Transparency and price formation

Kaya, Ayça; Liu, Qingmin Article Transparency and price formation Theoretical Economics Provided in Cooperation with: The Econometric Society Suggested Citation: Kaya, Ayça; Liu, Qingmin (2015) : Transparency and price formation, Theoretical Economics, ISSN 1555-7561, The Econometric Society, New Haven, CT, Vol. 10, Iss. 2, pp. 341-383, https://doi.org/10.3982/TE1566 This Version is available at: https://hdl.handle.net/10419/150252 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc/3.0/ Theoretical Economics 10 (2015), 341–383 1555-7561/20150341 Transparency and price formation Ayça Kaya Department of Economics, University of Miami Qingmin Liu Department of Economics, Columbia University We study the role that price transparency plays in determining the efficiency and surplus division in a sequential bargaining model of price formation with asymmetric information. Under natural assumptions on type distributions and for any discount factor, we show that the unobservability of past negotiations leads to lower prices and faster trading. Unobservability, therefore, enhances the “Coasian effect” by fostering efficiency and diverting more of the surplus to the player who possesses private information. In addition, we show that the equilibrium is unique and is in pure strategies in the nontransparent regime; this stands in sharp contrast to the existing literature and allows for a better understanding of the Coasian effect and price observability. Keywords. Coase conjecture, bargaining, durable goods monopoly, incomplete information, price formation, transparency. JEL classification. C61, C73, C78. 1. Introduction Sequential bargaining is not only a workhorse in analyzing bilateral interactions, with applications ranging from dispute resolution to labor contracting, but also a model of price formation and surplus division, which are of fundamental importance in economic theory. In dynamic trading environments, the details of information structures matter, because potential informational spillovers across players and over time introduce various channels through which incomplete information influences the price formation process. In bargaining games, a particular variation in the information structure is whether price offers are made publicly or in private. The goal of this paper is to investigate the effect of transparency in this sense on the price formation process in an important class of bargaining environments: Coasian bargaining. In our model, there is an impatient buyer and an infinite sequence of sellers. The first seller makes an offer to the buyer, which the buyer either accepts or rejects. If the Ayça Kaya: [email protected] Qingmin Liu: [email protected] We thank Martin J. Osborne, the anonymous referees, Yeon-Koo Che, Alexander Frankel, Brett Green, Hari Govindan, Philippe Jehiel, Tymofiy Mylovanov, John Roberts, Andrzej Skrzypacz, and Robert Wilson, as well as many seminar participants for helpful comments and suggestions. We thank Michael Borns for professional proofreading. Copyright ©2015 Ayça Kaya and Qingmin Liu. Licensed under the Creative Commons AttributionNonCommercial License 3.0. Available at http://econtheory.org. DOI: 10.3982/TE1566 342 Kaya and Liu Theoretical Economics 10 (2015) buyer accepts the seller’s offer, the game ends. If the buyer rejects the seller’s offer, the buyer moves to the next seller, who makes the buyer an offer. The game continues in the same fashion: if seller t’s offer is rejected, the buyer moves to seller t+1. The gains from trade are commonly known, but the buyer has private information about his willingness to pay. Formally, we amend the classic Coasian bargaining model with a sequence of sellers instead of one long-run seller. This model allows us to compare two different configurations of information flow among the short-run sellers. As in a standard Coasian model, with appropriate adjustments, our analysis remains valid when the roles and asymmetric information of the buyer and seller are switched. Several real-world markets—housing transactions, certain labor markets, corporate acquisitions, and overthe-counter derivatives trading—exhibit characteristics of this model with either a longrun informed buyer or a long-run informed seller. The observability of past rejected offers varies across these markets: among the examples listed, tender offers for corporate acquisitions are most often made publicly and are observed even when rejected, even though covert offers are not rare either. In housing markets and labor markets, offers are often covert, while rejected offers made public are not uncommon either. In over-thecounter markets as well, transactions are largely opaque and quotes are unobservable to subsequent traders (see, e.g., Zhu 2012).1Granted, all these markets have distinct characteristics and the market outcomes are the result of a complicated interaction of various institutional details. Our model, which is admittedly stark, captures one mechanism via which observability of past offers may impact the outcomes and, therefore, we believe that our results contribute to the understanding of such markets. We identify the role of price observability in determining the surplus distribution (measured by the equilibrium prices) and the efficiency of trade (measured by the amount of delay before trade takes place). More specifically, we compare the equilibrium price sequences and expected delay under two opposing specifications: one where past prices are observable to subsequent sellers (transparent regime)andonewhere they are not (nontransparent regime). Under natural restrictions on the distribution of the buyer’s valuations, we find that prices are uniformly lower in the nontransparent regime than in the transparent regime for each given discount factor of the long-run buyer. Moreover, even though an agreement is eventually reached in either regime, under stronger restrictions on the type distribution, we show that the expected delay is larger in the transparent regime. In the flip side of the model, when an informed longrun seller sequentially meets buyers who make offers, the transparent regime leads to lower prices and longer expected delay. All of our results are obtained for the arbitrary discount factors of the long-run buyer and not just for the case where the buyer is sufficiently patient. This feature makes our analysis of the effect of price transparency robust to market frictions, which, beyond its 1Indeed, even though the precise number of past quotes is unlikely to be known, the amount of time that traders spent searching for a particular deal might be observable. The same is true when an unemployed worker searches for jobs. Therefore, a deterministic arrival of players in our Coasian model is a plausible description of these markets. Theoretical Economics 10 (2015) Transparency and price formation 343 theoretical interest, is valuable in understanding interactions in real markets where frictions cannot be ignored. Indeed, for the frictionless limit, the outcomes of both regimes are degenerate: trade is efficient and the informed player captures all the surplus. This is consistent with the classic Coase conjecture. From this perspective, our results imply that, away from the frictionless limit, a lack of transparency enhances the “Coasian effect” by fostering efficiency and diverting more of the surplus to the informed player.2 Moreover, the comparison with the Coase conjecture implies that market frictions amplify the effect of transparency. In an infinite horizon bargaining game without parametric assumptions or closedform solutions, comparing equilibrium price paths in two different extensive forms is a rather challenging task. The observation that allowed us to make progress in this task comes from elementary demand theory: the comparison of equilibrium prices in two markets boils down to the comparison of demand elasticities in these markets. We identify conditions ensuring an appropriate demand elasticity ranking for our two dynamic bargaining environments. Our appeal to demand theory not only resolves the analytical difficulties, but also highlights the economic forces at play in dynamic bargaining problems. To gain some intuition, first recall the well known “skimming property” from Fudenberg et al. (1985): regardless of the regime and in any equilibrium, a price is accepted by the buyer if and only if his valuation is above an associated cutoff. This property allows one to interpret the buyer’s decisions as defining an endogenous demand curve that each seller faces in equilibrium, where the probability of trade at each price is interpreted as the quantity sold. Gul et al. (1986) point out that, with this interpretation, when both parties are long-run, their bargaining problem can equivalently be viewed as the problem of a durable goods monopolist lacking the power to commit to a price. In contrast to Gul et al. (1986), where a single durable goods monopolist competes with his future selves, in our model, a sequence of sellers compete with each other over time. Each seller faces a residual market characterized by a demand