{"http:\/\/dx.doi.org\/10.14288\/1.0067729":{"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool":[{"value":"Arts, Faculty of","type":"literal","lang":"en"},{"value":"Vancouver School of Economics","type":"literal","lang":"en"}],"http:\/\/www.europeana.eu\/schemas\/edm\/dataProvider":[{"value":"DSpace","type":"literal","lang":"en"}],"https:\/\/open.library.ubc.ca\/terms#degreeCampus":[{"value":"UBCV","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/creator":[{"value":"Wong, Chi Leung","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/issued":[{"value":"2009-09-30T17:53:37Z","type":"literal","lang":"en"},{"value":"2009","type":"literal","lang":"en"}],"http:\/\/vivoweb.org\/ontology\/core#relatedDegree":[{"value":"Doctor of Philosophy - PhD","type":"literal","lang":"en"}],"https:\/\/open.library.ubc.ca\/terms#degreeGrantor":[{"value":"University of British Columbia","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/description":[{"value":"This dissertation studies dynamic matching and bargaining games with two-sided private information bargaining. There is a market in which a large number of heterogeneous buyers and sellers search for trading partners to trade with. Traders in the market are randomly matched pairwise. Once a buyer and a seller meet, they bargain following the random-proposer protocol: either the buyer or the seller (randomly chosen) makes a take-it-or-leave-it offer to the other party. The traders leave once they successfully trade, and the market is continuously replenished with new-born buyers and sellers who voluntarily choose to enter. We study the steady state with positive entry. There are (except the asymmetric information) two kinds of frictions: time discounting and explicit search costs. Chapter 2 addresses existence and uniqueness of equilibrium. It provides a simple necessary and sufficient condition for the existence of a nontrivial steady-state equilibrium. The equilibrium is unique if the discount rate is small relative to the search costs. This chapter also analyzes how the composition of frictions affects the patterns of equilibria. It shows that if the discount rate is small relative to the search costs, in equilibrium every meeting results in trade. If the discount rate is relatively large, some meetings do not result in trade. Chapter 3 shows that private information typically deters entry. Because of search externalities, this entry-deterring effect may be socially desirable or undesirable. We provide and interpret a simple condition under which private information improves welfare. Chapter 4 studies the convergence properties of equilibria as frictions vanish. It not only shows that, as frictions vanish, the equilibrium price range collapses to the Walrasian price and the equilibrium welfare converges to the Walrasian welfare level, but also provides the rate of convergence. Under random-proposer bargaining, welfare converges at the fastest possible rate among all bargaining mechanisms. If we assume double auction instead of random-proposer bargaining, equilibria might converge at a slower rate or even not converge at all. These results also hold under full information bargaining. It suggests that private information does not affect asymptotic efficiency, but bargaining protocol might.","type":"literal","lang":"en"}],"http:\/\/www.europeana.eu\/schemas\/edm\/aggregatedCHO":[{"value":"https:\/\/circle.library.ubc.ca\/rest\/handle\/2429\/13397?expand=metadata","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/extent":[{"value":"957267 bytes","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/elements\/1.1\/format":[{"value":"application\/pdf","type":"literal","lang":"en"}],"http:\/\/www.w3.org\/2009\/08\/skos-reference\/skos.html#note":[{"value":"Essays on Dynamic Matching Markets by Chi Leung Wong B.Soc.Sci., The Chinese University of Hong Kong, 2000 M.Phil., The Chinese University of Hong Kong, 2002 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF Doctor of Philosophy in The Faculty of Graduate Studies (Economics) The University of British Columbia (Vancouver) September, 2009 c\u00b0 Chi Leung Wong 2009 Abstract This dissertation studies dynamic matching and bargaining games with two-sided private information bargaining. There is a market in which a large number of heterogeneous buyers and sellers search for trading partners to trade with. Traders in the market are randomly matched pairwise. Once a buyer and a seller meet, they bargain following the random- proposer protocol: either the buyer or the seller (randomly chosen) makes a take-it-or- leave-it offer to the other party. The traders leave once they successfully trade, and the market is continuously replenished with new-born buyers and sellers who voluntarily choose to enter. We study the steady state with positive entry. There are (except the asymmetric information) two kinds of frictions: time discounting and explicit search costs. Chapter 2 addresses existence and uniqueness of equilibrium. It provides a simple necessary and sufficient condition for the existence of a nontrivial steady-state equilibrium. The equilib- rium is unique if the discount rate is small relative to the search costs. This chapter also analyzes how the composition of frictions affects the patterns of equilibria. It shows that if the discount rate is small relative to the search costs, in equilibrium every meeting results in trade. If the discount rate is relatively large, some meetings do not result in trade. Chapter 3 shows that private information typically deters entry. Because of search externalities, this entry-deterring effect may be socially desirable or undesirable. We provide and interpret a simple condition under which private information improves welfare. Chapter 4 studies the convergence properties of equilibria as frictions vanish. It not only shows that, as frictions vanish, the equilibrium price range collapses to the Walrasian price and the equilibrium welfare converges to the Walrasian welfare level, but also provides the rate of convergence. Under random-proposer bargaining, welfare converges at the fastest possible rate among all ii Abstract bargaining mechanisms. If we assume double auction instead of random-proposer bargain- ing, equilibria might converge at a slower rate or even not converge at all. These results also hold under full information bargaining. It suggests that private information does not affect asymptotic efficiency, but bargaining protocol might. iii Table of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Dynamic matching and bargaining games . . . . . . . . . . . . . . . . . . . 1 1.2 Baseline model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.4 Other related literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2 Dynamic Matching and Two-sided Private Information Bargaining . . 