Supplement to “Trading Dynamics with Private Buyer Signals in the Market for Lemons” Ayc¸a Kaya∗ and Kyungmin Kim† January 2018
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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1
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 ˙ <0 how the two cases are combined, there must exist t such that p(t) ∈ (vL , p), at which point q(t) 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 h
q=
qe−λγH bt
hb t −λγH
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 +
Z
t
t+b t
h b h e−r(x−t) (cH − cL )d 1 − e−λγL (x−t) + e−(r+λγL )t (p∗ − cL ),
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 h
q=
qbe−λγH t qb and q = h qb + (1 − qb)e−λt qbe−λγH t + (1 − qb)e−λt
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 ˙ = q(t)(1 − q(t))λ(1 − γH ), and condition p(t) = p∗ , and q(t) increases according to q(t) • 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 ˙ = q(t)(1 − q(t))λ, and p(t) = p, and q(t) increases according to q(t)
• 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 q γ l cH − vL γ h cH − vL and = lL , = hL 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(γH ) = λ(γLh + γLl σS∗ ) ⇔ σS∗ =
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 h
q(t) =
h
qbe−λγH t
qbe−λγH t + (1 − qb)e−λ
Rt 0
h +γ l σ (x))dx (γL L S
If qb < q, then in any equilibrium,
∗ −t)
• if t < t∗ , then the buyer offers p(t) = cL + e−r(t
∈ [q, q], for any t ≥ 0.
(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., Taylor (1999); 6
¨ and Skrzypacz (2016) Zhu (2012) for the former case and Fuchs and Skrzypacz (2015); Fuchs, Ory 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 )) 6= 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 ).
7
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 ) ≡ Pn 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 ⇔
γL (sn∗ ) cH − vL qn∗ . (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) 6= 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 +
Z
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 There is an equilibrium in which
qbe−λ(1−ΓH (sn−1 ))t . = −λ(1−Γ (sn−1 ))t∗ H qbe + (1 − qb)e−λt∗
• 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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P • 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∗ +
Pn∗ +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, ′
qbe−λ(1−ΓH (sn ))tn = −λ(1−Γ (sn ))t′ . H n + (1 − q qbe b)e−λ(1−ΓL (sn ))t′n
qn+1
In addition, let p′n∗ ≡ vL and define p′n∗ −1 , ..., p′1 recursively, so that p′n
=
Z
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 )), P • 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∗ +
Pn∗ −1
′ l=n+1 tl ,
then the players play as in Lemma C1 and q(t) = q ∗ .
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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): γL (s(t)) cH − p(t) q(t) = . 1 − q(t) γH (s(t)) vH − cH 13
(16)
• The low-type seller’s reservation price: p(t) = cL +
Z
t′
e−r(x−t) (cH − cL )d(1 − e−λ
Rx t
(1−ΓL (s(y))dy
) + e−(r+λ
R 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 n is given byoΓ , 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 q d (t) to travel from qi+1 to qid in the equilibrium associated with increment size d. Finally, let i∗d P 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 + where Tk =
Z
Ti−1
d
d
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 ), Ti
(20)
Pk
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 analysis of the three-type case discussed in Section 6 in the main text.
Setup Extending the notations for the two-type case, for each a = L, M, H, we denote by ca and va the stock values of the type-a asset to the seller and buyers, respectively, and by γa the probability that each buyer receives signal h from type a. There are always gains from trade (i.e., ca < va for each a), but a higher type is more valuable to both the seller and buyers (i.e., cL < cM < cH and vL < vM < vH ). In order to highlight the effects of adverse selection, we assume that vL < cM and vM < cH . We also assume that a higher type is more likely to generate signal h (i.e., γL < γM < γH ). For each a = L, M, H, we denote by qa (t) the probability that the seller is of type a, by pa (t) the reservation price of the type-a seller, and by σa (t) the probability that the buyer offers pa (t) at time t conditional on s = h. In addition, we let q(t) denote buyers’ beliefs at time t (i.e., q(t) ≡ (qL (t), qM (t), qH (t))) and Eq,s [v] denote the expected buyer value of the asset conditional on belief q and signal s.
Stationary Path ∗ ∗ We begin by identifying a belief vector q ∗ = (qL∗ , qM , qH ) that can generate a stationary path. Denote by p∗a the type-a seller’s reservation price, by σa∗ the probability that each buyer offers
p∗a conditional on signal h, and by σS∗ the probability that the low-type seller accepts p∗L on the stationary path. For the same reasons as in the two-type case, p∗L = vL (otherwise, qL (t) increases or decreases) and p∗H = cH (by the Diamond paradox). Then, the conditions for the stationary path equilibrium variables, each of which is a natural extension of the corresponding condition in the two-type case, are given as follows.
15
• The low-type and the middle-type sellers’ reservation prices: ∗ ∗ r(vL − cL ) = λγL (σH (cH − vL ) + σM (p∗M − vL )),
(22)
∗ r(p∗M − cM ) = λγM σH (cH − p∗M ).
(23)
• Belief invariance: ∗ ∗ ∗ ∗ ∗ ∗ ∗ γH σH = γM (σH + σM ) = γL (σH + σM ) + (1 − γL (σH + σM ))σS∗ .