curve endogenously determined by the equilibrium strategies of all past and future sellers. With this interpretation at hand, the main exercise is to compare the demand curves faced by each seller in both regimes. First consider a hypothetical price change by seller 1. Transparency forces seller 2, who enters the game only when there is no trade in the first period, to respond with a price change in the same direction. In contrast, in the nontransparent regime, seller 2 cannot react to such a price change. In equilibrium, the buyer fully anticipates the reaction of seller 2 and, hence, is less sensitive to the price 2We believe this result may have something to add to the discussion of the design of certain markets. Since transparency affects both the surplus distribution and the speed of trading, the answer to the design question necessarily depends on the details of the markets and the designer’s objectives. If the market designer cares more about faster trading when there is common knowledge of gains from trade, then nontransparency should be preferred. For instance, in over-the-counter (OTC) markets, a trader often gets involved in many transactions and he is the more informed party on some and the less informed party on others. In this sense, perhaps the surplus distribution over different transactions might be averaged out for a given trader. In this case, maximizing the speed of transaction could be a plausible mechanism design objective. If this were the case, our results would give support to using a dark pool as opposed to an open-order book. 344 Kaya and Liu Theoretical Economics 10 (2015) change by seller 1 in the transparent regime. That is, the demand curve faced by seller 1 in the transparent regime is steeper than that in the nontransparent regime. However, the ranking of the slopes of the demand curves does not translate directly into the ranking of elasticities or the ranking of profit-maximizing prices. Indeed, since a seller’s demand curve is determined jointly by the strategic choices of all previous and future sellers, the relative positions of the two demand curves corresponding to the same seller in either regime is a priori unclear. Therefore, the above simple intuition is not enough given the subtleties of our problem. We show that a relatively benign regularity condition on the buyer’s type distribution—increasing hazard rate—pins down the relative positions of the demand curves and, therefore, allows an unambiguous comparison of prices in the two regimes. We next explore which regime leads to a larger delay in trade. Delay in the context of the bargaining model is related to the quantity traded in the analogous dynamic monopoly model with larger quantities corresponding to smaller delay. Typically, a less elastic demand curve implies a higher price, but, as is well understood in demand theory, elasticities alone do not determine the ranking of quantities. One needs to uncover additional details about the demand functions, which are endogenous equilibrium objects in our model. In spite of this, we are able to show that transparency entails more delay if the buyer’s type distribution is concave. Our contribution is not limited to the comparison of the two regimes. Even though the transparent regime in isolation is the focus of the Coasian bargaining and durable goods monopoly literature, the equilibrium characterization of the nontransparent regime is novel to this paper. We are able to show that the equilibrium outcome in the nontransparent regime is unique and is necessarily in pure strategies under the minimal assumption of increasing virtual valuation. This result, which does not require any genericity assumption, is in sharp contrast to the classic results of Coase bargaining where offers are publicly observable, in which case randomization can happen in the first period and may be necessary off the equilibrium path (see Fudenberg et al. 1985 and Ausubel et al. 2002). Gul et al. (1986) conjectured that a pure strategy equilibrium can be obtained for the no-gap case with continuous distributions, and they argued that this is one of the properties for an equilibrium to be “a salient predictor of market behavior.” Their conjecture remains open. The pure strategy property in our model is also surprising in view of the results on dynamic markets for lemons with unobservable offers where randomization is a generic property (see Hörner and Vieille 2009 and Fuchs et al. 2014). This equilibrium property allows for a characterization of the role of transparency that is not possible elsewhere. Related literature The role of observability has been investigated in different environments. Bagwell (1995) studies the connection between commitment power and observability and shows that the first-mover’s advantage can be eliminated if its action is not perfectly observed. Rubinstein and Wolinsky (1990) study random matching and bargaining. In their complete information environment, observability enlarges the equilibrium set by a folk theorem argument that is not at work in the presence of incomplete information. Swinkels Theoretical Economics 10 (2015) Transparency and price formation 345 (1999) analyzes a dynamic Spencian signaling model and obtains pooling equilibrium under private offers, while Nöldeke and van Damme (1990) previously obtained the Riley outcome in the case of public offers. Related to our study of the more standard Coasian bargaining with independent valuations is the strand of literature that studies bargaining with interdependent values; see Evans (1989), Vincent (1989), and Deneckere and Liang (2006). Our closest precursor is the work of Hörner and Vieille (2009), who study an interdependent-value model with a single long-run seller and a sequence of short-run buyers, and find that inefficiencies take different forms in the two opposing information structures. They show that in the hidden-offer case, multiple equilibria exist—all in mixed strategies—and an inefficient delay occurs even as the discount factor goes to 1, while in the public-offer case, remarkably, an inefficient impasse ensues beyond the first period. The question about the impact of price transparency on surplus division and the timing of trade for general discount factors is not addressed, however, and no clear-cut comparison of the two regimes in terms of price paths and the long-run player’s welfare is obtained. It is noteworthy that our independent-value model is not a limiting case of Hörner and Vieille’s model, and, hence, the qualitative divergence of results in terms of equilibrium structure, efficiency, and price comparisons is not completely surprising.3Beyond Hörner and Vieille’s (2009) initial exploration, the role of transparency has been extended to other settings. Kim (2012) presents a random matching model in which efficiency of trade may not be monotonic in the search friction. Bergemann and Hörner (2010) study the role of transparency on auction outcomes. In our model, efficiency is always obtained when the discounting friction vanishes. We emphasize that bilateral sequential bargaining, rather than other centralized mechanisms, is an appropriate model for thin markets in which trading opportunities do not arise frequently and, hence, discounting frictions are nonnegligible. Accordingly we focus on a comparison of price dynamics, surplus division, and the timing of trade in the two regimes that is robust to all discounting frictions, and this task requires new methods. We obtain an unambiguous comparison and show that the informed player has a clear-cut preference over nontransparent market information structure. Several bargaining models that feature discounting as a source of search friction are similar in structure to our model. Fudenberg et al. (1987) consider bargaining games where a seller can decide whether to switch to a new buyer or continue to bargain with an incumbent buyer. They show that a take-it-or-leave-it offer endogenously emerges as an equilibrium outcome. However, there are also other equilibria. The paper is organized as follows. Section 2 introduces the formal model. Section 3 considers a two-period example to demonstrate the forces driving our results. Section 4 establishes the existence and the uniqueness of equilibrium for the nontransparent regime. Sections 5and 6present our results concerning the comparison of prices 3The results of Hörner and Vieille (2009) rely crucially on the assumption that buyer–seller types are sufficiently interdependent and the discount factor is sufficiently large. Indeed, as the interdependence vanishes (i.e., the values of the uninformed short-run players become a constant), the lower bound required for the discount factor converges to 1, implying that the limiting case is not a well defined Coasian bargaining game. 346 Kaya and Liu Theoretical Economics 10 (2015) and speed of trade across two regimes, and Section 7 concludes. All omitted proofs are relegated to the Appendix. Additional material is available in a supplementary file on the journal website, http://econtheory.org/supp/1566/supplement.pdf. 2. Model A buyer bargains with a sequence of sellers. In each