Supplement to “Trading Dynamics with Private Buyer Signals in the Market for Lemons” Ayc¸a Kaya∗ and Kyungmin Kim† June 2015

Appendix A: Search-frictionless Limit This appendix presents and illustrates the limiting equilibrium outcome as the arrival rate of buyers λ tends to infinity. This case is of interest for two reasons. First, it allows us to isolate the effects due to information asymmetry from those due to search frictions. Second, it permits a direct comparison of our model to other models that assume away search frictions (i.e., the models in which the seller has an opportunity to trade at each instant). For each a = H, L, let Fa (·) represent the limit distribution of the random variable τa as λ approaches infinity. The following proposition provides the closed-form solutions for these distributions and the low-type seller’s expected payoff in the limit. Proposition A1 Let γH (si ) , i = 1, . . . , N; γL (si )

ρH =

γH (sN ) vL − cL r, γL (sN ) cH − vL

ψia =

1 − Γa (si ) , a = L, H, i = 1, . . . , N − 1. ΓL (si ) − ΓH (si )

and

li =

As λ tends to infinity, the probability that the type-a seller trades by time t converges to Fa (t) = 1 − (1 − Fa (0))e−ρH t , ∗ †

University of Miami. Contact: [email protected] University of Iowa. Contact: [email protected]

1

where if qb < q ∗ , then

FL (0) = 1 −

qb γH (sN ) vH − cH and FH (0) = 0, 1 − qb γL (sN ) cH − vL

(1)

and if qb ∈ (q n , qn−1 ) for some n = 2, ..., N, then Fa (0) = 1 −



ψa ψa N  Y 1 cH − vL 1 − qb n−1 li−1 i−1 × , a = L, H. ln vH − cH qb l i i=n+1

(2)

The low-type seller’s expected payoff converges to

 v , L p˜(b q) = v + F (0)(c − v ), L L H L

.

if qb ≤ q ∗ ,

if qb ∈ (q ∗ , q 1 ).

(3)

Proof. We derive the expression for Fa (t) in three claims. Claim 1 For any qb < q 1 , T (b q , q ∗ ) converges to 0 as λ tends to infinity.

Proof. Take qb ∈ (q n , qn−1 ) for some n = 2, ..., N. Then,

T (b q, q ∗ ) = T (b q , q n ) + T (qn , q n+1 ) + . . . + T (q N −1 , q N ),

where T (b q , qn ) and T (qn′ , qn′ +1 ) are determined by Bayes rule as follows:

and

qb −λ(ΓL (sn−1 )−ΓH (sn−1 ))T (bq ,qn ) qn qb e−λ(1−ΓH (sn−1 ))T (bq ,qn ) = e , = ) −λ(1−Γ (s ))T (b q ,q n−1 L n 1 − qn 1 − qb e 1 − qb q n′ +1 q n′ e−λ(1−ΓH (sn′ ))T (q n′ ,qn′ +1 ) qn′ −λ(ΓL (sn′ )−ΓH (sn′ ))T (q n′ ,qn′ +1 ) e = = . −λ(1−Γ (s ))T (q ,q ) L n′ n′ n′ +1 1 − q n′ +1 1 − q n′ e 1 − q n′

For each n′ ,

q n′ γL (sn′ ) cH − vL . = 1 − q n′ γH (sn′ ) vH − cH

(4)

(5)

(6)

Since each q n′ is independent of λ, it is clear from (4) and (5) that both T (b q , qn ) and T (q n′ , q n′ +1 ) approach 0 as λ tends to infinity.

2

Now consider qb < q N . Since q ∗ = qN , T (q, q ∗ ) is given by

qN 1 qb = . −λT 1 − qN 1 − qb e (bq,qN )

(7)

Again, since q N is independent of λ, it follows that T (q, q ∗ ) converges to 0 as λ goes to infinity. Claim 2 Whenever λ is sufficiently large that q ∗ = qN , the trading rate of each type on the stationary path is given by ρH = r

γH (sN ) vL − cL . γL (sN ) cH − vL

Proof. Recall that ρH is defined to be the trading rate of the high type on the stationary path. Since the low type should trade at the same rate as the high type, it is also the trading rate of the low type on the stationary path: note that ρL is the rate at which the low-type seller receives offer cH , not the rate at which she trades, because she also trades at vL with a positive probability (see Section 3.3.1). On the stationary path, the low-type seller’s reservation price must be equal to vL , and buyers offer cH with a positive probability only when their signal is sN . Therefore, r(vL − cL ) = λγL (sN )σB∗ (cH − vL ), which implies that σB∗ =

vL − cL r . λγL (sN ) cH − vL

Combining this with the fact that the high-type seller trades only at cH , ρH = λγH (sN )σB∗ = r

γH (sN ) vL − cL . γL(sN ) cH − vL

The above two claims establish that in the limit as λ tends to infinity, the game reaches the stationary path immediately and each seller type trades at a finite rate on the stationary path. This implies that the limit distribution of τa takes the following form: Fa (t) = 1 − (1 − Fa (0))e−ρH t . We now complete the derivation of the limit distribution by identifying Fa (0) (the probability that the type-a seller trades before reaching the stationary path) for each a = H, L. Claim 3 Fa (0) is given as in (1) and (2). 3

Proof. Suppose qb < q ∗ = q N . In this case, cH is never offered and, therefore, the high type never trades along the convergence path. This immediately implies that FH (0) = 0. The low type trades at rate λ on the convergence path. Therefore, the probability that she trades before the stationary ∗

path is given by 1 − e−λT (bq ,q ) . The explicit solution then follows from (6) and (7). Now consider qb ∈ (q n , qn−1 ) for some n = 2, ..., N. In this case, each seller type trades only

at cH along the convergence path. Therefore, the probability that trade does not take place before buyers’ beliefs reach q n is equal to e−λ(1−Γa (sn−1 ))T (bq ,qn ) . Combining this with (4) and (6), −λ(1−Γa (sn−1 ))T (b q ,qn )

e

=



 1−Γa (sn−1 )  ψa 1 cH − vL 1 − qb n−1 γL (sn ) cH − vL 1 − qb ΓL (sn−1 )−ΓH (sn−1 ) = . γH (sn ) vH − cH qb ln vH − cH qb

Similarly, the probability that trade does not take place while buyers’ beliefs travel between two cutoff levels qn′ and qn′ +1 is given by e−λ(1−Γa (sn′ ))T (q n′ ,qn′ +1 ) =



γL (sn′ +1 ) γH (sn′ ) γH (sn′ +1 ) γL (sn′ )

Γ

1−Γa (sn′ ) L (sn′ )−ΓH (sn′ )

=



ln ′ ln′ +1

ψ a ′ n

.

The desired result now follows from 1 − Fa (0) = e−λ(1−Γa (sn ))T (bq ,qn−1 ) · e−λ(1−Γa (sn ))T (q n ,qn+1 ) · · · e−λ(1−Γa (sN−1 ))T (q N−1 ,qN )   ψa ψna ψa  1 cH − vL 1 − qb n−1 lN −1 N−1 ln = ··· . · ln vH − cH qb ln+1 lN Finally, we turn to the derivation of the limit value of p˜(b q ). As explained in the main text, p˜(b q) can be calculated by assuming that the low-type seller accepts only cH . Moreover, the low-type seller’s reservation price on the stationary path is always equal to vL . If qb < q ∗ , then the low-type

seller never receives offer cH along the convergence path. Since the game reaches the stationary path immediately, it follows that p˜(b q ) = vL . Now suppose qb > q ∗ . In this case, the low-type seller receives offer cH with probability FL (0) immediately and then falls onto the stationary path. Therefore, p˜(b q ) = FL (0)cH + (1 − FL (0))vL .

It is well-recognized that in dynamic environments, real-time delay is unavoidable with severe adverse selection: for immediate trade, it is necessary that all seller types trade at an identical price. If adverse selection is sufficiently severe, however, buyers’ unconditional expected value of the asset falls short of the high-type seller’s reservation value, and thus there does not exist a price that yields non-negative payoffs to all seller types as well as buyers. The presence of real-time delay in our model does not immediately follow from this well-known wisdom. In the 4

presence of inspection which exogenously reveals type-dependent information, even if the lowtype seller exactly mimics the equilibrium behavior of the high-type seller, she cannot generate statistically identical outcomes. Since each arriving buyer generates a fixed amount of information about the seller’s type, in the search-frictionless limit, potentially there is sufficient information in the cumulative inspection outcomes to fully distinguish different seller types, which leaves no role for real-time delay. The failure of this mechanism (i.e., the persistence of real-time delay) in our model is due to the private nature of buyers’ signals. In other words, buyers only make inferences about previous buyers’ signals from the seller’s time-on-the-market, without directly observing them. Coupled with limited informativeness of their signals, this makes it impossible to fully deter the low-type seller from mimicking the high type, thereby rendering real-time delay to persist despite cumulatively sufficient information.

Appendix B: Uniform Garbling In this appendix, we further study the effects of varying the informativeness of buyers’ signals by imposing more structure on the garbling technology. Specifically, we restrict attention to the following form of Markov matrices: 

  M(ε) =   

1 − (N − 1)ε ε ... ε 1 − (N − 1)ε ... ... ... ... ε

ε

ε ε ...

... 1 − (N − 1)ε



  .  

This garbling technology embodies two special features. First, the number of signals N is preserved. This means that we can take the set of signals S as invariant and consider the effects of garbling on each signal sn . Second, noise to each signal is uniformly and symmetrically drawn from all other signals. This gives particular tractability to the analysis by summarizing the effects of garbling with one variable ε. Notice that ε = 0 corresponds to no garbling, while ε = 1/N indicates complete garbling (i.e., completely uninformative inspection technology). For tractability as well as to focus on purely informational issues, we restrict attention to the case where λ is sufficiently large (so that n∗ = N). A necessary and sufficient condition for this is that λ > λ∗N −1 with the original inspection technology (γL , γH ): as shown shortly, garbling always decreases γH (sN )/γL (sN ) and, therefore, preserves the inequality. Notice also that if buyers’ signals are sufficiently uninformative (i.e., γH (sN )/γL (sN ) is close to 1), then λ∗N −1 is necessarily close to λ∗0 , and thus this restriction incurs no loss of generality. Recall that general garbling puts a restriction only on q 1 and q N (that q1 decreases, while

5

q N increases as the informativeness of buyers’ signals decreases). Uniform garbling allows us to determine the behavior of other cutoff beliefs, q¯2 , ..., q¯N −1 , as the informativeness of buyers’ signals changes, as illustrated in the following result. Lemma A1 As ε increases, qn decreases if γH (sn )/γL (sn ) < 1, while it increases if γH (sn )/γL (sn ) > 1. In the limit as ε tends to 1/N, q 1 = ... = q N = (cH − vL )/(vH − vL ). Proof. Let ln (ε) denote the likelihood ratio for signal sn as a function of ε. By direct calculation, P ε k6=n γH (sk ) + (1 − (N − 1)ε)γH (sn ) ε + (1 − Nε)γH (sn ) , = ln (ε) = P ε k6=n γL (sk ) + (1 − (N − 1)ε)γL (sn ) ε + (1 − Nε)γL (sn ) and ln′ (ε) =

γL (sn ) γL (sn ) − γH (sn ) = (1 − ln (0)) . 2 (ε + (1 − Nε)γL (sn )) (ε + (1 − Nε)γL (sn ))2

The result is immediate if this is combined with the fact that qn 1 cH − vL . = 1 − qn ln (ε) vH − cH

The result is intuitive: signal sn is informative if its likelihood ratio γH (sn )/γL (sn ) is either sufficiently small (indicating the low quality) or sufficiently large (indicating the high quality). It is completely uninformative (equivalent to receiving no signal) when γH (sn )/γL (sn ) is equal to 1. Therefore, garbling brings γH (sn )/γL (sn ) closer to 1, whether γH (sn )/γL (sn ) is below or above 1. It follows that the equilibrium belief cutoffs q 1 , ..., q N become more concentrated as ε increases: the likelihood ratio γH (sn )/γL (sn ) is strictly increasing in n. Therefore, there exits n b such that q n decreases in ε if n ≤ n b, while it increases if n > n b. The following result is a uniform-garbling analogue to Proposition 2 in the main text.

q ) ∈ [0, 1/N) such that whenever Proposition A2 If qb < (cH − vL )/(vH − vL ), then there exists ε(b q), ε > ε(b • the low-type seller’s expected payoff p(b q ) decreases in ε, • the low-type seller’s expected time to trade E[τL ] increases in ε, and • the high-type seller’s time to trade τH increases in ε (first-order stochastic dominance). If qb > (cH − vL )/(vH − vL ), then there exists ε(b q ) ∈ [0, 1/N) such that whenever ε > ε(b q ), trade occurs with the first buyer at cH , and thus 6

