Trading Dynamics with Private Buyer Signals in the Market for Lemons ∗ Ayc¸a Kaya† and Kyungmin Kim‡ October 2017

Abstract We present a dynamic model of trading under adverse selection in which a seller sequentially meets buyers, each of whom receives a noisy signal about the quality of the seller’s asset and offers a price. We fully characterize the equilibrium trading dynamics and show that buyers’ beliefs about the quality of the asset can either increase or decrease over time, depending on the initial level. This result demonstrates how the introduction of private buyer signals enriches the set of trading patterns that can be accommodated within the framework of dynamic adverse selection, thereby broadening its applicability. We also examine the economic effects of search frictions and the informativeness of buyers’ signals in our model and discuss the robustness of our main insights in multiple directions. JEL Classification Numbers: C73, C78, D82. Keywords: Adverse selection; market for lemons; inspection; time-on-the-market.

1

Introduction

Buyers often draw inferences about the quality of an asset from its duration on the market. In the real estate market, a long time on the market is typically interpreted as bad news (see, e.g., Tucker et al., 2013; Dube and Legros, 2016). This is arguably the reason why some sellers reset ∗

We thank Dimitri Vayanos and three anonymous referees for various insightful comments and suggestions. We are also grateful to Raphael Boleslavsky, Michael Choi, Mehmet Ekmekci, H¨ulya Eraslan, William Fuchs, Martin Gervais, Sambuddha Ghosh, Seungjin Han, Ilwoo Hwang, Philipp Kircher, Stephan Lauermann, Benjamin Lester, Qingmin Liu, Tymofiy Mylovanov, Luca Rigotti, Santanu Roy, Galina Vereschagina, G´abor Vir´ag, Bumin Yenmez, and Huseyin Yildirim for many helpful comments and suggestions. † University of Miami. Contact: [email protected] ‡ University of Miami. Contact: [email protected]

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their days on the market by relisting their properties without any major repairs or renovations.1 In the labor market, it is well-known that unemployment duration affects a worker’s reemployment probability and reservation wage (see, e.g., Imbens and Lynch, 2006; Shimer, 2008). This duration dependence is often attributed to the “non-employment stigma,” which refers to the phenomenon that employers interpret a long unemployment spell as a bad signal about the worker’s productivity and, therefore, are reluctant to hire such a worker. Intriguingly, despite their intuitive appeal, these types of negative inferences have not garnered clear empirical support, with mostly “mixed and controversial” evidence (Ljungqvist and Sargent, 1998).2 Furthermore, most dynamic models of adverse selection, which seem the most natural framework to address such an inference problem, generates the opposite prediction, that average quality increases over time.3 We present a dynamic model of adverse selection that generates multiple dynamic patterns of trade. In particular, in our model, delay can be either good news or bad news about the quality of an asset, depending on market conditions. The trading environment is a familiar one: a seller has private information about the quality of her indivisible asset, which can be either high or low. Buyers arrive sequentially, observe the seller’s time-on-the-market, and make price offers. Our innovation is to introduce private buyer signals into this canonical environment: each buyer receives a private and imperfectly informative signal about the quality of the asset. Notice that such signals are often available to potential buyers in real markets, as they can be generated by common (home) inspections or (job) interviews. We show that this simple and plausible innovation suffices to enrich the set of dynamic trading patterns that can be accommodated within the framework of dynamic adverse selection. In order to understand when, and why, delay is perceived as good news or bad news, notice that there are three sources of delay in our model. First, delay could be just because of search frictions, that is, a seller may have been unlucky and not met any buyer yet. If this is the main source for delay, buyers’ inferences about the quality of an asset should be independent of the seller’s time1 This is a common practice, but its harmful effects are well-recognized. Blanton (2005) compares it to “resetting the odometer on a used car.” The real estate listing service in Massachusetts decided to prevent the practice in 2006. 2 There is an agreement over the negative relationship between duration and unconditional job-finding probability. However, it is not clear whether it is due to “true” duration dependence or unobserved heterogeneity, that is, whether each individual’s performance is indeed affected by his/her duration or not. Much effort has been put in to separate “true” duration dependence from unobserved heterogeneity. See Heckman and Singer (1984) for a fundamental econometric problem. Recent studies utilize a natural experiment (e.g., Tucker et al., 2013) or a field experiment (e.g., Oberholzer-Gee, 2008; Kroft et al., 2013; Eriksson and Rooth, 2014) in order to circumvent the identification problem. 3 See Evans (1989); Vincent (1989, 1990); Janssen and Roy (2002); Deneckere and Liang (2006); H¨orner and Vieille (2009) for some seminal contributions. In all of these papers, average quality increases over time. Daley and Green (2012) consider a model in which public news about the quality accumulates over time. Due to noise in the (Brownian) news process, buyers’ beliefs about the quality fluctuate over time. However, the expected quality weakly rises over time for the same reason as in other (deterministic) models. Note that there are several other theories for duration dependence, including depreciating human capital models (e.g., Acemoglu, 1995), duration-based ranking models (e.g., Blanchard and Diamond, 1994), and varying search intensity models (e.g., Coles and Smith, 1998; Lentz and Tranaes, 2005).

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on-the-market. Second, delay might be caused by adverse selection. A high-quality seller, due to her higher reservation value, is more willing to wait for a high price than a low-quality seller. In this case, delay conveys good news about the quality of the asset. Finally, previous buyers might have decided not to purchase after observing an unfavorable attribute. If this is the main driving force, then delay would be interpreted as bad news and buyers get more pessimistic about the quality of the asset over time. Our model intertwines these three forces and consequently accommodates different forms of trading dynamics. We show that whether delay is good or bad news depends on an asset’s initial reputation (i.e., buyers’ prior beliefs about the quality of the asset). If an asset enjoys a rather high reputation initially, the asset’s reputation, conditional on no trade, declines over time, while if an asset starts out with a low reputation, then the asset’s reputation improves over time. To understand these opposing patterns, first note that the higher an asset’s reputation is, the more likely buyers are to offer a high price. This implies that while enjoying a high reputation, even a low-quality seller would have a strong incentive to hold out for a future chance of a high price and, therefore, be reluctant to accept a low price. In this case, trade can be delayed only when buyers are unwilling to offer a high price despite the asset’s high reputation, which is the case when they receive sufficiently unfavorable inspection outcomes. Since a low-quality asset is more likely to generate such inspection outcomes, the asset is deemed less likely to be of high quality, the longer it stays on the market. In the opposite case when an asset suffers from a low reputation, a low-quality seller would be willing to settle for a low price, while a high-quality seller would still insist on a high price in order to recoup his higher cost. Since a high-quality asset would stay on the market relatively longer than a low-quality asset, the asset’s reputation improves over time. This result contributes to the existing literature mainly in two ways. First, it broadens the applicability of the theory of dynamic adverse selection. As introduced at the beginning, delay is perceived as bad news in several markets. Our analysis offers a simple and natural mechanism through which such negative inferences arise in this framework. Second, it provides a potential resolution for mixed empirical results. Whether delay is good news or bad news depends on market conditions. Therefore, it is natural that different studies report different empirical results. This finding further suggests that it may be fruitful to shift the focus of empirical study from a general qualitative question (whether delay is good news or bad news) to more sophisticated and quantitative ones (such as what market factors affect buyer inferences under what conditions, as exemplified by Kroft et al. (2013) and Eriksson and Rooth (2014)). By incorporating multiple dynamic patterns of trade, our model creates a potential for obtaining new insights regarding the effects of certain policies or changes in the economic environment. Indeed, we show that in our model, the economic effects of increasing the informativeness of buyers’

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signals, which can be interpreted as enhancing asset transparency, are in general ambiguous.4 If the initial reputation is rather low and, therefore, buyers’ beliefs increase over time, then an increase in the informativeness of buyers’ signals speeds up trade and also increases seller surplus. However, if the asset’s reputation is initially rather high and declines over time, then the same change can slow down trade and be harmful to market participants. This latter result holds precisely because delay can be caused by a lack of good signal realizations and, therefore, bad news about the quality. If buyers’ signals become more informative, then delay becomes a stronger indication of low quality, which reduces later buyers’ incentives to offer a high price and, therefore, adversely affects trade. The role of search frictions in equilibrium trading dynamics also deserves elaboration. First, although search frictions are neutral to the direction of the evolution of beliefs, they affect the speed of the evolution. Buyers can never exclude the possibility that the seller has been so unfortunate that no buyer has contacted her yet. This forces buyers’ beliefs to change gradually. Second, they indirectly influence the direction of the evolution of beliefs through their impact on the equilibrium structure. In particular, a reduction in search frictions makes the decreasing pattern more prevalent: the threshold initial reputation level decreases as search frictions reduce. Finally, search frictions are responsible only for a portion of delay: even if search frictions are arbitrarily small, the expected time to trade remains bounded away from zero. This is similar to the persistence of delay in other models of dynamic adverse selection, but differs in that it holds despite the fact that each buyer generates a constant amount of information and, therefore, an arbitrarily large amount of exogenous information is instantaneously generated about the quality of the asset in the search-frictionless limit.

Related Literature Most existing studies on dynamic adverse selection focus on the implications of the difference in different types’ reservation values and, therefore, feature only increasing beliefs. One notable exception is Taylor (1999). He studies a two-period model in which the seller runs a second-price auction with a random number of buyers in each period and the winner conducts an inspection, which can generate a bad signal only when the quality is low. He considers several settings that differ in terms of the observability of first-period trading outcomes (in particular, inspection outcome and reservation (list) price history) by second-period buyers. In all settings, buyers assign a 4

It is common wisdom that asset (corporate) transparency improves market efficiency by facilitating socially desirable trade. Such beliefs have been reflected in recent government policies, such as the Sarbanes-Oxley Act passed in the aftermath of the Enron scandal and the Dodd-Frank Act passed in the aftermath of the recent financial crises, both of which include provisions for stricter disclosure requirements on the part of sellers. Presumably, the main goal of such policies is to help buyers assess the merits and risks of financial assets more accurately. This naturally corresponds to an increase in the informativeness of buyers’ signals in our model.

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lower probability to the high quality in the second period than in the first period (that is, buyers’ beliefs decline over time). Despite various differences in modeling, the logic behind the evolution of beliefs is similar to our declining beliefs case: trade occurs only when the winner receives a good signal, and the high type is more likely to generate a good signal than the low type. Therefore, the asset remaining in the second period is more likely to be the low type.5 However, the opposite form of trading dynamics (in which buyers’ beliefs increase over time) is absent in his model.6 In addition, he addresses various other economic problems, such as the dynamics of reserve (list) prices and the effects of the observability of first-period reserve price and inspection outcome, while we focus on better understanding equilibrium trading dynamics. Two papers consider an environment similar to ours. Lauermann and Wolinsky (2016) investigate the ability of prices to aggregate dispersed information in a setting where, just like in our model, an informed player (buyer in their model) faces an infinite sequence of uninformed players, each of whom receives a noisy signal about the informed player’s type. Zhu (2012) studies a similar model, interpreted as an over-the-counter market, with an additional feature that the informed player can contact only a finite number of uninformed players. In both studies, in contrast to our model, uninformed players have no access to the informed player’s trading history. In particular, uninformed players do not observe the informed player’s time-on-the-market. This induces uninformed players’ beliefs and strategies to be necessarily stationary (i.e., their beliefs do not evolve over time). To the contrary, the evolution of uninformed players’ beliefs and the resulting trading dynamics are the main focus of this paper. Daley and Green (2012) study the role of exogenous information (“news”) about the quality of an asset in a setting similar to ours. The most crucial difference from ours is that news is public information to all buyers. This implies that buyers do not face an inference problem regarding other buyers’ signals, making their trading dynamics distinct from ours. Similarly to us, they also explore the effects of increasing the quality of news and find that it is not always efficiencyimproving. However, the mechanism leading to the conclusion is different from ours. In particular, the negative effect of increased informativeness stems from the strengthening of buyers’ negative inferences about other buyers’ signals in our model, while in Daley and Green (2012), it is due to its impact on the incentive of the high-type seller to wait for good news. The rest of the paper is organized as follows. We formally introduce the model in Section 2 and 5

Prior to Taylor (1999), this “screening” mechanism was discussed by Vishwanath (1989) and Lockwood (1991). However, they do not investigate the working of the mechanism in a full-blown strategic setting: in Vishwanath (1989), (stochastic) price offers are exogenously generated, while in Lockwood (1991), trade takes place only at one price, which is equal to the reservation value common to all worker types. 6 This is due to his assumption that there are no gains from trade of a low-quality asset. In this case, buyers have no incentive to offer a price that can be accepted only by the low type, and thus the low type can not trade faster than the high type. In the online appendix (Section B), we consider the comparable case and show that the same result holds in our model.

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provide a full characterization in Section 3. We analyze the effects of changing the informativeness of buyers’ signals in Section 4 and study the role of search frictions in Section 5. In Section 6, we demonstrate the robustness of our main insights in three dimensions: the number of seller types, the bargaining protocol, and the market structure. In Section 7, we conclude by providing several empirical implications and suggesting some directions for future research.

2

The Model

2.1 Physical Environment A seller wishes to sell an indivisible asset. Time is continuous and indexed by t ∈ R+ . The time the seller comes to the market is normalized to 0. Potential buyers arrive sequentially according to a Poisson process of rate λ > 0. Upon arrival, each buyer receives a private signal about the quality of the asset and offers a price to the seller. If the seller accepts the price, then they trade and the game ends. Otherwise, the buyer leaves, while the seller waits for the next buyer. All players discount future payoffs at rate r > 0. The asset is either of low quality (L) or of high quality (H). If the asset is of quality a = L, H, then the seller obtains flow payoff rca during her possession of the asset, while a buyer, once he acquires it, receives flow payoff rva indefinitely. The asset is more valuable to all players when its quality is high than when it is low: cL < cH and vL < vH . In addition, there are always gains from trade: cL < vL and cH < vH . However, the quality of the asset is private information of the seller, and adverse selection is severe in the sense that there is no price that always ensures trade: vL < cH . Finally, it is commonly known that the asset is of high quality with probability qb at time 0.7 Each buyer’s signal s takes one of two values, l or h. For each a = L, H and s = l, h, we let γas denote the probability that each buyer receives signal s from the type-a asset. Without loss l h < γLl ), so that buyers assign a higher of generality, we assume that γH > γLh (equivalently, γH probability to the asset being of high quality when s = h than when s = l. We also assume that γas > 0 for any a = L, H and s = l, h, so that no signal perfectly reveals the underlying type of the l asset. See Section 4.1 for the limiting equilibrium outcomes as γLh or γH tends to 0. We assume that buyers observe (only) how long the asset has been up for sale (i.e., time t).8 In many models of dynamic adverse selection, attention is restricted to the case where qb is so small (e.g., qbvH + (1 − qb)vL < cH ) that some inefficiency (delay) is unavoidable, that is, it cannot be an equilibrium that trade always occurs with the first buyer. We do not impose such a restriction, because the decreasing dynamics, which is the novel outcome of this paper, emerges only when qb is not sufficiently small. As explained in Section 3, the exact condition differs from, but is related to, the familiar inequality between qbvH + (1 − qb)vL and cH . 8 There is a sizable literature that studies the role of uninformed players’ (buyers’) information about the history in dynamic games with incomplete information. For example, in a closely related model to ours (but without buyer 7

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This enables us to focus on our main economic question, namely the relationship between timeon-the-market and economic variables. It also has a notable technical advantage. For any t, there is a positive probability (e−λt ) that no buyer has arrived and trade has not occurred. This means that there are no off-equilibrium-path public histories and, therefore, buyers’ beliefs at any point in time can be derived through Bayes’ rule.

2.2 Strategies and Equilibrium The offer strategies of buyers are represented by a Lebesgue-measurable right-continuous function σB : R+ ×{l, h}×R+ → [0, 1], where σB (t, s, p) denotes the probability that the buyer who arrives at time t and receives signal s offers price p to the seller. The offer acceptance strategy of the seller is represented by a Lebesgue-measurable right-continuous function σS : {L, H} × R+ × R+ → [0, 1], where σS (a, t, p) denotes the probability that the type-a seller accepts price p at time t. An outcome of the game is a tuple (a, t, p), where a denotes the seller’s type, t represents the time of trade, and p is the transaction price. All agents are risk neutral. Given an outcome (a, t, p), the seller’s payoff is given by (1 − e−rt )ca + e−rt p. The buyer who trades with the seller receives va − p, while all other buyers obtain zero payoff. We study perfect Bayesian equilibria of this dynamic trading game. Let q(t) represent buyers’ beliefs that the seller who has not traded until t is the high type. In other words, q(t) is the belief held by the buyer who arrives at time t prior to his inspection. A tuple (σS , σB , q) is a perfect Bayesian equilibrium of the game if the following three conditions hold. (i) Buyer optimality: σB (t, s, p) > 0 only when p maximizes a buyer’s expected payoff conditional on signal s and time t, that is, s p ∈ argmaxp′ q(t)γH σS (H, t, p′ )(vH − p′ ) + (1 − q(t))γLs σS (L, t, p′ )(vL − p′ ).

