Supplement to “Trading Dynamics with Private Buyer Signals in the Market for Lemons” Ayc¸a Kaya∗ and Kyungmin Kim† October 2017
Appendix A: Large Search Frictions This appendix provides a full characterization for the case when Assumption 1 (i.e., λγLh > ρL = r(vL − cL )/(cH − vL )) fails. We separate between the case when λγLh < ρL and the (non-generic) case when λγLh = ρL .
When λγLh < ρL Even with large search frictions, the interval-partitional equilibrium structure remains unchanged: there exist q and q such that the buyer never offers cH if q(t) < q, offers cH only when s = h if q(t) ∈ (q, q), and always offers cH if q(t) > q. The difference is that when q(t) ∈ (q, q), p(t) < vL and, therefore, q(t) increases over time by Lemma 1. Consequently, this case features only an increasing dynamics: if qb < q then q(t) increases and converges to q. If qb > q, then q(t) stays constant at qb. These, in turn, imply that only q can serve as a stationary belief level. See the left panel of Figure 1. Stationary Path We begin by identifying the value of q. Unlike in the small frictions case, we define it as the minimal value of qb such that it is an equilibrium that all buyers offer cH regardless of their signal. Under such offer strategies, q(t) remains equal to qb and the low-type seller’s reservation price, denoted by p, is equal to (rcL + λcH )/(r + λ). Note that p ≥ vL if and only if λ ≥ ρL . For this ∗ †
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strategy profile to be an equilibrium, it is necessary and sufficient that each buyer is indifferent between offering cH and min{vL , p} conditional on qb = q and s = l, that is, q γ l cH − min{vL , p} = lL . 1−q vH − cH γH
(1)
The following lemma corresponds to Lemma 3 in the main text. A proof is almost identical to that for Lemma 3 and therefore omitted. Lemma A1 In equilibrium, if q(t) < q, then p(t) < min{vL , p}, while if q(t) > q, then p(t) = p. This lemma implies that if qb > q, then it is the unique equilibrium that all buyers offer cH regardless of their signal. In addition, it suggests that q can indeed serve as a stationary path belief, as argued in the following lemma. Lemma A2 In equilibrium, if q(t) = q, then q(t′ ) = q for any t′ ≥ t. Proof. Suppose that q(t) = q, but there exists t′ > t such that q(t′ ) > q. This can arise only when the low type trades with a higher probability than the high type between t and t′ . This means that there exists x ∈ [t, t′ ) such that q(x) > q but p(x) < min{vL , p}, which contradicts Lemma A1. Now suppose that there exists t′ > t such that q(t′ ) < q. In this case, there must exist x ∈ (t, t′ ) such that q(x) < q but p(x) ≥ vL (so that trade occurs only at cH ), which again contradicts Lemma A1. Despite these similarities, the equilibrium behavior on the stationary path (i.e., when qb = q) can be different from that of the small frictions case in multiple ways. There are two different cases, depending on whether λ > ρL (in which case p > vL ) or not. When λ ≤ ρL . In this case, q(t) can stay constant at q only when all buyers offer cH regardless of their signal, because the low-type seller’s reservation price can never exceed vL and, therefore, she always trades with the first arriving buyer. When λ > ρL . In this case, there are two equilibria. One is such that all buyers offer cH regardless of their signal, which is an equilibrium as explained above. The other is similar to the stationary path equilibrium of the small frictions case and characterized as follows: each buyer offers cH with probability 1 conditional on s = h and with an interior probability, denoted by σB∗ , conditional on s = l, so that the low-type seller’s reservation price stays constant at vL , that is, r(vL − cL ) = λ(γLh + γLl σB∗ )(cH − vL ) ⇔ ρL = λ(γLh + γLl σB∗ ).
2
(2)
In addition, the low-type seller accepts vL with an interior probability, denoted by σS∗ , so that the two seller types trade at an identical rate, that is, h l λ(γH + γH σB∗ ) = λ(γLh + γLl σB∗ + γLl (1 − σB∗ )σS∗ ) ⇔ σS∗ =
h γH − γLh . γLl
(3)
One may think that these two equilibria can be combined in a certain way and, therefore, there may exist other equilibria. This is not the case with large search frictions, that is, there does not exist any other equilibrium: q(t) stays constant only when either buyers always offer cH (as in the first equilibrium) or buyers offer cH with probability σB∗ conditional on s = l and the low-type seller accepts vL with probability σS∗ (as in the second equilibrium). However, the low-type seller’s reservation price is equal to p(> vL ) in the first case and equal to vL in the second case. No matter how the two cases are combined, there must exist t such that p(t) ∈ (vL , p), at which point q(t) ˙ <0 unless buyers always offer cH . We summarize the results so far in the following lemma. Lemma A3 Suppose that qb = q, where q is given as in equation (1). If λ ≤ ρL , then there exists a unique equilibrium in which all buyers offer cH regardless of their signal. If λ > ρL , then there are two equilibria, one in which all buyers offer cH regardless of their signal and the other in which each buyer offers cH with probability 1 conditional on s = h and with probability σB∗ conditional on s = l, and the low-type seller accepts vL with probability σS∗ , where σB∗ and σS∗ are the values defined by equations (2) and (3). The Lower Cutoff q In order to characterize the equilibrium when qb < q, it suffices to identify the other cutoff belief q, at which the buyer is indifferent between offering cH and p(t) conditional on s = h. Let b t denote the length of time it takes for q(t) to move from q to q. Since the low-type seller trades regardless of signal s, while the high-type trades only when s = h, b t is given by the value that satisfies qe−λγH bt h