• the low-type seller’s expected payoff achieves the maximal value (rcL + λcH )/(r + λ) and • each seller type’s time to trade is minimized (first-order stochastic dominance). Proof. We use qn (ε) and γa (sN , ε) for each a = L, H to make clear their dependence on ε. Fix qb < (cH − vL )/(vH − vL ). Let ε(b q ) be the value such that q N (ε(b q )) = qb. Since q N (ε) q ) < 1/N. If ε > ε(b q ), then qb < qN (ε), strictly increases and converges to (cH − vL )/(vH − vL ), ε(b

and thus

p(b q) = (1 − e−rT (bq,qN (ε)) )cL + e−rT (bq,qN (ε)) vL ,

where q N (ε) =

qb . qb + (1 − qb)e−λT (bq ,qN (ε))

q N (ε) increases in ε, and thus T (b q, q N (ε)) also increases in ε. This implies that p(b q ) strictly decreases in ε. For the times to trade, first notice that provided that n∗ = N, the trading rate of both types on the stationary path is given by ρH (ε) = λγH (sN , ε)σB∗ =

r(vL − cL ) γH (sN , ε) = ρL lN (ε). cH − vL γL (sN , ε)

Since lN (ε) strictly decreases in ε, ρH (ε) always strictly decreases in ε. The high-type seller never trades until T (b q , qN (ε)) and trades at rate ρH thereafter. Since T (b q, q N (ε)) strictly increases, while ρH (ε) strictly decreases in ε, it is clear that the distribution of τH increases in the sense of first-order stochastic dominance. The low-type seller trades at rate λ until T (b q, q N (ε)) and trades at rate ρH thereafter. Although this change cannot be measured in terms of first-order stochastic dominance, E[τL ] =

Z

T (b q ,qN (ε)) −λt

td(1−e

−λT (b q,q N (ε))

)+e

0



1 T+ ρH (ε)



=

1 − e−λT (bq ,qN (ε)) e−λT (bq,qN (ε)) + . λ ρH (ε)

Notice that ρH (ε) = ρL lN (ε), while e−λT (bq ,qN (ε)) = Therefore, the expression shrinks to E[τL ] =

qb 1 − q N (ε) qb cH − vL = lN (ε) . 1 − qb q N (ε) 1 − qb vH − cH

1 − e−λT (bq ,qN (ε)) qb cH − vL 1 + . λ 1 − qb vH − cH ρL

Since T (b q , qN (ε)) strictly increases in ε, E[τL ] also strictly increases in ε. 7

Now fix qb > (cH − vL )/(vH − vL ). Let ε be the value such that q 1 (ε(b q )) = qb. Since q 1 (ε) q ) < 1/N. If ε > ε(b q ), then qb > q 1 (ε). decreases and converges to (cH − vL )/(vH − vL ), ε(b Therefore, trade occurs with the first buyer at cH .

Appendix C: Applications This online appendix provides formal descriptions and analyses of the applications discussed in Section 6.

Informationally Heterogeneous Buyers We modify our model to allow for buyer heterogeneity. In particular, we consider the case when some buyers receive informative signals, while the others do not. We refer to the former as informed buyers and the latter as uninformed buyers. Equilibrium with heterogeneous buyers Let b ∈ (0, 1) denote the probability that each buyer is informed (equivalently, the proportion of informed buyers in the population). We assume that the realization of each buyer’s type is independent of other buyers’ types as well as the seller’s time-on-the-market. This implies that informed buyers arrive at rate λb, while uninformed buyers arrive at rate λ(1 − b). The equilibrium characterization in Section 3 directly applies to this extended model. Buyers’ offers condition only on their posterior beliefs. Therefore, the extended model with buyer heterogeneity works just as the baseline model in which all buyers are ex ante identical but receive an uninformative signal (i.e., the signal whose likelihood ratio is equal to 1) with probability 1 − b. In other words, compound lotteries that first determine each buyer’s type and then his signal can be interpreted as simple lotteries that include an uninformative signal. Corollary A1 In the extended model with informationally heterogeneous buyers, there exists a generically unique equilibrium, which is described by a finite partition {qN +1 = 0, qN , ..., q1 , q 0 = 1}, uninformed buyers’ belief cutoff q u , and n∗ ∈ {1, ..., N} ∪ {u}: informed buyers behave as in Theorem 1, while uninformed buyers offer cH if q(t) > q u and p(t) if q(t) < q u . Hereafter, for tractability, we restrict attention to the case where λ is sufficiently large. Precisely, we assume that λbγL (sN ) > ρL . This ensures that the stationary belief level is determined

8

by signal sN (i.e., q ∗ = q N ). Each cutoff belief for informed buyers is then given by qn γL (sn ) cH − vL . = 1 − qn γH (sn ) vH − cH Similarly, the cutoff belief used by uninformed buyers, denoted by q u , is given by qu cH − vL . = 1 − qu vH − cH The resulting trading dynamics is as follows: if q(t) < q N then both uninformed buyers and informed buyers offer only p(t) until q(t) reaches q N . From that point on, only informed buyers with signal sN offer cH with a positive probability. Now suppose q(t) ∈ (q n+1 , qn ) for some n < N. Since uninformed buyers offer cH only when q(t) ≥ q u , the trading rate of the type-a seller is equal to λ(b(1 − Γa (sn )) + 1 − b) if q(t) > qu , while it is equal to λb(1 − Γa (sn )) if q(t) < q u . It follows that buyers’ beliefs evolve according to q(t) ˙ = q(t)(1 − q(t))λb(ΓH (sn ) − ΓL (sn )). Notice that this law of motion is independent of whether q(t) > q u or not, and the absolute value of q(t) ˙ is increasing in b. Both of these are due to the fact that uninformed buyers’ behavior does not reflect any new information and, therefore, buyers learn only from informed buyers’ behavior. Clearly, informed buyers perform better than uninformed buyers: their additional information (signals) helps them reduce both type I (not offering cH when the seller is the high type) and type II (offering cH when the seller is the low type) errors. We now study the effects of two policies regarding buyers’ signals on their expected payoffs. Decreasing the precision of informed buyers’ signals We first consider the policy that reduces the precision of informed buyers’ signals. We focus on uniform garbling introduced in Online Appendix B. Obviously, this helps leveling the playing field between uninformed and informed buyers. What is not a priori clear is how the convergence between the two buyer types occurs, in particular, whether such a policy would help or hurt uninformed buyers. The following proposition shows that under some regularity assumptions on γL (·) and γH (·), decreasing the precision of informed buyers’ signals increases uninformed buyers’ expected payoffs.1 1 We note that Proposition A3 is concerned with uninformed buyers’ expected payoffs conditional on the event that the asset is available by the time they arrive. Alternatively, one can consider their unconditional expected payoffs, taking into account the probability that the asset is available. The result in Proposition A3 does not generally hold for

9

Proposition A3 Suppose γL (sn ) is decreasing in n, while γH (sn ) is increasing in n. Then, uniform garbling weakly increases the expected payoff of an uninformed buyer who meets the seller at any time t (i.e., max{q(t)(vH − cH ) + (1 − q(t))(vL − cH ), 0}). Proof. Notice that uninformed buyers’ expected payoffs are equal to 0 whenever q(t) ∈ (qN , qu ). Therefore, we focus on the case where qb < q N or qb > q u . First, consider the case where qb < q N . In this case, garbling does not affect q(t) = but lowers p(t) for any t such that q(t) < q N . Formally,

qb , qb+(1−b q )e−λt

q, q N ). p(t) = cL + e−r(T (bq ,qN )−t) (vL − cL ), ∀t < T (b q, q N ) as well. As shown in Online Appendix B, uniform garbling increases q N and, therefore, T (b This strictly lowers p(t) for any t < T (b q , qN ). Suppose qb > q u . In this case, we show that garbling increases q(t) for any t such that q(t) >

q u . Let n be the value such that qb ∈ (q n+1 , qn ) and q n > q u . For notational simplicity, define q, q˜n+1 ), q˜n+i ≡ max{qn+i , q u }. Then, for any t < T (b qb −λb(ΓL (sn )−ΓH (sn ))t q(t) = e . 1 − q(t) 1 − qb

The monotonicity assumption on γL (·) and γH (·) implies that uniform garbling increases γH (sk ), while decreases γL (sk ) whenever γH (sk )/γL (sk ) < 1. Since n is such that γH (sn ) < γL (sn ) (this is equivalent to q n > q u ), it follows that γL (sk ) − γH (sk ) decreases for any k ≤ n, which implies that ΓL (sn ) − ΓH (sn ) also decreases. From the equation above, it is clear that q(t) increases. Now consider t ∈ (T (b q, q˜n+1 ), T (b q, q˜n+2 )). In this case, uniform garbling affects q(t) in two ways. First, as in the previous paragraph, ΓL (sn+1 ) − ΓH (sn+1 ) decreases, which slows down the decline of q(t). Second, it increases T (b q , q˜n+1 ), because qn+1 decreases and q(t) declines more slowly on the interval (q n+1 , q n ). The latter makes q(t) decrease even more slowly, because the monotonicity assumption on γL (·) and γH (·) also implies that ΓL (sn ) − ΓH (sn ) < ΓL (sn+1 ) − ΓH (sn+1 ) (recall that γL (sn+1 ) > γH (sn+1 ), provided that q n+1 > q u ). This argument can be recursively applied to any q(t) > q u . Therefore, we conclude that uniform garbling strictly increases q(t) for any t < T (b q, q u ). For the high initial belief (i.e., qb > q u ), the result is driven by the effects the policy has on buyers’ learning. If informed buyers’ signals become less precise, then buyers learn and update unconditional expected payoffs (i.e., the policy may lower some uninformed buyers’ unconditional expected payoffs). The monotonicity conditions on γL (·) and γH (·) are also important. Although they are sufficient, but not necessary, for the result, their violation may unravel the result for some uninformed buyers. For example, it is possible that garbling speeds up the decline of buyers’ beliefs and lowers some uninformed buyers’ expected payoffs.

10

their beliefs more slowly. When the initial belief is high, this means that q(t) stays high for a longer period of time. Conditional on offering cH , this is good news to uninformed buyers, increasing their expected payoffs. For the low initial belief (i.e., qb < q N ), the result is driven by another force. In this case, delay indicates only the seller’s insistence on the high price and does not reflect any information generated by informed buyers. Therefore, garbling does not affect the evolution of q(t). Still, uninformed buyers benefit from garbling for the following reason: a decrease in their signal precision makes informed buyers more reluctant to offer cH . Then, it takes longer for the seller to convince buyers to offer cH . This lowers the low-type seller’s reservation price, which is beneficial to uninformed buyers. The effects of the policy on informed buyers’ expected payoffs are not necessarily negative. To begin with, if qb < q N , then informed buyers also benefit from a decrease in p(t). If qb > q N , then early informed buyers clearly receive strictly lower expected payoffs, because they are mainly affected by the direct negative effect of the policy. However, late informed buyers enjoy the same benefits from slower learning as uninformed buyers. This indirect positive effect can outweigh the direct negative effect, and thus late informed buyers can benefit from the policy. Educating some uninformed buyers We now consider an alternative policy that transforms some uninformed buyers into informed. Such a policy clearly increases the expected payoffs of uninformed buyers who turn informed. However, it also has an indirect effect on the evolution of buyers’ beliefs, which can negatively affect other uninformed buyers. The following proposition shows that the indirect effect is indeed negative. Proposition A4 The expected payoff of an uninformed buyer at time t strictly decreases in b if q(t) ∈ [q u , q1 ), while it is independent of b otherwise. Proof. The result for q(t) < q u is straightforward, because the expected payoff of an uninformed buyer is equal to 0 if q(t) ∈ [q N , qu ] and independent of his type (whether he is informed or not) if q(t) < q N . q , qu ), q(t) strictly decreases in b, which Suppose qb ∈ [q u , q 1 ). We show that for any t < T (b

implies that q(t)(vH − cH ) + (1 − q(t))(vL − cH ) also strictly decreases. Using the same notation as in the proof of Proposition A3, suppose qb ∈ (˜ qn+1 , q n ). Then, q(t) qb −λb(ΓL (sn )−ΓH (sn ))t = e , ∀t ≤ T (b q , q˜n+1 ). 1 − q(t) 1 − qb

Given t, since ΓL (sn ) > ΓH (sn ), an increase in b always decreases q(t). Now, consider t ∈ 11

(T (b q, q˜n+1 ), T (b q, q˜n+2 )). In this case, q(t) qb −λb(ΓL (sn )−ΓH (sn ))T (bq ,˜qn+1 ) −λb(ΓL (sn+1 )−ΓH (sn+1 ))(t−T (bq ,˜qn+1)) = e e 1 − q(t) 1 − qb q n+1 −λb(ΓL (sn+1 )−ΓH (sn+1 ))(t−T (bq ,˜qn+1 )) e . = 1 − qn+1 In this case, q(t) decreases in b, not only because of b in the exponential term but also because T (b q, q˜n+1 ) decreases. The same argument can be recursively applied to any t < T (b q, q u ). The first negative result is due to the fact that an increase in b always speeds up the evolution of buyers’ beliefs. When qb ∈ [q u , q 1 ), q(t) falls faster over time. Therefore, given t such that q(t) > q u , a marginal increase in b necessarily lowers q(t) and, therefore, the uninformed buyer’s expected payoff as well. The second independence result derives from the fact that a buyer’s (ex ante) expected payoff is independent of his type if q(t) > q 1 or q(t) < q N , and uninformed buyers

obtain zero expected payoff if q(t) ∈ [q N , qu ]. Proposition A4 suggests that the policy of transforming some uninformed buyers into informed is appealing when qb ∈ (q N , qu ): uninformed buyers who become informed strictly benefit from the policy, while there is no negative effect on other uninformed buyers. If either qb > q 1 or qb ≤ q N , then the policy is neutral: information does not increase buyers’ expected payoffs, while there is also no negative effect on uninformed buyers. If qb ∈ (qu , q 1 ), then the policy generates conflicts among uninformed buyers. Information clearly helps buyers. However, uninformed buyers who fail to become informed necessarily suffer.