(ii) Seller optimality: σS (a, t, p) > 0 only when p is weakly greater than the type-a seller’s continuation payoff at time t, that is, [ ] p ≥ Eτ,p′ (1 − e−r(τ −t) )ca + e−r(τ −t) p′ |a, t , signals), H¨orner and Vieille (2009) show that all seller types eventually trade if (rejected) prices remain private (i.e., not observable to future buyers), while some seller types never trade if prices are public. Fuchs et al. (2016) demonstrate that the result crucially depends on the market structure: with two seller types, if buyers are competitive in each period, then the private-offers case and the public-offers case yield the same trading outcome. Combined with search frictions, our assumption that only t is observable ensures that buyers’ beliefs move continuously and smoothly (see Lemma 8 in the appendix). If, for example, buyers observe the number of previous buyers, then buyers’ beliefs change discontinuously (i.e., jump upon each buyer arrival). It is easy to verify that our main insights regarding the evolution of buyers’ beliefs carry over to such a setting.

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where τ (≥ t) and p′ denote the random time and price, respectively, at which trade takes place according to the strategy profile (σS , σB ). (iii) Belief consistency: q(t) is derived through Bayes’ rule, that is, ∫

q(t) =

∫

t qbe−λ 0 [

t qbe−λ 0 [

∑ s

∑ s

∫ s γH ( σB (x,s,p)σS (H,x,p)dp)]dx

∫ s γH ( σB (x,s,p)σS (H,x,p)dp)]dx

∫

t + (1 − qb)e−λ 0 [

∑ s

. ∫ s γL ( σB (x,s,p)σS (L,x,p)dp)]dx

2.3 Preliminaries Let p(t) denote the low-type seller’s reservation price (i.e., the price which the low-type seller is indifferent between accepting and rejecting) at time t. We restrict attention to strategy profiles in which each buyer offers either cH or p(t) at each point in time. This restriction incurs no loss of generality. First, for the same reasoning as in the Diamond paradox, buyers never offer a price strictly above cH .9 This implies that the high-type seller’s reservation price is always equal to her reservation value cH and in equilibrium she accepts cH with probability 1. Note that, due to the difference in flow payoffs (cL < cH ), p(t) is always smaller than cH : p(t) ≤ ∫∞ ((1 − e−rt )cL + e−rt cH )d(1 − e−λt ) < cH for any t. Second, it is strictly suboptimal for any 0 buyer to offer a price strictly between p(t) and cH . Finally, if in equilibrium a buyer offers a losing price (strictly below p(t)), then it suffices to set his offer to be equal to p(t) and specify the low type’s acceptance strategy σS (L, t, p(t)) to reflect her rejection of the buyer’s losing offer. This adjustment is feasible because the low-type seller is indifferent between accepting and rejecting p(t). The fact that all buyers offer either cH or p(t) implies that p(t) depends only on the rate at which the low type receives offer cH . This is because the low-type seller is indifferent between accepting and rejecting p(t) at any point in time and, therefore, her reservation price can be calculated as if Formally, let p denote the supremum among all equilibrium prices buyers offer in this game. Suppose p ≥ cH . Then, the best case scenario for the high-type seller is to receive p with probability 1 from the next buyer. This means that her reservation price at any point in time cannot exceed ∫ ∞ ( ) rcH + λp (1 − e−rt )cH + e−rt p d(1 − e−λt ) = . r+λ 0 9

Since no buyer has an incentive to offer more than (rcH + λp)/(r + λ), p ≤ (rcH + λp)/(r + λ). On the other hand, due to search frictions (i.e., λ < ∞), (rcH + λp)/(r + λ) ≤ p. Therefore, it must be that p = cH . Intuitively, search frictions endow each buyer with some monopsony power. If p > cH , then each buyer can undercut the price to (rcH + λp)/(r + λ) and still make sure that the offer is accepted. Knowing that no buyer would offer p and the highest price offer would be (rcH + λp)/(r + λ), each buyer can undercut the price even further. This process continues indefinitely as long as p > cH . Consequently, in equilibrium p cannot exceed cH .

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she would accept only cH .10 Formally, given any buyer strategy σB , p(t) is given by ∫ p(t) =

∞

) ( ∫x∑ s ((1 − e−r(x−t) )cL + e−r(x−t) cH )d 1 − e−λ t s γL σB (y,s,cH )dy .

t

Clearly, p(t) increases if buyers offer cH more frequently and decreases if they do so less frequently. We let q(t) ˙ and p(t) ˙ denote the right derivatives of q(t) and p(t), respectively, that is,11 q(t + ∆) − q(t) p(t + ∆) − p(t) and p(t) ˙ = lim . ∆→0+ ∆→0+ ∆ ∆

q(t) ˙ = lim

Both are straightforward to derive from the general equations for q(t) and p(t) above: q(t) ˙ = −q(t)(1−q(t))λ

( ∑

s γH σB (t, s, cH )

−

∑

s

and p(t) ˙ = r(p(t) − c) − λ

) γLs

(σB (t, s, cH ) + σB (t, s, p(t))σS (L, t, p(t))) ,

s

∑

γLs σB (t, s, cH )(cH − p(t)).

s

In the main text, we restrict attention to the case where search frictions are sufficiently small. Precisely, we maintain the following assumption: Assumption 1 ∫ vL < 0

∞

( ) rc + λγ h c r(vL − cL ) h L L H ⇔ λγLh > ((1 − e−rt )cL + e−rt cH )d 1 − e−λγL t = . h cH − vL r + λγL

This assumption ensures that if all subsequent buyers offer cH whenever s = h, then the lowtype seller’s reservation price p(t) exceeds vL (buyers’ willingness-to-pay for a low-quality asset). Clearly, it is necessary that γLh > 0 and, conditional on that, the inequality holds when λ is sufficiently large. We focus on this case, because otherwise, as formally shown in the online appendix (Section A), the model exhibits only the familiar increasing dynamics. 10

This does not rule out the possibility that the low-type seller accepts p(t). Indeed, as shown shortly, in the unique equilibrium of our model, she does accept p(t) after certain histories. We simply exploit the fact that the seller’s acceptance decision is not observable to future buyers and, therefore, her reservation price is independent of whether she accepts p(t) or not. This property fails, for example, if rejected prices are observable to future buyers. 11 The continuous-time specification and the presence of search frictions guarantee that the equilibrium objects p(·) and q(·) evolve continuously over time (Lemma 8 in the appendix) and both q(t) ˙ and p(t) ˙ are well-defined. The qualifier “right” is due to the fact that, as shown in Propositions 1 and 2, the left and the right derivatives of q(t) do not coincide at a finite number of points.

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3

Equilibrium Characterization

In this section, we characterize the unique equilibrium of the dynamic trading game. We begin by showing that there exists a belief level q ∗ and the corresponding equilibrium strategy profile such that buyers’ beliefs q(t) stay constant once they reach q ∗ , even though buyers continue to condition their offers on their signals. We then show that, unless buyers are so optimistic about the seller’s type at the beginning of the game (i.e., qb is so large) that it is an equilibrium for buyers to always offer cH , there is a unique equilibrium in which buyers’ beliefs continuously converge to q ∗ , whether starting from above or below. We also explain how these results generalize when there are more than two signals.

3.1 Stationary Path We explicitly construct the stationary equilibrium strategy profile in which q(t) stays constant at q ∗ even though buyers do not always offer cH . The following lemma, which is useful in later analysis as well, implies that a necessary condition for q(t) to stay constant is p(t) = vL . Lemma 1 In equilibrium, if p(t) < vL , then the low-type seller accepts p(t) with probability 1 and, therefore, q(t) ˙ ≥ 0. If p(t) > vL , then trade takes place only at cH and, therefore, q(t) ˙ ≤ 0. In both cases, the inequality holds strictly as long as the probability of trade conditional on buyer arrival is strictly between 0 and 1. Proof. See the appendix. Intuitively, if p(t) < vL , then the buyer can make sure that the low-type seller trades by offering slightly more than p(t). In this case, the low-type seller trades as long as there is a buyer, while the high-type seller insists on cH . Therefore, no trade (delay) is more likely when the seller is the high type. If p(t) > vL , then trade occurs only at cH , because no buyer would pay more than vL , knowing that it would be accepted only by the low type. Since buyers are more willing to offer cH when s = h than when s = l and the high type generates signal h more frequently than the low type, the high type is more likely to trade than the low type and q(t) decreases over time. Let ρL denote the constant rate at which the low-type seller receives offer cH on the stationary path. Then, her reservation price is given by ∫ p(t) = cL +

∞

e−rx (cH − cL )d(1 − e−ρL (x−t) ) =

t

For p(t) = vL , it must be that ρL = r

v L − cL . cH − vL 10

rcL + ρL cH . r + ρL

In other words, p(t) remains equal to vL if the low-type seller receives offer cH at a constant rate of ρL = r(vL − cL )/(cH − vL ). Assumption 1 (which is equivalent to λγLh > ρL ) implies that buyers must randomize between cH and p(t) conditional on s = h. Precisely, it is necessary and sufficient that conditional on s = h, buyers offer cH with probability σB∗ ≡

ρL r(vL − cL ) = . h λγL λγLh (cH − vL )

(1)

Clearly, buyers must be indifferent between offering cH and vL upon receiving signal h. This implies that no buyer offers cH when s = l because it would give him a negative payoff. We now determine q ∗ , using buyers’ indifference conditional on s = h. Consider a buyer who has prior belief q ∗ and receives signal h. By Bayes’ rule, his belief updates to h q ∗ γH . h q ∗ γH + (1 − q ∗ )γLh

At this belief, the buyer must be indifferent between offering cH and offering p(t) = vL . Therefore, h q ∗ γH (vH −cH )+(1−q ∗ )γLh (vL −cH ) = (1−q ∗ )γLh (vL −p(t)) = 0 ⇔

γLh cH − vL q∗ = . (2) h v −c 1 − q∗ γH H H

It remains to pin down the probability that the low-type seller accepts vL , which we denote by We use the fact that q(t) is time-invariant if and only if the two seller types trade at an identical rate. The high type accepts only cH . Therefore, given buyers’ offer strategies, her trading rate is h ∗ σB . If the low type accepts vL with probability σS∗ , then her trading rate is equal to equal to λγH λ(γLh σB∗ + (1 − γLh σB∗ )σS∗ ). The equilibrium value of σS∗ must equate the two rates, that is, σS∗ .

h ∗ λγH σB = λ(γLh σB∗ + (1 − γLh σB∗ )σS∗ ) ⇔ σS∗ =

h (γH − γLh )σB∗ . 1 − γLh σB∗

(3)

The following lemma summarizes all the findings on the stationary path. Note that the strategy profile is constructed so as to satisfy all players’ incentive constraints and, therefore, is an equilibrium. Lemma 2 Let q ∗ be the value defined by equation (2). Then, q ∗ is the unique belief level that supports an equilibrium in which for all t ≥ 0, (i) q(t) = q ∗ = qb and (ii) the probability of trade conditional on buyer arrival is strictly between 0 and 1. In the unique equilibrium, • all buyers offer cH with probability σB∗ and vL with probability 1 − σB∗ conditional on s = h and vL with probability 1 conditional on s = l, and 11

• the low-type seller accepts vL with probability σS∗ , where σB∗ and σS∗ are the values given by equations (1) and (3). Equation (3) well describes how the “skimming” effect (which stems from the fact that the low type has a lower reservation price than the high type and, therefore, drives up q(t)) and the signal effect (which originates from the fact that the high type generates good signals more frequently than the low type and, therefore, pushes down q(t)) manifest themselves and how they are balanced on the stationary path. The skimming effect is reflected in the fact that σS∗ > 0 (the low type accepts h ∗ not only cH but also p(t)), while the signal effect is materialized in the inequality γH σB > γLh σB∗ (the high type is more likely to receive cH than the low type). On the stationary path, σB∗ and σS∗ are such that the two effects cancel each other out and q(t) remains constant.

3.2 Equilibrium Dynamics We now construct an equilibrium for each value of qb ̸= q ∗ . We first consider the case where qb < q ∗ . In this case, early buyers are not willing to offer cH even when the inspection outcome is h, because whenever q(t) < q ∗ , h h q ∗ γH vH + (1 − q ∗ )γLh vL q(t)γH vH + (1 − q(t))γLh vL < = cH . h h q(t)γH + (1 − q(t))γLh q ∗ γH + (1 − q ∗ )γLh

This implies that as long as buyers’ beliefs remain in this range (i.e., until q(t) reaches q ∗ ), the high-type seller never trades, while p(t) < vL and, therefore, the low-type seller trades at rate λ. Given this observation and Lemma 2, an equilibrium can be immediately constructed, as formally stated in the following proposition. Proposition 1 If qb < q ∗ , then there is an equilibrium in which • whenever t < t∗ , the buyer offers p(t) regardless of his signal, the low-type seller accepts p(t) with probability 1, p(t) increases according to p(t) ˙ = r(p(t) − cL ) with the terminal condition p(t∗ ) = vL , and q(t) increases according to q(t) ˙ = q(t)(1 − q(t))λ, and • whenever t ≥ t∗ , the players play as in Lemma 2 and q(t) = q ∗ , where t∗ is defined by q∗ =

qb . qb + (1 − qb)e−λt∗

Proof. The result that the strategy profile is an equilibrium follows from the fact that p(t) < vL (which ensures that the low-type seller trades at rate λ) and q(t) < q ∗ (which implies that the buyer offers p(t)) whenever t < t∗ . 12

Now consider the case where qb > q ∗ . In order to distinguish between the case when buyers offer cH regardless of their signal and the case when they do so only when s = h, let q be the value such that l qγH vH + (1 − q)γLl vL q γLl cH − vL = c ⇔ = . (4) H l l v −c 1−q qγH + (1 − q)γLl γH H H In words, a buyer with prior belief q obtains zero expected payoff if he offers cH conditional on s = l. If qb > q, then there is an equilibrium in which each buyer offers cH regardless of his inspection outcome and, therefore, q(t) stays constant at qb. In this equilibrium, the low type seller’s reservation price p(t) remains constant at (rcL +λcH )/(r+λ), which strictly exceeds vL . This observation verifies that buyers’ offer strategies are indeed optimal. Next, suppose that qb ∈ (q ∗ , q). By the definitions of q ∗ and q, it is natural that buyers offer cH if and only if s = h until q(t) reaches q ∗ . Given this offer strategy, p(t) > vL , because the low-type seller receives cH at rate λγLh (> ρL ) until t∗ and at rate ρL thereafter. This implies that trade occurs only at cH , which conversely justifies buyers’ offer strategies. It also follows that q(t) decreases according to h q(t) ˙ = −q(t)(1 − q(t))λ(γH − γLh ) < 0 while q(t) ∈ (q ∗ , q).

(5)

As in the previous case, this allows us to explicitly calculate the length of time it takes for q(t) to reach q ∗ , which in turn can be used to derive p(t). Proposition 2 Let q be the value given in equation (4). If qb > q, then it is an equilibrium that all buyers offer cH regardless of their signal. In this case, p(t) = (rcL + λcH )/(r + λ) and q(t) = qb for any t ≥ 0. If qb ∈ (q ∗ , q), then there is an equilibrium in which • whenever t < t∗ , the buyer offers cH if s = h and p(t) if s = l, the low-type seller accepts only cH , p(t) decreases according to p(t) ˙ = r(p(t) − vL ) − λγLh (cH − p(t)) with the terminal condition p(t∗ ) = vL , and q(t) decreases according to equation (5), and • whenever t ≥ t∗ , the players play as in Lemma 2 and q(t) = q ∗ , where t∗ is defined by

h ∗

qbe−λγH t q = −λγ h t∗ h ∗. qbe H + (1 − qb)e−λγL t ∗

Proof. The result that the strategy profile for qb ∈ (q ∗ , q) is an equilibrium follows from the fact that p(t) > vL (which ensures that trade occurs only at cH ) and q(t) ∈ (q ∗ , q) (which implies that the buyer is willing to offer cH if and only if s = h) for any t < t∗ .