7 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.3 Nontrivial steady-state equilibria . . . . . . . . . . . . . . . . . . . . . . . . 13 2.4 Basic equilibrium properties . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.5 Full-trade equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.6 Uniqueness of equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.7 Necessary and sufficient condition for existence . . . . . . . . . . . . . . . . 37 2.8 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3 Role of Information Structure in Dynamic Matching Markets . . . . . 48 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 iv Table of Contents 3.2 Private information model . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.3 Full information (Mortensen-Wright) model . . . . . . . . . . . . . . . . . . 53 3.3.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.3.2 Basic equilibrium properties . . . . . . . . . . . . . . . . . . . . . . 56 3.3.3 Necessary and sufficient condition for existence . . . . . . . . . . . . 61 3.3.4 Full-trade equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . 62 3.4 No-discounting case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 3.5 Full-trade equilibria and bargaining efficiency . . . . . . . . . . . . . . . . . 68 3.6 Entry effect of private information . . . . . . . . . . . . . . . . . . . . . . . 70 3.7 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 4 Rate of Convergence towards Perfect Competition . . . . . . . . . . . . 80 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 4.2 The baseline model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 4.3 Rate of convergence of trading prices . . . . . . . . . . . . . . . . . . . . . 84 4.3.1 Convergence of full-trade equilibria . . . . . . . . . . . . . . . . . . 85 4.3.2 General convergence theorem . . . . . . . . . . . . . . . . . . . . . . 85 4.3.3 Proof of Theorem 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 4.3.4 Full information model . . . . . . . . . . . . . . . . . . . . . . . . . 91 4.4 Rate of convergence of welfare . . . . . . . . . . . . . . . . . . . . . . . . . 94 4.5 Results for k-double auction . . . . . . . . . . . . . . . . . . . . . . . . . . 101 4.6 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 5.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 5.2 Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 5.2.1 Continuous time, continuous types . . . . . . . . . . . . . . . . . . . 116 5.2.2 Symmetric pure strategies . . . . . . . . . . . . . . . . . . . . . . . 117 5.2.3 Random-proposer bargaining . . . . . . . . . . . . . . . . . . . . . . 117 v Table of Contents 5.2.4 Choice of friction space . . . . . . . . . . . . . . . . . . . . . . . . . 118 5.2.5 Constant-returns-to-scale matching function . . . . . . . . . . . . . 119 5.2.6 Continuum of traders . . . . . . . . . . . . . . . . . . . . . . . . . . 119 5.3 Further research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 Appendices A Additional Details for Existence of Nontrivial Steady-state Equilibrium 126 B Calculations for Section 3.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 vi List of Figures 2.1 Proposing and responding strategies in an equilibrium with overlapping sup- ports (which must be non-full-trade) . . . . . . . . . . . . . . . . . . . . . . 21 2.2 Proposing and responding strategies in a non-full-trade equilibrium with sep- arated supports . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.3 Proposing and responding strategies in a full-trade equilibrium . . . . . . . 24 2.4 Interpretation of \u03b60 and K (\u03b60) . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.5 Different patterns of equilibria in different regions of friction space . . . . . 37 2.6 Illustration of the idea behind the existence proof . . . . . . . . . . . . . . . 41 3.1 Equilibrium when discount rate is zero . . . . . . . . . . . . . . . . . . . . . 69 4.1 Construction of a double auction full-trade equilibrium . . . . . . . . . . . . 104 4.2 A two-step equilibrium under double auction . . . . . . . . . . . . . . . . . 108 vii Chapter 1 Introduction 1.1 Dynamic matching and bargaining games This dissertation contributes to the literature on dynamic matching and bargaining games (DMBG). This strand of literature stands in between two rather extreme paradigms in economic theory: the Walrasian theory and the bargaining theory. On one extreme, the Walrasian theory assumes that trading happens in a centralized market where every agent has no market power at all (i.e. every agent takes the market price as given). The concept of Walrasian equilibrium is usually justified by telling a story with a large number of buyers and sellers, perfect information, and\/or a Walrasian auctioneer enforcing the trading process. On the other extreme, the bargaining theory assumes that a small number of agents (say a pair of buyer and seller) strategically bargain over the economic outcome (say the quantity transacted and the trading price), possibly with private information. The methodology and equilibrium concepts in the arena of bargaining theory are game-theoretic. The literature on DMBG, which is in the middle, assumes that the market has a large number of buyers and sellers, but they cannot gather together to come up with trades. Instead, the quantities transacted and the trading prices are determined by a lot of small bargaining games, each of which is among a small coalition of agents (usually a pair of buyer and seller). The trading process is dynamic rather than a one-shot game. Also, all the agents in the market are connected through the way that the formation of bargaining coalitions varies over time, such that each agent is able to meet many other agents at different points of time. But this connection is only imperfect, because meeting other agents takes time, and it 1 Chapter 1. Introduction is costly due to either impatience, or a probability of death, or a fixed search cost, depending on the specific modeling. These \"costs of delay\" are called frictions of the market. And we call such a market with frictions a dynamic matching and bargaining market, or simply a dynamic matching market. The labor and the housing markets are often cited examples of this kind of markets. The nature of DMBG is suitable for economic theorists to build a game-theoretic foun- dation for the Walrasian theory. Indeed, a main focus of this strand of literature has been on the following question: as the frictions vanish, do the (game-theoretic) equilibrium out- comes of dynamic matching and bargaining games converge to the perfectly competitive outcome predicted by Walrasian equilibrium? Until very recently, most papers in the literature assume that the bargaining games are bilateral and under full information, i.e. a buyer and a seller bargain knowing each other\u2019s willingness-to-pay and cost. They include: Mortensen (1982), Rubinstein and Wolinsky (1985, 1990), Gale (1986a,b, 1987) and Mortensen and Wright (2002), among others.1 Sat- terthwaite and Shneyerov (2007) have recently introduced two-sided private information in a dynamic matching market where sellers use auctions, and have shown that the presence of private information does not prevent convergence to perfect competition.2 1A notable exception is the unpublished manuscript Butters (1979). Other papers that have incorporated private information in some form include Wolinsky (1988), De Fraja and Sakovics (2001) and Serrano (2002). 