Mandatory Disclosure of Known Defects We now analyze the effects of mandatory disclosure on welfare and efficiency. To keep the analysis focused as well as tractable, we restrict attention to the case where there are two signals (i.e., N = 2) and the high type never generates the low signal s1 (i.e., γH (s1 ) = 0). Our interpretation is that the quality of an asset is about whether the asset has a certain defect (low quality) or not (high quality), and a buyer may observe the defect, which can happen only when the asset is of low quality. We focus on the case where λ is sufficiently large, for the same reason as Assumption 1 in the main text, and the probability of detecting a defect from a low-quality asset (γL (s1 )) is sufficiently small, so that adverse selection is sufficiently problematic. Specifically, we assume that r(vL − cL ) < λ(γL (s1 )(cL − vL ) + γL (s2 )(cH − vL )).

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We first analyze the model of mandatory disclosure: once a buyer detects a defect (i.e., receives signal s1 ), the seller is required to report it to every subsequent buyer and, therefore, the low quality 12

of the asset becomes publicly known. We then compare the equilibrium outcome to that without mandatory disclosure (i.e., the equilibrium outcome of our main model). Subgame following signal s1 Once a buyer receives signal s1 , the low quality of the asset becomes publicly known. Assuming that subsequent buyers never update their beliefs thereafter,2 following the logic of the Diamond paradox, there is a unique equilibrium in which all buyers offer cL and the (low-type) seller accepts it immediately. In what follows, we take this subgame equilibrium outcome as given and characterize an equilibrium of the entire game. No decreasing pattern We now show that buyers’ beliefs q(t), conditional on no trade and no arrival of signal s1 , cannot strictly decrease over time in the model with mandatory disclosure. Using the same notation as in the main model, denote by σB (t, s, p) the probability that the buyer at time t offers p with signal s and by σS (a, t, p) the probability that the type-a seller accepts p at time t. The high-type seller trades at rate λσB (t, s2 , cH ) and never generates signal s1 , while the low-type seller generates signal s1 (and trades immediately) with probability γL (s1 ) or generates signal s2 , in which case he trades at either cH or p(t) at total rate σB (t, s2 , cH ) + (1 − σB (t, s2 , p(t)))σS (a, t, p(t)). Therefore, q(t) evolves according to q(t) ˙ = λq(t)(1−q(t))(γL (s1 )+γL (s2 )(σB (t, s2 , cH )+(1−σB (t, s2 , p(t)))σS (a, t, p(t)))−σB (t, s2 , cH )). This expression is always non-negative, and strictly positive unless σB (t, s2 , cH ) = 1. Therefore, q(t) either strictly increases or stays constant, the latter being the case if buyers’ strategy is to offer cH with probability 1 unless they receive signal s1 . High initial beliefs We first consider the case where qb is so large that it is an equilibrium that all buyers offer cH with probability 1 unless they receive signal s1 . To be formal, define q 2 as in the main model: q2 cH − vL γL (s2 ) cH − vL = γL (s2 ) . = 1 − q2 γH (s2 ) vH − cH vH − cH 2

This qualification is due to buyers’ off-the-equilibrium-path beliefs, in particular, after the event that a buyer first receives signal s1 , but the seller rejects his offer cL .

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Suppose qb ≥ q 2 . Consider a strategy profile in which all buyers offer cH with signal s2 and cL with signal s1 . In this strategy profile, q(t) stays constant at qb, as explained above. Since qb ≥ q 2 , in order to verify that this strategy profile is an equilibrium, it suffices to show that buyers have no

incentive to deviate to p(t). This follows from the fact that the low-type seller’s reservation price is equal to λ (γL (s1 )cL + γL (s2 )cH ) + rcL , p≡ r+λ which is strictly larger than vL under (8). Low initial beliefs We now consider the case where qb < q 2 . In this case, obviously, it cannot be an equilibrium that

buyers always offer cH with signal s2 . Therefore, q(t) must strictly increase over time. As in the main model, the low-type seller’s trading rate crucially depends on whether p(t) > vL or not. Therefore, we distinguish between the following two cases.3 (i) p(t) > vL : In this case, buyers are not willing to trade at p(t), as it would be accepted only by the low type. Since q(t) < q 2 , buyers are unwilling to offer cH even with the high signal. These imply that trade can occur only when a buyer receives signal s1 and, therefore, the low quality becomes publicly know. It follows that q(t) increases according to q(t) ˙ = λq(t)(1 − q(t))γL (s1 ).

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In addition, p(t) changes over time according to r(p(t) − cL ) = λγL (s1 )(cL − p(t)) + p(t). ˙

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(ii) p(t) < vL : In this case, as in the main model, the low-type seller must accept p(t) with probability 1. Since the low-type seller trades regardless of buyers’ signals, while the high-type seller never trades, q(t) increases according to q(t) ˙ = λq(t)(1 − q(t)).

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The evolution of the low-type seller’s reservation price p(t) is the same as in (10). 3

We do not separately consider the case when p(t) = vL . Unlike in the main model, q(t) strictly increases over time even if p(t) = vL and, therefore, the specification of the low-type seller’s acceptance behavior when p(t) = vL does not affect the overall equilibrium trading dynamics. It can be assumed that the low-type seller accepts p(t) = vL with any probability.

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Combining the results so far, it is clear that there exists a unique equilibrium and the following properties must hold: • Let q be the value such that p˜(q) = vL . More formally,
vL = 1 − e−λγL (s1 )T (q,q2 ) cL + e−λγL (s1 )T (q,q2 ) p, where T (q, q2 ) is defined to be the value such that q2 =

q . q + (1 − q)e−λγL (s1 )T (q,q2 )

• If q(t) ∈ [q, q2 ), then buyers never offer cH and the low-type seller does not accept p(t). q(t) increases according to (9), and p(t) increases from vL to p according to (10). • If q(t) < q, then buyers offer either p(t) (with signal s2 ) or cL (with signal s1 ), and the low-type seller always accepts p(t). q(t) increases according to (11), and p(t) increases to vL according to (10). Effects of mandatory disclosure The following proposition illustrates that mandatory disclosure tends to improve market efficiency. Proposition A5

1. If qb ≥ q 2 , then the time to trade for each seller type is smaller (in the sense

of first-order stochastic dominance) in the model with mandatory disclosure than in our main model. 2. If qb < q 2 and λ is sufficiently large, then the expected time to trade for each seller type is smaller in the model with mandatory disclosure than in our main model.

Proof. The first case where qb ≥ q 2 is clear from the equilibrium structure: In the main model, both types may fail to trade with the first buyer (note that the high-type seller may not trade with the first buyer on the stationary path), while in the model with mandatory disclosure, they always trade with the first buyer. Now suppose qb < q 2 . In this case, the times to trade cannot be ranked in terms of first-order

stochastic dominance: in the model with mandatory disclosure, the low-type seller trades at rate λγL (s1 )(< λ) while q(t) ∈ (q, q 2 ) but at rate λ(> ρH ) once q(t) becomes equal to q2 . In addition, it takes longer for q(t) to reach q 2 , and thus the high type’s times to trade also cannot be ranked in

terms of first-order stochastic dominance. Nevertheless, the result on the expected times to trade follows from the following observation: when λ is sufficiently large, T (q, q 2 ) is close to 0. Since 15

trade occurs immediately once q(t) becomes equal to q 2 , this means that the expected time to trade for each type is also close to 0. As shown in Online Appendix A, the expected time to trade for each type is bounded away from 0 in the main model even in the limit as λ tends to ∞. These give the result for qb ∈ [q, q2 ]. For qb < q, the result follows from the fact that the two models yield an identical trading rate for each seller type until q(t) reaches q. The effect on the low-type seller’s expected payoff is ambiguous. On the one hand, the low-type seller faces the direct risk of her quality being fully revealed, in which case she does not obtain any positive (net) payoff. On the other hand, conditional on no realization of signal s1 , buyers become more optimistic about the quality of the asset and, therefore, offer cH more frequently. Neither of these necessarily dominates the other and, therefore, mandatory disclosure may increase or decrease the low-type seller’s expected payoff. More precisely, if qb is sufficiently close to 1, the

+λcH low-type seller’s expected payoff is closer to rcLr+λ in the main model, which is strictly larger than her expected payoff in the model with mandatory disclosure, p. To the contrary, if qb is around q 2 (whether below or above), then the low-type seller’s expected payoff is necessarily higher under

mandatory disclosure, because it is around vL in the main model, which is strictly smaller than p.

Costly Inspection We now modify our main model so that each buyer can decide whether to conduct an inspection or not at some small, but positive, cost c. For simplicity, we restrict attention to the binary-signal case (i.e., N = 2). We characterize the equilibrium of the model and then study the effects of decreasing c. Equilibrium with costly inspection Since the equilibrium structure is similar to that of our main model, we only illustrate necessary adjustments. In addition, we focus on the case where λ is sufficiently large. It is easy to modify the characterization below for the case where λ is relatively small. We begin with a useful observation: a buyer conducts a costly inspection only when he would offer a different price depending on the realized signal. This immediately implies that buyers pay the inspection cost c if and only if q(t) ∈ [q 2 , q 1 ): if either q(t) < q 2 or q(t) ≥ q 1 , he would offer p(t) or cH , respectively, regardless of his signal. Therefore, he has no incentive to pay the inspection cost for more information. Equilibrium cutoff beliefs.

At q 1 , the buyer must be indifferent between offering cH without an

inspection and offering cH with an inspection but only after receiving signal s2 . Combining this

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with the fact that the low-type seller’s reservation price p˜(q 1 ) exceeds vL , and thus trade cannot occur if the seller receives signal s1 , q 1 (vH − cH ) + (1 − q 1 )(vL − cH ) = q1 γH (s2 )(vH − cH ) + (1 − q 1 )γL (s2 )(vL − cH ) − c, which is equivalent to q1 γH (s1 )(vH − cH ) + (1 − q 1 )γL (s1 )(vL − cH ) + c = 0.

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At q 2 , the buyer must be indifferent between offering p˜(q 2 ) = vL without an inspection and offering cH with an inspection but only after receiving signal s2 . Since her expected payoff is obviously equal to 0 in the former case, q2 is given by the value that satisfies q 2 γH (s2 )(vH − cH ) + (1 − q 2 )γL (s2 )(vL − cH ) − c = 0.

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Stationary path. When λ is sufficiently large, the stationary belief level q ∗ coincides with q 2 . The stationary path is sustained as follows: each buyer inspects the asset with probability σI ∈ (0, 1). He offers cH if the signal is s2 and vL if the signal is s1 . Without inspection, he offers vL . The low-type seller accepts vL with probability σS . The values of σI and σS are determined by the low-type seller’s indifference condition and the invariance of σS . Formally, since the low-type seller’s reservation price must be equal to vL , r(vL − cL ) = λσI γL (s1 )(cH − vL ) ⇒ σI =

r(vL − cL ) . λγL (s2 )(cH − vL )

For q(t) to stay constant, σI γH (s2 ) = σI γL (s2 ) + (1 − σI γL (s2 ))σS ⇒ σS = Convergence path.

