Imperfect durability and the Coase conjecture.

For a monopoly equilibrium to exist, it must be the case that when the state is x = (1 - [mu])[??], the monopolist prefers continuing to serve this replacement demand at the monopoly price P([??]) = [bar.v] to penetrating the whole market by charging P(1) and continuing by serving the larger replacement demand at the price P(1) forever, that is, [mu][??][bar.v] / 1 - [delta] [greater than or equal to] (1 - (1 - [mu])[??])P(1) + [delta][mu]P(1) / 1 - [delta]. Because P(1) [greater than or equal to] [v.bar], a necessary condition for a monopoly equilibrium to exist is that [mu] [greater than or equal to] [[mu].bar], where [[mu].bar] solves the equation [mu][??][bar.v] / 1 - [delta] = (1 - (1 - [mu])[??])[v.bar] + [delta][mu][v.bar] / 1 - [delta]. We prove that the condition [mu] [greater than or equal to] [[mu].bar] is also sufficient for a monopoly equilibrium to exist. Hence we have the following.

[FIGURE 2 OMITTED]

Theorem 2. There is at most one monopoly equilibrium. This equilibrium exists if and only if [mu] [greater than or equal to] [[mu].bar].

The proof of sufficiency is rather subtle. We construct a strictly increasing sequence of states [{[??].sub.k}.sup.m+1.sub.k=0] starting at [[??].sub.0] = (1 - [mu])[??] and ending at [[??].sub.m+1] [less than or equal to] 1 - [mu], such that when k [member of] {1, ..., m} and the state is [[??].sub.k], the monopolist is indifferent between bringing the next period state to [[??].sub.k-1] < [[??].sub.k] and staying at [[??].sub.k] forever. This construction is illustrated in Figure 2: for y [member of] ([[??].sub.k-1], [[??].sub.k]], we have P(y) = [[??].sub.k] and t((1 - [mu])y) = [??].sub.k-1]. The backward arrows indicate that when the state is y the state moves backward, that is, t((1 - [mu])y) < y. Note that for every state x [member of] ((1 - [mu])[??], (1 - [mu])], it will take at most m + 1 periods for the state to return to [??].sub.0] = (1 - [mu])[??]. Upon reaching [[??].sub.0], the monopolist then charges the monopoly price forever after.

If [mu] is sufficiently large, the monopolist will return to the monopoly steady state from any state above (1 - [mu])[??],so [??].sub.m+1] = 1 - [mu]. In this case there is a unique steady state, and buyers in the interval ([??], 1] purchase at a price exceeding their valuation, in order to make capital gains by reselling their good in the second-hand market at some later date. This is illustrated in the left panel of Figure 2. If [mu] falls below this threshold, then when the state equals [[??].sub.m+1], the monopolist is indifferent between going to [[??].sub.m] and fully penetrating the market by charging the price [v.bar], and remains there forever after. In this case [??].sub.m+1] < 1 - [mu] and [y.sub.s] = 1 is a second steady state. This is illustrated in the right panel of Figure 2.

In a monopoly equilibrium, from any state exceeding (1 - [mu])[??], it must take real time for the price to rise to the monopoly price. Letting [??] = [lim.sub.z[right arrow]0] [[??].sub.m(z)+1], this means that over the interval ([??], [??]], the state must move back slowly to [??]. This explains the next corollary.

Corollary 2. In the limit as the length of the time period z approaches zero, the monopoly equilibrium acceptance price converges to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Furthermore, when x < [??] the monopolist sells ([??] - x), when x > [??] she selects (1 - x), and when x [member of] ([??], [??]) she selects [??] = [lambda]x[(1 - [v.bar] / [bar.v] ([??] / q).sup.[lambda]+r / [lambda]).

[FIGURE 3 OMITTED]

[] The reputational equilibrium. Finally, we consider equilibria in which [y.sup.*] [less than or equal to] [??] is a steady state. In such equilibria, starting from the initial state x = 0 the monopolist charges the price [bar.v] in every period. She sells to all consumers y [less than or equal to] [y.sup.*] in the first period, and provides for the replacement demand [mu][y.sup.*] thereafter. In reputational equilibria, the seller's profits therefore fall short of monopoly profits. If the seller were to move the state beyond [y.sup.*] in an attempt to penetrate the market further, she would have to lower her price drastically. This is because for states y beyond [y.sup.*], play occurs according to the Coase Conjecture equilibrium. Consumers y > [y.sup.*] therefore expect future prices to be low, and hence are only willing to accept prices close to [v.bar]. In effect, by cutting the price, the monopolist "loses her reputation" for pricing high. The reputational equilibrium is illustrated in Figure 3.