2Several recent papers have explored convergence under private information in more detail: Satterthwaite and Shneyerov (2008) show convergence in the model that is a replica of Satterthwaite and Shneyerov (2007) except that it has exogenous exit rate. Lauermann (2008) shows convergence even if one side of the market has all the bargaining power, and Lauermann (2006b) shows that in that case, the welfare under private information may be higher than under full information. Atakan (2008) provides a generalization to multiple units. Lauermann (2006a) derives a set of general conditions for convergence. In addition, Hurkens and Vulkan (2006, 2007) study the role of privately observed deadlines in a matching and bargaining market. 2 Chapter 1. Introduction 1.2 Baseline model This dissertation studies a dynamic matching market, modeled as a DMBG. Our baseline model is a replica of the one in Mortensen and Wright (2002), modified with two-sided private information bargaining. It is roughly described as follows.3 There is a market in which a large number (more precisely, continuum) of risk-neutral buyers and sellers search for trading partners to trade with. Each buyer has a unit demand for a homogeneous and indivisible good; and each seller has a unit supply of the same good. The buyers and sellers are heterogeneous: different buyers have different valuations (or willingness-to-pay) and different sellers have different costs. Traders in the market are randomly matched pairwise. The mass of total matched pairs per unit time is determined by some unspecified Pissarides (2000) style matching function.4 Once a buyer and a seller meet, they bargain following the so-called random-proposer protocol: either the buyer or the seller (randomly chosen) makes a take-it-or-leave-it offer to the other party. The traders leave once they successfully trade, and the market is continuously replenished with new-born buyers and sellers who voluntarily choose to enter. We assume the market is in steady state and with positive mass of traders in it. From each trader\u2019s point of view, searching for a trading partner takes time, and it is costly both because traders are impatient (parameterized by a discount rate) and they have to spend other resources like money or effort to search (parameterized by explicit search costs). Thus, the \"costs of delay\", or frictions, are multi-dimensional in 3The modeling methods for DMBG can be divided into two classes: non-steady-state models and steady- state models. In a non-steady-state model (e.g. Moreno and Wooders (2002)), the market starts with a fixed number of agents and no more agent comes in later on. As time collapses, the number of agents left in the market decreases. On the other hand, in a steady-state model, new agents keep coming in, and attention is restricted to the steady-state equilibrium. Gale (1987) shows convergence for both versions of his model. We will use the steady-state approach throughout this dissertation. As a matter of fact, our model (or Mortensen-Wright model) is also a version of search models. 4A Pissarides-style matching function assigns a mass of total matched pairs for each combination of buyers\u2019 and sellers\u2019 masses currently participating in the market. With such a matching function, the precise matching process need not be specified, pretty much like a production function in macro models assigns a level of output for each combination of inputs, without specifying the precise production process. 3 Chapter 1. Introduction our model. 1.3 Summary Here we briefly summarize what will be seen in the following three chapters. More detailed summaries will be presented in the introduction sections of those chapters. Chapter 2 proves the existence of equilibrium for our baseline model, characterizes the equilibrium patterns, and develops a bunch of results that are useful for the subsequent two chapters. Chapter 3 analyzes the impacts of the private information in bargaining. Chapter 4 derives convergence properties of the equilibrium outcome as frictions vanish. Roughly speaking, Chapter 2 and Chapter 3 are concerned with markets with significant frictions; while Chapter 4 is concerned with markets with small frictions. More precisely, in Chapter 2 and Chapter 3 the level and composition of frictions are considered to be fixed; while frictions are considered to be vanishing in Chapter 4.5 A main interest throughout this dissertation is how the private information in bar- gaining games shapes the equilibrium outcome of a dynamic matching market. A recent paper by Satterthwaite and Shneyerov (2007) shows in a similar but different model that equilibrium outcome converges towards perfect competition as frictions vanish, even when traders hold private information. However their model might have no natural full infor- mation counterpart.6 Furthermore, even when private information in bargaining does not prevent convergence, would it make the convergence any slower? This question has not been addressed in the literature. On the other hand, if the private information has no impact 5This dichotomy is not strict. The core results of Chapter 4 apply to any level of frictions. 6The main differences between the model of Satterthwaite and Shneyerov (2007) and ours are that: in their model, time is discrete; every buyer is randomly matched with one seller in each period, so that a seller might be matched with several buyers, one buyer, or no buyer; and sellers sell their goods through first-price auctions without committed reserve price. Thus their model assumes a specific multi-lateral matching and bargaining process. In contrast, our model assumes bilateral matching and bargaining (which is common in the literature of DMBG and search models), with a general matching function and a general distribution of bargaining power. 4 Chapter 1. Introduction in the limit, one might wonder what impacts it has in the \"out-of-the-limit\" case. Is the private information always bad in the social point of view? The analyses in this dissertation shed some light on all these issues. 