Bertrand Duopoly Model Research Articles

Cournot–Bertrand Duopoly Model: Dynamic Analysis Based on a Computed Cost

In this paper, some mathematical properties and dynamic investigations of a Cournot–Bertrand duopoly game using a computed nonlinear cost are studied. The game is repeated and its evolution is presented by noninvertible map. The fixed points for this map are calculated and their stability conditions are discussed. One of those fixed points is Nash equilibrium, and the discussion shows that it can be unstable through flip and Neimark–Sacker bifurcation. The invariant manifold for the game’s map is analyzed. Furthermore, the case when both competing firms are independent is investigated. Due to unsymmetrical structure of the game’s map, global analysis gives rise to complicated basin of attraction for some attracting sets. The topological structure for these basins of attraction shows that escaping (infeasible) domain for some attracting sets becomes unconnected and the rise of holes is obtained. This confirms the existence of contact bifurcation.

Price response of the high fructose corn syrup industry in the United States: a Bertrand model application

ABSTRACT This paper uses a Bertrand duopoly model application and derives a high fructose corn syrup (HFCS) price reaction function that shows the response of the price of HFCS to changes in the price of sugar. It conducts cointegration analysis and assesses the performance of the HFCS price reaction model by estimating a Dynamic Ordinary Least-Squares model. It finds that the prices of HFCS-55, sugar, and corn are cointegrated over the period 1981:1 to 2017:12. The estimated long-run relationship shows that a one percent increase in the price of sugar increases the price of HFCS-55 by 0.19 percent. Additionally, the price of corn has a positive effect on the price of HFCS-55. The model and the results offer important insights for antitrust litigation.

PRICE-SETTING MIXED DUOPOLY, SUBSIDIZATION AND THE ORDER OF FIRMS’ MOVES: SUBSTITUTIVE, INDEPENDENT AND COMPLEMENTARY GOODS

This paper uses a mixed Bertrand duopoly model comprising a public firm and a private firm, and investigates the role that production subsidies play in the mixed duopoly model regarding privatization and efficiency. The paper considers substitutive, independent and complementary goods, and examines the following three games: (i) the public firm and the private firm simultaneously and independently set their own prices, (ii) the public firm acts as a Stackelberg leader, and (iii) both firms act simultaneously and independently as profit-maximizers. The paper solves the three games and demonstrates that under the optimal subsidy of each of substitutive, independent and complementary goods, both firms’ profits, prices, outputs and economic welfare are respectively identical regardless of the nature of the competition.

Self-Enforcing Price Leadership

A dynamic Bertrand-duopoly model where price leadership emerges in equilibrium is developed. In the price leadership equilibrium, a firm leads price changes and its competitor always matches in the next period. The firms produce a homogeneous product and are identical except for the information they possess about demand. The market size follows a two-state Markov process. Market size realizations are observed by one of the firms but not the other. Without explicit communication, price leadership allows firms to jointly approximate monopolistic profits in equilibrium as the market size becomes more persistent provided that firms are patient. In the presence of persistent market dynamics, the informed firm’s price serves as a signal of current and therefore future market conditions. In the proposed price leadership equilibrium, the informed firm could cut prices without being detected, but it does not do so because it would lead the uninformed to also lower their price in the following period.

The stability and duality of dynamic Cournot and Bertrand duopoly model with comprehensive preference

In this paper, a Cournot model and its dual Bertrand model where firms have comprehensive preferences are developed. The preference was proposed by Bowles (2004) based on the results of Rabin (1993) and Lebine (1998). Comparable static method is used to illustrate the impact of parameters on equilibrium. Three scenarios are classified by preference parameter: completely cooperative, hostile to each other and general Cournot or Bertrand model. In each situation, Cournot model and its dual model present different chaos phenomenon which have various and abundant strange attractors, while the shapes of the stability regions are similar in the same situation. In the static setting, the local stability analysis of equilibrium gives the bounded region of quantity or price. The one-dimensional Logistic mapping is applied to studying the dynamics of the models on invariant axes. The critical curves classify different regions according to the number of preimages (q1, q2) or (p1, p2). Besides, simulations give more intuitive results: the cycle attractor, chaotic attractor and the basin of attraction with “holes” are presented. The article also provides new findings that under the assumption of comprehensive preference, the Cournot model with substitutes (complements) and the Bertrand model with complements (substitutes) are still duality models in the dynamic settings.

Quantum Games with Unawareness with Duopoly Problems in View

Playing the Cournot duopoly in the quantum domain can lead to the optimal strategy profile in the case of maximally correlated actions of the players. However, that result can be obtained if the fact that the players play the quantum game is common knowledge among the players. Our purpose is to determine reasonable game outcomes when players’ perceptions about what game is actually played are limited. To this end, we consider a collection consisting of the classical and quantum games that specifies how each player views the game and how each player views the other players’ perceptions of the game. We show that a slight change in how the players perceive the game may considerably affect the result of the game and, in the case of maximally correlated strategies, may vary from the inefficient Nash equilibrium outcome in the classical Cournot duopoly to the Pareto optimal outcome. We complete our work by investigating in the same way the Bertrand duopoly model.

Split equilibrium problems for related games and applications to economic theory

ABSTRACTIn this paper we introduce the concept of split Nash equilibrium problems associated with two related noncooperative strategic games. Then we apply the Fan-KKM theorem to prove the existence of solutions to split Nash equilibrium problems of related noncooperative strategic games, in which the strategy sets of the players are nonempty closed and convex subsets in Banach spaces. As an application of this existence to economics, an example is provided that studies the existence of split Nash equilibrium of utilities of two related economies. As applications, we study the existence of split Nash equilibrium in the dual (playing twice) extended Bertrand duopoly model of price competition.