13

q(t)

q(t)

q q = q∗ q∗ = q

0

t

0

t

Figure 1: The evolution of buyers’ beliefs for different initial values of qb. The left panel is for the h case with informative buyer signals (γH = γLl = 2/3), while the right panel is for the case with h no (or uninformative) buyer signals (γH = γLl = 1/2). The other parameter values used for both panels are cL = 0, vL = 1, cH = 2, vH = 3, r = 0.25, and λ = 1. We note that the dashed curve in the left panel is not linear in the decreasing region. Figure 1 depicts three typical paths of buyers’ beliefs both for the case with informative buyer signals (left) and for the benchmark case without buyer signals (right). If qb is sufficiently large, then all buyers offer cH regardless of their signal and, therefore, q(t) stays constant at qb (the horizontal solid line above q in the left panel). If qb is rather low, buyers’ beliefs increase over time (the weakly increasing solid curve in the left panel). This dynamics prominently arises in the absence of buyer signals, as shown in the right panel, and is well-understood in the literature (e.g., Deneckere and Liang, 2006; H¨orner and Vieille, 2009; Kim, 2017): the low-type seller, due to her lower reservation value, accepts a wider range of prices than the high-type seller. Therefore, delay (no trade) is more likely when the quality is high, and q(t) increases over time. If qb ∈ (q ∗ , q), then buyers’ beliefs decrease over time (the weakly decreasing dashed curve in the left panel). This dynamics is in contrast to most existing work on dynamic adverse selection and precisely due to the introduction of private buyer signals, as is clear from the comparison between the two panels. At such beliefs, the low-type seller is optimistic about her prospect of receiving an offer of cH and unwilling to trade at a low price, similarly to the high type. Therefore, trade takes place only at a high price. Since buyers offer a high price only with a good inspection outcome, the high type trades at a higher rate than the low type. This drives down buyers’ beliefs over time, offsetting the usual skimming effect.

14

3.3 Equilibrium Uniqueness Propositions 1 and 2 present an equilibrium for each value of qb, except for the case where qb = q. If qb = q, then there is a continuum of equilibria. As for the case where qb > q, it is an equilibrium that all buyers offer cH regardless of their signal: each buyer obtains a strictly positive expected payoff if s = h but zero expected payoff if s = l. There is another equilibrium which is analogous to the equilibrium when qb ∈ (q ∗ , q): buyers offer cH if and only if s = h and trade occurs only at cH until q(t) reaches q ∗ . In addition, we can construct a continuum of equilibria by taking a convex combination of these two equilibria.12 The following result states that except for the above knife-edge case, there is a unique equilibrium. In other words, if qb ̸= q, then the equilibrium presented in Propositions 1 and 2 is the unique equilibrium in our model. Theorem 1 Unless qb = q, there exists a unique equilibrium. Proof. See the appendix. The equilibrium construction above utilizes the existence of a unique stationary path (Lemma 2) and the interval-partitional equilibrium structure (that buyers’ offer strategies can be described with respect to two cutoff beliefs, q(= q ∗ ) and q). Moreover, it is immediate that the constructed equilibrium is the unique equilibrium given the stationary equilibrium behavior in Lemma 2 and requiring the interval-partitional structure. This means that equilibrium uniqueness would follow once it is shown (i) that there is a unique equilibrium when q(t) = q ∗ (i.e., the strategy profile in Lemma 2 is the unique equilibrium when qb = q ∗ ) and (ii) that any equilibrium necessarily takes an interval-partitional structure. Both results derive from the following lemma. Lemma 3 In any equilibrium, q(t) ≤ q ∗ if, and only if, p(t) ≤ vL . Proof. See the appendix. This lemma is intuitive: the more optimistic buyers are about the seller’s type, the more frequently they would offer cH and, therefore, the higher would p(t) be. In particular, given that p(t) = vL when q(t) = q ∗ , it is natural that p(t) > vL if and only if q(t) > q ∗ . However, this conclusion is not immediate. In general, a buyer is willing to offer p(t) when s (1 − q(t))γLs (vL − p(t)) ≥ q(t)γH (vH − cH ) + (1 − q(t))γLs (vL − cH ).

Specifically, for any fixed t˜ ≥ 0, it is an equilibrium that buyers switch their behavior from the first equilibrium to the second one at t˜: buyers offer cH regardless of their signal until t˜ but start conditioning on their signal from t˜. Note that it is not an equilibrium to switch from the second type to the first type, because q(t) falls below qb = q¯ as soon as buyers employ the strategy of offering cH only when s = h. 12

15

Since the left-hand side depends on both q(t) and p(t), even when q(t) is higher (which lowers the left-hand side and raises the right-hand side), if p(t) is significantly lower (which increases the left-hand side), then he would be more reluctant to offer cH , which, in turn, would justify lower p(t). The main thrust of our proof of Lemma 3 is to show that such a possibility, which cannot be ruled out with a local argument, is not consistent with the long-run (global) dynamics. For example, if q(t) > q ∗ but p(t) < vL then, by Lemma 1, q(t) must increase over time. However, it cannot be increasing forever because the inequality above cannot hold if q(t) is sufficiently large. Lemma 3, combined with Lemma 1, implies that if q(t) hits q ∗ , it must stay constant thereafter: if q(t) becomes smaller than q ∗ , then p(t) < vL and, therefore, q(t) returns back to q ∗ . Likewise, if q(t) goes above q ∗ , then p(t) > vL , which pushes q(t) back to q ∗ . Then, by construction, the strategy profile in Lemma 1 is the unique equilibrium when q(t) = q ∗ . Lemma 3 also implies that a buyer never offers cH if q(t) < q ∗ and offers cH as long as it yields a positive payoff if q(t) > q ∗ . Combining the latter with the fact that a buyer’s expected payoff by offering cH is increasing in q(t), it follows that it is optimal for a buyer to offer cH conditional on s = h if and only if q(t) > q ∗ and conditional on s = l if and only if q(t) > q. All together, these imply that any equilibrium is interval-partitional, completing the uniqueness argument.

3.4 Beyond Binary Signals The assumption of binary signals allows us to illustrate the effects of private buyer signals in the simplest way possible but is not crucial for any qualitative aspect of the model. In this subsection, we illustrate how our equilibrium characterization can be generalized when there are more than two signals. We defer some details of construction as well as the formal statements of the results to the online appendix. 3.4.1

N Signals

Suppose that each buyer receives a signal from a finite set S = {s1 , ..., sN } and that the signal structure satisfies the usual monotone likelihood ratio property, so that sn+1 is a stronger indicator of high quality than sn for any n = 1, ..., N − 1. For each a = L, H, let Γa (sn ) denote the cumulative probability that each buyer receives a signal weakly below sn . Even in this general model, for a generic set of parameter values, there continues to exist a unique equilibrium, which exhibits the same qualitative properties as the unique equilibrium of the baseline binary-signal model. The (generically unique) equilibrium can be described by an integer n∗ and a finite partition {qN +1 = 0, qN , ...q1 , q0 = 1}. Here, n∗ is the integer that identifies the cutoff signal on the

16

q6 = 0

offer cH :

q5

never

q3 = q ∗

q4 {s5 }

{s4 , s5 }

q2

q1

q0 = 1

{s3 , s4 , s5 } {s2 , ..., s5 }

p(t) :

< vL , L accepts p(t)

> vL , L rejects p(t)

q(t) :

increase

decrease

always

=

rcL +λcH r+λ

constant

Figure 2: Equilibrium structure when there are 5 signals (N = 5) and n∗ = 3. stationary path and is determined by λ(1 − ΓL (sn∗ )) < ρL =

r(vL − cL ) < λ(1 − ΓL (sn∗ −1 )). cH − v L

These inequalities mean that the low-type seller’s reservation price p(t) falls short of vL if all subsequent buyers employ the strategy of offering cH if and only if s > sn∗ but exceeds vL if their strategy is to do so if and only if s ≥ sn∗ . Since the low-type seller’s reservation price must be equal to vL on the stationary path (for the same reason as in the baseline model), it follows that sn∗ must serve as the cutoff signal: each buyer offers cH with probability 1 if s > sn∗ and appropriately randomizes between cH and vL if s = sn∗ . In turn, this allows us to pin down the stationary belief level q ∗ (= qn∗ ), using the requirement that a buyer must be indifferent between offering cH and vL conditional on belief q ∗ and signal sn∗ . When q(t) ̸= q ∗ = qn∗ , buyers’ offer strategies are defined with respect to the partition {qN +1 = 0, qN , ...q1 , q0 = 1}: if q(t) ∈ (qn+1 , qn ), then the buyer offers cH if and only if s > sn (see the example in Figure 2). Combined with the inequalities above, this implies that the low-type seller’s reservation price exceeds vL if q(t) is larger than q ∗ and falls short of vL if q(t) is smaller than q ∗ . In turn, these together determine how buyers’ beliefs q(t) evolve over time: if q(t) ∈ (qn+1 , qn ) for n ≥ n∗ , then the low type accepts both p(t) and cH , while the high type trades if and only if s > sn , and thus q(t)e−λ(1−ΓH (sn ))dt q(t + dt) = , q(t)e−λ(1−ΓH (sn ))dt + (1 − q(t))e−λdt which yields q(t) ˙ = q(t)(1 − q(t))λΓH (sn ) > 0.

17

If q(t) ∈ (qn+1 , qn ) for n = 1, ..., n∗ − 1, then both seller types trade if and only if s > sn , and thus q(t + dt) =

q(t)e−λΓH (sn )dt , q(t)e−λΓH (sn )dt + (1 − q(t))e−λΓL (sn )dt

which implies q(t) ˙ = q(t)(1 − q(t))λ(ΓH (sn ) − ΓL (sn )) < 0. In both cases, q(t) eventually converges to q ∗ , just as in the baseline model. The cutoff beliefs, qN , ..., q1 , can be determined analogously to the baseline model. One complication is that, whereas the cutoffs above q ∗ (i.e., qn∗ −1 , ..., q1 ) can be found independently of p(t)(> vL ), the cutoffs below q ∗ (i.e., qN , ..., qn∗ +1 ) must be jointly determined with p(t): each qn is determined by the requirement that a buyer with belief qn must be indifferent between offering cH and p(t) conditional on signal sn . If p(t) > vL , then it is simply not accepted and, therefore, the indifference condition is independent of p(t). To the contrary, if p(t) < vL , then the buyer obtains a positive expected payoff even with p(t) and, therefore, qn depends on p(t). In fact, this complication arises even in the baseline model when Assumption 1 is violated (because q < q ∗ = q). We explain how to recursively construct the cutoffs below q ∗ in the online appendix (Section A for the baseline model and Section C for the general finite-signal model). 3.4.2

A Continuum of Signals

Now suppose that each buyer’s signal is drawn from the interval S = [s, s] according to the typedependent cumulative distribution function Γa with density γa and the monotone likelihood ratio property holds (i.e., γH (s)/γL (s) is strictly increasing in s). Equilibrium characterization proceeds just as in the general finite case above. Let s∗ be the unique value in S such that λ(1 − ΓL (s∗ )) = ρL =

r(vL − cL ) . cH − vL

Given s∗ , the stationary path can be fully constructed using buyers’ indifference between cH and p(t) = vL conditional on s∗ (which pins down q ∗ ) and belief invariance (which allows us to identify σS∗ ). Given the characterization of the unique stationary path, one can also show that q(t) gradually converges to q ∗ , whether from above or from below, by applying Lemmas 1 and 3, both of which extend to this case without modification. To be specific, let s(t) denote the cutoff signal above which buyers offer cH at time t. If q(t) > q ∗ , then trade occurs only at cH (because p(t) > vL ), and thus q(t)e−λ(1−ΓH (s(t)))dt q(t + dt) = , q(t)e−λ(1−ΓH (s(t)))dt + (1 − q(t))e−λ(1−ΓL (s(t)))dt 18

which leads to q(t) ˙ = q(t)(1 − q(t))λ(ΓH (s(t)) − ΓL (s(t))) < 0. If q(t) < q ∗ , then the low type accepts both p(t)(< vL ) and cH , and thus q(t + dt) =

q(t)e−λ(1−ΓH (s(t)))dt , q(t)e−λ(1−ΓH (s(t)))dt + (1 − q(t))e−λdt

which yields q(t) ˙ = q(t)(1 − q(t))λΓH (s(t)) > 0. Although this alternative specification has an advantage of purifying buyers’ offer strategies (i.e., all buyers, including those on the stationary path, play a simple cutoff strategy), the characterization of buyers’ offer strategies when q(t) < q ∗ is significantly more complicated. As explained also for the finite-signal case, if q(t) < q ∗ then p(t) < vL and, therefore, buyers’ offer strategies cannot be separately identified from p(t). Unlike in the finite case (where s(t) is a step function), s(t) varies continuously and, therefore, the three relevant equilibrium functions, s(t), q(t), and p(t), can be characterized only by the following system of equations (together with the law of motion for q(t) above): • Each buyer is indifferent between cH and p(t) conditional on s = s(t), and thus q(t) γL (s(t)) cH − p(t) = . 1 − q(t) γH (s(t)) vH − cH • The low-type seller’s reservation price changes over time according to r(p(t) − cL ) = λ(1 − ΓL (s(t)))(cH − p(t)) + p(t), ˙ Although a closed-form solution is not available, it can be shown that all equilibrium properties from the finite case carry over. In particular, if qb < q ∗ , then in equilibrium both p(t) and q(t) are necessarily increasing, while s(t) is decreasing (meaning that buyers offer cH more frequently), over time. See the online appendix for a formal analysis.

4

Informativeness of Buyers’ Signals

In this section, we analyze the effects of varying the informativeness of buyers’ signals. In particular, we study how an increase in the informativeness, which presumably helps mitigate information asymmetry in the market, affects market efficiency and seller surplus. For the former, we consider 19

the expected delay to trade, because inefficiency takes the form of delay in our dynamic environment.13 For the latter, we focus on the low-type seller’s expected payoff p(0), because the high-type seller never obtains a strictly positive expected payoff. For each a = L, H, we let τa denote the random time at which the type-a seller trades and Fa denote the corresponding distribution function, so that Fa (t) is the probability that the type-a seller trades before t. Note that the seller leaves the market only when she trades and, therefore, the hazard rate fa (t)/(1 − Fa (t)) coincides with the type-a seller’s trading rate at t. In addition, we let t∗ (b q ) denote the length of time it takes for q(t) to travel from qb to q ∗ in the unique equilibrium of the game, whether qb < q ∗ or not.

4.1 Blackwell Informativeness In our model, an inspection technology is described by a matrix ( Γ=

l γLl γH h γLh γH

)

( =

h 1 − γLh 1 − γH h γLh γH

) .

Applying Blackwell’s notion of informativeness (Blackwell, 1951) to our model, Γ is more informative than Γ′ if there exists a non-negative (Markov) matrix M = (mij )2×2 such that each row ∑ sums to 1 (i.e., j mij = 1) and Γ′ = M Γ. The following result shows that Blackwell informah l /γLh , in our model with binary /γLl and γH tiveness is fully summarized by the likelihood ratios, γH signals.14 Lemma 4 Γ is more informative than Γ′ , in the sense of Blackwell (1951), if and only if the likelihood ratio conditional on l is smaller, while that conditional on h is larger, under Γ than under Γ′ , that is, l l′ h h′ γH γH γH γH ≤ and ≥ . γLl γLl′ γLh γLh′ Proof. See the appendix. Intuitively, a more informative signal allows a decision-maker to take the right action (e.g., offering cH to the high type and p(t) to the low type) with a higher probability. This means that a more informative signal should bring the decision-maker’s posterior closer to 0 or 1, depending on An alternative is to consider expected social surplus from trade of each type, that is, E[e−rτa (va − ca )], where τa denotes the random time of trade when the seller’s type is a. We do not separately consider this alternative criterion, because it involves effectively the same arguments and leads to analogous economic conclusions. 14 In general, Blackwell informativeness regulates only the likelihood ratios of the two extreme signals, one with the lowest ratio and the other with the highest ratio (see, e.g. Ponssard, 1975). Most results in this section go through unchanged even with more than two signals and the resulting weaker implication on the likelihood ratios. See an earlier version of this paper for such a general treatment. 13

20

its realization. In other words, a signal is more informative if it induces more dispersed posterior beliefs. Lemma 4 stems from the fact that dispersion of posterior beliefs is determined by the likelihood ratios: given prior belief q(t) and signal s, the posterior is given by q(t, s) =

s s q(t)γH q(t, s) q(t) γH ⇔ = . s q(t)γH + (1 − q(t))γLs 1 − q(t, s) 1 − q(t) γLs