1.4 Other related literature The market games we analyze in this dissertation can also be counted as a search model. In- deed, our modeling choices include random matching, a Pissarides-style matching function, and steady state. All these are common features of classic labor search models (although recent developments allow directed search, on-the-job search, etc.). The literature on search models of labor market, surveyed for example in Mortensen and Pissarides (1999) and more recently Rogerson, Shimer, and Wright (2005), is large. While the literature on DMBG is micro-oriented, the literature on labor search models is macro-oriented. The latter studies topics like equilibrium unemployment, wage dispersions, and the constrained efficiency when the market is plagued by search frictions.7 Most if not all of the labor search literature neglects the private information in bar- gaining.8 Besides, many labor search models simply assume homogeneous workers and homogeneous firms, so that the information structure at the bargaining stages is irrelevant. This dissertation, in contrast, emphasizes the role of information structure at the bargaining stages. In this regard, this dissertation contributes to the search theory, by allowing private information. Another strand of literature related to this dissertation is the literature on static double auction. As we mentioned before, DMBG stands in between the Walrasian theory and bargain- ing theory, and hence theorists build the foundation of Walrasian equilibrium on it. The 7Search theory has also been applied to monetary models and marriage models. But the models in these areas are not as relevant to this dissertation as the labor search models. 8 In the labor search literature, when matching is random (rather than directed), typically the (generalized) Nash bargaining solution is assumed. 5 Chapter 1. Introduction literature on static double auction is very similar to DMBG in this aspect. A number of papers on static double auction ask whether the equilibrium outcomes converge to the Wal- rasian outcome as the number of traders n gets large. More importantly, this literature also looks at the rate of convergence. In particular, Rustichini, Satterthwaite, and Williams (1994) show robust convergence of double-auction equilibria in the symmetric class at the rate O (1\/n) for the bid\/ask strategies and the rate O \u00a1 1\/n2 \u00a2 for the ex-ante traders\u2019 wel- fare.9 Moreover, this literature also has asymptotic efficiency results: the double auction converges at the rate that is fastest among all incentive-compatible and individually ratio- nal mechanisms (Satterthwaite and Williams (2002); Tatur (2005)). Cripps and Swinkels (2005) substantially enrich the model by allowing correlation among bidders\u2019 valuations, and show convergence at the rate O \u00a1 1\/n2\u2212\u03b5 \u00a2 , where \u03b5 > 0 is arbitrarily small.10 In contrast, as far as I know, the rate of convergence has not been addressed in the literature of DMBG. Comparing with the rather sophisticated literature on static double auction, there is a gap in the DMBG literature.11 Our rate of convergence analysis in Chapter 4 takes a step to fill this gap. 9Other related papers include Gresik and Satterthwaite (1989), Satterthwaite and Williams (1989), Sat- terthwaite (1989), and Williams (1991). 10Reny and Perry (2006) allow interdependent values and show that it is almost efficient and almost fully aggregates information as n\u2192\u221e, although the rate of convergence is not addressed. 11Of course, the natures of convergence in these two strands of literature are different. For the static double auction, we let the number of traders get large. For DMBG, we let the level of frictions get small. However, they are analogous to each other, because both of them reflect increasing intensity of competition, as we will discuss in Chapter 4. After all, both of them lead to the convergence towards the perfect competition outcome. 6 Chapter 2 Dynamic Matching and Two-sided Private Information Bargaining 2.1 Introduction This chapter starts our formal analysis of dynamic matching and bargaining markets.12 We will study a replica of Mortensen and Wright (2002) model, modified with two-sided private information bargaining. There is a market in which continua of risk-neutral buyers and sellers search for trading partners to trade with. Each buyer has a unit demand for a homogeneous and indivisible good; and each seller has a unit supply of the same good. The buyers and sellers are heterogeneous: different buyers have different valuations v \u2208 [0, 1] and different sellers have different costs c \u2208 [0, 1]. Our model has features of a steady-state search-theoretic model, where the matching between buyers and sellers is pairwise, random, and described by a Pissarides-style matching function M (B,S) that gives the matching rate as a function of the masses of buyers B and sellers S currently participating in the market. In steady state and from the standpoint of a particular trader, matchings come up according to a Poisson process. The Poisson arrival rate of being matched is \u03b1B \u2261 M (B,S) \/B for a buyer, or \u03b1S \u2261 M (B,S) \/S for a seller. Since we assume that M exhibits constant returns to scale, the arrival rates \u03b1B and \u03b1S only depend on the buyer-seller ratio \u03b6 \u2261 B\/S. Our model also has the feature of two-sided asymmetric information bilateral bargaining. 12The chapter significantly includes the materials in my manuscript \"Bilateral Matching and Bargaining with Private Information\", which is joint with my dissertation co-supervisor Artyom Shneyerov. 7 Chapter 2. Dynamic Matching and Two-sided Private Information Bargaining The bargaining game between a pair of buyer and seller follows what we call the random- proposer protocol: with probability \u03b2B \u2208 (0, 1) the buyer makes a take-it-or-leave-it price offer to the seller; and with probability \u03b2S \u2261 1 \u2212 \u03b2B the seller makes a take-it-or-leave-it price offer to the buyer. The traders leave once they successfully trade, and the market is continuously replenished with new-born buyers and sellers who voluntarily choose to enter. Another important feature of our model is that our notion of frictions is multi-dimensional. There are two kinds of search frictions: searching for a trading partner takes time, parame- terized by an instantaneous discount rate r > 0, and also takes other resources (e.g. money, effort), parameterized by explicit search costs \u03baB > 0 for buyers and \u03baS > 0 for sellers, per unit time. The main purpose of this chapter is to prove the existence of equilibrium, and to un- derstand the equilibrium patterns and properties, under different combinations of frictions. This chapter will also be the foundation of the analyses of the next two chapters. Our fundamental result (Theorem 3) is: at least one nontrivial (i.e. with positive mass of traders participating) steady-state equilibrium exists if and only if \u03baB \u03b1B (\u03b60) + \u03baS \u03b1S (\u03b60) < 1, (2.1) where \u03b60 \u2261 \u03b2B\u03baS\/\u03b2S\u03baB, and \u03b1B (\u03b60) (resp. \u03b1S (\u03b60)) is a buyer\u2019s (resp. seller\u2019s) Poisson arrival rate of being matched when the (steady-state) buyer-seller ratio is \u03b60. An uninteresting trivial equilibrium, in which nobody participates, always exists. In- deed, if searching for trading partners is very costly, only the trivial equilibrium can exist. Roughly speaking, our fundamental existence result says that some nontrivial steady-state equilibrium also exists if the search costs \u03baB and \u03baS are moderate. To get more sense of the above necessary and sufficient condition (2.1) for the existence of some nontrivial steady-state equilibrium, let us restrict attention to the no-discounting case, i.e. r \u2192 0. In this case, equilibrium analysis becomes very tractable and it is easy to show that the equilibrium buyer-seller ratio must be \u03b60, which simply reflects the ratio of buyers\u2019 and sellers\u2019 bargaining powers, and the ratio of buyers\u2019 and sellers\u2019 per-unit-time search costs. Then for an unmatched buyer to bring himself matched, the expected total search costs is 8 Chapter 2. Dynamic Matching and Two-sided Private Information Bargaining \u03baB\/\u03b1B (\u03b60). Similarly, for an unmatched seller the expected total search costs is \u03baS\/\u03b1S (\u03b60). On the other hand, since we normalize the supports of buyers\u2019 valuations and sellers\u2019 costs to be [0, 1], the maximum gain of trade that a pair of buyer and seller can realize is 1. Therefore, condition (2.1) simply says that the maximum gain a buyer-seller pair can real- ize is greater than the expected total search costs they incur to get matched. While this existence condition is rather natural in the no-discounting case, our result shows that the existence condition does not change at all in the general case.13 We distinguish two kinds of nontrivial steady-state equilibrium: \"full-trade equilibrium\" (i.e. in which every meeting results in trade) and \"non-full-trade equilibrium\". Given that some nontrivial steady-state equilibrium exists, we make predictions on the equilibrium pattern (full-trade vs non-full-trade): there are two critical levels of discount rate r\u2217 and r, with 0 < r < r\u2217, such that a full-trade equilibrium exists if and only if r \u2264 r\u2217 (Theorem 1); and only a full-trade equilibrium, but no non-full-trade one, exists if r \u2264 r (Theorem 2).14 The formulas for r\u2217 and r are explicitly derived, in terms of parameters including (\u03baB, \u03baS). In particular, both r\u2217 and r are increasing in (\u03baB, \u03baS); and both r\u2217 and r tend to 0 as (\u03baB, \u03baS) \u2192 0. These results suggest that: in (nontrivial steady-state) equilibrium whether every meeting results in a trade mainly depends not on the level of frictions, but the relativity of the two kinds of frictions. More concretely, if r is small relative to (\u03baB, \u03baS), then in equilibrium every meeting results in a trade; if on the other hand r is large relative to (\u03baB, \u03baS), in equilibrium some meetings lead to bargaining breakdowns. Satterthwaite and Shneyerov (2007) also provide an existence theorem for a dynamic matching market with two-sided asymmetric information. Their existence theorem (SS existence theorem), in our language, is: there exists a full-trade equilibrium if \u03baB, \u03baS and r\/min {\u03baB, \u03baS} are sufficiently small. Their model involves one-to-many matchings 13 If r > 0, then both the left-hand side and the right-hand side of (2.1) are subject to discounting. Since the two effects cancel out, the generality of (2.1) is possible. 14Since there can be at most one full-trade equilibrium, r \u2264 r is also a sufficient condition for the uniqueness of nontrivial steady-state equilibrium. 9 Chapter 2. Dynamic Matching and Two-sided Private Information Bargaining and auctions, unlike ours.15 By switching to a bilateral matching and bargaining model like Mortensen and Wright (2002), we are able to prove much sharper results than theirs. Compared with our results, SS existence theorem has several limitations. First, how small \u03baB, \u03baS and r\/min {\u03baB, \u03baS} have to be is unknown. Second, when r is large relative to (\u03baB, \u03baS), it is unknown whether some nontrivial steady-state equilibrium exists. Third, SS existence theorem makes no prediction on the equilibrium pattern for any friction profile: small r relative to (\u03baB, \u03baS) does not imply full trade; and large r relative to (\u03baB, \u03baS) does not imply non-full trade. Our results in this chapter do not have these limitations. The aforementioned second limitation of SS existence theorem also brings about a limi- tation of their convergence theorem. Their convergence theorem (SS convergence theorem), again in our language, is: along any sequence of nontrivial steady-state equilibria associated with a sequence of (r, \u03baB, \u03baS) that tends to 0 proportionally, the set of transaction prices and the welfare of every agent must converge to their counterparts under perfect competition. The limitation of SS convergence theorem is: it does not preclude the possibility that, even when we let (r, \u03baB, \u03baS) tend to 0 proportionally, nontrivial steady-state equilibrium keeps absent. In contrast, our convergence results presented in Chapter 4, with the foundation of our existence results, do not have this limitation. The rest of this chapter proceeds as follows. Section 2.2 introduces the model. Section 2.3 defines the equilibrium concept. Section 2.4 analyzes the basic equilibrium properties. Section 2.5 studies full-trade equilibria and the condition under which this kind of equilibria exist. Section 2.6 proves that the equilibrium is unique when the discount rate is small. Section 2.7 presents and proves the \"general existence theorem\". Section 2.8 concludes. Additional details for the general existence proof is in Appendix A. 2.2 The model The agents in our model are potential buyers and sellers of a homogeneous, indivisible good. Each buyer has a unit demand for the good, while each seller has a unit supply. All traders 15For more details of Satterthwaite and Shneyerov (2007) model, see footnote 6. 