q=
hb −λγH t
qe
⇔ b
+ (1 − q)e−λt
q λ(1−γ h )bt q H . = e 1−q 1−q
(4)
Let p denote the low-type seller’s reservation price when q(t) = q. Since she receives cH at rate λγLh until q(t) reaches q, ∫ p = cL +
t+b t
( ) h h b e−r(x−t) (cH − cL )d 1 − e−λγL (x−t) + e−(r+λγL )t (p∗ − cL ),
t
3
(5)
where p∗ denotes her reservation price when q(t) converges to q from below (by continuity, p∗ = min{vL , p}). Finally, if q(t) = q, then the buyer must be indifferent between cH and p(t) conditional on s = h, and thus h qγH (vH − cH ) + (1 − q)γLh (vL − cH ) = (1 − q)γLh (vL − p) ⇔
q γ h cH − p = hL . 1−q γH vH − cH
(6)
We omit further derivations, but it is clear that these three equations uniquely determine the three unknowns, b t, q, and p. Proposition A1 summarizes the ongoing discussion and states the equilibrium strategies. It is straightforward from construction that the stated strategies constitute an equilibrium. We omit a proof for equilibrium uniqueness, because it is effectively identical to that for the small search frictions case. See the left panel of Figure 1 for a sample path of buyers’ beliefs q(t). Proposition A1 Let p∗ ≡ min{vL , (rcL + λcH )/(r + λ)} and q, p, and q be the values given by equations (1), (5), and (6). In addition, let t and t be the values such that qbe−λγH t qb and q = h qb + (1 − qb)e−λt qbe−λγH t + (1 − qb)e−λt h
q=
If qb > q, then it is the unique equilibrium that all buyers offer cH regardless of their signal, and p(t) = (rcL + λcH )/(r + λ) and q(t) = qb for any t ≥ 0. If qb ∈ [q, q), then there is a unique equilibrium in which • if t < t, then the buyer offers cH if s = h and p(t) if s = l, the low-type seller accepts both cH and p(t), p(t) increases according to p(t) ˙ = r(p(t)−cL )−λγLh (cH −cL ) with the terminal h condition p(t) = p∗ , and q(t) increases according to q(t) ), and ˙ = q(t)(1 − q(t))λ(1 − γH • if t ≥ t, then the players play as described in Lemma A3 (according to the unique equilibrium if λ ≤ ρL and according to the equilibrium in which p∗ = vL if λ > ρL ). If qb < q, then there is a unique equilibrium in which • if t < t, then the buyer offers p(t) regardless of his signal, the low-type seller accepts p(t) with probability 1, p(t) increases according to p(t) ˙ = r(p(t)−cL ) with the terminal condition p(t) = p, and q(t) increases according to q(t) ˙ = q(t)(1 − q(t))λ, and • if t ≥ t, then the players play as in the above case with qb = q.
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q(t)
q(t)
q∗ = q
q
q q
qb
qb 0
t
t+b t
t
0
t∗
t
Figure 1: The evolution of buyers’ beliefs q(t) with large search frictions. The parameter values for both panels are identical to those for Figure 1 in the main text, except that λ = 0.4 (so that λγLh < ρL ) in the left panel and λ = 0.75 (so that λγLh = ρL ) in the right panel.
When λγLh = ρL In the non-generic case where λγLh = ρL , equilibrium uniqueness fails. This follows from the fact that any q ∗ ∈ [q, q] can be supported as a stationary belief level (the gray area in the right panel of Figure 1). To be precise, let q and q be the values such that q γ h cH − v L q γ l cH − vL = hL and , = lL 1−q 1−q γH vH − cH γH vH − cH
(7)
so that a buyer breaks even when he offers cH conditional on belief q and signal h or conditional on belief q and signal l. Fix any q ∗ ∈ [q, q] and assume that all buyers offer cH if and only if s = h. Since λγLh = ρL , p(t) = vL . Given q ∗ ∈ [q, q] and p(t) = vL , buyers’ offer strategies are optimal. Finally, for q(t) to stay constant at q ∗ , it suffices to set σS∗ (the probability that the low-type seller accepts p(t) = vL ), so that h ) = λ(γLh + γLl σS∗ ) ⇔ σS∗ = λ(γH
h γH − γLh . 1 − γLh
In fact, q(t) does not even need to converge to a certain level, because it can fluctuate in an arbitrary manner within the interval [q, q]. This arises because, unlike buyers’ offer strategies that are determined by the equilibrium requirement that p(t) = vL , the low-type seller’s acceptance
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strategy of vL is indeterminate. For instance, the low-type seller may accept vL with probability 1 until q(t) reaches q and then with a constant probability so that q(t) stays constant. Or, she may reject vL with probability 1, until q(t) hits q. q(t) may even keep oscillating between q and q (or between any other pair of beliefs in the interval). Nevertheless, all equilibria have crucial properties in common, all of which can be shown just as in the generic case. First, within the range [q, q], the low-type seller’s reservation price is necessarily equal to vL . Second, if qb < q, then q(t) gradually converges to the interval [q, q] (see the solid curve in the right panel of Figure 1). Finally, given the first two properties, it follows that for any initial belief qb, all the equilibria are payoff-equivalent. The only difference among the equilibria is the low-type seller’s trading rates while q(t) ∈ [q, q], as they also depend on the low-type seller’s acceptance strategy. Proposition A2 Let q and q be the values given by equation (7). If qb > q, then it is the unique equilibrium that all buyers offer cH regardless of their signal, and p(t) = (rcL + λcH )/(r + λ) and q(t) = qb for any t ≥ 0. If qb ∈ [q, q], then in any equilibrium, • all buyers offer cH if s = h and vL if s = l, and • the low-type seller’s acceptance strategy, which is represented by the probability σS (t) that she accepts vL , is such that qbe−λγH t h
q(t) =
qbe−λγH t + (1 − qb)e−λ h
∫t
h l 0 (γL +γL σS (x))dx
∈ [q, q], for any t ≥ 0.