σI (γH (s2 ) − γL (s2 )) . (1 − σI γL (s2 ))

While q(t) ∈ (q2 , q1 ), all buyers inspect and offer cH only when they receive

signal s2 . Trade occurs only at cH , and thus q(t) strictly decreases according to q(t) ˙ = λq(t)(1 − q(t))(γL (s2 ) − γH (s2 )) (the same rate as in our main model). If qb < q 2 , then all buyers offer

p(t) without inspection. The low-type seller accepts p(t) with probability 1, and thus q(t) strictly increases according to q(t) ˙ = λq(t)(1 − q(t)) (again, the same rate as in our main model). In both cases, once q(t) arrives at q 2 , it stays there forever.

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Effects of reducing the inspection cost We now study the effects of reducing the inspection cost c. Equilibrium structure.

From (12) and (13), it is clear that q 1 increases, while q 2 decreases as c

decreases. This is natural given that buyers inspect the asset only when q(t) ∈ [q 2 , q1 ): they are more willing to inspect as the inspection cost decreases. In the limit as c tends to 0, the equilibrium cutoff beliefs converge to those in our main model. The limiting equilibrium does not exactly coincide with the equilibrium of our main model, because buyers who receive signal s2 offer cH with probability 1. Still, the equilibria are outcome-equivalent, because the inspection probability σI is equal to the probability of offering cH conditional on s2 (i.e., σB∗ ) in our main model. Effects on efficiency and welfare. The following result reports the effects of decreasing c on the time to trade for each type and the low-type seller’s expected payoff. Notice that the results are similar to those for increasing the informativeness of buyers’ signals reported in Section 5 in the main text. This is natural, because a decrease in c allows more buyers to acquire information. Proposition A6 1. If qb < q 2 , then a marginal decrease in c increases the time to trade for the low type but decreases that for the high type, in the sense of first-order stochastic dominance. It increases the low-type seller’s expected payoff.

2. If qb ∈ (q 2 , q 1 ), then a marginal decrease in c decreases the time to trade for the high type

and increases the low-type seller’s expected payoff. Its effect on the low type’s time to trade is ambiguous.

3. If qb is just above q1 , then a decrease in c increases the time to trade for each type and decreases the low-type seller’s expected payoff.

Proof. The first two sets of results follow from the following observations. First, a marginal decrease in c lowers q 2 . Therefore, it makes the time for q(t) to reach q 2 shorter from below (i.e., if qb < q 2 ), while longer from above (i.e., if qb ∈ (q 2 , q 1 )). In addition, the trading rate on the stationary path (λσI γH (s2 )) and the rate at which the low-type seller receives cH (λσI γL (s2 )) are independent of c. Finally, the high type’s trading rate is higher when q(t) ∈ (q 2 , q1 ) (λγH (s2 )) than

on the stationary path (λσI γH (s2 )). The result on the low-type seller’s time to trade is ambiguous, because her trading rate may be lower when q(t) ∈ (q 2 , q 1 ) (λγL (s2 )) than on the stationary path (λσI γH (s2 )). The last set of results is immediate from the equilibrium structure and the fact that q1 decreases in c.

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Appendix D: Formal Analysis of the Three-type Case This appendix formally introduces the model with three types and explicitly constructs an equilibrium described in Section 7.

Setup There are three types of assets, low (L), middle (M), and high (H). For each a = L, M, H, we denote by ca the stock value of the asset to the seller, by va the stock value to a buyer, and by γa (sn ) the probability that a buyer receives signal sn from type a. Similarly to the two-type case, we assume that a higher type is more valuable to both the seller and buyers (i.e., cL < cM < cH and vL < vM < vH ) and there are always positive gains from trade (i.e., ca < va for each a). In order to highlight the effects of adverse selection, we also assume that vL < cM and vM < cH . Finally, we assume that there does not exist a signal that fully reveals the seller’s type (i.e., γa (sn ) > 0 for all a = L, M, H and n = 1, ..., N) and the signal-generating process exhibits MLRP (monotone likelihood ratio property): both γM (sn )/γL (sn ) and γH (sn )/γM (sn ) are strictly increasing in n. For notational simplicity, we also use γa to denote the probability that a buyer receives signal sN from a type-a asset (i.e., γa ≡ γa (sN ) for each a = L, M, H). For each a = L, M, H, denote by pa (t) the reservation price of the type-a seller at time t. The assumption cL < cM < cH guarantees that pL (t) < pM (t) < pH (t). By the same reasoning as in the two-type case, no buyer offers strictly more than cH , which implies that the high-type seller’s reservation price is always equal to cH (i.e., pH (t) = cH for any t) and she always accepts cH . As in the two-type case, without loss of generality, we restrict attention to the equilibria in which each buyer offers either pL (t), pM (t), or pH (t) = cH . We denote by σa (sn , t) the probability that the buyer at time t offers pa (t) when his signal is sn and by σS (t) the probability that the low type accepts pL (t) at time t. It is not necessary to define the middle type’s acceptance probability of her reservation price, because pM (t) is never offered if pM (t) ≥ vM (note that it is also accepted by the low type and, therefore, leads to a negative payoff), while it must be accepted by the middle type with probability 1 if pM (t) < vM : otherwise, a buyer would strictly prefer offering slightly above pM (t). We focus on the case where λ is sufficiently large. This ensures that in any equilibrium, buyers eventually offer pM (t) and pH (t) only when they receive the best signal sN . As in the two-type case, this gives more tractability to the analysis. In addition, we restrict attention to the set of parameter values in which the following two assumptions are satisfied: let q˜L and q˜M be the unique values that solve   q˜M (γH − γM )(vM − vL ) γH q˜L γL (vL − cM ) + q˜M γM (vM − cM ) cH − vL + = (vL − cL ), q˜L γL (cH − vL ) + q˜M γM (cH − vM ) q˜L γL + q˜M γM γL 19

and q˜L (γL (cH − vL ) + γH (vH − cH )) + q˜M (γM (cH − vM ) + γH (vH − cH )) = γH (vH − cH ). If q˜M = 0, then it is clear that the left-hand side is strictly smaller than the right-hand side in the first equation. We assume that the right-hand side is smaller if q˜L = 0. Equivalently, Assumption 1 q˜L > 0. As shown later, this ensures the existence of a stationary path (a point in the belief space that can be time-invariant). In addition, we assume that Assumption 2 For any qL ∈ (0, q˜L ), qL γL qM γM γM 2 (qL γL2 (cH − vL ) + qM γM (cH − vM )) − (γM − γL ) (vM − vL ) > 0, γH qL γL + qM γM where qL (γL (cH − vL ) + γH (vH − cH )) + qM (γM (cH − vM ) + γH (vH − cH )) = γH (vH − cH ). This assumption guarantees the convergence of buyers’ beliefs toward the convergence point along Path 2, described below. The parameter space that satisfies Assumptions 1 and 2 is not empty: q˜L can be arbitrarily close to 0, while Assumption 2 is automatically satisfied if q˜L is sufficiently close to 0. We first characterize the point at which buyers’ beliefs can stay constant (i.e., q ∗ in Figure 6 in ∗ ∗ ∗ the main text). Next, we consider the points that lie between q ∗ and either of qLM , qLH , or qM H (i.e., Paths 1, 2, and 3 in Figure 6). Finally, we examine all other points (i.e., Areas A, B, and C in Figure 6).

Stationary Path We begin with an observation regarding the low-type seller’s reservation price on the stationary path: as in the two-type case, it must be equal to vL . Suppose it is strictly smaller than vL . Then, the low type must accept pL (t) with probability 1 and trade whenever a buyer arrives: otherwise, a buyer would strictly prefer offering slightly above pL (t). But then, unless buyers always offer cH with probability 1, the probability of the low type necessarily decreases over time relative to the other two types. To the contrary, suppose that the low-type seller’s reservation price at q ∗ is strictly

20

greater than vL . In this case, again unless all buyers offer cH with probability 1, the probability of the low type, relative to the other two types, necessarily increases over time. Given the above observation, the relevant equilibrium variables for the stationary path are as follows: • p∗M : the middle type’s reservation price at q ∗ . • σa∗ for a = H, M: the probability that each buyer offers the type-a seller’s reservation price conditional on signal sN . – When λ is sufficiently large, buyers must offer p∗M or cH only when they receive the best signal sN : otherwise, the low type’s and the middle type’s reservation prices would exceed vL and vM , respectively. • σS∗ : the probability that the low-type seller accepts her reservation price vL . • qa∗ for a = H, M, L: the probability that the seller is of type a. The equilibrium variables must satisfy the following conditions. We omit the interpretations of the conditions, because they are identical to those in the main text. • Each seller type’s reservation price: ∗ ∗ r(vL − cL ) = λγL (σH (cH − vL ) + σM (p∗M − vL )),

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∗ (cH − p∗M ). r(p∗M − cM ) = λγM σH

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

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• Buyers’ indifference over vL , p∗M , and cH with signal sN : ∗ 0 = qL∗ γL (vL − p∗M ) + qM γM (vM − p∗M ) ∗ ∗ = qL∗ γL (vL − cH ) + qM γM (vM − cH ) + qH γH (vH − cH ).

∗ ∗ From (16), σM = (γH − γM )σH /γM . Plugging this into (14) and combining with (15),

γM (cH − p∗M ) p∗M − cM  . = γH −γM vL − cL ∗ γL cH − vL + γM (pM − vL ) 21

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If p∗M = cM , then the left-hand side is strictly smaller than the right-hand side. On the other hand, Assumption 1 guarantees that the left-hand side is larger when p∗M = vM .4 Finally, the left-hand side is strictly increasing in p∗M , while the right-hand side is strictly decreasing. Therefore, there exists a unique value of p∗M ∈ (cM , vM ) that satisfies the equation. ∗ ∗ Given p∗M ∈ (cM , vM ), σH can be obtained from (15). Note that σH is well-defined in (0, 1), ∗ ∗ ∗ provided that λ is sufficiently large. Then, σM and σS∗ follow from (16). Finally, qL∗ , qM , and qH follow from (17): p∗M ∈ (cM , vM ) guarantees that all three values are well-defined in (0, 1). ∗ ∗ To sum up, there exist a unique point q ∗ ≡ (qL∗ , qM , qH ) at which buyers’ beliefs can stay ∗ ∗ constant in equilibrium. The corresponding equilibrium strategy profile, identified by p∗M , σH , σM , ∗ and σS , is also unique.

∗ Path 1: Straight Line between q ∗ and qLH ∗ Consider the points q = (qL , qM , qH ) that lie on the straight line between q ∗ and qLH (Path 1 in Figure 6). If a buyer with signal sN offers cH , then he obtains zero expected payoff both at q ∗ and ∗ at qLH . Therefore, at each point on Path 1, it is also the case that

qL γL (vL − cH ) + qM γM (vM − cH ) + qH γH (vH − cH ) = 0. ∗ In addition, clearly, qM < qM . We show that there is an equilibrium in which buyers’ beliefs converge to q ∗ along Path 1.

We first describe the equilibrium structure and then characterize the equilibrium. Along Path 1, buyers offer either cH or vL , the former only when the signal is sN . The low type’s reservation price stays constant at vL . Buyers offer cH and the seller accepts vL with just enough probabilities so that buyers’ beliefs stay on the path. Since the middle type trades only at cH and, therefore, less frequently than the high type, the probability of the middle type increases over time and converges ∗ to qM .

Buyers’ offer strategies.

Let σ ˜H be the value that satisfies r(vL − cL ) = λγL σ ˜H (cH − vL ).

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In words, if each buyer offers cH with probability σ ˜H when, and only when, they receive signal sN ∗ (until buyers’ beliefs reach q ), then the low type’s reservation price stays constant at vL . 4

∗ ∗ ∗ ∗ ∗ ∗ The relationship to Assumption 1 becomes more evident if p∗M is replaced with (qL γL vL + qM γM vM )/(qL γL +

∗ ∗ qM γM ).

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Low-type seller’s acceptance strategy. For buyers’ beliefs to stay on Path 1, it must be that q˙L γL (vL − cH ) + q˙M γM (vM − cH ) + q˙H γH (vH − cH ) = 0.