10 Chapter 2. Dynamic Matching and Two-sided Private Information Bargaining are risk neutral. Potential buyers are heterogeneous in their valuations (or types) v of the good. Potential sellers are also heterogeneous in their costs (or types) c of providing the good. For simplicity, we assume v, c \u2208 [0, 1]. Time is continuous and infinite horizon. The instantaneous discount rate is r > 0. The details of the model are described as follows: \u2022 Entry: Potential buyers and sellers are continuously born at rates b and s respectively. We normalize the aggregate born rate to be 1, i.e. b+ s = 1. The type of a new-born buyer is drawn i.i.d. from the c.d.f. F (v) and the type of a new-born seller is drawn i.i.d. from the c.d.f. G(c). Each trader\u2019s type will not change once it is drawn. Entry (or participation, or being active) is voluntary. Each potential trader decides whether to enter the market once he is born. Those who do not enter will get zero payoff. Those who enter must incur the search cost continuously at the rate \u03baB for buyers and \u03baS for sellers, until they leave the market. \u2022 Matching: Active buyers and sellers are randomly and continuously matched pairwise at a flow rate given by a Pissarides (2000) style matching function M(B,S), where B and S are the masses of active buyers and active sellers currently in the market. \u2022 Bargaining: Once a pair of buyer and seller are matched, they bargain without ob- serving each other\u2019s type. The bargaining protocol is what we call random-proposer bargaining: with probability \u03b2B \u2208 (0, 1), the buyer makes a take-it-or-leave-it price offer to the seller, then the seller chooses either to accept or reject. And with prob- ability \u03b2S \u2261 1 \u2212 \u03b2B the seller proposes and the buyer responds. (The \"bargaining weights\" \u03b2B and \u03b2S can be interpreted as the buyer\u2019s and seller\u2019s relative bargaining powers.) We also assume the market is anonymous, so that bargainers do not know their partners\u2019 market history, e.g. how long they have been in the market, what they proposed previously, and what offers they rejected previously. \u2022 If a type v buyer and a type c seller trade at a price p, then they leave the market with (current value) payoff v \u2212 p and p \u2212 c respectively. If the bargaining between the matched pair breaks down, both traders can either stay in the market waiting for 11 Chapter 2. Dynamic Matching and Two-sided Private Information Bargaining another match (and incur the search costs) as if they were never matched, or simply exit and never come back. We make the following assumptions on the primitives of our model. Assumption 1 (distributions of inflow types) The cumulative distributions F (v) and G(c) of inflow types have densities f(v) and g(c) on (0, 1), bounded away from 0 and \u221e: 0 < f \u2264 f(v) \u2264 f\u0304 <\u221e, 0 < g \u2264 g (c) \u2264 g\u0304 <\u221e. Assumption 2 (matching function) The matching function M is continuous on R2+, nondecreasing in each argument, exhibits constant returns to scale (i.e. homogeneous of degree one), and satisfies M (B,S) = 0 if B = 0 or S = 0. Given the current mass of buyers B > 0 and the mass of sellers S > 0, trading op- portunities for a buyer come at the Poisson arrival rate M (B,S) \/B.16 Similarly, trading opportunities for a seller come at the Poisson arrival rate M (B,S) \/S. It is more convenient to work with a normalized matching function. Let \u03b6 \u2261 B\/S be the ratio of buyers to sellers (or market tightness), and define m(\u03b6) \u2261M(\u03b6, 1). Since the matching technology is assumed to exhibit constant returns to scale, it is easy to see that m(\u03b6) is also equal to M (B,S) \/S, which is a seller\u2019s Poisson arrival rate of being 16That is, M(B,S)\/B is the probability that a buyer is matched over a short time period of length dt divided by the length dt. 12 Chapter 2. Dynamic Matching and Two-sided Private Information Bargaining matched. Similarly, m (\u03b6) \/\u03b6 is equal toM (B,S) \/B, a buyer\u2019s Poisson arrival rate of being matched. We denote these two arrival rates as \u03b1B (\u03b6) and \u03b1S (\u03b6): \u03b1B (\u03b6) \u2261 m (\u03b6) \u03b6 , \u03b1S (\u03b6) \u2261 m (\u03b6) . Assumption 2 implies that 1. \u03b1B (\u03b6) is continuous and nonincreasing; 2. \u03b1S (\u03b6) is continuous and nondecreasing; and 3. \u03b1B (\u221e) = \u03b1S (0) = 0. 2.3 Nontrivial steady-state equilibria Throughout this dissertation we will restrict attention to steady-state equilibria, i.e. ones in which the market distribution of active traders and the agents\u2019 strategies are time-invariant. Like other DMBG in the literature, our model always has an uninteresting perfect Bayesian equilibrium, in which no potential trader enters. Indeed, if no potential trader enters, not to enter is optimal to every potential trader. Throughout this dissertation we will only consider nontrivial equilibria, i.e. ones in which positive entry occurs (or equivalently, positive trade occurs, or the steady-state market mass of active traders is positive). We thus call our equilibrium notion nontrivial steady-state equilibrium, which will be formally defined in a moment. Let WB,WS : [0, 1] \u2192 R+ be the value functions for buyers and sellers: WB(v) is the continuation payoff of a type v buyer whenever he has not traded and is unmatched; and WS(c) is the continuation payoff of a type c seller whenever she has not traded and is unmatched. Let NB, NS : [0, 1]\u2192 R+ be the (stock) market distribution functions: NB (v) is the mass of buyers in the market with valuations less than or equal to v; and NS (c) is the mass of sellers with costs less than or equal to c. (In this notation, B = NB (1) and S = NS (1).) Let \u03c7B, \u03c7S : [0, 1] \u2192 {0, 1} be the entry strategies: buyers with valuation v enter if and only if \u03c7B (v) = 1; sellers with cost c enter if and only if \u03c7S (c) = 1. Also 13 Chapter 2. Dynamic Matching and Two-sided Private Information Bargaining let pB, pS : [0, 1] \u2192 [0, 1] be the proposing strategies: buyers with valuation v propose the take-it-or-leave-it price offer pB (v); sellers with cost c propose pS (c). Sequential optimality requires that the value functions in steady state satisfy the fol- lowing Bellman equations. For a type v buyer, rWB(v) = max \u03c7\u2208{0,1} \u03c7 \u00b7 {\u03b1B(\u03b6)[\u03b2B\u03c0B (v) (2.2) +\u03b2S Z {c:v\u2212pS(c)\u2265WB(v)} (v \u2212 pS(c)\u2212WB(v)) dNS(c) S ]\u2212 \u03baB} where \u03c0B (v) is the buyer\u2019s capital gain when he becomes a proposer: \u03c0B (v) \u2261 max p\u2208[0,1] \u23a7 \u23aa\u23a8 \u23aa\u23a9 Z {c:p\u2212c\u2265WS(c)} (v \u2212 p\u2212WB(v)) dNS(c) S \u23ab \u23aa\u23ac \u23aa\u23ad . (2.3) The buyer\u2019s equilibrium entry strategy \u03c7B (v) must be an optimal value of \u03c7 in (2.2), and his equilibrium proposing strategy pB (v) must be an optimal value of p in (2.3). The intuition is that, contingent on entry, a buyer\u2019s flow value of search rWB(v) is equal to the expected capital gain due to matching a partner, net of the flow search cost. Specifically, the buyer\u2019s proposed price pB (v) is accepted by the seller if her trade surplus is weakly greater than the value of search, i.e. if pB(v) \u2212 c \u2265 WS (c). The seller\u2019s proposed price pS (c) is accepted by the buyer if his trade surplus is weakly greater than his value of search, i.e. if v\u2212 pS(c) \u2265WB(v). When the buyer trades, the capital gain