Split Equilibrium Problems for Related Games and Applications to Economic Theory

In this paper we introduce the concept of split Nash equilibrium problems associated with two related noncooperative strategic games. Then we apply the Fan–KKM theorem to prove the existence of solutions to split Nash equilibrium problems of related noncooperative strategic games, in which the strategy sets of the players are nonempty closed and convex subsets in Banach spaces. As an application of this existence to economics, an example is provided that studies the existence of split Nash equilibrium of utilities of two related economies. As applications, we study the existence of split Nash equilibrium in the dual (playing twice) extended Bertrand duopoly model of price competition.

THE DYNAMICS OF BERTRAND PRICE COMPETITION WITH COST‐REDUCING INVESTMENTS

AbstractWe extend the classic Bertrand duopoly model of price competition to a dynamic setting where competing duopolists invest in a stochastically improving production technology to “leapfrog” their rival and attain temporary low‐cost leadership. We find a huge multiplicity of Markov‐perfect equilibria (MPE) and show that when firms move simultaneously the set of all MPE payoffs is a triangle that includes monopoly payoffs and a symmetric zero mixed strategy payoff. When firms move asynchronously, the set of MPE payoffs is strictly within this triangle, but there still is a vast multiplicity of MPE, most of which involve leapfrogging.

Price competition with differentiated goods and incomplete product awareness

We investigate the effects of increased transparency on prices in the Bertrand duopoly model. Market transparency is defined as the proportion of consumers that are fully informed about the market and thus not captive to one firm. We consider two main cases of strategic interaction, prices as strategic complements and as strategic substitutes. For the former class of games, conventional wisdom concerning prices is confirmed, in that they decrease with market transparency. Consumer welfare always increases with higher transparency but changes in firms’ profits are ambiguous. For the latter class of games, an increase in market transparency may lead to an increase in one of the prices, which implies ambiguous effects on both consumer welfare and firms’ profits. An example with linear demand for differentiated products is also investigated. The results of the paper shed light on the mixed evidence concerning the effects of the Internet on retail markets and may illuminate some of the ongoing related public policy debates.

Konkurencja typu Bertranda przy kwadratowych funkcjach kosztów

The authors focus on a model of competition known in economics as Bertrand competition. They investigate how the outcome of Bertrand competition changes when linear cost functions are replaced by quadratic functions in the model. Named after French mathematician Joseph Louis Francois Bertrand (1822-1900), Bertrand competition describes interactions among firms and a market situation in which firms make their output and pricing decisions based on the assumption that their competitors will not change their own prices. Prokop, Ramsza and Wiśnicki show that the introduction of quadratic cost functions in the model leads to a qualitative change in competition between firms. In this case, the standard assumption that a firm with a lower price is interested in taking over the entire market is not only unrealistic (even in the absence of capacity constraints), but also irrational from the profit-maximization viewpoint, the authors say. Therefore the results of previous studies are misleading, according to Prokop, Ramsza and Wiśnicki. They argue that the so-called Bertrand duopoly model should be adjusted to better capture the behavior of firms under quadratic cost functions. The authors relax the assumption that the output of firms is determined by existing demand. They offer a modified Bertrand duopoly model in which firms competing in prices are free to choose their level of production depending on the market demand for their products at specific prices. “Under the modified assumptions, the static price competition game of identical duopolists has no symmetric equilibrium in pure stra- tegies,” the authors conclude. Their research shows that, under quadratic cost functions, it is possible to expect price fluctuations on oligopolistic markets rather than a stable equilibrium situation described by the standard model of price competition.

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.

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.

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.

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.

Informative advertising with spillover effects

This paper studies the effects of advertising in a Bertrand duopoly model, which extends Grossman and Shapiro (1984) by introducing the spillover advertising effect on top of the direct advertising effect. It is shown that the spillover effects would make the environment more monopolistic: higher prices and lower demand.

Antidumping Regulations: Anti-Competitive and Anti-Export

In a Bertrand duopoly model, it is shown that an antidumping regulation can be strategically exploited by the home firm to reduce the degree of competition in the home market. The home firm commits not to export to the foreign market which gives the foreign firm a monopoly in its own market. As a result the foreign firm will increase its price allowing the home firm to increase its price and its profits. If the products are sufficiently close substitutes then the higher profits in the home market are large enough to compensate for the loss of profits on exports.

STRATEGIC EXPORT POLICY IN VERTICALLY RELATED MARKETS

ABSTRACTThis paper analyses how strategic export policies are affected by introducing an imperfectly competitive intermediate good into a Bertrand duopoly model with product differentiation, where a home and a foreign final‐good firm export to a third‐country market. It is shown that when the home and foreign markets for the intermediate good are segmented, the optimal export policy towards the final good is a tax. In contrast, under integrated markets, the optimal export intervention is a subsidy. Whether bilateral export intervention is welfare improving compared with free trade, depends on the degree of product differentiation between the home and foreign final goods.

Public Goods Provision in the African Periphery: A Bertrand Duopoly Model

Until now, corruption has been treated as the “block-box” of African political economy. It can hinder economic growth and thwart political development but very little information exists on it. This paper disaggregates corruption in the context of a regional trade model in addressing a long-standing conundrum in political science which centers on why the African state offers weak provision of public goods to its rural regions. In a quest for answers, comparisons to European state formation highlight Africa’s sparsely-populated and inhospitable hinterland that does not warrant international competition so attracts little in the form of coercion or capital concentration. If the state finds it too expensive to generate tax revenue in these regions it will then be slow to provide public goods essential to economic development. With the introduction of international competition to this model, states will seek to provide public goods at a lower marginal cost as it did in Europe, as it is a signal that there is a welfare benefit from this hinterland. In addition, international competition requires continual efforts towards efficiency and proficiency while bribes fund these state investments.