(19)

Given σ˜H and σS (t) (the probability that the low type accepts vL ),5 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 )) . If γL σ ˜H + (1 − γL σ ˜H )σS (t) = γH σ ˜H , then the left-hand side in (19) is strictly less than 0: note that in that case, q˙L /qL = q˙H /qH < q˙M /qM . Therefore, the left-hand side is equal to q˙H qL γL (vL − cH ) + q˙M γM (vM − cH ) + q˙H γH (vH − cH ) qH q˙H = (qL γL (vL − cH ) + qH γH (vH − cH )) + q˙M γM (vM − cH ) qH q˙H (−qM γM (vM − cH )) + q˙M γM (vM − cH ) = qH   q˙H q˙M = qM γM (cH − vM ) − < 0. qH qM Intuitively, for this range of beliefs, if the low type and the high type trade at the same rate, while the middle type trades at a lower rate, then buyers’ unconditional expected value falls, and thus offering cH becomes strictly unprofitable. Now suppose σS (t) = 1. In this case, the left-hand side in (19) is strictly greater than 0: when λ is sufficiently large, σ ˜H is sufficiently small, and thus q˙L ≃ −λqL (qM + qH ) = −λqL + λqL qL , q˙M ≃ λqM qL , and q˙H ≃ λqH qL . Therefore, the left-hand side becomes close to −λqL (vL − cH ) + λqL (qL γL (vL − cH ) + qM γM (vM − cH ) + qH γH (vH − cH )) = λqL (cH − vL ) > 0. Intuitively, if the low type accepts her reservation price with probability 1, then buyers’ unconditional expected value strictly increases, and thus offering cH gives a buyer a strictly positive expected payoff. Since q˙L is strictly decreasing in σS (t), while both q˙M and q˙H are strictly increas5

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

23

+ qH (t)e−λγH σ˜H

.

ing in σS (t), there exists a unique σS (t) that satisfies (19). Incentive compatibility. By construction, each seller type’s acceptance strategy is optimal. The only potential incentive problem is buyers’ incentives to offer pM (t). To show that no buyer has ∗ an incentive to offer pM (t), notice that σ ˜H > σH : along the stationary path, the low type receives ∗ not only cH , but also pM with a positive probability, while on Path 1, she receives only the former. Therefore, for her reservation price to be equal to vL , she must receive cH more frequently on Path ∗ 1 than at q ∗ . This implies that pM (t) > p∗M . Since, in addition, qL (t)/qM (t) > qL∗ /qM , it follows that ∗ qL (t)γL (vL − pM (t)) + qM (t)γM (vM − pM (t)) < qL∗ γL (vL − p∗M ) + qM γM (vM − p∗M ) = 0,

that is, a buyer obtains a strictly negative expected payoff if he offers pM (t) on Path 1. ∗ Path 2: Straight Line between q ∗ and qM H ∗ Now we consider the points on the straight line between q ∗ and qM H (Path 2 in Figure 6). At each of those points, again we have

qL γL (vL − cH ) + qM γM (vM − cH ) + qH γH (vH − cH ) = 0, and qL < qL∗ . Similarly to the previous case, we show that there is an equilibrium in which buyers’ beliefs converge to q ∗ along Path 2. Along Path 2, buyers offer either cH and pM (t) with positive probability only when they receive signal sN . For notational simplicity, let σa (t) ≡ σa (sN , t) for each a = M, H. The low type’s reservation price strictly exceeds vL , and thus in equilibrium the low type rejects pL (t) with probability 1 (i.e., buyers make losing offers). The probability of the low type decreases over time, and buyers’ beliefs converge to q ∗ . Buyers offer cH and pM (t) with just enough probabilities so that buyers’ beliefs stay on Path 2. An additional complication, relative to Path 1, is that pM (t) changes over time, and thus σH (t) is also time-dependent. Equilibrium conditions. Given buyers’ strategies σH (t) and σL (t) and the fact that the low type rejects pL (t) with probability 1, buyers’ beliefs evolve as follows: q˙L = λqL (qM (γM − γL )(σH (t) + σM (t)) + qH (γH σH (t) − γL (σH (t) + σM (t)))) , q˙M = λqM (qL (γL − γM )(σH (t) + σM (t)) + qH (γH σH (t) − γM (σH (t) + σM (t)))) ,

24

q˙H = λqH (qL (γL (σH (t) + σM (t)) − γH σH (t)) + qM (γM (σH (t) + σM (t)) − γH σH (t)) . For buyers’ beliefs to stay on Path 1, again, (19) must hold at each point in time. In addition, a buyer with signal sN must be indifferent between offering cH and pM (t). On Path 2, the former gives 0 expected payoff to the buyer. Therefore, it must be that pM (t) =

qM (t) γM vM − pM (t) qL (t)γL vL + qM (t)γM vM ⇔ = 1. qL (t)γL + qM (t)γM qL (t) γL pM (t) − vL

(20)

Given σH (t), the law of motion governing pM is given by r(pM (t) − cM ) = λγM σH (t)(cH − pM (t)) + p˙ M (t). Existence.

(21)

In order to show that there exists a strategy profile that satisfies the above conditions,

we first argue that for any given σH (t), the value of σM (t) is uniquely determined by (19). Suppose σM (t) is such that γM (σH (t) + σM (t)) = γH σH (t). In this case, the left-hand side in (19) is strictly negative, because the middle and the high type trade at the same rate, while the low type trades at a lower rate. On the other hand, if σM (t) is close to 1, then the left-hand side in (19) is strictly positive, because the middle and the low type trade much faster than the high type. In addition, the left-hand side is strictly increasing in σM (t). Therefore, there exists a unique value of σM (t) that makes buyers’ beliefs stay on Path 2 for a given value of σH (t). Notice that σM (t) is linear in σH (t): each q˙a is linear in both σH (t) and σM (t). Therefore, if σH (t) doubles, then σM (t) also has to double in order to satisfy (19). In addition, p˙ M (t) is also linear in σH (t): from (20), p˙ M (t)



vM

1 1 + − pM (t) pM (t) − vL



=

q˙M (t) q˙L (t) − . qM (t) qL (t)

The result follows from the fact that both q˙M (t)/qM (t) and q˙L (t)/qL (t) are linear in σH (t). In (21), the left-hand side is independent of σH (t), while the right-hand side is linear in σH (t). In addition, under Assumption 2, the right-hand side is necessarily positive on any point on Path 2. Therefore, there exists a unique value of σH (t) (and associated σM (t)) that satisfies all the equilibrium conditions. Incentive compatibility. It suffices to show that the low type’s reservation price exceeds vL . ∗ ∗ This follows from the fact that σH (t) > σH (because pM (t) > p∗M ) and σM (t) > σM (because ∗ σH (t) > σH and, for buyers’ beliefs to stay on Path 2, qM (t) needs to decrease faster than at q ∗ ).

25

∗ Path 3: (Not Straight) Curve between q ∗ and qLM ∗ We now consider the points on the curve connecting q ∗ and qLH (i.e., Path 3 in Figure 6). Unlike

in the previous two cases, the curve is not necessarily straight and needs to be constructed as well. Therefore, no priori condition (corresponding to (19) for Path 1 and Path 2) is available, and the existence of such a path must be established as well. Nevertheless, we show that there exist such a path and an associated equilibrium such that in the equilibrium buyers’ beliefs converge to q ∗ along the path. Along Path 3, the low type’s reservation price stays constant at vL . Buyers offer either pM (t) or vL , the former with a positive probability only when they receive signal sN . The probability of the high type increases over time, and buyers’ beliefs converge to q ∗ . Buyers offer pM (t) and the low type accepts vL with just enough probabilities so that buyers’ beliefs move along the curve. As hinted above, a new complication arises because the path itself needs to be identified. Equilibrium conditions.

Given σM (t), σH (t) = 0 (no buyer offers cH ), σS (t) , buyers’ beliefs

evolve as follows: q˙L = λqL (qM (γM σM (t)−(γL σM (t)+(1−γL σM (t))σS (t)))−qH (γL σM (t)+(1−γLσM (t))σS (t))), q˙M = λqM (qL ((γL σM (t) + (1 − γL σM (t))σS (t)) − γM σM (t)) − qH γM σM (t)), q˙H = λqH (qL (γL σM (t) + (1 − γL σM (t))σS (t)) + qM γM σM (t)). At each point on Path 3, a buyer with signal sN must be indifferent between offering vL and pM (t). This implies that pM (t) =

qM (t) γM vM − pM (t) qL (t)γL vL + qM (t)γM vM ⇔ = 1. qL (t)γL + qM (t)γM qN (t) γL pM (t) − vL

(22)

In addition, since σH (t) = 0 until buyers’ beliefs reach q ∗ , pM (t) strictly increases over time according to r(pM − cM ) = p˙ M . Finally, for the low type’s reservation price to stay constant at vL , buyers with signal sN must offer pM (t) with probability σM (t) such that r(vL − cL ) = λγL σM (t)(pM (t) − vL ). Existence.

(23)

It is not possible to directly construct a continuous-time strategy profile that satisfies

all the equilibrium conditions. We prove the existence of the strategy profile by discretizing the problem: we construct an equilibrium in a discrete-time setting with period length ∆ and let ∆ tend to 0.

26

Fix ∆ and let t∗ be the time such that q(t∗ ) = q ∗ . We use backward induction to construct a path of play for t = t∗ − n∆, n = 1, 2, ..., so that the above equilibrium conditions are satisfied along the path. At t∗ − ∆, the middle type’s reservation price is such that pM (t∗ − ∆) − cM = e−r∆ (p∗M − cM ). For the low type’s reservation price at t∗ − 2∆ to be equal to vL , σM (t∗ − ∆) must be such that (1 − e−r∆ )(vL − cL ) = e−r∆ (1 − e−λ∆ )γL σM (t∗ − ∆)(pM (t∗ − ∆) − vL ). It remains to pin down qL (t∗ − ∆), qM (t∗ − ∆), and σS (t∗ − ∆). One of three necessary conditions is (22). The other two conditions come from the fact that buyers’ beliefs must be equal to q ∗ in the next period. Formally, given σM (t∗ − ∆) and σS (t∗ − ∆), qL (t∗ − ∆)(1 − (1 − e−λ∆ )(γL σM (t∗ − ∆) + (1 − γL σM (t∗ − ∆))σS (t∗ − ∆))) qL∗ = , 1 − qL∗ qM (t∗ − ∆)(1 − (1 − e−λ∆ )γL σM (t∗ − ∆)) + qH (t∗ − ∆) ∗ qM qM (t∗ − ∆)(1 − (1 − e−λ∆ )γL σM (t∗ − ∆)) = . ∗ 1 − qM qL (t∗ − ∆)(1 − (1 − e−λ∆ )(γL σM (t∗ − ∆) + (1 − γL σM (t∗ − ∆))σS (t∗ − ∆))) + qH (t∗ − ∆)

It is clear that qL (t∗ −∆) is strictly increasing in σS (t∗ −∆), while qM (t∗ −∆) is strictly decreasing in σS (t∗ − ∆). In addition, if σS (t∗ − ∆) = 0, then the right-hand side is larger in (22), while if σS (t∗ − ∆) = 1, then the left-hand side is larger. Therefore, there exists a unique value of σS (t∗ − ∆) that satisfies the conditions. The uniqueness of qL (t∗ − ∆) and qM (t∗ − ∆) also follows. We recursively apply this procedure to construct the whole sequence of the equilibrium variables. Given pM (t∗ − n∆) and q(t∗ − n∆), we can go one more period backward and determine (pM (t∗ − (n + 1)∆), σM (t∗ − (n + 1)∆), σS (t∗ − (n + 1)∆), q(t∗ − (n + 1)∆)). Notice that pM (t) converges to cM as we go further backwards (increasing n). It then follows that the path ∗ necessarily converges to qLM . This construction works for any (small) ∆. It is then clear that the limit path as ∆ tends to 0 exists, and the limit strategy profile satisfies all the necessary conditions for the continuous-time

problem. Incentive compatibility. The only potential incentive problem in this strategy profile is buyers’ incentive to offer cH , instead of pM (t) or vL . The result is straightforward, however, because at any point on Path 3, buyers’ unconditional expected value is strictly smaller than cH : notice that ∗ ∗ Path 3 necessarily lies to the left of the line that connects qLH and qM H.

27

Area A: High qL We now consider the points at which the probability of the low type is relatively high (Area A in Figure 6). In this region, we show that it is an equilibrium that each buyer offers pL (t) regardless of his signal. Under this strategy profile, the probability of the low type decreases over time and buyers’ beliefs reach either Path 1 or Path 3, from which point on they converge to q ∗ along Path 1 or Path 3. Since only the low type trades in this region, qM (t)/qH (t) stays constant. Buyers’ beliefs move along the straight line that starts from (1, 0, 0) and passes through the initial belief. Equilibrium conditions.

If each buyer offers pL (t) regardless of his signal, then buyers’ beliefs

evolve according to the following laws of motion in Area A: q˙L = λqL (−qM − qH ) = −λqL (1 − qL ), q˙M = λqM qL , q˙H = λqH qL . Denote by t∗ the time at which buyers’ beliefs reach Path 1 or Path 3. From the laws of motion of buyers’ beliefs, it is clear that t∗ is well-defined. Under the strategy profile, ∗ −t)

pL (t) − cL = e−r(t

∗ −t)

(pL (t∗ ) − cL ), pM (t) − cM = e−r(t

(pM (t∗ ) − cM ).

