Stability of Equilibrium Prices in a Dynamic Duopoly Bertrand Game with Asymmetric Information and Cluster Spillovers

Bounded rationality, asymmetric information and spillover effects are widespread in the economic market, and had been studied extensively in oligopoly games, but few references discussed incomplete information in a duopoly market with rationality expectations. Considering the positive externalities brought by the spillover effect between enterprises in a cluster, a duopoly Bertrand game with bounded rationality and asymmetric information is proposed in this paper. In our model, a firm with private information, high or low marginal cost, is introduced. Interestingly, our theoretical analysis reveals that: (1) In a dynamic duopoly Bertrand game with perfect rationality and asymmetric information, the equilibrium price is positively correlated with product substitution rate and the probability of a high marginal cost, while it is negatively correlated with the cluster spillover. (2) In a dynamic duopoly Bertrand game with asymmetric information and adaptive expectation adopted by both firms, the Nash equilibrium prices are always asymptotically stable. (3) In a dynamic duopoly Bertrand game with heterogenous expectation and asymmetric information, where two firms use adaptive expectation and boundedly rational expectation respectively, the Nash equilibrium prices are locally stable under certain conditions. Furthermore, results indicate that, high product substitution rate or large probability of high marginal cost for firm 2 with private information may make the market price unstable, bifurcating or even falling into chaos, while high technology spillover is conducive to stabilize the market by contrast. It is also shown that the chaos can be controlled by a hybrid control strategy with the state variables feedback and parameter variation. Our research has an important theoretical and practical significance to the price competition in oligopoly markets.

Research on Pricing Decision of Direct Sales Green Supply Chain Based on Consumers’ Valuation

Consumer’s valuation of merchandise is an important factor affecting consumer buying behavior. When the consumer’s valuation exceeds the price of product, the consumer generally makes a decision to purchase the product; conversely, when the consumer’s estimate is lower than the price of product, the consumer will usually refuse to buy the product. From the perspective of consumer product valuation, this study assumed that the consumer’s product valuation obeys a uniform distribution, and a novel consumer demand function was proposed. On this basis, we studied enterprises’ pricing decisions in the supply chain of green agricultural products and obtained the equilibrium prices and optimal profits of the enterprises in several different scenarios, including Vertical Nash game model (VNM), firm A Stackelberg game model (FASM), firm B Stackelberg game model (FBSM), and cooperative game model (CM). In addition, the influence of parameters, such as green level, green preference payment coefficient, and green cost on the optimal profit, was discussed based on game theory and numerical simulation analysis. It was found that equilibrium prices always existed in several different scenarios, and when consumer’s green preference payment coefficient was large enough, the optimal profit of firm B was greater than the optimal profit of firm A. Furthermore, in CM, the sum of optimal profit of firm A and optimal profit of firm B is maximum for four scenarios. Finally, in the three competitive scenarios, green level, green preference payment coefficient, and green cost, have a positive or negative effect on the optimal profits of firm A or firm B. The research conclusions of this study provided theoretical support for the decision-making of enterprises and related management departments.

Integration of Complementary Multiproduct Firms

Two firms offer product series from which multiple complementary pairs are formed. The firms engage in a price- or quantity-choosing game in the market. It is found that the integration of the two firms may not necessarily lower the equilibrium prices because it precludes “indirect competition” in the market. Therefore, the integration, which may appear as a vertical integration, could be an antitrust concern even in the absence of exclusionary purpose.

