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...extraction, the monopolist has an incentive to subsidize learning by charging price that results in information revelation. Nonetheless, in the long run, the monopolist generally induces herding by either selling to all buyers or exiting the market.
1. Introduction
* We study optimal pricing by a monopolist in a market in which buyers learn from each other's purchases. We consider buyers who have limited first-hand information about the product's value and who can freely observe the decisions made by other buyers in the past. For example, readers deciding which book to buy often consult the list of best sellers, (1) licensees of a new technology look at the behavior of other potential adopters, and employers assess job applicants on the basis of their employment history. In these situations, buyers decide without taking into account the value of the information revealed to the future buyers who observe these decisions. An informational externality is therefore present.
When buyers face fixed prices, this externality easily leads to pathological outcomes, such as herd behavior. As shown by Banerjee (1992) and Bikhchandani, Hirshleifer, and Welch (1992), henceforth BHW, public information quickly swamps private information, and late buyers end up imitating the decisions of a few initial buyers who act alike. Since these late buyers behave in the same way regardless of their private information, nothing can be inferred from their behavior. These informational cascades result in the loss of a potentially large amount of private information collectively possessed by late buyers.
This article investigates what happens when prices are instead variable, being set by a monopolist. The monopolist is a long-run player and can charge different prices over time and thereby affect the process of information aggregation. We examine what happens when prices are allowed to adjust in response to the buyers' choices; how the monopolist influences the buyers' learning process; and, finally, whether monopoly power improves or worsens herd behavior.
In our model, on the demand side of the market there is a sequence of potential buyers, one in each period. All buyers have unit demand and identical (ex post) valuations of the product. The common value of the good is either high or low and is unknown to buyers and seller. Each buyer observes a discrete private signal partially informative about the good's value, as well as the decisions made by all previous buyers. On the supply side, in each period the monopolist posts a price without observing the private signal of the current buyer, who then decides whether to accept or reject the offer. The seller and the subsequent buyers observe the public history of posted prices and purchase decisions, but they do not directly observe the private signals on the basis of which these decisions were taken. We assume that the monopolist has no private information about the good's value and is learning along with the buyers.
In this setting, prices have two functions: immediate rent extraction and information screening. On the one hand, the current price allows the monopolist to extract rent from the current buyer. On the other hand, the inference of future buyers about the current buyer's signal depends on the price the current buyer is charged. That is, the monopolist also uses the price as a screening device, affecting how much of the current buyer's information is made publicly available and, thus, how much future buyers are willing to pay for the product.
As a building block to our analysis, we isolate the immediate rent extraction role of prices by considering static monopoly pricing with a partially informed buyer. In Section 4 we revisit the classic monopoly tradeoff between price and quantity, for the case in which the buyer's willingness to pay depends on the realization of a discrete signal about the good's value. Since long-run outcomes depend on what happens when beliefs are extreme, we are specifically interested in the situation of extreme prior beliefs. We find that when the prior belief about the good's value is favorable enough, it is optimal for the monopolist to sell even to the buyer with the most unfavorable signal. (2) When the prior belief is instead sufficiently unfavorable, the monopolist either exits the market by setting a price so high that no buyer type purchases (if the low value of the good is below its cost) or sells to all buyer types (if the low value of the good is above its cost). In the borderline case in which the low value of the good exactly equals its cost, the seller never exits the market and may find it optimal, even at extremely unfavorable beliefs, to demand a price such that only buyers with sufficiently high signals purchase the good.
In Section 5 we analyze the effect of prices on information screening and expected future profits. We start by examining how prices affect the information available to future buyers. In each period, any given price induces a bipartition of the set of signal realizations. Buyers purchase if and only if they observe a private signal above a certain threshold. If the price is so low that all buyer types purchase, or so high that no type purchases, no information is revealed publicly, as in an informational cascade. Intermediate prices instead allow future buyers to infer some information, and they result in more information than do extreme prices. (3) The question of interest is whether the monopolist benefits from charging an information-revealing price. To answer this, we first show that the monopolist's static profit is convex in the prior belief. Since revelation of public information corresponds to a mean-preserving spread in the belief distribution, it increases expected profits, Hence, the monopolist's expected future profits are maximized at a price that reveals information to future buyers. Of course, the monopolist risks that ex post the information turns out to be unfavorable. However, ex ante this loss is more than compensated by the gain realized if the information is favorable. Being a long-run player, the monopolist has an incentive to partially internalize the informational externality.
