Selling a Lemon under Demand Uncertainty∗ Kyungmin Kim† and Sun Hyung Kim‡ August 2017

Abstract We study a dynamic pricing problem of the seller who has private information about the quality of her good but is uncertain about the arrival rate of buyers. Restricting attention to the equilibria in which the high-quality seller insists on a constant price, we show that the lowquality seller’s expected payoff and equilibrium pricing strategy crucially depend on buyers’ knowledge about the demand state. If they are also uncertain about demand, then demand uncertainty increases the low-quality seller’s expected payoff, and her optimal pricing strategy is to offer a high price initially and drop it to a low price later. If buyers know the demand state, then demand uncertainty does not affect the low-quality seller’s expected payoff, and a simple cutoff pricing strategy cannot constitute an equilibrium. In the latter case, we show that there exists an equilibrium in which the low-quality seller begins with a low price, switches up to a high price, and eventually reverts back to the low price. JEL Classification Numbers: C73, C78, D82, D83, L15. Keywords: Adverse selection; market for lemons; demand uncertainty; dynamic pricing.

1 Introduction Consider a seller who wishes to sell a used car. Due to her experience with the car, she is likely to be better informed about the car than potential buyers. This adverse selection produces risks for buyers, thereby complicating their purchase decisions. Meanwhile, the seller is likely to have uncertainty about her demand. She may not have precise information about the aggregate state of the economy or the general popularity of her car. This is arguably the reason why most used car ∗

We thank Harold Cole and three anonymous referees for various helpful suggestions and comments. We are also grateful to Raphael Boleslavsky, Ayc¸a Kaya, David Kelly, Lucas Maestri, Latchezar Popov, Christian Traeger, and seminar audiences at various places. † University of Miami. Contact: [email protected] ‡ University of Iowa. Contact: [email protected]

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sellers refer to price information services such as Kelly Blue Book. Both adverse selection and demand uncertainty have been extensively, but separately, studied in the literature. The goal of this paper is to understand the interplay between adverse selection and demand uncertainty and investigate its implications for price dynamics in a tractable dynamic trading environment. We study a dynamic pricing problem facing a seller who wishes to sell an indivisible object. She sets a price and can adjust it at any point in time at no cost. Buyers arrive sequentially, observe the posted price, and decide whether to purchase the good or not. The seller has private information about the quality of the good, which is either high or low. In the meantime, she faces uncertainty about demand. Specifically, she is uncertain whether the arrival rate of buyers is high or low. Buyers may or may not be informed about the demand state. We focus on the equilibria in which the seller insists on a constant price if her good is of high quality. Among others, this implies that at any point in time, the low-quality seller effectively chooses only between two prices, one which is charged by the high-quality seller and the other which is optimal conditional on her type being revealed. This incurs no loss of generality in the benchmark environment without demand uncertainty, as elaborated at the end of Section 3. In the presence of demand uncertainty, it imposes a rather strong restriction. However, it ensures that the model remains tractable. In particular, it allows us to analyze the low-quality seller’s nonstationary pricing strategy, which is impossible to characterize in general. In addition, as shown shortly, despite such a strong restriction, a rich set of dynamic pricing patterns emerge in our model. There is still a continuum of prices that can be employed as the high price. Instead of selecting a particular equilibrium, we characterize all such equilibria.1 We first consider the case where buyers are also uncertain about demand (symmetric demand uncertainty). In that case, we show that the low-quality seller’s optimal pricing strategy is a simple switching-down strategy: she begins with the high price and switches to the low price once she fails to trade for a while. This is a familiar result in the literature on demand uncertainty (experimentation). Intuitively, as she fails to sell, she becomes more pessimistic about demand and eventually switches to the low price, which speeds up trade and, therefore, is optimal with low demand. An important difference from the existing literature on demand uncertainty is that in our model, the low-quality seller always begins with the high price, no matter how small the initial probability of high demand is. This is because effective demand is endogenously determined by the presence of adverse selection in our model. To see this more clearly, suppose that the low-quality seller never 1

Most selection criteria, such as the intuitive criterion (Cho and Kreps, 1987) and universal divinity (Banks and Sobel, 1987), are not effective in our model. One exception is undefeatedness by Mailath et al. (1993). In the benchmark model without demand uncertainty, it selects a unique equilibrium in which the high-quality seller offers the maximal price (that is equal to buyers’ willingness-to-pay for the high-quality asset) and buyers accept the price with a just enough probability such that the low-type seller is indifferent between the two prices. In the other models with demand uncertainty, however, it does not yield such a sharp prediction and typically produces an interval of prices, which makes comparisons among different environments rather problematic.

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offers the high price. Buyers then believe that the high price is offered only by the high-quality seller and, therefore, accept the price. This, of course, provides an incentive for the low-quality seller to deviate and offer the high price. This shows that the low-quality seller must offer the high price with a positive probability. Combining this with the fact that she becomes more pessimistic over time, it follows that the low-quality seller always plays a switching-down pricing strategy. We also show that symmetric demand uncertainty is always beneficial to the low-quality seller: her expected payoff is strictly higher under symmetric demand uncertainty than without demand uncertainty.2 This contrasts well with the conventional wisdom that (demand) uncertainty inhibits a seller’s optimization and, therefore, learning is always valuable. This is, again, due to endogenous demand. Buyers adjust their purchase behavior, depending on whether there is demand uncertainty or not. Therefore, the conventional wisdom, which applies to a seller’s optimal decision problem, does not apply to our strategic environment. The payoff dominance under symmetric demand uncertainty is due to the fact that, although the seller faces the same constraint as buyers initially, she has the advantage of learning about demand and adjusting her price accordingly over time. Notice that this shows that (private) learning is still valuable in our environment. We then consider the case where buyers are informed about demand (asymmetric demand uncertainty) and demonstrate that the results are markedly different from those under symmetric demand uncertainty. Asymmetric demand uncertainty has no effect on the seller’s expected payoff: both seller types’ expected payoffs in each state are identical to those without demand uncertainty. In addition, the low-quality seller’s switching-down pricing strategy can never be a part of equilibrium. We show that there exists an equilibrium in which the low-quality seller adopts a switching-up-and-down pricing strategy: she begins with the low price, switches up to the high price at some point, and finally reverts back to the low price. The payoff result is, again, due to endogenous demand. We show that if buyers know the demand state, then in equilibrium they adjust their purchase strategies, so that the low-quality seller is indifferent between the high price and the low price in both demand states. This implies that, although the seller still learns about demand over time, learning is simply of no value to her. Notice that this can never arise with exogenous demand, unless the decision problem itself is trivial. More intuitively, the seller does not enjoy an informational advantage over buyers and, therefore, cannot extract more surplus. The necessity of a complicated dynamic pricing structure highlights strategic aspects of our model. In the model with asymmetric demand uncertainty, as argued above, the low-quality seller is indifferent between the high price and the low price in both states, and thus any pricing strategy is the low-quality seller’s best response to buyers’ equilibrium purchase strategies. However, equi2

The effect of symmetric demand uncertainty on the high-quality seller’s expected payoff is ambiguous in general. This shows that the low-quality seller’s gain is not associated with the high-quality seller’s loss.

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librium imposes a restriction on the set of feasible pricing strategies, because the pricing strategy affects buyers’ incentives, which must be adjusted so that they play a particular purchase strategy. The resulting set of restrictions can be jointly resolved only by a rather sophisticated form of pricing strategy, such as a switching-up-and-down strategy. Although our model is fairly abstract and not tailored to a specific application, our results provide new perspectives into certain empirical facts. For example, Merlo and Ortalo-Magne (2004) present a set of stylized facts about dynamic patterns of list prices in the real estate market: a significant portion of sellers adjust their list price at least once and some of them do multiple times. Most sellers lower their price, after waiting for 11 weeks on average, but some of them increase their price. As discussed by Merlo and Ortalo-Magne (2004), most existing theories can explain a subset of these facts but are not rich enough to accommodate all of them.3 Our analysis shows that a combination of adverse selection and demand uncertainty significantly enriches the set of admissible dynamic pricing patterns and, therefore, provides a potential resolution to those facts. Our paper mainly contributes to the literature on demand uncertainty. A non-exhaustive list of seminal contributions includes Rothschild (1974); Easley and Kiefer (1988); Mirman et al. (1993); Keller and Rady (1999). All existing studies we are aware of in this literature consider an agent’s dynamic decision problem with exogenous demand and do not endogenize demand through adverse selection, as we do in this paper. Particularly close to ours is Mason and V¨alim¨aki (2011). They consider a similar dynamic pricing problem under demand uncertainty, but with exogenous demand: in their model, each buyer has private value for the seller’s good and accepts any price below his value. As explained above, this leads to various different results. For example, if the probability of high demand is sufficiently small, then the seller immediately settles on the low price in their model, while the low-quality seller still begins with the high price in our model with symmetric demand uncertainty. Nevertheless, we significantly benefit from their analysis. In particular, in the model with symmetric demand uncertainty, given buyers’ purchase strategies, the low-quality seller’s optimal pricing problem is formally identical to their problem, and thus their explicit solution to the binary case applies unchanged to our environment (see Proposition 2). Our paper also contributes to the literature on dynamic adverse selection. See Evans (1989); Vincent (1989, 1990); Taylor (1999); Janssen and Roy (2002); Deneckere and Liang (2006); H¨orner and Vieille (2009) for some seminal contributions. Most papers in this literature consider the case where uninformed players make price offers to informed players, mainly in order to avoid equilibrium multiplicity due to signaling. We are aware of three exceptions, Lauermann and Wolinsky (2011), Palazzo (2017), and Gerardi et al. (2014), each of which studies the opposite case where informed players make price offers to uninformed players. The first two focus on un3

See Merlo et al. (2015) for a behavioral model that can accommodate several prominent patterns in Merlo and Ortalo-Magne (2004). They also provide a more detailed discussion about the limitations of previous studies.

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defeated equilibria (Mailath et al., 1993), while the last one characterizes the set of all equilibrium payoffs. To our knowledge, we are the first to introduce demand uncertainty into a dynamic trading environment with adverse selection. Dynamic pricing has been more extensively studied in the context of durable goods monopoly with multiple units or without capacity constraint. See, e.g., Garrett (2016), Dilme and Li (2016), and the references therein. An important question in that literature is to rationalize (or generate) fluctuating pricing behavior that is often observed in reality. All existing explanations hold exactly because of multi-unit supply. For example, in Garrett (2016), the monopolist has an incentive to offer temporary price discounts because high-value consumers trade quickly, while low-value consumers accumulate over time. In Dilme and Li (2016), price occasionally jumps because the monopolist has a finite number of units to sell and, therefore, competition among consumers intensifies as sales occur. Our pricing result for asymmetric demand uncertainty complements these results by showing that fluctuating prices can emerge even with unit supply. The rest of the paper is organized as follows. We introduce the model in Section 2 and studies the benchmark case without demand uncertainty in Section 3. We then analyze the case of symmetric demand uncertainty in Section 4 and the case of asymmetric demand uncertainty in Section 5. We conclude by discussing three related issues in Section 6.

