A differential game in a duopoly with instantaneous incentives

In this paper, we study a differential game in which two competing firms exploit a public renewable resource that is relevant from a landscape point of view. We consider a policy maker that provides an instantaneous incentive to the firms in order to prevent the resource exhaustion during the whole extraction period, which coincides with the harvesting license period. We compute an open-loop Nash equilibrium of the differential game, showing that it coincides with a linear feedback Nash equilibrium. Finally, we compute the value of the incentive that leads to the maximization of social welfare considering the incentive both as a pure transfer and as a cost.     

On nonrenewable resource oligopolies: The asymmetric case

We give a full characterization of the open-loop Nash equilibrium of a nonrenewable resource game between two types of firms differing in extraction costs. We show that (i) there almost always exists a phase where both types of firms supply simultaneously, (ii) when the high cost mines are exploited by a number of firms that goes to infinity the equilibrium approaches the cartel-versus-fringe equilibrium with the fringe firms acting as price takers, and (iii) the cheaper resource may not be exhausted first, a violation of the Herfindahl rule, that may be detrimental to social welfare.     

Convergence to Walrasian equilibrium with minimal information

We consider convergence to Walrasian equilibrium in a situation where firms know only market price and their own cost function. We term this a situation of minimal information. We model the problem as a large population game of Cournot competition. The Nash equilibrium of this model is identical to the Walrasian equilibrium. We apply the best response (BR) dynamic as our main evolutionary model. This dynamic can be applied under minimal information as firms need to know only the market price and the their own cost to compute payoffs. We show that the BR dynamic converges globally to Nash equilibrium in an aggregative game like the Cournot model. Hence, it converges globally to the Walrasian equilibrium under minimal information. We extend the result to some other evolutionary dynamics using the method of potential games.

     

Endogenous mechanisms and Nash equilibrium in competitive contracting games

We model strategic competition in a market with asymmetric information as a noncooperative game in which each firm competes for the business of a buyer of unknown type by offering the buyer a catalog of products and prices. The timing in our model is Stackelberg: in the first stage, given the distribution of buyer types known to all firms and the deducible, type-dependent best responses of the agent, firms simultaneously and noncooperatively choose their catalog offers. In the second stage the buyer, knowing his type, chooses a single firm and product-price pair from that firm’s catalog. By backward induction, this Stackelberg game with asymmetric information reduces to a game over catalogs with payoff indeterminacies. In particular, due to ties within catalogs and/or across catalogs, corresponding to any catalog profile offered by firms there may be multiple possible expected firm payoffs, all consistent with the rational optimizing behavior of the agent for each of his types. The resolution of these indeterminacies depends on the tie-breaking mechanism which emerges in the market. Because each tie-breaking mechanism induces a particular game over catalogs, a reasonable candidate would be a tie-breaking mechanism which supports a Nash equilibrium in the corresponding catalog game. We call such a mechanism an endogenous Nash mechanism. The fundamental question we address in this paper is, does there exist an endogenous Nash mechanism—and therefore, does there exist a Nash equilibrium for the catalog game? We show under fairly mild conditions on primitives that catalog games naturally possess tie-breaking mechanisms which support Nash equilibria.     

Imitation and price competition in a differentiated market

We study the stable market outcome that evolves in a spatially differentiated market when price-competing firms choose actions by imitation of the most profitable firm. We compare and contrast the stable outcomes under two imitation procedures: one, where each firm immediately imitates the most profitable firm, and the other when a firm imitates another firm only if it is more profitable while being “sufficiently similar” (in context of the market segment it operates in) or “sufficiently close”. In either case, the symmetric pure strategy Nash equilibrium is always a stable outcome. However, when imitation of the most profitable firm is immediate and market differentiation is ‘moderate’, states with prices lower than the Nash equilibrium are also stable. In contrast, when imitation of the most profitable firm is more gradual and market differentiation is below a threshold, states with prices above the Nash equilibrium are also stable. Thus, while competitive evolutionary pressure in this imitation based model does result in the Nash equilibrium always being stable, other outcomes may be stable as well. Interestingly, the states that are stable under gradual imitation give the firms a higher profit than the stable states under immediate imitation.     

Location decisions and welfare under spatial price discrimination

The welfare effects of regulation are of crucial importance to policy makers. To this end, we present a model of n firms with differentiated costs competing in a linear market within the framework of spatial price discrimination. We prove that the Nash equilibrium locations of firms are always socially optimal irrespective of the number of competitors, the distribution of consumers, firms' cost heterogeneity, the level of privatization, and the number and/or the varieties of the produced goods. We also provide an algorithm on how to find the unique Nash equilibrium in the case of uniformly distributed consumers.     

