A Convergent Adjustment Process for Firms in Competition

Abstract

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.. The Econometric Society is collaborating with JSTOR to digitize, preserve and extend access to Econometrica. This paper describes a market in which firms vary their quantities of production according to a new adjustment process. Each firm bases its new production entirely upon a knowledge of its own previous productions and profits. It has no knowledge of the payoff functions of the market. Numerical analysis of the process indicates an approach to equilibrium for all initial states. The set of allowed limit points is rigorously characterized, and determined explicitly in the case of two firms. Some exact solutions are found. The process can be regarded as a way of playing a continuous game with a minimum of information.Models may be loosely classified into those which attempt to model what firms are currently doing [1, 6, 9, and 13] and those which attempt to provide an improved strategy for firms [2, 11, 12, 14, and 16]. (It is ironical that any success with the latter may make the former more difficult to construct.) Our model is of the first type, and we are primarily interested here in the large scale properties of the market, rather than the benefits of individuals.The process described here may also be regarded as a way of playing an n-person game with incomplete information, in the sense of Harsanyi [8]. In each play of the game the only information which a player has is the history of his own strategies and payoffs in previous games. This past history is then used to determine his strategy in the next play. ACCOUNTING PROCEDUREAs in references [4 and 5], we suppose that the firm has no control over prices, but can decide what quantity of goods to produce in a given business period. It tries to choose this quantity so as to maximize its profits.At the end of each business period, at time t, the firm tries to estimate how its profit per good, mr (cIt), varies with the quantity o-of goods it produces. We call this the estimated average profit function (EAPF), and suppose, for convenience, that t takes integer values. The firm constructs this function by fitting a straight line to the data from its previous business periods, viz. its outputs v-(1),

Learning cycles in Bertrand competition with differentiated commodities and competing learning rules

“…In this paper we generalize some results of Kirman (1983) to the case of n rms under a nonlinear demand specication. Gates et al (1977) consider least squares learning in a Cournot oligopoly. Each rm regresses its average prots on its outputs and uses the estimated function to determine the output for the next period.…”

“…The learning method the authors consider diers from ours in two respects. First, each observation has the same weight in our model whereas rms weigh observations dierently in Gates et al (1977). Second, the rms' action is specied as the action that maximizes the one-period expected prot in our paper.…”

“…Second, the rms' action is specied as the action that maximizes the one-period expected prot in our paper. In Gates et al (1977) the next action is the weighted average of the previous action and the action that maximizes the estimated prot function. Tuinstra (2004) considers the same kind of model that is studied in this paper.…”

Learning and Pricing with Models That Do Not Explicitly Incorporate Competition

“…The paper shows that although any pair of prices can be part of a pseudo equilibrium, only a certain subset of price pairs can be part of such a limit pseudo equilibrium. It also conjectures that the only possible limit prices are those in the identified subset, and cites the simulations of Kirman (1975) and Gates et al (1977) as showing that prices converge from arbitrary initial conditions. It leaves open the issue of proving such convergence.…”

The coexistence of stable equilibria under least squares learning

“…However, since the true relationship is unknown, it might occur that agents use a misspecied functional form in the regression. The eect of such misspecication was analyzed in the Industrial Organization literature, see Kirman (1975Kirman ( , 1983, Gates et al (1977), Brousseau and Kirman (1992), Kopányi (2013) and Anufriev et al (2013). Misspecied LSL leads to a so-called self-sustaining equilibrium in which rms do not learn the true demand function correctly but they end up in a situation where 1. they are choosing the prot-maximizing price or quantity subject to their estimated function and 2. in the equilibrium their estimation is correct in the sense that the price or demand they expect to get (based on their estimation) coincides with the actual realization.…”