The effect of a finite time horizon in the durable good monopoly problem with atomic consumers

TL;DR

This paper investigates how finite time horizons affect the profits of a monopolist in durable goods markets with atomic consumers, showing profits are bounded and exploring equilibrium properties.

Contribution

It demonstrates that, unlike the infinite horizon case, duropoly profits with finite horizons are limited to at most double static monopoly profits, and analyzes equilibrium properties.

Findings

Duropoly profits are at most twice static monopoly profits in finite horizons.

Existence of equilibria not satisfying the skimming property for atomic consumers.

At least one profit-maximizing equilibrium with two periods satisfies the skimming property.

Abstract

A durable good is a long-lasting good that can be consumed repeatedly over time, and a duropolist is a monopolist in the market of a durable good. In 1972, Ronald Coase conjectured that a duropolist who lacks commitment power cannot sell the good above the competitive price if the time between periods approaches zero. Coase's counterintuitive conjecture was later proven by Gul et al. (1986) under an infinite time horizon model with non-atomic consumers. Remarkably, the situation changes dramatically for atomic consumers and an infinite time horizon. Bagnoli et al. (1989) showed the existence of a subgame-perfect Nash equilibrium where the duropolist extracts all the consumer surplus. Observe that, in these cases, duropoly profits are either arbitrarily smaller or arbitrarily larger than the corresponding static monopoly profits -- the profit a monopolist for an equivalent consumable good could generate. In this paper we show that the result of Bagnoli et al. (1989) is in fact driven by the infinite time horizon. Indeed, we prove that for finite time horizons and atomic agents, in any equilibrium satisfying the standard skimming property, duropoly profits are at most an additive factor more than static monopoly profits. In particular, duropoly profits are always at least static monopoly profits but never exceed twice the static monopoly profits. Finally we show that, for atomic consumers, equilibria may exist that do not satisfy the skimming property. For two time periods, we prove that amongst all equilibria that maximize duropoly profits, at least one of them satisfies the skimming property. We conjecture that this is true for any number of time period.