Optimal Mechanism for the Sale of a Durable Good

Abstract

We show that posted prices are the optimal mechanism to sell a durable good to a privately informed buyer when the seller has limited commitment in an infinite horizon setting. We provide a methodology for mechanism design with limited commitment and transferable utility. Whereas in the case of commitment, subject to the buyer's truthtelling and participation constraints, the seller's problem is a decision problem, in the case of limited commitment, the seller's problem corresponds to an intrapersonal game, where different "incarnations" of the seller represent the different beliefs he may have about the buyer's valuation. Pollak (1968). Our contribution to this literature is to show that optimal mechanisms under limited commitment can be obtained as a solution to an intrapersonal game, even when both the seller and the buyer have time consistent preferences. In this way, we extend the range of applications of this literature, which has already seen applications ranging from growth and development (Bernheim et al. (2015)) to saving decisions (Harris and Laibson (2001)).Organization The rest of the paper is organized as follows. Section 2 describes the model; Section 2.1 summarizes the results in Doval and Skreta (2018) used to simplify the analysis that follows. Section 3 formally states the main result and describes the main steps in the proof of Theorem 1. Section 3.1 derives the recursive formulation that is the basis of the intrapersonal game; Section 3.2 characterizes the unique equilibrium of the intrapersonal game; Section 3.3 uses the intrapersonal equilibrium to build a PBE assessment that delivers the revenue-maximizing PBE. Section 4 concludes. ModelPrimitives: Two players, a seller and a buyer, interact over infinitely many periods. The seller owns one unit of a durable good to which he attaches value 0. The buyer has private information: before her interaction with the seller starts, she observes her valuation v ∈ {v L , v H } ≡ V, with 0 ≤ v L < v H . Let µ 0 denote the probability that the buyer's valuation is v H at the beginning of the game. In what follows, we denote by ∆(V) the set of distributions on V.An allocation in period t is a pair (q, x) ∈ {0, 1} × R, where q indicates whether the good is traded (q = 1) or not (q = 0), and x is a payment from the buyer to the seller. The game ends the first time the good is traded.Payoffs are as follows. If in period t, the allocation is (q, x), the flow payoffs are u B (q, x, v) = vq − x and u S (q, x) = x for the buyer and the seller, respectively. The seller and the buyer maximize the expected discounted sum of flow payoffs. They share a common discount factor δ ∈ (0, 1). Mechanisms:To introduce the timing of the game, we first define the action space of the seller in each period. In each period, the seller offers the buyer a mech-3. µ(h t ) is derived via Bayes' rule where possible (see Definition 4 in Section A.1). 10 9 While there is no output message when the buyer does not participate in the mechanism, we denote this by s = ∅ to keep the length...

Mechanism Design With Limited Commitment

“…That beliefs in the support of the mechanism are determined via Bayes' rule has one practical implication that we exploit throughout our analysis of Example 1 and in our concurrent work (Doval and Skreta (2020, 2021)): the mechanism's disclosure rule together with the principal's belief about the agent's type induce a Bayes' plausible distribution over posteriors. As a consequence, we can apply tools from information design to derive qualitative properties of the principal's problem (see Proposition 1).…”

“…However, other aspects of our analysis may not carry immediately to the analysis of renegotiation: indeed, Proposition B.1 uncovered that PBE outcomes have a recursive structure. In games with time‐separable payoffs, Proposition B.1 implies PBE payoffs can be characterized by relying on self‐generating techniques as in Abreu, Pearce, and Stacchetti (1990) and Athey and Bagwell (2008), reducing the analysis of a complex dynamic game essentially to a series of static problems (see Doval and Skreta (2020) for an application).…”

“…This avoids having to consider complicated mixed strategies on the part of the agent (see Laffont and Tirole (1988), Bester and Strausz (2001)) and transforms it instead into a program that combines elements of mechanism design and information design. Indeed, we exploit the information design elements to derive properties of the mechanisms (see Proposition 1 and our companion work, Doval and Skreta (2020, 2021)).…”

“…In games with time‐separable payoffs, public PBE payoffs are amenable to self‐generation techniques, as in Abreu, Pearce, and Stacchetti (1990). See Doval and Skreta (2020).…”

Storable Good Market With Intertemporal Cost Variations

“…The commitment issue of a durable good monopolist has also been addressed from a mechanism design perspective. In a setting where a seller of a durable good faces a privately informed buyer, Doval and Skreta (2019) characterize the revenue-maximizing equilibrium when the seller cannot commit to the mechanism offered to the buyer in the case of no trade.…”