Imperfect durability and the Coase conjecture.

The article that is perhaps most closely related to the present work is Karp (1996). Karp considers a continuous time model, and shows that for any positive depreciation rate there exists a continuum of stationary equilibria, only one of which satisfies the Coase Conjecture. Our models are similar, and our results are complementary to Karp's. However, there are important differences. First, we analyze a discrete time model, and this allows us to uncover interesting results that are unobtainable in Karp's continuous time framework. For example, in our model, when players become arbitrarily patient (i.e., the discount rate r converges to zero) and the depreciation factor is sufficiently large, the monopoly outcome is the unique stationary equilibrium outcome. In contrast, Karp's model predicts that any stock below the competitive stock can be a steady state. Second, Karp's analysis is limited to strictly decreasing continuous demand functions, and hence explicitly rules out discontinuities and flat sections in the monopolist's demand function. Our article shows how these can be handled. Third, and most importantly, it is well known that the continuous time method can introduce spurious equilibria. For example, in the durable goods monopoly model with learning by doing analyzed by Deneckere and Liang (2006), the continuous time model has a continuum of qualitatively distinct equilibria, only one of which is the limit of the unique stationary equilibrium of the corresponding discrete time model. This raises the real possibility that many of the equilibria uncovered by Karp (1996) are an artifact of the continuous time approach. Indeed, our results imply that most of Karp's "strong Markov perfect equilibria" are fragile to small perturbations in the model. More precisely, there is a continuum of qualitatively distinct SMPE emanating from any steady state; we show that only one of them can be the limit of discrete time equilibria of arbitrarily fine step function approximations to the monopolist's demand function. Thus, we provide a method for determining which continuous time equilibria are reasonable. Finally, our results reveal that Karp's analysis failed to uncover an important class of equilibria, the class of "stationary reputational equilibria."

8. Conclusion

* In this article, we have developed a discrete time model of a monopolist selling a new imperfectly durable good. We characterized its stationary equilibria as a function of three parameters, the common discount rate of the seller and the consumers, the depreciation rate of the good, and the length of the time period between successive price changes. We have shown that for any given positive period length, there is a unique equilibrium outcome when the good is either sufficiently perishable or sufficiently durable. In the former case, the only equilibrium outcome is the monopoly outcome, whereas in the latter case the unique outcome is the Coase Conjecture outcome. For intermediate values of the depreciation rate, equilibria with steady states ranging from the competitive output to the monopoly output, and even below the monopoly output, can coexist. These results persist even if players become arbitrarily patient; however, the range of depreciation rates for which only a Coase Conjecture equilibrium exists shrinks to zero. In the limit, a monopoly equilibrium always exists, whereas a Coase Conjecture equilibrium exists only if the depreciation rate is sufficiently small. Taken together, these results show that the equilibrium outcome set varies continuously from the competitive outcome to the monopoly outcome as the depreciation rate varies from zero to infinity. Furthermore, when players are very patient, the monopoly outcome can be obtained unless the good has almost infinite durability.

To keep our model tractable, and facilitate comparison with the existing literature, we stuck with a traditional Wicksellian formulation of the durable goods market. In particular, we assumed that new and used durables are perfect substitutes and that there exists a perfect second-hand market. Whereas the second-hand market is never active along the equilibrium outcome path of a stationary equilibrium, it is active following off-equilibrium behavior of the monopolist. Thus, the operation of the second-hand market plays an important role in sustaining equilibrium behavior. For this reason, it would be interesting to extend our analysis to cases where the second-hand market operates imperfectly, either because of adverse selection (as in Hendel and Lizzeri, 1999a; Janssen and Roy, 2002; Johnson and Waldman, 2003; House and Leahy, 2004; Hendel, Lizzeri and Siniscalchi, 2005), or because of transaction costs (Konishi and Sandfort, 2002; Stolyarov, 2002). Relaxing the assumption of perfect substitutability between new and used durables (as in Rust, 1985; Waldman, 1996; or Hendel and Lizzeri, 1999b) might also be a fruitful avenue of research. The intuition put forth in our article strongly suggests that our results are robust to these variations, but a formal analysis must await future work.

Appendix

* Proof of Lemma 1. First, if [y.sub.s] is the stock after trade in a steady state, then P([y.sub.s]) = f([y.sub.s]). The reason is that in the steady state, we have t((1 - [mu])[y.sub.s]) = [y.sub.s]. Hence, the consumer's arbitrage equation f([y.sub.s]) P([y.sub.s]) = [rho][f([y.sub.s]) - P(t((1 - [mu])[y.sub.s]))] implies P([y.sub.s]) = f([y.sub.s]).