One immediate but crucial implication of Lemma 4 is that as Γ becomes more informative, the two cutoff beliefs, q ∗ and q, move in opposite directions (see Figure 3) and the seller trades faster on the stationary path, as formally stated in the following result. Corollary 1 If Γ becomes more informative, then q ∗ decreases, while q increases. In addition, the seller’s trading rate on the stationary path, denoted by ρ, increases. Proof. The results are immediate from Lemma 4 and the following closed-form solutions: h r(vL − cL ) q∗ γLh cH − vL q γLl cH − vL γH , , and ρ = , = = h l h ∗ 1−q γH vH − cH 1 − q γH vH − cH γL cH − vL

where the first two come from equations (2) and (4), while the last follows from the fact that h ∗ σB (the high type’s trading rate on the stationary path) and equation (1). ρ = λγH To understand this result, recall that q ∗ is the point at which a buyer obtains zero expected payoff when he offers cH conditional on s = h, and q is the corresponding point conditional on s = l. If Γ becomes more informative, then h becomes a stronger signal of high quality, while l becomes a weaker signal of high quality. That is, the buyer becomes more confident about the quality of the asset conditional on s = h but less confident conditional on s = l. This pushes down q ∗ but drives up q, because the prior belief level q s necessary for the buyer to break even with cH s is inversely related to the likelihood ratio γH /γLs , that is, s q s γH (vH − cH ) + (1 − q s )γLs (vL − cH ) = 0 ⇔

1 cH − v L qs = s s . s 1−q γH /γL vH − cH

For the result on the seller’s trading rate ρ on the stationary path, notice that an increase in the informativeness of Γ makes the high type generate signal h even more frequently relative to the low type. This, together with the fact that the rate at which the low type receives cH on the stationary path must remain unchanged at ρL = r(vL − cL )/(cH − vL ), implies that the high type’s trading rate (at price cH ) increases. Since the seller’s trading rate is independent of her type on the stationary path, the low-type seller also trades faster. The following limiting cases further clarify Corollary 1. First, consider the (uninformative) h limit as the difference γH − γLh disappears. In this case, both q ∗ and q converge to (cH − vL )/(vH − 21

vL ). In the limit, since the interval [q ∗ , q) collapses, the model does not generate the decreasing l belief dynamics (see the right panel of Figure 1). Second, suppose that γH tends to 0, so that ∗ signal l becomes an arbitrarily strong signal of low quality. In this case, q decreases to γLh (cH − vL )/(γLh (cH − vL ) + vH − cH ), while q converges to 1. The latter is intuitive, because in the limit buyers would never offer cH when s = l, no matter how high their prior beliefs are. Finally, suppose that γLh tends to 0, so that signal h indicates high quality for sure. In this case, q rises toward (cH − vL )/(cH − vL + γLl (vH − cH )), while the lower cutoff approaches 0, because buyers always strictly prefer offering cH to p(t) conditional on s = h.15

4.2 Pessimistic Initial Beliefs If qb < q ∗ then, as shown in Proposition 1, buyers offer p(t) even with s = h and the low-type seller accepts p(t) with probability 1 until t∗ (b q ) (i.e., until q(t) reaches q ∗ ). After t∗ (b q ), both seller types trade at rate ρ and p(t) stays constant at vL . Therefore, p(0) = cL + e−rt and q(t) =

∗ (b q)

(p(t∗ (b q )) − cL ) = cL + e−rt

∗ (b q)

(vL − cL )

qb for all t < t∗ (b q ). qb + (1 − qb)e−λt

An increase in the informativeness of Γ affects this dynamic outcome in two ways. First, since q ) decreases (see the left panel of Figure 3). Second, since ρ q ∗ falls (by Corollary 1 above), t∗ (b q ). The following result is increases (again by Corollary 1), both seller types trade faster after t∗ (b immediate once these effects are applied to the equilibrium outcome. Proposition 3 Suppose that qb < q ∗ . If Γ becomes more informative, then τH decreases in the sense of first-order stochastic dominance, E[τL ] decreases, and p(0) increases. Proof. See the appendix. Proposition 3 is fairly intuitive. An increase in the informativeness of buyers’ signals reduces information asymmetry in the market. This makes buyers become less reluctant to offer cH when they receive a good signal and, therefore, start offering cH from an earlier time. This is beneficial to the low-type seller, who receives cH at the constant rate of ρL (independent of Γ) on the stationary path. Since the high-type seller’s trading rate on the stationary path increases (by Corollary 1), this also means that the high-type seller clearly trades faster. At the same time, such an adjustment 15 Assumption 1 is violated if γLh becomes sufficiently small. Therefore, the equilibrium dynamics characterized in the online appendix (Section A), not in Section 3, applies in this case. However, the result that the lower cutoff belief approaches 0 is independent of this necessary change.

22

q(t)

q(t)

q′ q

q∗ q ∗′

0

q) t∗′ (b q ) t∗ (b

t

0

t

Figure 3: The effects on the evolution of buyers’ beliefs when Γ becomes more informative, from h h γH = γLl = 2/3 (the dashed lines, q ∗ , and q) to γH = γLl = 3/4 (the solid lines, q ∗′ , and q ′ ). The other parameter values used are identical to those for Figure 1. increases the low-type seller’s incentive to reject p(t) and wait for cH , which is why τL does not decrease in the sense of first-order stochastic dominance: in the left panel of Figure 3, the lowtype seller’s trading rate decreases from λ to ρ over the interval [t∗′ (b q ), t∗ (b q )). Nevertheless, E[τL ] unambiguously decreases because this indirect negative effect cannot outweigh the direct positive effect of trading faster on the stationary path.

4.3 Optimistic Beliefs Now we consider the case when qb ∈ (q ∗ , q). In this case, as shown in Proposition 2, until t∗ (b q) ∗ (i.e., until q(t) reaches q ), buyers offer cH if s = h and p(t)(> vL ) if s = l. Since trade occurs only at cH , buyers’ beliefs decrease according to qbe−λγH t qb q(t) = −λγ h t ht = h h . −λγ qbe H + (1 − qb)e L qb + (1 − qb)eλ(γH −γL )t h

The length of time it takes for q(t) to reach q ∗ is given by h ∗

h qb γH vH − cH γLh cH − vL q∗ qb e−λγH t (bq) h −γ h )t∗ (b λ(γH q) L ⇔ e = = . = h t∗ (b h v −c h ∗ −λγ q ) 1−q 1 − qb e L 1 − qb γL cH − vL γH H H

With optimistic beliefs, Γ determines not only the length of the convergence path t∗ (b q ) but 23

also each seller type’s trading rate λγah on the path. This implies that the information content of time-on-the-market is also influenced by a change in Γ. The evolution of buyers’ beliefs, however, h h is determined by the difference γH − γLh , not by the ratio γH /γLh , as shown in the equation for q(t) above. This suggests that without further restrictions, various different results may emerge s s depending on how we vary Γ, because Blackwell informativeness disciplines γH /γLs but not γH − h h h γLs in general. For instance, if both γH and γLh decrease, then γH − γLh can fall when γH /γLh rises (and vice versa). In what follows, in order to further discipline variations in Γ and get clean insights, we focus h on the symmetric signal structure such that γLl = γH = γ for some γ ∈ (1/2, 1), that is, ( Γ=

γ 1−γ 1−γ γ

) .

Naturally, γ measures the informativeness of buyers’ signals: Γ is more informative, in the sense h of Blackwell (1951), if and only if γ is higher. In addition, both the likelihood ratio γH /γLh = h − γLh = 2γ − 1 always increase in γ. γ/(1 − γ) and the difference γH Under the symmetry restriction, the equation for q(t) above simplifies to q(t) =

qb . qb + (1 − qb)eλ(2γ−1)t

(6)

Clearly, for any t < t∗ (b q ), q(t) decreases in γ. Intuitively, when qb ∈ (q ∗ , q), delay is mainly caused by the failure to generate signal h and q(t) reflects the expected difference in the frequency of signal h between the two seller types. If this difference grows due to an increase in the informativeness of Γ, then delay becomes a stronger indicator of low quality and, therefore, q(t) decreases faster (see the right panel of Figure 3). The equation for t∗ (b q ) above reduces to ∗ (b q)

eλ(2γ−1)t

=

qb qb 1 − q ∗ γ vH − cH = . ∗ 1 − qb q 1 − qb 1 − γ cH − vL

(7)

h The left-hand side captures the effect of the speed of belief evolution (i.e., λ(γH − γLh )) on t∗ (b q ), ∗ while the right-hand side reflects the distance between qb and q . As γ increases, q(t) falls faster, which shortens t∗ (b q ). In the meantime, as shown in Corollary 1, q ∗ decreases and, therefore, becomes further apart from qb, which lengthens t∗ (b q ) (see the right panel of Figure 3). In general, t∗ (b q ) can both increase or decrease in γ. The following lemma provides a necessary and sufficient condition under which t∗ (b q ) increases in γ.

Lemma 5 Let qb∗ ≡ (cH − vL )/(vH − vL ) ∈ [q ∗ , q]. If qb ≤ qb∗ , then t∗ (b q ) increases in γ. Otherwise, 24

qb/(1 − qb)

q(γ) 1−q(γ)

τH E[

]↓

qb∗ 1−b q∗

]↓ E [τ L

γ(b q)

q ∗ (γ) 1−q ∗ (γ)

γ

1/2

Figure 4: The effects of increasing the informativeness of Γ on the equilibrium outcome. The gray area is the parameter region in which both E[τH ] and E[τL ] increase as Γ becomes more informative. E[τH ] decreases in γ outside the gray area, while E[τL ] decreases in γ if and only if (γ, qb) lies below the solid curve. The dashed line represents γ(b q ) that is defined in Lemma 5. The parameter values used for this figure are cL = 0, vL = 1, cH = 2, vH = 3, r = 0.05, and λ = 10. there exists γ(b q ) ∈ (1/2, 1) such that t∗ (b q ) increases in γ if and only if γ > γ(b q ). Proof. See the appendix. Intuitively, if qb is close to q ∗ , then t∗ (b q ) is close to 0. In this case, a marginal change of the speed of belief evolution over [0, t∗ (b q )) has a negligible impact, while a decrease in q ∗ has the first-order effect on t∗ (b q ). Therefore, t∗ (b q ) increases in γ. If qb is considerably larger than q ∗ , then the relative strength of the two effects depends on γ, because the marginal effect of the speed of convergence (captured by the term 2γ − 1) is independent of γ, while that of q ∗ (captured by the term γ/(1 − γ)) increases in γ. Therefore, t∗ (b q ) decreases in γ if γ is close to 1/2 but increases if ∗ γ is close to 1. In Figure 4, t (b q ) decreases in γ if and only if (γ, qb) lies above the dashed line. In order to understand the economic effects of these changes, first consider τH . By Proposition 2, the high-type seller’s trading rate is given as follows:  λγ

fH (t) = 1 − FH (t) ρ

if t < t∗ (b q ), if t ≥ t∗ (b q ).

An increase in γ raises the high-type seller’s trading rates both on the convergence path (λγ) and on

25

the stationary path (ρ), where the latter follows from Corollary 1. Since λγ > ρ, if it also increases t∗ (b q ), then the overall effect is clear: τH decreases in the sense of first-order stochastic dominance. If t∗ (b q ) decreases, instead, the overall effect is ambiguous. Still, since all the variables change continuously, it is natural that E[τH ] increases as long as t∗ (b q ) does not decrease sufficiently fast. Now consider the low-type seller’s expected payoff p(0). Recall that p(0) depends only on the rate at which the low type receives cH . Letting ρL (t) denote the rate at each t, by Proposition 2,  λ(1 − γ) ρL (t) = ρ = r(v − c )/(c − v ) L L L H L

if t < t∗ (b q ), if t ≥ t∗ (b q ).

In contrast to the high type’s corresponding rates, λ(1 − γ) falls in γ, and ρL is independent of γ. Since λ(1 − γ) > ρL , an increase in γ clearly lowers p(0) if it decreases t∗ (b q ). Otherwise, the ∗ overall effect is ambiguous, but p(0) would decrease as long as t (b q ) does not increase so fast that the negative effect due to lower λ(1 − γ) outweighs the positive effect due to higher t∗ (b q ). For the effects on τL , recall that the low-type seller’s trading rate is given as follows, again by Proposition 2:  λ(1 − γ) if t < t∗ (b q ), fL (t) = 1 − FL (t) ρ if t ≥ t∗ (b q ). An increase in γ lowers λ(1 − γ) but raises ρ (by Corollary 1). Therefore, regardless of whether t∗ (b q ) increases or decreases, the overall effect is ambiguous: γL does not change in the sense of first-order stochastic dominance. Nevertheless, it is clear that if qb is so close to q ∗ that t∗ (b q ) is sufficiently small, then the former negative effect is dominated by the latter positive effect, and thus E[τL ] decreases. In the opposite case when qb is considerably larger than q ∗ , the former effect can be significant and dominate the latter effect, in which case E[τL ] increases. We summarize the results so far in the following proposition. Roughly, it states that improving the informativeness of buyers’ signals may be harmful to efficiency and seller surplus when qb > qb∗ = (cH − vL )/(vH − vL ), which is the case when trade is fully efficient in the absence of buyer signals and, therefore, typically excluded in other models of dynamic adverse selection. Proposition 4 Suppose that qb ∈ (q ∗ , q) and consider the symmetric signal structure such that h γH = γLl = γ for some γ ∈ (1/2, 1). If qb is sufficiently close to q ∗ , then both E[τL ] and E[τH ] decrease, while p(0) increases, in γ. If qb is sufficiently close to q and γ is sufficiently close to 1/2, then both E[τL ] and E[τH ] increase, while p(0) decreases, in γ. Proof. See the appendix. For the intuition behind the possibly adverse economic consequences of more informative sig26

nals, recall that if qb ∈ (q ∗ , q), then time-on-the-market t contains negative information about the seller’s type: a seller’s availability reflects how unlikely she is to generate signal h (the signal effect), not how much she insists on a high price (the skimming effect). When Γ becomes more informative, this negative information contained in t is amplified, which makes buyers more pessimistic and, therefore, weakens their incentives to offer a high price. When qb is close to q (in which case t∗ (b q ) is significant), this negative effect is particularly strong and may even outweigh the general positive effects of more informative signals. If that happens, market efficiency deteriorates and the seller loses out. Our efficiency result is particularly related to Daley and Green (2012), who study the effects of introducing public news in a model with competitive buyers. They find that introducing news necessarily improves efficiency if a static lemons condition holds (translated as vL < cH in our model) but weakly reduces efficiency if the condition fails. This is qualitatively consistent with our result that introducing private buyer signals (i.e., increasing γ from 1/2) contributes to efficiency if and only if qb < qb∗ = (cH − vL )/(vH − vL ). The mechanism behind their result is different from ours: their result is driven by the high-type seller’s incentive to wait for more favorable public news, which strengthens as news quality improves, not by buyers’ inferences about previous buyers’ signals. Nevertheless, both results highlight the subtle role of informative signals in the market for lemons and call for caution on the conventional wisdom that transparency necessarily helps market efficiency.

5

The Role of Search Frictions

In this section, we investigate the role of search frictions in our dynamic trading environment. In particular, we study whether, and how, an increase in the arrival rate of buyers λ can improve market efficiency and seller surplus. An increase in λ has a direct positive effect on both efficiency and seller surplus: if the players’ strategies were to remain unchanged, then trade would occur faster and the low-type seller would obtain a higher expected payoff. However, the players do adjust their strategies in response. In particular, the low-type seller becomes more willing to wait for cH , which induces buyers to adjust their offer behavior accordingly. In order to systematically assess the overall effects, we separately consider the effects on the stationary path (after q(t) reaches q ∗ ) and those on the convergence path (before q(t) reaches q ∗ ). Recall that the seller’s trading rate on the stationary path (which is by definition independent

27

of the seller’s type) is given by ρ=

h ∗ λγH σB

h h γH γH r(vL − cL ) = h ρL = h . γL γL cH − vL

(8)

Notice that ρ is independent of λ, that is, an increase in λ has no effect on the seller’s trading rate on the stationary path. Technically, this is because p(t) = vL on the stationary path, which can be sustained only when γLh σB∗ (the probability that each buyer offers cH to the low-type seller) proportionally decreases as λ increases. Intuitively, an increase in λ strengthens the low-type seller’s incentive to reject p(t) and wait for cH , which weakens buyers’ incentives to offer cH . In equilibrium, buyers decrease their probability of offering cH up to the point where p(t) remains equal to vL . Naturally, the stationary belief q ∗ is also independent of λ, because it is determined by the requirement that a buyer must break even with offer cH conditional on belief q ∗ and signal h, for which the buyer arrival rate λ is irrelevant. In order to evaluate the impact on the convergence path, recall from Propositions 1 and 2 that ∗ t (b q ) satisfies the following equation in each case: if qb < q ∗ then qb q = ⇔ λt∗ (b q ) = log qb + (1 − qb)e−λt∗ (bq) ∗

(

qb 1 − q ∗ 1 − qb q ∗

) ,

(9)

while if qb ∈ (q ∗ , q) then h ∗

1 qbe−λγH t (bq) ⇔ λt∗ (b q) = − h log q = −λγ h t∗ (bq) h t∗ (b −λγ q ) γH − γLh qbe H + (1 − qb)e L ∗

(

qb 1 − q ∗ 1 − qb q ∗

) .