(24)
• Buyers’ indifference over vL , p∗M , and cH conditional on signal h: 0 = qL∗ γL (vL − p∗L ) ∗ = qL∗ γL (vL − p∗M ) + qM γM (vM − p∗M )
(25)
∗ ∗ = qL∗ γL (vL − cH ) + qM γM (vM − cH ) + qH γH (vH − cH ).
(26)
For q ∗ to be well-defined, it is necessary that p∗M < vM (otherwise, qL∗ ≤ 0). In fact, this is also sufficient, because given p∗M (< vM ), all other variables can be explicitly derived from the above conditions and shown to be well-defined. The following lemma uses this observation to provide a necessary and sufficient condition under which q ∗ is well-defined. ∗ ∗ ) ≫ 0 which supports an equilibrium in , qH Lemma D1 There exists a belief vector q ∗ = (qL∗ , qM
which for all t ≥ 0, (i) q(t) = q ∗ and (ii) the probability of trade conditional on buyer arrival is strictly between 0 and 1 if and only if γM (cH − vM ) vM − cM . > vL − cL M γL cH − vL + γHγ−γ (v − v ) M L M
(27)
∗ ∗ Proof. In order to prove the necessity of condition (27), first observe that σM = (γH − γM )σH /γM (from (24)). Plugging this into (22) and combining it with (23) yield
γM (cH − p∗M ) p∗M − cM . = γH −γM vL − cL ∗ (p − v ) γL cH − vL + γM L M The solution p∗M to this equation is strictly less than vM if and only if (27) holds: the left-hand side is necessarily smaller than the right-hand side when p∗M = cM . In addition, the left-hand side is increasing, while the right-hand side is decreasing, in p∗M . Therefore, it is necessary and sufficient that the left-hand side is larger when p∗M = vM , which is equivalent to (27) in the lemma. Since 16
p∗M < vM is necessary for the existence of a vector q ∗ , (27) is also necessary. For sufficiency when λ is sufficiently large, we show that given (27) which ensures p∗M < vM , ∗ all other equilibrium variables are well-defined. From (23), σH = (r/λγM )((p∗M −cM )/(cH −pM )), which is positive because p∗M < vM and is less than 1 when λ is sufficiently large. From (24), ∗ ∗ ∗ σM = (γH − γM )σH /γM , which is well-defined when λ is sufficiently large because σH → 0 as ∗ λ → ∞. σM and γH > γM . (24) yields a unique value of σS∗ and also guarantees that σS∗ ∈ (0, 1), ∗ ∗ because γM > γL (which implies σS∗ > 0) and γH σH < 1 (which implies σS∗ < 1). Finally, qH can be obtained and similarly shown to be well-defined with equation (26).
If condition (27) fails, then q ∗ that solves the above stationary path conditions does not lie inside the probability simplex. In particular, qL∗ ≤ 0. This arises, for example, when γL is sufficiently small. In this case, the low-type seller is so unlikely to receive p∗M or cH that her reservation price p∗L falls short of vL and, therefore, qL (t) continues to decrease. It also happens when vM − cM is sufficiently small. In this case, buyers will have little incentive to offer pM (t). Equilibrium would be similar to the equilibrium in the model with the low type and the high type only.
Convergence Path Our main characterization result for the three-type case states that unless Eqb,l [v] ≥ cH (in which case it is an equilibrium that buyers always offer cH ), there is an equilibrium in which buyers’ beliefs conditional on no trade converge to q ∗ , starting from any initial belief.
Proposition D1 In the model with three seller types, suppose that condition (27) holds and λ is sufficiently large. For any qb such that Eqb,l [v] < cH , there exists an equilibrium in which q(t) converges to q ∗ . Path LH ∗ On path LH, the equilibrium behavior mimics that at qLH , which corresponds to the stationary belief in the model with types L and H only. Buyers randomize between cH and vL when s = h and offer vL when s = l. The low-type seller accepts vL with probability σS (t) ∈ (0, 1). Specifically,
σH (t) is pinned down by the requirement that pL (t) = vL : r(vL − cL ) = λγL σH (t)(cH − vL ) ⇔ σH (t) = σ ˜H ≡
17
r(vL − cL ) . λγL (cH − vL )
Given σ˜H and σS (t), q(t) evolves according to1 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 )) . It remains to find σS (t). We use the fact that for q(t) to stay on Path LH, q˙L γL (vL − cH ) + q˙M γM (vM − cH ) + q˙H γH (vH − cH ) = 0.
(28)
If γL σ ˜H + (1 − γL σ ˜H )σS (t) = γH σ ˜H , then the left-hand side in (28) 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 (28) 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 increas1
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 . qL (t)e−λ(γL σ˜H +(1−γL σ˜H )σS (t))dt + qM (t)e−λγM σ˜H + qH (t)e−λγH σ˜H
18
ing in σS (t), there exists a unique value of σS (t) that satisfies (28). 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
(29)
pM (t) evolves according to r(pM (t) − cM ) = λγM σH (t)(cH − pM (t)) + p˙ M (t).
(30)
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 (29) and (30) 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 ∗ to offer pL (t)). Notice that σH (t) decreases and converges to σH as q(t) approaches q ∗ (otherwise,
19
∗ pM (t) decreases, which ultimately violates equation (29)). 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
(31)
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 ).
