Imperfect durability and the Coase conjecture.

The intuition for why a Coase Conjecture equilibrium always exists when the period length is allowed to vanish, no matter how large the depreciation rate [lambda], is as follows. If the monopolist instantly saturates the market with the competitive output, consumers will be willing to pay no more than the competitive price; with such expectations, the monopolist in turn cannot do any better than instantly saturating the market. However, Corollary 4 reveals that in order for Coase's conjecture to hold in a world of imperfect durability, it no longer suffices to let the discount factor between periods converge to one. Unlike in a world of perfect durability, it matters whether this is accomplished by letting the period length vanish or whether this is accomplished by letting players become infinitely patient. In the first case, replacement demand becomes very small in any given period, whereas in the second case replacement demand can be substantial in a period. In order for the Coase Conjecture to hold in some stationary equilibrium, the monopolist must be able to revise his price frequently enough over any real length of time. Even when this is the case, however, a monopoly equilibrium (and a reputational equilibrium) will still exist when the depreciation rate [lambda] is sufficiently high. For example, when [v.bar]/[bar.v] = .6, [??] = .8, and [delta] = .95, then [[mu].bar]([delta]) = 2.9% and [bar.[mu]]([delta]) = 31.8%. Thus, when the real interest rate is 5% per year, the monopoly equilibrium will exist if the turnover is less than 34 years, and will be the unique equilibrium if the turnover is less than 3 years. Figure 5 illustrates the equilibrium profits for these parameter values as a function of the depreciation rate [mu].

5. The N-step case

* In this section, we analyze general "neoclassical" demand functions, for which buyers' valuations take on a finite number of values [v.sub.1] > [v.sub.2] > ... > [v.sub.N] > 0. Thus, letting 0 = [q.sub.0] < [q.sub.1]

< ... < [q.sub.N] = 1, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

We discuss below how the techniques used to analyze the two-step demand ease can be applied to this larger class, and argue that the qualitative properties we discovered for this ease are general.

Lemma 1 continues to hold as stated: in any stationary equilibrium, there exists at least one steady state [y.sub.s] > 0, and P([y].sub.s]) = f([y.sub.s]). Furthermore, the unique construction of a stationary triplet {P, R, t} emanating from a steady state remains valid. For any stationary equilibrium, if the acceptance price is below the demand price near the steady state, the expectation of a drop in the future price has to be confirmed by moving the state variable forward. We call this a declining price path. On the other hand, if the acceptance price is above the demand price near the steady state, the expectation of an increase in the future price has to be confirmed by moving the state variable backward. We call this a rising price path. Thus, for generic parameter values, only a declining price path can reach a steady state from the left. However, to the right hand of a steady state, there are two possibilities. Either there is a declining price path that reaches the next steady state, in which case the equilibrium is supported by concerns for reputation, or as in the monopoly equilibrium for the two-step ease, there is a rising price path that reaches the steady state from the right. Note that compared to the two-step case, there may now be more than two (but no more than N) steady states in a stationary equilibrium.

A typical equilibrium therefore has the following structure. There are J [greater than or equal to] 1 steady states, 0 < [y.sup.*.sub.1] < [y.sup.*.sub.2] < ... < [y.sup.*.sub.j]. To the left of the smallest steady state, the acceptance function lies everywhere (weakly) below the demand function. Letting h(y) = t((1 - [mu])y), the monopolist then reaches the steady state [y.sup.*.sub.1] in [m.sub.1] < [infinity] steps, by successively selecting [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (the construction of the stationary equilibrium for states y [less than or equal to] [y.sup.*.sub.1] is described in detail in the proof of Theorem 4). (16) If [y.sup.*.sub.1] is supported by reputational concerns (which is the case whenever [y.sup.*.sub.1] [member of] ([q.sub.k-1], [q.sub.k]) for some k), then whenever the initial state x = (1 - [mu])y lies in the interval ((1 - [mu])[y.sup.*.sub.1], (1 - [mu])[y.sup.*.sub.2], the monopolist reaches the steady state [y.sup.*.sub.2] in m [less than or equal to] [m.sub.2] < [infinity] steps, by successively selecting h(y) < [h.sup.2](y) < ... < [h.sup.m](y) = [y.sup.*.sub.2]. Furthermore, at x = (1 [mu])[y.sup.*.sub.1], the monopolist is indifferent between serving replacement demand at [y.sup.*.sub.1] forever and selling to h([y.sup.*.sub.1] + [epsilon]) and continuing optimally thereafter. If [y.sup.*.sub.1] = [q.sub.k] for some k < N, then there exists [??] [member of] [(1 - [mu])[y.sup.*.sub.1] (1 - [mu])[y.sup.*.sub.2] such that when the initial state x (1 - [mu])y [member of] ((1 [mu])[y.sup.*.sub.1], [??], the monopolist reaches the steady state [y.sup.*.sub.1] in [m'.sub.2] < [infinity] steps, by selling to [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] = [y.sup.*.sub.1], so market penetration decreases over time (the construction of the stationary equilibrium for states y [member of] ([y.sup.*.sub.1], [??]] is described in detail in the proof of Theorem 4). (17) When x [member of] ([??], (1 - [mu])[y.sup.*.sub.2]], the monopolist reaches [y.sup.*.sub.2] in a finite number of steps. For any j s.t. 2 [less than or equal to] j < J, and initial states in ((1 - [mu])[y.sup.*.sub.j], (1 - [mu])[y.sup.*.sub.j+1]], the equilibrium between [(1 - [mu])[y.sup.*.sub.j], (1 [mu])[y.sup.*.sub.j+1]) can again have one of two possible structures, depending upon whether or not [y.sup.*.sub.j] is a discontinuity point of f. Finally, for initial states x [member of] ((1 - [mu])[y.sup.*.sub.j], 1 - [mu]], the monopolist returns to the steady state [y.sup.*.sub.j] in a finite number of steps.