(10)

In either case, λt∗ (b q ) is independent of λ, that is, t∗ (b q ) proportionally decreases as λ increases. This means that when λ increases, the probability that each seller type trades on the convergence path remains constant but, since t∗ (b q ) decreases, trade occurs faster on average. Combining these two results leads to the following conclusion: the indirect effect associated with an increase in λ cannot outweigh the direct positive effect and, therefore, an increase in λ improves both market efficiency and seller surplus. Proposition 5 If λ increases, then both τL and τH decrease in the sense of first-order stochastic dominance and p(0) increases. Proof. From the characterization results in Section 3, if qb < q ∗ then   ∗  0 if t < t∗ (b λ if t < t (b q ) q ), fH (t) fL (t) and = = 1 − FL (t) ρ otherwise, 1 − FH (t) ρ otherwise,

28

fL (t)/(1 − FL (t))

FL (t) 1

λ′ γLh

q(t) = q ∗

λγLh ρL

0

⇐ t∗ (b q)

0

t

t

Figure 5: The effects of increasing the arrival rate of buyers from λ = 1.5 (dashed) to λ′ = 2 (solid) on the rate at which the low-type seller trades (left) and on the cumulative probability with which the low-type seller trades (right). The two shaded areas in the left panel are of equal size. The dotted line in the right panel is for the case where λ is sufficiently large (λ = 20). The parameter h = γLl = 2/3, and values used for this figure are cL = 0, vL = 1, cH = 2, vH = 3, r = 0.35, γH qb = 0.55. while if qb > q ∗ then for both a = L, H,  λγ h fa (t) a = 1 − Fa (t) ρ

if t < t∗ (b q ), otherwise.

Using the fact that both λt∗ (b q ) and ρ are independent of λ, one can directly show that for both a = L, H, and whether qb < q ∗ or qb ∈ (q ∗ , q), Fa (t) strictly decreases in the sense of first-order stochastic dominance as λ increases. The payoff result can be established by showing that the distribution of the random time by which the low-type seller receives cH also decreases in λ in the sense of first-order stochastic dominance, whose proof is analogous to the one above. Figure 5 illustrates the logic behind Proposition 5. An increase in λ does not affect the seller’s trading rate (ρ) and the low-type seller’s expected payoff (p(t) = vL ) on the stationary path. However, it shortens the length of time it takes for q(t) to reach q ∗ . When qb ∈ (q ∗ , q), this means that the seller receives cH more frequently at earlier times (see the left panel of Figure 5). This reduces the expected delay to trade and, due to discounting, increases seller surplus. When qb < q ∗ , an increase in λ directly speeds up trade of the low-type seller before t∗ (b q ). In addition, since t∗ (b q) decreases, the seller starts receiving cH earlier, which implies both that the high type also trades ∗ faster and that the low-type seller’s expected payoff p(0) = cL + e−rt (bq) (vL − cL ) increases. 29

Proposition 5 is in stark contrast to a result by Fuchs and Skrzypacz (2017). They consider a discrete-time model with competitive buyers and show that under a regularity condition, the most infrequent trading (restricting trade to take place only at time 0) maximizes expected gains from trade. Their result follows because a seller who needs to wait longer until the next trading opportunity has a weaker incentive to delay trade and, therefore, is more willing to trade. This effect is present in our model as well, reflected in the fact that p(t) increases in λ. The main difference lies in the flexibility of timing design. Whereas we vary only the Poisson arrival rate λ, they consider a more general timing design problem in which, effectively, buyers’ arrival times can be chosen by the designer. In our model, the arrival time of the first buyer, as well as the frequency of subsequent arrivals, depends on λ, which offsets the aforementioned effect of increased λ and ultimately leads to the conclusions in Proposition 5. We note that although an increase in λ improves market efficiency, it does not eliminate inefficiency even in the limit as λ tends to infinity. For any qb < q, t∗ (b q ) approaches 0 and the seller trades almost immediately with a positive probability. However, as shown above, the trading rate on the stationary path ρ is independent of λ and, therefore, real-time delay persists even as λ grows unboundedly. The dotted line in the right panel of Figure 5 illustrates this limiting outcome. When λ is sufficiently large, Fa (t) reaches a point at which q(t) = q ∗ almost immediately. However, its hazard rate fa (t)/(1 − Fa (t)) remains constant at ρ thereafter, which generates real-time delay.

6

Robustness

Our model is parsimonious in various dimensions. This allows us to deliver our main insights in a particularly simple fashion as well as analyze the effects of key policy variables. However, it also raises the question of the robustness of our findings. In this section, we consider three alternative environments, which differ from our baseline environment in one of the following three aspects: the cardinality of seller types, the bargaining protocol, and the market structure. We show that our main insights continue to hold in all three environments. We briefly discuss our exercises and discuss the main lessons, while relegating all formalities to the online appendix. We note that all the results in this section are for the case when search frictions are sufficiently small (i.e., λ is sufficiently large).

6.1 Three Seller Types In this subsection, we consider the case of three types, where the asset can be either of low quality (L), of middle quality (M ), or of high quality (H). Although still restrictive, the subsequent analysis illustrates general complications that arise when there are more than two seller types. 30

Nevertheless, it also suggests that the main insights from our two-type analysis are likely to hold in a more general environment. Extending the notations for the two-type case, for each a = L, M, H, we denote by ca and va the stock values of the type-a asset to the seller and buyers, respectively, and by γa the probability that each buyer receives signal h from type a. There are always gains from trade (i.e., ca < va for each a), but a higher type is more valuable to both the seller and buyers (i.e., cL < cM < cH and vL < vM < vH ). In order to highlight the effects of adverse selection, we assume that vL < cM and vM < cH . We also assume that a higher type is more likely to generate signal h (i.e., γL < γM < γH ). For each a = L, M, H, we denote by qa (t) the probability that the seller is of type a, by pa (t) the reservation price of the type-a seller, and by σa (t) the probability that the buyer offers pa (t) at time t conditional on s = h. In addition, we let q(t) denote buyers’ beliefs at time t (i.e., q(t) ≡ (qL (t), qM (t), qH (t))) and Eq,s [v] denote the expected buyer value of the asset conditional on belief q and signal s. ∗ ∗ We begin by identifying a belief vector q ∗ = (qL∗ , qM , qH ) that can generate a stationary path. ∗ ∗ Denote by pa the type-a seller’s reservation price, by σa the probability that each buyer offers p∗a conditional on signal h, and by σS∗ the probability that the low-type seller accepts p∗L on the stationary path. For the same reasons as in the two-type case, p∗L = vL (otherwise, qL (t) increases or decreases) and p∗H = cH (by the Diamond paradox). Then, the conditions for the stationary path equilibrium variables, each of which is a natural extension of the corresponding condition in the two-type case, are given as follows. • The low-type and the middle-type sellers’ reservation prices: ∗ ∗ r(vL − cL ) = λγL (σH (cH − vL ) + σM (p∗M − vL )),

(11)

∗ r(p∗M − cM ) = λγM σH (cH − p∗M ).

(12)

• Belief invariance: ∗ ∗ ∗ ∗ ∗ ∗ ∗ γH σH = γM (σH + σM ) = γL (σH + σM ) + (1 − γL (σH + σM ))σS∗ .

(13)

• Buyers’ indifference over vL , p∗M , and cH conditional on signal h: 0 = qL∗ γL (vL − p∗L ) ∗ = qL∗ γL (vL − p∗M ) + qM γM (vM − p∗M )

(14)

∗ ∗ = qL∗ γL (vL − cH ) + qM γM (vM − cH ) + qH γH (vH − cH ).

(15)

For q ∗ to be well-defined, it is necessary that p∗M < vM (otherwise, qL∗ ≤ 0). In fact, this is also 31

sufficient, because given p∗M (< vM ), all other variables can be explicitly derived from the above conditions and shown to be well-defined. The following lemma uses this observation to provide a necessary and sufficient condition under which q ∗ is well-defined. ∗ ∗ Lemma 6 There exists a belief vector q ∗ = (qL∗ , qM , qH ) ≫ 0 which supports an equilibrium in ∗ which for all t ≥ 0, (i) q(t) = q and (ii) the probability of trade conditional on buyer arrival is strictly between 0 and 1 if and only if

vM − cM γM (cH − vM ) ( ). > vL − cL M γL cH − vL + γHγ−γ (v − v ) M L M

(16)

Proof. See the appendix. If condition (16) fails, then q ∗ that solves the above stationary path conditions does not lie inside the probability simplex. In particular, qL∗ ≤ 0. This arises, for example, when γL is sufficiently small. In this case, the low-type seller is so unlikely to receive p∗M or cH that her reservation price p∗L falls short of vL and, therefore, qL (t) continues to decrease. It also happens when vM − cM is sufficiently small. In this case, buyers will have little incentive to offer pM (t). Equilibrium would be similar to the equilibrium in the model with the low type and the high type only. Our main characterization result for the three-type case states that unless Eqb,l [v] ≥ cH (in which case it is an equilibrium that buyers always offer cH ), there is an equilibrium in which buyers’ beliefs conditional on no trade converge to q ∗ , starting from any initial belief. Proposition 6 In the model with three seller types, suppose that condition (16) holds and λ is sufficiently large. For any qb such that Eqb,l [v] < cH , there exists an equilibrium in which q(t) converges to q ∗ . Figure 6 depicts how buyers’ beliefs, which can be represented by a two-dimensional simplex, evolve over time in the current three-type case.16 The whole space is divided into three areas. In Area a = L, M, H, trade occurs only at pa (t). Its effects on the evolution of buyers’ beliefs, however, differ across different areas. pL (t) is accepted only by the low type. Therefore, in Area L, qL /qM and qL /qH decrease over time, while qM /qH stays constant. pM is accepted by the low type and the middle type but is more likely to be offered to the middle type than to the low type. Therefore, in Area M, qM /qL , qM /qH , and qL /qH decrease over time. Finally, pH (t) is accepted 16 We follow the standard interpretation of a two-dimensional simplex. Each vertex corresponds to buyers’ degenerate beliefs. For example, the top vertex (L) is the point at which buyers assign probability 1 to the event that the seller is the low type. The probability that the seller is of type a is constant on any straight line that is parallel to the line between buyers’ degenerate beliefs for the other types. For example, the probability of the low type is constant on any horizontal line. The probability of a particular type decreases along the line that connects from buyers’ degenerate beliefs for that type to the center of their degenerate beliefs for the other two types.

32

L

LM pL (t) or pM (t)

LH pL (t) or pH (t). L pL (t)

∗ qLM b

qb ∗ M pM (t) M

∗ qb LH

H pH (t) Always cH

b

H

∗ qM H

MH pM (t) or pH (t)

Figure 6: The evolution of buyers’ beliefs and their equilibrium offer strategies with three types. ∗ qab represents the equilibrium stationary belief level in the model with types a and b only. by all three types. In Area H, since γH > γM > γL , qH /qL , qH /qM , and qM /qL decrease over time. Generically, q(t) arrives at one of the three dividing curves. From that point on, trade occurs at two relevant prices. The players’ trading strategies are such that q(t) converges to q ∗ following the path. Relegating a full analysis to the online appendix, we explain how to construct an equilibrium ∗ /qL∗ (so that q(t) later follows for the (simplest) case where qb belongs to Area L and qbH /b qL > q H Path LH, rather than Path LM). Since the seller types’ reservation prices can be derived only backward in time, we characterize Path LH first. On this path, the equilibrium behavior mimics ∗ that at qLH , which corresponds to the stationary belief in the model with types L and H only. Buyers randomize between cH and vL when s = h and offer vL when s = l. The low-type seller accepts vL with probability σS (t) ∈ (0, 1). Specifically , σH (t) is pinned down by the requirement that pL (t) = vL : r(vL − cL ) = λγL σH (t)(cH − vL ) ⇔ σH (t) = σ ˜H ≡

33

r(vL − cL ) . λγL (cH − vL )

Given σ ˜H and σS (t), q(t) evolves according to17 q˙L = λqL (qM (γM σ ˜H − (γL σ ˜H + (1 − γL σ ˜H )σS )) + qH (γH σ ˜H − (γL σ ˜H + (1 − γL σ ˜H )σS ))) , q˙M = λqM (qL (γL σ ˜H + (1 − γL σ ˜H )σS − γM σ ˜H ) + qH (γH σ ˜H − γM σ ˜H )) , q˙H = λqH (qL (γL σ ˜H + (1 − γL σ ˜H )σS − γH σ ˜H ) + qM (γM σ ˜H − γH σ ˜H )) . In addition, σS (t) can be found from the requirement that q(t) should stay on Path LH: q˙L γL (vL − cH ) + q˙M γM (vM − cH ) + q˙H γH (vH − cH ) = 0. In Area L, as stated before, buyers offer pL (t) regardless of their signal. Therefore, buyers’ beliefs evolve according to q˙L = −λqL (qM + qH ) = −λqL (1 − qL ), q˙M = λqM qL , q˙H = λqH qL . Then, it is possible to calculate the length of time it takes for q(t) to reach Path LH. This, together with the fact that only pL (t) is offered in Area L, allows us to explicitly calculate pL (t). The optimality of the seller’s acceptance strategy immediately follows. In addition, by construction, buyers are indifferent between cH and vL conditional on s = h on Path LH. It remains to show that buyers have no incentive to offer pM (t) on Path LH or in Area L and cH in Area L. The latter is immediate because Eq,h [v] < cH whenever q is in Area L. For the former, note that ∗ , which implies that pM (t) > p∗M on Path LH. Intuitively, pL (t) is equal to vL both on the σ ˜ H > σH stationary path and on Path LH. However, the low type receives both cH and p∗M on the former but only cH on the latter. Therefore, it must be that she is more likely to receive cH on Path LH than ∗ /qL∗ on Path LH, it follows on the stationary path. Combining this with the fact that qM /qL < qM that ∗ q ∗ γL vL + qM γM vM qL (t)γL vL + qM (t)γM vM pM (t) > p∗M = L ∗ > . ∗ qL γL + qM γM qL (t)γL + qM (t)γM In Area L, cH is never offered. This means that pM (t) increases until q(t) reaches Path LH. This provides a potential incentive for buyers to offer pM (t). However, the fact that pM (t) ≥ p∗M and the strict incentive on Path LH can be used to show that buyers still have no incentive to offer pM (t) in Area L. 17

For example, the first differential equation can be obtained from qL (t + dt) =

qL (t)e−λ(γL σ˜H +(1−γL σ˜H )σS (t))dt −λ(γ σ ˜ +(1−γ ˜H )σS (t))dt + q (t)e−λγM σ ˜H L H Lσ qL (t)e M

34

+ qH (t)e−λγH σ˜H

.

Proposition 6 suggests that the central lessons from our model go beyond the simple twotype case. Depending on the initial belief, the reputation of an asset can evolve in various different directions. With more than two types, an asset’s reputation cannot be measured by a single variable any longer. Still, there is a sense in which reputation evolves in a monotone way: the probability of each type tends to decrease if it is relatively large and increases if it is relatively small until it converges to a certain point. The underlying economic forces are, reassuringly, similar to those in the two-type case. If buyers initially assign a large probability to the low type (Area L), then they offer only pL (t), which is accepted only by the low type. Therefore, delay is mainly attributed to higher types’ resistance to accept a low price, and thus the reputation of the asset improves over time. If the initial probability of the high type is relatively large (Area H), then buyers offer pH (t), unless they observe a particularly bad signal. Therefore, delay mainly conveys negative information about the quality of the asset, and thus the reputation deteriorates over time. When the initial probability of the middle type is large (Area M), delay is interpreted as a mixture of these two effects. On the one hand, it indicates the high type’s unwillingness to trade at a mediocre price pM (t), thereby increasing the probability of the high type. On the other hand, it also suggests the possibility that all previous buyers have received sufficiently bad signals about the quality of the asset, thereby increasing the relative probability of the low type as well. To the extent that these economic forces are sensible, our results are likely to carry over to a more general environment, although exponentially increasing technical difficulties do not allow us to formally obtain those results with more types.

6.2 Alternative Bargaining Protocols In our main model, (uninformed) buyers make price offers to the (informed) seller. Although this bargaining protocol is most commonly adopted in the literature, it exhibits some properties that may be considered undesirable or implausible. In particular, the high(est) seller type never obtains a positive expected payoff and, therefore, does not play an active role in the model. We demonstrate that the central lessons from our main model are not subject to this particular property by considering an alternative bargaining protocol. We study the case in which the seller makes a price offer to each arriving buyer.18 Due to the signaling nature of the seller’s price offers, this model admits a plethora of equilibria. For the purpose of the current exercise, instead of fully characterizing the set of all equilibria or delving into equilibrium selection, we construct a class of simple equilibria and show that they behave qualita18

In an earlier version, we studied two additional bargaining protocols: simultaneous announcement bargaining, as in Wolinsky (1990) and Blouin and Serrano (2001), and random proposals bargaining, as in Compte and Jehiel (2010) and Lauermann and Wolinsky (2016). Although each case requires separate analysis, the main qualitative equilibrium properties hold unchanged with both bargaining protocols.