(32)
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−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)). 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 ). 20
It remains to pin down qL (t∗ − ∆), qM (t∗ − ∆), and σS (t∗ − ∆). One of three necessary conditions is (31). 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 (31), 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. For Path LM, the optimality of buyers’ offers is straightforward, because Eq,h [v] < cH at any point on the path. Area L In Area L, buyers offer pL (t) regardless of their signal. Therefore, buyers’ beliefs evolve according to q˙L = −λqL (qM + qH ) = −λqL (1 − qL ), q˙M = λqM qL , q˙H = λqH qL . It is clear that no buyer would offer cH . We prove that in Area A, no buyer has an incentive to offer pM (t) as well. 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 ),
21
and
qM (t∗ ) qM (0) . = −λt∗ ∗ qL (0) qL (t ) e
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
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. 22
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 pL (t) ≥ vL and qL (t)γL vL + qM (t)γM vM pM (t) ≥ qL (t)γL + qM (t)γM on both Path LH and Path MH.
Appendix E: Alternative Bargaining Protocols In this appendix, we study the following three bargaining protocols. 1. Simultaneous announcement game: when the seller meets a buyer, the following normal form game is played: the buyer and the seller simultaneously choose whether to play T (tough) or S (soft). Their choices lead to the trading outcomes summarized in the following table: Buyer
Seller
S
S Trade at pM
T Trade at pL
T
Trade at pH
No trade
The prices pH ∈ (cH , vH ), pM ∈ (vL , cH ) and pL ∈ (cL , vL ) are exogenously given and fixed across different meetings. 2. Random proposals bargaining: in each meeting, nature draws a price from an exogenously given distribution function and proposes the price to both parties. Both the seller and the
23
buyer can decide only whether to accept the price or not. For tractability, we consider a simplified version with only two prices, pL ∈ (cL , vL ) and pH ∈ (cH , vH ).2 3. Price offers by the seller: the seller, who is the informed player, makes price offers to buyers (uninformed players). For the same reason as in our main model, we restrict attention to the case where 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 lowtype 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 h. Denote by σB∗ the probability that each buyer plays S when his signal is h on the stationary path. Finally, we denote by p∗ the low-type seller’s reservation price on the stationary path. As clarified shortly, p∗ does not have to coincide with vL , unlike in our main model. For the low-type seller to be indifferent between T and S, r(p∗ − cL ) = λγLh σB∗ (pH − p∗ ) = λ(γLh σB∗ (pM − p∗ ) + (1 − γLh σB∗ )(pL − p∗ )).
(33) (34)
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 between S (left) and T (right) conditional on s = h, h q ∗ γH (vH − pH ) + (1 − q ∗ )γLh (σS∗ (vL − pM ) + (1 − σS∗ )(vL − pH )) = (1 − q ∗ )γLh σS∗ (vL − pL ). 2
Compte and Jehiel (2010) and Lauermann and Wolinsky (2016) consider a more general case where p is drawn from a continuous distribution. The stationarity of their environments gives tractability to their analyses. The analysis becomes significantly more complicated in our non-stationary environment.
24
Arranging the terms, q∗ γLh σS∗ (pM − pL ) + (1 − σS∗ )(pH − vL ) = h . 1 − q∗ vH − pH γH
(35)
Finally, q(t) remains equal to q ∗ if and only if both seller types trade at the same rate. Since the high-type seller always plays T and the low-type seller trades for sure when she plays S, h ∗ γH σB = γLh σB∗ + (1 − γLh σB∗ )σS∗ .
(36)
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 (33) and (34): p∗ =
rcL + λγLh σB∗ pH rcL + λγLh σB∗ pM + λ(1 − γLh σB∗ )pL . = r+λ r + λγLh σB∗
The existence and uniqueness of a solution follows from the facts: the second term is strictly convex in σB∗ , while the last term is linear in σB∗ . In addition, the right-hand side exceeds the lefthand side when σB∗ = 0, while the opposite holds when σB∗ = 1. Given σB∗ , σS∗ can be obtained from (36). Given σS∗ , q ∗ follows from (35). 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 (33) and (34). High initial beliefs We now consider the case where qb > q ∗ . Again, the equilibrium structure is similar to that of the
main model. Both seller types play only T (analogously to accepting only cH ) until q(t) reaches 25
q ∗ . In order to describe buyers’ strategies, let q be the value such that l qγH (vH
− pH ) + (1 −
q)γLl (vL
q γLl pH − vL − pH ) = 0 ⇔ = l . 1−q γH vH − pH
If qb > q, then it is the unique equilibrium that all buyers play S. In this case, q(t) stays constant at qb. If qb < q, then buyers play S if and only if s = h. In this case, buyers’ beliefs decrease according to
h q(t) ˙ = −q(t)(1 − q(t))λ(γH − γLh ).
