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The Coexistence of Stable Equilibria under Least Squares Learning
Abstract
In this paper we consider firms that learn about market conditions by estimating the demand function using past market data. We show that learning may lead to suboptimal outcomes even when the estimated demand function perfectly fits the observations used in the regression and firms thus perceive to have learned the demand function correctly.
We consider the Salop model with three firms and two types of consumers that differ in their sensitivity to product differences. Firms do not know the demand structure and they apply least squares learning to learn the demand function. In each period, firms estimate a linear perceived demand function and they play the perceived best response to the previous-period price of the other firms.
This learning rule can lead to three different outcomes: a self-sustaining equilibrium, the Nash equilibrium or an asymmetric learning-equilibrium. In this last equilibrium one firm underestimates the demand for low prices and it attracts consumers with high sensitivity only. This type of equilibrium has not been found in the literature on least squares learning before. Both the Nash equilibrium and the asymmetric learning-equilibrium are locally stable therefore the model has coexisting stable equilibria.
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In this paper, we propose and compare three heterogeneous Cournotian duopolies, in which players adopt best response mechanisms based on different degrees of rationality. The economic setting we assume is described by an isoelastic demand function with constant marginal costs. In particular, we study the effect of the rationality degree on stability and convergence speed to the equilibrium output. We study conditions required to converge to the Nash equilibrium and the possible route to destabilization when such conditions are violated, showing that a more elevated degree of rationality of a single player does not always guarantee an improved stability. We show that the considered duopolies exhibit either a flip or a Neimark-Sacker bifurcation. In particular, in heterogeneous oligopolies models, the Neimark-Sacker bifurcation usually arises in the presence of a player adopting gradient-like decisional mechanisms, and not best response heuristic, as shown in the present case. Moreover, we show that the cost ratio crucially influences not only the size of the stability region, but also the speed of convergence toward the equilibrium.
We consider a price adjustment process in a model of monopolistic compe-tition. Firms have incomplete information about the demand structure. When they set a price they observe the amount they can sell at that price and they observe the slope of the true demand curve at that price. With this information they estimate a linear demand curve. Given this estimate of the demand curve they set a new optimal price. We investigate the dynamical properties of this learning process. We Þnd that, if the cross-price effects and the curvature of the demand curve are small, prices converge to the Bertrand-Nash equilibrium. The global dynamics of this adjustment process are analyzed by numerical sim-ulations. By means of computational techniques and by applying results from homoclinic bifurcation theory we provide evidence for the existence of strange attractors.
We propose an oligopoly game where quantity setting firms have incomplete information about the demand function. At each time step they solve a profit maximization problem assuming a linear demand function and ignoring the effects of the competitors’ outputs. Despite such a rough approximation, that we call “Local Monopolistic Approximation” (LMA), the repeated game may converge at a Nash equilibrium of the game played under the assumption of full information. An explicit form of the dynamical system that describes the time evolution of oligopoly games with LMA is given for arbitrary differentiable demand functions, provided that the cost functions are linear or quadratic. In the case of isoelastic demand, we show that the game based on LMA always converges to a Nash equilibrium. This result, compared with “best reply” dynamics, shows that in this particular case less information implies more stability.
The Chamberlinian monopolistically competitive equilibrium has been explored and extended in a number of recent papers. These analyses have paid only cursory attention to the existence of an industry outside the Chamberlinian group. In this article I analyze a model of spatial competition in which a second commodity is explicitly treated. In this two-industry economy, a zero-profit equilibrium with symmetrically located firms may exhibit rather strange properties. First, demand curves are kinked although firms make "Nash" conjectures. If equilibrium lies at the kink, the effects of parameter changes are perverse. In the short run, prices are rigid in the face of small cost changes. In the long run, increases in costs lower equilibrium prices. Increases in market size raise prices. The welfare properties are also perverse at a kinked equilibrium.