35

tively in the same way as the unique equilibrium of our baseline model. Specifically, we restrict attention to equilibria in which the high-type seller always offers a fixed price p ∈ [cH , vH ).19 This significantly simplifies the analysis by effectively restricting the low-type seller to choose between p and vL at each point in time: it is strictly dominant for buyers to accept a price below vL , and thus the low-type seller can trade for sure if she offers vL . The high-type seller’s behavior implies that p is the only other price available to the low-type seller. A key observation for equilibrium characterization (and a distinguishing feature from our main model) is that the low-type seller must offer p and trade with a positive probability at all points in time. This is because in the current environment, a buyer updates his belief based not only on his signal, but also on the seller’s offer, and there does not exist a fully separating equilibrium. To be precise, let σS (t) denote the probability that the low-type seller offers vL . Then, the buyer’s belief conditional on offer p and signal s is given by q(t, p, s) ≡

s s q(t)γH q(t, p, s) q(t) 1 γH ⇔ = . s q(t)γH + (1 − q(t))(1 − σS (t))γLs 1 − q(t, p, s) 1 − q(t) 1 − σS (t) γLs

If the low-type seller never offered p (i.e., if σS (t) = 1), then the buyer would believe that the seller who offers p is the high type for sure (i.e., q(t, p, s) = 1) and, therefore, accept it with probability 1. If so, the low-type seller would strictly prefer offering p to vL , which is a contradiction. If p was rejected for sure, the low-type seller would have no incentive to offer p, which leads to the same contradiction. Stationary path: Let q ∗ denote the stationary path belief. In addition, denote by σS∗ the probability that the low-type seller offers vL and by σB∗ the probability that each buyer accepts p conditional on s = h on the stationary path. For similar reasons to those for the main model (see Section 3.1), these values can be derived from the following equilibrium conditions: • The low-type seller is indifferent between offering vL (left) and p (right), and thus rcL + λvL rcL + λγLh σB∗ p . = r+λ r + λγLh σB∗

(17)

• The buyer is indifferent between accepting and rejecting p conditional on s = h, and thus p=

h 1 γLh p − vL q ∗ γH vH + (1 − q ∗ )(1 − σS∗ )γLh vL q∗ = . ⇔ h h v −p 1 − q ∗ 1 − σS∗ q ∗ γH γH + (1 − q ∗ )(1 − σS∗ )γLh H

(18)

19 It is easy to support this behavior by appropriately specifying buyers’ off-the-equilibrium-path beliefs. For example, it suffices to assume that buyers believe that a seller who offers a price different from p is the low type for sure.

36

q(t)

q

q q∗

t

0

Figure 7: The evolution of buyers’ beliefs with the alternative bargaining protocol. The parameter values used for this figure are cL = 0, vL = 1, cH = 1.25, vH = 1.75, r = 0.25, λ = 0.8, h = γLl = 2/3, and p = 1.3. γH • The two seller types trade at an identical rate, and thus h ∗ γH σB = (1 − σS∗ )γLh σB∗ + σS∗ .

(19)

Equilibrium dynamics: Let q and q be the values such that p=

h qγH vH + (1 − q)γLh vL q γLh p − vL = , ⇔ h h v −p 1−q qγH + (1 − q)γLh γH H

(20)

p=

l qγH vH + (1 − q)γLl vL q γLl p − vL ⇔ = , l l v −p 1−q qγH + (1 − q)γLl γH H

(21)

and

respectively. In words, q is the minimal prior belief such that the buyer is willing to accept p conditional on s = h even if the low-type seller offers p with probability 1 (i.e., σS (t) = 0) and q h l is the corresponding value conditional on s = l. Since σS∗ > 0 and γLh /γH < 1 < γLl /γH , it is clear ∗ that q < q < q. Suppose that q(t) < q. In this case, the low-type seller must offer vL with a positive probability, because otherwise q(t, p, h) < q and, therefore, the buyer would reject p for sure. In addition, the buyer must randomize between accepting and rejecting p so as to keep the low type indifferent between the two offers. The two indifference conditions imply that the buyer, conditional on 37

s = h, accepts p with the same probability σB∗ as on the stationary path. In addition, σS (t) is determined by q q(t) 1 γ h p − vL = hL = . (22) 1 − q(t) 1 − σS (t) 1−q γH vH − p It then follows that q(t) evolves according to ∗

q(t)e−λγH σB dt q(t + dt) = , h ∗ h ∗ q(t)e−λγH σB dt + (1 − q(t))e−λ(σS (t)+(1−σS (t))γL σB )dt h

which is equivalent to h ∗ σB − σS (t) − (1 − σS (t))γLh σB∗ ). q(t) ˙ = −q(t)(1 − q(t))λ(γH

(23)

Importantly, q(t) increases over time (i.e., q(t) ˙ > 0) if and only if q(t) < q ∗ , which is because the probability 1 − σS (t) that the low type offers p increases in q(t) by equation (22) and q(t) ˙ = 0 if ∗ q(t) = q . Now suppose that q(t) > q. In this case, the buyer accepts p (at least) conditional on s = h, regardless of the seller’s offer strategy. It is then optimal for the low-type seller to offer p with probability 1. In turn, this implies that the buyer’s acceptance strategy depends only on his prior q(t): by the definition of q, he accepts p only conditional on s = h if q(t) ∈ (q, q) and independent of s if q(t) > q. Clearly, q(t) decreases according to q(t)e−λγH dt h q(t + dt) = ˙ = −q(t)(1 − q(t))λ(γH − γLh ) < 0 h dt h dt ⇔ q −λγH −λγL q(t)e + (1 − q(t))e h

in the former case and stays constant in the latter case. We summarize all the results so far in the following proposition. For conciseness, we present only how q(t) evolves over time in each region, omitting a lengthy but straightforward description of the equilibrium strategy profile corresponding to each qb. Proposition 7 Suppose that the seller makes price offers. For each p ∈ [cH , vH ), let q ∗ , q, and q be the values defined by equations (18), (20), and (21). There exists an equilibrium in which the high-type seller always offers p and the following properties hold: • If q(t) > q, then trade occurs at p independent of s and q(t) stays constant. • If q(t) ∈ (q, q), then trade occurs only at p and if and only if s = h. In this case, q(t) h decreases according to q(t) ˙ = −q(t)(1 − q(t))λ(γH − γLh ). • If q(t) ≤ q, then the low-type seller offers vL with probability σS (t) and p with probability 38

1 − σS (t), and the buyer accepts p with probability σB∗ , where σB∗ and σS (t) are defined by equations (17) and (22). In this case, q(t) changes according to equation (23), which yields q(t) ˙ > 0 if q(t) < q ∗ and q(t) ˙ < 0 if q(t) > q ∗ . Figure 7 illustrates how q(t) evolves in the equilibrium given in Proposition 7. Clearly, its dynamic patterns are similar to those of our main model. In particular, unless qb is sufficiently large (above q), q(t) monotonically converges to q ∗ , whether from above or below. One notable difference is that q(t) converges to q ∗ only asymptotically. This is because the low-type seller’s trading rate, which in the baseline model jumps at q ∗ = q, varies continuously around q ∗ in the current model. As q(t) tends to q ∗ , the difference between the two seller types’ trading rates h ∗ (γH σB − σS (t) − (1 − σS (t))γLh σB∗ ) vanishes and, therefore, q(t) cannot reach q ∗ in finite time.

6.3 Competitive Market Structure In our main model, the seller faces at most a single buyer at each time. In other words, each meeting is bilateral and, therefore, each buyer possesses temporary monopsony power. This is another driving force for the Diamond paradox (see footnote 9). We now introduce instantaneous competition among buyers, which is another way to overcome the Diamond paradox (Burdett and Judd, 1983), and demonstrate that our main insights continue to hold under this alternative market structure. We consider the case where a fixed number of buyers arrive simultaneously, observe a common signal, and offer prices competitively. This specification allows us to take a reduced-form approach about buyers’ offer strategies, as in, e.g., Daley and Green (2012) and Fuchs and Skrzypacz (2015). Let pa (t) denote the type-a seller’s reservation price at time t and ps (q) denote the expected buyer value of the good conditional on prior belief q and signal s, that is, ps (q) ≡

s vH + (1 − q)γLs vL qγH . s qγH + (1 − q)γLs

By the standard Bertrand competition logic, in equilibrium all buyers obtain zero expected payoff. This implies that trade can occur either at vL , if only the low-type seller is willing to trade, or at ps (q(t)), if both seller types are willing to trade. The following lemma provides a condition that distinguishes the two cases. Lemma 7 The competitive offer (i.e., bidding equilibrium outcome) is equal to ps (q(t)) if ps (q(t)) > pH (t) and equal to vL if ps (q(t)) < pH (t). Proof. Suppose that ps (q(t)) > pH (t) but there is a positive probability that the competitive offer is less than ps (q(t)) − ε for ε > 0. Then, a buyer can obtain a strictly positive expected payoff by 39

bidding a price between max{ps (q(t)) − ε, pH (t)} and ps (q(t)), which cannot arise in equilibrium. Conversely, if ps (q(t)) < pH (t), then the winning buyer obtains a non-negative expected payoff only when his bid is less than vL , but the competitive bid cannot be strictly lower than vL . In the current competitive environment, ps (q(t)) > cH when q(t) is sufficiently large and, therefore, the high-type seller may obtain a positive expected payoff. This difference, however, does not qualitatively change the equilibrium structure. We demonstrate this by constructing an equilibrium that behaves just as the unique equilibrium of our baseline model. We first identify a unique stationary path q ∗ and then construct an equilibrium strategy profile in which, unless qb is sufficiently large, q(t) converges to q ∗ whether from above or below. Stationary path: Given a stationary path belief q ∗ , the corresponding equilibrium behavior can be characterized as in Section 3.1, using the buyers’ indifference between ph (q ∗ ) and vL conditional on s = h, the low-type seller’s indifference between accepting and rejecting vL , and the equality of the two seller types’ trading rates. The only potential difference from the main model is that ph (q ∗ ) may not coincide with cH . This difference, however, does not materialize in equilibrium (i.e., ph (q ∗ ) = cH ) and, therefore, the corresponding equilibrium strategy profile is exactly identical to that of our main model (given in Lemma 2). To see this, observe that on any stationary path, the high-type seller’s reservation price is determined by r(pH (t) − cH ) = λγH σB (t)(ph (q ∗ ) − pH (t)). Therefore, ph (q ∗ ) > cH if and only if ph (q ∗ ) > pH (t). By Lemma 7, the competitive price conditional on s = h is ph (q ∗ ) with probability 1. But then, the low-type seller’s reservation price pL (t) exceeds vL , in which case q(t) cannot stay constant. Equilibrium dynamics: Given the unique stationary path, an equilibrium can be constructed as in the main model. If q(t) < q ∗ , then ph (q(t)) < cH , and thus only the low-type seller trades. In this case, q(t) increases according to q(t) ˙ = q(t)(1 − q(t))λ. The difference from our main model is that buyers’ (competitive) offers remain constant at vL along this path. Letting t∗ denote the time at which q(t) reaches q ∗ , the low-type seller’s reservation price at t < t∗ is given by ∫ pL (t) = cL +

t∗

∗ −t)

e−r(s−t) (vL − cL )d(1 − e−λ(s−t) ) + e−(r+λ)(t

(vL − cL ) < vL .

t

Therefore, the low-type seller has a strict incentive to trade until q(t) reaches q ∗ . There also exists q(> q ∗ ) such that if qb > q, then the seller trades with the first arriving buyers with probability 1. Unlike in the main model, the transaction price depends on the signal: it is 40

equal to ph (b q ) if s = h and equal to pl (b q ) if s = l. Clearly, this immediate trade outcome arises if and only if pH (t) ≤ pl (b q ). Since q is the lowest value with the property, it is characterized by the following indifference condition: pH (t) =

r λ l l p (q)) = pl (q). cH + (γ h ph (q) + γH r+λ r+λ H

(24)

In our main model, when q(t) ∈ (q ∗ , q), trade occurs if and only if s = h. This is not the case in the current competitive environment. Suppose qb is slightly smaller than q. If trade were to occur only when s = h, then q(t) would reach q ∗ in finite time and pH (0) would be strictly smaller than pl (b q ). But then, Lemma 7 implies that trade must occur when s = l as well, which is a contradiction. On the other hand, trade cannot occur with probability 1 conditional on s = l, because if so, q(t) would stay constant and pH (0) > pl (b q ). The above arguments suggest that if q(t) is not significantly smaller than q, then trade must occur conditional on s = l with an interior probability. Lemma 7 implies that this can arise only when pH (t) = pl (q(t)), so that the buyers may optimally randomize between pH (t) and vL . In such a case, pL (t) > vL and, therefore, trade takes place only at pH (t). Let σBl (t) denote the probability that the competitive offer is equal to pH (t) conditional on s = l. Since trade occurs with probability 1 conditional on s = h and with probability σBl (t) conditional on s = l, q(t) evolves according to q(t)e−λ(γH +γH σB (t))dt q(t + dt) = , h l l h l l q(t)e−λ(γH +γH σB (t))dt + (1 − q(t))e−λ(γL +γL σB (t))dt h

l

l

which reduces to h q(t) ˙ = −q(t)(1 − q(t))λ(γH − γLh )(1 − σBl (t)).

(25)

Both pH (t) and pl (q(t)) decrease over time. When σBl (t) ∈ (0, 1), they decrease at the same rate and, therefore, stay identical. In fact, σBl (t) is determined so as to preserve the equality between pH (t) and pl (q(t)). Naturally, there exists q ∈ (q ∗ , q) such that σBl (t) ∈ (0, 1) if q(t) ∈ (q, q), while σBl (t) = 0 if q(t) ∈ (q ∗ , q]. The identification of q is rather involved and, therefore, relegated to the appendix. We summarize all the results in the following proposition. As in the case of Proposition 7, we present only how q(t) evolves, avoiding a full description of the equilibrium strategy profile for each qb. Proposition 8 Suppose that the seller receives competitive price offers at Poisson rate λ and Assumption 1 holds. Let q ∗ be the same value as in Lemma 2 and q be the value defined by equation (24). There exist q ∈ (q ∗ , q) and an equilibrium such that the following properties hold: 41

ph (q(t)) pl (q(t)), pH (t)

ph (q ∗ ) = cH

pl (q ∗ ) vL 0

t1

t2

t

Figure 8: The evolution of pH (t) and ps (q(t)) under the competitive market structure when qb ∈ (q, q). The two dashed lines depict ph (q(t)) (upper) and pl (q(t)) (lower), while the solid line represents pH (t). The parameter values used for this figure are identical to those for Figure 1. • If q(t) > q, then trade occurs with the first arriving buyers and, therefore, q(t) stays constant. • If q(t) ∈ (q ∗ , q), then trade occurs with probability 1 conditional on s = h and with probability σBl (t) ∈ (0, 1) conditional on s = l. In this case, q(t) decreases according to equation (25). If q(t) ∈ (q, q) then σBl (t) ∈ (0, 1), while if q(t) ∈ (q ∗ , q] then σBl (t) = 0. • If q(t) < q ∗ , then trade occurs if and only if the seller is the low type and, therefore, q(t) increases according to q(t) ˙ = q(t)(1 − q(t))λ. • If q(t) = q ∗ , then trade occurs as in Lemma 2 and q(t) stays constant. Proof. See the appendix. Figure 8 illustrates equilibrium dynamics under the competitive market structure. It depicts how different prices evolve over time when qb ∈ (q, q).20 Until q(t) reaches q (i.e., until t1 in the figure), pl (q(t)) and pH (t) coincide, which is possible due to buyers’ randomization between pH (t)(= pl (q(t))) and vL conditional on s = l, which slows down the fall of q(t) relative to the case when trade occurs only when s = h. Once q(t) falls below q (i.e., between t1 and t2 ), as in the baseline model, trade occurs if and only if s = h and, therefore, q(t) continues to decrease. The evolution of prices when qb does not lie in (q, q) is analogous to that in the baseline model. For all cases, q(t) can be readily recovered from ps (q(t)). 20

42

During this period, ph (q(t)) falls faster than pH (t). They meet when q(t) reaches q ∗ (i.e., at t2 ), after which the game unfolds just as on the stationary path of the baseline model: buyers randomize between pH (t)(= cH ) and vL conditional on s = h, the low-type seller accepts vL with a positive probability, and q(t) stays constant at q ∗ .

7

Conclusion

We conclude by discussing various empirical implications and potential directions for future research.

