Imperfect durability and the Coase conjecture.

(ii) For every [delta] < 1 there exists [bar.[mu]]([delta]) < 1 such that for all [mu] [member of] ([bar.[mu]]([delta]), 1], every stationary equilibrium is of the monopoly type.

(iii) For every [delta] < 1 there exists ([[mu].sub.L]([delta]), [[mu].sub.H]([delta])) [subset] ([[mu].bar]([delta]), [bar.[mu]]([delta])) such that for all [mu] [member of] ([[mu].sub.L]([delta]), [[mu].sub.H]([delta])), there is a reputational equilibrium whose smallest steady state falls below the monopoly quantity [q.sup.*].

(iv) If the monopoly quantity [q.sup.*] cannot be a steady state, then no q < [q.sup.*] can be a steady state either.

6. Endogenous choice of durability

* Suppose now that the monopolist has the ability to choose the durability of the product she sells. For simplicity, let demand be given by (2), and let production occur at a constant marginal cost c([lambda]) > 0, satisfying rc(0) < b. We assume that the rental cost associated with the depreciation rate [lambda], (r + [lambda])c([lambda]), is strictly increasing in [lambda]. (18,19) Because the flow benefit from the services of a unit of the durable good is constant, the social optimum requires [lambda] = 0, and because rc(0) < b, both consumer types are served in the social optimum.

As far as stationary equilibria are concerned, we will focus on the limiting case where z [right arrow] 0. A condition analogous to Corollary 4 (ii) then holds: there exists a unique value [[lambda].sub.0] such that the Coase Conjecture equilibrium is the unique equilibrium whenever [lambda] < [[lambda].sub.0], and such that all three types of equilibria coexist whenever [lambda] [greater than or equal to] [[lambda].sub.0]. (20)

First, let us consider the incentive to distort the level of durability in the Coase Conjecture equilibrium. To guarantee that this equilibrium is necessarily played, we assume that the user cost (r + [[lambda].sub.0])c([[lambda].sub.0) is sufficiently large that the monopolist will never select a depreciation rate [lambda] [greater than or equal to] [[lambda].sub.0]. Specifically, we require that (r + [[lambda].sub.0])c([[lambda].sub.0]) > [bar.c] [equivalent to] a[??]-b+rc(0)/[??] for all [lambda] [greater than or equal to] [[lambda].sub.0]. (21)

Proposition 1. Suppose that (r + [[lambda].sub.0])c([[lambda].sub.0]) > [bar.c]. Then in the limit as z [right arrow] 0 the monopolist selects the socially optimal level of durability.

Bulow (1986) argued that a monopoly seller of an infinitely durable good has an incentive to build in obsolescence. Indeed, the Coase Conjecture undermines the monopolist's implicit commitment power; reducing durability from the efficient level allows her to regain some of this commitment power. But Bulow failed to take this argument to its logical conclusion: when the length of time period becomes vanishingly small, according to Proposition 1 this distortion must disappear. The reason is as follows: when z [right arrow] 0, the Coase Conjecture forces the benefit the monopolist receives from any given reduction in durability to zero. Meanwhile, the monopolist's cost of providing the equilibrium stream of services (which, by the Coase Conjecture, is fixed at the competitive level) strictly increases.

Nevertheless, our results imply that there is scope for the monopolist to distort the level of durability below the socially optimal level when (r + [[lambda].sub.0])c([[lambda].sub.0]) < [bar.c].

Proposition 2. Suppose that (r + [[lambda].sub.0])c([[lambda].sub.0]) [bar.c], and that the monopoly equilibrium is played for all [lambda] [greater than or equal to] [[lambda].sub.0]. Then in the limit as z [right arrow] 0 the monopolist selects [lambda] = [[lambda].sub .0].

The impetus behind this "planned obsolescence" is quite different from Bulow's. In the Coase Conjecture equilibrium, when z [right arrow] 0, the durability of the good no longer affects the monopolist's ability to exercise her monopoly power. In contrast, by selecting [lambda] > [[lambda].sub.0], the monopolist can affect her ability to exercise monopoly power, even as z [right arrow] 0. Lowering durability allows the monopolist to credibly commit to a higher steady-state price, leading to a distortion in the choice of durability.

Let us now briefly address the incentive to distort durability when z > 0. The major difference with the limiting case is that now the monopolist is able to guarantee that the monopoly equilibrium is played, by selecting a level of durability equal to [mu]. That such a strategy can be profitable is illustrated in Figure 5. More generally, whenever players are sufficiently patient (i.e., for sufficiently small r), selecting [mu] = [bar.[mu]] will dominate selecting any [mu] that leads to a Coase Conjecture equilibrium.

7. Related literature

* Sobel (1991) considers a market for an infinitely durable good in which demand grows over time. Thus, like in our model, the monopolists' demand never dries up. However, if the monopolist were to keep on charging the static monopoly price forever, she would accumulate a backlog of low-valuation consumers that grows as time passes on. This backlog eventually becomes so large that it becomes attractive to lower the price. As a consequence, contrary to the depreciation case, all stationary equilibria of Sobel's model satisfy the Coase Conjecture.

Bond and Samuelson (1984) study a market with linear demand for an exponentially depreciating durable good. Using a procedure similar to Stokey (1981) and Sobel and Takahashi (1983), they construct a linear-quadratic equilibrium, and show that it satisfies the Coase Conjecture for all finite depreciation rates. Bond and Samuelson's analysis fails to uncover other stationary equilibria for their model, and therefore suggests that the limiting outcome must be the competitive outcome. Our article shows that this is not the case. (22)

Bond and Samuelson (1987) revisit the linear demand example, extending Ausubel and Deneckere's (1989) construction of reputational equilibria to markets for products with limited durability. These equilibria are qualitatively very different from the reputational equilibria discovered in the present article. All of our equilibria are stationary, whereas Bond and Samuelson's equilibria are nonstationary. This distinction is important, for several reasons. First, nonstationary equilibria rely on punishments for deviating from a prespecified outcome path. In contrast, in our reputational equilibria, the seller's profits are (Lipschitz) continuous in the state. Thus, if the seller deviates from equilibrium by expanding sales beyond the steady-state level, she receives no punishment. Second, stationary equilibria embody the Markovian restriction that players only pay attention to the portion of the history that is payoff relevant, which many authors regard as desirable. (23) Finally, nonstationary equilibria permit virtually any price path to arise in equilibrium when the length of the time period is sufficiently small, and hence have little predictive power. In contrast, the requirement of stationarity significantly reduces the size of the equilibrium set. To see this, note that for the case of a "gap" analyzed in the present article, the seller's profit in any stationary equilibrium is bounded below by the Coase Conjecture payoff, which is no lower than [v.sub.N]. In contrast, the worst nonstationary payoff converges to zero as the length of the time period shrinks to zero. (24)