However, unlike the equilibria in Ausubel and Deneckere (1989), the equilibria we construct here are stationary. In particular, because depreciation of the stock allows the state to move backward over time, the monopolist always has the ability to regain her reputation for toughness by keeping sales low over a length of real time sufficient to bring the state back below [y.sup.*]. (15) Also, unlike in the nonstationary equilibria of Ausubel and Deneckere (1989), the seller's profits are a continuous function of the state.

Continuity implies that when the state is (1 - [mu])[y.sup.*], the monopolist's replacement profits [mu][y.sup.*][bar.v] / 1 - [delta] are equal to the Coase Conjecture profits at [y.sup.*]. Because the former are increasing in [y.sup.*] and the latter are decreasing in [y.sup.*], we may conclude that for every value of [mu] there is at most one [y.sup.*] [less than or equal to] [??] such that [y.sup.*] is a steady state. Hence, we have the following:

Theorem 3. There is at most one reputational equilibrium. This equilibrium exists if and only if [[mu].bar] < [mu] [less than or equal to] [[bar.[mu].

The reason that a reputational equilibrium exists only when a monopoly equilibrium exists is that the profits in the steady state of a reputational equilibrium fall short of the steady-state profits in a monopoly equilibrium. Because the temptation to expand sales beyond the steady state (and thereby earn Coase Conjecture profits) is larger in the reputational equilibrium, a reputational equilibrium cannot exist when a monopoly equilibrium fails to exist. From Corollary 1, we conclude:

Corollary 3. Let [y.sup.*] = [lambda][v.bar] + r[v.bar] / [lambda][bar.v] + r[v.bar]. Then, in the limit as the length of the time period z approaches zero, the reputational equilibrium acceptance price converges to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

[FIGURE 4 OMITTED]

[] Coexistence of stationary equilibria. We now turn to the question of when stationary equilibria are unique and when they are not. To settle the issue, we first need an auxiliary result.

Lemma 3. The functions [bar.[mu]]([delta]) and[[mu].bar]([delta]) are decreasing, with [bar.[mu]](0) = [[mu].bar](0) = (1 - [??])[v.bar]/([bar.v] - [v.bar])[??], [lim.sub.[delta][right arrow]1] [[mu].bar]([delta]) = 0, and [lim.sub.[delta][right arrow]1] [bar.[mu]]([delta]) > 0.

Figure 4 illustrates the situation. At [delta] = 0, a monopoly equilibrium cannot coexist with a Coase Conjecture equilibrium. For any [delta] > 0, the Coase Conjecture equilibrium is the unique equilibrium when [mu] < [[mu].bar]([delta]). For [mu] [member of] [[[mu].bar]([delta]), [bar.[mu]]([delta])], all three equilibrium types coexist. Finally, for [mu] > [bar.[mu]]([delta]), there is only a monopoly equilibrium. The economic force behind this result is that as the depreciation factor increases, staying with monopoly replacement profits becomes more and more attractive relative to further market penetration.

Another way to look at Figure 4 is that for any given value of [mu], a monopoly equilibrium always exists for sufficiently large [delta]. This is because as long as [mu] > 0, no matter how small, the difference in net present value of replacement profits at y = [??] and at y = 1 overshadows the one-time gain in capturing additional customers. We also see that when the monopolist discounts the future less, a Coase Conjecture equilibrium is less likely to exist. This is because the acceptance price at [??], P([??]) = (1 - [rho])[bar.v] + [rho][v.bar] is increasing in [delta], so staying at [??] becomes more attractive relative to further market penetration.

Figure 4 also reveals what happens when the length of the time period is allowed to vanish. As z converges to 0 the point ([delta], [mu]) then approaches the upper left corner of Figure 4 for all values of r > 0 and [lambda] < [infinity]. Because [mu]([delta]) = 1 - [e.sup.-[lambda]z] = 1 - [[delta].sup.[lambda]/r], the function [mu]([delta]) will lie below the graph of [[mu].bar]([delta]) when [lambda]/r is small, and above it otherwise. The cutoff value [[lambda].sub.0] below which the Coase equilibrium is the unique equilibrium at z = 0 is given by [[lambda].sub.0] = r (1 - [??])[v.bar]/[bar.v][??] - [v.bar]. We summarize the previous discussion by the following:

[ILLUSTRATION OMITTED]

Corollary 4

(i) Fix any [mu] > [bar.[mu]](1) . Then for sufficiently low r the monopoly equilibrium is the unique equilibrium.

(ii) Let [[lambda].sub.0] = r (1-[??])[v.bar]/[bar.v][??]-[v.bar]. Then in the limit as the length of the time interval converges to zero, a Coase Conjecture equilibrium exists for all values of [lambda] < [infinity]. When [lambda] < [[lambda].sub.0] it is the unique equilibrium. When [lambda] > [[lambda].sub.0] all three types of equilibria coexist.