We modify the Salop (1979) model of price competition with differentiated products by assuming that consumers are loss averse relative to a reference point given by their recent expectations about the purchase. Consumers' sensitivity to losses in money increases the price responsiveness of demand—and hence the intensity of competition—at higher relative to lower market prices, reducing or eliminating price variation both within and between products. When firms face common stochastic costs, in any symmetric equilibrium the markup is strictly decreasing in cost. Even when firms face different cost distributions, we identify conditions under which a focal-price equilibrium (where firms always charge the same "focal" price) exists, and conditions under which any equilibrium is focal. (JEL D11 , D43, D81, L13)
In this paper we analyze a dynamic game of Cournot competition with heterogeneous firms choosing between two different adaptive behavioral rules in deciding output strategies. The underling oligopoly structure is standard: using a constant returns to scale technology, N firms produce homogeneous goods, which are sold in a market characterized by constant price elasticity. In this setup, we assume that a fraction of firms employs a quite rough rule of thumb, the so-called Local Monopolistic Approximation (LMA), whereas the complementary fraction plays Best Reply (BR), a more demanding strategy in terms of information and computation requirements. The model is first considered with exogenously fixed fractions of firms in the two complementary groups. Then it is generalized by considering an endogenous evolutionary switching process between the two behavioral strategies based on profit-driven replicator dynamics. The role of the number of firms, information costs and inertia (or anchoring attitude) in production decisions is analyzed, as well as the influence in the evolutionary process of random noise in the demand function and memory of past profits. Global properties of the oligopoly with evolutionary pressure between behavioral rules are discussed, with particular regard to cases in which the Nash equilibrium is unstable.
The paper describes quality/price equilibrium and welfare effects of R&D spillovers when firms are located around the circumference of a Salop (1979) circle and the extent of the spillover may depend on the geographic proximity between firms. In particular, in contrast to previous related contributions that studied the relationship between spatial competition and quality provision, we show that an increase in competition (i.e. additional entry on the circle) may have a positive effect on the provision of quality and firms' profits. We also extend the model allowing a multinational enterprise, MNE, to locate at the centre of the circle. In this scenario it is important to understand the interplay of local R&D spillovers with spillovers that propagate from and to the centre.
This paper stresses the importance of heterogeneity in learning. We consider a Bertrand oligopoly with firms using either least squares learning or gradient learning for determining the price. We demonstrate that convergence properties of the rules are strongly affected by heterogeneity. In particular, gradient learning may become unstable as the number of gradient learners increases. Endogenous choice between the learning rules may induce cyclical switching. Stable gradient learning gives higher average profit than least squares learning, making firms switch to gradient learning. This can destabilize gradient learning which, because of decreasing profits, makes firms switch back to least squares learning.
We consider a Cournot oligopoly game, where firms produce an homogenous good and the demand and cost func-tions are nonlinear. These features make the classical best reply solution difficult to be obtained, even if players have full information about their environment. We propose two different kinds of repeated games based on a lower degree of rationality of the firms, on a reduced information set and reduced computational capabilities. The first adjustment mechanism is called ''Local Monopolistic Approximation'' (LMA). First firms get the correct local estimate of the demand function and then they use such estimate in a linear approximation of the demand function where the effects of the competitorsÕ outputs are ignored. On the basis of this subjective demand function they solve their profit maxi-mization problem. By using the second adjustment process, that belongs to a class of adaptive mechanisms known in the literature as ''Gradient Dynamics'' (GD), firms do not solve any optimization problem, but they adjust their pro-duction in the direction indicated by their (correct) estimate of the marginal profit. Both these repeated games may con-verge to a Cournot–Nash equilibrium, i.e. to the equilibrium of the best reply dynamics. We compare the properties of the two different dynamical systems that describe the time evolution of the oligopoly games under the two adjustment mechanisms, and we analyze the conditions that lead to non-convergence and complex dynamic behaviors. The paper extends the results of other authors that consider similar adjustment processes assuming linear cost functions or linear demand functions.
