Price vs quantity in a duopoly supergame with Nash punishments

We examine the endogenous choice between price and quantity behaviour in a duopoly supergame with product differentiation. We find that (i) if cartel profits are evenly split between firms, then only symmetric equilibria s obtained; (i) if instead the additional profits available through collusion are split according to the Nash bargaining solution, there are parameter regions where all subgame perfect equilibria are asymmetric, with firms colluding in price–quantity supergames.     

NASH BARGAINING AND CO-OPERATIVES

This paper develops the generalised Nash bargaining solution for a bargaining co-operative selling its output to a single buyer. Three assumptions for co-operative members' behaviour are examined: profit maximisation, co-operative surplus maximisation and maximising members' price. Solutions are compared and comparative statics presented for these alternative assumptions and the bargaining over price and quantity transacted. The most striking feature of the results is that pursuing the objective of maximising members' price does not necessarily lead to the highest members' price     

The Nash bargaining solution in general n-person cooperative games

We present a noncooperative foundation for the Nash bargaining solution for an n-person cooperative game in strategic form. The Nash bargaining solution should be immune to any coalitional deviations. Our noncooperative approach yields a new core concept, called the Nash core, for a cooperative game based on a consistency principle. We prove that the Nash bargaining solution can be supported (in every subgame) by a stationary subgame perfect equilibrium of the bargaining game if and only if the Nash bargaining solution belongs to the Nash core.     

Collective choice with endogenous reference outcome

A collective choice problem—essentially a bargaining problem without disagreement outcome—is studied. An extended solution, which determines a solution and a reference point simultaneously, is characterized. The unique extended solution that meets the extended versions of Pareto-optimality, independence of irrelevant alternatives, symmetry, and scale invariance maximizes the Nash product with respect to both the solution and the reference point.     

Profit sharing as entry deterrence mechanism

In a right-to-manage framework, this paper analyzes the optimal choice of the pay scheme (profit sharing vs. fixed wage) in a unionized duopoly with potential market entry and decentralized bargaining. The paper shows that, depending on the institutional features, both pay systems can arise as equilibria in Nash strategies. Under duopoly with committed bargaining, the fixed wage is the Nash equilibrium; with flexible bargaining, an agreement between the incumbent firm and its union about profit sharing arises as Nash equilibrium, if the union is not too strong. A monopoly with threat of entry reinforces the selection of profit sharing as a deterrent mechanism.     

Wage bargaining and induced technical change in a linear economy: Model and application to the US (1963–2003)

In a simple one-sector, two-class, fixed-proportions economy operating at full capacity, wages are set through generalized axiomatic bargaining à la Nash (1950). As for choice of technology, firms choose the direction of factor-augmenting innovations to maximize the rate of unit cost reduction (Kennedy, 1964, Funk, 2002). The aggregate environment resulting by self-interested decisions made by economic agents is described by a two-dimensional dynamical system in the employment rate and output/capital ratio. The economy converges cyclically to a long-run equilibrium involving a Harrod-neutral profile of technical change, a constant rate of employment of labor, and constant input shares. The type of oscillations predicted by the model matches qualitatively the available data on the United States (1963–2003). Institutional change, as captured by variations in workers’ bargaining power, has a positive effect on the rate of output growth but a negative effect on employment.     

On price taking behavior in a nonrenewable resource cartel–fringe game

We consider a nonrenewable resource game with one cartel and a set of fringe members. We show that (i) the outcomes of the closed-loop and the open-loop nonrenewable resource game with the fringe members as price takers (the cartel–fringe game à la Salant, 1976) coincide and (ii) when the number of fringe firms becomes arbitrarily large, the equilibrium outcome of the closed-loop Nash game does not coincide with the equilibrium outcome of the closed-loop cartel–fringe game. Thus, the outcome of the cartel–fringe open-loop equilibrium can be supported as an outcome of a subgame-perfect equilibrium. However the interpretation of the cartel–fringe model, where from the outset the fringe is assumed to be price taker, as a limit case of an asymmetric oligopoly with the agents playing Nash–Cournot, does not extend to the case where firms can use closed-loop strategies.     