period t=12, a new seller enters the game. We refer to the seller at period tas seller t. Each seller has one unit to sell for which his reservation value is normalized to 0. The buyer has demand for one unit. The buyer discounts future payoffs at rate δ∈(01)and has private information about his valuation, v,whichwerefertoashistype. The prior cumulative distribution of buyer types is F, which has support [¯ v ¯ v],with¯ v>¯ v>0, and admits density f.We assume that there exists a constant m>0such that 1/m < f(v) < m for any v∈[ ¯ v ¯ v].4 Throughout, we assume that Fhas increasing “virtual valuation”; that is, v−1−F(v) f(v) is increasing This assumption is standard in the mechanism design literature. Bulow and Roberts (1989) point out that this assumption is equivalent to the monotonicity of the marginal revenue of a monopolist seller facing an inverse demand curve 1−F. The bargaining within each period tis as follows. Seller tproposes a price ptto the buyer. The buyer may choose to accept or reject this offer. If the price is accepted, the transaction takes place at this price and the bargaining game ends; the buyer obtains a payoff of δt−1(v−pt), while the seller tobtains a payoff of pt. If the price is turned down, seller tleaves the market and the game proceeds to period t+1. We refer to the information structure in which past rejected offers are observable to subsequent sellers as the transparent regime and the structure in which these offers are unobservable as the nontransparent regime. We consider the perfect Bayesian equilibria of the two specifications of the bargaining game.5The first thing to notice is that in both regimes, the “skimming property” is satisfied. That is, after any history, on or off the equilibrium path, if a price offer pis accepted by a type v, then it is also accepted by all types v>v. This allows us to cast 4That is, we focus on the so-called gap case; see Section 7 for additional discussion. We emphasize that the assumption that fis bounded below by a strictly positive number is important for the gap case. For instance, if F(v)=(v −¯ v)2/(¯ v−¯ v)2, then the model behaves like a no-gap case even though ¯ v>0. 5A perfect Bayesian equilibrium for finite games with observed actions is defined in Fudenberg and Tirole (1991, p. 333). The extension of this concept to our bargaining games is natural. A perfect Bayesian equilibrium in either regime specifies history-dependent sequences of the sellers’ price offers, the acceptance and rejection decisions of all buyer types, and the sellers’ beliefs about the buyer’s types such that the strategies are best responses given the beliefs. The beliefs are updated at each observable history from the strategies by Bayes’ rule whenever possible. After a seller’s deviation in the transparent regime, all future sellers’ beliefs are consistent with the buyer’s strategies upon this deviation; in the nontransparent regime, an off-path event occurs only when the buyer remains on the market beyond the last period in which trade happens with a positive probability on the equilibrium path. The off-path belief here turns out to be irrelevant. See also Hörner and Vieille (2009,p.33)forarelateddiscussion. Theoretical Economics 10 (2015) Transparency and price formation 347 the problem of each seller choosing a price as a problem of each seller choosing a cutoff type kto trade with (or a probability of trade).6 In the equivalent dynamic monopoly interpretation of Gul et al. (1986), a sequence of sellers face an inverse demand function 1−F. Each seller has unlimited supplies and can serve a fraction of the market at some transaction price. The game is prolonged not because all previous prices are rejected (in each period some prices can be offered and accepted), but because the market is not fully penetrated. 3. An example We first consider a two-period version of our model. For further simplicity, we assume that buyer types are uniformly distributed with support [01].7 By the skimming property, for each onor off-equilibrium-path history, there exists k1such that seller 2 believes that buyer types higher than k1trade with seller 1 and the remaining types are [0k1]. Since the second period is the final period, regardless of the regime, a remaining buyer type kaccepts seller 2’s offer p2if p2is below k. Therefore, seller 2’s problem in either regime can be cast as choosing p2=kto solve max kk(k1−k) Then, regardless of the regime, when the remaining types are [0k1], seller 2 charges a price p2=1 2k1and trades with buyer types [1 2k1k1]. The two regimes differ in the formation of beliefs off the equilibrium path: whereas an off-path price of seller 1 in the nontransparent regime does not affect the belief of seller 2, who cannot observe this deviation, it does do so in the transparent regime. To be more specific, in the nontransparent regime, seller 2 believes that the highest remaining buyer type is a fixed constant k∗ 1, even when the actual cutoff of seller 1 is different. Therefore, seller 2’s price in the second period is a fixed constant equal to 1 2k∗ 1. If seller 1 wishes to sell to types [k1], the highest price he can charge is p1(k) =(1−δ)k +1 2δk∗ 1(1) This is the price that makes the marginal type kindifferent between buying at a price p1(k) now and waiting until the second period for the constant price 1 2k∗ 1. Hence, seller 1 in the nontransparent regime solves the problem max k(1−k)p1(k) (2) 6See, for example, Fudenberg and Tirole (1991, p. 406). The proof for the skimming property does not rely on the assumption of observability; the crucial elements are price posting by the seller and single-unit demand by the buyer. In the nontransparent regime, it can be shown that the buyer uses a reservation price strategy, which is stronger than the skimming property. 7Even though our general model assumes ¯ v>0, assuming ¯ v=0greatly simplifies the computation in the example. The intuition highlighted in this example applies to the nontrivial gap case where trade takes more than one period to complete. 348 Kaya and Liu Theoretical Economics 10 (2015) Period 1 cutoff Period 2 cutoff Period 1 price Period 2 price Nontransparent 1 2(1−δ 4−3δ)1 4(1−δ 4−3δ)( 1 2−1 4δ)(1−δ 4−3δ)1 4(1−δ 4−3δ) Transparent 1 2 1 4 1 2−1 4δ1 4 Table 1. Comparison of the two regimes. Period 1 cutoff Period 2 cutoff Period 1 price Period 2 price Nontransparent 0000 Transparent 1 2 1 4 1 4 1 4 Table 2. Comparison of the two regimes as δ→1. It follows that k∗ 1=1 21−1 2 δ 1−δk∗ 1 Hence, k∗ 1=1 21−δ 4−3δp ∗ 1=1 2−1 4δ1−δ 4−3δp ∗ 2=1 41−δ 4−3δ In contrast, in the transparent regime, if seller 1 sells to buyer types [k 1]for any k, seller 2 will correctly anticipate the remaining types to be [0k]and charge a price 1 2k accordingly. Moreover, seller 2’s response of setting price 1 2kto seller 1’s deviation is fully anticipated by the buyer, implying that the highest price that seller 1 can charge and sell to buyer types [k 1]is p1(k) =(1−δ)k +1 2δk =k1−1 2δ(3) Now, seller 1’s problem is given by (2), where p1(k) is specified by (3). Simple algebra shows that k∗ 1=1 2p ∗ 1=1 21−1 2δp ∗ 2=1 4 In this example, the demand curves faced by seller 1 are linear (where the quantity sold is 1−k). The demand curve of the nontransparent regime, (1), is flatter than the demand curve of the transparent regime, (3). We summarize our finding in Table 1. The contrast becomes more apparent if we take δ→1,asshowninTable 2. This example illustrates the following qualitative results which we generalize later. The prices are uniformly higher in the transparent regime and, hence, the lack of transparency diverts more surplus to the informed long-run buyer. In addition, the expected delay in trade, that is, the expected value of 1−δτ(k),whereτ(k) is the period in which type ktrades, is higher in the nontransparent regime and, hence, the lack of transparency fosters efficiency. Theoretical Economics 10 (2015) Transparency and price formation 355 Notice that the two results, NTR ≥TR and kTR 1(pTR 1)≤kNTR 1(pTR 1), would immediately imply (4)if,say,Fwere the uniform distribution. However, for an arbitrary F,the ranking of the numerators in (4) is not clear, simply because the relevant intervals of discouraged types typically do not share end points and, therefore, a smaller-sized interval (TR) can pack a larger measure under Fthan can the larger interval (NTR). This would imply that the price increase to pNTR 1would reduce the quantity sold by seller 1 in the transparent regime by a larger absolute amount. Then the ranking of the percentage changes would be ambiguous. This ambiguity is resolved for Fsatisfying the increasing hazard rate property. Now, the increasing hazard rate property is precisely that the quantity (F(k +) −F(k))/(1−F(k)) is increasing in k(see Lemma 11 in Appendix B). This quantity is also increasing in by the monotonicity of F. Therefore, (4) follows from the two observations (i) NTR ≥TR and (ii) kTR 1(pTR 1)≤kNTR 1(pTR 1)for Fsatisfying the increasing hazard rate property. To summarize, (4) means that the percentage change in quantity in response to a given price change (from pTR 1to pNTR 1) is larger in the nontransparent regime; i.e., at this range of prices, the demand curve faced by the monopolist in the nontransparent regime is more elastic. Nevertheless, this monopolist strictly prefers the higher price pNTR 1to the lower price pTR 1, since pNTR 1is the unique solution to his profit-maximizing problem. There, the monopolist in the transparent regime, facing a less elastic demand, should also have the same preference over these two prices, which contradicts the optimality of pTR 1. This establishes that pTR 1≥pNTR 1. Since the price paths in the two regimes that the long-run buyer faces are uniformly ranked, it follows immediately that the buyer has a clear-cut preference over the two regimes. Corollary 1. Fix any δ∈(01). Suppose that Fexhibits an increasing hazard rate. Then the long-run buyer is better off in the nontransparent regime. 