Imperfect durability and the Coase conjecture.

--. "Durable Goods Theory for Real World Markets." Journal of Economic Perspectives, Vol. 17 (2003), pp. 131-154.

(1) Our results also admit an alternative interpretation of the evidence. We show that a monopoly equilibrium always exists when players are sufficiently patient. Thus, when this condition holds, durable goods producers need not earn any less than their nondurable counterparts.

(2) We assume that a consumer's value of possessing an extra unit is zero when the first unit is still working. Nevertheless, consumers may still wish to hold more than one unit, so as to guard against the possibility that the good would cease to perform before the next trading opportunity arises. However, if the length of the time period is sufficiently small and the marginal cost of production is positive, consumers will never choose to do so. This is because the expected value of holding a second unit converges to zero as the length of the time period shrinks to zero. When the length of the time period is large, the expected value of holding a second unit may exceed the marginal cost of providing it. In this case, we assume that the storage costs for holding a second unit are infinite. Alternatively, we could treat the demand for the second unit as a demand for the first unit by a fictitious consumer.

(3) In this case, the monopolist's maximization problem in equation (3) below becomes R(x) = [max.sub.y[member of][x,1]] {P(y) (y - x) + [delta]R((1 - [mu]y)}.

(4) If the monopolist is not allowed to buy back, and no competitive second-hand market exists, the game would be much more complicated, as the state variable then becomes the distribution of current holdings, rather than just the size of current stock. Importantly, the qualitative nature of the equilibria changes, as they necessarily become cyclical whenever the good does not depreciate too quickly (Deneckere and Liang, 2005). However, the main message of the current article survives: when the depreciation rate is sufficiently large, monopoly is the unique equilibrium outcome, and when the depreciation rate is sufficiently low, the Coase Conjecture equilibrium is the unique equilibrium.

(5) Because T(x) is nondecreasing, there are at most a countable number of points for which T(x) is multivalued. It follows from the consumer indifference equation (4) that in equilibrium, in any period but the initial one, the monopolist must select t(y), even when T(y) is not single-valued (for details, see Ausubel and Deneckere, 1989).

(6) The left-continuity of t(x) follows from the upper hemi-continuity of T(x), the monotonicity of T(x), and the definition of t(x).

(7) Indeed, if the consumer does not currently hold the good, he must purchase it at a price P(q). Next period, if the good is still alive, which occurs with probability (1 - [mu]), he can sell it at a price P(t((1 - [mu])q)). If the consumer currently holds the good, then he could sell it at a price P(q), but by doing so would forego the opportunity of selling it next period at the price P(t((1 - [mu])q)), whenever the good is still alive. Thus, the right-hand side of (4) still represents the correct opportunity cost of holding the good for one period.

(8) In the nongeneric situation where T(0) is multiple valued, the monopolist may select randomly from the set P(T(0)).

9 We cannot have y = 0 as a steady state, because in any stationary equilibrium the monopolist can always guarantee herself a positive profit by charging the price [v.bar] in every period.

(10) Let h(x) = (1 - [mu])t(x); then the optimal trajectory of the depreciated stock is [{[h.sup.n](x)}.sup.[infinity].sub.n=0]. If h(x) > x, then the sequence {[h.sup.n](x)} is increasing, and if h(x) < x, then it is decreasing.

(11) The hairline case in which [v.bar] = [??] [bar.v] can be handled as a limit of the case [v.bar] < [??][bar.v].

(12) The intuition for why this is true is as follows. First note that if the acceptance price of type q [member of] ([??], [??] + [epsilon]) exceeded [v.bar], then q would expect to make capital gains by purchasing, that is, q would expect the price to increase next period. Second, because [??] is not a steady state, it must be that type [??]'s acceptance price is below [bar.v], implying that type [??] expects the price to go down next period. But this would mean that the price has to remain constant if the state is [??], which is impossible because [??] is not a steady state.

(13) Because P([??]) < [bar.v], equation (4) implies that P([??]) > P(t((1 - [mu])[??])), and so t((1 - [mu])[??]) > [??]. Because P(q) = [v.bar] for q > [??] we have t((1 - [mu])[??]) = 1.

(14) More precisely, there exists a unique stationary triplet supporting y = 1 as the unique steady state. Generically, the outcome path associated with this stationary triplet is unique. Nonuniqueness of the outcome path arises only if [[bar.x].sub.m] = 0, in which case the monopolist may randomize between selling to [[bar.y].sub.m-1], and [[bar.y].sub.m-2] in the initial period.

(15) However, in the reputational equilibrium of our model, the monopolist never finds it optimal to regain her reputation once it is lost: once the state moves beyond [y.sup.*] it moves forward until the second steady state [y.sub.s] = 1 is reached.

(16) See the definition of the sequence {[[bar.x].sup.i.sub.k]}, where [[bar.x].sup.i.sub.0] = (1 [mu])[y.sup.*.sub.1].

(17) See the definition of the sequence {[[??].sup.i.sub.k]}, where [[??].sup.i.sub.0] = (1 - [mu])[y.sup.*.sub.1].

(18) This allows c([lambda]) to be decreasing in [lambda], but at a rate sufficiently low that the user cost is minimized at [lambda] = 0.

(19) The monotonicity condition can be relaxed to the requirement that (r + [lambda])c([lambda].) is uniquely minimized at [lambda] = 0. In Proposition 1, we then need to require [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (r + [lambda])c([lambda]) > [bar.c], and in Proposition 2, we conclude that the monopolist selects [lambda] [greater than or equal to] [[lambda].sub.0].

(20) Specifically, [[lambda].sub.0] is the unique solution to [lambda] = r (b - (r + [lambda])c([lambda]))(1 - [??])/ [??](a - (r + [lambda])c([lambda])) - (b - (r + [lambda])c([lambda])).

(21) Under this condition, the monopolist prefers the Coase profits at [lambda] = 0 to the monopoly profits at [lambda] = [[lambda].sub.0]. More precisely, b - rc(0) > (a - (r + [[lambda].sub.0])c([[lambda].sub.0]))[??].

(22) For smooth demand curves, Theorem 5 (ii) generalizes to the requirement that all stationary equilibria must be close to the monopoly equilibrium when [mu] is sufficiently large. Bond and Samuelson's equilibrium satisfies this property. However, unlike in our monopoly equilibrium, their equilibrium approaches the competitive outcome when the time between offers is allowed to vanish, no matter how large the depreciation rate.

(23) The Markovian restriction is implied by Harsanyi and Selten's (1988) principle of subgame consistency, which holds that behavior in any subgame should only depend upon the structure of that subgame.

(24) As an additional illustration of this cutting power, note that the two-type model has at most three stationary equilibria but a continuum of nonstationary equilibria (when the length of the time period is sufficiently small).

Raymond Deneckere *

and

Meng-Yu Liang **

* University of Wisconsin-Madison; rjdeneck@wiscmail.wisc.edu.

** Academia Sinica; myliang@econ.sinica.edu.tw.