Wage bargaining as an optimal control problem: a dynamic version of the efficient bargaining model

Decisions in Economics and Finance - In this paper, I develop a dynamic version of the efficient bargaining model grounded on optimal control in which a firm and a union bargain over the wage in a...     

Stable and efficient bargaining networks

Abstract. We examine a bargaining game among players connected by a network, where successively potential partners negotiate over terms of a project. For any network structure and ordering of selection of bargaining partners, there is a unique subgame perfect equilibrium outcome which coincides with the standard Rubinstein split. Based on this, we derive expected payoffs for players as a function of the network. We the characterize the pairwise stable and efficient bargaining networks. These two sets do not always coincide and in many situations pairwise stable networks are over-connected. Received: 10 April 2001 / Accepted: 17 January 2002 I am indebted to Matthew Jackson for suggesting this paper. I also thank Bernard Caillaud, Sanjeev Goyal, and the seminar participants at Universidad de Alicante, Universidad Carlos III de Madrid, CEMFI (Madrid), Universitat Autònoma de Barcelona, Erasmus University of Rotterdam, Universitat Pompeu Fabra (Barcelona) and CERAS (Paris) for helpful comments. This paper was written while at Universitat Pompeu Fabra and CERAS. Financial support from the Spanish Ministry of Education through research grant DGESIC PB96-0302, and from the Ecole Nationale des Ponts et Chaussées, Paris is gratefully acknowledged. All errors are of course mine.     

Wage bargaining,productivity growth and long-run industry structure

This paper studies the innovation dynamics of an oligopolistic industry. The firms compete not only in the output market but also by engaging in productivity enhancing innovations to reduce labor costs. Rent sharing may generate productivity dependent wage differentials. Productivity growth creates intertemporal spillover effects, which affect the incentives for innovation at subsequent dates. Over time the industry equilibrium approaches a steady state. The paper characterizes the evolution of the industry's innovation behavior and its market structure on the adjustment path.     

Wage bargaining,union membership,and the organization of unemployment insurance

The paper offers a theoretical analysis of a labor market institution known as the Gent system, which is a system where unions run unemployment insurance (UI) through government-subsidized UI funds. This system is practiced in four Nordic countries with comparatively very high unionization rates. The analysis shows that the Gent system is more conducive to unionization than a compulsory UI system if the Gent system is heavily subsidized by the government or if workers are strongly risk averse. Moreover, a rise in the share of benefits financed by union members is likely to reduce wages as well as union membership.     

Labor force participation and pair-wise efficient contracts with search and bargaining

A “constant” wage is pair-wise inefficient in a standard search model when workers endogenously separate from employment. We derive a pair-wise efficient employment contract that involves workers paying a hiring fee (or bond) upon the formation of a match. We estimate the constant wage and pair-wise efficient contract assuming the hiring fee is unobservable, and find evidence to reject the pair-wise efficient contract in favor of the constant wage rule. A counterfactual experiment reveals the current level of labor force participation to be 9.6% below the efficient level, and a structural shift to the pair-wise efficient contract improves welfare by roughly 3.5%.     

External bargaining versus internal manipulation: A theory of managerial delegation

Does the competition mode influence the delegation decisions of the firm owners? By constructing a vertical negotiation game model, we find that under Cournot competition in the downstream market, the downstream firm's owner will not choose delegation, whereas under Bertrand competition, the downstream firm's owner will choose delegation. If the product substitution is relatively large, the adoption of delegation management by the owners of downstream firms under Bertrand competition will bring higher profits. It further shows that compared with the situation of no delegation, delegation management may reverse the social welfare ranking under Bertrand and Cournot competitions.     

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.     

Credible collusion in multimarket oligopoly

This article refines an established explanation of how multimarket contact facilitates collusion when firms enjoy reciprocal advantages across markets: When there are reciprocal asymmetries between firms, multimarket contact allows them not only to develop spheres of influence, but also to implement attractively simple strategies that are subgame perfect and weakly renegotiation proof. Hence, collusive equilibria are supported by fully credible punishments. A significant implication is, multimarket contact involving reciprocal differences between firms may be more facilitating to their cooperative efforts than multimarket contact based on other factors. The article discusses existing empirical work as it relates to this implication. Copyright © 2007 John Wiley & Sons, Ltd.     

Incentive pay and rent in oligopoly

Our purpose is to examine strategic delegation in nonlinear Cournot oligopoly. The findings generalize earlier results and show that managerial contracts reward sales under the condition of a fixed input price. Alternatively, under a variable input price, owners might punish sales even when goods are strategic substitutes. We conclude that optimal strategic motivation depends critically on the input price. For example, motivation that supports positive owner profit under a fixed input price nullifies owner-profit if an upstream monopolist with convex costs sets the input price. In a vertical relationship between a duopoly and an upstream monopolist, strategic delegation punishes sales.     

Implementation of social optimum in oligopoly

This paper presents a decentralized mechanism implementing socially optimal output choices by non-cooperatively acting oligopolists. A decentralized mechanism is a vector of balanced transfers among firms determined as a function of firms’ output choices. The mechanism is devised by a regulator with a full knowledge of demand and without any knowledge of the firms’ cost functions. Restricting the set of admissible demand and cost functions such that the firms always have an incentive to produce, it turns out that the socially optimal solution is implementable insofar as the demand function is a polynomial of at most (n−1)th degree, n being the number of firms in the industry. The author is indebted to a referee and an associate editor for their helpful comments.     