7.1 Empirical Implications Our model environment is stylized, abstracting away from many important details in real markets. As always, a certain degree of abstraction is unavoidable to obtain clean and fundamental economic insights. On the other hand, taking such a model to data requires additional steps to account for various factors that are not present in the model. For instance, brokers and list prices play an important role in the real estate market (see, e.g., Horowitz, 1992; Merlo and Ortalo-Magn´e, 2004; Hendel et al., 2009). Unemployment durations are also affected by other factors, such as skill depreciation and worker discouragement (see, e.g., Pissarides, 1992; Gonzalez and Shi, 2010). Nevertheless, our model generates some novel and robust predictions regarding market outcomes, some of which are potentially testable. It is beyond the scope of this paper to develop a complete empirical strategy. We provide a list of potentially testable predictions of our model and discuss each of them briefly. As shown in Sections 3 and 6, buyers’ beliefs and (reservation) prices typically move in the same direction. In what follows, we say that the equilibrium trading dynamics exhibits the increasing (decreasing) pattern if they increase (decrease) over time. Prediction 1 The trading dynamics exhibits the decreasing pattern if an asset’s initial reputation is high and the increasing pattern if the initial reputation is low. This prediction restates our main result. Its simplicity is desirable for empirical purposes. One potential obstacle lies in the difficulty of measuring (initial) reputations. Although this is a non-trivial task in itself, there typically exist observable characteristics that are related to an asset’s (seller’s) reputation and, therefore, can be used to construct a reputation variable. For instance, the neighborhood, vintage, building company, and owner history provide information about a property’s quality. Similarly, a worker’s education and prior employment histories would affect his reputation in the labor market. 43

Our next prediction links the pattern of trading dynamics to the evolution of trading probability (equivalently, volume) over time. Prediction 2 If the trading dynamics exhibits the decreasing (increasing) pattern, then the overall trading probability also decreases (increases) initially. The result follows from the fact that the frequency with which buyers offer cH is an increasing function of their beliefs. If qb > q ∗ , then trade occurs only at cH until q(t) reaches q ∗ . This immediately implies that trade occurs less frequently over time. If qb < q ∗ , then the low-type seller trades at a constant rate of λ until q(t) reaches q ∗ , while the trading rate of the high-type seller weakly increases. This co-movement of trading pattern and trading probability does not extend into the full time horizon: it is valid along the convergence path, but not at the moment of convergence (i.e., when q(t) reaches q ∗ ). The trading rate of the high-type seller is always monotone over time. However, when qb < q ∗ (i.e., the increasing pattern), the trading rate of the low type jumps down from λ to ρ at the end of the convergence path. If qb > q ∗ (i.e., the decreasing h ∗ σB . Depending on the parameter pattern), then the trading rate changes from λγLl to ρ = λγH values, the latter can exceed the former. We now relate the pattern of trading dynamics to three market characteristics. Prediction 3 The decreasing pattern is more likely to arise with little gains from trade of low quality (i.e., relatively small vL − cL ), small search frictions (i.e., large λ), and a good inspection technology (i.e., informative buyer signals). Each of these observations is immediate from the three main characterization sections (Sections 3-5). These results could be useful in interpreting both cross-sectional data and time series. For example, if both the search technology and the inspection technology have improved over time, then the decreasing pattern is more likely to arise in recent data than in old data. Our final prediction is concerned with the relationship between trading dynamics and the nature of inspection. Specifically, we compare the case when inspection is mainly about finding a fatal flaw (red flag) and the case when inspection may reveal a particular merit of the asset (green flag). Formally, we compare the following two cases: for γ, ϵ > 0, Red flag l γH = ϵ, h γH = 1 − ϵ,

Green flag l = 1 − ϵ, γH = 1 − γ, h h γL = ϵ, γH = γ.

γLl = γ, γLh = 1 − γ,

γLl

If ϵ is sufficiently small, then l (red flag) is a sufficiently informative signal about low quality in the case represented by the left-hand column, while h (green flag) is a sufficiently good signal about high quality in the case represented by the right-hand column. Assuming that γ is not particularly 44

large (precisely, λ(1 − γ) > ρL ), the following prediction is straightforward to obtain from the characterization results in Section 3 and Section A in the online appendix. See Section E in the online appendix for formal arguments. Prediction 4 The decreasing pattern is more likely to arise when the inspection technology is of a red-flag kind than when it is of a green-flag kind.

7.2 Directions for Future Research Our findings suggest several directions for future research. One of our maintained assumptions is that buyers’ offers are private and not observable to future buyers. H¨orner and Vieille (2009) show that it is a crucial assumption and the equilibrium dynamics dramatically changes if buyers’ offers are public. In particular, with public offers, “bargaining impasse” can arise. As also suggested by H¨orner and Vieille (2009), it could be interesting to allow for buyer inspection in the model with public offers and investigate its impact on bargaining outcome. Our model is a dynamic trading model with random search, but there is a growing literature on directed search with adverse selection (see, e.g., Guerrieri et al., 2010; Guerrieri and Shimer, 2014; Chang, 2017). Introducing buyer inspection into models of directed search might also lead to interesting insights or predictions. Finally, we assume, crucially, that buyers are short-lived. This assumption can be relaxed in various ways. For example, one can consider a market environment in which there are many sellers and buyers and all agents go through sequential search until they trade (see, e.g., Wolinsky, 1990; Blouin and Serrano, 2001; Moreno and Wooders, 2010). Such a model, whether stationary or non-stationary, would embed our model into a market setting and endogenize buyers’ outside options. Another possibility is to introduce and endogenize buyers’ optimal timing decisions (i.e., when to arrive and make an offer to the seller). Clearly, it would influence the informational content of time-on-the-market and, therefore, potentially make buyers’ inference problems even more intriguing.

Appendix: Omitted Proofs Proof of Lemma 1. If p(t) < vL , then in equilibrium the low-type seller accepts p(t) with probability 1: otherwise, the buyer could offer a slightly higher price than p(t), which would increase the low-type seller’s acceptance probability to 1 and, therefore, give a strictly higher expected payoff to the buyer. Since σS (L, t, p(t)) = 1, the law of motion for q(t) is given by ( ) ∑ s q(t) ˙ = −q(t)(1 − q(t))λ γH σB (t, s, cH ) − 1 ≥ 0. s

45

∑ s Notice that q(t) ˙ > 0 as long as s γH σB (t, s, cH ) < 1 (i.e., the probability that the high type trades conditional on buyer arrival is less than 1). If p(t) > vL , then trade takes place only at cH : p(t) is accepted only by the low-type seller, but no buyer would be willing to pay more than vL for a low-quality asset. Therefore, such p(t) must be rejected in equilibrium. In this case, q(t) evolves according to ∑ s q(t) ˙ = −q(t)(1 − q(t))λ (γH − γLs )σB (t, s, cH ). s

For the buyer’s optimality, he should offer cH with a higher probability when s = h than when h s = l (i.e., σB (t, h, cH ) ≥ σB (t, l, cH )). Combining this with the fact that γH > γLh , it follows that q(t) ˙ ≤ 0 and the inequality holds strictly as long as σB (t, l, cH ) ̸= 1 (i.e., unless buyers always offer cH and, therefore, the probability of trade conditional on buyer arrival is equal to 1). Lemma 8 Given any strategy profile (such that all buyers’ offers are between cL and vH ), q(·) and p(·) are continuous. Proof. Fix t and consider ∆ > 0. q(t) increases fastest when the low-type seller trades whenever a buyer arrives (i.e., at rate λ), while the high-type seller does not trade at all and decreases fastest when the opposite holds. Therefore, q(t) q(t)e−λ∆ ≤ q(t + ∆) ≤ . −λ∆ q(t)e + (1 − q(t)) q(t) + (1 − q(t))e−λ∆ As ∆ tends to 0, both bounds converge to q(t). Therefore, lim∆→0 q(t + ∆) = q(t). An analogous argument can be used also to show that lim∆→0 q(t − ∆) = q(t). Given p(t + ∆), p(t) is maximized when the low-type seller receives the highest possible price vH whenever a buyer arrives between t and t + ∆ and minimized when she does not receive any offer between t and t + ∆. Therefore, ∫ ∆ −r∆ e (p(t + ∆) − cL ) ≤ p(t) − cL ≤ e−r(s−t) (vH − cL )d(1 − e−λ(s−t) ) + e−r∆ (p(t + ∆) − cL ). 0

From these bounds, it follows that lim∆→0 p(t + ∆) = p(t). Again, an analogous argument can be used also to show that lim∆→0 p(t − ∆) = p(t). Proof of Lemma 3. We establish the result in three steps. (1) If q(t) < q ∗ , then p(t) < vL . Suppose q(t) < q ∗ , but p(t) > vL . First we establish that there exists t′ > t such that p(t′ ) ≤ vL . Suppose, towards a contradiction, that for all t′ > t, p(t′ ) > vL . Then, by Lemma 1, q(·) cannot be strictly increasing and, therefore, q(t′ ) < q ∗ for all t′ > t. This implies that trade can occur neither at cH (because q(t′ ) < q ∗ ) nor at p(t′ ) (because p(t′ ) > vL ). But then, p(t) = cL < vL , which is a contradiction. Let t′ = inf{t′′ ≥ t|p(t′′ ) ≤ vL }, so that for any x ∈ (t, t′ ), p(x) > vL . By Lemma 1, for any such x, q(x) ≤ q(t) < q ∗ and, therefore, the buyer never offers cH . This, together with p(t′ ) ≤ vL , implies that p(t) < vL , which is a contradiction. Now suppose q(t) < q ∗ , but p(t) = vL . By continuity of q(·), there exists ε > 0 such that for all t′ ∈ (t, t + ε], q(t′ ) < q ∗ . Since no buyer offers cH between t and t + ε, vL = p(t) = 46

(1 − e−rε )cL +e−rε p(t + ε) < p(t + ε). This means that q(t + ε) < q ∗ while p(t + ε) > vL , which was ruled out above. (2) If q(t) > q ∗ , then p(t) > vL . Suppose q(t) > q ∗ , but p(t) < vL . First we establish that there exists t′ > t such that p(t′ ) ≥ vL . Suppose, towards a contradiction, that for all t′ > t, p(t′ ) < vL . Then, letting ρa (t) denote the rate at which type-a seller trades at time t, we have ρL (t′ ) = λ for all t′ > t. Moreover, by Lemma 1, q(·) is non-decreasing over time, which implies that there exists q∞ such that limt→∞ q(t) = q∞ . Such convergence can occur only if limt→∞ ρH (t)/ρL (t) = 1. Since ρL (t′ ) = λ for all t′ > t, this implies that limt→∞ ρH (t) = λ. Since the high type trades only at price cH , this implies that the unconditional (on signal realization) probability of each seller offering cH converges to 1, which in turn implies that conditional on each signal, this probability approaches 1. But then, p(t) → (rcL + λcH )/(r + λ) > vL , where the inequality follows by Assumption 1, which is a contradiction. Let t′ = inf{t′′ ≥ t|p(t′′ ) ≥ vL }, so that for any t′′ ∈ (t, t′ ), p(t′′ ) < vL . By Lemma 1, and also noting that buyers always offering cH leads to a contradiction to p(t) < vL , for all t′′ ∈ (t, t′ ), we have q(t′′ ) > q(t) > q ∗ . Consider a buyer arriving at such t′′ < t′ with s = h. When t′′ is sufficienly close to t′ , this buyer’s payoff from offering p(t′′ ) is almost 0, as p(t′′ ) is almost vL , by continuity of p(·). In contrast, such a buyer’s payoff from offering cH is bounded away from 0, since q(t′′ ) > q(t) > q ∗ . Therefore, there exists t′′′ < t′ such that for all t′′ ∈ (t′′′ , t′ ), a buyer arriving at t′′ with s = h offers cH with probability 1. Then, ′′′

∫

p(t ) = cL +

t′ −t′′′

′

′′′

e−rx (cH − cL )d(1 − e−λγL x ) + e−(r+λγL )(t −t ) (vL − cL ) > vL , h

h

0

where the last inequality follows by Assumption 1. This is a contradiction establishing that if q(t) > q ∗ , then p(t) ≥ vL . Now suppose q(t) > q ∗ , but p(t) = vL . By continuity of q(·), there exists ε > 0 such that for all t′ ∈ (t − ε, t + ε), q(t′ ) > q ∗ . Next we claim that there exists t′′ ∈ (t, t + ε) such that p(t′′ ) < vL . Suppose not. Then any buyer arriving at t′′′ ∈ (t, t + ε) offers cH at least when s = h, as this generates a positive expected payoff while offering p(t′′′ ) ≥ vL generates a non-positive expected payoff. But then, by Assumption 1, p(t) > vL , which is a contradiction. It then follows that there exists t′′ ∈ (t, t + ε) such that p(t′′ ) < vL while q(t′′ ) > q ∗ , whose possibility was ruled out above. (3) If q(t) = q ∗ , then p(t) = vL . Suppose that q(t) = q ∗ but p(t) > vL . Without loss of generality, let t be the first time at which trade occurs with a positive probability. The buyer at t with s = l makes a losing offer since offering p(t) or cH generates a negative expected payoff. Then, the probability of trade at time t is strictly less than 1. Thus, by Lemma 1, q(t) ˙ < 0. By continuity of q(·) and p(·), there exists ′ ′ ∗ ′ t such that q(t ) < q and p(t ) > vL , which was ruled out above. Now suppose that q(t) = q ∗ but p(t) < vL . In this case, the buyer strictly prefers offering p(t) to cH , regardless of his signal. By Lemma 1, q(t) must be decreasing at t. By continuity of q(·) and p(·), there exists t′ such that q(t′ ) > q ∗ but p(t) < vL , which was also ruled out above. Proof of Theorem 1. We first argue that if q(t) = q ∗ after any history, then for all t′ > t, q(t′ ) = q ∗ . It suffices to show that q(t) ˙ = 0 whenever q(t) = q ∗ . For a contradiction, suppose q(t) ˙ > 0 (respectively, 47

q(t) ˙ < 0). Then, by continuity of q(·) there exits ε such that for all t′ ∈ (t, t + ε), q(t′ ) > q ∗ (respectively, q(t′ ) < q ∗ ). Moreover, for all such t′ , by Lemma 3 we have p(t′ ) > vL (respectively, ˙ ′ ) ≤ 0 (respectively, q(t ˙ ′ ) ≥ 0). This implies that p(t′ ) < vL ), which implies by Lemma 1 that q(t ′ ∗ limε′ →0 q(t + ε ) ≥ q(t + ε/2) > (limε′ →0 ≤ q(t + ε/2) <) q , contradicting the continuity of q(·). Hence we have established that if q(t) = q ∗ after any history, then for all t′ > t, q(t′ ) = q ∗ . Then, the construction described in Section 3.1 uniquely pins down the subsequent equilibrium behavior. Now, by the previous argument and the continuity of q(·), if qb < q ∗ , then q(t) ≤ q ∗ for all t. Let t∗∗ = inf{t|q(t) = q ∗ } if {t|q(t) = q ∗ } ̸= ∅. Otherwise, set t∗∗ = ∞. Then, by Lemma 3, for any t < t∗∗ , p(t) < vL . Moreover, by the definition of q ∗ , for any t < t∗∗ the buyer is not willing to offer cH . Then the buyer at t < t∗∗ offers p(t) regardless of his signal. Then, the low-type seller’s reservation price is uniquely pinned down as in Proposition 1 and t∗∗ = t∗ as defined in the same Proposition. Again by the above argument and continuity of q(·), if q > qb > q ∗ , then q > q(t) ≥ q ∗ for all t. Let t∗∗ = inf{t|q(t) = q ∗ } if {t|q(t) = q ∗ } ̸= ∅. Otherwise, set t∗∗ = ∞. Then, by Lemma 3, for any t < t∗∗ , p(t) > vL . Then, for any t < t∗∗ trade never takes place at p(t). Then by the definitions of q and q ∗ , the buyer at t < t∗∗ offers cH if and only if the realized signal is h. Then, the low-type seller’s reservation price is uniquely pinned down as in Proposition 2 and t∗∗ = t∗ as defined in the same proposition. Finally, assume qb > q. We show that for all t, q(t) > q. Suppose for a contradiction that there exists t with q(t) ≤ q. Then, by continuity of q(·), there exists a minimum t∗ such that q(t∗ ) = q. For all t < t∗ , q(t) > q, and thus p(t) > vL (by Lemma 3). Then, for such t, it is optimal for the buyer to offer cH regardless of his signal. This means that both seller types trade at the same rate and, therefore, q(t) = qb > q until t∗ , which contradicts the continuity of q(·) at t∗ . This argument also establishes that if qb > q, then q(t) = qb for all t and therefore the equilibrium strategies are necessarily as claimed. Proof of Lemma 4. If Γ is more informative than Γ′ , then there exists a non-negative Markov matrix M such that Γ′ = M Γ. Without loss of generality, let ( ) 1 − εl εl M= . 1 − εh εh Then, ( ′

Γ = It follows that

and

l′ γLl′ γH h′ γLh′ γH

)

( = MΓ =

h l + εl γH (1 − εl )γLl + εl γLh (1 − εl )γH l h εh γLl + (1 − εh )γLh εh γH + (1 − εh )γH

) .

l′ l l h l γH (1 − εl )γH γH + εl γH (1 − 2εl )γH + εl = ≥ , = γLl′ (1 − εl )γLl + εl γLh (1 − 2εl )γLl + εl γLl h′ l h h h γH εh γH + (1 − εh )γH εh + (1 − 2εh )γH γH = = ≤ . γLh′ εh γLl + (1 − εh )γLh εh + (1 − 2εh )γLh γLh

l h The inequalities are due to the fact that γH /γLl < 1 < γH /γLh (MLRP). l l l′ l′ h h h′ Now suppose that γH /γL ≤ γH /γL and γH /γL ≥ γH /γLh′ . Then, there exist εl and εh such that

48

εl , εh ∈ [0, 1] and l′ l h h′ l h γH (1 − εl )γH + εl γH γH εh γH + (1 − εh )γH = and = . γLl′ (1 − εl )γLl + εl γLh γLh′ εh γLl + (1 − εh )γLh

(

It then suffices to set M=

1 − εl εl εh 1 − εh

) .