On both Path 1 and Path 3, the low-type seller’s reservation price is equal to vL (i.e., pL (t∗ ) = vL ), which implies that pL (t) < vL for any t < t∗ and all buyers before t∗ obtain a strictly positive expected payoff. Incentive compatibility. At any point in Area A, qL γL (vL − cH ) + qM γM (t)(vM − cH ) + qH γH (vH − cH ) < 0. Therefore, it is clear that no buyer has an incentive to offer cH . It suffices to show that offering pM (t) is unprofitable even when a buyer receives the highest signal sN . We show that for any t < t∗ , pM (t) ≥

qM (t) γM vM − pM (t) qL (t)γL vL + qM (t)γM vM ⇔ ≤ 1. qL (t)γL + qM (t)γM qL (t) γL pM (t) − vL

(24)

Notice that qL (t)/qM (t) is bounded away from 0 in Area A. This implies that (24) is necessarily satisfied for pM (t) close to vM . Let p˜M < vM be the maximum value of pM (t) such that (24) holds for any q in Area A. Now it suffices to show that the left-hand side of the second inequality in (24) is increasing in t whenever pM (t) ≤ p˜M , because (24) holds at t∗ and pM (t) is increasing over time in Area A. Differentiating the left-hand side and arranging the terms yields q˙L q˙M − − p˙ M qM qL



vM

1 1 + − pM pM − vL



= λ − r(pM − cM )

28



vM

1 1 + − pM pM − vL



.

Since pM (t) ≤ p˜M < vM , it is clear that this expression is strictly positive for λ sufficiently large.

Area B: High qM We now consider the points at which the probability of the middle type is relatively high (Area B in Figure 6). In this case, we show that it is an equilibrium that each buyers offers either pM (t) or a losing price (i.e., pL (t) is offered but rejected with probability 1 because pL (t) > vL ). Under this strategy profile, the middle type trades faster than the other two types. In particular, the probability of the high type necessarily increases over time, and buyers’ beliefs reach either Path 2 or Path 3. Evolution of buyers’ beliefs. Suppose buyers offer pM (t) if and only if their signal is weakly above sn . Then, their beliefs evolve as follows: −1 −1 q˙L = λqL (qM (Γ−1 L (sn ) − ΓM (sn )) − qH (1 − ΓL (sn ))), −1 −1 q˙M = λqM (qL (Γ−1 M (sn ) − ΓL (sn )) − qH (1 − ΓM (sn ))), −1 q˙H = −λqH (qL (1 − Γ−1 L (sn )) + qM (1 − ΓM (sn ))).

Equilibrium construction. Denote by ΞN the set of all points in Area B and by ∂ΞN its partial boundary that consists only of Path 2 and Path 3 (i.e., ∂ΞN does not include the points at which qL = 0 or qH = 0). First, assume that at any q ∈ ΞN , buyers offer pM (t) only when their signal is sN . Then, for each point q, there exist a unique convergence point on ∂ΞN , and the finite time it takes for buyers’ beliefs to reach the convergence point can be calculated. Denote by t∗ the latter. Then, ∗ pM (0) = cL + e−rt (pM (t∗ ) − cL ) < pM (t∗ ) and qL (t∗ )γL (sN )(vL − pM (t∗ )) + qM (t∗ )γM (sN )(vM − pM (t∗ )) = 0. Together with qL (t)/qM (t) < qL (t∗ )/qM (t∗ ) for any t < t∗ , these imply that qL (t)γL (sN )(vL − pM (t)) + qM (t)γM (sN )(vM − pM (t)) > 0. In other words, along the convergence path, each buyer with signal sN obtains a strictly positive expected payoff by offering pM (t). Notice that if qL (t)/qM (t) is sufficiently small, then the buyer with signal sN −1 also would want to offer pM (t), instead of a losing price pL (t). To accommodate this, let ΞN −1 be the subset

29

of ΞN such that for each point q(0) in ΞN −1 , qL (0)γL (sN −1 )(vL − pM (0)) + qM (0)γM (sN −1 )(vM − pM (0)) ≥ 0. Clearly, ΞN −1 is a convex and closed set. Given ΞN −1 , assume that at any point in ΞN −1 , buyers offer pM (t) if and only if their signal is weakly above sN −1 . Then, as above, for each q ∈ ΞN −1 , there exist a convergent point on ∂ΞN −1 (the boundary of Ξ excluding the points at which qL = 0 or qH = 0) and the time it takes for buyers’ beliefs to converge to the point. From this point on, one can show that at any point in ΞN −1 , a buyer with a signal weakly above sN −1 obtains a positive expected payoff. Furthermore, as above, the smaller set ΞN −2 , defined in an analogous manner to ΞN −1 , can be obtained. The equilibrium construction can be completed by recursively applying the same procedure and finding the smaller sets ΞN −2 , ..., Ξ0 . If q ∈ Ξn − Ξn−1 , then buyers offer pM (t) if and only if their signal is weakly above sn until their beliefs reach ∂Ξn . Then, they offer pM (t) if and only if their signal is weakly above sn+1 until ∂Ξn+1 . This process continues until buyers’ beliefs reach ∂ΞN (i.e., Path 2 or Path 3). Incentive compatibility. At each point in Area B, qL γL (vL −cH )+qM γM (vM −cH )+qH γH (vH − cH ) < 0. Therefore, obviously, no buyer has an incentive to offer cH . It suffices to show that pL (t) ≥ vL . This directly follows from the fact that in Area B every buyer offers pM (t) ≥ cM with probability 1 at least when his signal is sN , and the low-type seller’s reservation price is greater than or equal to vL on Path 2 and Path 3.

Area C: High qH Finally, we consider the points at which the probability of the high type is relatively large (i.e., Area C in Figure 6). In this region, we show that there is an equilibrium in which every buyer offers either cH or a losing price (i.e., pL (t) > vL ). Precisely, at each point q in Area C, it is an equilibrium that a buyer with signal sn offers cH if and only if qL γL (sn )(vL − cH ) + qM γM (sn )(vM − cH ) + qH γH (sn )(vH − cH ) ≥ 0. Naturally, Area C is divided into N regions, depending on buyers’ cutoff signal. In equilibrium, the high type trades faster than the other two types, and buyers’ beliefs reach either Path 1 or Path 2, as long as they do not start from the region on which buyers offer cH with probability 1 regardless of their signal.

30

Incentive compatibility. We show that no buyer has an incentive to offer pL (t) or pM (t). The former is straightforward: Each buyer offers cH with probability 1 at least when his signal is sN . In addition, the low-type seller’s reservation price is weakly above vL at any point on Path 1 or Path 2. Therefore, as in Area B, the low-type seller’s reservation price strictly exceeds vL . For the latter, we show that at any point in Area C, offering pM (t) yields a negative payoff to a buyer. A sufficient condition for this is that even when the highest signal is received, the expected payoff from offering pM (t) is non-positive, i.e., qM (t) γM vM − pM (t) ≤ 1. qL (t) γL pM (t) − vL If pM (t) ≥ vM , then this inequality is trivially satisfied. From now on, we suppose pM (t) < vM . Since the inequality holds on Path 2 and Path 3, the proof is complete once we establish that the left-hand side of the inequality is increasing over time, that is,   q˙M (t) q˙L (t) 1 1 ≥ 0. − − p˙ M (t) + qM (t) qL (t) vM − pM (t) pM (t) − vL Suppose at q(t), the cutoff signal is sn (that is, buyers offers cH if and only if their signal is weakly above sn ). Then, − − − q˙L = λqL (qM (Γ− L (sn ) − ΓM (sn )) + qH (ΓL (sn ) − ΓH (sn ))), − − − q˙M = λqM (qL (Γ− M (sn ) − ΓL (sn )) + qH (ΓM (sn ) − ΓH (sn ))),

p˙ M (t) = r(pM (t) − cM ) − λ(1 − Γ− M (sn ))(cH − pM (t)). Plugging these into the above inequality and arranging the terms,   − λ (1 − ΓM (s(t)))(cH − pM (t))

vM

  1 1 − − − (ΓL (s(t)) − ΓM (s(t))) ≥ r(pM (t)−cM ). + − pM (t) pM (t) − vL

Since the right-hand is clearly bounded, the inequality holds for λ sufficiently large, as long as the expression in the curly bracket is positive. A sufficient condition for this requirement is (cH − pM (t)) ×



vM

 γL 1 1 ≥1− . + − pM (t) pM (t) − vL γM

The assumption that cH > vM guarantees that this sufficient condition is always satisfied.

31

Appendix E: Alternative Bargaining Protocols This appendix provides a formal analysis of the model with each alternative bargaining protocol, introduced in Section 7.2. In all three cases, we explicitly construct an equilibrium (or a class of equilibria) for the case when λ is sufficiently large.

Simultaneous Announcement Game The high-type seller’s optimal strategy is clear: she always plays T . In what follows, we represent the low-type seller’s strategy by a function σS : R+ → [0, 1], where σS (t) is the probability that the low-type seller plays S at time t. We also denote by p(t) the low-type seller’s reservation price at time t. Stationary path We begin by identifying the unique stationary path. It is clear that buyers’ beliefs cannot stay constant if the low-type seller always plays S or T . Denote by σS∗ the probability that the low-type seller plays S on the stationary path. By the same logic as in the main model, it must be the case that in equilibrium buyers play S with a positive probability only when they receive signal sN . Denote by σB∗ the probability that each buyer plays S when his signal is sN on the stationary path. Finally, we denote by p∗ the low-type seller’s reservation price on the stationary path. It is clear shortly that, unlike in the main model, p∗ does not have to coincide with vL . For the low-type seller to be indifferent between T and S, r(p∗ − cL ) = λγL (sN )σB∗ (pH − p∗ ) = λ(γL (sN )σB∗ (pM − p∗ ) + (1 − γL (sN )σB∗ )(pL − p∗ )).

(25) (26)

The first term represents the low-type seller’s expected payoff when she keeps playing T , while the second term is when she keeps playing S. For the indifference of the buyer with signal sN between S (left) and T (right), q ∗ γH (sN )(vH −pH )+(1−q ∗ )γL (sN )(σS∗ (vL −pM )+(1−σS∗ )(vL −pH )) = (1−q ∗ )γL (sN )σS∗ (vL −pL ). Arranging the terms, γL (sN ) σS∗ (pM − pL ) + (1 − σS∗ )(pH − vL ) q∗ . = 1 − q∗ γH (sN ) vH − pH

(27)

Finally, q(t) remains equal to q ∗ if and only if both seller types trade at the same rate. Since 32

the high-type seller always plays T and the low-type seller trades for sure when she plays S, γH (sN )σB∗ = γL (sN )σB∗ + (1 − γL (sN )σB∗ )σS∗ .

(28)

The four equilibrium variables, p∗ , σB∗ , σS∗ , and q ∗ , can be found from the above 4 equations. Specifically, p∗ and σB∗ can be derived from (25) and (26): p∗ =

rcL + λγL (sN )σB∗ pM + λ(1 − γL (sN )σB∗ )pL rcL + λγL (sN )σB∗ pH . = r + λγL (sN )σB∗ r+λ

The uniqueness follows from the fact that the second term is strictly convex in σB∗ , while the last term is linear in σB∗ , and the right-hand side exceeds the left-hand side when σB∗ = 0. Given σB∗ , σS∗ can be obtained from (28). Given σS∗ , q ∗ follows from (27). Low initial beliefs We first consider the case where qb < q ∗ . The equilibrium structure is similar to that of the main model. Until q(t) reaches q ∗ , buyers play only T (analogously to offering only p(t)), and the low-

type seller plays only S (analogously to accepting p(t)). Since the low-type seller trades whenever a buyer arrives, while the high-type seller never trades, q(t) increases according to q(t) ˙ = q(t)(1 − q(t))λ. To verify that this strategy profile is indeed an equilibrium, it suffices to show that the low-type seller’s reservation price p(t) falls short of pL while q(t) < q ∗ . This is immediate from the fact that p(t) < p∗ < pL if q(t) < q ∗ : the first inequality is due to the fact that p(t) is an increasing function of the frequencies of buyers’ playing S, while the second inequality follows from the equality of (25) and (26). High initial beliefs We now consider the case where qb > q ∗ . Again, the equilibrium structure is similar to that of the main model. If q(t) ∈ (q n+1 , q n ), then buyers play S (analogously to offering cH in the main model) if and only if they receive a signal strictly above sn . Since both seller types only play T (analogously to accepting only cH ), trade occurs only at pH . The high type, who generates good signals more frequently, trades faster than the low type. Buyers’ beliefs decrease according to q(t) ˙ = q(t)(1 − q(t))λ(ΓH (sn ) − ΓL (sn )).

