Imperfect durability and the Coase conjecture.

* Our method of constructing stationary equilibria and proving existence must overcome several difficulties that are absent in the model with infinite durability. First, when there is depreciation, the state need not always move forward (and indeed, in some equilibria it does not: in some periods, the monopolist sells less than the amount the stock depreciates). Second, there may be multiple stationary equilibria that coexist for given values of the parameters, and even in a single equilibrium there may exist multiple steady states. Finally, the usual method of backward induction on the state fails because given a stationary equilibrium on a subset of states, the extension to a stationary equilibrium on the whole state space is not unique.

Our solution to these difficulties is to backward induct from a steady state, carefully keeping track of any multiplicities in continuation that may exist. Depending on the case, backward induction on the state leads us to extend forward in the state space or backward in the state space. To prove existence, we show that only three possible equilibria can arise, and characterize the parameter values for which each type occurs.

To backward induct, we first need to characterize the set of steady states that can be associated with any stationary equilibrium. Formally, a steady state is defined as a stock level [y.sub.s] > 0 satisfying t((1 - [mu])[y.sub.s]) = [y.sub.s]. (9) Our first result establishes that such a steady state always exists.

Lemma 1. In any stationary equilibrium, there exists at least one steady state, that is, there exists [y.sub.s] [member of] (0, 1] such that t((1 - [mu])[y.sub.s]) = [y.sub.s]. Furthermore, the steady-state price must satisfy P([y.sub.s]) = f ([y.sub.s]).

The intuition behind the existence of a steady state is that because the monopolist's objective function is supermodular, the policy function t(x) is nondecreasing. As a consequence, optimal trajectories must be monotone, which rules out cycles, and guarantees the existence of a steady state. (10)

When selling a perfectly durable good, the monopolist cannot resist penetrating the market fully, resulting in a steady state [y.sub.s] = 1. Indeed, if there existed a steady state [y.sub.s] < 1, the monopolist could profitably deviate by cutting the price to [v.bar], thereby leading all remaining customers in the interval ([y.sub.s], 1] to purchase. However, when the good depreciates, the monopolist may choose not to fully penetrate the market, leading to the existence of a steady state [y.sub.s] < 1. The reason the monopolist may be willing to do this is that selling to replacement demand at a high price may be more profitable than cutting the price in an effort to increase sales.

Lemma 2 below establishes that there are only three possible steady states: [y.sub.s] = 1, [y.sub.s] = [??], and [y.sub.s] = [y.sup.*] (where [y.sup.*] [member of] (0, [??]) will turn out to be uniquely defined). This considerably narrows down our search for stationary equilibria. In particular, if [y.sub.s] = 1 is the smallest steady state, then we end up with an equilibrium satisfying the Coase Conjecture. If [y.sub.s] = [??] is the smallest steady state, then from the initial state x = 0, the monopoly steady state is reached. An equilibrium with this property will be referred to as a monopoly equilibrium. In a monopoly equilibrium, depending upon the depreciation rate [mu], there may or may not be an additional steady state at y = 1. However, we will later establish that only one of these two possibilities can occur for any given set of parameters. Finally, if [y.sub.s] = [y.sup.*] is the smallest steady state, we will obtain a reputational equilibrium.

Lemma 2. Consider any stationary equilibrium, and let S denote the associated set of steady states. Then one of the following holds:

(i) S = {1} (Coase Conjecture equilibrium);

(ii) S = {[??]} or S = {[??], 1} (monopoly-type equilibrium);

(iii) S = {[y.sup.*], 1}, for some [y.sup.*] [member of] (0, [??]) (reputational equilibrium).

The intuition for Lemma 2 is that in any given stationary equilibrium, the acceptance price at a steady state [y.sub.s] must be equal to f([y.sub.s]). Because there cannot exist two distinct steady states with the same acceptance price, and because the number of steps in f equals two, at most two steady states can coexist in equilibrium.

In the analysis of stationary equilibria below, we confine attention to the interesting case where [v.bar] < [??] [bar.v]. (11) When [v.bar] > [??] [bar.v] holds, there is a unique stationary equilibrium, in which the monopolist charges [v.bar] in every period. Indeed, the static monopoly price on the demand curve f(x) then equals [v.bar], so a monopolist with perfect commitment power would charge [v.bar] in every period. Because in any stationary equilibrium, consumers will accept the price [v.bar] (see Fudenberg, Levine, and Tirole, 1985), a monopolist without commitment power is then able to earn the commitment profits.