If qb < q, then in any equilibrium, ∗
• if t < t∗ , then the buyer offers p(t) = cL + e−r(t −t) (cH − cL ), the low-type seller accepts p(t) with probability 1, and q(t) increases according to q(t) ˙ = q(t)(1 − q(t))λ, and • if t ≥ t∗ , then the players play as in the above case with qb = q, where t∗ is the value such that q=
qb . qb + (1 − qb)e−λt∗
Appendix B: No Gap at the Bottom or at the Top We have assumed that there are positive gains from trade for both seller types (i.e., va > ca for both a = L, H). Although this case has been more widely considered in the literature, the cases with no gap at the bottom (vL = cL ) or at the top (vH = cH ) also have been studied: see, e.g., 6
¨ and Taylor (1999); Zhu (2012) for the former case and Fuchs and Skrzypacz (2015); Fuchs, Ory Skrzypacz (2016) for the latter case. To make comparisons to those papers more transparent, we explain what happens if there is no gap at the bottom or at the top in our model. If there is no gap at the top (i.e., vH = cH ), the result is trivial in our model. No buyer has an s incentive to offer cH , because q(t)γH (vH − cH ) + (1 − q(t))γLs (vL − cH ) < 0 as long as q(t) < 1. Given this, the equilibrium is essentially identical to that of the complete-information case in which the seller is known to be the low type. Due to the Diamond paradox, all buyers offer only cL . The low-type seller trades with the first buyer, while the high-type seller never trades. We note that this triviality is driven by our two-type restriction, and the existing papers with the no-gap-at-the-top assumption consider a continuous type space. If there is no gap at the bottom (i.e., vL = cL ), then buyers have no incentive to target only the low-type seller and offer p(t). This means that the problem shrinks to when buyers have an incentive to offer cH . The equilibrium is again characterize by two cutoffs, q and q, each of which takes the same value as in the main case with vL > cL . The equilibrium behavior when qb > q is also similar to that in Proposition 2: if qb > q, then all buyers offer cH regardless of their signal and q(t) remains constant at qb. If qb ∈ (q, q), then buyers offer cH if and only if s = h until q(t) reaches q. A crucial difference is that when q(t) = q at some t, buyers never offer cH and, therefore, gains from trade are never realized thereafter. This is when the probability of the high type is so low that buyers’ expected value of the asset does not exceed cH even conditional on signal h. Since buyers also have no incentive to trade with the low-type seller, there is no scope for trade and the market essentially breaks down. If qb < q, then there are equilibria in which the low-type seller trades at vL = cL with a positive probability (insofar as q(t) stays below q), but cH is never offered in any equilibrium.
Appendix C: Beyond Binary Signals Finite Signal Space This appendix provides a formal characterization of the model with a general finite signal space discussed in Section 3.4 of the main text. We restrict attention to the generic case where λ(1 − ΓL (sn )) ̸= ρL for any n. In addition, we focus on the case where λ > ρL . This latter assumption ensures that the low-type seller’s reservation price on the stationary path is equal to vL , thereby significantly reducing notational burden. The case when the inequality fails can be characterized just as in Appendix A (in particular, the case when p < vL ).
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The signal structure: Each buyer receives a signal from a finite set S = {s1 , ..., sN }. For each a = L, H and n = 1, ..., N , let γa (sn ) denote the probability that each buyer receives sn from the type-a asset and Γa (sn ) represent the corresponding cumulative probability (i.e., Γa (sn ) ≡ ∑n k=1 γa (sk )). Without loss of generality, assume that the likelihood ratio γH (sn )/γL (sn ) is strictly increasing in n, so that buyers assign a higher probability to the high type when s = sn+1 than when s = sn .
Equilibrium Construction Just as in the case with two signals, we first construct an equilibrium and then argue its uniqueness. Stationary Path The unique stationary path is as described in the main text. Let n∗ be the unique integer such that λ(1 − ΓL (sn∗ )) < ρL < λ(1 − ΓL (sn∗ −1 )).