6. Expected delay As is well understood in the Coasian bargaining literature, as the buyer becomes extremely patient, the outcome becomes efficient. However, the literature so far has had little to say about the delay and efficiency when δis bounded away from 1.Instead,the literature has focused on the limiting case of δ→1to study “real delay” in various environments. Studying the equilibrium outcomes in the limiting case is not only conceptually important for our understanding of commitment power but also facilitates definite conclusions, such as the limiting efficiency result for the gap case. Yet, to understand fully the applications in real-market environments, it is necessary to consider discount factors that are bounded away from 1. This section is concerned with the question of which regime leads to a longer delay in trade for any buyer discount factor δ∈(01). Under our interpretation, which identifies the probability of sale with the quantity sold by a residual monopolist, the expected delay in sale is smaller if the sales are more “front-loaded,” that is, if earlier sellers cover a larger share of the market. In the example of Section 3, a comparison of the two regimes in this dimension was immediate from 356 Kaya and Liu Theoretical Economics 10 (2015) Figure 3. Representative demand curves of seller 1 for the infinite horizon model. Note that the two demand curves intersect at a price lower than the first-period equilibrium price of the unobservable regime. This is because when the horizon is longer than two periods, the game starting in the second period onward is no longer identical across the two regimes. the fact that the first seller in the nontransparent regime serves a larger share of the market (targets a smaller cutoff type). In the general model, however, the comparison of quantities sold by seller 1—or subsequent sellers, for that matter—in the two regimes is not possible and should not be expected. This is because the demand curves faced by seller 1 in either regime typically are related to each other in the manner shown in Figure 3. Therefore, even though the ranking of the elasticities is possible, it implies only that seller 1 in the transparent regime chooses a price above pNTR 1, and not necessarily above ¯ p. Intuitively, when the game has a longer horizon, since the future prices are also expected to be lower in the nontransparent regime, the buyer may have a stronger incentive to wait, reducing the incentives of the seller to increase prices. Nevertheless, we are able to establish the result for the general model under the additional assumption that the buyer’s type distribution is concave. To formally state our result, let {kTR t}be a realization of an equilibrium cutoff sequence in the transparent regime and let {kNTR t}be the unique equilibrium cutoff sequence in the nontransparent regime with the convention that ki 0=¯ vand ki t=¯ vfor t>T i,whereTiis the last period such that trade takes place with positive probability along the realized path. Given these sequences, for each type v<¯ v, and for either i=NTRTR, there is a unique tsuch that ki t−1>v≥ki t.Letτi(v) represent this t. Then a measure of the delay that type vexperiences is 1−δτi(v)−1, which is the portion of the payoff lost due to the delay in reaching an agreement. Therefore, the expected delay in regime iis ¯ v ¯ v (1−δτi(v)−1)dF(v)=1−¯ v ¯ v δτi(v)−1dF(v) Theoretical Economics 10 (2015) Transparency and price formation 357 Notice that, ex ante, the probability that the trade will take place at period tis F(ki t−1)− F(ki t). Therefore, the above expectation can alternatively be expressed as 1− ∞  t=1 δt−1(F(ki t−1)−F(ki t)) which simplifies to (1−δ) ∞  t=1 δt−1F(ki t) (5) Proposition 2. Fix any δ∈(01). Assume that pTR t≥pNTR tfor any t,where{pTR t}is a realization of the equilibrium price sequence of any equilibrium in the transparent regime and {pNTR t}is the unique equilibrium sequence of prices in the nontransparent regime. Then, for a concave F, the expected delay in the transparent regime is larger than the expected delay in the nontransparent regime. The complete proof is presented in Appendix C. To gain some intuition into how the ranking of prices helps and what role the concavity of the type distribution Fplays, note that for each i,wehave 15 pTR t=(1−δ) ∞  l=t δl−tkTR l≥(1−δ) ∞  l=t δl−tkNTR l=pNTR t(6) In words, the discounted sum of the tails of the cutoff sequence from period ton (which is equal to the price in that period) is larger for the transparent regime than for the nontransparent regime. It is clear that when Fis applied to each ki tto obtain the expression for the expected delay in (5), this ranking need not be preserved. Proposition 2 shows that this ranking is preserved when Fis concave. The intuition can most easily be gleaned from the following thought experiment: suppose that in each regime, trade takes place in the second period at the latest. Then (6)impliesthat (1−δ)kNTR 1+δkNTR 2≤(1−δ)kTR 1+δkTR 2and kNTR 2≤kTR 2 This means that one of the following two rankings must hold: (i) kNTR 1≤kTR 1and kNTR 2≤kTR 2. (ii) kNTR 1>k TR 1≥kTR 2≥kNTR 2. 15Since a pure strategy is played on the equilibrium path after period 1 (Fudenberg et al. 1985), for any t≥1, the indifference condition of the cutoff buyer type ktis given by pi t=(1−δ)ki t+δpi t+1 where pi t+1is deterministic. Iterating this indifference condition, we obtain that prices are discounted sums of the cutoff types. Indeed, this is the only equilibrium condition that is invoked in the proof. Proposition 2 holds for all ranked sequences of prices for which the corresponding sequences of cutoff types are decreasing. 358 Kaya and Liu Theoretical Economics 10 (2015) If the ranking in (i) obtains, (5) follows immediately from the monotonicity of Fwithout referring to concavity. Under the ranking in (ii), the cutoffs in the transparent regime are “less spread out”—as well as, on average, higher—than those in the nontransparent regime, which implies that, evaluated under a concave and increasing function F,their expectation is larger. To see how this intuition is generalized to longer horizons, consider an alternative interpretation of our model as a bargaining game with a stochastic deadline: suppose that δ, instead of representing the discount factor of the buyer, represents the probability with which the bargaining ends before the next period, conditional on the fact that it has not yet ended. It is well understood that such a game is strategically equivalent to the game we have analyzed so far. Then “the smallest buyer type that gets to trade” in either regime is a random variable assigning probability δt−1to ki t. If the realization of this random variable is k, then the realized probability (or quantity) of sale is 1−F(k). Under this interpretation, 1 minus the expression in (5) is the expected quantity sold or, equivalently, the expected probability of sale in either regime. What (6) allows us to show is that the random variable determining the smallest buyer type that trades in the transparent regime second-order stochastically dominates its nontransparent regime counterpart. That is why the expectation of this random variable evaluated at concave F is larger in the transparent regime, and so there is a lower expected probability of trade or, equivalently, a higher expected delay, when past rejected prices are observable. 