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Description
We generalize the standard repeated-games model of dynamic oligopolistic competition to allow for consumers who are long-lived and forward looking. Each period leaves some residual demand to future periods and pricing in one period affects consumers' expectations about future prices. We analyze this setting for an indivisible durable good with price-setting firms and overlapping cohorts of consumers. The model nests the repeated-game model and the Coasian durable-goods model as its two extreme cases. The analysis is mostly focused on constant-price collusion but conditions for collusive recurrent sales are also identified.
1. Introduction
* A central question in oligopoly theory is the viability of collusion: When can firms sustain prices above the competitive price? This question is often asked specifically for the monopoly outcome: When can firms sustain the monopoly price? The answer is well understood for the infinitely-repeated-game case, that is, when the firms face the same demand curve in each of infinitely many time periods. With identical price-setting firms that discount future profits by the same factor, the monopoly outcome--in which all firms set the monopoly price in each period--is a subgame-perfect equilibrium outcome in an infinitely repeated game, granted the discount factor is large enough, as indeed is posting any constant price above marginal cost. (1)
Another important--and complementary--question was posed by Coase (1972): Can a monopolist who introduces a new durable good sustain the monopoly price over time, when consumers are long-lived and forward looking? Coase's original insight, formalized by Stokey (1979, 1981) and Gul, Sonnenschein, and Wilson (1986), was that the monopolist is not able to sustain the monopoly price in such a setting because he would be tempted to offer a discount after the first period to reel in consumers who passed at the monopoly price. Consumers with valuations above the monopoly price would anticipate this and thus, if sufficiently patient, postpone their purchases beyond the first period to take advantage of that price cut. Indeed, it was shown that, as the time between offers goes to zero, all equilibrium outcomes converge to marginal-cost pricing. (2) Put differently, repeated-games oligopoly and the Coase monopoly model offer two extremes on the sustainability of monopoly prices--and the dramatic difference in outcomes is driven by the presence of long-lived and forward-looking consumers in the Coase model.
Two important subsequent papers, Gul (1987) and Ausubel and Deneckere (1987), brought collusion back to Coasian analysis. In particular, they studied an oligopoly with long-lived and forward-looking consumers, just as in the mentioned papers. They identified two forces supporting collusion--"competitor-induced discipline" and "consumer-induced discipline." Competitor-induced discipline is the standard punishment rife in repeated-game analysis--if an oligopoly firm drops its price today, competitors discipline it by significantly dropping their prices tomorrow and onward, thus washing out the benefit to the deviator of his first-period profit boost. Consumer-induced discipline is complementary; because forward-looking consumers anticipate competitor discipline from tomorrow onward, they do not rush to buy from a firm that has just undercut the market. Rather, they wait for an ensuing price war, thereby making even the first period of a deviation not very profitable for the deviator. Both of these forms of discipline make price cuts unattractive and hence support collusion) Consequently, collusive profits can be sustained in a Coasian oligopoly--and indeed profits arbitrarily close to the monopoly profit can be sustained as the discount factor goes to one. Does that mean Coase was wrong in believing that forward-looking consumers would rein in the monopoly outcome or, more generally, collusion? Is patience among forward-looking firms--high discount factor--all that is needed for collusion, regardless of consumer behavior? The point of this paper is to show that the answer is no; Coase was, in fact, right. Collusion is indeed harder to sustain when consumers are forward looking and long-lived.
Specifically, we study the following model--which nests at its two extremes the two canonical models mentioned above, the repeated-game model and the Coasian model. We want to emphasize, though, that while the Coase-conjecture literature is focused on the transient market for a new product, for which our model also allows, we are here primarily concerned with the steady-state market conditions for a product that has already been around for a long time. In every one of infinitely many periods, the demand emanates from two subpopulations--"newborns" or young, who constitute a proportion [beta], and "past-borns" or old, who constitute the remaining proportion, 1--[beta]. To keep the total population constant, the proportion [beta] of the population in a period "dies" before the next period begins. (4) The lower the birth-and-death or consumer turnover rate [beta] is, the more long-lived are the consumers; at [beta] = 1, each consumer lives only in one period, while their expected life span tends to infinity as [beta] [right arrow] (see Section 2). In each period, there is a market for a durable good that every consumer buys at most once during her lifetime. Consumers have an intrinsic valuation for the good and they differ in these valuations. All consumers are forward looking, i.e., whether young or old, time their purchase on the basis of (a) the current price, (b) their expectation of future prices and (c) their time preference. (5) The supply side of the market is made up of n identical price-setting firms with constant marginal cost and a common discount factor. When [beta] = 1, we thus have the standard repeated-game model of intertemporal Bertrand competition, with the population turning over every period, while [beta] = is the oligopolistic Coasian model, with no population turnover.
