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Dynamic Pricing of Network Goods With Boundedly Rational Consumers

Roy Radner, Arun Sundararajan

Last revised: January 2005

Abstract:   An important simplifying assumption made when analyzing goods that display positive network effects is that potential consumers can form a rational expectation of the equilibrium demand for the good, and that they all form the same expectation, which is then fulfilled based on their consumption choices - sometimes called a rational expectations equilibrium (REE). We examine whether the results of these models are robust to the relaxation of this assumption. In our model, consumers differ in their marginal utility of total demand (intensity of the network effect), which varies according to a given distribution (the distribution of consumer "types"), and are boundedly rational in two ways. First, only a fraction of consumers "pay attention" to price announcements in any interval of time. Second, those consumers who pay attention make their consumption choices based on a boundedly rational expectation of future demand. Our benchmark model is of myopic expectations, although we show how our results generalize (1) to a case in which the population of consumers contains both those who are myopic and those who are "fully rational," and (2) to a case in which consumers have expectations that are partly "stubborn". We base our analysis on a continuous-time approximation of an underlying discrete-time model. Under this approximation, the instantaneous choices of consumers continuously influence the rate at which demand adjusts over time, and a monopolist chooses a price trajectory to maximize profit. First, we show that, under fairly general assumptions about the distribution of types, the profit-maximizing rational expectations equilibrium is not a steady state of the optimal trajectory with boundedly rational consumers. Our second theorem shows that if consumer types are uniformly distributed and consumers form myopic (or more rational) expectations, the monopolist's optimal pricing trajectory is generated by a "target policy" with the following properties: when current demand is below the target, the price is zero; when current demand is above the target, the price is the maximum possible, and when current demand is at the target, the price is chosen to keep demand stationary. We also show that the optimal demand target with boundedly rational consumers is always strictly lower than the equilibrium level of demand predicted by a model with rational expectations. Furthermore, the difference between the target demand and the rational expectations demand is higher when consumers pay attention to the monopolist's price announcements at a lower rate. We generalize the results from this example in two ways. Our third theorem examines the case of myopic consumers and strictly concave distributions of consumer types. To find an optimal policy one must expand the set of controls to include measure-valued controls. The optimal policy is similar to the target policy of Theorem 2, except that when current demand is at the target, the monopolist chooses the "mixture" between a price of zero and the maximum possible price that keeps demand stationary. For convex consumer type distributions in the neighborhood of the uniform distribution, we give a heuristic argument to support a conjecture that the monopolist continues to choose a demand target lower than the rational expectations demand, but varies price gradually in the neighborhood of the demand target. Finally, for uniformly distributed types and consumer expectations that are both myopic and "stubborn", we show that the monopolist's optimal pricing trajectory is generated by a target policy with the same properties as those in Theorem 2, although with a target that is strictly lower, and that increases as consumers become progressively less stubborn.
(This paper is part of a program of research whose broad objective is to explore the conditions under which the assumption of unbounded rationality in economic models is a reasonable one).

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