To prove that a steady state must exist, we first show that if P and f cross, then there always is a steady state. Then we establish that P and f necessarily cross.

We claim that if S = {q:P(q) = f(q)} is nonempty, then q' = max S is a steady state. To see this, observe that q' is the maximum state to have acceptance price P(q'). Indeed, if there existed q" > q' with P(q") = P(q'), then from the definition of q' we must have f(q") < f(q'). Furthermore, because t(.) is nondecreasing, and because P(.) is nonincreasing, we have P(t((1 - [mu])q")) [less than or equal to] P(t((1 - [mu])q')). This implies the contradiction P(q") = (1 - [rho])f(q") + [rho]P(t((1 - [mu])q")) < (1 - [rho])f(q') + [rho] P(t((1 - [mu])q')) = P(q'). Hence following the offer P(q'), all q [less than or equal to] q' accept and all q > q' reject. Now from P(q') = (1 - [rho])f(q') + [rho] P(t((1 - [mu])q')) and P(q') = f(q'), we have P(t((1 - [mu] )q')) = P(q').

Thus, when the state before trade is ( 1 - [mu])q', the monopolist's price P(t(( 1 - [mu])q')) = P(q') leads to a state after trade equal to q', that is, t((l - [mu])q') = q'.

Suppose now that there is a stationary equilibrium which does not have any steady state. We claim that this implies P(q) < [bar.v] for q [member of] [0, [??]] and P(q) > [v.bar] for q [member of] ([??], 1]. To see this, note that because there is no steady state, it follows from the previous paragraph that the set S is empty, that is, P(q) [not equal to] [bar.v] for any q [member of] [0, [??]] and P(q) [not equal to] [v.bar] for any q [member of] ([??], 1]. An argument similar to Fudenberg, Levine, and Tirole (1985) establishes that P(q) [greater than or equal to] [v.bar] for all q [member of] [0, 1]. Hence we necessarily have P(q) > [v.bar] for all q [member of] ([??], 1]. Furthermore, we cannot have P(q) > [bar.v] for some q [member of] [0, [??]]. Otherwise, because P(.) is nonincreasing, we would have P(0) > [bar.v]. But then P(0) = (1 - [rho])[bar.v] + [rho]P(t(0)) implies P(t(0)) > P(0). This is a contradiction, as t(0) [greater than or equal to] 0 and P(.) is a nonincreasing function.

Next, we show that P(q) < [bar.v] for q [member of] [0, [??]] and P(q) > [v.bar] for q [member of] ([??], 1] imply that the total stock is increasing for q [member of] [0, [??]] and decreasing for q [member of] ([??], 1], and that this yields a contradiction.

By the consumer's arbitrage equation, we have [bar.v] - P(q) = [rho][[bar.v] - P(t(1 - [mu])q)] for all q [member of] [0, [??]]. Became [rho] [member of] (0, 1) for all z > 0 and [bar.v] - P(q) > 0, we have [bar.v] - P(q) < [bar.v] - P(t(1 - [mu])q), which implies P(t((1 [mu])q)) < P(q) for all q [member of] [0, [??]]. Because P(*) is decreasing, we have t((l - [mu])q) > q for all q [member of] [0, [??]].

A similar argument also establishes that t((1 - [mu])([??] + [epsilon])) < [??] + [epsilon] for all [epsilon] [member of] (0, 1 - [??]]. Hence, [[lim.sub.[epsilon][right arrow]0] t((1 [mu])([??] + [epsilon])) [less than or equal to] [??] < t((1 - [mu])[??]). Because T(.) is upper hemi-continuous, [lim.sub.[epsilon][right arrow]0] t((1 - [mu])([??] + [epsilon])) [member of] T((1 - [mu])[??]. This contradicts the definition of t(*) = min T(*). Q.E.D

Lemma 4. If f(q) = f(q') and P(q) = P(q') for some q < q', then q cannot be a steady state.

Proof. If q were a steady state, then when the state after depreciation is (1 - [mu])q, the monopolist will charge P(q). All buyers in ((1 - [mu])q, q'] will accept this, contradicting that t((1 - [mu])q) = q. Q.E.D

Proof of Lemma 2. First, we show that y [member of] ([??], 1) cannot be a steady state. Suppose not; then Lemma 1 implies that P(y) = [v.bar]. Became P(*) is a decreasing function, we have P(q) [less than or equal to] [v.bar] for q [greater than or equal to] y. Similar to Fudenberg, Levine, and Tirole (1985), we have P(q) [greater than or equal to] [bar.v] for all q [member of] [0, 1]. Hence, P(q) = [v.bar] for all q [member of] [y, 1]. Lemma 4 then shows that y cannot be a steady state.