Consistent Conjectural Variations Research Articles

Bilevel Optimization and Conjectural Equilibrium: Theoretical Results and Numerical Algorithms (An Invited Tutorial Paper)

We study a mixed oligopoly model, including a nonprivate firm that maximizes the convex combination of domestic social surplus and net profit, under the concepts of consistent conjectural variations (CCVE), Cournot–Nash, and perfect competition equilibria and analyze the behavior of these equilibria. We compare the results to develop an optimality criterion for a parameter, decided by the nonprivate firm, known as socialization level. To work with the CCVE, we need to define the concept of consistency; hence, we also consider a classic oligopoly, which only includes private firms that maximize net profit, and reformulated it as a non-cooperative game, named meta-game, where the players (the private firms) select a conjecture as their strategies; then, we prove that the Nash equilibrium for the meta-game implies the CCVE. After that, due to its similarities with the meta-game, we study the Tolls Optimization Problem (TOP). Taking advantage of the sensitivity analysis for convex quadratic optimization and the filled function method; then, we present an efficient algorithm to solve the TOP. Finally, we apply these results to a financial model in which a group of sectors buy, sell, or exchange a set of instruments; each sector faces some uncertainty when selecting its assets and liabilities to optimize its portfolio structure. The problem is modeled using the concepts of conjectural variations and meta-game and we present results concerning the CCVE.

Consistent Conjectural Variations Equilibria: Characterization & Stability for a Class of Continuous Games

Consistent Conjectural Variations Equilibria: Characterization & Stability for a Class of Continuous Games

Consistent Conjectural Variations Equilibrium for a Financial Model

Consistent Conjectural Variations Equilibrium for a Financial Model

Consistent Conjectural Variations Equilibrium in a Mixed Oligopoly Model With a Labor-Managed Company and a Discontinuous Demand Function

In this work, we examine a mixed oligopoly model within the framework of consistent conjectural variations, in which one of the producers is a labor-managed company competing with other private firms, and consumer demand is given by a discontinuous function. Private firms attempt to maximize their net profit while the labor-managed company attempts to maximize the convex combination of its net profit with an income-per-worker function, which describes the mixed nature of the model. Each producer conjectures the dependence of the market-clearing price on its own production volume and then selects its most suitable conjecture based on a verification procedure. We introduce the notions of exterior and interior equilibrium and prove the existence and uniqueness of the conjectural variation equilibrium, as well as the existence of the particular equilibrium states known as consistent. Finally, we analyze the behavior of the market’s consistent equilibrium state in response to changes in consumer demand.

Consistent Conjectural Variations Equilibrium in the Semi-Mixed Oligopoly

We study a variant of the mixed oligopoly model with conjectural variations equilibrium, in which one of the producers maximizes not his net profit but the convex combination of the latter with the domestic social surplus. The coefficient of this convex combination is named socialization level. The producers’ conjectures concern the price variations depending upon their production output variations. In this work, we extend the models studied before, considering the case of the producers’ cost functions being convex but not necessarily quadratic. The notion of exterior and interior equilibrium is introduced (similarly to previous works), developing a consistency criterion for the conjectures. Existence and uniqueness theorems are formulated and proven. Results concerning the comparison between conjectural variations, perfect competition, and Cournot equilibriums are provided. Based on these results, we formulate an optimality criterion for the election of the socialization level. The existence of the optimal socialization level is proven under the condition that the public company cannot be too weak as compared to the private firms.

Special Issue on Variational Inequalities: Consistent Conjectural Variations Coincide with the Nash Solution in the Meta-Model

We keep investigating the properties of consistent conjectural variations equilibrium (CCVE) developed for a single-commodity oligopoly. Although, in general, the consistent conjectures are distinct from those of Cournot-Nash, in our previous papers, we established the following remarkable fact. Define a meta-model as such where the players are the same agents as in the original oligopoly but now using the conjectures as their strategies. Then the Cournot-Nash equilibrium in the meta-model generated the consistent conjectural variations equilibrium in the original oligopoly. In this paper, we study the conditions under which the inverse is also true, that it, every consistent CVE provides for the Cournot-Nash optimal strategies for the meta-model. This equivalence allows one to extend the concept of CCVE to other kinds of economic and financial models lacking the oligopoly structure.