7. Concluding remarks To conclude, we emphasize several aspects of our model that we feel deserve further elaboration. We can compare our model with classical models of oligopoly. In particular, the bargaining model where previous prices are observable to future sellers is reminiscent of the “price leadership” in oligopoly, as later sellers observe and react to choices that earlier sellers have “committed to.” The nontransparent regime, alternatively, is suggestive of “Bertrand competition,” because each seller makes a price choice without observing the choices of any other seller. Obviously, the analogy requires that the products of various sellers be vertically differentiated, since in either regime, for a given price, each type of buyer prefers to buy from an earlier seller rather than wait for a later seller—because of discounting. However the analogy is superficial. The extensive form of our sequential bargaining game implies that later sellers face an endogenously determined residual market, while in the standard oligopoly literature, there is a single demand curve common to all sellers. The implication of this is that a given seller in the oligopoly model, by lowering his price, can steal buyers from higher quality (earlier) sellers as well as lower quality (later) sellers, whereas in the dynamic bargaining model, the buyers of higher quality (earlier) sellers are locked in, so that a deviation to a lower price can lure buyers only from lower quality (later) sellers. This discrepancy creates very distinct incentives to deviate and leads to different equilibrium outcomes. A second implication of this distinction is that, while in our model, quantity competition is equivalent to price competition, this is not true in an oligopoly model. Therefore, even though the forces we Theoretical Economics 10 (2015) Transparency and price formation 359 uncover that lead to the comparison of the two regimes in our model are also present in models of oligopoly, they are obscured by other mechanisms and, therefore, are not highlighted in that literature. In fact, to our knowledge, the industrial organization literature has not studied analogous questions to those we asked in this paper. In addition, most papers look at simple models with closed-form solutions. Our result might indeed shed light on the oligopoly literature. Another “competitive” benchmark with which we can contrast the performance of the two regimes is one where there is “within-period competition” between the short-run sellers; that is, the buyer meets two (or more) sellers each period and price is determined by competitive bidding. In this case, it is easy to see that the prices will immediately be zero in either regime. Therefore, our results suggest that the nontransparent regime leads to outcomes that are “closer” to this benchmark. Rather than studying within-period competitive bidding, which would more reasonably describe a “thick” market, our purpose in studying one-to-one bargaining is precisely to unlock the strategic aspect of price formation in a thin market. We consider take-it-or-leave-it offers by the uninformed players. Introducing different bargaining protocols would be an interesting exercise. If the informed buyer makes all the offers, in both regimes an equilibrium is one where the buyer always offers the price 0and all sellers accept it. In this equilibrium, the buyer gets all the surplus. This is intuitive. In the complete information of this game, when the long-run buyer is making offers, in the unique equilibrium he would always offer 0regardless of his valuation. If, in addition, the buyer’s type is his private information, he cannot do worse than in the case of complete information when his type is known. This observation is made by Ausubel and Deneckere (1989b, Theorem 4). Indeed, they show that this is the unique equilibrium. This suggests that in our context, the uninformed player must have some bargaining power for the impact of transparency to be present. If the bargaining protocol is such that both parties can make offers, then the game becomes complicated due to the signaling effect. The literature makes strong refinement assumptions. Ausubel et al. (2002, Theorem 7) show that Coase conjecture holds in alternating offer bargaining games under a refinement they call “assuredly perfect equilibrium.” This refinement requires that when an off-equilibrium action is observed, the players believe that it is more likely to come from low buyer types, giving a strong structure to the off-path beliefs. We believe the same logic works in our transparent regime. Similarly, two-sided uncertainty will also give rise to signaling issues; see, for example, Cramton (1984), Chatterjee and Samuelson (1987), and Abreu and Gul (2000). Our bargaining model corresponds to the “gap” case—well known in the bargaining literature. In fact, the Coase conjecture fails in the no-gap case with short-run sellers: we can employ the insights of Ausubel and Deneckere (1989a) to construct multiple nonstationary equilibria, where the long-run buyer builds a “reputation” for having a low willingness to pay. Nevertheless, the gap case nicely captures the insights of imperfect competition by silencing the reputation effect. Some interesting questions within our exact model remain unresolved as well. In particular, we do not consider the ranking—across the two regimes—of ex ante sums of 360 Kaya and Liu Theoretical Economics 10 (2015) expected payoffs or of cutoff types that trade each period. If we were to be able to establish the ranking of cutoffs, the ranking of the discounted sum of payoffs would immediately follow. Yet, this is a challenging exercise: To obtain a ranking of the cutoffs and, hence, the sum of payoffs, one needs to be able to reason about the magnitude of price differences/ratios across regimes—and not only their ordinal rankings. As the price path is an endogenous object in an infinite horizon model, this becomes rather involved and our current proof strategies do not work. In the supplementary material, we present our results on the special cases of power function distributions and two-period models for general type distributions. In these two cases, we are able to obtain unambiguous rankings of cutoffs and, therefore, efficiency. In addition, we show in these two cases that we can drop the concavity assumption in Proposition 2. Moreover, in the two-period case, we are able to dispense with the increasing hazard rate condition for Theorem 2.16 Appendix A: Proof of Theorem 1 We first prove that any equilibrium must be in pure strategies (Lemma 1). We then apply this property to prove equilibrium existence and uniqueness. By virtue of the skimming property (Fudenberg and Tirole 1991, p. 407) and the fact that the only observable history in the nontransparent regime is rejection, in any equilibrium we can identify seller t’s price ptwith the infimum buyer type who accepts pt. For any fixed equilibrium, since the seller in period tcan potentially randomize, let Kt be the support of the cutoffs in period tin this equilibrium and write K= T  t=1 Kt For each t,define ¯ kt:= supKtand ¯ kt:= infKt Hence, after period t, the largest possible interval of remaining types is [¯ v ¯ kt), while the smallest such interval is [¯ v ¯ kt).Define ¯ k t:= supKt\{¯ kt} By convention, sup ∅=−∞. Therefore, [¯ v ¯ k t)is the second largest possible interval after a (potential) seller randomization in period t. It is possible a priori that ¯ k t=¯ kt.All these variables just defined depend on the fixed equilibrium. We suppress the dependence for notational convenience. A.1 Preliminary results In our dynamic environment, the distribution of types varies over time. We first make the observation that a lower truncation of Finherits the monotone marginal revenue (virtual valuation) property from F. 16Note that power function distributions satisfy the monotone hazard rate assumption. Theoretical Economics 10 (2015) Transparency and price formation 361 Lemma 2. Assume that k−(1−F(k))/f (k) is strictly increasing. Then k−(α−F(k))/f (k) is strictly increasing in kwhenever F(k)<α≤1. Proof.Considerk<kand F(k)<α.Wewanttoshow k−α−F(k) f(k) >k −α−F(k) f(k)(7) Define L(α) =k−k+F(k)−F(k) f(k)−(α −F(k))1 f(k) −1 f(k) Inequality (7) is equivalent to L(α) > 0. Notice that L(1)>0since Fhas increasing marginal revenue (virtual valuation). For α<1,wehavetwocasestoconsider:if 1/f (k) −1/f (k)≤0,thenL(α) > 0follows immediately by the definition of L(α);if 1/f (k) −1/f (k)>0, then L(α) is decreasing in αand, hence, L(α) > L(1)>0. By standard arguments, in any equilibrium, ¯ kt≥¯ vfor any tand the game ends in finite time with a price equal to ¯ v.ThisisformalizedinLemma 3. Lemma 3. In any equilibrium of the nontransparent regime, there exists 0<T<∞such that trade takes place with probability 1within Tperiods. Proof. We proceed in the following steps. Step 1. A seller never makes a price offer below ¯ v. The argument is standard: all buyer types will accept a price of (1−δ)¯ vimmediately, which is better than waiting for a price of 0next period; but then (1−δ2)¯ vwill be accepted for sure because the best price in the next period is bounded below by (1−δ)¯ v; iterating this argument shows that a seller will never make a price offer below (1−δn)¯ vfor any n, and the claim follows. Step 2. Suppose to the contrary that there is an equilibrium in which there is some positive measure of types that never trade for some history. Then in this equilibrium, ¯ kt>¯ vfor all t>0.Inthiscase{¯ kt}is a decreasing and, hence, convergent sequence: if, however, ¯ kt<¯ kt+1for some t, then seller t+1makes a profit of 0by making an offer close to ¯ kt+1; but he can make a strictly positive profit by offering ¯ vaccording to the previous step. Consequently, |¯ kt−¯ kt+1|→0. Thus, the profit of seller tconverges to 0as t→∞. Moreover, it must be that ¯ kt↓¯ v. To see this, suppose that limt→∞ ¯ kt=k∗>¯ v. Then seller twillgetaprofitcloseto0if tis large enough, but any seller can deviate to charge a price ¯ v, which, by the previous claim, guarantees a