It remains to show that when q(t) ∈ (q ∗ , 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 . A necessary and sufficient condition for this incentive compatibility is γLl (pL − p(t)) + γLh (pM − p(t)) ≤ γLh (pH − p(t)), which is equivalent to γLh (pH − pM ) ≥ γLl (pL − p(t)). This inequality is guaranteed from equations (33) and (34) 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 (2016). 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 s q(t)γH (vH
− pH ) + (1 −
q(t))γLs (vL
γLs pH − vL q(t) ≥ s . − pH ) ≥ 0 ⇔ 1 − q(t) γH vH − pH
26
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 pH only when he receives signal h 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: γLh pH − vL q∗ . = h 1 − q∗ vH − pH γH • The low-type seller’s reservation price: r(pL − cL ) = λαγLh σB∗ (pH − pL ). • Belief invariance: h ∗ αγH σB = αγLh 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 ∗ . Let q be the value such that γ l pH − vL q . = lL 1−q γH vH − pH 27
It is then clear that each buyer’s optimal strategy is to accept pH regardless of his signal if q(t) > q and only when s = h if q(t) ∈ (q ∗ , q). Given this, the low-type seller’s reservation price exceeds pL and, therefore, trade occurs only at pH whenever q(t) > q ∗ . It then follows that q(t) remains equal to qb if qb > q and decreases according to
h q(t) ˙ = −q(t)(1 − q(t))λα(γH − γLh )
if q(t) ∈ (q ∗ , q). Notice that the only difference from the baseline model is the appearance of α in this expression.
Price Offers by the Seller We now study the case in which the seller makes a price offer to each arriving buyer. Due to the signaling nature of the seller’s price offers, this model admits a plethora of equilibria. For the purpose of the current exercise, instead of fully characterizing the set of all equilibria or delving into equilibrium selection, we construct a class of simple equilibria and show that they behave qualitatively in the same way as the unique equilibrium of our baseline model. Specifically, we restrict attention to equilibria in which the high-type seller always offers a fixed price p ∈ [cH , vH ).3 This significantly simplifies the analysis by effectively restricting the low-type seller to choose between p and vL at each point in time: it is strictly dominant for buyers to accept a price below vL , and thus the low-type seller can trade for sure if she offers vL . The high-type seller’s behavior implies that p is the only other price available to the low-type seller. A key observation for equilibrium characterization (and a distinguishing feature from our main model) is that the low-type seller must offer p and trade with a positive probability at all points in time. This is because in the current environment, a buyer updates his belief based not only on his signal, but also on the seller’s offer, and there does not exist a fully separating equilibrium. To be precise, let σS (t) denote the probability that the low-type seller offers vL . Then, the buyer’s belief conditional on offer p and signal s is given by q(t, p, s) ≡
s s q(t)γH q(t, p, s) q(t) 1 γH ⇔ = . s q(t)γH + (1 − q(t))(1 − σS (t))γLs 1 − q(t, p, s) 1 − q(t) 1 − σS (t) γLs
3
It is easy to support this behavior by appropriately specifying buyers’ off-the-equilibrium-path beliefs. For example, it suffices to assume that buyers believe that a seller who offers a price different from p is the low type for sure. If p = vH , then there can exist only fully separating equilibria: if the low-type seller offers p = vH with a positive probability, then the buyer has a strict incentive to reject it, which then eliminates the low-type seller’s incentive to offer p. It is easy to see that there are multiple equilibria. It is an equilibrium that buyers never accept p = vH and, therefore, the low-type seller offers only vL . There is also an equilibrium which can be interpreted as the limit of the equilibria characterized below (with p < vH ) as p tends to vH : in that equilibrium, the low-type seller never offers p = vH but buyers accept p with a positive probability. Any convex combination of these two equilibria is also an equilibrium, because all players face the same incentives in the two equilibria.
28
If the low-type seller never offered p (i.e., if σS (t) = 1), then the buyer would believe that the seller who offers p is the high type for sure (i.e., q(t, p, s) = 1) and, therefore, accept it with probability 1. If so, the low-type seller would strictly prefer offering p to vL , which is a contradiction. If p was rejected for sure, the low-type seller would have no incentive to offer p, which leads to the same contradiction. Stationary path:
Let q ∗ denote the stationary path belief. In addition, denote by σS∗ the probabil-
ity that the low-type seller offers vL and by σB∗ the probability that each buyer accepts p conditional on s = h on the stationary path. For similar reasons to those for the main model, these values can be derived from the following equilibrium conditions: • The low-type seller is indifferent between offering vL (left) and p (right), and thus rcL + λγLh σB∗ p rcL + λvL = . r+λ r + λγLh σB∗
(37)
• The buyer is indifferent between accepting and rejecting p conditional on s = h, and thus p=
h q∗ γLh p − vL 1 q ∗ γH vH + (1 − q ∗ )(1 − σS∗ )γLh vL ⇔ . = h h v −p 1 − q ∗ 1 − σS∗ q ∗ γH + (1 − q ∗ )(1 − σS∗ )γLh γH H
(38)
• The two seller types trade at an identical rate, and thus h ∗ γH σB = (1 − σS∗ )γLh σB∗ + σS∗ .