Constrained Pseudo-Market Equilibrium

We propose a pseudo-market solution to resource allocation problems subject to constraints. Our treatment of constraints is general: including bihierarchical constraints due to considerations of diversity in school choice, or scheduling in course allocation; and other forms of constraints needed to model, for example, the market for roommates, combinatorial assignment problems, and knapsack constraints. Constraints give rise to pecuniary externalities, which are internalized via prices. Agents pay to the extent that their purchases affect the value the of relevant constraints at equilibrium prices. The result is a constrained-efficient market-equilibrium outcome. The outcome is fair to the extent that constraints treat agents symmetrically. (JEL D47, D61, D63, I11, I21)

Brand premia driven by perceived vertical differentiation in markets with information disparity and optimistic consumers

We have considered a duopoly with perceived vertical differentiation, information disparity and optimistic consumers. When firms compete for informed and uninformed consumers, the former contribute to raise product quality, while equilibrium prices increase with optimistic misperception of the latter, in our first equilibrium. Brand premium includes a quality premium and a misperception rent. In our second equilibrium, informed consumers buy low-quality goods and minimum product differentiation without Bertrand competition occurs. The brand premium is just a misperception rent, however, an increase of the informed consumers share implies price re-balancing and rent reduction. Consumers externalities arise in both equilibria. Firms compete only for informed consumers within our third and fourth equilibrium, as uninformed ones are passive and represent a captive market. Uninformed consumers in one case are overoptimistic, they buy the high quality good and can be cheated in equilibrium. Uninformed consumers approach the real quality differential in the fourth equilibrium, and the model reduces to standard vertical differentiation with perfect information.

The Transformation Problem of Values into Prices of Production: Marx Errors or an Inattentive Reading of 'Capital'?

The author presents the results of the discovery in Marx's works of the disparate elements of the theory of the original transformation of value into prices and the establishment of the general rate of profit. These results show:(a) Marx's tables in Chapter 9 of Volume III of Capital do not represent the usual interrelated branches of the economy, but particular spheres of production, exempt from the double-counting of profits and wages, and producing only final commodities. The total value of these commodities is equal to the net social product.(b) Marx carried out the original transformation of values into prices under the condition that wages remain unchanged. As a result, the first (chief) macroeconomic equality is fulfilled—the sum of the production prices for all net social products must be equal to the sum of its values. Also is fulfilled the second macroeconomic equality—the sum of profits of all sectors forming separate spheres of production must be equal to the sum of surplus values.(c) Marx assumed that the original transformation takes place in two stages: in the first stage, average rates of profit are formed in separate spheres of production, comprising two sectors of production: A and B. Sector A produced of constant capital for the sphere's own need. Sector B releases the final product for an exchange with other particular spheres. In the second stage, is established the general rate of profit in sectors B. A property of the original conversion is some change in the level of real wages, especially noticeable when using numerical models with a few spheres of commodity production. Therefore, Marx introduces the hypothesis of mutual compensation of positive and negative deviations of prices from the values of commodities. The hypothesis is fully confirmed under the conditions of the law of large numbers.(d) Marx also explains that non-equilibrium original prices of production, in which demand and supply of final goods do not coincide, can be transformed into equilibrium prices of production. For this to happen, corresponding changes in monetary wages, prices of constant capital, and the general rate of profit are necessary. However, the attainment of equilibrium prices was regarded by Marx as a secondary issue. At equilibrium prices, only the first (chief) macroeconomic equality is fulfilled.The author in developing alternative methods of transforming value into original and equilibrium prices of production uses all of the above elements of the theory of transformation of values into production prices. First, he restores the double counting of profits and wages in Marx's table. Second, he applies an iterative procedure of sequentially establishing the average and general rate of profit in the sectors and spheres of commodity production.The paper proposes new iterative calculation algorithms in the Excel program for the original and equilibrium transformation of values into production prices. The author tested the algorithms using the Wolfram Mathematica software. He also developed a method for converting the equilibrium production prices of goods back to their initial absolute values. This method refutes the well-known “eraser algorithm” by P. Samuelson. Ultimately, the author argues that Marx does not have the errors of transformation that his critics have attributed to him for so long.

The Transformation Problem of Values into Prices of Production: Marx Errors or an Inattentive Reading of 'Capital'?