The dynamically optimal prices depend on the interplay of immediate rent extraction and information screening (Section 6). The price that maximizes the expected present period's profit generally does not also maximize expected future profits. The monopolist may therefore sacrifice immediate rent extraction to ensure that information is revealed to future buyers. The resolution of this tradeoff naturally depends on the discount factor and the public belief about quality. For any fixed public prior belief, a sufficiently patient seller is exclusively interested in information screening and so demands a price that reveals information. However, given any discount rate, immediate rent extraction dominates when the belief is sufficiently extreme. The reason is that the value of information screening goes to zero when the belief that the value is high is either sufficiently optimistic or pessimistic. This is because the posterior belief is very close to the prior belief when the prior is very high or very low. Since the value of information goes to zero, while its cost is bounded away from zero, the optimal price is the one that maximizes immediate rent extraction.
We then turn to the long-run predictions of the model, for a fixed discount factor. (4) Provided that the cost does not equal the low value of the good, the monopolist eventually stops the learning process by inducing an informational cascade or exiting, thereby preventing asymptotic learning of the good's true value. Instead, if the cost equals the low value of the good, herding is still triggered at high public beliefs, but it may be optimal to charge an informative price for all low beliefs. Only in this case and only if the value of the good is actually low, may buyers asymptotically learn the good's true value. In all other cases, learning stops before the true value is learned.
The article proceeds as follows. Section 2 discusses the related literature. Section 3 formulates the model, Section 4 analyzes immediate rent extraction, and Section 5 considers the effect of information screening on future rent extraction. Section 6 gives results for the general dynamic model, focusing on the long-term outcomes and consequences for information aggregation. Section 7 concludes by discussing the role played by competition for the efficiency of the process of information aggregation. The Appendix collects the proofs of all the results.
2. Related literature
* Despite the pervasiveness of social influence on economic decisions, its implications for pricing in markets have been largely overlooked in economics until recently. Becker (1991) considered a competitive market in which each buyer's demand for a good depends directly on the demand by other buyers. Our model provides a foundation for such dependence based on informational externalities. In our model, the payoff to a buyer depends on the decisions of others only indirectly, through the information revealed about the good's value.
In the herding literature, other articles have analyzed the role of prices. Welch (1990) considered pricing by a monopolist (IPO issuer) in a market with a sequence of partially informed buyers (investors). In this model, the monopolist is constrained to choose a fixed price for all buyers and finds it optimal to induce an immediate informational cascade. (5) In this article, we instead characterize the optimal dynamic pricing strategy for the monopolist. We find that the monopolist benefits from inducing social learning in the market and typically delays the occurrence of informational cascades. In the conclusion, we discuss why we obtain such sharply different predictions.
Avery and Zemsky (1995) also have introduced history-dependent prices in the herding model. In their sequential trading model, in each period a privately informed trader places an order on either side of the market. Prices are set competitively by market makers and so incorporate retrospectively the information revealed by past trades. In that setting, herd behavior is impossible when agents have unidimensional information. In our model, a monopolist instead sets prices taking into account the prospective effect of information to be revealed in the future.
The article closest in spirit to ours is Caminal and Vives (1996). (6) In their two-period model, there is a continuum of buyers with normally distributed private signals about the quality of two competing products. (7) Second-period buyers infer quality from the observable quantity sold in the first period but do not directly observe first-period prices. Our model is instead designed to study the effect of monopoly pricing on informational cascades, and so it differs from theirs in a number of ways. First, in our model there is a single buyer in each period with a discrete signal about the binary value of the good, as in BHW. (8) Second, we consider the case of monopoly rather than duopoly. (9) Third, we assume that past prices are observed publicly in order to avoid the additional signal-jamming effect that they identify. (10)
This article also relates to the literature on learning and experimentation in markets, pioneered by Rothschild (1974) and further developed by Easley and Kiefer (1988) and Aghion et al. (1991), among others. As in those models, in our model the price charged by the monopolist affects what is learned about the demand curve. But while learning in those models is one-sided, in our model the seller as well as the buyers have the opportunity of becoming better informed about the quality of the good. The effect of current prices on future demand curves through learning is typically absent in those models, but it plays a key role in our model.
Our analysis of the effect of endogenous pricing on the outcomes of bilateral learning is in the spirit of Bergemann and Valimaki (1996, 2000). While they focused on situations in which ex post information (such as experience) about product quality is revealed publicly over time, in our model buyers make purchase decisions on the basis of pre-existing (ex ante) private information. (11) In Bergemann...
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