Durable goods with quality differentiation
Introduction
The paper spells out the dynamic problem faced by a monopolist who faces a market of fixed size with heterogeneous consumers and can each period adjust both the price and the quality of his goods. Consequently, we combine two important strands of the literature on price discrimination: Second-degree price discrimination with commitment to a single offer (cf. Mussa and Rosen, 1978, Maskin and Riley, 1984) and intertemporal price discrimination in the “durable good” commitment problem (cf. Coase, 1972, Stokey, 1981, Bulow, 1982, Fudenberg et al., 1985, Gul et al., 1986). In our model with low- and high-valuation consumers, we thus allow a monopolistic firm to offer each period a range of qualities. Our key finding is that as the firm becomes sufficiently flexible in adjusting qualities and prices over time, it serves the whole market in the very first period, possibly even incurring a loss with the low-type consumers. Serving the low-type consumers by offering a low-quality version potentially at a loss in the first period represents a commitment that ensures higher profits from high-valuation consumers.
In the one-stage (second-degree price discrimination) commitment game, the firm would never sell below cost. (In fact, the virtual surplus with the low type must be strictly positive to ensure that the whole market is covered.) This feature clearly also holds in the standard durable-goods problem with only a single feasible quality level. (On the other hand, below-cost pricing also features in different contexts such as, for instance, in the presence of network externalities as in Farrell and Saloner, 1986 or with price wars as in the recent contribution by Marx and Shaffer, 2000.) In the standard durable-goods model, while the real time in which the market is served goes to zero as the time between periods shrinks, the number of periods it takes to clear the market also increases.
Other papers have also looked at the interaction of time-inconsistency problems and price discrimination. Some papers consider the successive introduction of goods of different qualities, e.g., as a sequence of upgrades (e.g., Villas-Boas, 1999). Typically, in these models the market operates for only two periods. A two-period version with quality differences is also considered in Takeyama (2002), which also generates below-cost pricing but cannot offer a comparison with the standard durable-goods case.1 An open time horizon is considered in a (menu) bargaining game by Wang (1998). As he only allows for a very specific payoff function, for which the virtual surplus is always strictly positive, the game always ends in the first period, with an outcome as under the optimal commitment mechanism.
Section snippets
The model
We consider a market for an infinitely durable and indivisible good. There is a continuum of mass one of infinitely-lived consumers. Consumers come in two types, t ∈ T = {1, 2} with respective masses qt0 > 0. We call consumers of type t = 2 the ‘high-valuation’ segment and consumers of type t = 1 the ‘low-valuation’ segment. Consumers want to buy at most a single good. The market is served by a monopolistic firm, which chooses prices and qualities. Quality is measured by a real-valued variable x. If a
Analysis
We state next our main result. Proposition There exists a unique equilibrium in which the durable-goods monopolist serves the whole market in the first period by offering the range of qualities and prices {(xtc, ptc)}t∈T characterized in Eq. (1) if either of the following conditions holds: the initial fraction of high-valuation consumers falls below the threshold υ¯, or, regardless of the composition of the market, the real time between two consecutive periods, z, becomes sufficiently small.
If the fraction of
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