The central result of this paper is the following: the longer consumers live (the smaller [beta] is), the harder it is to sustain prices above marginal cost. In particular, we show that, fixing all other parameters, collusive monopoly pricing is not a subgame-perfect equilibrium in a steady state if the consumer turnover rate [beta] is below a certain positive cutoff. Hence, in a range of parameter values, including the limiting case of the Coasian model, the monopoly price cannot be sustained indefinitely. For certain combinations of values of the other parameters, collusive monopoly pricing is sustainable as a steady-state subgame perfect equilibrium for all consumer turnover rates that exceed the mentioned cutoff. (6) Indeed, the same qualitative results are valid for any collusive price [p.sup.*] above marginal cost. (7)
A natural question to ask is how this result squares with the Gul (1987) and Ausubel and Deneckere (1987) analyses. After all, these authors showed that consumer-induced discipline adds to competitor-induced discipline--the only discipline present in the standard repeated-game model--making deviation profits lower when there are forward-looking consumers. Indeed, we are able to show more along those lines in any steady state: we show that the deviation profits are increasing in [beta], being lowest in the Coase model and highest in the repeated-game model. In that sense, collusion would seem more likely in the Coasian model because there is less to be gained by deviating. The point, though, is that equilibrium profits are also increasing in [beta], i.e., the benefit of posting the monopoly price period after period, is increasing in [beta]. The profits from such pricing are lowest in the Coasian setting where consumers are infinitely lived and no new consumers enter the market. Whoever has a valuation above the monopoly price buys in the first period. Hence, if there are no price cuts, there is no more trade. By contrast, the profits from constantly posting the monopoly price are highest in the repeated-game setting because then every period consists entirely of a fresh cohort of consumers, some of whom have valuations above the monopoly price and thus buy at that price (when constant). As Coase correctly pointed out, although there is less to be gained from deviating when consumers are long-lived, there is also less to be gained by not deviating. What is not immediate--but turns out to be true in a wide range of cases--is that the on-equilibrium profits increase faster in the consumer turnover rate [beta] than the off-equilibrium profits. Hence, the result that collusion is sustainable if and only if [beta] is high enough. (8)
A second result of the paper is the following: Keep [beta] fixed and vary instead consumer patience. Consumer-induced discipline (see above) is greater, the greater is consumer patience. Fully impatient consumers will immediately buy and not wait for the price war to star, but patient consumers will wait. Hence, collusion is easier to sustain with greater consumer patience. Putting the two insights together, we obtain that collusion increases in consumer turnover and in consumer patience, being least in the Coasian world of zero turnover if, additionally, consumers are impatient, and being highest in the repeated-game world with patient consumers.
There is one other interesting consequence of having forward-looking consumers. In the standard Bertrand model, if a firm were to undercut a collusive price not exceeding the monopoly price, then it would do so by undercutting ever so marginally. The reason for that is that any price below the going price would guarantee the whole market for the deviant, and the industry revenue is increasing in prices below the going price (assuming that the industry revenue function is single peaked). In the current model, by contrast, the most profitable undercutting price may be substantially lower than the collusive price. Such significant undercutting can be profitable because it attracts past-born consumers with low valuations. Only a significant price cut can bring in such consumers because those among them who have not yet bought have valuations below the going price. (9) Because they represent a positive fraction of the potential buyers, a deviant firm, when the collusive price is at or near the monopoly price, will find it optimal to capture their demand. This is a third result of the paper.
The fourth result in this paper has to do with equilibrium sales. Firms in Conlisk, Gerstner, and Sobel (1984) and Sobel (1984, 1991) cannot resist dropping a collusive price, that is, have sales, because the residual demand from consumers with low valuations grows beyond any bound. By contrast, firms in our model can sustain the same collusive price in equilibrium. Indeed, most of our analysis is focused on such equilibria. However, we also show that, under certain circumstances, equilibrium sales are possible also in our model, that is, with a constant size of the consumer population. It may be profitable for the firms to now and then run a coordinated sale, in equilibrium, and thereby increase their profits above constant monopoly pricing. By assumption, such sales are anticipated by consumers with perfect foresight and will hence reduce profits in the periods just preceding the sale, but this may be compensated by the profits made during the sale because of the accumulated residual demand among old low-value consumers. During the sale, consumers correctly anticipate reversion to "normal" pricing next period, and hence have no reason to postpone their purchases. This contrasts sharply with the unanticipated price deviations mentioned above, where the demand facing the undercutting firm is dampened by consumers' anticipation of an ensuing price war. Such equilibrium sales can be viewed as a form of intertemporal price discrimination.
A fifth result has to do with optimal punishments. Because the market price in the first period after a unilateral price cut, if anticipated by consumers, will affect their demand in the defection period, the "punishment" of a unilateral price cut not only affects the defector's future profits but also its profit in the defection period itself. Because of this effect, absent in repeated games, punishments that give even weaker incentives to undercut than do grim-trigger strategies are possible if the marginal cost is positive; namely, to force the defector to price below marginal cost in the postdeviation period, thus bringing down the profit in the defection period below what it would have been under grim-trigger strategies. We identify and analyze a class of such "generalized trigger" strategies. Any constant collusive price that can be supported in subgame-perfect equilibrium can also be supported by subgame-perfect equilibria in such strategies, so this approach allows us to explore the full range of stationary subgame-perfect equilibrium outcomes.
Finally, a couple of modeling novelties might be worth remarking on. In each period, the newly arrived consumers' valuations are distributed according to some fixed cumulative distribution function, while the remaining old consumers are divided into two groups: those who already bought a unit and those who did not yet do so. The valuation distribution in the latter group depends on the history of prices and price expectations. The existing valuation distribution constitutes a state variable in a game played by the firms--and what we have, therefore, is a stochastic game and not a purely repeated one.
The bulk of our analysis is focused on the case of consumers with perfect foresight. However, this paper is not a plea that analysts should always assume all economic agents to have perfect foresight. We believe that consumers and firms may more realistically be modeled as having less than perfect foresight, and we indeed show how our model applies to consumers with behaviorally more plausible expectations. Our position is rather that the contrast in repeated-game models of dynamic oligopolistic competition between, on the one hand, the intertemporal substitution possibilities, sophistication and expectations coordination ascribed to firms and, on the other hand, the complete lack of intertemporal substitution ascribed to consumers, should be replaced by a milder contrast. Even taking a small step in this... |
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