(39)
We note that q ∗ may or may not exceed the counterpart of our main model: in equation (38), if σS∗ = 0 (i.e., the low-type seller offers only p) and p = cH , then q ∗ is identical to that of our main model (see equation (4) in the main text). However, q ∗ increases in p and decreases in σS∗ . Intuitively, a buyer must be more optimistic about the quality of the asset if he needs to accept a higher price or the low-type seller must be more likely to offer p (less likely to offer vL ). Equilibrium dynamics:
and
Let q and q be the values such that
h vH + (1 − q)γLh vL qγH q γLh p − vL p= , = h ⇔ h 1−q qγH + (1 − q)γLh γH vH − p
(40)
l vH + (1 − q)γLl vL qγH γLl p − vL q ⇔ = , p= l l v −p 1−q qγH + (1 − q)γLl γH H
(41)
29
respectively. In words, q is the minimal prior belief such that the buyer is willing to accept p conditional on s = h even if the low-type seller offers p with probability 1 (i.e., σS (t) = 0) and q h l is the corresponding value conditional on s = l. Since σS∗ > 0 and γLh /γH < 1 < γLl /γH , it is clear that q ∗ < q < q. Suppose that q(t) < q. In this case, the low-type seller must offer vL with a positive probability, because otherwise q(t, p, h) < q and, therefore, the buyer would reject p for sure. In addition, the buyer must randomize between accepting and rejecting p so as to keep the low type indifferent between the two offers. The two indifference conditions imply that the buyer, conditional on s = h, accepts p with the same probability σB∗ as on the stationary path. In addition, σS (t) is determined by q γ h p − vL q(t) 1 . (42) = hL = 1 − q(t) 1 − σS (t) 1−q γH vH − p It then follows that q(t) evolves according to h
q(t + dt) =
∗
q(t)e−λγH σB dt , h ∗ h ∗ q(t)e−λγH σB dt + (1 − q(t))e−λ(σS (t)+(1−σS (t))γL σB )dt
which is equivalent to h ∗ q(t) ˙ = −q(t)(1 − q(t))λ(γH σB − σS (t) − (1 − σS (t))γLh σB∗ ).
(43)
Importantly, q(t) increases over time (i.e., q(t) ˙ > 0) if and only if q(t) < q ∗ , which is because the probability 1 − σS (t) that the low type offers p increases in q(t) by equation (42) and q(t) ˙ = 0 if q(t) = q ∗ . Now suppose that q(t) > q. In this case, the buyer accepts p (at least) conditional on s = h, regardless of the seller’s offer strategy. It is then optimal for the low-type seller to offer p with probability 1. In turn, this implies that the buyer’s acceptance strategy depends only on his prior q(t): by the definition of q, he accepts p only conditional on s = h if q(t) ∈ (q, q) and independent of s if q(t) > q. Clearly, q(t) decreases according to h
q(t)e−λγH dt h ⇔ q˙ = −q(t)(1 − q(t))λ(γH − γLh ) < 0 q(t + dt) = h h q(t)e−λγH dt + (1 − q(t))e−λγL dt in the former case and stays constant in the latter case. We summarize all the results so far in the following proposition. For conciseness, we present only how q(t) evolves over time in each region, omitting a lengthy but straightforward description of the equilibrium strategy profile corresponding to each qb. Proposition E1 Suppose that the seller makes price offers. For each p ∈ [cH , vH ), let q ∗ , q, and q 30
be the values defined by equations (38), (40), and (41). There exists an equilibrium in which the high-type seller always offers p and the following properties hold: • If q(t) > q, then trade occurs at p independent of s and q(t) stays constant. • If q(t) ∈ (q, q), then trade occurs only at p and if and only if s = h. In this case, q(t) h decreases according to q(t) ˙ = −q(t)(1 − q(t))λ(γH − γLh ). • If q(t) ≤ q, then the low-type seller offers vL with probability σS (t) and p with probability 1 − σS (t), and the buyer accepts p with probability σB∗ , where σB∗ and σS (t) are defined by equations (37) and (42). In this case, q(t) changes according to equation (43), which yields q(t) ˙ > 0 if q(t) < q ∗ and q(t) ˙ < 0 if q(t) > q ∗ . Proof. If qb ≤ q or qb > q, then the optimality of each player’s strategy is straightforward from the
construction of the equilibrium strategy profile. For the case when qb ∈ (q, q), let t∗ be the value such that h ∗ qbe−λγH t q = −λγ h t∗ h ∗. qbe H + (1 − qb)e−λγL t
Then, for any t < t∗ , the low-type seller offers only p, and the buyer accepts p with probability 1 conditional on s = h. If t ≥ t∗ , then the players play as in the case when qb ≤ q. In the latter region, the players’ incentives are straightforward: the low-type seller is indifferent between offering p and vL , and each buyer is indifferent between accepting and rejecting p conditional on s = h. Therefore, we focus on the former region. The optimality of the buyer’s acceptance strategy is immediate from the fact that q(t) ∈ (q, q) and σS (t) = 0 (which together imply that q(t, p, s) = q(t) ∈ (q, q)). For the seller’s optimality, let pL (t) denote the low-type seller’s continuation payoff at time t. Given the strategy profile, pL (t∗ ) =
rcL + λvL rcL + λγLh σB∗ p = r+λ r + λγLh σB∗
and for each t < t∗ , pL (t) = cL +
Z
t∗ −t
h
h
∗ −t)
e−rx (p − cL )d(1 − e−λγL x ) + e−(r+λγL )(t
(pL (t∗ ) − cL ) > pL (t∗ ),
0
where the inequality follows from σB∗ < 1 (i.e., buyers accept p with a lower probability after t∗ ). Conditional on facing a buyer at time t < t∗ , the low-type seller’s expected payoff is equal to γLh p + (1 − γLh )pL (t) if he offers p (because p is accepted only when s = h) and equal to vL if he offers vL (because vL is always accepted). It suffices to show that the former payoff is larger than
31
q(t)
q
q q∗
0
t
Figure 2: The evolution of buyers’ beliefs with the seller-offer bargaining protocol. The parameter values used for this figure are cL = 0, vL = 1, cH = 1.25, vH = 1.75, r = 0.25, λ = 0.8, h γH = γLl = 2/3, and p = 1.3. the latter payoff. The result follows from γLh p + (1 − γLh )pL (t) > γLh p + (1 − γLh )pL (t∗ ) > γL σB∗ p + (1 − γL σB∗ )pL (t∗ ) = vL where the second inequality is due to the fact that p > pL (t∗ and σB∗ < 1, while the equality derives from equation (37). Figure 2 illustrates how q(t) evolves in the equilibrium given in Proposition E1. Clearly, its dynamic patterns are similar to those of our main model. In particular, unless qb is sufficiently
large (above q), q(t) monotonically converges to q ∗ , whether from above or below. One notable difference is that q(t) converges to q ∗ only asymptotically. This is because the low-type seller’s trading rate, which in the baseline model jumps at q ∗ = q, varies continuously around q ∗ in the current model. As q(t) tends to q ∗ , the difference between the two seller types’ trading rates h ∗ (γH σB − σS (t) − (1 − σS (t))γLh σB∗ ) vanishes and, therefore, q(t) cannot reach q ∗ in finite time.