Consistent Conjectural Variations Equilibrium in a Semi-Mixed Duopoly

This paper considers conjectural variations equilibrium (CVE) in the one item market with a mixed duopoly of competitors. The duopoly is calledsemi-mixedbecause one (semi-public) company’s objective is to maximize a convex combination of her net profit and domestic social surplus (DSS). The two agents make conjectures about fluctuations of the equilibrium price occurring after their supplies having been varied. Based on the concepts of theexteriorandinterior equilibrium, as well as the existence theorem for the interior equilibrium (a.k.a. the consistent CVE, or the exterior equilibrium withconsistent conjectures) demonstrated in the authors’ previous papers, we analyze the behavior of the interior equilibrium as a function of the semi-public firm’s level of socialization. When this parameter reflected by the convex combination coefficient tends to 1, thus transforming the semi-public company into a completely public one, and the considered model into the classical mixed duopoly, two trends are apparent. First, for the private company, the equilibrium with consistent conjectures (CCVE) becomes more attractive (lucrative) than the Cournot-Nash equilibrium. Second, there exists a (unique in the case of an affine demand function) value of the convex combination coefficient such that the private agent’s profit is the same in both of the above-mentioned equilibrium types, thus making no subsidy to the producer or to the consumers necessary. Numerical experiments with various mixed duopoly models confirm the robustness of the proposed algorithm for finding the optimal value of the above-mentioned combination coefficient (a.k.a. the semi-public company’s socialization level).

Consistent conjectures are optimal Cournot-Nash strategies in the meta-game

In this paper, we investigate the properties of consistent conjectural variations equilibrium developed for a single-commodity oligopoly. Although, in general, the consistent conjectures are distinct from those of Cournot-Nash, we establish the following remarkable fact. Define a meta-game as such where the players are the same agents as in the original oligopoly but now using the conjectures as their strategies. Then the consistent conjectures of the original oligopoly game provide for the Cournot-Nash optimal strategies in the meta-game.

Finding Consistent Conjectural Variations Equilibrium in a Human Migration Model

Abstract In this paper, we develop a human migration model using the concept of conjectural variations equilibrium (CVE). In contrast to previous works we extend the model to the case where the conjectural variations coefficients may be not only constants, but also (continuously differentiable) functions of the total population at the destination and of the group's fraction in it. Moreover, we allow these functions to take distinct values at the abandoned location and at the destination. As an experimental verification of the proposed model, we develop a specific form of the model based upon relevant population data of a three-city agglomeration at the boundary of two Mexican states: Durango (Dgo.) and Coahuila (Coah.) Namely, we consider the 1980-2015 dynamics of population growth in the three cities: Torreon (Coah.), Gomez Palacio (Dgo.) and Lerdo (Dgo.), and propose utility functions of four various kinds for each of the three cities. After having collected necessary information about the average movement and transportation costs for each pair of the cities, we apply the above-mentioned human migration model to this example. Numerical experiments have been conducted revealing interesting results concerning the consistency of probable conjectural variations equilibrium states.

Extending Cournot: When Does Insight Dissipate?

The Cournot model makes assumptions about the knowledge of the players, which can lead to counterintuitive results when extending the basic model. Closed-loop equilibria and models with consistent conjectural variations are problematic in that they reduce or eliminate economic rents while the players have more knowledge than under standard Cournot assumptions. The loss of profit arises because the information is about reactions and not about equilibria, the players do not deduce or carry over relevant information from one stage of the game to the next, and they operate in a reactive decision space rather than reframing the game proactively. We show how the equilibrium changes and profits improve as players enlarge their strategies and point out the problems associated with each improvement in their strategies. However, the equilibrium is dependent on the starting point. Critical to modeling oligopolies is identifying the knowledge levels and the decision spaces of the players. Our conclusion is that, although the basic Cournot model provides insights into oligopolies, extensions that rely on the Cournot assumption that the other player does not react are probably mathematical artifacts that do not contribute to understanding how markets operate. Oligopoly theory should focus more on alternative descriptions of players' knowledge and their strategic alternatives, especially on how the players frame the behavior of others in their industries, rather than rely on a few standard equilibria. This makes the case for an agent-based approach.

Structure of demand and consistent conjectural variations equilibrium (CCVE) in a mixed oligopoly model

We study conjectured variations equilibrium (CVE) in a model of mixed oligopoly with not necessarily continuous demand functions. The agents’ conjectures concern the price variations depending upon their production output increase or decrease. We establish the existence and uniqueness results for the CVE (called exterior equilibrium) for any set of feasible conjectures. To introduce the notion of interior equilibrium, we develop a consistency criterion for the conjectures (referred to as influence coefficients) and prove the existence theorem for the interior equilibrium (understood as CVE with consistent conjectures, or CCVE). In addition, we also examine the behavior of the consistent conjectures as functions of a parameter representing the demand’s derivative with respect to the market price. The latter results allow one to predict the behavior of groups of consumers with different consumption abilities. The proposed techniques permit one to develop a qualitative description of the dependence of the market price on the active demand component, too. It should be noticed that due to the non-smoothness of the demand function, there is possibly a path dependency and indeterminacy of equilibria in certain cases. This is, on the one hand, a theoretically inconvenient result (multiple equilibria), but on the other hand, it may happen to be extremely useful for applications. Indeed, the latter multiplicity of equilibria might serve as a rationale for the regulatory intervention to induce a change of equilibrium whenever the total welfare could be improved (cf., for example, the Keynesian stimulus).