is his trade surplus minus the value of search. The Bellman equation for the sellers has a similar form: for a type c seller, rWS(c) = max \u03c7\u2208{0,1} \u03c7 \u00b7 {\u03b1S(\u03b6)[\u03b2S\u03c0S (c) + (2.4) \u03b2B Z {v:pB(v)\u2212c\u2265WS(c)} (pB (v)\u2212 c\u2212WS(c)) dNB(v) B ]\u2212 \u03baS} where \u03c0S (c) \u2261 max p\u2208[0,1] \u23a7 \u23aa\u23a8 \u23aa\u23a9 Z {v:v\u2212p\u2265WB(v)} (p\u2212 c\u2212WS (c)) dNB(v) B \u23ab \u23aa\u23ac \u23aa\u23ad . (2.5) 14 Chapter 2. Dynamic Matching and Two-sided Private Information Bargaining The seller\u2019s equilibrium entry strategy \u03c7S (c) must be an optimal value of \u03c7 in (2.4), and her equilibrium proposing strategy pS (c) must be an optimal value of p in (2.5). It is convenient to define the trading probabilities in a given meeting, qB (v) for buyers and qS (c) for sellers: qB (v) \u2261 \u03b2B Z {c:pB(v)\u2212c\u2265WS(c)} dNS(c) S + \u03b2S Z {c:v\u2212pS(c)\u2265WB(v)} dNS(c) S , qS (c) \u2261 \u03b2S Z {v:v\u2212pS(c)\u2265WB(v)} dNB(v) B + \u03b2B Z {v:pB(v)\u2212c\u2265WS(c)} dNB (v) B . In steady state, the rate of inflow of the traders of each type is equal to the rate of the outflow due to trading:17 b\u03c7B (v) dF (v) = \u03b1B(\u03b6)qB (v) dNB(v), (2.6) s\u03c7S (c) dG(c) = \u03b1S(\u03b6)qS (c) dNS(c). (2.7) We now formally define nontrivial steady-state equilibrium.18 Definition 1 A tuple (WB,WS , \u03c7B, \u03c7S , pB, pS, NB,NS) is a nontrivial steady-state equilib- rium if B \u2261 NB (1) > 0, S \u2261 NS (1) > 0, equations (2.2), (2.4), (2.6) and (2.7) hold, and \u03c7B, pB, \u03c7S , pS solve the optimization problems in (2.2), (2.3), (2.4) and (2.5) respectively. 2.4 Basic equilibrium properties Our characterization of equilibrium patterns begins with showing that the slopes of equi- librium value functions WB (v) and WS (c) are the corresponding \"ultimate probabilities of trade\", which can be defined as the present value of one dollar to be received at the time of next successful trade. Since every active trader must recover their search costs, these 17Exiting without trade never occurs in steady-state equilibrium. 18We implicitly assume that traders use symmetric pure strategies. But this is essentially without loss of generality and merely for simplicity of exposition. We will come back to this point in the conclusion chapter. 15 Chapter 2. Dynamic Matching and Two-sided Private Information Bargaining ultimate probabilities of trade must be strictly positive on the active regions, i.e. the sup- ports of NB and NS. Therefore the active regions must be intervals [v, 1] and [0, c\u0304] for some v and c\u0304. Furthermore, we show that WB and WS are convex, which implies that trading probabilities qB and qS are monotonic.19 Lemma 1 In any nontrivial steady-state equilibrium, there are marginal entering types v, c\u0304 \u2208 (0, 1) such that the supports of NB and NS are [v, 1] and [0, c\u0304] respectively. Marginal entrants (i.e. type v buyers and type c\u0304 sellers) are indifferent between entering or not, while the entry preferences of all others are strict. {v : \u03c7B (v) = 1} is either [v, 1] or (v, 1]. {c : \u03c7S (c) = 1} is either [0, c\u0304] or [0, c\u0304). WB is absolutely continuous, convex, nondecreasing on [0, 1], strictly increasing on [v, 1], with WB (v) = 0; whenever differentiable, W 0B(v) = \u03c7B (v) \u03b1B (\u03b6) qB (v) r + \u03b1B (\u03b6) qB (v) . (2.8) WS is absolutely continuous, convex, nonincreasing on [0, 1], strictly decreasing on [0, c\u0304], with WS (c\u0304) = 0; whenever differentiable, W 0S(c) = \u2212\u03c7S (c) \u03b1S (\u03b6) qS (c) r + \u03b1S (\u03b6) qS (c) . (2.9) The trading probability qB is strictly positive and nondecreasing on [v, 1], while qS is strictly positive and nonincreasing on [0, c\u0304]. Proof. We prove the results for buyers only. We use an argument from mechanism design. For any v \u2208 [0, 1], define tB (v) \u2261 \u03b2B Z {c:pB(v)\u2212c\u2265WS(c)} pB (v) dNS(c) S +\u03b2S Z {c:v\u2212pS(c)\u2265WB(v)} pS(c) dNS(c) S . The buyers\u2019 Bellman equation (2.2) implies for any v, v\u0302 \u2208 [0, 1] and any \u03c7 \u2208 {0, 1}, rWB (v) \u2265 \u03c7 \u00b7 {\u03b1B [qB (v\u0302) v \u2212 tB (v\u0302)\u2212 qB (v\u0302)WB (v)]\u2212 \u03baB} 19Lemma 1 is generally true for any bargaining protocol, as long as the bargainers\u2019 types are private information. We therefore provide a proof that can easily be generalized. 16 Chapter 2. Dynamic Matching and Two-sided Private Information Bargaining or equivalently WB (v) \u2265 \u03c7 \u00b7 uB (v, v\u0302) where uB (v, v\u0302) \u2261 \u03b1B [qB (v\u0302) v \u2212 tB (v\u0302)]\u2212 \u03baB r + \u03b1BqB (v\u0302) . And the inequality becomes equality if v\u0302 = v and \u03c7 = \u03c7B (v). Let UB (v)\u2261maxv\u0302\u2208[0,1] uB (v, v\u0302). We then have WB (v) = \u03c7B (v)uB (v, v) = \u03c7B (v)UB (v) = max {UB (v) , 0}. For any v\u0302, uB (v, v\u0302) is affine and nondecreasing in v. Milgrom and Segal (2002) Envelope Theorem im- plies UB (v) is absolutely continuous, convex, nondecreasing, and with slope \u03b1BqB (v) \/(r+ \u03b1BqB (v)) whenever differentiable. The same properties are inherited by WB (v), except that its slope becomes \u03c7B (v)\u03b1BqB (v) \/(r + \u03b1BqB (v)). Obviously UB (0) < 0. Let v \u2261 sup {v \u2208 [0, 1] : UB (v) < 0}. By continuity of UB, we have v > 0 and UB (v) \u2264 0. But UB (v) < 0 is impossible in nontrivial equilibrium because it implies \u03c7B (v) = 0 \u2200v \u2208 [0, 1] and hence B = 0. Thus UB (v) = WB (v) = 0. By monotonicity of UB, for all v < v, we have UB (v) < 0 and hence \u03c7B (v) = WB (v) = 0. Moreover, qB (v) > 0 for all v \u2265 v. It is because for all v \u2265 v, the fact UB (v) \u2265 0 implies \u03b1BqB (v) \u2265 \u03baB > 0. It furthermore implies U 0B (v+) \u2265 \u03b1BqB (v+) \/(r + \u03b1BqB (v+)) > 0. Thus for all v > v, we have UB (v) > 0 and hence \u03c7B (v) = 1 and WB (v) = UB (v). From steady-state equation (2.6), [v, 1] is the support of NB. Since the inflow distribution F does not have atom point, neither does NB. Hence B > 0 implies v < 1. Finally, the convexity of UB implies that qB is nondecreasing on [v, 1]. We call the v and c\u0304 in Lemma 1 the buyers\u2019 and sellers\u2019 marginal entering types, and call the traders with such types marginal entrants. Since the flow and stock masses of marginal entrants (who are indifferent between entering or not) are zero anyway, we will without loss of generality assume throughout they enter, i.e. \u03c7B (v) = \u03c7S (c\u0304) = 1. Before providing further equilibrium properties, let us make a note on the traders\u2019 bargaining strategies (i.e. proposing and responding strategies) by introducing a pair of important notions. Define \u03c1B (v) \u2261 v \u2212WB (v) , (2.10) 17 Chapter 2. Dynamic Matching and Two-sided Private Information