2 The Model 2.1 Physical Environment A seller wishes to sell an indivisible object. Time is continuous and indexed by t ∈ R+ . The time the seller arrives at the market is normalized to 0. At each point in time, the seller posts a price. Buyers arrive sequentially according to a Poisson process of rate λ. Upon arrival, each buyer observes the posted price and decides whether to purchase the good or not. If the buyer purchases, then trade takes place between the buyer and the seller, and the game ends. If not, the buyer leaves, while the seller continues the game. The common discount rate is given by r > 0. The seller’s good is either of high quality (H) or of low quality (L). If the good is of quality a = H, L, then it yields flow payoff rca to the seller (while she retains it) and flow payoff rva to each buyer (once he acquires it). Note that the (reservation) value of the good is ca to the seller and va to buyers. A high-quality unit is more valuable to both the seller and buyers (i.e., cH > cL and vH > vL ). There are always positive gains from trade (i.e., va > ca for both a = H, L), but the quality of the good is known only to the seller. It is common knowledge that the seller’s good is of high quality with probability q0 at the beginning of the game. Without loss of generality, we normalize cL to 0. 5

The seller is uncertain about the arrival rate of buyers, which is either λh (high demand) or λl (low demand), where λh > λl > 0. It is commonly known that the good is in high demand (i.e., λ = λh ) with probability µ0 at the beginning of the game, and the realization of the demand state is independent of the quality of the good. We consider two cases that differ in terms of buyers’ knowledge about the demand state. We say that demand uncertainty is symmetric if the demand state is also unknown to buyers, and refer to the opposite case as asymmetric demand uncertainty.4 All agents are risk-neutral and maximize their expected utility. If a buyer purchases the good at price p at time t and the good is of quality a, then the buyer receives payoff va − p, while the seller obtains (1 − e−rt )ca + e−rt p. All other buyers receive zero payoff.

2.2 Strategies and Equilibrium We consider the following information structure:5 the seller does not observe buyer arrivals, while each buyer observes only the price posted at the time of his arrival. The former implies that the seller cannot tell whether the failure of sale is due to no arrival of buyers or due to buyers’ refusal to accept the posted price. The latter implies that buyers’ beliefs and strategies are independent of their arrival time and, therefore, stationary over time from the seller’s viewpoint.6 Both assumptions give tractability to the analysis. Under the information structure, the seller’s (pure) offer strategy is a function p : {L, H} × R+ → R+ , where p(a, t) represents the price the type-a seller posts at time t. For buyers’ beliefs and strategies, denote by Λ the information set of buyers regarding the demand state. The set Λ is a singleton with symmetric demand uncertainty (i.e., if buyers cannot distinguish between high demand λh and low demand λl ), while it is isomorphic to the set {λh , λl } with asymmetric demand uncertainty. Buyers’ beliefs about the seller’s type are represented by a function q : Λ × R+ → [0, 1], where q(λ, p) denotes the probability that buyers assign to the high type conditional on demand state λ and price p. Similarly, their (mixed) purchase strategies are a function σB : Λ × R+ → [0, 1], where σB (λ, p) denotes the probability that buyers accept p in state λ. A tuple (p, q, σB ) is a perfect Bayesian equilibrium of the dynamic trading game if the following three conditions hold. 4

Asymmetric demand uncertainty captures situations in which buyers have superior (preference or economic) information about other buyers to sellers. For example, (home) buyers may belong to a younger generation than sellers and, therefore, understand other buyers’ preferences better. In Appendix A, we analyze another asymmetric case in which the demand state is known to the seller, but not to buyers. 5 It is well-known that uninformed players’ (buyers’) information about informed players’ (sellers’) histories plays a crucial role in dynamic environments (see, e.g., Swinkels, 1999; Taylor, 1999; H¨orner and Vieille, 2009; Kim, 2017). The same applies to our model. See Section 6 for a relevant discussion and Appendix B for the characterization of the case in which buyers observe the full price history. 6 This is a common modeling assumption in the literature on dynamic adverse selection (see, e.g., Zhu, 2012; Lauermann and Wolinsky, 2016). For different approaches, which give rise to non-stationary dynamics, see, e.g., H¨orner and Vieille (2009) and Kim (2017).

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• Seller optimality: for each a = H, L, the type-a sellers’ pricing strategy p(a, ·) maximizes her expected payoff, that is, p(a, ·) ∈ argmaxp′ (a,·) E[(1 − e−rτ )ca + e−rτ p′ (a, τ )], where τ is the (random) time at which a buyer purchases the good. • Buyer optimality: for each λ ∈ Λ and p ∈ R+ , each buyer accepts price p only when his expected payoff by doing so is non-negative. In other words, σB (λ, p) > 0 only when q(λ, p)vH + (1 − q(λ, p))vL − p ≥ 0. • Belief consistency: for each λ ∈ Λ and p ∈ R+ , q(λ, p) is obtained from p and σB by Bayes’ rule whenever possible.

2.3 Assumptions We focus on the case where adverse selection is so severe that some inefficiency is unavoidable. Specifically, we maintain the following assumption, which is common in the literature. Assumption 1 q0 vH + (1 − q0 )vL < cH . The left-hand side is buyers’ unconditional expected value of the good, while the right-hand side is the high-type seller’s reservation value. This assumption ensures that there does not exist an equilibrium in which both seller types always offer a price in [cH , q0 vH + (1 − q0 )vL ] and buyers always accept the price, which is an efficient market outcome. Note that, since vH > cH , the assumption holds only when vL is strictly less than cH and q0 is sufficiently small. Because price signals the seller’s type, the game suffers from severe equilibrium multiplicity. In order to focus on economic insights stemming from the model, as well as for tractability, we restrict attention to the equilibria of the following structure: Assumption 2 The high-type seller plays a constant strategy of offering pH ∈ [cH , vH ) after all histories. This strategy of the high-type seller can be supported, for example, by assuming that buyers believe that all other prices are offered only by the low-type seller. This assumption excludes the trivial equilibria in which the high-type seller always offers a losing price (above vH ). More importantly,

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it does not allow the high-type seller to dynamically adjust her price. As shown shortly, in the absence of demand uncertainty, this incurs no loss of generality in characterizing the set of equilibria. For the cases of demand uncertainty in Sections 4 and 5, this significantly simplifies the analysis. Assumption 2 implies that there are effectively two prices, vL and pH : the high type always offers pH , while the low type chooses between vL and pH at each point in time.7 With a slight abuse of notation, in what follows, we describe the low-type seller’s offer strategy by a function σS : R → [0, 1], where σS (t) denotes the probability that the low-type seller offers pH at time t. In addition, we use σB to denote the probability that each buyer accepts pH (i.e., σB (λ) ≡ σB (λ, pH ) from now on).

3 No Demand Uncertainty We first study the benchmark case in which there is no demand uncertainty (i.e., λ is commonly known). This allows us to identify the effects due to demand uncertainty as well as explain some basic concepts and tools used in the subsequent sections.

3.1 Buyers’ Beliefs In our model, buyers’ beliefs about the seller’s type depart from their prior beliefs q0 for two reasons. First, the very fact that a buyer meets the seller provides information about the seller’s type. The low-type seller trades faster than the high-type seller, because the former may offer vL (which is accepted with probability 1), while the latter insists on pH (which may not be accepted with probability 1). This means that the high type stays longer than the low type, and thus the seller who is still available on the market is more likely to be the high type. We denote by q I buyers’ beliefs at this stage and refer to them as their interim beliefs. Second, the posted price also conveys information about the seller’s type. Since buyers’ beliefs (and optimal purchase decisions) following vL are trivial, we focus on their beliefs conditional on (being offered) pH . We refer to those beliefs as buyers’ ex post beliefs and denote by q ∗ . Given σS (t) and σB , the trading (exit) rate of the high-type seller is equal to λσB , while that of the low-type seller is equal to λ(σS (t)σB + 1 − σS (t)). This means that the probability that the ∫t high-type seller stays on the market until time t is equal to e− 0 λσB dt , while that of the the low-type ∫t seller is equal to e− 0 λ(σS (t)σB +1−σS (t))dt . Since a seller can be interpreted to be randomly drawn from the space {L, H} × R+ (i.e., the areas below the solid lines in Figure 1, with total weights q0 7

It is a strictly dominant strategy for each buyer to accept p < vL , because his expected value is bounded below by vL . Therefore, the low-type seller’s optimal price conditional on her type being revealed is equal to vL , and it must be accepted by buyers with probability 1 in equilibrium.

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High type

1

Low type

1

e−λσB t

e−

∫t 0

λ(σS (x)σB +1−σS (x))dx

vL

pH pH

t

0

t

0

Figure 1: The probability that each seller type does not trade by time t. The dashed line in the right panel depicts the∫probability that the low-type seller does not trade by time t and offers pH t at time t (i.e., σS (t)e− 0 λ(σS (x)σB +1−σS (x))dx ). The right panel is drawn for the case where σS (t) is independent of t. to the high type and 1 − q0 to the low type), buyers’ interim beliefs can be calculated as follows:8 I

q =

q0

∫∞ 0

e−

∫t 0

q0 λσB dx

∫∞ 0

e−

∫t 0

dt + (1 − q0 )

λσB dx

∫∞ 0

dt

e−

∫t 0

λ(σS (x)σB +1−σS (x))dx

,

which is equivalent to ∫ ∞ − ∫ t λσ dx e 0 B dt qI q0 0 ∫t . = ∫ 1 − qI 1 − q0 ∞ e− 0 λ(σS (x)σB +1−σS (x))dx dt

(1)

0

Since σS (t)σB + 1 − σS (t) ≥ σB at any t, q I is necessarily larger than q0 . As explained above, the low-type seller trades faster than the high-type seller, and thus a seller who is available is more likely to be the high type. Now we incorporate the signaling aspect of posted price and derive buyers’ ex post beliefs q ∗ . The high-type seller insists on pH , while the low-type seller offers pH with probability σS (t) at time t. Since time t is not observable to buyers, it is necessary to derive the probability that the low-type seller offers pH unconditional on t.9 Applying the fact that that a seller is randomly 8

In our model, there are infinitely many buyers who arrive in a uniform random order. This means that the probability that a particular buyer arrives (i.e., is sampled by the seller) is equal to zero and, therefore, q I is potentially not well-defined. This problem has been well-recognized and understood in the literature. We omit a detailed discussion and simply apply the standard updating rule. One way to resolve the issue is to consider the case where there are n potential buyers and let n tend to infinity. For further discussions and formal derivations, see Lauermann and Wolinsky (2016). 9 If time t were observable by buyers, then the probability would be simply σS (t), and thus buyers’ ex post beliefs

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drawn from the entire population (i.e., the space {L, H} × R+ ) and the probability that the seller ∫t has not traded by time t is equal to e− 0 λ(σS (x)σB +1−σS (x))dx , the unconditional probability that the low-type seller offers pH is equal to ∫∞ 0

σS (t)e−

∫∞ 0

e−

∫t 0

∫t 0

λ(σS (x)σB +1−σS (x))dx

λ(σS (x)σB +1−σS (x))dx

dt

dt

.