Nonlinear pricing in spatial oligopoly

A model of duopoly competition in nonlinear pricing when firms are imperfectly informed about consumer locations is analyzed. A continuum of consumers purchase a variable amount of a product from one of two firms located at the endpoints of the market. At the Nash equilibrium in quantity-outlay schedules, consumers buy the same quantities as they would from the same firm if it were a monopolist facing the same informational asymmetries, but they receive greater surplus. Hence, no efficiency gains result from competition. If consumers have the option to reveal their locations and have the firms deliver the goods, all consumers choose to reveal their locations in equilibrium. Thus, the inefficiencies from information asymmetries may not arise because firms can deliver the good to consumers. In contrast, with a monopoly seller, consumers have no incentives to reveal their locations.     

Tax Competition with Heterogeneous Firms

Abstract

This paper studies tax competition in an economic geography model that allows for agglomeration economies with trade costs and heterogeneous firms. We find that the Nash equilibrium involves a large country charging a higher tax than a small nation. Lower trade costs lead to an intensification of competition, a drop in Nash tax rates and a narrowing of the gap. Since large, productive firms are naturally more sensitive to tax differences in our model, large firms are the crux of tax competition in our model. This also means that tax competition has consequences for the average productivity of big and small nations' industries; by lowering tax rates, a small nation can attract high-productivity firms.     

How to get firms to invest: A simple solution to the hold-up problem in regulation

Abstract. Many governmental programs are effective only if firms make costly investments. The inability of authorities to precommit to a regulatory scheme creates incentives for firms not to invest and to hold-up the regulator. This paper describes a simple subsidy/tax scheme embedded in a four-stage mechanism that solves the hold-up problem. We design a self-financing subsidy/tax scheme which benefits a complying firm at the expense of a non-complying firm. In order to be credible, the subsidy and tax rates must maximize social welfare for any combination of investment decisions. We show that there exists a unique subgame perfect equilibrium in which all firms invest and no actual implementation with subsidies and taxes is required. We discuss in which cases the mechanism can work under incomplete information. Received: 30 December 1998 / Accepted: 12 October 2001     

Dynamic R&D with spillovers: A comment

Cellini and Lambertini [2009. Dynamic R&D with spillovers: competition vs cooperation. J. Econ. Dyn. Control 33, 568–582] study a dynamic R&D game with spillovers. This comment demonstrates that, contrary to what is claimed in their paper, the game is not state redundant and the open-loop Nash equilibrium is not subgame perfect.     

A Note on Managerial Delegation with Asymmetric and Convex Costs

In this note, we investigate if the standard result by the managerial delegation literature, i.e., the sub‐game perfect Nash equilibrium is not Pareto‐optimal from the firms' viewpoint, still applies when asymmetric and convex costs are introduced into the analysis. In such a framework, the managerial delegation choice still represents a sub‐game Nash perfect equilibrium, but the more efficient firm may obtain higher profits provided that the degree of cost asymmetry between firms is sufficiently large. Copyright © 2015 John Wiley & Sons, Ltd.     

Bounds for best response functions in binary games

This paper studies the identification of best response functions in binary games without making strong parametric assumptions about the payoffs. The best response function gives the utility maximizing response to a decision of the other players. This is analogous to the response function in the treatment-response literature, taking the decision of the other players as the treatment, except that the best response function has additional structure implied by the associated utility maximization problem. Further, the relationship between the data and the best response function is not the same as the relationship between the data and the response function in the treatment-response literature. We focus especially on the case of a complete information entry game with two firms. We also discuss the case of an entry game with many firms, non-entry games, and incomplete information. Our analysis of the entry game is based on the observation of realized entry decisions, which we then link to the best response functions under various assumptions including those concerning the level of rationality of the firms, including the assumption of Nash equilibrium play, the symmetry of the payoffs between firms, and whether mixed strategies are admitted.     

Divisionalization and Delegation in Oligopoly

We consider a model where oligopolistic firms create independent divisions or franchises, which subsequently delegate output decisions to managers. We show that the number of firms required to make divisionalization privately profitable is greater in our model than in previous pure divisionalization models. However, in contrast with pure delegation models, we show that the subgame perfect Nash equilibrium approaches perfect competition as divisionalization costs tends to zero, even with a small fixed number of firms.     