(8)
These inequalities mean that the low-type seller’s reservation price p(t) falls short of vL when all subsequent buyers offer cH if and only if s > sn∗ but exceeds vL when they do so if and only if s ≥ sn∗ . Then, qn∗ plays the same role as the stationary belief level q ∗ in the main model: if q(t) reaches qn∗ , then q(t) stays constant thereafter. Buyers offer cH with probability σB∗ ∈ (0, 1) conditional on s = sn∗ , so that λ(1 − ΓL (sn∗ ) + γL (sn∗ )σB∗ ) = ρL =
r(vL − cL ) . cH − vL
(9)
The stationary belief q ∗ = qn∗ is pinned down by the requirement that each buyer must be indifferent between offering cH and offering p(t) = vL conditional on s = sn∗ : qn∗ γH (sn∗ )(vH − cH ) + (1 − qn∗ )γL (sn∗ )(vL − cH ) = 0 ⇔
qn∗ γL (sn∗ ) cH − vL = . (10) 1 − qn ∗ γH (sn∗ ) vH − cH
Finally, the low-type seller accepts p(t) = vL with probability σS∗ ∈ (0, 1), so that the two seller types trade at an identical rate, that is, λ(1 − ΓH (sn∗ ) + γH (sn∗ )σB∗ ) = λ(1 − ΓL (sn∗ ) + γL (sn∗ )σB∗ + (γL (sn∗ )(1 − σB∗ ) + ΓL (sn∗ −1 ))σS∗ ). Consequently, σS∗ =
ΓL (sn∗ ) − ΓH (sn∗ ) + (γH (sn∗ ) − γL (sn∗ ))σB∗ . γL (sn∗ )(1 − σB∗ ) + ΓL (sn∗ −1 ) 8
(11)
Lemma C1 Let n∗ , σB∗ , q ∗ , and σS∗ be the values given by equations (8)-(11). If qb = q ∗ , then there is an equilibrium in which • each buyer offers cH with probability 1 if s > sn∗ , with probability σB∗ if s = sn∗ , and with probability 0 if s < sn∗ , • the low-type seller accepts p(t) = vL with probability σS∗ , and • q(t) stays constant at q ∗ . Convergence Path In the equilibrium we are constructing, there exists a partition {qN +1 = 0, qN , ...q1 , q0 = 1} which informs equilibrium behavior when q(t) ̸= q ∗ = qn∗ . These cutoff beliefs are such that at qn , the buyer is indifferent between offering cH and p(t) conditional on s = sn . Moreover, p(t) is smaller than vL if q(t) < q ∗ (pessimistic beliefs) and larger than vL if q(t) > q ∗ (optimistic beliefs). By Lemma 1 (which applies unchanged even with more than two signals), q(t) increases if q(t) < q ∗ and decreases if q(t) > q ∗ (unless q(t) > q1 ). Specifically, if q(t) ∈ (qn+1 , qn ) for n ≥ n∗ , then the low-type seller accepts both p(t) and cH , while the high-type seller trades if and only if s ≥ sn . Therefore, q(t) increases according to q(t) ˙ = q(t)(1 − q(t))λΓH (sn ) > 0. If q(t) ∈ (qn+1 , qn ) for n = 1, ..., n∗ − 1, then both seller types trade if and only if s > sn . Since 1 − ΓH (sn ) > 1 − ΓL (sn ), q(t) decreases according to q(t) ˙ = q(t)(1 − q(t))λ(ΓH (sn ) − ΓL (sn )). The cutoff beliefs above q ∗ (i.e., qn∗ −1 , ..., q1 ) are pinned down by the fact that p(t) > vL whenever q(t) > q ∗ and, therefore, it suffices that the buyer breaks even with offer cH conditional on prior qn and signal sn : qn γH (sn )(vH − cH ) + (1 − qn )γL (sn )(vL − cH ) = 0 ⇔
γL (sn ) cH − vL qn = . 1 − qn γH (sn ) vH − cH
(12)
The cutoff beliefs below q ∗ (i.e., qN , ..., qn∗ +1 ), which are harder to obtain because p(t) < vL , can be found through the following recursive procedure (which generalizes the characterization for the large frictions case in Appendix A): let tn denote the length of time it takes for q(t) to travel from qn to qn−1 conditional on the low type trading at rate λ and the high type trading at rate
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λ(1 − ΓH (sn−1 )): qn−1 =
qn e−λ(1−ΓH (sn−1 )tn . qn e−λ(1−ΓH (sn−1 )tn + (1 − qn )e−λtn
(13)
Let pn∗ = vL and for each n > n∗ , define pn recursively as follows: ∫ p n = cL +
tn
e−rt (cH − cL )d(1 − e−λ(1−ΓL (sn−1 ))t ) + e−(r+λ(1−ΓL (sn−1 )))tn pn−1 .
(14)
0
In other words, pn is the low-type seller’s reservation price when she expects to receive cH at rate λ(1 − ΓL (sn−1 )) for tn length of time and her reservation price then becomes equal to pn−1 . It then suffices to find qn ’s recursively, so that a buyer is indifferent between offering cH and pn conditional on s = sn : qn γL (sn ) cH − pn = . (15) 1 − qn γH (sn ) vH − cH Given (qn−1 , . . . , q ∗ ) and (pn−1 , . . . , p∗ = vL ), the pair (qn , pn ) is uniquely determined and qn < qn−1 (which implies pn < pn−1 ). The uniqueness follows from the fact that equations (13) and (14) yield a continuous and strictly increasing relationship between pn and qn , while equation (15) defines a continuous and strictly decreasing relationship between these variables. qn < qn−1 is due to the fact that in equation (15), the left-hand side is larger than the right-hand side if qn = qn−1 (because pn = pn−1 but γL (sn )/γH (sn ) < γL (sn−1 )/γH (sn−1 )), while the opposite obviously holds if qn = 0. Next, we present Propositions C1 and C2 which generalize the Propositions 1 and 2 in the main text. Their proofs are immediate from the above construction and therefore omitted. Pessimistic beliefs. Recall that tn represents the length of time it takes for q(t) to move from qn to qn−1 conditional on the event that the low type trades at rate λ and the high type trades at rate λ(1 − ΓH (sn−1 )). In addition, pn is the low-type seller’s reservation price when q(t) = qn . Proposition C1 Suppose qb ∈ [qn , qn−1 ) for some n > n∗ . Let t∗ be the value such that ∗
qn−1
qbe−λ(1−ΓH (sn−1 ))t = −λ(1−Γ (sn−1 ))t∗ . H qbe + (1 − qb)e−λt∗
There is an equilibrium in which • if t < t∗ , then the buyer offers cH if s > sn−1 and p(t) if s ≤ sn−1 , the low-type seller accepts p(t) with probability 1, p(t) increases according to p(t) ˙ = r(p(t) − cL ) − λ(1 − ΓL (sn−1 ))(cH − vL ) with the terminal condition p(t∗ ) = pn−1 , and q(t) increases according to q(t) ˙ = q(t)(1 − q(t))λΓH (sn−1 ),
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∑ • if t ∈ [t∗ , t∗ + kl=n−1 tl ) for some k ∈ [n∗ +1, n−1], then the buyer offers cH if s > sk−1 and p(t) if s ≤ sk−1 , the low-type seller accepts p(t) with probability 1, p(t) increases according to p(t) ˙ = r(p(t) − cL ) − λ(1 − ΓL (sk−1 ))(cH − vL ) with the terminal condition p(t∗ ) = pk−1 , and q(t) increases according to q(t) ˙ = q(t)(1 − q(t))λΓH (sk−1 ), • if t ≥ t∗ +
∑n∗ +1
l=n−1 tl ,
then the players play as in Lemma C1 and q(t) = q ∗ .