In the two-step case, we established the existence of a stationary equilibrium in an indirect way, by characterizing all values of the parameters ([delta], [mu]) for which each type of equilibrium exists. Because the number of possible equilibria is at least of order [2.sup.N], enumerating all possible equilibria for the N-step case quickly becomes unwieldy, so instead we construct a stationary equilibrium for any fixed set of parameter values. The equilibrium is constructed so that its acceptance function dominates the acceptance function in any other possible stationary equilibrium.

Theorem 4. Let f be any demand function taking on a finite number of values. Then for any 0 [less than or equal to] [delta] < 1 and [less than or equal to] [mu] < 1, there exists at least one stationary equilibrium.

From here on, suppose that there is a unique monopoly quantity [q.sup.*] on the demand curve f(*) (note that this will generically be the case). We define a stationary equilibrium to be a Coase Conjecture equilibrium if the smallest steady state [y.sup.*.sub.1] = 1. As in the two-type case, such an equilibrium is necessarily unique. We say that a stationary equilibrium is of the monopoly type if [y.sup.*.sub.1] = [q.sup.*]. In a monopoly-type equilibrium, starting from the initial state x = 0, the steady state [q.sup.*] is always reached in a single step. However, unlike in the case N = 2, several qualitatively distinct monopoly-type equilibria may now coexist for a given set of parameter values (these equilibria thus have different acceptance functions over the interval ([q.sup.*], 1]). Finally, we define a stationary equilibrium to be of the reputational type if [y.sup.*.sub.1] is not a discontinuity point of f(*).

Our next result shows that, just like in the two-step case, when the good is sufficiently durable only a Coase Conjecture equilibrium can exist, and when the good is sufficiently perishable only monopoly-type equilibria can exist. Furthermore, reputational equilibria with a steady state below the monopoly quantity always exist for intermediate values of the parameter [mu]. Importantly, as in the two-type case, if monopoly-type equilibria do not exist, the equilibrium outcome will never produce lower surplus than static monopoly surplus.

Showing that only a Coase Conjecture equilibrium can exist when durability is sufficiently high is straightforward, as all we need to rule out is the possibility of a rising price path in a right neighborhood of any potential steady state. For this, it suffices to check that a deviation to full market saturation always pays. However, showing that no equilibria other than the monopoly-type equilibria exist when durability is sufficiently low is complicated, for we need to establish that any declining price path reaching a steady state beyond the monopoly quantity [q.sup.*] cannot emanate from a state below (1 - [mu])[q.sup.*]. To establish the existence of reputational equilibria with [y.sup.*.sub.1] < [q.sup.*], we first show there exists a value of the depreciation rate [mu] = [[mu].sub.L] such that at [[mu].sub.L] there is a monopoly-type equilibrium in which h(y) > y for all y [member of] ([q.sup.*], [y.sup.*.sub.2]). We then show that this "borderline" reputational equilibrium becomes a reputational equilibrium when [mu] is perturbed upward.

Theorem 5

(i) For every [delta] < 1 there exists [bar.[mu]]([delta]) > 0 such that for [mu] [member of] [0, [[bar.[mu]]([delta])), the Coase Conjecture equilibrium is the unique stationary equilibrium.