Rational expectations equilibrium and the strategic choice of costly information

This paper studies costly information acquisition in one-good production economies when agents acquire private information and prices transmit information. Before asset markets open, agents choose the quality of their private information. After this information stage, agents trade assets in sequentially complete markets taking into account their private information and the information revealed by equilibrium prices (rational expectations equilibrium, (Radner, R., 1979. Rational expectations equilibrium: generic existence and the information revealed by prices, Econometrica 47, 655–678.)). An overall equilibrium in asset and information market is defined as a Nash equilibrium of the information game in which agents’ actions are information choices and their utility payoffs are the ex-ante expected utilities of the corresponding rationale expectations equilibrium. This paper shows that for a generic set of economies parameterized by endowments and productivity shocks, an overall equilibrium in information and asset market (a Nash equilibrium of the induced information game) with costly information acquisition and fully-revealing prices exists. In other words, informational efficiency is in general consistent with costly information acquisition.     

Feedback equilibria in a dynamic renewable resource oligopoly: Pre-emption,voracity and exhaustion

We revisit a recent literature on productive asset exploitation describing a differential oligopoly game of resource extraction under static, linear feedback and nonlinear feedback strategies, where we explicitly allow for the possibility of resource exhaustion. We show that (i) feedback rules entail resource exhaustion for a finite number of firms; and (ii) feedback strategies are more aggressive than static ones as long as the resource stock is large enough, in accordance with the acquired view based on the traditional pre-emption argument associated with feedback information.     

Workers’ enterprises are not perverse: differential oligopoly games with sticky price

We take a differential game approach to study the dynamic behaviour of labour managed (LM) firms, in the presence of price stickiness. We find that the oligopoly market populated by LM firms reaches the same steady state equilibrium allocation as the oligopoly populated by profit-maximising (PM) firms, provided that the LM membership and the PM labour force are set before the market game starts. The conclusion holds under both the open-loop solution and the closed-loop solution. The result confirms the point made by Sertel (Eur Econ Rev 31:1619–1625, 1987) in a static framework.     

Product Differentiation, Strategic Divisionalization, and Persistence of Monopoly

This paper analyzes the strategic incentive of oligopolists to create autonomous rival divisions when products are differentiated. We consider a two-stage game where firms choose the number of autonomous divisions in the first stage and all the divisions engage in Cournot competition in the second. It is shown that product differentiation ensures the existence of an interior subgame perfect Nash equilibrium (SPNE), and the equilibrium number of divisions increases with the degree of substitution among products and the number of firms. Further, if divisions are allowed to divide further, they always will, which leads to total rent dissipation. Thus, parent firms have incentives to unilaterally restrict their divisions from further dividing. In the free-entry equilibrium, it is found that the possibility of setting up autonomous divisions is a natural barrier to entry. Incumbents may persistently earn abnormally high profits. In the cases where product differentiation is difficult, the only pure-strategy free-entry SPNE is the monopoly outcome even if the entry cost is relatively low.     

Heterogeneous players in a Cournot model with differentiated products

In this article, we analyse a duopolistic Cournotian game with firms producing differentiated goods, marginal costs are constant and demand functions are microfounded. We consider firms adopting different decisional mechanisms which are based on a reduced degree of rationality. In particular, we assume that a firm adopts the local monopolistic approximation approach, while the rival adjusts its output level according to the gradient rule. We provide conditions for the stability of the Nash equilibrium and investigate some bifurcation scenarios as parameters vary. The main finding of the article is that both a high level and a low level in goods differentiation may have a destabilising role in the system.     

The relationship between mixed strategies and strategic groups

This paper identifies a theoretical basis for the existence of strategic groups. We represent the competitive marketplace by means of a game-theoretic model and motivate the existence of markets wherein the only Nash solution may be in mixed strategies. Given deterministic behavior on the part of the firms, we interpret this solution in terms of a strategic groups equilibrium. The major implication of this analysis is that strategic groups can exist even if there is no inherent asymmetry (in terms of skills) between firms.     

On pure-strategy Nash equilibria in price–quantity games

This paper examines the existence and characteristics of pure-strategy Nash equilibria in oligopoly models in which firms simultaneously set prices and quantities. Existence of a pure-strategy equilibrium is proved for a class of price–quantity games. If the demand function is continuous, then the equilibrium outcome is similar to that of a price-only model. With discontinuous demand and limited spillover, there are rationing equilibria in which combined production falls short of market demand. Moreover, there might again be an equilibrium reflecting the outcome of a price game. Competition in price and quantity thus yields Bertrand outcomes under a variety of market conditions.