Optimistic beliefs. For qb > q ∗ , let t′n denote the length of time it takes for q(t) to travel from qn to qn+1 conditional on the event that the type-a seller trades at rate λ(1 − Γa (sn )), that is, ′
qn+1
qbe−λ(1−ΓH (sn ))tn . = −λ(1−Γ (sn ))t′ H n + (1 − q qbe b)e−λ(1−ΓL (sn ))t′n
In addition, let pn′ ∗ ≡ vL and define p′n∗ −1 , ..., p′1 recursively, so that p′n
∫ =
t′n
′
e−rt (cH − cL )d(1 − e−λ(1−ΓL (sn )t) ) + e−(r+λ(1−ΓL (sn )))tn p′n+1 .
0
Since n < n∗ , p′n > vL . It is also clear that p′n is decreasing in n. Proposition C2 Suppose qb ∈ [qn+1 , qn ) for some n < n∗ . Let t∗ be the value such that ∗
qn+1
qbe−λ(1−ΓH (sn ))t . = −λ(1−Γ (sn ))t∗ H qbe + (1 − qb)e−λ(1−ΓL (sn ))t∗
There is an equilibrium in which • if t < t∗ , then the buyer offers cH if s > sn and p(t) if s ≤ sn , the low-type seller accepts p(t) with probability 0, p(t) decreases according to p(t) ˙ = r(p(t)−cL )−λ(1−ΓL (sn ))(cH −vL ) with the terminal condition p(t∗ ) = p′n+1 , and q(t) decreases according to q(t) ˙ = q(t)(1 − q(t))λ(ΓH (sn ) − ΓL (sn )), ∑ • if t ∈ [t∗ , t∗ + kl=n+1 t′l ) for some k ∈ [n + 1, n∗ − 1], then the buyer offers cH if s > sk and p(t) if s ≤ sk , the low-type seller accepts p(t) with probability 0, p(t) decreases according to p(t) ˙ = r(p(t) − cL ) − λ(1 − ΓL (sk ))(cH − vL ) with the terminal condition p(t∗ ) = p′k+1 , and q(t) decreases according to q(t) ˙ = q(t)(1 − q(t))λ(ΓH (sk ) − ΓL (sk )), • if t ≥ t∗ +
∑n∗ −1
′ l=n+1 tl ,
then the players play as in Lemma C1 and q(t) = q ∗ .
11
Equilibrium Uniqueness The arguments for the following results are minor modifications of their counterparts for the binary signal case in the main text. Therefore, we simply re-state them for reference. In any equilibrium, 1. q(t) ≤ q ∗ if and only if p(t) ≤ vL (Lemma 3 in the main text), and 2. if q(t) reaches q ∗ , then it stays constant thereafter. Now we are ready to prove the uniqueness result: Theorem C1 Unless qb = q1 , there exists a unique equilibrium. Proof. For the case when qb > q1 , the same argument as for the case when qb > q in the main model applies unchanged. Assume that q1 > qb > q ∗ . Then, p(t) > vL and, therefore, trade takes place only at cH until q(t) reaches q ∗ . Consequently, each buyer’s unique optimal strategy is to offer cH if and only if it yields a non-negative payoff, as given in Proposition C1. The equilibrium uniqueness then follows from the explicit and unique equilibrium construction given above. Now assume that qb < q ∗ . The following lemma shows that p(t) is non-decreasing over time in this case. Lemma C2 In any equilibrium, if q(t) < q ∗ , then p(·) is strictly increasing in t. Proof. Suppose there exists t such that q(t) < q ∗ , but p(t) is weakly decreasing (i.e., p(t) ˙ ≤ 0). By Lemmas 1 and 3 in the main text, p(t) is strictly smaller than vL and eventually converges to vL . Since p(·) is also continuous, there exists t′ such that t′ > t and p(t′ ) = p(t). Without loss of generality, assume that p(x) ≤ p(t) for any x ∈ (t, t′ ) and p(x) ˙ > 0 for any x > t′ such that q(x) < q ∗ (if p(·) is not strictly increasing until it reaches vL , there always exist t and t′ that satisfy these properties). For x ∈ (t, t′ ), p(x) ≤ p(t′ ), while q(x) < q(t′ ). This implies that the cutoff signal used by the buyer at x ∈ (t, t′ ) must be at least as large as that used by the buyer at t′ . To the contrary, whenever x > t′ , p(x) > p(t′ ) and q(x) > q(t′ ). Therefore, the cutoff signal used by the buyer at x > t′ must be no larger than that used by the buyer at t′ . Combining these observations with the fact that p(·) is strictly increasing from t′ , it follows that p(t) < p(t′ ), which is a contradiction. Since both p(·) and q(·) are strictly increasing over time, there is a one-to-one and increasing mapping between them until q(t) reaches q ∗ . In addition, the cutoff signal is weakly increasing over time. As in the main model, this suffices to establish that any equilibrium has an intervalpartitional equilibrium structure. By construction, the equilibrium in Propositions C1 and C2 is 12
the unique equilibrium exhibiting the interval-partitional structure. Together with the uniqueness of equilibrium behavior when q(t) = q ∗ , it follows that the equilibrium described in Propositions C1 and C2 is the unique equilibrium of the model.