strictly positive profit (F(k∗)−F(¯ v))¯ v,a contradiction. Step 3. Now from Step 2, for each ε>0,thereexiststsuch that ¯ v< ¯ kt<¯ v+ε.Then we claim that there exists εsuch that for any k∈(¯ v ¯ v+ε) and any k∈(¯ vk], (F(k) −F(k))k<F(k) ¯ v (8) 362 Kaya and Liu Theoretical Economics 10 (2015) To see this, note that the left-hand side is differentiable in kand its derivative is −f(k)k+F(k)−F(k).Now −f(k)k+F(k)−F(k)<−1 m¯ v+F(k)−F(k) <−1 m¯ v+F(¯ v+ε) −F(¯ v) <−1 m¯ v+mε Hence, when ε<¯ v/m2, then −f(k)k+F(k)−F(k)<0,and(8) follows immediately. Step 4. Notice that the left-hand side of (8) is the highest possible payoff a seller can obtain when facing buyer types [¯ vk]if he wants to sell to the types [kk](it assumes that a price equal to kwill be accepted by all types above k), while the right-hand side of (8), by Step 1, is the seller’s exact payoff from making a price offer ¯ v. Therefore, (8) implies that if ¯ kt<¯ v+¯ v/m2, it is an ex post strictly dominant strategy for seller t+1 to make a price offer equal to ¯ vfor each realization of kt∈(¯ v ¯ kt]. Therefore, kt+1=¯ v is an ex ante strictly dominant strategy for seller tas long as ¯ v< ¯ kt<¯ v+¯ v/m2.This contradicts the supposition that ¯ kt+1>¯ vfor each t. We next argue that the upper bound of the support of a seller’s potential randomization in the fixed equilibrium is strictly decreasing over the periods during which trade takes place with positive probability. Lemma 4. In any equilibrium in which Tis the last period in which trade takes place with a positive probability, we have ¯ v> ¯ kt>¯ kt+1for any t<T. Proof.If ¯ kt≤¯ kt+1, then seller t+1gets 0profit. He can get positive profit by charging ¯ v. Moreover, if ¯ kt=¯ v, then seller tcan charge ¯ vand get a strictly higher profit.  A.2 Pure strategy: Proof of Lemma 1 Lemma 5. In any equilibrium, K1∩[¯ k2¯ k1]={¯ k1}. Proof. To prove this claim, note that by the definition of ¯ k2, the buyer type k∈[¯ k2¯ k1] is guaranteed to trade at or before period 2. Therefore, by choosing a marginal type k∈[¯ k2¯ k1], seller 1 would sell with probability 1−F(k).Thepricep1(k) is such that the marginal type k, who will buy for sure next period, is indifferent between buying now or waiting, k−p1(k) =δ(k −E[p2]) where E[p2]is the expected price in period 2 (seller 2 could potentially randomize). Hence, p1(k) =(1−δ)k +δE[p2]and, therefore, seller 1’s problem is max k(1−F(k))[(1−δ)k +δE[p2]] Theoretical Economics 10 (2015) Transparency and price formation 363 The first-order derivative of the objective function can be calculated to be −f(k)(1−δ)k−1−F(k) f(k) +δE[p2](9) Since k−(1−F(k))/f(k) is strictly increasing by assumption, (9) is strictly increasing over the interval [¯ k2¯ k1]. Now, since ¯ k1maximizes seller 1’s profit (or types arbitrarily close to ¯ k1if ¯ k1=supK1is not achieved by any k∈K1)and ¯ k1is in the interior of [¯ v ¯ v], it must be that (9)is0at k=¯ k1. Moreover, since (9) is strictly increasing, it must be negative for any k∈[¯ k2¯ k1).Hence,nok∈[¯ k2¯ k1)is optimal. Therefore, K1∩[¯ k2¯ k1]= {¯ k1}. Lemma 5 does not imply that seller 1 must play a pure strategy. It does not rule out thecasethatK1contains points that are not in [¯ k2¯ k1],thatis,thecaseK1\[¯ k2¯ k1]=∅. However, we are able to successively narrow down K1.ThisisdoneinLemma 1 of the main text, which is repeated as follows. Lemma 1. Fix any equilibrium in which the game ends for sure at T. For any τ= 1T −1,(τ t=1Kt)∩[¯ kτ+1¯ k1]={¯ k1¯ k2 ¯ kτ}. Proof. The proof is by induction. We proceed in the following steps. Step 1. First note that K1∩[¯ k2¯ k1]={¯ k1}.ThisiswhatweprovedinLemma 5.This step shows that ¯ k1is an isolated point in K1. Step 2. Next we argue that for 1≤τ+2≤T,if τ  t=1 Kt∩[¯ kτ+1¯ k1]={¯ k1¯ k2 ¯ kτ}(10) then τ+1  t=1 Kt∩[¯ kτ+2¯ k1]={¯ k1¯ k2 ¯ kτ+1} In words, we want to show inductively that ¯ ktis an isolated point in the support of seller t’s cutoffs and no seller will ever set a cutoff in the interval (¯ kt¯ kt+1). The induction step is illustrated in Figure 4. From the induction hypothesis, (τ t=1Kt)∩[¯ kτ+1¯ k1]={¯ k1¯ k2 ¯ kτ}. Recall that ¯ k t=supKt\{¯ kt}.Takethesmallest t∗such that ¯ k t∗=sup{¯ k t|t=1τ+1}.Thatis, ¯ k t∗is the highest among the “second highest equilibrium cutoffs” in periods up to τ+1. By the induction hypothesis, for any t≤τ, ¯ k t∗≤¯ kτ+1<¯ kt(11) If Kt\{¯ kt}=∅for all t=1τ+1, then the proof is complete already. Suppose that this is not the case. If ¯ k t∗<¯ kτ+2, then the induction is complete as well. Now suppose that ¯ k t∗≥¯ kτ+2. Step 3. We establish the following claims. 364 Kaya and Liu Theoretical Economics 10 (2015) Figure 4. The left panel depicts the induction hypothesis; the right panel depicts the induction step. Claim A.1. There exists ε>0such that (¯ k t∗¯ k t∗+ε) ∩K=∅. That is, there is no (future or past) cutoff immediately above ¯ k t∗.Hence,inanyperiodtafter any history, buyer types [¯ k t∗¯ k t∗+ε) must be either entirely in the support of the posterior or entirely outside of the support of the posterior. Proof.By(11), we have either ¯ k t∗=¯ kτ+1or ¯ k t∗<¯ kτ+1.By(11), in the former case we have ¯ kτ+2<¯ k t∗=¯ kτ+1; in the latter case, we have ¯ kτ+2≤¯ k t∗<¯ kτ+1 It follows immediately from Lemma 4 that there is no offer (buyer cutoff) within (¯ k t∗¯ k t∗+ε) in all periods t=1T. Claim A.2. There exists ε>0such that (¯ kt∗−ε ¯ kt∗+ε) ∩K={¯ kt∗}.Thatis,thereisno (future or past) cutoff in an ε-neighborhood of ¯ kt∗. Proof.Ift∗≤τ, the claim follows from the induction hypothesis (10). If t∗=τ+1,the problem arises only when ¯ kτ+1=¯ k τ+1because then ¯ kτ+1is not an isolated point. This means that there exists kn τ+1↑¯ kτ+1. Then it must be that there exists ¯ t<τ+1with equilibrium cutoffs kn ¯ t∈K¯ tsuch that kn ¯ t↑¯ kτ+1; otherwise, by the same line of arguments in Step 1 that establishes Lemma 5, seller τ+1will not offer both kn τ+1and ¯ kτ+1.Butthen we must have ¯ k ¯ t=¯ k t∗. Since ¯ t<τ+1=t∗, this contradicts the definition of t∗. Claim A.3. ¯ k t∗= ¯ kt∗. In addition, (¯ k t∗¯ kt∗)∩Kt=∅for all t≤t∗. That is, (¯ k t∗¯ kt∗) includes no past cutoffs. Theoretical Economics 10 (2015) Transparency and price formation 371 for all s, the cutoff chosen by the sth seller with belief ˆ ks−1is at most bN−s. Therefore, the price charged by the second seller is at most (1−δ)(bN−2+δbN−3+···+δN−2b1)+δN−1 ¯ v On the other hand, the cutoff k∗ schosen by seller sin the equilibrium of Lemma 8 is strictly greater than bN−s, and, therefore, the price is strictly above (1−δ)(bN−2+δbN−3+···+δN−2b1)+δN−1 ¯ v But this is a contradiction since k(b p) is decreasing in p. Now suppose that ˆ k1>b N.Letsbe the first period when the cutoff ˆ ks≤bN.Then it must be that ˆ ks≤bN−s,becauseforanyk∈(bN−sbN], there is a unique continuation equilibrium that lasts at least N−s+1periods. Now, consider the equilibrium constructed in Lemma 8, starting from initial belief βN−s(ˆ ks). Then, by construction, the seller with this belief chooses ˆ ks. Moreover, by the induction hypothesis, the continuation of this equilibrium coincides with the continuation of the other equilibrium where ˆ ks−1chooses ˆ ks. This implies that the next period price is the same in both equilibria. Call this price p. Note that ˆ ks−1>b N>β N−s(ˆ ks). But this is a contradiction since k( ˆ ks−1p) > k(βN−s(ˆ ks)p). Step 3. Step 2 establishes that for b∈(bNbN+1], all equilibria last exactly N+1 periods. We show that the equilibrium is unique, which is the one we constructed in Lemma 8. Suppose, by contradiction, that there is another equilibrium (in addition to the one constructed in the proof of Lemma 8) that lasts exactly N+1periods. Let k∗be the first period cutoff of the equilibrium constructed in the proof of Lemma 8.Thenk∗≤bN.Let kbe the first cutoff of the other equilibrium. Then it must be that k∗= kbecause there is a unique Nperiod equilibrium following cutoff k∗by the induction hypothesis. Now, if k≤bN,itmustbethatk>b N−1, since otherwise the equilibrium lasts at most N−1periods. Suppose, without loss of generality (w.l.o.g.), that k∗>k .Then it must be the case that the second-period price in equilibrium of Lemma 8 is higher than the second-period price following k. This is because, in the unique continuation equilibrium, all cutoffs are increasing in the initial belief, since the functions βs(·)are increasing; and because, after each of these cutoffs, the equilibrium lasts exactly Nadditional periods. But this leads to a contradiction since k(b p) is decreasing in p. Now suppose k∗>b N. We shall use an argument similar to the one used to establish that all equilibria last at least N+1periods. Let sbe the first period when the cutoff ˆ ks≤bN. Then it must be that bN−s≤ˆ ks≤bN−s+1, because that is the only way that the equilibrium will have N−s+1additional periods. Now consider the equilibrium constructed in Claim A.1, starting from initial belief βN−s(ˆ ks). Then, by construction, the seller with this belief chooses ˆ ks. Moreover, by the induction hypothesis, the continuation of this equilibrium coincides with the continuation of the other equilibrium where ˆ ks−1chooses ˆ ks. This implies that the next period price is the same in both