It then follows that buyers’ ex post beliefs are given by qI q∗ = 1 − q∗ 1 − qI

1

∫ − 0t λ(σS (x)σB +1−σS (x))dx dt 0 σS (t)e∫ ∫ ∞ − t λ(σ (x)σ +1−σ (x))dx S B S 0 dt 0 e

∫∞

.

(2)

Since σS (t) ≤ 1 at any t, q ∗ is always greater than q I . Intuitively, the high-type seller always offers pH , while the low-type seller may offer vL . Therefore, the seller who offers pH is more likely to be the high type. Combining equations (1) and (2) yields ∫ ∞ − ∫ t λσ dx e 0 B dt q∗ q0 0 ∫t . = ∫ 1 − q∗ 1 − q0 ∞ σS (t)e− 0 λ(σS (x)σB +1−σS (x))dx dt

(3)

0

This equation gives a unique value of q ∗ as a function of the low-type seller’s pricing strategy σS (·) and buyers’ purchase strategies σB . In other words, the equilibrium requirement that buyers’ beliefs must be consistent with a strategy profile reduces to this equation.

3.2 Equilibrium Characterization We complete equilibrium characterization by deriving two other equilibrium conditions and combining them with condition (3). The two other equilibrium conditions are (i) that buyers must randomize between accepting and rejecting pH (i.e., σB ∈ (0, 1)) and (ii) that the low-type seller must offer both vL and pH with positive probabilities. To understand the first condition, suppose that buyers always accept pH . Then, clearly, the low-type seller strictly prefers offering pH to vL . Since both seller types play an identical strategy, q ∗ = q I = q0 . But then, Assumption 1 implies that buyers’ expected payoffs are strictly negative, which cannot arise in equilibrium. Now suppose that buyers always reject pH . If so, the low-type seller strictly prefers offering vL to pH . This implies that buyers would believe would be equal to

q∗ qI 1 = . ∗ I 1−q 1 − q σS (t)

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that the seller who offers pH is the high type with probability 1 (i.e., q ∗ = 1) and, therefore, would accept pH (< vH ) with probability 1, which is a contradiction. The same argument can be used for the second equilibrium condition that the low-type seller must randomize between vL and pH . Formally, the two equilibrium conditions are q∗ p H − vL q (vH − pH ) + (1 − q )(vL − pH ) = 0 ⇔ = , ∗ 1−q vH − p H ∗

and

∫

∞

e 0

−rt

pH d(1 − e

∗

−λσB t

λσB pH = )= r + λσB

∫ 0

∞

e−rt vL d(1 − e−λt ) =

λ vL . r+λ

The first equation corresponds to buyers’ indifference between accepting and rejecting pH , while the second one corresponds to the low-type seller’s indifference between pH (left) and vL (right). We combine and summarize all the results in the following proposition. Proposition 1 In the model without demand uncertainty, for each pH ∈ [cH , vH ), there exists a continuum of payoff-equivalent equilibria. In any equilibrium, q ∗ = (pH − vL )/(vH − vL ) and σB = rvL /((r + λ)pH − λvL ), while the low-type seller’s strategy σS (t) supports an equilibrium if and only if it satisfies equation (3). The low-type seller’s expected payoff is equal to λvL /(r + λ), while the high-type seller’s expected payoff is equal to (rcH + λσB pH )/(r + λσB ). To see equilibrium multiplicity more concretely, consider the following three simple pricing strategies. • Time-invariant strategy: the low-type seller randomizes between pH and vL with a constant probability over time (i.e., σS (t) is independent of t). • Switching-down strategy: the low-type seller begins with pH , and switches down to vL at a certain time TD (i.e., σS (t) = 1 if t < TD and σS (t) = 0 otherwise). • Switching-up strategy: the low-type seller begins with vL , and switches up to pH at a certain time TU (i.e., σS (t) = 0 if t < TU and σS (t) = 1 otherwise). It is easy to see that each of these pricing strategies can be a part of equilibrium, by identifying proper values of σS , TD , and TU . In the next two sections, we show that the introduction of demand uncertainty has a dramatic impact on equilibrium multiplicity and the structure of equilibrium pricing strategy. Observe that the low-type seller’s expected payoff is independent of pH , while the high-type seller’s expected payoff strictly increases in pH . The former holds because in equilibrium, the low-type seller should be indifferent between vL and pH , independently of pH . The latter is due to a single crossing property: the high-type seller, due to her higher reservation value, is more willing 11

to trade off slower trade (i.e., lower purchase probability) for a higher price. Since the low-type seller is indifferent among different values of pH , the high-type seller obtains a higher expected payoff as pH increases. The payoff result implies that Assumption 2 incurs effectively no loss of generality in the model without demand uncertainty.10 Suppose that the high-type seller offers both pH and p′H with positive probabilities, where cH < pH < p′H < vH . It can arise only when she is indifferent between pH and p′H . Meanwhile, for the same reason as above, the low-type seller should be indifferent between vL and pH . Then, due to the single crossing property, the low-type seller strictly prefers vL and pH to p′H . This, however, implies that buyers would accept p′H with probability 1 and, therefore, the high-type seller would prefer p′H to pH , which is a contradiction. It also follows that in the model without demand uncertainty, the unique undefeated equilibrium outcome, in the sense of Mailath et al. (1993), is the limiting outcome as pH tends to vH in Proposition 1. The low-type seller’s expected payoff is equal to λvL /(r + λ) in any equilibrium, whether Assumption 2 is imposed or not, while the high-type seller’s expected payoff is maximized when pH = vH and buyers accept vH with probability rvL /((r + λ)vH − λvL ). As shown in the next section, however, this strong selection result fails under symmetric demand uncertainty: the lowtype seller’s expected payoff varies according to pH and, in fact, is minimized when pH = vH (see footnote 12). In addition, the argument above regarding no loss of generality of Assumption 2 no longer applies. Therefore, in what follows, instead of restricting attention to undefeated equilibria, we continue to consider the set of all equilibria that satisfy Assumption 2.

4 Symmetric Demand Uncertainty We now introduce demand uncertainty into the model: the arrival rate of buyers is λh with probability µ0 and λl with probability 1 − µ0 . We begin with the case in which both the seller and buyers are uncertain about the demand state.

4.1 Optimal Pricing In the presence of demand uncertainty, the seller learns about demand over time, and thus her problem is no longer stationary. In particular, conditional on no trade, she becomes increasingly pessimistic. Naturally, it is optimal for her to begin with the high price pH , which yields higher 10

“Effectively” is due to the presence of fully separating equilibria in which the high-type seller offers only prices weakly above vH and buyers accept them with a sufficiently small probability (0 if pH > vH and a probability below rvL /((r + λ)vH − λvL ) if pH = vH ). Notice that the limiting equilibrium as pH tends to vH in Proposition 1 is the most efficient equilibrium among all these separating equilibria, because it maximizes the probability that buyers accept pH = vH subject to the low-type seller’s incentive constraint.

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profits if accepted but has a lower acceptance probability. Once she becomes sufficiently pessimistic, she switches to the low price vL , which results in lower profits but guarantees faster trade. To systematically analyze the low-type seller’s optimal pricing problem, let µ(t) denote the probability that she assigns to demand state h at time t. The evolution of µ(t) depends on her pricing strategy as well as buyers’ purchase strategies. Suppose buyers accept pH with probability σB . If the low-type seller offers pH , then µ(t) evolves according to µ(t + dt) =

µ(t)e−λh σB dt , µ(t)e−λh σB dt + (1 − µ(t))e−λl σB dt

which can be reduced to µ(t) ˙ = −µ(t)(1 − µ(t))(λh − λl )σB .

(4)

Similarly, if the offer is vL , then trade occurs as long as a buyer arrives, and thus µ(t + dt) =

µ(t)e−λh dt . µ(t)e−λh dt + (1 − µ(t))e−λl dt

As before, this expression can be reduced to µ(t) ˙ = −µ(t)(1 − µ(t))(λh − λl ).

(5)

Clearly, the low-type seller’s belief decreases faster in the latter case. Intuitively, when the price is vL , a sale does not occur only when no buyer has arrived, while with price pH , it may be also because of a buyer’s refusal to accept the price. Therefore, no sale is a stronger signal about low demand when the price is vL . The seller’s optimal pricing strategy is a standard cutoff rule: there is a threshold belief µ such that the low-type seller offers pH if and only if her belief exceeds µ. Since her belief µ(t) always decreases over time, this means that there exists a cutoff time at which the low-type seller switches from pH to vL , and the low-type seller never reverts back to pH . Clearly, the cutoff time is the point at which the seller’s belief is equal to µ(t) = µ. To explicitly derive the low-type seller’s optimal pricing strategy, let V (µ) denote her expected payoff when her belief is equal to µ. If µ > µ, then her optimal price is pH . Therefore, the

13

continuous-time Bellman equation is given as follows:11 rV (µ) = (µλh + (1 − µ)λl )σB (pH − V (µ)) + V˙ (µ). Combining this equation with equation (4) yields rV (µ) = (µλh + (1 − µ)λl )σB (pH − V (µ)) − V ′ (µ)µ(1 − µ)(λh − λl )σB . If µ ≤ µ, then the low-type seller offers vL . Since her belief constantly decreases, she never switches to pH and continues to offer vL , as if she commits to vL . Therefore, ∫

∞

V (µ) = µ

−rt

e

−λh t

vL d(1−e

∫ )+(1−µ)

0

∞

e−rt vL d(1−e−λl t ) = µ

0

λh λl vL +(1−µ) vL . r + λh r + λl

This form of optimal stopping problem is familiar in the literature on experimentation. In particular, given buyers’ purchase strategies σB , the problem is effectively identical to the binary case in Mason and V¨alim¨aki (2011). They study a dynamic pricing problem in which buyers have private values (implying no inference problem on the seller’s type) and the distribution of their values is exogenously given (i.e., in the context of our model, σB is exogenously given). We translate their solution into our context and report all the results in the following proposition. Proposition 2 Given buyers’ purchase strategies σB , the low-type seller’s optimal pricing strategy is to offer pH if her belief exceeds µ and offer vL otherwise, where    1, if µ= 0, if  

λh σ B p H λh vL ≤ r+λ , r+λh σB h λl σB pH λl v L ≥ r+λl , r+λl σB −λl (r+λh )(σB λl (pH −vL )+r(σB pH −vL )) , (λh −λl )(r(σB λl (pH −vL )+r(σB pH −vL ))+σB λh (pH −vL )(r+λl ))

otherwise.

