همگرایی یادگیری بیزی برای تعادل عمومی در مدل های اشتباه مشخص شده

A central unanswered question in economic theory is that of price formation in disequilibrium. This paper lays the groundwork for a model that has been suggested as an answer to this question in, particularly, Arrow [Toward a theory of price adjustment, in: M. Abramovitz, et al. (Ed.), The Allocation of Economic Resources, Stanford University Press, Stanford, 1959], Fisher [Disequilibrium Foundations of Equilibrium Economics, Cambridge University Press, Cambridge, 1983] and Hahn [Information dynamics and equilibrium, in: F. Hahn (Ed.), The Economics of Missing Markets, Information, and Games, Clarendon Press, Oxford, 1989]. We consider sellers that monopolistically compete in prices but have incomplete information about the structure of the market they face. They each entertain a simple demand conjecture in which sales are perceived to depend on the own price only, and set prices to maximize expected profits. Prior beliefs on the parameters of conjectured demand are updated into posterior beliefs upon each observation of sales at proposed prices, using Bayes’ rule. The rational learning process, thus, constructed drives the price dynamics of the model. Its properties are analysed. Moreover, a sufficient condition is provided, relating objectively possible events and subjective beliefs, under which the price process is globally stable on a conjectural equilibrium for almost all objectively possible developments of history.

In economic theory, a key role in the coordination of behavior is played by prices. As a consequence, the so-called price mechanism is much debated, and the need for it operating freely often stressed. Yet, there are many open research questions on the matter of prices, especially on how they come to take on equilibrium values. For one thing, it is generally left unexplained whose business it actually is to call and change prices. Particularly in models in which price-taking behavior is assumed, this is a pressing question. Reliance on a unique price vector indicates it is left to a single person or institution, and a number of models has been presented in which the central person is in fact an altruistic auctioneer, e.g. in the tâtonnement process, the Edgeworth process, and the Hahn process.1 Apart from the fact that it seems odd, if not plainly inconsistent, to model all behavior but that of the auctioneer as resulting from constrained rational choice, at least two things meet the eye in these explanations. First, these processes need an exogenous central coordinator to explain the rise of equilibria that are meant to be the outcome of decentralized competitive economies. Second, the conditions these processes need for convergence on equilibrium price values for arbitrary initial prices, i.e. for global stability of the disequilibrium process—have been found to be pretty strong. A number of suggestions has been made to study the disequilibrium behavior of prices more seriously. An early one is in Arrow (1959), in which Arrow proposed to make price a choice variable of individual firms, that consequently need to come equipped with some local monopoly power, at least as a disequilibrium phenomenon. To Arrow, the construct of perfect competition did not allow for an explanation of price behavior. More recently, Fisher (1983) develops an elaborate model of disequilibrium behavior in which there is clarity on who is setting prices. It is done by dealers, who specialize in differentiated goods, which gives them the local monopoly power to act as a coordinator and set prices. How prices are adjusted with changes in perceptions, however, is not discussed in depth in the monograph, yet indicated as an area of promising further research. Finally, in Hahn (1989) several partial examples are given of perception changes and associated behavior that may indeed be plausible for monopolistically competing price-setters to develop—including a rudimentary version of the behavior we study in this paper. Yet, the consequences of such behavior, particularly when performed in general equilibrium settings, are only hinted upon. When prices are choice variables of firms, the way firms perceive their market position, and especially changes in these perceptions, can account for the dynamics of prices. This idea is used in the present paper to construct a model of individually rational price adjustment and study its limit behavior, particularly its stability properties. In the present model, each of a number of firms is in monopolistic price competition, but does not have perfect information on the market demand it faces. At each moment in time, based on its information to date on past prices and sales, each firm entertains a demand conjecture instead. Naturally, this conjecture has a structural form different from that of objective demand. Particularly, we consider the most extreme case where firms only consider their own price as an explanatory variable, and do not consider the price effects of competing products. Within their conjectured structures, firms learn in a Bayesian way about the value of the demand parameters it has modeled. A fleshed out conjecture serves as a basis for an optimal price through expected profit maximization. It is shown that for initial beliefs that do not assign zero probability to developments of prices and sales that can actually happen, the incomplete beliefs converge to a finite limit, and, therefore, prices converge as well. This is called ‘No Statistical Surprise’. Convergence takes place on a set of ‘conjectural equilibria’. Under ‘No Statistical Surprise’, therefore, the price process is globally stable in that it reaches an equilibrium for every initial belief-structure. Which particular equilibrium is reached depends on the initial beliefs. This path-dependency result runs solely over beliefs, since the model assumes absence of trade at disequilibrium prices. The stability result does not rely on specific conditions on the structure of objective demand. Instead, the condition of ‘No Statistical Surprise’ is sufficient for the perceived structure to absorb all price effects on objective demand. The literature on Bayesian or rational learning is quite recent and large. Our paper builds on several of its results. One focus has been the concern to justify the use of rational expectations equilibria. Particularly Bray and Savin (1986), and Bray and Kreps (1987) work in this direction, and establish convergence results for myopic Bayesian learners on rational expectations equilibrium in versions of the cobweb-model. Early work by Blume and Easley, 1982 and Blume and Easley, 1986 is also concerned with the influence learning has on the eventual equilibrium situation reached, but in a general equilibrium setting. Particularly, they focus on conditions under which Bayesian learners will identify the true model among several models. In partial equilibrium models of single firms learning their demand, Easley and Kiefer (1988) among others, study the influence of active learning on firms’ optimization problems. Actively learning firms are aware of the fact that their behavior influences their options for learning. In a discrete game theoretical setting, Kalai and Lehrer, 1993 and Kalai and Lehrer, 1995 have obtained results for rational learning behavior. The former reference, Kalai and Lehrer (1993), considers learning in a correctly specified structure, and states conditions under which it converges to a Nash equilibrium of the perfect information game that are similar to ours. Another, much less extensively traveled route has been to study the influence of structural mis-specification on the convergence process and its equilibria. Kirman, 1975, Kirman, 1983 and Kirman, 1995 sets up an early example of two firms learning, in a least squares way, in a mis-specified structure of their game. However, he does not establish general convergence results. Nyarko (1991) constructs an example of a single, actively learning monopolist whose beliefs do not settle, due to a very particular structural specification error. Kalai and Lehrer (1995) extends the 1993 convergence conditions to structurally mis-specified models to identify the usable notion of equilibrium. However, their article does not present explicit convergence results. This paper is organized as follows. Section 2 presents the model structure. Section 3 introduces the way in which information is processed, as well as an associated equilibrium concept. 4 and 5 introduce the convergence result, the nature of which is subsequently discussed in Section 6. Section 7 presents the global stability of the price process on the equilibria of the model, introducing the concept of ‘No Statistical Surprise’. Section 8 closes with some concluding remarks on possible extensions of the model.