The author presents the results of the discovery in Marx's works of the disparate elements of the theory of the original transformation of value into prices and the establishment of the general rate of profit. These results show:(a) Marx's tables in Chapter 9 of Volume III of Capital do not represent the usual interrelated branches of the economy, but particular spheres of production, exempt from the double-counting of profits and wages, and producing only final commodities. The total value of these commodities is equal to the net social product.(b) Marx carried out the original transformation of values into prices under the condition that wages remain unchanged. As a result, the first (chief) macroeconomic equality is fulfilled—the sum of the production prices for all net social products must be equal to the sum of its values. Also is fulfilled the second macroeconomic equality—the sum of profits of all sectors forming separate spheres of production must be equal to the sum of surplus values.(c) Marx assumed that the original transformation takes place in two stages: in the first stage, average rates of profit are formed in separate spheres of production, comprising two sectors of production: A and B. Sector A produced of constant capital for the sphere's own need. Sector B releases the final product for an exchange with other particular spheres. In the second stage, is established the general rate of profit in sectors B. A property of the original conversion is some change in the level of real wages, especially noticeable when using numerical models with a few spheres of commodity production. Therefore, Marx introduces the hypothesis of mutual compensation of positive and negative deviations of prices from the values of commodities. The hypothesis is fully confirmed under the conditions of the law of large numbers.(d) Marx also explains that non-equilibrium original prices of production, in which demand and supply of final goods do not coincide, can be transformed into equilibrium prices of production. For this to happen, corresponding changes in monetary wages, prices of constant capital, and the general rate of profit are necessary. However, the attainment of equilibrium prices was regarded by Marx as a secondary issue. At equilibrium prices, only the first (chief) macroeconomic equality is fulfilled.The author in developing alternative methods of transforming value into original and equilibrium prices of production uses all of the above elements of the theory of transformation of values into production prices. First, he restores the double counting of profits and wages in Marx's table. Second, he applies an iterative procedure of sequentially establishing the average and general rate of profit in the sectors and spheres of commodity production.The paper proposes new iterative calculation algorithms in the Excel program for the original and equilibrium transformation of values into production prices. The author tested the algorithms using the Wolfram Mathematica software. He also developed a method for converting the equilibrium production prices of goods back to their initial absolute values. This method refutes the well-known “eraser algorithm” by P. Samuelson. Ultimately, the author argues that Marx does not have the errors of transformation that his critics have attributed to him for so long.

Even in Simple Economic Systems, Equilibrium Can Be Non-Unique: An Example

According to economics, prices are determined by the relationship between supply and demand: they correspond to the equilibrium in which supply is exactly equal to the demand, i.e., for which the optimal amount that the seller is willing to sell at this price is exactly equal to the optimal amount that the buyer is willing to pay. In many situations, the corresponding prices are indeed uniquely determines by the supply-demand relation. From the purely mathematical viewpoint, there are situations when the equilibrium is not unique, but most economists believe that in practical situations, equilibrium prices are usually uniquely determined. In this paper, we provide a simple but realistic example in which the equilibrium is not unique.

Seeding the Herd: Pricing and Welfare Effects of Social Learning Manipulation

This paper is motivated by the recent emergence of various interference tactics employed by sellers attempting to manipulate social learning. We revisit the classic model of observational social learning and extend it to allow for (i) asymmetric information on product value between the seller and the consumers and (ii) the ability of the seller to “seed” the observational learning process with a fake purchase, in an attempt to manipulate consumer beliefs. We examine the interaction between social learning manipulation and equilibrium market outcomes as well as the impact of antimanipulation measures aimed at detecting and punishing misconduct. The analysis yields three main insights. First, we show that increasing the intensity of antimanipulation measures can have unintended consequences, often inducing higher levels of manipulation as well as higher equilibrium prices. Second, we find that although measures of high intensity can completely deter misconduct, such measures do not lead to any improvement in either seller or consumer payoffs, relative to the case where no measures are present. Third, we demonstrate that in many cases, measures of intermediate intensity can leverage seller manipulation to simultaneously improve both seller and consumer payoffs. This paper was accepted by Jayashankar Swaminathan, operations management.