The low-type seller’s expected payoff as a function of her belief is equal to  ) ( ) r+λL σB (   µ λh σB + (1 − µ) λl σB p + C(1 − µ) 1−µ (λH −λL )σB , H r+λl σB µ V (µ) = ( r+λh σB )  λ λ  µ h + (1 − µ) l v , r+λh

11

r+λl

L

if µ > µ, if µ ≤ µ,

Heuristically, this Bellman equation can be obtained from the following recursive equation: V (µ(t)) = (µ(t)λh + (1 − µ(t))λl )σB dt · pH + (1 − (µ(t)λh + (1 − µ(t))λl )σB dt)e−rdt V (µ(t + dt)).

It suffices to subtract e−rdt V (µ(t)) and divide both sides by dt after appropriately arranging the terms.

14

V (µ)

µ µ

0

1

Figure 2: The low-type seller’s expected payoff as a function of her belief about the demand state, given buyers’ purchase strategies σB . where µ(λh − λl )σB C= µ(λh − λl )σB + λl σB

(

r(λh − λl )σB pH rvL (λh − λl ) − (r + λh σB )(r + λl σB ) (r + λh )(r + λl )

)(

1−µ µ

)− (λr+λ L σB −λ )σ H

L

B

.

Figure 2 depicts the value function V (µ). The two dashed lines represent the low-type seller’s expected payoffs when she posts only vL (the line that coincides with V (µ) when µ ≤ µ) or pH (the line that meets V (µ) when µ = 1). As is well-known, the flexibility to adjust the price (in particular, the option to decrease the price from pH to vL ) is valuable to the seller, which is reflected in the fact that VB (µ) stays uniformly above the two dashed lines. For later use, let T (σB ) denote the length of time it takes for the low-type seller’s belief to reach µ, assuming that she follows the optimal pricing strategy in Proposition 2. The following result provides an explicit solution for T (σB ). Corollary 1 Given buyers’ purchase strategies σB , the low-type seller offers pH until time T (σB ) and vL thereafter, where {

1 T (σB ) = max 0, ln (λh − λl )σB

15

(

µ0 1 − µ 1 − µ0 µ

)} .

4.2 Buyers’ Beliefs We now derive buyers’ equilibrium beliefs q ∗ (the probability that each buyer assigns to the high type conditional on observing pH ). As shown in the previous section, the trading rates of each seller type affect q ∗ . Since the trading rates also depend on the demand state, it is necessary to determine buyers’ beliefs about the demand state, although those beliefs do not directly influence their purchase decisions. We denote by µ∗ buyers’ (ex post) beliefs about the demand state (conditional on observing pH ). We first take µ∗ as given and determine q ∗ . Given µ∗ and σB , buyers’ beliefs about the seller’s type q ∗ can be derived as in the previous section. As shown above, given σB , the low-type seller’s optimal pricing strategy is uniquely determined. Therefore, unlike in Section 3, it suffices to derive q ∗ that corresponds to the lowtype seller’s pricing strategy in Proposition 2: the low-type seller offers pH until time T (σB ) and switches to vL . Following the same steps as in the previous section (and skipping the derivation of q I ), ∫∞ ∫∞ µ∗ 0 e−λh σB t dt + (1 − µ∗ ) 0 e−λl σB t dt q∗ q0 = ∫ ∫ 1 − q∗ 1 − q0 µ∗ T (σB ) e−λh σB t dt + (1 − µ∗ ) T (σB ) e−λl σB t dt 0

q0 = 1 − q0

µ∗ λh σ B

µ∗ (1−e−λh σB T (σB ) ) λh σ B

+

+

(6)

0

1−µ∗ λl σ B

(1−µ∗ )(1−e−λl σB T (σB )) ) λl σ B

.

Now, q ∗ departs from q0 for three reasons. First, the high-type seller stays on the market relatively longer than the low-type seller (which pushes up q I above q0 ). Second, the high-type seller is more likely to offer pH than the low-type seller (which pushes up q ∗ above q I ). Finally, buyers’ beliefs about the demand state also influence q ∗ . This last effect is unclear at this stage, because µ∗ is also an endogenous variable. We now determine µ∗ . Similarly to the relationship between q ∗ and q0 , there are two reasons why µ∗ departs from µ0 . First, trade occurs faster in demand state h than in demand state l. Therefore, the very fact that the seller is still available makes buyers assign a lower probability to demand state h. Second, for a given length of time, the seller meets relatively more buyers in demand state h than in demand state l. Therefore, a buyer is more likely to arrive (i.e., be born) in demand state h than in demand state l, which increases the probability of demand state h. Applying similar arguments to those used to derive q ∗ above, ∫∞ ∫ T (σ ) µ0 λh q0 0 e−λh σB t dt + (1 − q0 ) 0 B e−λh σB t dt µ∗ = ∫ ∫ 1 − µ∗ 1 − µ0 λl q0 ∞ e−λl σB t dt + (1 − q0 ) T (σB ) e−λl σB t dt 0

0

−λh σB T (σB )

=

µ0 q0 + (1 − q0 )(1 − e ) . −λ σ T (σ ) B B l 1 − µ0 q0 + (1 − q0 )(1 − e ) 16

(7)

The second term in the right-hand side (λh /λl ) captures the second effect in the previous paragraph, while the last term represents the first effect. From the final expression, it follows that buyers’ beliefs about demand µ∗ exceed their prior beliefs µ0 (which means that the second effect necessarily dominates the first effect), as long as T (σB ) is finite. Combining equations (6) and (7) yields the following result. Lemma 1 Given buyers’ purchase strategies σB (and the low-type seller’s optimal response to σB , as characterized in Proposition 1), buyers’ ex post beliefs about the seller’s type q ∗ are uniquely determined by q0 q∗ = ∗ 1−q 1 − q0

µ0 (q λh 0

+ (1 − q0 )κh (σB )) +

µ0 κ (σ )(q0 λh h B

+ (1 − q0 )κh (σB )) +

1−µ0 (q0 + (1 − q0 )κl (σB )) λl , 1−µ0 κl (σB )(q0 + (1 − q0 )κl (σB )) λl

where κd (σB ) = 1 − e−λd σB T (σB ) , for each d = h, l.

4.3 Equilibrium Characterization We complete equilibrium characterization by endogenizing buyers’ purchase strategies σB . A necessary condition, once again, comes from the fact that in equilibrium, buyers must be indifferent between accepting and rejecting pH . If they always accept pH , then the low-type seller prefers offering pH to vL , independently of her belief about the demand state. In this case, q ∗ = q0 , but then buyers’ expected payoffs become strictly negative, due to Assumption 1. To the contrary, if buyers always reject pH , then the low-type seller always prefers vL to pH . In this case, q ∗ = 1, and thus buyers strictly prefer accepting pH to rejecting it, which is a contradiction. This leads to the last equilibrium condition: q ∗ (vH − pH ) + (1 − q ∗ )(vL − pH ) = 0 ⇔

p H − vL q∗ = . ∗ 1−q vH − p H

(8)

Given Proposition 2, Lemma 1, and equation (8), equilibrium characterization reduces to finding a value of σB that satisfies q0 1 − q0

µ0 (q λh 0

+ (1 − q0 )κh (σB )) +

µ0 κ (σ )(q0 λh h B

+ (1 − q0 )κh (σB )) +

1−µ0 (q0 + (1 − q0 )κl (σB )) λl 1−µ0 κl (σB )(q0 + (1 − q0 )κl (σB )) λl

=

p H − vL . vH − pH

(9)

Proposition 3 For each pH ∈ [cH , vH ), there exists a unique equilibrium in the model with symmetric demand uncertainty. In the equilibrium, the probability that each buyer accepts pH , σB , is such that µ(σB ) < µ0 (i.e., T (σB ) > 0). 17

Proof. See Appendix C. Proposition 3 implies that under symmetric demand uncertainty, the low-type seller necessarily plays a switching-down strategy: she offers pH until time T (σB )(> 0) and then switches down to vL . Importantly, this property holds even if the prior probability of high demand µ0 is sufficiently small. This is in stark contrast to the result in Mason and V¨alim¨aki (2011): in their model with exogenous demand, the cutoff belief µ is independent of the prior probability µ0 , and thus the seller immediately offers the low price vL whenever µ0 is below µ. In our environment with endogenous demand, the cutoff belief µ depends on µ0 . In particular, if µ0 is small, then µ becomes even smaller, and thus the low-type seller always begins with the high price pH .12 Proposition 3 has a significant implication for the low-type seller’s payoff, as formally stated in the following corollary and graphically illustrated in Figure 3.13 Corollary 2 The low-type seller’s expected payoff V (µ0 ) is always higher under symmetric demand uncertainty than under no demand uncertainty: for any µ0 ∈ (0, 1), ( ) λh λl V (µ0 ) > µ0 + (1 − µ0 ) vL . r + λh r + λl A key to understanding this result, again, lies in the fact that demand is endogenously determined in our model. With exogenous demand, demand uncertainty always lowers the seller’s expected payoff for a standard reason, namely that it creates the possibility that the seller takes a wrong action. With endogenous demand, demand uncertainty also influences buyers’ purchase strategies, and thus the standard argument no longer applies. Going one step further, the low-type seller benefits from symmetric demand uncertainty, because it allows her to utilize her own ability to dynamically adjust the price. In the absence of demand uncertainty, she faces a stationary problem, and thus the ability is of no value to her. Demand uncertainty transforms her problem into a non-stationary one, in which the ability is endowed with a positive value. To be more concrete, consider the case where the low-type seller is deprived of the ability: she can choose between vL and pH at the beginning of the game but is not allowed to change her price later. By the usual argument, in equilibrium the low-type seller must play vL with 12

In the limit as pH tends to vH , µ becomes equal to µ0 , so that the low-type seller switches to vL immediately. This result follows from equation (9): the right-hand side becomes arbitrarily large as pH tends to vH . For the left-hand side to grow arbitrarily, κd (σB ) must converge to 0, which can be the case if and only if T (σB ) approaches 0 (see Corollary 1). Intuitively, buyers never accept vH unless they are certain that the seller is the high type. Therefore, in equilibrium, the low-type seller should never offer pH = vH . Notice that this implies that the low-type seller’s expected payoff is equal to (µ0 λh /(r + λh ) + (1 − µ0 )λl /(r + λl )) vL and, therefore, minimized when pH = vH . 13 The result for the high-type seller is ambiguous in general. Typically, the high-type seller’s expected payoff is higher under symmetric demand uncertainty than under no demand uncertainty if pH is rather small, while the opposite result holds if pH is rather large.