33

Given the equilibrium structure, each q n can be obtained from the following indifference condition of the buyer with signal sn : q n γH (sn )(vH − pH ) + (1 − q n )γL (sn )(vL − pH ) = 0 ⇔

γL (sn ) pH − vL qn . = 1 − qn γH (sn ) vH − pH

It remains to show that when q(t) > q ∗ , the low-type seller has no incentive to deviate to S, which gives her a low price if the buyer plays S but allows her to trade even if the buyer plays T . When q(t) ∈ (qn+1 , q n ), a necessary and sufficient condition for the incentive compatibility is ΓL (sn )(pL − p(t)) + (1 − ΓL (sn ))(pM − p(t)) ≤ (1 − ΓL (sn ))(pH − p(t)). Equivalently, (1 − ΓL (sn ))(pH − pM ) ≥ ΓL (sn )(pL − p(t)). This inequality is guaranteed from the equality of (25) and (26) and the fact that p(t) > p∗ when q(t) > q ∗ (because the low-type seller can trade both at pH and pM with higher probabilities than on the stationary path).

Random Proposals Bargaining We adopt the same timing assumption as Lauermann and Wolinsky (2015). Upon meeting the seller, each buyer first receives a private signal s about the seller’s type. Then, nature draws a (suggested) price p for trade. Let α ∈ (0, 1) denote the probability that p = pH for each meeting. Once p is drawn, the buyer first decides whether to accept or reject it. If he accepts, then the seller decides whether to accept or reject the price. Trade occurs at the proposed price if and only if both players accept the price. Otherwise, the buyer leaves, and the seller waits for the next buyer. The high-type seller’s optimal strategy is straightforward: she accepts only pH . It is also clear that the low-type seller always accepts pH . From now on, we let σS (t) denote the probability that the low-type seller accepts pL at time t. Since pL < vL , it is strictly dominant for buyers to accept pL . If the suggested price is pH , then the buyer accepts it only when q(t)γH (s)(vH − pH ) + (1 − q(t))γL (s)(vL − pH ) ≥ 0 ⇔

q(t) γL (s) pH − vL = . 1 − q(t) γH (s) vH − pH

Stationary path For the same reason as in the baseline model, q(t) can stay constant if and only if the low-type seller’s reservation price is equal to pL , so that she randomizes between accepting and rejecting pL . When λ is sufficiently large, this can be the case only if each buyer plays the strategy of accepting 34

pH only when he receives signal sN and, even then, with probability less than 1. Similarly to the baseline model, let σB∗ denote the probability that he accepts pH and σS∗ denote the probability that the low-type seller accepts pL on the stationary path. Then, we obtain the following three conditions for q ∗ , σB∗ , and σS∗ : • Buyers’ indifference condition: γL (sN ) pH − vL q∗ = . ∗ 1−q γH (sN ) vH − pH • The low-type seller’s reservation price: r(pL − cL ) = λαγL (sN )σB∗ (pH − pL ). • Belief invariance: αγH (sN )σB∗ = αγL (sN )σB∗ + (1 − α)σS∗ . It is clear that there exists a unique tuple (q ∗ , σB∗ , σS∗ ) that satisfies these three conditions. Low initial beliefs Suppose qb < q ∗ . In this case, buyers never accept pH until q(t) reaches q ∗ . Given this, the lowtype seller’s reservation price falls short of pL , and thus she accepts pL with probability 1 (i.e., σS (t) = 1 whenever t < T (b q , q ∗ )). These imply that trade occurs on the convergence path only when pL is drawn, and thus q(t) increases according to q(t) ˙ = q(t)(1 − q(t))λ(1 − α). Notice that the only difference from the main model is the appearance of 1 − α in the expression. High initial beliefs Now we consider the case where qb > q ∗ . For each n = 1, ..., N, let q n be the value such that γL (sn ) pH − vL qn . = 1 − qn γH (sn ) vH − pH

It is then clear that each buyer’s optimal strategy is to accept pH only when his prior belief is above q n and his signal is above sn (i.e., if q(t) ∈ (q n+1 , qn ), he accepts pH whenever he receives a higher signal than sn ). Given this, the low-type seller’s reservation price exceeds pL and, therefore, trade 35

occurs only at pH whenever q(t) > q ∗ . It then follows that buyers’ beliefs evolve according to q(t) ˙ = q(t)(1 − q(t))λα(ΓH (sn ) − ΓL (sn )), over the interval (q n+1 , qn ). Notice that the only difference from the baseline model is the appearance of α in this expression. Unless qb > q 1 , buyers’ beliefs conditional on no trade continuously decrease and converge to q ∗ in finite time.

Price Offers by the Seller We restrict attention to strategy profiles in which the high-type seller insists on a single price pH ∈ [cH , vH ) (i.e., she offers pH after any history). It is easy to support this behavior of the high type by appropriately specifying buyers’ off-the-equilibrium-path beliefs. Now notice that it is dominant for buyers to accept any price offer below vL . This implies that the low-type seller’s problem effectively reduces to whether to offer pH or vL at each time. We now use σS (t) to denote the probability that the low-type seller offers vL at time t (conditional on the arrival of a buyer). Given this, each buyer accepts pH only when q(t)γH (s)(vH −pH )+(1−q(t))(1−σS (t))γL (s)(vL −pH ) ≥ 0 ⇔

γL (s) pH − vL q(t) . ≥ (1−σS (t)) 1 − q(t) γH (s) vH − pH

The term (1/(1 − σS (t))) reflects a signaling aspect of the bargaining protocol: a buyer updates his belief about the seller’s type from her pricing behavior. The high type always offers pH , while the low type may offer vL . Therefore, the seller is (weakly) more likely to be the high type if she offers pH . Stationary path We use the following notations for the stationary path equilibrium variables: • q ∗ : the stationary (prior) belief level. • p∗ : the low-type seller’s reservation price. • σS∗ : the probability that the low-type seller offers vL at each meeting. • σB∗ : the probability that each buyer accepts pH conditional on signal sN . Then, the equilibrium conditions for the stationary path translate into the following conditions with the current bargaining protocol:

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• The low-type seller’s indifference between offering vL and pH : r(p∗ − cL ) = λ(vL − p∗ ), = λγL (sN )σB∗ (pH − p∗ ).

(29) (30)

This condition holds in equilibrium for the following reasons: if the low-type seller always offers vL , then buyers would accept pH with probability 1, in which case the low-type seller strictly prefers offering pH to vL , which is a contradiction. If the low-type seller always offers pH , then the high type trades faster than the low type, and thus q(t) strictly decreases over time. • Buyers’ indifference between accepting and rejecting pH conditional on signal sN : q∗ ∗ γL (sN ) pH − vL = (1 − σ ) . S 1 − q∗ γH (sN ) vH − pH

(31)

This condition holds in equilibrium for the same reason as in the main model. • Belief invariance: the high type always offers pH and, therefore, trades only when a buyer receives signal sN and, even then, with probability σB∗ . The low type trades when either she offers pH and a buyer receives signal sN and accepts pH or she offers vL . Therefore, q(t) stays constant when γH (sN )σB∗ = (1 − σS∗ )γL (sN )σB∗ + σS∗ .

(32)

There exists a unique tuple (q ∗ , p∗ , σS∗ , σB∗ ) that satisfies the four equations: p∗ is pinned down from equation (29). σB∗ is then obtained from equation (30), which then yields σS∗ together with equation (32). Finally, given σS∗ , q ∗ is obtained from equation (31). High initial beliefs We begin by characterizing the case when qb > q ∗ . In this case, the low-type seller always offers

pH on the convergence path (i.e., σS (t) = 0 for any t < T (b q , q ∗ )). Given this, buyers’ optimal acceptance strategies are, once again, characterized by the cutoff belief levels, q 1 , ..., qN : for each n, let qn be the value such that qn γL (sn ) pH − vL . = 1 − qn γH (sn ) vH − pH

37

If q(t) ∈ (qn+1 , qn ), then each buyer accepts pH if and only if his signal is above sn . It then follows that buyers’ beliefs decrease, just as in the main model, according to q(t) ˙ = q(t)(1 − q(t))λ(ΓH (sn ) − ΓL (sn )). Low initial beliefs We now consider the case when qb < q ∗ . Unlike in the main model, the low-type seller does not trade with probability 1 even if a buyer arrives before q(t) reaches q ∗ . To see this, suppose the

low-type seller offers vL with probability 1. This implies that conditional on receiving offer pH , buyers assign probability 1 to the high type and, therefore, accept pH with probability 1. The low type then obviously prefers pH to vL . It also cannot be the case that the low-type seller always offers pH , because qb < q ∗ , and thus buyers would never accept pH . In equilibrium, the low-type seller must randomize between vL and pH at each point in time.

This implies that the low-type seller’s reservation price is always equal to p∗ : formally, r(p(t) − cL ) = λ(vL − p(t)), and thus p(t) = p∗ for each t. The indifference between vL and pH further implies that r(p∗ − cL ) = λ(vL − p∗ ) = λγL (sN )σB (t)(pH − p∗ ), where σB (t) denote the probability that the buyer at time t accepts pH conditional on signal sN . Notice that this condition holds if and only if σB (t) = σB∗ , that is, each buyer accepts pH with the same probability on the convergence path as on the stationary path. This, in turn, implies that each buyer must be indifferent between accepting and rejecting pH conditional on signal sN at each point on the stationary path: γL (sN ) pH − vL q(t) . = (1 − σS (t)) 1 − q(t) γH (sN ) vH − pH This equation yields the equilibrium offer strategy of the low-type seller, σS (t). Notice that σS (t)

is a strictly decreasing function of q(t), which means that the low-type seller is more likely to offer pH when buyers assign a higher probability to the high type. The high-type seller insists on pH . Therefore, she trades at rate λγH (sN )σB∗ . The low-type seller offers pH with probability 1 − σS (t), in which case her trading rate is equal to λγL (sN )σB∗ , and vL with probability σS (t), in which case her trading rate is equal to λ. Therefore, while q(t) < q ∗ , buyers’ beliefs evolve according to q(t) ˙ = q(t)(1 − q(t))λ((1 − σS (t))γL (sN )σB∗ + σS (t) − γH (sN )σB∗ ).

38

This expression is strictly positive because σS (t) > σS∗ , and thus (1 − σS (t))γL (sN )σB∗ + σS (t) > (1 − σS∗ )γL (sN )σB∗ + σS∗ = γH (sN )σB∗ .

Appendix F: Competitive Buyers This appendix formally presents a model of competitive buyers and demonstrates that our central insights extend to such an alternative market structure.

Setup We consider the same single seller search environment as in the main model. The only difference is that now the seller faces multiple buyers each time. This can be formally modelled as follows: there is a continuum of homogeneous buyers available in the market. The seller’s opportunity to sample a finite (and fixed) number of buyers arrives at Poisson rate λ. Let M denote the number of buyers the seller samples each time. The buyers in each sample observe a common signal about the seller’s type and simultaneously offer prices. If a price is accepted, then the game ends. Otherwise, the buyers leave, and the seller waits for the next opportunity to sample M other buyers. The model in Section 2 can be interpreted as the case when M = 1. We now consider the case where M > 1. We focus on the case where λ is sufficiently large and construct a set of equilibria.

Buyers’ Bidding Behavior For notational simplicity as well as analytical tractability, we take a reduced-form approach about buyers’ pricing behavior, as in, e.g., Daley and Green (2012) and Fuchs and Skrzypacz (2015). Specifically, we assume that buyers’ behavior is summarized as follows: • Denote by pa (t) the type-a seller’s reservation price at time t for each a = L, H. • Denote by pn (q) buyers’ expected value of the asset conditional on signal sn and prior q. Formally, for each n and q, pn (q) ≡

qγH (sn )vH + (1 − q)γL (sn )vL . qγH (sn ) + (1 − q)γL (sn )

• If pn (q(t)) > pH (t), then the (winning) offer to the seller is pn (t), which is accepted by both seller types.

39

• If pn (q(t)) < pH (t), then the final (best) offer to the seller is max{pL (t), vL }. The offer is accepted by the low-type seller if pL (t) < vL and rejected if pL (t) > vL (in this case, pL (t) is a losing offer, as in the main model). If pL (t) = vL , then the low-type seller’s acceptance probability is determined by other equilibrium conditions. • If pn (q(t)) = pH (t), then the final (best) offer to the seller is either pH (t) or max{pL (t), vL }. pH (t) is accepted by both seller types, while the low-type seller’s acceptance probability of max{pL (t), vL } depends on whether pL > vL or not, as in the previous case. This specification of buyers’ behavior reflects, among others, the zero profit condition for buyers: notice that the winning buyer’s expected payoff is always equal to 0.6 This reduced-form approach allows us to avoid a full specification and derivation of buyers’ bidding strategies, which is unnecessarily cumbersome. On the other hand, it is clear that any bidding equilibrium must satisfy the above properties.