Appendix F: Competitive Market Structure In this appendix, we study the competitive market structure introduced in Section 6 in the main text. Specifically, we consider the case where a fixed number of buyers arrive simultaneously, observe a common signal, and offer prices competitively. 32
Let pa (t) denote the type-a seller’s reservation price at time t and ps (q) denote the expected buyer value of the good conditional on prior belief q and signal s, that is, ps (q) ≡
s qγH vH + (1 − q)γLs vL . s qγH + (1 − q)γLs
By the standard Bertrand competition logic, in equilibrium all buyers obtain zero expected payoff. This implies that trade can occur either at vL , if only the low-type seller is willing to trade, or at ps (q(t)), if both seller types are willing to trade. The following lemma provides a condition that distinguishes the two cases. Lemma 1 The competitive offer (i.e., bidding equilibrium outcome) is equal to ps (q(t)) if ps (q(t)) > pH (t) and equal to vL if ps (q(t)) < pH (t). Proof. Suppose that ps (q(t)) > pH (t) but there is a positive probability that the competitive offer is less than ps (q(t)) − ε for ε > 0. Then, a buyer can obtain a strictly positive expected payoff by bidding a price between max{ps (q(t)) − ε, pH (t)} and ps (q(t)), which cannot arise in equilibrium. Conversely, if ps (q(t)) < pH (t), then the winning buyer obtains a non-negative expected payoff only when his bid is less than vL , but the competitive bid cannot be strictly lower than vL . In the current competitive environment, ps (q(t)) > cH when q(t) is sufficiently large and, therefore, the high-type seller may obtain a positive expected payoff. This difference, however, does not qualitatively change the equilibrium structure. We demonstrate this by constructing an equilibrium that behaves just as the unique equilibrium of our baseline model. We first identify a unique stationary path q ∗ and then construct an equilibrium strategy profile in which, unless qb is sufficiently large, q(t) converges to q ∗ whether from above or below.
Stationary path: Given a stationary path belief q ∗ , the corresponding equilibrium behavior can be characterized as in our main model (in particular, see Section 3.1 in the main text), using the buyers’ indifference between ph (q ∗ ) and vL conditional on s = h, the low-type seller’s indifference between accepting and rejecting vL , and the equality of the two seller types’ trading rates. The only potential difference from the main model is that ph (q ∗ ) may not coincide with cH . This difference, however, does not materialize in equilibrium (i.e., ph (q ∗ ) = cH ) and, therefore, the corresponding equilibrium strategy profile is exactly identical to that of our main model (given in Lemma 2 in the main text). To see this, observe that on any stationary path, the high-type seller’s reservation price is determined by r(pH (t) − cH ) = λγH σB (t)(ph (q ∗ ) − pH (t)).
33
Therefore, ph (q ∗ ) > cH if and only if ph (q ∗ ) > pH (t). By Lemma 1, the competitive price conditional on s = h is ph (q ∗ ) with probability 1. But then, the low-type seller’s reservation price pL (t) exceeds vL , in which case q(t) cannot stay constant. Equilibrium dynamics: Given the unique stationary path, an equilibrium can be constructed as in the main model. If q(t) < q ∗ , then ph (q(t)) < cH , and thus only the low-type seller trades. In this case, q(t) increases according to q(t) ˙ = q(t)(1 − q(t))λ. The difference from our main model is that buyers’ (competitive) offers remain constant at vL along this path. Letting t∗ denote the time at which q(t) reaches q ∗ , the low-type seller’s reservation price at t < t∗ is given by pL (t) = cL +
Z
t∗ ∗ −t)
e−r(s−t) (vL − cL )d(1 − e−λ(s−t) ) + e−(r+λ)(t
(vL − cL ) < vL .