Continuum of signals Now we consider the case where the signal space is continuous: each buyer’s signal is drawn from the interval S = [s, s] according to the type-dependent distribution function Γa , where Γa (s) denotes the probability that each buyer receives a signal below s from the type-a asset. For tractability, assume that for both a = L, H, Γa admits a positive and continuously differentiable density γa and γH (s)/γL (s) is strictly increasing in s (MLRP). The existence and uniqueness of a stationary path in this environment is already shown in the main text. Let s∗ and q ∗ be the stationary cutoff signal and belief, respectively, as defined in the main text. The following properties can be established analogously to their counterparts for the finitesignal case: 1. p(·), q(·), and s(·) are continuous. 2. p(t) ≥ vL if and only if q(t) ≥ q ∗ . 3. p(t) is strictly increasing if q(t) < q ∗ . 4. q(t) is increasing in t if p(t) < vL and decreasing if p(t) > vL . These properties immediately imply that in any equilibrium, if the initial belief qb is above q ∗ , then the high type trades at a higher rate than the low type and, therefore, q(t) decreases over time. To the contrary, if qb is below q ∗ , then the low type trades at rate λ, while the high type trades at a lower rate λ(1 − ΓH (s(t))), and thus q(t) increases over time. Moreover, monotonicity of p(t) guarantees that the cutoff belief above which each buyer offers cH increases over time if qb < q ∗ and decreases over time if qb > q ∗ . Therefore, the main economic insights from the final signal case continue to hold in this environment. The existence of equilibrium is again by construction. Since the case when qb ≥ q ∗ is fully discussed in the main text, we focus on the case where qb < q ∗ , for which the equilibrium conditions are given as follows. • The buyer’s indifference between cH and p(t) conditional on s = s(t): q(t) γL (s(t)) cH − p(t) = . 1 − q(t) γH (s(t)) vH − cH 13
(16)
• The low-type seller’s reservation price: ∫
t′
p(t) = cL +
e−r(x−t) (cH − cL )d(1 − e−λ
∫x t
(1−ΓL (s(y))dy
) + e−(r+λ
∫ t′ t
(1−ΓL (s(y))dy))
t
p(t′ ). (17)
• The evolution of beliefs : q(t) ˙ = q(t)(1 − q(t))λΓH (s(t)).
(18)
To show that there exist functions s(·), p(·), and q(·) that satisfy these equilibrium conditions, we consider a sequence with finite signals that converge to the given continuous signal structure. Fix d > 0 and define N (d) ≡ (¯ s − s)/d. Assume that for any d considered, N (d) is an integer. Define Sd ≡ {s, s + d, s + 2d, . . . , s¯}. For notational simplicity, we assume that (s∗ − s)/d is an integer, so that s∗ ∈ Sd , for any d. Let Γda (s) ≡ Γa (s+id) and γad (s) = Γa (s+id)−Γa (s+(i−1)d) for all s ∈ (s + (i − 1)d, s + id]. Clearly, Γda and γad converge to Γa and γa , respectively, as d → 0. d By the analysis in section C1, when the signal structure { is given by}Γ , there exists a (generically) d d unique equilibrium with associated cutoff beliefs q1d , q2d , ·, qN (d) , reservation price p (t), and d , qid ). In addition, let s∗d belief q d (t). Define sd (t), so that sd (t) ≡ s + id whenever q d (t) ∈ [qi+1 and qd∗ represent the stationary belief level and cutoff signal for the case with Γd . By construction, the values of s∗d and qd∗ are independent of d. d Fix qb < q ∗ and let bi be the value such that qb ∈ [qbi+1 , qbid ). Let tbdi denote the time it takes for q d (t) to travel from qb to qbid . In addition, for each i < bi, let tdi denote the length of time it takes for d to qid in the equilibrium associated with increment size d. Finally, let i∗d q d (t) to travel from qi+1 ∑ be the value such that si∗d = s∗d . Then, by construction, for t ∈ T d ≡ { ij=bi tdj |i = bi, . . . , i∗d }, the following properties hold for any d. • The buyer’s indifference between cH and pd (t) conditional on s = sd (t): q d (t) γLd (sd (t)) cH − pd (t) = . d 1 − q d (t) γH (sd (t)) vH − cH
(19)
• The low-type seller’s reservation price: ∫ p(Ti ) = cL +
Ti−1
e−r(x−Ti ) (cH −cL )d(1−e−λ(1−ΓL (si ))(Ti−1 −Ti ) )+e−(r+λ(1−ΓL (si ))(Ti−1 −Ti ) p(Ti−1 ), d
Ti
where Tk =
d
(20)
∑k
d j=bi tj .