Implementations of the Nash solution based on its Walrasian characterization

Summary. The present paper provides three different support results for the Nash bargaining solution of -person bargaining games. First, for any bargaining game there is defined a non-cooperative game in strategic form, whose unique Nash equilibrium induces a payoff vector that coincides with the Nash solution of the bargaining game. Next this game is modified in such a way that the unique Nash equilibrium that supports the Nash solution is even in dominant strategies. After that an -stage game in extensive form is presented whose unique subgame perfect equilibrium supports the Nash solution of the bargaining game. Finally, the support results are shown to induce implementation results in the sense of mechanism theory. Received: October 3, 1999; revised version: October 26, 1999     

Optimal cartel trigger price strategies

A dynamical model of industry equilibrium is described in which a cartel deters deviations from collusive output levels by threatening to produce at Cournot quantities for a period of fixed duration whenever the market price falls below some trigger price. In this model firms can observe only their own production level and a common market price. The market demand curve is assumed to have a stochastic component, so that an unexpectedly low price may signal either deviations from collusive output levels or a “downward” demand shock.     

Regulating a duopoly by a Pretend-but-Perform Mechanism

In a linear duopoly, participants are required to declare their costs under a Pretend—but—Perform Mechanism (PPM) which enforces accordance of performance with pretended (declared) cost.Specifically, under the PPM, each firm's output in reaction to rival output is to maximize profit according to declared cost. Compared with the Cournot solution before PPM Regulation, Nash implementation of the PPM induces a Cournot solution, among the make—believe firms, giving greater industry output and efficiency along with increased consumer surplus even after compensating the duopolists for decreased profits, except in a trivial case which — educatively — thruthful declarations are confined.     

Oil market share dynamics: A Markov chain analysis of consumer and producer adjustments

We develop an “optimal market share rule” model of cartel behavior which when applied to the OPEC cartel appears capable of explaining its stability and responses to changed market events. In particular, by attaching importance to market shares based approximately on costs, OPEC members can by maintaining optimal shares deter deviant member attempts to break cartel rules. After a thorough discussion of the theory, the model is tested empirically using a Markov probability model. The estimated Markov transition matrix is further decomposed into what Theil has called the exchange matrix and the mean passage matrix. Dynamic adjustment processes in the market are revealed by the latter while an emerging pattern of OPEC member surveillance of consumers is revealed by the former which facilitates cartel stability. Inspection of these matrixes further suggests that after the formation of OPEC there is evidence of less potential for producer conflict while there appears more evidence for consumer conflict. While these results must be tentative in view of the fact that they have been estimated using a simplified two consumer — two producer model and limited data, it is argued that the results are highly suggestive and the approach in this study can be extended to cover all producer and consumers, and can be integrated into a complete model of the world oil market.     

Bargaining and Boldness

We study a multiperson bargaining problem with general risk preferences through the use of Shaked's game of cycling offers with exogenous breakdown. If preferences are “smooth,” then as the risk of breakdown vanishes, the limiting outcome is one in which bargainers are equally marginally bold; where a bargainer's marginal boldness measures his willingness to risk disagreement in return for a marginal improvement in his position. Under smoothness, any (ordinal-)Nash solution is an equally marginally bold outcome. However, unlike the concept of the (ordinal-)Nash solution, a unique equally marginally bold outcome exists in natural cases—in particular, if all bargainers have risk-averse preferences of the rank-dependent expected utility type. For these preferences, the equally marginally bold outcome maximizes a “bargaining power”-adjusted (asymmetric) Nash product where the degree of asymmetry is determined by the disparity in the marginal valuation of certainty among bargainers. Journal of Economic Literature Classification Numbers: C72, C78, D81.     