t
Therefore, the low-type seller has a strict incentive to trade until q(t) reaches q ∗ . There also exists q(> q ∗ ) such that if qb > q, then the seller trades with the first arriving buyers with probability 1. Unlike in the main model, the transaction price depends on the signal: it is
equal to ph (b q ) if s = h and equal to pl (b q ) if s = l. Clearly, this immediate trade outcome arises if l and only if pH (t) ≤ p (b q ). Since q is the lowest value with the property, it is characterized by the following indifference condition: pH (t) =
λ r l l h h p (q)) = pl (q). cH + (γH p (q) + γH r+λ r+λ
(44)
In our main model, when q(t) ∈ (q ∗ , q), trade occurs if and only if s = h. This is not the case in the current competitive environment. Suppose qb is slightly smaller than q. If trade were to occur only when s = h, then q(t) would reach q ∗ in finite time and pH (0) would be strictly smaller than pl (b q ). But then, Lemma 1 implies that trade must occur when s = l as well, which is a contradiction. On the other hand, trade cannot occur with probability 1 conditional on s = l, because if so, q(t) would stay constant and pH (0) > pl (b q ). The above arguments suggest that if q(t) is not significantly smaller than q, then trade must occur conditional on s = l with an interior probability. Lemma 1 implies that this can arise only when pH (t) = pl (q(t)), so that the buyers may optimally randomize between pH (t) and vL . In such a case, pL (t) > vL and, therefore, trade takes place only at pH (t). Let σBl (t) denote the probability that the competitive offer is equal to pH (t) conditional on s = l. Since trade occurs with probability 1 conditional on s = h and with probability σBl (t) conditional on s = l, q(t) evolves according to h
q(t + dt) =
l
l
q(t)e−λ(γH +γH σB (t))dt , h l l h l l q(t)e−λ(γH +γH σB (t))dt + (1 − q(t))e−λ(γL +γL σB (t))dt 34
which reduces to h q(t) ˙ = −q(t)(1 − q(t))λ(γH − γLh )(1 − σBl (t)).
(45)
Both pH (t) and pl (q(t)) decrease over time. When σBl (t) ∈ (0, 1), they decrease at the same rate and, therefore, stay identical. In fact, σBl (t) is determined so as to preserve the equality between pH (t) and pl (q(t)). Naturally, there exists q ∈ (q ∗ , q) such that σBl (t) ∈ (0, 1) if q(t) ∈ (q, q), while σBl (t) = 0 if q(t) ∈ (q ∗ , q]. The identification of q is rather involved and, therefore, relegated to the appendix. We summarize all the results in the following proposition. As in the case of Proposition E1, we present only how q(t) evolves, avoiding a full description of the equilibrium strategy profile for each qb.
Proposition F1 Suppose that the seller receives competitive price offers at Poisson rate λ and λγLh > r(vL − cL )/(cH − vL ) (i.e., Assumption 1 in the main text holds). Let q ∗ be the value such
h that q ∗ /(1 − q ∗ ) = γLh (cH − vL )/(γH (vH − cH )) (as in equation (2) in the main text) and q be the value defined by equation (44). There exist q ∈ (q ∗ , q) and an equilibrium such that the following properties hold:
• If q(t) > q, then trade occurs with the first arriving buyers and, therefore, q(t) stays constant. • If q(t) ∈ (q ∗ , q), then trade occurs with probability 1 conditional on s = h and with probability σBl (t) ∈ (0, 1) conditional on s = l. In this case, q(t) decreases according to equation (45). If q(t) ∈ (q, q) then σBl (t) ∈ (0, 1), while if q(t) ∈ (q ∗ , q] then σBl (t) = 0. • If q(t) < q ∗ , then trade occurs if and only if the seller is the low type and, therefore, q(t) increases according to q(t) ˙ = q(t)(1 − q(t))λ. • If q(t) = q ∗ , then trade occurs as in Lemma 2 in the main text and q(t) stays constant. Proof. Given the analysis above, we begin by identifying q. If q(t) ∈ (q ∗ , q), then trade occurs if and only if s = h. Let t∗ be the value such that h ∗
qe−λγH t
∗
q =
h ∗
h ∗
qe−λγH t + (1 − q)e−λγL t
⇔
q −λ(γ h −γ h )t∗ q∗ L H = . e ∗ 1−q 1−q
If qb = q, then pH (t) decreases according to
h r(pH (t) − cH ) = λγH (ph (q(t)) − pH (t)) + p˙ H (t) whenever t < t∗ .