Consistent Conjectural Variations Equilibrium In A Bilevel Human Migration Model

In this paper, we develop a bi-level human migration model using the concept of conjectural variations equilibrium (CVE). In contrast to previous works, here we construct a bi-level programming model. The upper level agents are municipalities of competing locations, whose strategies are investments into infrastructures of the locations (cities, towns, etc.). These investments aim at making the locations more attractive for both the residents and potential migrants from other locations. At the lower level of the model, the present residents (grouped into professional communities) are also considered as potential migrants to other locations. They make their decision where to migrate (if at all) comparing the expected values of the utility functions of the destinations and original locations, which are estimated by taking into account their group's conjectures concerning equilibrium migration flows between the involved locations. Applying a special technique to verify the consistency of the conjectures (influence coefficients), the existence and uniqueness results for the consistent conjectural variations equilibrium (CCVE) are established.

Consistent Conjectural Variations Equilibrium in a Mixed Duopoly

In this paper, we consider a model of mixed duopoly with Conjectured Variations Equilibrium (CVE). The agents’ conjectures concern the price variations depending on the increase or decrease in their production outputs. We establish existence and uniqueness results for the conjectured variations equilibrium (called an exterior equilibrium) for any set of feasible conjectures. To introduce the notion of an interior equilibrium, we develop a consistency criterion for the conjectures (referred to as influence coefficients) and prove the existence theorem for the interior equilibrium (understood as a CVE with consistent conjectures). To prepare the base for the extension of our results to the case of non-differentiable demand functions, we also investigate the behavior of the consistent conjectures in dependence upon a parameter representing the demand function derivative with respect to the market price.

Duopsony models with consistent conjectural variations

In this note we develop a consistent conjectural variation model that generalizes Bresnahan's (1981) results to a duopoly-duopsony setting. This is the first duopsony model in which firms are constrained to have consistent conjectural variations, and two interesting results emerge. First, Bertrand conjectures occur when inputs are homogeneous and when input demand functions are horizontal. Second, the consistent conjectures equilibrium approaches Cournot when input demand functions are vertical or when the firm is a monopolist-monopsonist. This implies that firms will behave more competitively in a duopsony setting as inputs and outputs become more homogeneous and as input demand functions become relatively more elastic.

Consistent conjectural variations in a two-dimensional spatial market

In this paper we calculate the consistent conjectural price variation on the basis of spatial competition models for alternative market shapes. It can be shown that these variation coefficients depend on the geographic size as well as on the shape of the market area. The use of consistent conjectural variation coefficients reveals two important results. On the one hand, we can show that Hotelling/Smithies competition allows a sufficient approximation to the consistent results, and on the other hand we demonstrate that hexagonal market areas create the highest welfare effects under consistent solutions.

Consistency of conjectures and reactions?

This note discusses consistency in conjectural variations models for oligopoly theory. It is argued that consistency of conjectures and reactions is not an appropriate equilibrium concept, since reaction functions do not have a role in conjectural variations models. Furthermore, the reaction functions used in the concept of a consistent conjectural variations equilibrium are based on unobservable information sets. A weaker type of consistency, that is, consistency of conjectures and actions, is proposed as an alternative. This consistent conjectures equilibrium has for each set of conjectures one of a variety of competitive and collusive outcomes as a solution.

Optimal tariffs in consistent conjectural variations equilibrium

This paper analyzes the determination of the optimal tariff under the assumption of consistent conjectural variations (CCV). A general characterization of the CCV equilibrium is given. We show that (i) there are, in general, a multiplicity of each equilibria, and (ii) under certain restrictions, the Cournot equilibrium, which is based on the assumption of no retaliation, can also be a CCV equilibrium. By contrast, free trade is never a CCV equilibrium. Finally, the CCV equilibrium is solved explicitly in a simple example.

Oligopoly and Consistent Conjectural Variations

This article examines conjectural variations in an oligopoly model with a homogeneous product. A conjectural variation is consistent if it is equivalent to the optimal response of the other firms at the equilibrium defined by that conjecture. When the number of firms is fixed, we find that competitive behavior is consistent when marginal costs are constant, but that when marginal costs are rising, the consistent conjectural variation will be between competitive and Cournot behavior. Finally, if we allow free entry and redefine consistency to account for such, then only competitive behavior will be consistent.