18

V (µ0 ) λh r+λh vL

λl r+λl vL

0

1

µ0

Figure 3: The low-type seller’s ex ante expected payoffs with symmetric demand uncertainty (solid) and without demand uncertainty (dashed). a positive probability, which implies that her expected payoff would be equal to that in Proposition 1 in each state. Importantly, this result holds regardless of the presence of demand uncertainty. This demonstrates that Corollary 2 is driven precisely by the interplay between symmetric demand uncertainty and the low-type seller’s dynamic pricing ability. It is worth noting that Corollary 2 does not deny the value of learning. In fact, as shown in Figure 2, given σB , the low-type seller’s value function V (µ) is convex, and thus learning is valuable to the seller. It is crucial to distinguish between ex ante demand uncertainty (corresponding to initial belief µ0 ) and interim demand uncertainty (corresponding to the seller’s updated belief µ(t) over time). The payoff result (that demand uncertainty increases the low-type seller’s expected payoff) is concerned with the former, while the learning result (that the value function V (µ) is convex in Proposition 2, and thus learning is valuable) is related to the latter. With exogenous demand, the distinction between the two is inconsequential: the seller’s expected payoff is the same whether she begins with belief µ0 or reaches the same level µ0 after some time. With endogenous demand, the distinction is crucial, because ex ante demand uncertainty affects buyers’ purchase strategies σH , while interim demand uncertainty does not. To put it differently, public learning (reducing ex ante demand uncertainty) is not valuable to the seller, while private learning (learning about demand over time) is always valuable.

19

5 Asymmetric Demand Uncertainty In this section, we analyze the case of asymmetric demand uncertainty: the seller is still uninformed about the demand state, while buyers know whether the arrival rate of buyers is high or low. This case is of interest, not only because it represents a plausible situation in which buyers have better information about other buyers than sellers, but also because it provides another benchmark for the results in the previous section. For notational simplicity, we use σd to denote the probability that each buyer accepts pH and qd∗ to denote buyers’ beliefs about the seller’s type conditional on price pH when the demand state is d = h, l. In other words, σd = σB (λd ) and qd∗ = q(λd , pH ) for each d = h, l.

5.1 Buyers’ Beliefs We begin by deriving buyers’ ex post beliefs about the seller’s type in each demand state. Fix the low-type seller’s pricing strategy σS (·) and buyers’ purchase strategies (σh , σl ). Following the same steps as in Section 3.1, for each demand state d = h, l, ∫ ∞ − ∫ t λ σ dx e 0 d d dt qd∗ q0 0 ∫t = . ∫ 1 − qd∗ 1 − q0 ∞ σS (t)e− 0 λd (σS (x)σd +1−σS (x))dx dt

(10)

0

There is no general relationship between qh∗ and ql∗ : each can be larger than the other, depending on agents’ strategies. For example, suppose that the low-type seller plays a simple switching-down pricing strategy with cutoff time TD . In that case, for each d = h, l, q0 qd∗ = ∗ 1 − qd 1 − q0

1 λd σ d 1−e−λd σd TD λd σ d

=

1 q0 . −λ 1 − q0 1 − e d σd TD

(11)

Therefore, qh∗ < ql∗ if λh σh > λl σl , while the opposite is true if λh σh < λl σl . If buyers’ purchase strategies were independent of the demand state (i.e., σh = σl ), then the former would be necessarily the case (because λh > λl ). However, there is no priori reason why buyers play an identical purchase strategy in both states. Indeed, as shown shortly, in equilibrium buyers’ purchase strategies do depend on the demand state (i.e., σh ̸= σl ).

5.2 Buyers’ Equilibrium Purchase Strategies For the same reason as in the previous sections, it cannot be an equilibrium that the low-type seller offers only one price. Under asymmetric demand uncertainty, this means that there are only the following three possibilities:

20

• The low-type seller’s optimal price is pH in demand state h and vL in demand state l: λh λl σl λl λh σh pH > vL , while pH < vL . r + λh σh r + λh r + λl σl r + λl • The low-type seller’s optimal price is vL in demand state h and pH in demand state l: λh σh λh λl σl λl pH < vL , while pH > vL . r + λh σh r + λh r + λl σl r + λl • The low-type seller is indifferent between pH and vL in both demand states: λh σh λh λl σl λl pH = vL , and pH = vL . r + λh σh r + λh r + λl σl r + λl Consider the first possibility, which is similar to the equilibrium under symmetric demand uncertainty. In this case, the low-type seller’s optimal pricing strategy is a simple switching-down strategy: she begins with pH and drops the price to vL once she becomes sufficiently pessimistic. But then, buyers assign a lower probability to the high type in demand state h than in demand state l (i.e., qh∗ < ql∗ ), which follows from the fact that λh σ h λh λl λl σ l pH > vL > vL > pH ⇒ λh σh > λl σl r + λh σh r + λh r + λl r + λl σ l and the argument given above in association with equation (11). This means that buyers have a stronger incentive to accept pH in demand state l than in demand state h, in which case the initial two inequalities cannot materialize. For an intuition, consider the case where λh is sufficiently large, while λl is sufficiently close to 0. Given TD (> 0) that is independent of the demand state, qh∗ would be close to q0 (because both seller types are likely to trade before TD ), while ql∗ would be close to 1 (because both seller types are likely to trade after TD ). But then, buyers would surely reject pH in demand state h and accept pH in demand state l, which is a contradiction. In the second case, the opposite reasoning applies. Given the low-type seller’s optimal price in each state, she plays either a switching-up pricing strategy or a version of constant (time-invariant) strategy.14 In both cases, it can be shown that buyers assign a higher probability to the high type in demand state h than in demand state l (i.e., qh∗ > ql∗ ) and, therefore, it cannot be that the low-type seller’s optimal price is vL in demand state h and pH in demand state l. For an intuition, again, consider the case where λh is sufficiently large, while λl is sufficiently close to 0, and suppose that As shown in the proof of Proposition 4, the former case arises if λh σh ≥ λl σl , while the latter case arises if λh σh < λl σl (which implies that the seller’s belief, conditional on no trade, increases if the price is pH but decreases if the price is vL ). 14

21

the low-type seller plays a switching-up strategy with cutoff time TU . Then, the low-type seller is likely to trade before TU (at price vL ) in demand state h but after TU (at price pH ) in demand state l. Therefore, qh∗ would be close to 1, while ql∗ would be close to q0 , which implies that the low-type seller’s optimal price would be pH in demand state h and vL in demand state l, opposite to the initial supposition. These results suggest that only the last case can arise in equilibrium, as formally stated in the following proposition. Proposition 4 In the model with asymmetric demand uncertainty, buyers’ equilibrium purchase strategies must be given by σd ≡

rvL , for each d = h, l. (r + λd )pH − λd vL

Proof. See Appendix C. As explained above, a contradiction arises if σd differs from σ d . Conversely, if σd = σ d for both d = h, l, then the low-type seller’s optimality is straightforward: she is indifferent between pH and vL conditional on each state and, therefore, independently of her belief. It then suffices to specify her pricing strategy σS (·), so that buyers’ purchase strategies in Proposition 4 are optimal as well. Before we characterize the low-type seller’s equilibrium pricing strategy, we discuss a significant implication of Proposition 4 that is formally reported in the following corollary. Corollary 3 In the model with asymmetric demand uncertainty, conditional on each demand state d = h, l, the low-type seller’s expected payoff is equal to λd vL /(r+λd ), while the high-type seller’s expected payoff is equal to (rcH + λd σ d )/(r + λd σ d ). In other words, asymmetric demand uncertainty has no payoff consequence: in each demand state, each seller type obtains the same expected payoff as in the model without demand uncertainty. Together with Proposition 3, this result highlights the subtlety of the effects of demand uncertainty on the seller’s expected payoff. If buyers are also uninformed about demand, then demand uncertainty necessarily increases the low-type seller’s expected payoff. It may or may not increase the high-type seller’s expected payoff. If buyers are informed about demand, then demand uncertainty has no impact on the seller’s expected payoff: each seller type obtains the same expected payoff as in the model without demand uncertainty.

5.3 Equilibrium Pricing Strategy Proposition 4 suggests that any pricing strategy is the low-type seller’s best response to buyers’ equilibrium purchase strategies. This does not mean that any pricing strategy can be a part of 22

equilibrium (thus, equilibrium pricing strategy, rather than optimal pricing strategy). Recall that in the model without demand uncertainty, the low-type seller is also indifferent between pH and vL , but there is an equilibrium restriction on her pricing strategy σS (t): ∫ ∞ − ∫ t λσ dx e 0 B dt q0 q∗ pH − vL 0 ∫ = = . ∫∞ t 1 − q0 1 − q∗ vH − p H σS (t)e− 0 λ(σS (x)σB +1−σS (x))dx dt 0

The following proposition presents an analogous condition for the model with asymmetric demand uncertainty. Proposition 5 In the model with asymmetric demand uncertainty, a strategy profile (σS (·), σh , σl ) is an equilibrium if and only if for both d = h, l, σd = σ d and ∫ ∞ − ∫ t λ σ dx e 0 d d dt q0 qd∗ p H − vL 0 ∫t = . = ∫ ∞ ∗ 1 − qd 1 − q0 vH − pH σS (t)e− 0 λd (σS (x)σd +1−σS (x))dx dt

(12)

0

Proof. The result immediately follows from equation (10) and Proposition 4. We first show that a simple pricing strategy cannot constitute an equilibrium. • Time-invariant strategy: suppose σS (t) = σ bS for all t. Then, qd∗ q0 = ∗ 1 − qd 1 − q0 Notice that σh =

1 λd σ d σ bS σ bS σ d +1−b σS

=

q0 σ bS bS σ d + 1 − σ . 1 − q0 σ bS σ d

rvL rvL < = σl . rpH + λh (pH − vL ) rpH + λl (pH − vL )

Applying this to the above equation, it follows that qh∗ > ql∗ . • Switching-down strategy: suppose for some TD ≥ 0, σS (t) = 1 if t < TD , while σS (t) = 0 if t ≥ TD . Then, as shown above, 1 qd∗ q0 . = ∗ −λ 1 − qd 1 − q0 1 − e d σd TD Notice that λh σ h > λl σ l , because λh σ h λh λl λl σ l pH = vL > vL = pH . r + λh σ h r + λh r + λl r + λl σ l It is then immediate that qh∗ < ql∗ .