The optimality of nominal contracts

Summary. This paper presents a model in which agents choose to use money as a medium of exchange, a means of payment, and a unit of account. The paper defines conditions under which nominal contracts, promising future payment of a fixed number of units of fiat money, prove to be the optimal contract form in the presence of either relative or aggregate price risk. When relative prices are random, nominal contracts are optimal if individuals have ex ante similar preferences over future consumption. When the aggregate price level is random, whether from shocks to the money supply or aggregate output, nominal contracts (perhaps coupled with equity contracts) lead to optimal risk-sharing if individuals have the same degree of relative risk aversion. Finally, nominal contracts may be optimal if the repayment of contracts is subject to a binding cash-in-advance constraint. In this case, a contingent contract increases the risk of holding excessive cash balances. Received: March 29, 1996; revised version: February 25, 1997     

OPTIMAL CARTEL PRICING IN THE PRESENCE OF AN ANTITRUST AUTHORITY*

The dynamic behavior of a price‐fixing cartel is explored when it is concerned about creating suspicions that a cartel has formed. Consistent with preceding static theories, the cartel's steady‐state price is decreasing in the damage multiple and the probability of detection. However, contrary to those theories, it is independent of the level of fixed fines. It is also shown that the cartel prices higher when a more competitive benchmark price is used in calculating damages.     

Risk sensitivity in bargaining with more than two participants

In two-person bargaining over riskless outcomes where the outcome is chosen according to the Nash solution or the Kalai-Smorodinsky solution, an increase in a player's risk aversion is to the advantage of the opponent. It is shown in this note that for the n-person Nash solution, an increase in risk aversion is to the player's own disadvantage but needs not be advantageous to all the opponents. For the n-person Kalai-Smorodinsky solution, the increase is to the player's own disadvantage and to the advantage of all the opponents.     

Debt Concentration and Bargaining Power: Large Banks,Small Banks,and Secondary Market Prices

Commerical bank debts of developing countries are held by large international banks and smaller domestic banks. This paper investigates how debt concentration—the proportion of a country's debt held by large banks relative to small banks—affects the secondary market price for these loans. We find that countries with higher concentrations have higher secondary-market prices. We explain this empirical finding in a bargaining model that endogenizes the maximum penalty that banks can credibly impose on a recalcitrant debtor. We show that the banks' bargaining power increases with the degree of debt concentration, thus increasing repayment and secondary-market prices.     

STICKY PRICES AND COSTLY CREDIT

We develop a theory of money and credit as competing payment instruments, then put it to work in applications. Agents use cash and credit because the former (latter) is subject to the inflation tax (transaction costs). Frictions that make the choice of payment method interesting also imply equilibrium price dispersion. We derive closed-form solutions for money demand, and show how to simultaneously account for the price-change facts, cash–credit shares in micro data, and money-interest correlations in macro data. The effects of inflation on welfare, price dispersion and markups are discussed, as are nonstationary equilibria with dynamics in the price distribution.     

A NOTE ON MARKET COVERAGE IN VERTICAL DIFFERENTIATION MODELS WITH FIXED COSTS

With fixed costs of quality improvement, we find that a covered market outcome with an interior solution in the price stage is not a Nash equilibrium. When the degree of consumer heterogeneity is high (low) enough, an uncovered market outcome (a covered market outcome with a corner solution in the price stage) is the only Nash equilibrium. When the degree of consumer heterogeneity is moderate, both of the two market outcomes are Nash equilibria, but an uncovered market outcome yields higher social welfare than a covered market outcome with a corner solution in the price stage.     

Von Neumann-Morgenstern stable sets, discounting, and Nash bargaining

We establish a link between von Neumann-Morgenstern stable set and the Nash solution in a general n-player utility set. The stable set-solution is defined with respect to a dominance relation: payoff vector u dominates v if one player prefers u even with one period delay. We show that a stable set exists and, if the utility set has a smooth surface, any stable set converges to the Nash bargaining solution when the length of the period goes to zero.     

On the Interpretation of Nonsymmetric Bargaining Solutions and Their Extension to Nonexpected Utility Preferences

In a recent paper, Rubinstein, Safra, and Thomson reexamine the foundations of the axiomatic bargaining theory as formulated by Nash. In it they reinterpret the Nash bargaining solution and extend it to a family of nonexpected utility preferences, providing an interpretation of this solution concept and an axiomatization by means of a suitable translation of Nash axioms. This paper extends the work of these authors to the family, characterized by Kalai, of nonsymmetric solutions that appear from the Nash bargaining solution when the symmetry axiom is dropped. This extension requires an axiom relating the solution outcomes of isomorphic games. Journal of Economic Literature Classification Number: C78.