We consider repeated pricing games in which two competing sellers use mathematical models to choose the prices of their products. Over the sequence of games, each seller attempts to estimate the values of the parameters of a demand model that expresses demand as a function only of its own price using data comprised only of its own past prices and demand realizations. Thus, as is often the case in practice, the sellers' models do not explicitly account for other sellers. We study the behavior of the sellers' prices and parameter estimates under various assumptions regarding the sellers' knowledge and estimation procedures. We identify situations in which (a) the sellers' prices converge to the Nash equilibrium associated with knowledge of the correct demand model, (b) the sellers' prices converge to the cooperative solution, and (c) the sellers' prices converge to other values that are neither the Nash equilibrium nor the cooperative solution and that depend on the initial prices.
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.
The stability of the rational expectations equilibrium of a simple asset market model is studied in a situation where a group of traders learn about the relationship between the price and return on the asset using ordinary least squares estimation, and then use their estimates in predicting the return from the price. The model which they estimate is a well-specified model of the rational expectations equilibrium, but a misspecified model of the situation in which the traders are learning. It is shown that for appropriate values of a stability parameter the situation converges almost surely to the rational expectations equilibrium.
This paper studies the impact of hospital competition on waiting times. We use a Salop-type model, with hospitals that differ in (geographical) location and, potentially, waiting time, and two types of patients: high-benefit patients who choose between neighbouring hospitals (competitive segment), and low-benefit patients who decide whether or not to demand treatment from the closest hospital (monopoly segment). Compared with a benchmark case of monopoly, we find that hospital competition leads to longer waiting times in equilibrium if the competitive segment is sufficiently large. Given a policy regime of hospital competition, the effect of increased competition depends on the parameter of measurement: Lower travelling costs increase waiting times, higher hospital density reduces waiting times, while the effect of a larger competitive segment is ambiguous. We also show that, if the competitive segment is large, hospital competition is socially preferable to monopoly only if the (regulated) treatment price is sufficiently high.
This paper surveys the literature which examines the stability of the expectations that agents are assumed to have in a rational expectations equilibrium (REE). This issue is more complex than the usual statistical estimation problem because the relationship between observable variables and payoff relevant variables is endogenous. One approach taken in the literature yields convergence to a REE but requires agents to have extensive knowledge about the structure and dynamics of the model that prevails while they learn. A second approach does not assume that agents have correctly specified likelihood functions and finds that REE may not be stable. Peer Reviewed http://deepblue.lib.umich.edu/bitstream/2027.42/24021/1/0000270.pdf
This paper investigates the existence of oligopoly equilibria when firms arrive, through local price experiments, at a correct estimation of their demand curves in a neighborhood of a given statu s quo. The author provides sufficient conditions for the existence of a local Nash equilibrium, defined as a point where each firm is at a local maximum of its profit function, given the prices charged by th e other firms. He also provides two examples, one of a duopoly with l ocal Nash equilibria but no Nash equilibria and the other of a duopol y with no local Nash equilibria. Copyright 1988 by Economics Department of the University of Pennsylvania and the Osaka University Institute of Social and Economic Research Association.
We study a class of models in which the law of motion perceived by agents influences the law of motion that they actually face. We assume that agents update their perceived law of motion by least squares. We show how the perceived law of motion and the actual one may converge to one another, depending on the behavior of a particular ordinary differential equation. The differential equation involves the operator that maps the perceived law of motion into the actual one.
In a standard model of oligopoly with differentiated products, the existence of an equilibrium at which the first-order conditions for profit maximisation are simultaneously satisfied for all firms is proved and this is done without imposing any restrictions on the demand functions. This is an equilibrium in the following sense: although some firms may not necessarily be maximising their profits, nevertheless if each firm's knowledge of demand is limited to the linear approximation of its own demand curve, then it will believe that it is indeed maximising its profits.
We study the dynamical system of expectations generated by a simple general equilibrium model of an exchange economy in which each agent considers a finite collection of models, each of which specifies a relationship between payoff-relevant information and equilibrium prices. One of the models under consideration is a correct description of the rational expectations equilibrium. We find that under a Bayesian type of learning process the rational expectations equilibrium is locally stable, but that nonrational equilibria may also be locally stable.
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.