23

• Switching-up strategy: suppose for some TU ≥ 0, σS (t) = 0 if t < TU , while σS (t) = 1 if t ≥ TU . Then, 1 q0 q0 qd∗ 1 λd σ d = . −λd TU = ∗ −λ e 1 − qd 1 − q0 1 − q0 e d TU λd σ d

Therefore, it is necessarily the case that qh∗ > ql∗ . The following proposition presents one form of equilibrium pricing strategy, which has the simplest structure but contrasts well with the unique equilibrium pricing strategy under symmetric demand uncertainty. We note that there are many other equilibria, because the function σS (t) can be adjusted in various ways to satisfy equation (12). Proposition 6 In the model with asymmetric demand uncertainty, there exists a (switching-upand-down) equilibrium in which the low-type seller begins with vL , switches up to pH at some TU (> 0), and reverts back to vL at some TD (> TU ). Proof. Given TU and TD , buyers’ beliefs, conditional on pH , are as follows: for each d = h, l, qd∗ q0 = ∗ 1 − qd 1 − q0

1 λd σ d e−λd TU (1−e−λd σd TD ) λd σ d

=

q0 1 . −λ T U d 1 − q0 e (1 − e−λd σd TD )

For each TU ≥ 0, let ϕ(TU ) be the value such that ql∗ p H − vL q0 1 = = . ∗ σ ϕ(T ) −λ T −λ 1 − ql 1 − q0 e l U (1 − e l l U ) vH − pH The function ϕ(TU ) is well-defined as long as 1 pH − vL q0 ≤ . −λ T U l 1 − q0 e vH − p H

(13)

It is clear that the function is continuous and strictly increasing over the relevant range. Now notice that if TU = 0, then (as in the switching-down case above) qh∗ 1 q0 1 ql∗ q0 < = = . 1 − qh∗ 1 − q0 1 − e−λh σh ϕ(0) 1 − q0 1 − e−λl σl ϕ(0) 1 − ql∗ To the contrary, if TU is the maximal value that satisfies the inequality (13), then ϕ(TU ) = ∞, and thus (as in the switching-up case above) 1 q0 1 ql∗ q0 qh∗ > = = . 1 − qh∗ 1 − q0 e−λh TU 1 − q0 e−λl TU 1 − ql∗

24

By the continuity and monotonicity, there exists a unique value of TU that satisfies qh∗ q0 1 pH − vL . = = ∗ −λ T −λ σ ϕ(T ) 1 − qh 1 − q0 e l U (1 − e l l U ) vH − p H

6 Discussion We conclude by discussing three relevant points to the main results, which can be summarized by the following table: Demand uncertainty Low type’s payoff Equilibrium pricing Example

No λ equal to r+λ vL arbitrary subject to (3) constant, switching-once...

Symmetric λ larger than r+λ vL unique switching-down

Asymmetric λ equal to r+λ vL arbitrary subject to (12) switching-up-and-down...

Relative amount of information about demand. Our result that the low-type seller benefits from symmetric demand uncertainty, but not from asymmetric demand uncertainty seems to suggest that the relative amount of information about demand determines seller surplus. Although fairly plausible, this intuition does not hold in general. In Appendix A, we consider the other asymmetric case in which λ is known to the seller but not to buyers, so that the seller has an additional information advantage over buyers. We show that whether the seller benefits from this type of demand uncertainty depends on the prior probability µ0 (see Proposition 7 and Corollary 4 in Appendix A). Specifically, there exists µ0 such that if µ < µ0 , then both seller types are strictly better off than in the benchmark case without demand uncertainty. However, if µ > µ0 , then the high-type seller receives a strictly lower expected payoff than in the benchmark case, while the low-type seller obtains the same expected payoff. Non-stationary purchase strategies. In all the cases considered in this paper, buyers’ purchase strategies are assumed to be independent of their arrival time, which derives from the assumption that buyers receive no information about the history of the game. The property ensures that buyers’ purchase behavior is stationary from the seller’s viewpoint and, therefore, the analysis remains tractable: among other things, in the model with symmetric demand uncertainty, it is impossible to provide a general characterization for the low-type seller’s optimal pricing strategy if buyers’ purchase strategies are history-dependent. However, as is often the case with dynamic models of 25

adverse selection, it is responsible for some of our results. In particular, as explained in Section 4, the result that the low-type seller benefits from symmetric demand uncertainty is precisely due to the fact that buyers’ purchase strategies are stationary, while the low-type seller can learn about demand and dynamically adjust her price. The seller’s dynamic pricing ability may be of no value if buyers can also dynamically adjust their purchase strategies. In Appendix B, we analyze a non-stationary model in which each buyer observes the full price history of the seller.15 We show that demand uncertainty, whether symmetric or asymmetric, does not affect the low-type seller’s expected payoff in that setting: her expected payoff is equal to λvL /(r + λ) in all three relevant cases. The result is driven by a strong restriction that buyers’ observability of the full price history imposes on the low-type seller’s dynamic pricing: in equilibrium, the low-type seller chooses between vL ad pH at the beginning of the game and sticks to it forever, regardless of the form of demand uncertainty (see Proposition 8 in Appendix B). This does not mean that demand uncertainty has no consequence on equilibrium strategies. In particular, we demonstrate that a simple time-invariant purchase strategy (in which all buyers accept pH with an identical probability) constitutes an equilibrium without demand uncertainty, but not with symmetric demand uncertainty. It should also be noted that the fact that the low-type seller’s expected payoff is independent of demand uncertainty in our specific non-stationary model implies by no means that the same result would hold in other non-stationary models as well. Alternative pricing mechanisms. Despite its undesirable property regarding equilibrium multiplicity, we consider the price-posting mechanism for two reasons. First, it is ideally suited to address one of our main economic questions of what is the seller’s optimal dynamic response to demand uncertainty. Second, combined with the assumption that the seller does not directly observe buyer arrivals, it ensures necessary tractability. To see the latter tractability problem more clearly, consider a more common trading mechanism in which (uninformed) buyers make price offers to the (informed) seller. In general, in order to determine buyers’ optimal (offer) strategies, it is necessary to determine the joint distribution of the seller’s intrinsic type (H or L) and reservation price. In the absence of demand uncertainty (so no learning), the seller’s reservation price depends only on her intrinsic type and, therefore, it suffices to derive the distribution of the seller’s intrinsic type, which is often not particularly demanding as illustrated in Section 3.1. When the seller faces demand uncertainty and learns about demand over time, the seller’s reservation price also depends on her belief about demand and, therefore, it 15

The case in which buyers observe only the seller’s time-on-the-market (or the number of previous buyers) is often more tractable than the case in which buyers also observe past prices (see,e.g. Swinkels, 1999; H¨orner and Vieille, 2009; Kim, 2017). It does not apply to the current environment with demand uncertainty, because the former gives rise to a complicated updating problem regarding the evolution of beliefs (not only on the seller’s type, but also on the demand state), while the problem becomes almost trivial in the latter, as shown in Lemma 2 in Appendix B.

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becomes more difficult to determine the joint distribution. The problem could remain manageable if the seller’s belief evolves in a simple fashion and each seller type’s trading behavior is also sufficiently simple, as is the case in Section 4.2. When buyers make price offers, however, the seller’s belief about demand stochastically changes over time: it continuously decreases (without buyer arrival) but occasionally jumps (upon each arrival). The process itself is hard to aggregate, but the fact that the seller’s trading probability also depends on her belief complicates the derivation of the joint distribution even further. Although we are unable to provide a concrete characterization for the above alternative trading mechanism, our main analysis offers some useful insights. As explained in Section 4.3, the potential benefit of symmetric demand uncertainty to the seller stems from her ability to learn about demand and adjust her strategy accordingly, to which who makes price offers is irrelevant. This does not mean that the low-type seller’s expected payoff would be necessarily higher under symmetric demand uncertainty than under no demand uncertainty, but it is clear that the relevant effect (benefit) should be present in the alternative model as well. Unlike in our main model, it is no longer the case that buyers play the same strategy under asymmetric demand uncertainty as under no demand uncertainty. This is because buyers can use their bargaining power to exploit the most pessimistic seller: suppose λl > rvL /(cH − vL ), so that the low-type seller’s reservation price is equal to vL in the model without demand uncertainty (see Kim, 2017). The most pessimistic low-type seller’s reservation price should be strictly lower than vL , because otherwise, trade would occur only at cH , which cannot arise in equilibrium under Assumption 1.

Appendix A: The Case When λ is Known Only to the Seller In this appendix, we consider another asymmetric case in which the demand state is known to the seller, but not to buyers. The setup is straightforward to interpret, so we avoid its detailed description. We begin with a lemma that characterizes the low-type seller’s best response to buyers’ purchase strategies σB . The values of σ h and σ l are as defined in Proposition 4. Recall that σ h < σ l . Lemma 2 Suppose each buyer accepts pH with probability σB . • If σB ≤ σ h , then the low-type seller weakly prefers offering vL to pH in demand state h and strictly prefers vL to pH in demand state l. • if σB ∈ (σ h , σ l ), then the low-type seller strictly prefers offering pH to vL in demand state h and strictly prefers vL to pH in demand state l.

27

• If σB ≥ σ l , then the low-type seller strictly prefers offering pH to vL in demand state h and weakly prefers pH to vL in demand state l. Proof. The result is straightforward given that the low-type seller knows the demand state and for each d = h, l, she is indifferent between pH and vL in demand state d when each buyer accepts pH with probability σ d . For the same reason as in the other cases, it cannot be an equilibrium that the low-type seller offers only one price in both states. Therefore, in equilibrium, buyers’ purchase strategies must be such that σB ∈ [σ h , σ l ]. We focus on one particular pricing strategy where the low-type seller offers only pH in demand state h and vL in demand state l and identify a condition under which that pricing strategy can be an equilibrium. The condition is then used to identify the parameter range in which σB = σ h and the parameter range in which σB = σ l . Given the seller’s pricing strategy, as in Section 4.2, buyers’ beliefs about the seller’s type, conditional on pH , are given by q∗ q0 = ∗ 1−q 1 − q0

µ∗ λh σ B

+

1−µ∗ λl σ B

µ∗ λh σ B

q0 = 1 − q0

( ) 1 − µ ∗ λh 1+ . µ ∗ λl

Buyers’ beliefs about the demand state, conditional on pH , are given by µ∗ µ0 = 1 − µ∗ 1 − µ0

q0 λh σ B

+

1−q0 λh σ B

q0 λl σ B

=

µ0 λl . 1 − µ0 q0 λh

Combining the two equations and imposing another equilibrium condition that q ∗ (vH − pH ) + (1 − q ∗ )(vL − pH ) = 0, ( ) p H − vL q0 1 − µ0 λ2h = 1+ q0 2 . (14) vH − pH 1 − q0 µ0 λl Since the right-hand side is strictly decreasing from infinity to q0 /(1 − q0 ) as µ0 increases from 0 to 1, under Assumption 1 (which ensures that the right-hand side is smaller than the left-hand side when µ0 is sufficiently close to 1), there exists a unique interior value of µ0 that satisfies the equation. The following result is straightforward from the characterization above. Proposition 7 Let µ0 denote the unique value of µ0 that satisfies equation (14). • If µ0 < µ0 , then in equilibrium each buyer accepts pH with probability σ l , and the low-type seller offers only pH in demand state h but both pH and vL in demand state l. 28