If q is equal to q ∗ , then pH (0) = cH = ph (q) > pl (q). 35
On the other hand, if q is close to 1, then pH (0) <
h λγH vH + rcH < vH ≈ pl (q). h r + λγH
Therefore, there exists q ∈ (q ∗ , 1) such that pH (0) = pl (q). We define q to be the lowest such value, so that pH (t) > pl (q(t)) for any t. Given q, the unique equilibrium strategy profile for the case when qb ∈ (q ∗ , q) can be derived as in our main model (in particular, see Proposition 2 in the main text). As in the above case, the only difference is that the seller receives ph (q(t)), not cH , conditional on s = h. Next, we determine q. As explained in the main text, it is defined by equation (44). In order to show that q is well-defined, first notice that λ rcH + λvH r h h l l cH + (γH p (1) + γH p (1)) = < vH = pl (1), r+λ r+λ r+λ and
r λ h h l l cH + (γH p (q) + γH p (q)) > pl (q). r+λ r+λ
The second inequality comes from the fact that when qb = q, pH (0) = pl (q) and ph (q(t)) < ph (q)
for any t > 0. Finally, the above equation can be rewritten as
h ph (q) − cH r + λγH . = h pl (q) − cH λγH
The left-hand side takes an inverted-U shape over [0, 1]. By the above two inequalities, the lefthand side is larger if q = q and smaller if q = 1. Therefore, there exists a unique value of q ∈ (q, 1) that satisfies equation (44). It is clear that if qb > q, then it is the unique equilibrium that the competitive offer is always ps (b q ) and, since pH (t) < pl (b q ), both seller types trade whether the price is ph (b q ) or pl (b q ). Finally, we consider the case when qb ∈ (q, q). In this region (until q(t) reaches q), pH (t) = l p (q(t)) and q(t) decreases according to equation (45). Given pH (t) = pl (q(t)) when q(t) = q, it
suffices to show that whenever q(t) ∈ (q, q), there exists σBl (t) ∈ (0, 1) such that p˙ H (t) = p˙ l (q(t)). For notational simplicity, let ps = ps (q(t)). Then, −p˙l (q(t)) = pl′ (q(t))q(t) ˙ =
(vH − pl )(pl − vL ) h λ(γH − γLh )(1 − σBl (t)). vH − vL
36
while h −p˙H (t) = λγH (ph − pl ) − r(pl − cH ) =
h λγH (γH − γLh )(vH − pl )(pl − vL ) − r(pl − cH ). h l h vH − vL γLl γH (pl − vL ) + γH γL (vH − pl )
Obviously, −p˙ l (q(t)) < −p˙H (t) if σBl (t) = 1. We show that σBl (t) = 0 implies −p˙l (q(t)) > −p˙H (t) whenever q(t) ∈ (q, q). Define H(pl ) =
h (vH − pl )(pl − vL ) λγH (γH − γLh )(vH − pl )(pl − vL ) h λ(γH −γLh )− −r(pl −cH ). h l h vH − vL vH − vL γLl γH (pl − vL ) + γH γL (vH − pl )
q )). Define For our purpose, it suffices to show that H(pl ) > 0 whenever pl ∈ (pl (q), pl (¯ ˜ l ) ≡ γ l γ h (pl − vL ) + γ l γ h (vH − pl ) H(pl ). H(p L H H L ˜ is a cubic function of pl and, therefore, it has at most three roots. Now observe that Note that H(·) ˜ L ) < 0, H(p ˜ l (q)) > 0 (because pH (t) > pl (q(t)) whenever q(t) ∈ (q ∗ , q)), and limpl →∞ = H(v ˜ is a cubic function) imply that there are two roots less −∞. These (together with the fact that H ˜ H ) > 0 implies that there is the last root above vH . All together, than pl (q). In addition, H(v ˜ l ) > 0 whenever pl ∈ (pl (q), pl (¯ q )) ⊂ (pl (q), vH ), from which it follows that we know that H(p h l h H(pl ) > 0 whenever pl ∈ (pl (q), pl (¯ q )), because γLl γH (pl − vL ) + γH γL (vH − pl ) > 0.
Figure 3 illustrates equilibrium dynamics under the competitive market structure. It depicts how different prices evolve over time when qb ∈ (q, q).4 Until q(t) reaches q (i.e., until t1 in
the figure), pl (q(t)) and pH (t) coincide, which is possible due to buyers’ randomization between pH (t)(= pl (q(t))) and vL conditional on s = l, which slows down the fall of q(t) relative to the case when trade occurs only when s = h. Once q(t) falls below q (i.e., between t1 and t2 ), as in the baseline model, trade occurs if and only if s = h and, therefore, q(t) continues to decrease. During this period, ph (q(t)) falls faster than pH (t). They meet when q(t) reaches q ∗ (i.e., at t2 ),
after which the game unfolds just as on the stationary path of the baseline model: buyers randomize between pH (t)(= cH ) and vL conditional on s = h, the low-type seller accepts vL with a positive probability, and q(t) stays constant at q ∗ . 4
The evolution of prices when qb does not lie in (q, q) is analogous to that in the baseline model. For all cases, q(t) can be readily recovered from ps (q(t)).
37
ph (q(t)) pl (q(t)), pH (t)
ph (q ∗ ) = cH
pl (q ∗ ) vL 0
t1
t2
t
Figure 3: The evolution of pH (t) and ps (q(t)) under the competitive market structure when qb ∈ (q, q). The two dashed lines depict ph (q(t)) (upper) and pl (q(t)) (lower), while the solid line represents pH (t). The parameter values used for this figure are cL = 0, vL = 1, cH = 2, vH = 3, h r = 0.25, λ = 1, and γH = γLl = 2/3.
Appendix G: 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 ǫ 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 q γ cH − vL qR∗ 1 − γ cH − vL = , and = . = ∗ 1−q ǫ vH − cH 1 − qR 1−q 1 − ǫ vH − cH When ǫ 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 4 for a graphical representation.
38
q(t)
q(t)
q
∗ =q qG
∗ =q qR
q 0
t
0
t
Figure 4: 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.
Green Flag Since ǫ is sufficiently close to 0, λγLh = λǫ < ρL . Therefore, the equilibrium is given as in Section A in this online appendix. It then follows that ∗ q 1 − ǫ cH − vL qG = = . ∗ 1 − qG 1−q 1 − γ vH − cH
From the equilibrium structure, it is also clear that if ǫ 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 4 for a graphical representation.
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