14
• The evolution of beliefs : q˙d (t) = q d (t)(1 − q d (t))λΓH (sd (t)).
(21)
It is clear that as d → 0, equations (19), (20) and (21) converge to (16), (17) and (18), respectively. Then, the pointwise limit of the functions sd (·), q d (·), pd (·) as d → 0 satisfy (16), (17) and (18), establishing the desired existence result.
Appendix D: The Three-type Case This appendix provides a formal proof for Proposition 6 in the main text. In particular, we explicitly construct an equilibrium for any qb such that Eqb,l [v] < cH .
Path LH The strategy profile for this case is described in the main text. In particular, on this path pL (t) = vL and each buyer randomizes between offering cH and vL when s = h. The latter implies the following indifference condition: qL γL (vL − cH ) + qM γM (vM − cH ) + qH γH (vH − cH ) = 0. ∗ In addition, clearly, qM < qM . 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 ∗ . As in the main text, let σ ˜H be the value that satisfies qM
r(vL − cL ) = λγL σ ˜H (cH − vL ).
(22)
For buyers’ beliefs to stay on Path LH, it must be that q˙L γL (vL − cH ) + q˙M γM (vM − cH ) + q˙H γH (vH − cH ) = 0.
(23)
Reproducing the equations describing the evolution of q(t), 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 )) . 15
If γL σ ˜H + (1 − γL σ ˜H )σS (t) = γH σ ˜H , then the left-hand side in (23) is strictly less than 0: 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 (23) 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 increasing in σS (t), there exists a unique value of σS (t) that satisfies (23).
Path MH On Path MH, buyers randomize between cH and pM (t) conditional on s = h, and trade never occurs at pL (t). Buyers’ offer probabilities of cH and pM (t) are such that q(t) stays on Path MH. Since the low type trades more slowly than the middle type, q(t) converges to q ∗ . For these to hold, as on Path LH, Eq(t),h [v] = cH . In addition, pM (t) =
qL (t)γL vL + qM (t)γM vM . qL (t)γL + qM (t)γM
(24)
pM (t) evolves according to r(pM (t) − cM ) = λγM σH (t)(cH − pM (t)) + p˙M (t).
16
(25)
q(t) changes according to 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)))) , q˙H = λqH (qL (γL (σH (t) + σM (t)) − γH σH (t)) + qM (γM (σH (t) + σM (t)) − γH σH (t)) . Relative to Path LH, a complication is that pM (t) changes over time and, therefore, σH (t) is also time-varying. In order to establish the existence of such a strategy profile, first observe that given σH (t), there exists a unique σM (t) which induces q˙L γL (vL − cH ) + q˙M γM (vM − cH ) + q˙H γH (vH − cH ) = 0. This can be directly established from the law of motion for q(t). Intuitively, if σM (t) is such that γM (σH (t) + σM (t)) = γH σH (t), then qM /qH stays constant, while qL (t) increases, in which case Eq(t+dt),h [v] < cH . If σM (t) is sufficiently large, then qH /qL and qH /qM increase, in which case Eq(t+dt),h [v] > cH . Applying this result, q(t) can be expressed as a function of (only) σH (t). Combining this with equations (24) and (25) yields a differential equation for σH (t). Given the ∗ ∗ boundary conditions that σH (t) = λγM (cH − vM )/(r(vM − cM )) if q(t) = qM H and σH (t) = σH when q(t) = q ∗ , the general existence theorem applies. For incentive compatibility, it suffices to show that pL (t) > vL (so that buyers have no incentive ∗ as q(t) approaches q ∗ (otherwise, to offer pL (t)). Notice that σH (t) decreases and converges to σH ∗ pM (t) decreases, which ultimately violates equation (24)). This implies both σH (t) > σH and ∗ ∗ σM (t) > σM (since σH (t) > σH , buyers’ beliefs can stay on Path MH only when qM (t) needs to decrease faster than at q ∗ ). Combining this with pM (t) > p∗M , it follows that pL (t) > vL at any point on Path MH.
Path LM On Path LM, buyers randomize between pL (t) and pM (t) conditional on s = h. For the usual reason, pL (t) = vL . Since buyers are indifferent between vL and pM (t) (as on Path MH), pM (t) =
qL (t)γL vL + qM (t)γM vM . qL (t)γL + qM (t)γM
17
(26)
Since cH is never offered until q(t) arrives at q ∗ , pM (t) increases according to r(pM − cM ) = p˙M . For pL (t) to stay constant at vL , r(vL − cL ) = λγL σM (t)(pM (t) − vL ).
(27)
Finally, given σM (t), and σS (t) , buyers’ beliefs evolve according to q˙L = λqL (qM (γM σM (t)−(γL σM (t)+(1−γL σM (t))σS (t)))−qH (γL σM (t)+(1−γL σM (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)). Unlike the other two paths, there is no candidate trajectory for q(t). Therefore, the law of motion for q(t) cannot be used to discipline equilibrium trading behavior. In what follows, we construct Path LM itself. It is not possible to directly construct a continuous-time strategy profile that satisfies all the equilibrium conditions. We solve this problem by discretizing the model: we construct an equilibrium in a discrete-time setting with period length ∆ and let ∆ tend to 0. 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 (26). 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∗ qL (t∗ − ∆)(1 − (1 − e−λ∆ )(γL σM (t∗ − ∆) + (1 − γL σM (t∗ − ∆))σS (t∗ − ∆))) = , 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 (26), while
18
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. For Path LM, the optimality of buyers’ offers is straightforward, because Eq,h [v] < cH at any point on the path.
