Imperfect durability and the Coase conjecture.

Surprisingly, the literature on monopoly in markets for new products with imperfect durability is relatively sparse, consisting mainly of contributions by Bond and Samuelson (1984, 1987), Sobel (1991), and Karp (1996). These papers overlap with ours to varying degrees, but none of them provide a complete analysis of the set of stationary equilibria. We defer a detailed discussion of the similarities and differences between the various models and their results to Section 7.

The rest of the article is organized as follows. Section 2 describes a two-type model where resale is allowed, and introduces the equilibrium concept. Section 3 characterizes the steady states of stationary equilibria. Section 4 describes the construction of the stationary equilibria, and delineates when each type of equilibrium exists. Section 5 generalizes the analysis to general N-step demand functions. Section 6 considers endogenous choice of durability. Section 7 discusses the related literature, and Section 8 concludes. Important proofs are relegated to the Appendix; the remaining proofs are available as an online Appendix (idv.sinica.edu.tw/mliang/papers.htm).

2. The model

* Consider a market for an imperfectly durable good which depreciates stochastically along a continuous time path, but is offered for sale at discrete points in time, spaced a length of time z > 0 apart. The durable good is indivisible and provides either full services or no service at all. The probability that the good is still working after a length of time t [member of] [R.sub.+] equals [e.sup.-[lambda]1]. Letting [mu] denote the probability that the good depreciates within one period, we therefore have

[mu] = 1 - [e.sup.-[lambda]z]. (1)

The market is populated by a continuum of infinitely lived consumers, indexed by q [member of] I = [0, 1]. All consumers are risk neutral and have the same discount rate r. Each consumer wishes to possess at most one unit of the durable good. (2) We assume that the flow benefit of the services consumer q derives from owning one unit of the durable good is described by the following inverse demand function:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Our reason for initially focusing on the two-type case is twofold. First, in order to construct stationary equilibria and analyze their properties, we use the method of "backward induction on the state", starting from any potential steady state. Unlike in the model without depreciation, this method only works for the finite-type case. In addition, the two-type model allows us to provide simple closed-form solutions for the equilibria, and to bring out the economics in the most transparent way possible. We analyze the general N-type case in Section 5.

Let f (q) denote consumer q's willingness to pay for the privilege of a one-time opportunity of acquiring one unit of the durable good. That is,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (2)

where [bar.v] = a / r + [lambda]] and [v.bar] = b / r + [lambda]. Thus, if the price at time t is p, then by purchasing or selling a unit of the durable good (and never transacting thereafter), consumer q can derive a net surplus of [e.sup.-rt] (f(q) - p) or [e.sup.-rt](p - f(q)), respectively.

A consumer is allowed to access the market as often as she wishes. Consumers seek to maximize the present value of their expected net surplus over all possible trading decisions, as a function of their holding status.

The market is served by a monopolist whose marginal cost of production, c, is constant and less than b / r + [lambda]. Without loss of generality, we normalize c to zero. The monopolist seeks to maximize the net expected present value of profits, using the same discount rate as consumers, r.

Sales occur at times t = 0, z, 2z, ..., nz, ..., and neither the monopolist nor consumers are allowed to trade at any time t [member of] (nz, (n + 1)z). We will refer to the time t = nz as "period n." The timing of play within each period is as follows. Before trade, the monopolist selects a price, p. Then consumers can trade (buy or sell) with the monopolist at the price/), or choose not to trade. After trade occurs, a time interval of length z elapses, after which play is repeated.

Note that the monopolist is allowed to repurchase some of the stock of the durable. However, in the class of stationary equilibria (to which the analysis in the article is confined), the monopolist always has net positive sales in every time period, even when the price is rising over time. Hence, all of our results hold for a model in which the monopolist is not allowed to buy back, and instead a competitive second-hand market is operative at trade time in every period. (3,4)

Following Gul, Sonnenschein, and Wilson (1986), we are interested in stationary equilibria of this game. For our context, we define a stationary equilibrium to be a subgame perfect equilibrium in which consumers' strategies in each period depend only on the current market price. Note that this implies that a consumer's decision on whether or not to hold the durable for the next period is unrelated to his current holding status. Thus, in a stationary equilibrium, a consumer's holding status after trade is described by a nonincreasing left-continuous acceptance function P(*), with consumer q choosing to hold a durable in the current period if and only if the monopolist's current price satisfies p [less than or equal to] P(q).

Because in our model the durable good depreciates randomly at the individual level, the set of remaining buyers before trade will (with probability one) not be an interval. However, stationarity implies that the set of remaining buyers after trade is an interval of the form (q, 1].

The acceptance function P(*) acts as a static demand curve for the monopolist, who faces a tradeoff between less intertemporal price discrimination and delaying trade. Let x [less than or equal to] 1 - [mu] denote the total stock in the market before trade, and for x [member of] [0, 1 - [mu]] let R(x) denote the monopolist's net present value of profits when the acceptance behavior of consumers is governed by P(x); then this tradeoff is captured by the dynamic programming equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (3)

where [delta] = [e.sup.-rz] is the discount factor between periods. To understand equation (3), note that when the monopolist brings the state after trade to y by charging the price P(y), then her net sales are given by y - x. The total stock in the market at the beginning of the next period will be (1 - [mu])y, because (by the law of large numbers) a fraction [mu] of the stocky will have depreciated. The monopolist then receives the continuation value R((1 - [mu])y)) with one period delay.

Let T(x) be the argmax correspondence in (3). By the generalized theorem of the maximum (Ausubel and Deneckere, 1993) and the contraction mapping theorem, there exists a unique continuous function R(x) satisfying (3), and T(*) is a nonempty and compact-valued correspondence. Furthermore, the supermodularity of the objective function in (3) implies that T(*) is a nonincreasing correspondence. Let t(x) = min T(x) denote the monopolist's equilibrium choice. (5) Note that t(*) : [0, 1 - [mu]] [right arrow] [R.sub.+] is a left-continuous nondecreasing function. (6) Let

[rho] = [delta](1 - [mu]).

In order for the consumer's acceptance function to be consistent with consumer optimization, the following indifference equation must hold:

f(q) - P(q) = [rho](f(q) - P(t((1 - [mu])q))), for all q [member of] [0, 1]. (4)

To interpret (4), observe that the above equation implies (1 - [rho]) f(q) = P(q) - [rho] P(t((1 - [mu])q)). The right-hand side of this equation is the consumer's cost of holding the durable for one period when the monopolist's current price is p(q). (7) Meanwhile, for any consumer q', the benefit of holding the good for one period is (1 - [rho])f (q'), which is monotone in q'. Thus, if q' < q consumer q' is willing to hold the good for one period, whereas if q' > q, consumer q' is willing to forego holding the good for one period.

The triplet {P(*), R(*), t(*)} completely describes a stationary equilibrium. A stationary equilibrium path has the following structure. In the initial period, the monopolist selects a price P([y.sub.0]) = P(t(0)) and all consumers q [less than or equal to] [y.sub.0] purchase. (8) At the beginning of the next period, a stock [x.sub.1] = (1 - [mu])[y.sub.0] remains, the monopolist selects a price P([y.sub.1]) = P(t([x.sub.1]])), and all consumers q [less than or equal to] [y.sub.0] decide to hold the durable (either by purchasing, or holding onto the unit they currently hold). This process continues until the stock reaches a steady-state level [y.sub.s], at which point the monopolist ceases any attempt to further penetrate the market, and continues by fulfilling the replacement demand [mu][y.sub.s].

3. Characterization of steady states