the parameter variation control method. The conclusion has great significance in theory innovation and practice.

A Feedback Effect from Stock Market Trading to Innovations in a Bertrand Duopoly

Knowledge spillover often influences firms' innovation decisions and consequently the technological advance of an industry. Spillover may be caused by involuntary or strategic disclosures of firms' intermediary R&D progress, it can also occur due to the feedback from stock prices to decisions about innovation investment. In this paper, I endogenize the probability of knowledge spillover, which is no longer the consequence of disclosure strategies. I show that changes in stock price can amount to information leakage about the success probability of an innovation, which induces the interdependence between the fundamentals and the amount of information produced in the stock market. When providing additional information to competing firms in a Bertrand duopoly, this feedback effect may also result in fewer innovations and decreased consumer welfare, provided that the innovation requires low investment and has a low success rate. The overall impact also depends on the incentives of traders in the stock market.

Oligopolistic Market Structure in the Japanese Pistachio Import Market

Abstract Iran is the major producer of pistachio nuts in the world. Iran dominated the international pistachio export markets until 1982, when a new competitor, the United States, emerged in the international markets. Currently, both Iran and the United States are major pistachio exporters to Japan. In this study, we address the following empirical question: Is the Japanese pistachio import market best characterized by Cournot or Bertrand duopoly competition or some other form of game? We use two historical data sets on pistachio prices and quantities imported by Japan from Iran and the United States and estimate the demand functions for these two main exporters. Considering two different strategic variables, quantity and price, full information maximum likelihood (FIML) resulted in different estimates. We used normalized likelihood ratio statistics for the model comparison. Having proper game-specific nonlinear-equation restrictions enabled us to form six alternative models. The model that best fit the data was the Stackelberg model with US quantity leadership.

Perverse environmental effects of ambient charges in a Bertrand duopoly

This paper investigates the effect of an increase in ambient charges on total pollution in a Bertrand-type duopolistic market. We analyse two distinct models. In the first model, the regulator announces the ambient charge and then both firms set their prices simultaneously. In the second model, after the regulator announces the ambient charge, both firms simultaneously choose their pollution abatement technologies in the first stage and prices in the second stage. An interesting conclusion of this paper is that in both models the strategic interaction between the firms can lead to a “perverse” ambient charge effect on total pollution. That is, an increase in the ambient charge may lead to greater pollution.

Chaotic dynamics and chaos control in differentiated Bertrand model with heterogeneous players

PurposeThe purpose of this paper is to investigate the chaotic dynamics behaviours and chaos control of differentiated Bertrand model.Design/methodology/approachThe paper analyzes a nonlinear differentiated Bertrand duopoly game by using the theory of bifurcations of dynamical system, where players have heterogeneous expectations and nonlinear cost function: two types of players are considered – bounded rational and naive expectation. The equilibrium point and local stability of the duopoly game are investigated.FindingsThe paper demonstrates that as some parameters of the game are varied, the stability of Nash equilibrium is lost through period doubling bifurcation. The chaotic features are justified numerically via computing Lyapunov exponents, and sensitive dependence on initial conditions. By using the delay feedback control method, the authors have made the system form chaotic state to stable state.Originality/valueThis paper fulfils an identified need to study the diversity of expectations and how to lead to rich dynamics and complexity.

Stability Analysis in a Bertrand Duopoly with Different Product Quality and Heterogeneous Expectations

We study the local stability properties of a nonlinear Bertrand duopoly with vertical differentiation and heterogeneous players under the hypotheses of both covered and uncovered markets. In the former case, the unique pure strategy Nash equilibrium can undergo a flip bifurcation when the extent of consumer’s heterogeneity increases. In the latter, the quality differential determines the stability of prices over time. Numerical evidence is provided to show the occurrence of endogenous fluctuations.