Area L See the main text for the equilibrium strategy profile. We prove that no buyer has an incentive to offer pM (t) in Area L. Fix q in Area L and let t∗ denote the length of time it takes for q(t) to reach Path LM or Path LH. Since only the low type trades until t∗ , ∗
pM (0) = cM + e−rt (pM (t∗ ) − cM ), and
qM (0) qM (t∗ ) = −λt∗ . ∗ qL (t ) e qL (0)
The latter implies that ∗
qL (0)γL vL + qM (0)γM vM qL (t∗ )eλt γL vL + qM (t∗ )γM vM = qL (0)γL + qM (0)γM qL (t∗ )eλt∗ γL + qM (t∗ )γM Recall that on both path LM and path LH, pM (t∗ ) ≥
qL (t∗ )γL vL + qM (t∗ )γM vM . qL (t∗ )γL + qM (t∗ )γM
Then, it is clear that if λ is sufficiently large, then for any t∗ , pM (0) >
qL (0)γL vL + qM (0)γM vM , qL (0)γL + qM (0)γM
19
and thus, buyers have no incentive to offer pM (t).
Area M Suppose that buyers offer pM (t) conditional on s = h at any point in Area M. Then, for each point qb, there exists a unique convergence path to either Path LM or Path MH. Let t∗ denote the length of time it takes for q(t) to reach Path LM or Path MH. Since cH is never offered along the path, ∗
pM (0) = cL + e−rt (pM (t∗ ) − cL ) < pM (t∗ ). Let Ξ denote the set of qb’s such that qbL (1 − γL )(vL − pM (0)) + qbM (1 − γM )(vM − pM (0)) ≥ 0. In other words, if qb ∈ Ξ, then buyers are willing to offer pM (0) even conditional on s = l. This set is non-empty (as the inequality holds whenever qbM is sufficiently large) and connected (as pM (0) changes continuously as qb varies). Now assume that buyers offer pM (t) independent of s if q(t) ∈ Ξ. This, of course, changes the law of motion for q(t). In particular, both qM (t) and qL (t) decrease at the same rate, while qH (t) increases. It is easy to recalculate pM (t) based on this change in the law of motion. By construction, if qb ∈ / Ξ, then it stays outside Ξ and reaches Path LM or Path MH. If qb ∈ Ξ, then q(t) escapes Ξ in finite time, stays in M Xi for a while, and eventually reaches Path LM or Path MH. The optimality of buyers’ offer strategies is straightforward. Whenever q(t) is in Area M, Eq(t),h [v] < cH , and thus buyers have no incentive to offer pH (t) = cH . In addition, the low type receives pM (t)(≥ cM ) at least at rate λγL . Combining this with the fact that pL (t) ≥ vL on both Path LM and Path LH, it follows that pL (t) > vL everywhere in Area M.
Area H In Area H, buyers offer cH conditional on s = h if Eq(t),l [v] < cH and independent of s if Eq(t),l [v] ≥ cH . In the latter case, q(t) stays constant and the seller trades with the first arriving buyer independent of her type. In the former case, since γH > γM > γL , the high type trades faster than the middle type, who in turns trades faster than the low type. q(t) converges to either Path LH or Path MH. In Area H, the type-a seller receives cH at least at rate λγa . Incentive compatibility (that buyers have no incentive to offer pL (t) and pM (t)) follows once this is combined with the fact that
20
q(t)
q(t)
q
∗ =q qG
∗ =q qR
q t
0
0
t
Figure 2: 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, ϵ = 0.02, r = 0.1, and λ = 1.5. pL (t) ≥ vL and pM (t) ≥
qL (t)γL vL + qM (t)γM vM qL (t)γL + qM (t)γM
on both Path LH and Path MH.
Appendix E: Red Flag vs. Green Flag In this appendix, we provide a formal analysis for Prediction 4 in the main text. 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 ϵ is sufficiently small.
Red Flag Since λγLh = λ(1 − γ) > ρL , the unique equilibrium is characterized by Propositions 1 and 2 in the main text. This implies that q γ cH − vL qR∗ q 1 − γ cH − vL = = , and = . ∗ 1−q ϵ vH − cH 1 − qR 1−q 1 − ϵ vH − cH
21
When ϵ is close to 0, q is close to 1, while qR∗ = q is bounded away from both 0 and 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 2 for a graphical representation.
Green Flag Since ϵ is sufficiently close to 0, λγLh = λϵ < ρL . Therefore, the equilibrium is given as in Section A in this online appendix. It then follows that ∗ qG q 1 − ϵ cH − v L = = . ∗ 1 − qG 1−q 1 − γ v H − cH
From the equilibrium structure, it is also clear that if ϵ is sufficiently close to 0, then q 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 signal h. Since the low-type seller always trades, q(t) strictly increases over time. See the right panel of Figure 2 for a graphical representation.
References Fuchs, William and Andrzej Skrzypacz, “Government interventions in a dynamic market with adverse selection,” Journal of Economic Theory, 2015, 158, 371–406. ¨ , Aniko Ory, and Andrzej Skrzypacz, “Transparency and distressed sales under asymmetric information,” Theoretical Economics, 2016, 11 (3), 1103–1144. Taylor, Curtis R., “Time-on-the-market as a sign of quality,” Review of Economic Studies, 1999, 66 (3), 555–578. Zhu, Haoxiang, “Finding a good price in opaque over-the-counter markets,” Review of Financial Studies, 2012, 25 (4), 1255–1285.
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