Proof of Proposition 3. By Proposition 1, { { 1 − e−λt if t < t∗ (b q ), 0 FL (t) = and FH (t) = ∗ −λt∗ (b q )−ρ(t−t∗ (b q )) 1−e otherwise, 1 − e−ρ(t−t (bq))

if t < t∗ (b q ), otherwise.

The result for τH is immediate from the fact that ρ increases, while t∗ (b q ) decreases (which implies FH (t) weakly increases at any t). τL does not decrease in the sense of first-order stochastic dominance because FL (t) decreases when t = t∗ (b q ). Consider the expected value of τL : ∫

t∗ (b q)

td(1 − e

E[τL ] =

−λt

∫

−λt∗ (b q)

∞

)+e

t∗ (b q)

0

td(1 − e

−ρ(t−t∗ (b q ))

1 − e−λt )= λ

∗ (b q)

∗

e−λt (bq) + ρ

∗

The second term e−λt /ρ is independent of Γ, because ρ=

h h qb 1 − q ∗ qb γH γH r(vL − cL ) cH − vL −λt∗ (b q) and e = = . h c −v h ∗ 1 − qb q 1 − qb γL vH − cH γL H L

Then, the desired result (that E[τ ] decreases as Γ becomes more informative) follows from the fact ∗ that the first term (1 − e−λt (bq) )/λ increases in t∗ (b q ). The result on p(0) is straightforward from the ∗ −λt (b q) fact that p(0) = cL + e (vL − cL ). Proof of Lemma 5. From equation (7), e

2λt∗ (b q)

( =

where C=

γ C 1−γ

1 ) γ−1/2

,

qb vH − cH . 1 − qb cH − vL

Then, ∗

de2λt = dγ

( C

γ 1−γ

1 ) γ−1/2 (

) ) ( 1 1 1 γ − + . log C (γ − 1/2)2 1−γ γ − 1/2 γ(1 − γ)

49

In order to find the condition under which dt∗ (b q )/dγ > 0, it suffices to find the condition for ( ) γ − 1/2 γ > logC + log . (26) γ(1 − γ) 1−γ Define a function H : [1/2, 1) → R so that γ − 1/2 H(γ) = − log γ(1 − γ)

(

γ 1−γ

) .

Then, H(1/2) = 0 and ) ( 1 1 γ 2 − (1 − γ)2 > 0 whenever γ > . H (γ) = γ − 2 2 2 γ (1 − γ) 2 ′

It then follows that inequality (26) holds for any γ > 1/2 if C ≤ 1. If C > 1, then there exists γ(b q ) such that H(γ) > log C (and, therefore, t∗ (b q ) increases in γ) if and only if γ > γ(b q ). Notice ∗ ∗ that qb is defined to be the value such that C = 1 when qb = qb . Proof of Proposition 4. For notational simplicity, we denote t∗ (b q ) simply by t∗ . Recall that if qb ∈ (q ∗ , q), then { { 1 − e−λ(1−γ)t if t < t∗ , 1 − e−λγt if t < t∗ , FL (t) = and F (t) = H ∗ ∗ ∗ ∗ 1 − e−λ(1−γ)t −ρ(t−t ) otherwise, 1 − e−λγt −ρ(t−t ) otherwise, and ρ=

γ r(vL − cL ) . 1 − γ cH − vL

From thesecolorblue and using the definition of C in the proof of Lemma 5, we get [ ] 1 1 1 E[τH ] = PH (γ) − , + ρ λγ λγ [ ] 1 1 1 E[τL ] = PL (γ) − + , and ρ λ(1 − γ) λ(1 − γ) p(0) = (1 − D(γ))(P¯ (γ) − vL ) + vL , , where ( PH (γ) =

1 1−γ C γ

γ ) 2γ−1

( , PL (γ) =

1 1−γ C γ

50

1−γ ) 2γ−1

( , D(γ) =

1 1−γ C γ

)A(γ) ,

with A(γ) = (r +λ(1−γ))/(λ(2γ −1), and P¯ (γ) = (rcL +λ(1−γ)cH )/(r +λ(1−γ)). Therefore, ( ) ( ) ∂E[τH ] ∂PH (γ) 1 1 ∂ρ/∂γ −1 1 = − + PH (γ) − − 2 − 2, 2 ∂γ ∂γ ρ λγ ρ λγ λγ ( ) ( ) ∂E[τL ] ∂PL (γ) 1 1 ∂ρ/∂γ 1 1 = − + PL (γ) − − + , 2 2 ∂γ ∂γ ρ λ(1 − γ) ρ λ(1 − γ) λ(1 − γ)2 ) ∂ P¯ (γ) ∂p0 ∂D(γ) ( ¯ = − P (γ) − vL + (1 − D(γ)). ∂γ ∂γ ∂γ Finally, we note that ( ) ∂PH (γ) 1 log(PH (γ)) 1 = − PH (γ) + , ∂γ 2γ − 1 γ 1−γ ( ) ∂PL (γ) 1 log(PL (γ)) 1 = − PL (γ) + , and ∂γ 2γ − 1 1−γ γ ( ) ∂D(γ) 1 2r + λ r + λ(1 − γ) = − D(γ) log(D(γ)) + . ∂γ 2γ − 1 r + λ(1 − γ) λγ(1 − γ) First consider the case when qb is sufficiently close to q. As qb → q, PH (γ), PL (γ), D(γ) → 1. Therefore, ( ) 1 1 1 1 ∂E[τH ] ∂ρ/∂γ →− − . − ∂γ 2γ − 1 1 − γ ρ λγ ρ2 Then, the claim for τH follows by noting that ∂ρ/∂γ > 0 and ρ < λγ. Also, ( ) ∂E[τL ] 1 1 1 1 ∂ρ/∂γ →− − − , ∂γ 2γ − 1 γ ρ (1 − γ)λ ρ2 which simplifies to 1 − (1 − γ)(2γ − 1)

(

1 1 − ρ λ

) .

Then, the claim for τL follows because ρ < λ. Finally, 1 r + λ(1 − γ) ¯ ∂p(0) → (P (γ) − vL ). ∂γ 2γ − 1 λγ(1 − γ) The result follows simply by P¯ (γ) > vL . Next, consider the case when qb is sufficiently close to q¯ and γ is sufficiently close to 1/2. As qb → q¯, we have 1/C × (1 − γ)/γ → ((1 − γ)/γ)2 . Then, when also γ → 1/2, we have PH (γ) → exp(−2), PL (γ) → exp(−2), D(γ) → exp(−2(2r + λ)/λ). H ] ∂E[τL ] , ∂γ → +∞ with positive coefficients (note The results for τH and τL follow because ∂E[τ ∂γ that for the case of τL , the coefficient is positive if and only if r(vL − cL ) < (1/2)λ(cH − vL ), which is exactly requirement (1), when γ = 1/2) and all other terms remain finite. For the result on p(0), we note that since (1 − D(γ)∂ P¯ )/∂γ remain finite in these limits, and P¯ − vL > 0, it

51

is sufficient to argue that ∂D(γ)/∂γ → +∞. Substituting the expression for ρ and rearranging terms, we find that the coefficient of the term 1/(2γ − 1) in the expression for ∂D(γ)/∂γ > 0 approaches 2 2r+λ > 0, establishing that ∂D(γ)/∂γ → +∞ as γ → 1/2 and qb → q¯. λ ∗ Proof of Lemma 6. In order to prove the necessity of condition (16), first observe that σM = ∗ (γH − γM )σH /γM (from (13)). Plugging this into (11) and combining it with (12) yield

p∗M − cM γM (cH − p∗M ) ( ). = v L − cL ∗ M (p − v ) γL cH − vL + γHγ−γ L M M The solution p∗M to this equation is strictly less than vM if and only if (16) holds: the left-hand side is necessarily smaller than the right-hand side when p∗M = cM . In addition, the left-hand side is increasing, while the right-hand side is decreasing, in p∗M . Therefore, it is necessary and sufficient that the left-hand side is larger when p∗M = vM , which is equivalent to (16) in the lemma. Since p∗M < vM is necessary for the existence of a vector q ∗ , (16) is also necessary. For sufficiency when λ is sufficiently large, we show that given (16) which ensures p∗M < vM , ∗ = (r/λγM )((p∗M −cM )/(cH −pM )), all other equilibrium variables are well-defined. From (12), σH which is positive because p∗M < vM and is less than 1 when λ is sufficiently large. From (13), ∗ ∗ ∗ → 0 as /γM , which is well-defined when λ is sufficiently large because σH = (γH − γM )σH σM ∗ ∗ ∗ λ → ∞. σM and γH > γM . (13) yields a unique value of σS and also guarantees that σS ∈ (0, 1), ∗ ∗ can < 1 (which implies σS∗ < 1). Finally, qH because γM > γL (which implies σS∗ > 0) and γH σH be obtained and similarly shown to be well-defined with equation (15). Proof of Proposition 7. If qb ≤ q or qb > q, then the optimality of each player’s strategy is straightforward from the construction of the equilibrium strategy profile. For the case when qb ∈ (q, q), let t∗ be the value such that h ∗

q=

qbe−λγH t h ∗ h ∗. qbe−λγH t + (1 − qb)e−λγL t

Then, for any t < t∗ , the low-type seller offers only p, and the buyer accepts p with probability 1 conditional on s = h. If t ≥ t∗ , then the players play as in the case when qb ≤ q. In the latter region, the players’ incentives are straightforward: the low-type seller is indifferent between offering p and vL , and each buyer is indifferent between accepting and rejecting p conditional on s = h. Therefore, we focus on the former region. The optimality of the buyer’s acceptance strategy is immediate from the fact that q(t) ∈ (q, q) and σS (t) = 0 (which together imply that q(t, p, s) = q(t) ∈ (q, q)). For the seller’s optimality, let pL (t) denote the low-type seller’s continuation payoff at time t. Given the strategy profile, pL (t∗ ) = and for each t < t∗ , ∫ pL (t) = cL +

t∗ −t

rcL + λγLh σB∗ p rcL + λvL = r+λ r + λγLh σB∗

∗ −t)

e−rx (p − cL )d(1 − e−λγL x ) + e−(r+λγL )(t h

0

52

h

(pL (t∗ ) − cL ) > pL (t∗ ),

where the inequality follows from σB∗ < 1 (i.e., buyers accept p with a lower probability after t∗ ). Conditional on facing a buyer at time t < t∗ , the low-type seller’s expected payoff is equal to γLh p + (1 − γLh )pL (t) if he offers p (because p is accepted only when s = h) and equal to vL if he offers vL (because vL is always accepted). It suffices to show that the former payoff is larger than the latter payoff. The result follows from γLh p + (1 − γLh )pL (t) > γLh p + (1 − γLh )pL (t∗ ) > γL σB∗ p + (1 − γL σB∗ )pL (t∗ ) = vL where the second inequality is due to the fact that p > pL (t∗ and σB∗ < 1, while the equality derives from equation (17). Proof of Proposition 8. As explained in the main text, q ∗ is given as in Lemma 2, and the equilibrium strategy profile for the case when qb < q ∗ can be immediately constructed. We now identify q. If q(t) ∈ (q ∗ , q), then trade occurs if and only if s = h. Let t∗ be the value such that h ∗ qe−λγH t q −λ(γ h −γ h )t∗ q∗ ∗ H L q = −λγ h t∗ ⇔ = e . h ∗ 1 − q∗ 1−q qe H + (1 − q)e−λγL t If qb = q, then pH (t) decreases according to h r(pH (t) − cH ) = λγH (ph (q(t)) − pH (t)) + p˙H (t) whenever t < t∗ .

If q is equal to q ∗ , then pH (0) = cH = ph (q) > pl (q). On the other hand, if q is close to 1, then pH (0) <

h vH + rcH λγH < vH ≈ pl (q). h r + λγH

Therefore, there exists q ∈ (q ∗ , 1) such that pH (0) = pl (q). We define q to be the lowest such value, so that pH (t) > pl (q(t)) for any t. Given q, the unique equilibrium strategy profile for the case when qb ∈ (q ∗ , q) can be derived as in Proposition 2. As in the above case, the only difference is that the seller receives ph (q(t)), not cH , conditional on s = h. Next, we determine q. As explained in the main text, it is defined by equation (24). In order to show that q is well-defined, first notice that r λ rcH + λvH h h l l cH + (γH p (1) + γH p (1)) = < vH = pl (1), r+λ r+λ r+λ and

r λ l l p (q)) > pl (q). cH + (γ h ph (q) + γH r+λ r+λ H

The second inequality comes from the fact that when qb = q, pH (0) = pl (q) and ph (q(t)) < ph (q)

53

for any t > 0. Finally, the above equation can be rewritten as h ph (q) − cH r + λγH = . h pl (q) − cH λγH

The left-hand side takes an inverted-U shape over [0, 1]. By the above two inequalities, the lefthand side is larger if q = q and smaller if q = 1. Therefore, there exists a unique value of q ∈ (q, 1) that satisfies equation (24). It is clear that if qb > q, then it is the unique equilibrium that the competitive offer is always s p (b q ) and, since pH (t) < pl (b q ), both seller types trade whether the price is ph (b q ) or pl (b q ). Finally, we consider the case when qb ∈ (q, q). In this region (until q(t) reaches q), pH (t) = pl (q(t)) and q(t) decreases according to equation (25). Given pH (t) = pl (q(t)) when q(t) = q, it suffices to show that whenever q(t) ∈ (q, q), there exists σBl (t) ∈ (0, 1) such that p˙H (t) = p˙l (q(t)). For notational simplicity, let ps = ps (q(t)). Then, −p˙l (q(t)) = pl′ (q(t))q(t) ˙ =

(vH − pl )(pl − vL ) h − γLh )(1 − σBl (t)). λ(γH vH − vL

while h −p˙H (t) = λγH (ph − pl ) − r(pl − cH ) =

h − γLh )(vH − pl )(pl − vL ) (γH λγH − r(pl − cH ). l h l h l l vH − vL γL γH (p − vL ) + γH γL (vH − p )

Obviously, −p˙l (q(t)) < −p˙H (t) if σBl (t) = 1. We show that σBl (t) = 0 implies −p˙l (q(t)) > −p˙H (t) whenever q(t) ∈ (q, q). Define H(pl ) =

h (vH − pl )(pl − vL ) − γLh )(vH − pl )(pl − vL ) (γH λγH h −r(pl −cH ). λ(γH −γLh )− l h h vH − vL vH − vL γLl γH γL (vH − pl ) (pl − vL ) + γH

For our purpose, it suffices to show that H(pl ) > 0 whenever pl ∈ (pl (q), pl (¯ q )). Define [ ] h l h ˜ l ) ≡ γLl γH H(p (pl − vL ) + γH γL (vH − pl ) H(pl ). ˜ is a cubic function of pl and, therefore, it has at most three roots. Now observe that Note that H(·) ˜ ˜ l (q)) > 0 (because pH (t) > pl (q(t)) whenever q(t) ∈ (q ∗ , q)), and limpl →∞ = H(vL ) < 0, H(p ˜ is a cubic function) imply that there are two roots less −∞. These (together with the fact that H l ˜ than p (q). In addition, H(vH ) > 0 implies that there is the last root above vH . All together, ˜ l ) > 0 whenever pl ∈ (pl (q), pl (¯ we know that H(p q )) ⊂ (pl (q), vH ), from which it follows that h l h H(pl ) > 0 whenever pl ∈ (pl (q), pl (¯ q )), because γLl γH (pl − vL ) + γH γL (vH − pl ) > 0.

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