• If µ0 = µ0 , then in equilibrium each buyer accepts pH with probability σB ∈ [σ h , σ l ], and the low-type seller offers only pH in demand state h and vL in demand state l. • If µ0 > µ0 , then in equilibrium each buyer accepts pH with probability σ h , and the low-type seller offers both pH and vL in demand state h, but only vL in demand state l. Proof. If µ0 < µ0 , then q ∗ > (pH − vL )/(vH − vL ) whenever the low-type seller offers vL with a positive probability in demand state h (which implies that she offers only vL in demand state l). Therefore, she must offer only pH in demand state h. Given this, it is clear that she must offer both pH and vL in demand state l, which can be the case only when σB = σ l . If µ0 = µ0 , then, as shown above, q ∗ = (pH − vL )/(vH − vL ) when the low-type seller offers only pH in demand state h and vL in demand state l. This can be the case if and only if σB ∈ [σ h , σ l ]. If µ0 > µ0 , then, opposite to the first case, q ∗ < (pH − vL )/(vH − vL ) whenever the low-type seller offers pH with a positive probability in demand state l (which implies that she offers only pH in demand state h). Therefore, she must offer only vL in demand state l. Given this, she must offer both pH and vL in demand state h, which can be the case only when σB = σ h . In each case, the equilibrium condition q ∗ = (pH − vL )/(vH − vL ) imposes a restriction on the behavior of the low-type seller’s pricing strategy. The restriction can be derived as in the other models and, therefore, omitted. Proposition 7 implies that the seller’s expected payoff may or may not be larger than in the model without demand uncertainty, depending on whether µ0 is above or below µ0 , as formalized in the following corollary. Corollary 4 Suppose the demand state is known to the seller, but not to buyers. • If µ0 < µ0 , then in any equilibrium, the low-type seller’s expected payoff is equal to µ0

λl λh σ l pH + (1 − µ0 ) vL , r + λh σ l r + λl

and the high-type seller’s expected payoff is equal to ) ( λl σ l λh σ l + (1 − µ0 ) (pH − cH ). µ0 r + λh σ l r + λl σ l In this case, both seller types obtain higher expected payoffs than in the model without demand uncertainty.

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• If µ0 > µ0 , then in any equilibrium, the low-type seller’s expected payoff is equal to µ0

λh λl vL + (1 − µ0 ) vL , r + λh r + λl

and the high-type seller’s expected payoff is equal to ( ) λh σ h λl σ h µ0 (pH − cH ). + (1 − µ0 ) r + λh σ h r + λl σ h In this case, the low-type seller obtains the same payoff as in, but the high-type seller receives a lower payoff than, in the model without demand uncertainty.

Appendix B: A Non-stationary Model We consider the same physical environment as in the baseline (stationary) model. The only difference is that now buyers observe the full price history of the seller. It is still assumed that the seller does not observe buyer arrivals and each buyer does not know how many buyers had arrived (and when) before him. These assumptions simplify belief updating (as the seller’s belief depends only on her pricing history, not on stochastic arrivals of buyers) and ensure the congruence between the seller’s and buyers’ beliefs after all histories (if the seller’s past prices are not observable, then buyers may have different beliefs about the demand state than the seller in case the seller plays a mixed strategy or deviates from a pure strategy). Since the problem is clear, we omit a cumbersome formal description of agents’ strategies and equilibrium.

The Low-type Seller’s Offer Strategy The observability of the full price history gives extra tractability to the analysis because, given Assumption 2 (that the high-type seller always offers pH ), the low type is fully revealed if the seller ever offers vL . In other words, if the low-type seller deviates from pH at one point, then she cannot revert back to pH , and thus her problem reduces to finding an optimal time to change the price from pH to vL . In addition, it suffices to derive buyers’ beliefs about the seller’s type conditional on the event that the seller has offered only pH . The following lemma, which applies regardless of the presence of demand uncertainty, shows that buyers must be indifferent between accepting and rejecting pH at all points. Proposition 8 Let q(t) denote the probability that the buyer at time t assigns to the high type, conditional on the event that the seller has offered only pH by t. In equilibrium, q(t) = q ∗ ≡ (pH − vL )/(vH − vL ) for any t ≥ 0. 30

Proof. Since the high-type seller insists on pH , while the low-type seller may switch to vL but never reverts back to pH , q(t) must be non-decreasing. First, it is clear that q(t) must reach q ∗ at some point: otherwise, buyers would never accept pH , in which case the low-type seller would offer vL immediately and, therefore, q(0) = 1 > q ∗ . In what follows, we show that q(t) can never strictly exceed q ∗ and reaches q ∗ immediately. Suppose q(t) strictly exceeds q ∗ from t′ + dt. From the point on, all subsequent buyers strictly prefer accepting pH to rejecting it. However, it can happen only when the low-type seller may change the price to vL at t′ . Knowing that pH will be accepted for sure after t′ , she obviously has no incentive to do so, which is a contradiction. Let t∗ be the first time at which q(t∗ ) = q ∗ , and suppose t∗ > 0. Since q(0) < q(t∗ ), it must be an optimal strategy for the low-type seller to change the price to vL at some t ∈ (0, t∗ ]. However, buyers never accept pH until t∗ , and thus the strategy is dominated by offering vL immediately, which is a contradiction. Proposition 8 implies that the low-type seller either offers vL immediately or insists on pH forever. In other words, in equilibrium, the low-type seller chooses between vL and pH at the beginning and sticks to it through the game. The fact that q(0) = q ∗ allows us to determine the probability that the low-type seller takes each strategy, as formally reported in the following corollary. Corollary 5 In equilibrium, the low-type seller chooses pH , and insists on it forever, with probability q0 /(1 − q0 )(vH − pH )/(pH − vL ) and offers vL immediately with the complementary probability. Proof. Let σS denote the probability. The result is straightforward from the fact that for limt→0+ q(t) = q ∗ to hold, pH − vL q∗ q0 1 = = . vH − p H 1 − q∗ 1 − q0 σS

Notice that this result holds in any model, whether there is demand uncertainty or not. This immediately implies that, unlike in the baseline model, demand uncertainty has no impact on the low-type seller’s expected payoff.

Buyers’ Acceptance Strategies It remains to determine buyers’ acceptance strategies. Note that Proposition 8 implies that all buyers must be indifference between accepting and rejecting pH but does not pin down with what probability each buyer accepts pH .

31

Let σB (t) denote the probability that the buyer who arrives at t accepts pH (conditional on the event that the seller has offered only pH ). In addition, let p(t) denote the expected payoff of the low-type seller at time t, conditional on insisting on pH , and p˜(t) denote her expected payoff when she switches to vL at time t. How these functions interact one another depend on the specific model (i.e., the presence and form of demand uncertainty). Nevertheless, in any model, the following two conditions are necessary and sufficient for σB to constitute an equilibrium. • p(0) = p˜(0): this condition stems from the fact that the low-type seller must be indifferent between pH and vL at time 0. • p(t) ≥ p˜(t) for any t > 0: this is a necessary and sufficient condition for the low-type seller to never switch to vL when she begins with pH . In what follows, we explain what restrictions these two conditions impose on σB (t) in each case. No Demand Uncertainty Suppose λ is commonly known among all agents. In this case, p˜(t) =

λ vL , r+λ

and p(t) evolves according to rp(t) = λσH (t)(pH − p(t)) + p(t). ˙ There are many ways to satisfy the above necessary and sufficient conditions. Here, we explicitly construct two equilibria, each with a particularly simple structure. • Stationary equilibrium: suppose σB (t) stays constant over time and let σB denote its constant value. In this case, p(t) ˙ = 0, and thus the two equilibrium conditions reduce to p(t) =

λ λσH = p˜(t) = vL , for all t. r + λσH r+λ

The value σB = rvL /((r + λ)pH − λvL ) is well-defined in (0, 1), because pH > vL . • Simple pure-strategy equilibrium: suppose there exists t∗ such that buyers never accept pH before t∗ and always accept pH after t∗ . In this case, p(t) = e−r max{t

32

∗ −t,0}

λ pH . r+λ

Since p(t) strictly increases until t∗ and stays constant thereafter (while p˜(t) is time-invariant), it is necessary and sufficient that ∗

p(0) = e−rt

λ λ pH = p˜(0) = vL . r+λ r+λ

The value t∗ is well-defined, again, because pH > vL . In fact, this way of equilibrium construction can be easily extended to construct a continuum of equilibria: for each σB′ ∈ [σB , 1], it suffices to find t∗ (σB′ ) ∈ [0, t∗ ] such that ∗ (σ ′ ) B

p(0) = e−rt

′ λσH λ pH = p˜(0) = vL . ′ r + λσH r+λ

Symmetric Demand Uncertainty Let µ(t) denote the probability that the low-type seller assigns to the high demand state conditional on the event that she has offered only pH . By Bayesian updating, ∫t

µ(t) µ0 e− 0 λh σB (x)dx µ0 − ∫ t (λh −λl )σB (x)dx ∫t , = = e 0 1 − µ(t) 1 − µ0 e− 0 λl σB (x)dx 1 − µ0 which implies that µ(t) ˙ = −µ(t)(1 − µ(t))(λh − λl )σB (t). Since λh > λl , µ(t) necessarily decreases over time. This is natural because no sale is more likely to occur in the low demand state and, therefore, always bad news about the demand state. Given µ(t), p˜(t) and p(t) are given as follows: ( ) λh λl p˜(t) = µ(t) + (1 − µ(t)) vL , r + λh r + λl and rp(t) = (µ(t)λh + (1 − µ(t))λl )σB (t)(pH − p(t)) + p(t). ˙ Unlike in the case without demand uncertainty, p˜(t) decreases over time. In addition, if σB (t) is constant over time, p(t) falls over time. Both of these are, of course, because µ(t) decreases. As before, there are various ways to satisfy the necessary and sufficient equilibrium conditions. We again construct two simple equilibria. • Constant-continuation-payoff equilibrium: Suppose p(t) ˙ = 0 for any t. Then, (µ(t)λh + (1 − µ(t))λl )σB (t) p(t) = pH = p˜(0) = r + (µ(t)λh + (1 − µ(t))λl )σB (t) 33