Imitators and optimizers in Cournot oligopoly

We analyze a symmetric n-firm Cournot oligopoly with a heterogeneous population of optimizers and imitators. Imitators mimic the output decision of the most successful firms of the previous round à la Vega-Redondo, F., [1997. The evolution of Walrasian behavior. Econometrica 65, 375–384]. Optimizers play a myopic best response to the opponents’ previous output. Firms make mistakes and deviate from their decision rules with a small probability. Applying stochastic stability analysis, we find that the long run distribution converges to a recurrent set of states in which imitators are better off than are optimizers.     

Compensation Schemes and Labor Market Competition: Piece Rate versus Wage Rate

We investigate the choice of compensation scheme by firms. Our basic model shows that the unique equilibrium choice for profit maximizing duopsonists in a labor market is for one firm to offer a wage rate and for the other to offer a piece rate. This result arises because the firms recognize that, by offering different compensation schemes, they induce self-selection among workers, which thereby decreases the intensity of competition in the labor market. We find this asymmetry to be robust to allowing for firing, free entry, and a class of more general compensation schemes. When we broaden our model to permit firms to be differentiated in the eyes of workers (either geographically or by "other working conditions," e.g.), we find that our results are preserved when differentiation is low, but that both firms choose to offer a piece rate when differentiation is high.     

Multi-agent bilateral bargaining and the Nash bargaining solution

This paper studies a bargaining model where n   players negotiate how to share a pie through (n−1)$(n-1)$ bilateral bargaining sessions. In each session, two players bargain for a partial agreement that specifies who exits and who moves on to the next session (if there is any) via the alternating-proposal framework of Rubinstein [Rubinstein, A., 1982. Perfect equilibrium in a bargaining model. Econometrica 50, 97–109]. We consider two bargaining procedures under which the subgame perfect equilibrium outcomes converge to the Nash [Nash, J., 1950. The bargaining problem. Econometrica 18, 155–162] bargaining solution for the corresponding bargaining problem as the players’ discount factor goes to one. Hence, the model studied here provides a non-cooperative foundation for the Nash cooperative bargaining solution in the multilateral case.     

Plant-level bargaining in France

Institutionalised bargaining and formal collective agreements at plant level are a rarity in France. However, research shows that some form of unofficial bargaining does take place. Edward Slack illustrates this phenomenon using evidence obtained from a chemical plant and an industrialised dairy in 1974.     

Technology choice in an evolutionary oligopoly game

In this paper, we propose and analyze a two-stage oligopoly game in which firms first simultaneously choose production technologies and in the second stage simultaneously choose production quantities. After characterizing the Nash equilibrium of the game, we cast our static model in a dynamic setting exploring the stability properties of the market equilibrium in two different cases: (i) exogenously distributed technologies and Cournot adjustments and (ii) endogenously distributed technologies in an infinite population game with Cournot–Nash equilibrium outputs. The main aim of the paper is that of extending the results about Cournot oligopoly stability in an evolutionary setting of heterogeneous decreasing returns-to-scale technologies. We show how the interplay between production decisions and R&D decisions can generate endogenous market fluctuations leading to complex dynamic phenomena.     

-learning agents in a Cournot oligopoly model

Q-learning is a reinforcement learning model from the field of artificial intelligence. We study the use of Q-learning for modeling the learning behavior of firms in repeated Cournot oligopoly games. Based on computer simulations, we show that Q-learning firms generally learn to collude with each other, although full collusion usually does not emerge. We also present some analytical results. These results provide insight into the underlying mechanism that causes collusive behavior to emerge. Q-learning is one of the few learning models available that can explain the emergence of collusive behavior in settings in which there is no punishment mechanism and no possibility for explicit communication between firms.     

Nash-equilibria in a heterogeneous oligopoly with fuzzy information

We consider a model of an oligopolistic market with heterogeneous firms and products where neither the cost nor the demand functions are common knowledge. Instead, each firm only has some vague ideas about the price strategies adopted by its competitors which is modelled by a fuzzy set. In analogy to the notion of an "equilibrium of actions and beliefs" we define and characterize a generalized Nash-equilibrium and show its existence under general conditions. Furthermore, the impact of the fuzzy information on the equilibrium outcome is analyzed by means of a comparative static analysis within a particular model framework.Received: 28 May 2002, Accepted: 25 December 2002, JEL Classification: D43, D80, L13We wish to thank Bernhard F. Arnold, the editor Murat Sertel , and two anonymous referees for helpful comments. Of course, we are responsible for all remaining errors.     

Nash bargaining in ordinal environments

We analyze the implications of Nash’s (Econometrica 18:155–162, 1950) axioms in ordinal bargaining environments; there, the scale invariance axiom needs to be strenghtened to take into account all order-preserving transformations of the agents’ utilities. This axiom, called ordinal invariance, is a very demanding one. For two-agents, it is violated by every strongly individually rational bargaining rule. In general, no ordinally invariant bargaining rule satisfies the other three axioms of Nash. Parallel to Roth (J Econ Theory 16:247–251, 1977), we introduce a weaker independence of irrelevant alternatives (IIA) axiom that we argue is better suited for ordinally invariant bargaining rules. We show that the three-agent Shapley–Shubik bargaining rule uniquely satisfies ordinal invariance, Pareto optimality, symmetry, and this weaker IIA axiom. We also analyze the implications of other independence axioms.