Stationary Path We begin by identifying a stationary path, on which all agents’ strategies do not vary over time. ∗ Denote by qN buyers’ beliefs on the stationary path and by p∗a the type-a seller’s reservation price for each a = L, H. Finding the equilibrium variables. For the same reason as in the main model, p∗L = vL on the stationary path: otherwise, the low-type seller must either accept or reject p∗L with probability 1, ∗ and thus buyers’ beliefs drift away from qN . In addition, since λ is sufficiently large, it must be that buyers offer a high price (acceptable to both seller types) only when they receive the best signal sN and, even then, only with probability less than 1 (which is why we denote buyers’ beliefs on ∗ the path by qN in the first place). Together with the properties of buyers’ bidding behavior, these imply that the following two equations characterize each seller type’s reservation price: ∗ r(vL − cL ) = λγL (sN )σB∗ (pN (qN ) − vL ),

(33)

∗ r(p∗H − cH ) = λγH (sN )σB∗ (pN (qN ) − p∗H ),

(34)

and

∗ where σB∗ denotes the probability that buyers offer pN (qN ) to the seller conditional on signal sN . 6

The zero-profit condition is due to the fact that not only all buyers are ex ante identical, but also the buyers in each sample observe a common signal. If the latter part is relaxed (i.e., each buyer receives a private signal), then the bidding problem becomes a common-value first-price auction. Our reduced-form approach is no longer valid in such an environment. The alternative model is certainly interesting but beyond the scope of this paper.

40

Denote by σS∗ the probability that the low-type seller accepts vL on the stationary path. For buyers’ beliefs to be invariant over time, as in the main model, γH (sN )σB∗ = γL (sN )σB∗ + (1 − γL (sN )σB∗ )σS∗ .

(35)

∗ Fix qN . The three equations give the unique values of σB∗ , p∗H , and σS∗ . σB∗ is immediate from (33). Given σB∗ , p∗H can be obtained from (34), while σS∗ can be derived from (35).

Equilibrium conditions. Unlike in the main model, the stationary path is not (generically) unique: now we have 4 variables, but only 3 equations (recall that in the main model p∗H = cH ). Still, there are two further equilibrium conditions that restrict the admissible set of stationary paths. ∗ : since pN (·) is an increasing First, clearly, pN (q ∗ ) ≥ p∗H ≥ cH . This gives a lower bound of qN function, γL (sN )(cH − vL ) ∗ qN ≥ qN ≡ . γH (sN )(vH − cH ) + γL (sN )(cH − vL )

Second, on the stationary path, buyers should be willing to offer a price above p∗H only when they receive signal sN . In other words, their expected value cannot exceed p∗H if they receive any other signal. A necessary and sufficient condition for this is ∗ ∗ pN −1 (qN ) < p∗H ≤ pN (qN ). ∗ ) < p∗H if and only if It is easy to show that there exists a unique value of q N such that pN −1 (qN ∗ ∗ ∗ qN < q N . Since pN (qN ) = p∗H = cH if qN = qN , it is clear that q N < q N . ∗ ∗ There is no further restriction on qN . Therefore, for any qN ∈ [q N , q N ), there exists a corre∗ sponding stationary path.7 From now on, we fix qN ∈ [q N , qN ) and, as in the main model, construct ∗ an equilibrium that converges from qb to qN , for any qb not sufficiently large.

Low Initial Beliefs ∗ We first consider the case where qb < qN . The equilibrium construction is particularly easy in this

case. As in the main model, buyers offer only a low price for a while. Only the low-type seller accepts this low price, which increases buyers’ beliefs q(t) over time. Formally, until q(t) reaches ∗ qN , q(t) increases according to qb q(t) = . qb + (1 − qb)e−λt 7

This equilibrium multiplicity with competitive bidding in models of dynamic adverse selection is well-known in the literature. See, for example, Vincent (1990), Daley and Green (2012), Fuchs and Skrzypacz (2015), and Fuchs, ¨ and Skrzypacz (2014). Ory

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Unlike in the main model, and precisely because of competitive bidding, buyers’ offers are constant at vL on this region (recall that buyers’ offers are increasing on the corresponding region in the main model). This, in turn, affects the low-type seller’s reservation price, which is no longer identical to ∗ 8 buyers’ offers. For any t < T (b q, qN ),

Z

∗ ) T (b q ,qN




∗ (1 − e−r(s−t) )cL + e−r(s−t) vL d 1 − e−λ(s−t) + e−λ(T (bq ,qN )−t) vL t
∗ ∗ (r + λe−λ(T (bq ,qN )−t) )cL + λ 1 − e−λ(T (bq ,qN )−t) vL ∗ + e−λ(T (bq ,qN )−t) vL . = r+λ

pL (t) =

∗ Clearly, pL (t) is strictly increasing and becomes identical to vL at time T (b q, qN ). Since the high∗ type seller never trades before T (b q, qN ), it also follows that ∗

∗

pH (t) = (1 − e−r(T (bq ,qN )−t) )cH + e−r(T (bq,qN )−t) p∗H . In order to verify that this strategy profile is indeed an equilibrium, it suffices to show that buyers do not offer a price above pH (t) even with signal sN , that is, pH (t) > pN (q(t)) for any t < ∗ T (b q, qN ). In the main model, the high-type seller’s reservation is always equal to cH . Therefore, ∗ the result is immediate from the fact that q(t) < qN . In the current model, pH (t) is also strictly

increasing. This makes the result not straightforward, because buyers may be willing to offer ∗ pH (t) if it is sufficiently low, even though q(t) < qN . Nevertheless, the result can be established as follows: first observe that p˙ H (t) = r(pH (t) − cH ), while p˙ N (q(t)) =

γH (sN )γL (sN )(vH − vL ) q(t)(1 − q(t))λ. (q(t)γH (sN ) + (1 − q(t)) + γL (sN ))2

It is clear that for λ sufficiently large, the latter is necessarily larger than the former. Combining ∗ ∗ this with the fact that pH (T (b q , qN )) = pN (q(T (b q, qN ))), it follows that pH (t) > pN (q(t)) for any ∗ t < T (b q , qN ).

High Initial Beliefs ∗ Now we consider the case where qb > qN . Unlike in the main model, pH (t) varies over time, and thus the cutoff belief levels and pH (t) must be simultaneously determined. Still, there is

an equilibrium whose structure is similar to that of the main model. In particular, we construct an ∗ ∗ equilibrium that is characterized by a finite partition of the belief space, {q0∗ = 1, ..., qN , qN +1 = 0}

We define T (q, q ′ ) as in the main model. Namely, T (q, q ′ ) denotes the length of time it takes for buyers’ beliefs to move from q to q ∗ . 8

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∗ such that if q(t) ∈ (qn+1 , qn∗ ], then buyers offer a price above pH (t) if and only if they receive a signal strictly above sn . Unlike in the main model, buyers’ offers conditional on being above pH (t) ∗ are not deterministic, but depend on the signal. For example, if q(t) ∈ (qn+1 , qn∗ ], then the offer is

pn+1 (q(t)) if the signal is sn+1 , pn+2 (q(t)) if the signal is sn+2 , and so on. ∗ ∗ ∗ We first identify qN −1 . At each t such that q(t) ∈ (qN , qN −1 ), trade occurs if and only if buyers receive signal sN . Therefore, q(t) decreases according to q(t) ˙ = −q(t)(1 − q(t))λ(γH (sN ) − γL (sN )). The high-type seller’s reservation price decreases according to r(pH (t) − cH ) = λγH (sN )(pN (q(t)) − pH (t)) + p˙ H (t). It is clear that pH (t) < pN (q(t)), which establishes buyers’ incentives to offer pN (q(t)) with signal sN . It is also clear that pL (t) > vL , and thus trade never occurs at pL (t). ∗ To determine qN −1 , we use the fact that buyers’ expected value conditional on signal sN −1 must ∗ coincide with the high-type seller’s reservation price at that point, that is, pN −1 (qN −1 ) = pH (t): ∗ this is a necessary condition for trade to occur with signal sN −1 whenever q(t) > qN −1 . The ∗ uniqueness of such a point follows from the facts: (i) if q(t) = qN , then pN −1 (q(t)) < pH (t), (ii) if q(t) is sufficiently close to 1, then pN −1 (q(t)) > pH (t), and (iii) if λ is sufficiently large, pH (t) is close to pN (q(t)), and thus

γH (sN −1 )γL (sN −1 )(vH − vL ) q(t)(1 − q(t))λ(γH (sN ) − γL (sN )) (q(t)γH (sN −1 ) + (1 − q(t)) + γL (sN −1 ))2 < p˙ H (t) = r(pH (t) − cH ) − λγH (sN )(pN (q(t)) − pH (t)). p˙ N −1 (q(t)) = −

All other cutoff values can be found by recursively applying the above procedure: if q(t) ∈ then q(t) decreases according to

∗ (qn+1 , qn∗ ),

q(t) ˙ = q(t)(1 − q(t))λ(ΓH (sn ) − ΓL (sn )), and pH (t) decreases according to r(pH (t) − cH ) = λ

N X

γH (sk )(pk (q(t)) − pH (t)) + p˙H (t).

k=n+1 ∗ Given qn+1 , these two equations can be used to solve for qn∗ such that if q(t) = qn∗ , then pn (q(t)) = pH (t). As above, by construction, pn (q(t)) > pH (t) if and only if q(t) > qn∗ . Since, obviously,

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q(t)

q(t)

q1

∗ qG

∗ qR

q2 0

t

0

t

Figure 1: The evolution of buyers’ beliefs in the red-flag case (left) and in the green-flag case (right). The parameter values used for both panels are vH − cH = cH − vL = vL − cL , γ = 0.5, ǫ = 0.02, r = 0.1, and λ = 1.5. ∗ pL (t) > vL whenever q(t) > qN (which implies q(t) > qn∗ ), it follows that the constructed strategy profile is indeed an equilibrium.

Appendix G: Red Flag vs. Green Flag In this appendix, we provide a formal analysis for Prediction 4 in the main text. We apply Theorem 1 to each case and derive relevant quantities. To distinguish between the two cases, we denote by ∗ qR∗ the stationary belief level for the red-flag case and by qG that for the green-flag case. Note that we focus on the case where λ(1 − γ) > ρL and ǫ is sufficiently small.

Red Flag Since λγL (s2 ) = λ(1 − γ) > ρL , it is clear that n∗ = 2 (i.e., qR∗ = q 2 ). This implies that q1 q2 γ cH − vL 1 − γ cH − vL , and . = = 1 − q1 ǫ vH − cH 1 − q2 1 − ǫ vH − cH When ǫ is close to 0, q1 is close to 1, while q 2 is bounded away from 1. Therefore, effectively, there are only two cases: if qb > qR∗ , then q(t) decreases over time, while if qb < qR∗ , then q(t) increases

over time. See the left panel of Figure 1 for a graphical representation.

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Green Flag Since ǫ is sufficiently close to 0, λγL (s2 ) = λǫ < ρL . Therefore, n∗ = 1 and ∗ qG q1 1 − ǫ cH − vL = . = ∗ 1 − qG 1 − q1 1 − γ vH − cH

From the equilibrium structure, it is also clear that if ǫ is sufficiently close to 0, then q 2 is also ∗ close to 0. These imply that, again, there are effectively two cases: if qb > qG , then all buyers offer

∗ cH and, therefore, q(t) stays constant over time. If qb < qG , then buyers offer cH only when they receive the green flag s2 . Since the low-type seller always trades, q(t) strictly increases over time. See the right panel of Figure 1 for a graphical representation.

References Daley, Brendan and Brett Green, “Waiting for news in the dynamic market for lemons,” Econometrica, 2012, 80 (4), 1433–1504. Fuchs, William and Andrzej Skrzypacz, “Government interventions in a dynamic market with adverse selection,” Journal of Economic Theory, 2015, forthcoming. ¨ , Aniko Ory, and Andrzej Skrzypacz, “Transparency and distressed sales under asymmetric information,” mimeo, 2014. Lauermann, Stephan and Asher Wolinsky, “Search with adverse selection,” mimeo, 2015. Vincent, Daniel R., “Dynamic auctions,” Review of Economic Studies, 1990, 57 (1), 49–61.

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