( ) λh λl µ0 + (1 − µ0 ) vL . r + λh r + λl

Since µ(t) strictly decreases over time, σB (t) must strictly increase over time. Intuitively, buyers should be increasingly more likely to accept pH in order to compensate the lowtype seller’s growing pessimism. It can be shown that σB (t) for each t is well-defined in (rvL /((r + λh )pH − λh vL ), rvL /((r + λl )pH − λl vL )). By construction, p(0) = p˜(0). The condition that p(t) ≥ p˜(t) is also automatically satisfied because p˜(t) decreases over time. • Simple pure-strategy equilibrium: as in the case without demand uncertainty, suppose there exists t∗ such that buyers never accept pH before t∗ and always accept pH after t∗ . In this case, µ(t) = µ0 if t < t∗ , while µ(t) ˙ = −µ(t)(1 − µ(t))(λh − λl ) if t > t∗ . This implies that p˜(t) stays constant, while p(t) strictly increases, until t∗ . Since, obviously, p(t) > p˜(t) if t ≥ t∗ , it suffices to set t∗ so that p(0) = e

−rt∗

( ) ( ) λh λl λh λl µ0 + (1 − µ0 ) pH = p˜(0) = µ0 + (1 − µ0 ) vL . r + λh r + λl r + λh r + λl

The value t∗ > 0 is well-defined, again, because pH > vL . Asymmetric Demand Uncertainty A full characterization of this case is unnecessarily cumbersome and repetitive given the characterization of the main model. We simply argue that asymmetric demand uncertainty effectively has no impact on the equilibrium outcome. Specifically, we show that it is an equilibrium under asymmetric demand uncertainty that in each state, buyers behave as in the case of no demand uncertainty. For each d = h, l, fix an equilibrium acceptance function σd (t) in the model without demand uncertainty (when λ = λd ). In addition, let pd (t) denote the corresponding continuation payoff function. Suppose buyers accept pH according to σd (t) in state d = h, l. Given the low-type seller’s belief about the demand state µ(t) (which can be obtained through Bayes’ rule), ( ) λh λl p˜(t) = µ(t) + (1 − µ(t)) vL r + λh r + λl and p(t) = µ(t)ph (t) + (1 − µ(t))pl (t). The two equilibrium conditions are immediate from the fact that for each d = h, l, pd (t) constitutes an equilibrium in the model without demand uncertainty, and thus pd (t) ≥ pd (0) =

34

λd vL . r + λd

Appendix C: Omitted Proofs Proof of Proposition 3. Let σ B be the value such that µ(σ B ) = µ0 (i.e., the maximal value of σB such that T (σB ) = 0). In addition, let σ B be the value such that µ(σ B ) = 0 (i.e., the minimal value of σB such that T (σB ) = ∞). We show that equilibrium σB necessarily lies in (σ B , σ B ). Whenever σB ≤ σ B , kd (σB ) = 0 for both d = h, l. Therefore, the left-hand side is necessarily larger than the right-hand side in equation (9). To the contrary, if σB ≥ σ B , then κd (σB ) = 1 for both d = h, l. This implies that the left-hand side in equation (9) reduces to q0 /(1 − q0 ), which is strictly less than the right-hand side by Assumption 1. Finally, the left-hand side is continuous and strictly decreasing on (σ B , σ B ). Therefore, there exists a unique value of σB that satisfies (9) on (σ B , σ B ). Proof of Proposition 4. In demand state a, the low-type seller is indifferent between offering pH and vL when σd = σ d . Therefore, she strictly prefers pH to vL if σd > σ d , while the opposite is true if σd < σ d . We divide the proof into three cases. The first case is when the low-type seller prefers pH in demand state h and vL in demand state l, while the latter two cases correspond to the case where the low-type seller prefers vL in demand state h and pH in demand state l (i) σh > σ h , while σl < σ l . In this case, there exists a belief level µ such that the seller’s optimal price is pH if and only if µ > µ. In addition, λh σh > λl σl , because λh σh λh λl λl σl > vL > vL > pH . r + λh σh r + λh r + λl r + λl σl This means that conditional on no trade, the seller’s belief decreases over time, whether she offers pH or vL . It follows that the low-type seller’s optimal pricing strategy is a simple switching-down strategy: she begins with pH and then switches down to vL at some TD (≥ 0). But then, 1 1 qh∗ q0 ql∗ q0 = < = . 1 − qh∗ 1 − q0 1 − e−λh σh TD 1 − q0 1 − e−λl σl TD 1 − ql∗ This implies that either σh = 0 (if σl = σ l ) or σl = 1 (if σh = σ h ), which is a contradiction to the supposition that σh > σ h , while σl < σ l . (ii) σh < σ h , σl > σ l , and λh σh ≥ λl σl . In this case, there exists a belief level µ such that the seller’s optimal price is pH if µ < µ and vL if µ > µ. In addition, since λh σh > λl σl , conditional on no trade, the seller’s belief decreases according to µ˙ = −µ(1 − µ)(λh σh − λl σl ) ≤ 0, 35

while she offers pH . It follows that the low-type seller’s optimal pricing strategy is a simple switching-up strategy: she begins with vL and then switches up to pH at some TU (≥ 0). But then, qh∗ q0 1 q0 1 ql∗ = > = . 1 − qh∗ 1 − q0 e−λh TU 1 − q0 e−λl TU 1 − ql∗ Therefore, either σh = 1 (if σl = σ l ) or σl = 0 (if σh = σ h ), which is a contradiction to the supposition that σh < σ h , while σl > σ l . (iii) σh < σ h , σl > σ l , and λh σh < λl σl . In this case, as in (ii), there exists µ such that the seller’s optimal price is pH if µ < µ and vL if µ > µ. Different from (ii), conditional on no trade, the seller’s belief strictly increases according to µ˙ = −µ(1 − µ)(λh σh − λl σl ) > 0, while she offers pH . Since her belief conditional on no trade strictly decreases while she offers vL , this means that her belief stays constant once it reaches µ (i.e., µ is an absorbing state). For this to happen, the low-type seller must offer pH at rate σS (t) = σ bS =

λh − λl < 1. (λh − λl ) − (λh σh − λl σl )

In other words, every time she offers vL , she needs to offer pH (λh − λl )/(λl σl − λh σh ) times.16 Suppose µ0 ≤ µ. In this case, the low-type seller offers pH until some T and then follows the above pricing strategy. Then, for each d = h, l, qd∗ q0 = ∗ 1 − qd 1 − q0

1−e−λd σd T λd σ d

1 λd σ d e−λd σd T σ bS + λd (b σS σd +1−b σS )

=

q0 1 −λ σ 1 − q0 1 − e d d T σb

1−b σS σS S σd +1−b

.

Since λl σl > λh σh , it follows that qh∗ > ql∗ . As in (ii), this implies that either σh = 1 or σl = 0, which contradicts the supposition that σh < σ h , while σl > σ l . Now suppose µ0 > µ. In this case, the low-type seller offers vL until some T and then follows the above pricing strategy. Then, for each d = h, l, qd∗ q0 = ∗ 1 − qd 1 − q0

1 λd σ d e−λd T σ bS λd (b σS σd +1−b σS )

=

q0 eλd T (b σS σd + 1 − σ bS ) . 1 − q0 σ bS σd

Therefore, again, qh∗ > ql∗ , which leads to same contradiction as above.

16

This means that in discrete time, the low-type seller’s belief asymmetrically oscillates around µ.

36

References Banks, Jeffrey S and Joel Sobel, “Equilibrium selection in signaling games,” Econometrica: Journal of the Econometric Society, 1987, pp. 647–661. Cho, In-Koo and David M Kreps, “Signaling games and stable equilibria,” The Quarterly Journal of Economics, 1987, 102 (2), 179–221. Deneckere, Raymond and Meng-Yu Liang, “Bargaining with interdependent values,” Econometrica, 2006, 74 (5), 1309–1364. Dilme, Francesc and Fei Li, “Revenue management without commitment: dynamic pricing and periodic fire sales,” mimeo, 2016. Easley, David and Nicholas M. Kiefer, “Controlling a stochastic process with unknown parameters,” Econometrica, 1988, 56 (5), 1045–1064. Evans, Robert, “Sequential bargaining with correlated values,” Review of Economic Studies, 1989, 56 (4), 499–510. Garrett, Daniel F, “Intertemporal price discrimination: Dynamic arrivals and changing values,” The American Economic Review, 2016, 106 (11), 3275–3299. Gerardi, Dino, Johannes H¨orner, and Lucas Maestri, “The role of commitment in bilateral trade,” Journal of Economic Theory, 2014, 154, 578–603. H¨orner, Johannes and Nicolas Vieille, “Public vs. private offers in the market for lemons,” Econometrica, 2009, 77 (1), 29–69. Janssen, Maarten C. W. and Santanu Roy, “Dynamic trading in a durable good market with asymmetric information,” International Economic Review, 2002, 43 (1), 257–282. Keller, Godfrey and Sven Rady, “Optimal experimentation in a changing environment,” Review of Economic Studies, 1999, 66 (3), 475–507. Kim, Kyungmin, “Information about sellers’ past behavior in the market for lemons,” Journal of Economic Theory, 2017, 169, 365–399. Lauermann, Stephan and Asher Wolinsky, “Search with adverse selection,” mimeo, 2011. and

, “Search with adverse selection,” Econometrica, 2016, 84 (1), 243–315.

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Mailath, George J., Masahiro Okuno-Fujiwara, and Andrew Postlewaite, “Belief-based refinements in signalling games,” Journal of Economic Theory, 1993, 60, 241–276. Mason, Robin and Juuso V¨alim¨aki, “Learning about the arrival of sales,” Journal of Economic Theory, 2011, 146 (4), 1699–1711. Merlo, Antonio and Francois Ortalo-Magne, “Bargaining over residential real estate: evidence from England,” Journal of urban economics, 2004, 56 (2), 192–216. , Franc¸ois Ortalo-Magn´e, and John Rust, “The home selling problem: Theory and evidence,” International Economic Review, 2015, 56 (2), 457–484. Mirman, Leonard J., Larry Samuelson, and Amparo Urbano, “Monopoly experimentation,” International Economic Review, 1993, 34 (3), 549–563. Palazzo, Francesco, “Search costs and the severity of adverse selection,” Research in Economics, 2017, 71 (1), 171–197. Rothschild, Michael, “A two-armed bandit theory of market pricing,” Journal of Economic Theory, 1974, 9 (2), 185–202. Swinkels, Jeroen, “Education signalling with preemptive offers,” Review of Economic Studies, 1999, 66 (4), 949–970. Taylor, Curtis R., “Time-on-the-market as a sign of quality,” Review of Economic Studies, 1999, 66 (3), 555–578. Vincent, Daniel R., “Bargaining with common values,” Journal of Economic Theory, 1989, 48 (1), 47–62. , “Dynamic auctions,” Review of Economic Studies, 1990, 57 (1), 49–61. Zhu, Haoxiang, “Finding a good price in opaque over-the-counter markets,” Review of Financial Studies, 2012, 25 (4), 1255–1285.

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