The groundwork laid in this paper for modeling individually rational disequilibrium price adjustment by introducing elements of imperfect competition, imperfect and incomplete information and learning from self-generated signals, allows for a fairly strong global stability condition for general equilibrium models, that of ‘No Statistical Surprise’. Apart from being intuitively appealing, and doing away with the deus ex machina approach to disequilibrium behavior, this condition extends quite naturally on ‘No Favorable Surprise’, the global stability condition established in Fisher (1983). Our approach also calls for a number of extensions. In the present model, firms gradually estimate the parameters of their conjectured demand. Naturally, provided they have some monopsony power as well, firms could likewise be taken to learn about supply, proposing purchase prices in the process. ‘No Statistical Surprise’ is likely to be strong enough to obtain convergence results in such a dealer-model as well. A more demanding extension of the model would be to further specify the relationship between objective demand and supply structures and their subjective counterparts on which behavior is based. The present conjectures consider only the own price effect. Typically, firms would take the prices of several of their nearest competitors into account, applying econometric techniques in which the costs of including additional explanatory variables, or sharper functional forms, are weighted against the expected benefits of more precise predictions, thus, determining the best structural specification to work with. Such an approach would lead to an optimal level of mis-specification and introduce interesting problems concerning the strategic behavior towards rivals. Related to this is the concept of active learning, where firms reckon with the fact that their prices will provide future information that can be used to increase profits. The type of non-myopic price setting that results from this has been studied in a partial setting Easley and Kiefer (1988) and Kiefer and Nyarko (1989), where convergence results similar to the ones obtained here are established. The present price adjustment model would benefit greatly from an extension of dealer behavior in this direction, even though we expect ‘No Statistical Surprise’ to be powerful enough to again assure almost sure convergence. The model presented in this paper relies on a specific and exogenously given structure of the market. Certain firms make it their business to act as intermediaries in the trade of a particular good. Casually, this setup has been defended by an appeal to product differentiation and transaction costs. It is to represent a socially accepted shopping-area structure. Although the identification of commodities with firms, which naturally leads to this market structure, seems quite appropriate in many markets, further specification of these underlying properties of markets is called for. Particularly, the consequences of entry and exit, and the possibility to compete for locally dominant dealerships raises interesting questions. For one thing, efforts to endogenize the market structure may well result in entry conditions that have the model sound more than presently like a disequilibrium story with a competitive ending. Finally, the present model is altogether silent on the issue of social consequences of the disequilibrium dynamics modeled. In accordance with the observations by Arrow (1959) referred to earlier, it trades efficiency of competitive equilibrium for global stability by introducing monopolistic competition as an essential disequilibrium phenomenon. The precise welfare consequences of this seem a promising area of further research. In the appendices we have collected those parts of the theoretical framework needed in the paper that would disrupt the flow of the argument too much when presented in the main body of the paper. Appendix A provides a detailed and complete proof of the continuity of the Bayes operator. Appendix B provides proofs concerning the support of some of the probability measures used in the text. These two appendices are largely based on Easley and Kiefer (1988).