equi- 372 Kaya and Liu Theoretical Economics 10 (2015) libria. Call this price p. Note that ˆ ks−1>b N>β N−s(ˆ ks). But this is a contradiction since k( ˆ ks−1p) > k(βN−s(ˆ ks)p). Appendix B: Proof of Proposition 1 B.1 Preliminary results Before presenting our induction argument, we present some preliminary results that facilitate the ensuing discussion. The first set of results are technical and they do not rely on equilibrium conditions. B.1.1 Technical results The first result establishes a strong form of monotonicity for the solution of the profit-maximization problem of a seller facing a truncation of F.In particular, the profit-maximizing price of any seller is nondecreasing in the highest type ¯ kthat he believes to be remaining, regardless of the continuation play. The solution to this maximization problem may not be unique (in the transparent regime). Therefore, it is necessary to make precise the notion of monotonicity for the set of profit-maximizing prices. The appropriate definition in this context is as follows. Definition 1. Consider two sets XY ⊂R. We say that the set Xis greater than the set Yif and only if for all x∈Xand y∈Y,x≥y. This set order is stronger than the strong set order (see Topkis 1998), which allows for nonsingleton intersections of two sets. We shall use the set order defined in Definition 1 when we refer to monotonicity of sets. Remark 1. Unlike in the nontransparent regime in which there is a unique equilibrium in pure strategies, the equilibrium in the transparent regime may involve mixed strategies (especially off the equilibrium path) and there may be multiple equilibria. The multiplicity of the equilibria is not an issue as we fix an arbitrary equilibrium in the transparent regime and compare it with the unique pure strategy equilibrium in the nontransparent regime. To deal with mixed strategies, we consider sets of prices and introduce a strong notion of monotonicity of sets. In addition, since the current period cutoff buyer type might face a random price by delaying trade to the next period, we need to consider the expectation of this random price to study this cutoff buyer’s incentives. In our game, when a short-run seller faces buyer types [¯ v ¯ k], he chooses a price pto maximize [F(¯ k) −F(ki(p))]p,whereki(p) is the cutoff buyer type in regime i.Itturns out that regardless of the properties of ki(p), we can show that the set of optimal prices argmaxp[F(¯ k) −F(ki(p))]pis nondecreasing in ¯ kin the sense of Definition 1. Lemma 10. For any real-valued function k(p),argmaxp[F(¯ k)−F(k(p))]pis nondecreasing in ¯ kin the sense of Definition 1. Theoretical Economics 10 (2015) Transparency and price formation 373 Proof. Notice that the objective function has increasing differences in ¯ kand p.Itfollows from Topkis’ theorem that the solution set is nondecreasing in ¯ kin the strong set order (Topkis 1998). We strengthen this conclusion below. Suppose to the contrary that the ordering in Definition 1 does not hold. That is, there exist p∈arg max[F(¯ k) −F(k(p))]p and p∈argmax[F(¯ k)−F(k(p))]p,¯ k< ¯ kbut p>p . Then it follows from Topkis’ theorem that pand pare maximizers for both objective functions: F(¯ k) −F(k(p))p=F(¯ k) −F(k(p))p F(¯ k)−F(k(p))p=F(¯ k)−F(k(p))p Subtracting the second equation from the first, we have [F(¯ k) −F(¯ k)]p=[F(¯ k) −F(¯ k)]p It follows immediately that p=p, a contradiction.  Next, we establish a direct implication of the increasing hazard rate property for F. Specifically, we show that a truncation of Finherits the hazard rate property from F.We present the version that is actually used in our proof. Lemma 11. Suppose that v<v+≤k. Then if Fhas increasing hazard rate, that is, f(v)/(1−F(v))is nondecreasing, then (F(v +) −F(v))/(F(k) −F(v))is nondecreasing in v. Proof. First note that f(v) F(k)−F(v) is nondecreasing in v. To see this, simply note that f(v) F(k)−F(v) =f(v) 1−F(v) ·1−F(v) F(k)−F(v) =f(v) 1−F(v)1−F(k) F(k)−F(v) +1 and both terms are increasing in v. Now note that F(v+) −F(v) F(k)−F(v) =1−F(k)−F(v+) F(k)−F(v)  Thus, ∂ ∂vF(v+) −F(v) F(k)−F(v) =f(v+) F(k)−F(v) −F(k)−F(v+) (F(k) −F(v))2f(v) =F(k)−F(v+) F(k)−F(v) f(v+) F(k)−F(v+) −f(v) F(k)−F(v) ≥0 This completes the proof.  374 Kaya and Liu Theoretical Economics 10 (2015) B.1.2 Preliminary equilibrium characteristics Next we establish two intuitive, but not immediate, properties of equilibria in the transparent regime. The first result establishes that for seller 1, the probability of sale decreases in his price offer. The second establishes that a higher price in the first period leads to higher expected continuation prices. To make these statements precise, it is necessary to introduce additional notation. Notice that under the transparent regime, the second-period price (on or off the equilibrium path) can depend only on the first-period price, as this is the only observable history. Following an off-equilibrium first-period price, seller 2 may play a mixed strategy. Let ˆ pTR 2(p) be the expected second-period price if the first-period seller in the transparent regime chooses p.Foranypricepthat is accepted with a positive probability in equilibrium, there exists a unique kthat satisfies the indifference condition: k−p=δ(k −ˆ pTR 2(p)) That is, the buyer type kis indifferent between buying at pin this period versus waiting for a (random) price with an expectation of ˆ pTR 2(p) in the next period. This indifference condition is well defined because, in equilibrium, if kis the highest type in the next period, all prices in the support of the seller’s equilibrium strategy must induce the acceptance of k; that is, all of these prices are lower than k. We reformulate the indifference condition as follows for future reference: p=(1−δ)k +δˆ pTR 2(p) Denote the unique cutoff type defined by this indifference condition by kTR 1(p) =p−δˆ pTR 2(p) 1−δ For future reference, let kNTR 1(p) be the cutoff type defined by the indifference condition of the nontransparent regime, p=(1−δ)kNTR 1(p) +δpNTR 2 where pNTR 2is the unique equilibrium price in period 2 (in pure strategy). Lemma 12. We have that kTR 1(p) is nondecreasing in p. Proof.Takep>p and suppose that kTR 1(p) < kTR 1(p).Then,byLemma 10, p=(1−δ)kTR(p) +δˆ p2(p) ≤(1−δ)kTR(p)+δˆ p2(p)=p a contradiction.  The next lemma shows that in the transparent regime, a deviation by the first seller to a higher price weakly increases the expected price in the second period. Lemma 13. Let pTR 1be any price in the support of seller 1’s strategy. If pTR 1<p, then the expected price in period 2 satisfies ˆ pTR 2(pTR 1)≤ˆ pTR 2(p). Theoretical Economics 10 (2015) Transparency and price formation 375 Proof.Takep>p TR 1.ThenLemma 12 implies that kTR 1(pTR 1)≤kTR 1(p).Moreover, kTR 1(pTR 1)=kTR 1(p) contradicts the optimality of pTR 1. Therefore, kTR 1(pTR 1)<k TR 1(p). Then the claim follows from Lemma 10. Fudenberg et al. (1985) prove that the equilibrium price in the second period is a pure strategy, that is, ˆ pTR 2(pTR 1)=pTR 2. Our proof above does not utilize this result. B.2 The induction proof of Proposition 1 Recall that Tiis the last period during which trade takes place with a positive probability in regime ion the given equilibrium path. If max{TTRTNTR}=1, then the prices in the two regimes must equal ¯ vand the claim in Proposition 1 is vacuously satisfied. Induction hypothesis. Fix a distribution function Fthat exhibits an increasing hazard rate. For any ¯ v≥¯ kTR ≥¯ kNTR ≥¯ vand any realization of the price sequence {pi}of any equilibrium in regime i=TRNTR, where the buyer’s type distribution is a truncation of Fwith support [¯ v ¯ ki], assume pTR t≥pNTR tfor all twhenever max{TTRTNTR}≤τ,where τ≥1. Remark 2. We shall show that if ¯ kTR ≥¯ kNTR ≥¯ vis such that max{TTRTNTR}=τ+1in some equilibria of the two regimes, the prices can be ranked. We do this in three steps: (i) We show that under the induction hypothesis, the second-period price of the nontransparent regime is smaller than any possible second-period price of the transparent regime. (ii) Then we show that, under the induction hypothesis and using (i), the firstperiod price in the transparent regime must be larger than the equilibrium price of the nontransparent regime. (iii) Finally, we complete the proof by showing that the prices in the later periods must also be ranked as claimed. Our discussion in the main text refers to step (ii). We start with (i) mentioned in Remark 2. Lemma 14. Fix any ¯ kTR ≥¯ kNTR ≥¯ v. Fix any equilibrium in the transparent regime and let pTR 1be any price in the support of seller 1’s equilibrium strategy. Suppose that max{TTRTNTR}=τ+1.Thenˆ pTR 2(pTR 1)≥pNTR 2. That is, the expected second-period price in the transparent regime following any equilibrium path history is no less than the second-period price in the nontransparent regime. Proof. We abuse notation slightly by writing ki 1as ki 1(pi 1).Thatis,ki 1is the equilibrium marginal type that purchases at the given realized equilibrium price pi 1in period 1 in regime i. Suppose to the contrary that the claim is false: ˆ pTR 2(pTR 1)<p NTR 2. Then we claim that the highest buyer type at the beginning of period 2 following pi 1must