4. Characterization of stationary equilibria

* In this section, we characterize the stationary equilibria corresponding to the three possible configurations of steady states enumerated in Lemma 2. We show that for a given demand curve and fixed parameter values ([delta], [mu]), there is at most one equilibrium of each type. We also characterize the range of parameter values ([delta], [mu]) for which each type occurs.

[] The Coase Conjecture equilibrium. First, we consider equilibria in which [y.sub.s] = 1 is the unique steady state. In such equilibria, the acceptance price of any buyer type with valuation [v.bar] necessarily equals [v.bar]. (12) Furthermore, because [??] is not a steady state, we have t(x) = 1 for all x > (1 - [mu])[??] [equivalent to] [[bar.x].sub.1]. (13) Thus, the fact that [y.sub.s] = 1 is the unique steady state allows us to completely pin down the associated stationary triplet for x [member of] ([[bar.x].sub.1], 1 - [mu]] and y [member of] ([??], 1].

[FIGURE 1 OMITTED]

We then define a sequence of states [{[[bar.x].sub.k]}.sup.m+1.sub.k=2] such that when the state is [[bar.x].sub.k], the monopolist is indifferent between selling to [[bar.y].sub.k-1] = [[bar.x].sub.k-1] / 1 - [mu] at the price [p.sub.k-1], and to [[bar.y].sub.k-2] = [[bar.x].sub.k-2] / 1 - [mu] at the price [p.sub.k-2], where the sequence [{[p.sub.k]}.sup.m+1.sub.k=0] is defined such that the high-valuation consumer is indifferent between purchasing at [p.sub.k] now and waiting one period to purchase at [p.sub.k-1]. This construction is illustrated in Figure 1: for y [member of] ([[bar.y].sub.k], [[bar.y].sub.k-1]], we have P(y) = [p.sub.k-1] and t((1 - [mu])y) = [[bar.y].sub.k-2]. The arrows in the figure indicate at any state y the direction of movement of the state. As illustrated, in the Coase Conjecture equilibrium the movement is forward, that is, t((1 - [mu])y) > y.

Our proof establishes that a necessary and sufficient condition for a Coase Conjecture equilibrium to exist is that the sequence [{[[bar.x].sub.k]}.sup.m+1.sub.k=1] be strictly decreasing and satisfy [[bar.x].sub.m+1] < 0 [less than or equal to] [[bar.x].sub.m]. We also establish that this condition holds if (and only if) [mu] falls below a threshold level [bar.[mu]]. We therefore have the following.

Theorem 1. There is at most one Coase Conjecture equilibrium. (14) This equilibrium exists if and only if [mu] < [bar.[mu]], for some 0 < [bar.[mu]] < 1.

Intuitively, the reason why a Coase Conjecture equilibrium does not exist when [mu] is large is that when the state equals [[bar.x].sub.1], selling to replacement demand [mu][??] at the price P([??]) = [p.sub.1] = (1 - [rho])[bar.v] + [rho][v.bar] becomes preferable to fully penetrating the market and selling to replacement demand at the price [v.bar]. This is because both the replacement demand [mu][??] and the replacement price P([??]) are then large.

In a Coase Conjecture equilibrium, the seller fully penetrates the market in m periods. We show that m has a finite limit [??] as z approaches zero. Thus, the real time that elapses before the price drops to [v.bar] converges to zero as z approaches zero. Because a high-valuation consumer who purchases in the initial period has the option of postponing his purchase decision until time [??]z to purchase at price [v.bar], the initial price must be close to [v.bar]. Hence, we conclude:

Corollary 1. In the Coase Conjecture equilibrium, the initial price converges to the lowest buyer valuation [v.bar] as the length of the time period z vanishes.

[] The monopoly equilibrium. Next, we consider equilibria in which [y.sub.s] = [??] is a steady state. In such equilibria, the monopolist initially charges the static monopoly price [bar.v], selling to all consumers q [less than or equal to] [??]. In subsequent periods, she continues to charge the monopoly price. Hence, from then, on she serves only replacement demand, which equals [mu][??].