satisfy kTR 1<k NTR 1. To see this, first note that from the second period onward, all trade takes place in at most τperiods in the continuation equilibrium in either regime. If, however, kTR 1≥kNTR 1,it 376 Kaya and Liu Theoretical Economics 10 (2015) follows from the induction hypothesis (with ¯ kTR =kTR 1and ¯ kNTR =kNTR 1on the continuation game) that ˆ pTR 2(pTR 1)≥pNTR 2, a contradiction. For each k∈[ ¯ v ¯ ki],define pi 1(k) := sup{p:ki 1(p) ≤k} That is, pi 1(k) is the highest price that seller 1 can charge so that buyer type kbuys in period 1. Since pNTR 1is seller 1’s unique optimal price in the nontransparent regime, the following inequality must hold: (F( ¯ kNTR)−F(kTR 1))pNTR 1(kTR 1)<(F(¯ kNTR)−F(kNTR 1))pNTR 1(kNTR 1) (19) The left-hand side of (19) is seller 1’s profit in regime NTR if he targets a cutoff type kTR 1 with a price pNTR 1(kTR 1); the right-hand side is seller 1’s profit by following the unique pure strategy equilibrium, that is, targeting equilibrium cutoff type kNTR 1with the equilibrium price pTR 1=pNTR 1(kNTR 1).17 By the previous claim, kTR 1<k NTR 1. Inequality (19) can be rewritten as F(¯ kNTR)−F(kTR 1) F(¯ kNTR)−F(kNTR 1)<pNTR 1(kNTR 1) pNTR 1(kTR 1) and further as F(kNTR 1)−F(kTR 1) F(¯ kNTR)−F(kNTR 1)<pNTR 1(kNTR 1)−pNTR 1(kTR 1) pNTR 1(kTR 1)(20) by substracting 1from each side.18 We now argue that in regime TR, increasing the cutoff from kTR 1to kNTR 1strictly increases the payoff of seller 1. This will lead to the desired contradiction. The idea is to show that in the transparent regime, this change of cutoff types leads to a smaller percentage decrease in trading probability (the left-hand side of (20)) but is accompanied by an even larger percentage increase in price than in the nontransparent regime (the right-hand side of (20)). We first compare the percentage changes in trading probability in the two regimes. Since ¯ kTR ≥¯ kNTR by assumption, and kTR 1<k NTR 1byapreviousclaim,itisimmediate that F(kNTR 1)−F(kTR 1) F(¯ kTR)−F(kNTR 1)≤F(kNTR 1)−F(kTR 1) F(¯ kNTR)−F(kNTR 1)(21) 17To avoid introducing further notation, we slightly abuse notation. 18In words, (20) has the following interpretation: seller 1 in regime NTR faces a set of buyer types [¯ v ¯ ki];if seller 1’s targeted type increases from kTR 1to kNTR 1, the trading probability decreases by a percentage factor of (F(kNTR 1)−F(kTR 1))/(F( ¯ kNTR)−F(kNTR 1)), but this is accompanied by a larger percentage increase in price of (pNTR 1(kNTR 1)−pNTR 1(kTR 1))/pNTR 1(kTR 1). Therefore, increasing the cutoff type from kTR 1to kNTR 1is desirable (recall that kNTR 1is the equilibrium cutoff level). Theoretical Economics 10 (2015) Transparency and price formation 377 We now compare the percentage changes in price in the two regimes. Note that pTR 1(kTR 1)=(1−δ)kTR 1+δˆ p2(pTR 1) <(1−δ)kTR 1+δpNTR 2(22) =pNTR 1(kTR 1) where inequality (22) follows from the supposition that ˆ p2(pTR 1)<p NTR 2. Note also that since kTR 1<k NTR 1, it follows from Lemma 10 that pTR 1(kTR 1)≤ pTR 1(kNTR 1)and, hence, by Lemma 13, ˆ pTR 2(pTR 1(kTR 1)) ≤ˆ pTR 2(pTR 1(kNTR 1)) (23) Therefore, pNTR 1(kNTR 1)−pNTR 1(kTR 1) =[(1−δ)kNTR 1+δpNTR 2]−[(1−δ)kTR 1+δˆ p2(pTR 1)] =(1−δ)(kNTR 1−kTR 1) (24) ≤(1−δ)(kNTR 1−kTR 1)+δˆ pTR 2(pTR 1(kNTR 1)) −ˆ pTR 2(pTR 1(kTR 1)) =(1−δ)kNTR 1+δˆ pTR 2(pTR 1(kNTR 1))−(1−δ)kTR 1+δˆ pTR 2(pTR 1(kTR 1)) =pTR 1(kNTR 1)−pTR 1(kTR 1) where inequality (24) follows from (23). It then follows from (22)and(24) that pNTR 1(kNTR 1)−pNTR 1(kTR 1) pNTR 1(kTR 1)<pTR 1(kNTR 1)−pTR 1(kTR 1) pTR 1(kTR 1)(25) Combining (20), (21), and (25), we have F(kNTR 1)−F(kTR 1) F(¯ kTR)−F(kNTR 1)<pTR 1(kNTR 1)−pTR 1(kTR 1) pTR 1(kTR 1)(26) This says that, in regime TR, increasing the cutoff from kTR 1to kNTR 1leads to a smaller percentage change in trading probability than in price. Applying the same argument between (19)and(20), we can rewrite (26)as (F( ¯ kTR)−F(kTR 1))pTR 1(kTR 1)<(F(¯ kTR)−F(kNTR 1))pTR 1(kNTR 1) This inequality says that in regime TR, seller 1 can be strictly better off by targeting the cutoff type kNTR 1rather than the equilibrium cutoff type kTR 1, a contradiction.  The next lemma establishes (ii) mentioned in Remark 2. That is, it shows that the first-period equilibrium price of the transparent regime is larger than the first-period equilibrium price in the nontransparent regime. 378 Kaya and Liu Theoretical Economics 10 (2015) Lemma 15. Fix any ¯ kTR ≥¯ kNTR ≥¯ v. Fix any equilibrium in the transparent regime and let pTR 1be any realized first-period equilibrium price. Let pNTR 1be the unique firstperiod equilibrium of the nontransparent regime. Suppose max{TTRTNTR}=τ+1.Then pTR 1≥pNTR 1. Proof. For a contradiction, suppose that there exists pTR 1in the support of seller 1’s equilibrium strategy in the transparent regime such that pTR 1<p NTR 1. Since seller 1 in the nontransparent regime has a unique optimal strategy, we have, as in (19), F(¯ kNTR)−F(kNTR 1(pTR 1))pTR 1<F(¯ kNTR)−F(kNTR 1(pNTR 1))pNTR 1 which can be rewritten as F(kNTR 1(pNTR 1)) −F(kNTR 1(pTR 1)) F(¯ kNTR)−F(kNTR 1(pTR 1)) <pNTR 1−pTR 1 pNTR 1 (27) Now we compare F(kNTR 1(pNTR 1)) −F(kNTR 1(pTR 1)) F(¯ kNTR)−F(kNTR 1(pTR 1)) =F(kNTR 1(pTR 1)+NTR)−F(kNTR 1(pTR 1)) F(¯ kNTR)−F(kNTR 1(pTR 1)) to F(kTR 1(pNTR 1)) −F(kTR 1(pTR 1)) 1−F(kTR 1(pTR 1)) =F(kTR 1(pTR 1)+TR)−F(kTR 1(pTR 1)) 1−F(kTR 1(pTR 1))  where i≡ki 1(pNTR 1)−ki 1(pTR 1) We make the following two claims. Claim B.1. We have kNTR 1(pTR 1)≥kTR 1(pTR 1). Proof.IfpTR 1is offered by seller 1 in regime NTR, then kNTR 1(pTR 1)is the cutoff buyer type; the indifference condition for kNTR 1(pTR 1)is pTR 1=(1−δ)kNTR 1(pTR 1)+δpNTR 2(28) If pTR 1is offered by seller 1 in regime TR, then kTR 1(pTR 1)is the cutoff buyer type; the indifference condition for kTR 1(pTR 1)is pTR 1=(1−δ)kTR 1(pTR 1)+δˆ pTR 2(pTR 1) (29) By Lemma 14,pNTR 2≤ˆ pTR 2(pTR 1). Taken together, equations (28)and(29)implythat kNTR 1(pTR 1)≥kTR 1(pTR 1). Claim B.2. We have NTR ≥TR. Theoretical Economics 10 (2015) Transparency and price formation 379 Proof. The indifference condition for the cutoff type kNTR 1(pNTR 1)in regime NTR is given by pNTR 1=(1−δ)kNTR 1(pNTR 1)+δpNTR 2(30) The indifference condition for the cutoff type kTR 1(pNTR 1)in regime TR is given by pNTR 1=(1−δ)kTR 1(pNTR 1)+δˆ pTR 2(pNTR 1) (31) Therefore, kNTR 1(pNTR 1)−kNTR 1(pTR 1)=pNTR 1−pTR 1 1−δ ≥pNTR 1−pTR 1−(ˆ pTR 2(pNTR 1)−ˆ pTR 2(pTR 1)) 1−δ =kTR 1(pNTR 1)−kTR 1(pTR 1) where the first line follows from (28)and(30), the second line follows from Lemma 13 and the supposition that pTR 1<p NTR 1, and the third line follows from (29)and(31). This establishes the observation.  With these two claims, we are ready to prove the lemma. Note that F(kNTR 1(pTR 1)+NTR)−F(kNTR 1(pTR 1)) F(¯ kNTR)−F(kNTR 1(pTR 1)) ≥F(kNTR 1(pTR 1)+TR)−F(kNTR 1(pTR 1)) F(¯ kNTR)−F(kNTR 1(pTR 1)) ≥F(kTR 1(pTR 1)+TR)−F(kTR 1(pTR 1)) F(¯ kNTR)−F(kTR 1(pTR 1)) ≥F(kTR 1(pTR 1)+TR)−F(kTR 1(pTR 1)) F(¯ kTR)−F(kTR 1(pTR 1))  where the first inequality follows from Claim B.2, the second inequality follows from Claim B.1 and Lemma 11, and the third inequality follows from the assumption that ¯ kTR ≥¯ kNTR. Combining this with (27), we get F(kTR 1(pTR 1)+TR)−F(kTR 1(pTR 1)) F(¯ kTR)−F(kTR 1(pTR 1)) <pNTR 1−pTR 1 pNTR 1  which, after substituting in TR, can be rewritten as F(¯ kTR)−F(kTR 1(pNTR 1))pNTR 1>p TR 1F(¯ kTR)−F(kTR 1(pTR 1)) This says that in regime TR, pNTR 1gives seller 1 a larger profit than the equilibrium price pTR 1, a contradiction.  We now complete the induction proof of Proposition 1. This is step (iii) mentioned in Remark 2. 380 Kaya and Liu Theoretical Economics 10 (2015) Proof of Proposition 1. We have already shown that pTR 1≥pNTR 1. Suppose to the contrary that for some s≤τ+1,wehavepTR t≥pNTR tfor all t<s,butpTR s<p NTR s. By the induction hypothesis, this is only possible if kTR s−1(pTR s−1)<k NTR s−1(pNTR s−1).But then the indifference condition of buyer type kTR s−1(pTR s−1)in period s−1in regime TR is pTR s−1=(1−δ)kTR s−1(pTR s−1)+δpTR s and the indifference condition of buyer type kNTR s−1(pNTR s−1)in period s−1in regime NTR is pNTR s−1=(1−δ)kNTR s−1(pNTR s−1)+δpNTR s Since kTR s−1(pTR s−1)<k NTR s−1(pNTR s−1)and pTR s<p NTR s, we have, from the above two indifference conditions, that pTR s−1<p NTR s−1, a contradiction.  Appendix C: Proof of Proposition 2 Let {ki t}be a realization of an equilibrium cutoff sequence in any equilibrium in regime i, with the convention that ki t=¯ vfor t>T i,whereTiis the latest period in which trade takes place with a positive probability. Define a random variable, xi, that takes values in {ki t}, with a cumulative distribution Gidefined as Gi(k) =Pr(xi≤k) =δτi(k) where τi(k) is the unique number that satisfies k∈[ki τi(k)ki τi(k)−1). In words, the support of xiis the equilibrium cutoff in regime i. The marginal types trading at time tor earlier in each regime have a total probability of δt−1under the relevant random variable. Lemma 16. The variable xTR second-order stochastically dominates xNTR.Thatis, ∀k:k ¯ v Pr(xTR ≤x) dx ≤k ¯ v Pr(xNTR ≤x) dx Proof. k ¯ v Pr(xTR ≤x) dx = ∞  t=τTR(k) (1−δ)δt−1(k −kTR t) and k ¯ v Pr(xNTR ≤˜ k)d ˜ k= ∞  t=τNTR(k) (1−δ)δt−1(k −kNTR t)