Bargaining \u03c1S (c) \u2261 c+WS (c) . (2.11) We call \u03c1B (v) type v buyers\u2019 dynamic valuation, and \u03c1S (c) type c sellers\u2019 dynamic cost. Both of them are called dynamic types. The reason is that, as far as we are concerned with bargaining between a buyer and a seller in our dynamic model, \u03c1B (v) and \u03c1S (c) play the same roles as v and c do in a static bargaining game. Indeed, as captured in our equilibrium definition, a type v buyer is willing to accept a price offer p if and only if p \u2264 \u03c1B (v); and a type c seller is willing to accept a price offer p if and only if p \u2265 \u03c1S (c). Thus the dynamic types fully characterize the responding strategies played in equilibrium. Furthermore, the proposing problems in (2.3) and (2.5) are nothing more than static take- it-or-leave-it problems with types replaced by dynamic types. Lemma 1 implies that \u03c1B and \u03c1S are absolutely continuous and increasing: \u03c10B(v) = r r + \u03b1B (\u03b6) qB (v) > 0 a.e. v \u2208 [v, 1] (2.12) \u03c10S(c) = r r + \u03b1S (\u03b6) qS (c) > 0 a.e. c \u2208 [0, c\u0304] . (2.13) Since WB(v) = WS(c\u0304) = 0, the marginal entering types are equal to the corresponding dynamic types: \u03c1S (c\u0304) = c\u0304, \u03c1B (v) = v. Thus the buyers\u2019 lowest and highest reservation prices are v and \u03c1B (1). The sellers\u2019 lowest and highest reservation prices are \u03c1S (0) and c\u0304. We will see in the next lemma that the proposing strategies pB and pS are also monotonic on the active intervals [v, 1] and [0, c\u0304]. Thus the lowest and highest price offers by buyers are pB (v) and pB (1). The lowest and highest price offers by sellers are pS (0) and pS (c\u0304). Lemma 2 In any nontrivial steady-state equilibrium, (a) for all v \u2208 [v, 1], \u03c1B (v) > pB (v) \u2208 [\u03c1S (0) , c\u0304]; for all c \u2208 [0, c\u0304], \u03c1S (c) < pS (c) \u2208 [v, \u03c1B (1)]; 18 Chapter 2. Dynamic Matching and Two-sided Private Information Bargaining (b) the proposing strategies pB(v) and pS(c) are nondecreasing on [v, 1] and [0, c\u0304] re- spectively; (c) \u03b1B (\u03b6)\u03b2B\u03c0B (v) = \u03baB and \u03b1S (\u03b6)\u03b2S\u03c0S (c\u0304) = \u03baS. Proof. Step 1: Suppose, by way of contradiction, pB(v) > c\u0304 for some v \u2208 [v, 1]. Then pB(v) is accepted by any active seller (because \u03c1S (c) is increasing in c). A type v buyer can lower his offer without losing acceptance probability. But then pB (v) does not solve the proposing problem in (2.3). Therefore pB(v) \u2264 c\u0304 for all v \u2208 [v, 1]. Similarly pS (c) \u2265 v for all c \u2208 [0, c\u0304]. Step 2: The buyers with type v cannot get positive bargaining surplus when he is a responder, i.e. the second term inside the square bracket of (2.2), evaluated at v = v, is 0. It is because, from step 1, v \u2212WB (v) = v is no higher than pS (c) proposed by any active seller. Then, since WB(v) = 0 from Lemma 1, the Bellman equation (2.2) evaluated at v = v implies \u03b1B(\u03b6)\u03b2B\u03c0B (v)\u2212\u03baB = 0. It follows that \u03c0B (v) > 0 and hence \u03c0B (v) > 0 for all v \u2208 [v, 1] (because any buyer can choose p = pB (v) in his proposing problem in (2.3)). Similarly, we can prove \u03b1S (\u03b6)\u03b2S\u03c0S (c\u0304)\u2212 \u03baS = 0 and \u03c0S (c) > 0 for all c \u2208 [0, c\u0304]. Step 3: Fix any v \u2208 [v, 1]. From \u03c0B (v) > 0 given by step 2, we have v\u2212pB(v) > WB(v) (or equivalently \u03c1B(v) > pB(v)) and pB(v) \u2212 c \u2265 WS (c) for some c. The last result is equivalent to pB(v) \u2265 \u03c1S (0) because \u03c1S (c) is increasing in c. Similarly we can prove for all c \u2208 [0, c\u0304], \u03c1S (c) < pS (c) \u2264 \u03c1B (1). Step 4: Let \u0393S (p) \u2261 R {c:p\u2212c\u2265WS(c)} dNS(c) S . Obviously \u0393S is nondecreasing. Then the buyers\u2019 proposing problem in (2.3) can be written as \u03c0B (v) = maxp\u2208[0,1][v \u2212 WB (v) \u2212 p]\u0393S (p). Pick any v1, v2 \u2208 [v, 1]. Let p1 \u2261 pB (v1) and p2 \u2261 pB (v2). Revealed preference implies [v1 \u2212WB (v1)\u2212 p1]\u0393S (p1) \u2265 [v1 \u2212WB (v1)\u2212 p2]\u0393S (p2) (2.14) and [v2 \u2212WB (v2)\u2212 p2]\u0393S (p2) \u2265 [v2 \u2212WB (v2)\u2212 p1]\u0393S (p1) . 19 Chapter 2. Dynamic Matching and Two-sided Private Information Bargaining Sum these two inequalities and then simplify. We obtain [(v2 \u2212WB (v2))\u2212 (v1 \u2212WB (v1))] \u00b7 [\u0393S (p2)\u2212 \u0393S (p1)] \u2265 0. Suppose, by way of contradiction, v2 > v1 and p2 < p1. Then the above inequality implies \u0393S (p2) \u2265 \u0393S (p1) and the monotonicity of \u0393S implies \u0393S (p2) \u2264 \u0393S (p1). We thus have \u0393S (p2) = \u0393S (p1) > 0, where the last inequality is from step 2. Substitute back into (2.14), we have p2 \u2265 p1, a contradiction. The intuition is, in equilibrium, the marginal entrants do not get bargaining surplus in responding stages (the worst types in the market do not have information rent) so that these marginal entrants must earn positive surpluses in proposing stages, otherwise they cannot recover their search costs. Since even the marginal entrants earn positive proposing surplus, all entrants do as well. Then any buyer\u2019s offer must be lower than his dynamic valuation and within the support of sellers\u2019 reservation prices. Of course, a symmetric argument can be made by switching the roles of buyers and sellers. We thus have part (a) of Lemma 2. Part (b), the monotonicity of proposing strategies, is due to standard revealed-preference argument. Part (c), the marginal type equations, simply says that, for the marginal entrants to be indifferent between entering or not, their expected gain from searching in the market, net of search cost, must be zero. (Recall that marginal entrants makes positive bargaining surplus only when they are proposer.) In equilibrium, it could be the case that v \u2264 c\u0304, or v > c\u0304. If the former one is the case, we say it is an equilibrium with overlapping supports. If the latter one is the case, we say it is an equilibrium with separated supports. Figure 2.1 and Figure 2.2 visualize the pattern of proposing and responding strategies of a possible equilibrium of each kind. Before closing this section, we compare the equilibrium price range with the Walrasian price. Define Walrasian price p\u2217 as the price that clears the flow demand and flow supply:20 b[1\u2212 F (p\u2217)] = sG (p\u2217) . 20This is the appropriate concept of market-clearing price in the steady-state context, as first pointed out by Gale (1987). 20 Chapter 2. Dynamic Matching and Two-sided Private Information Bargaining ,c v 0 1 1 )(vpB ( )vB\u03c1 )(cS\u03c1 )(cpS cv Responding interval Proposing interval Figure 2.1: Proposing and responding strategies in an equilibrium with overlapping supports (which must be non-full-trade) ,c v 0 1 1 )(vpB ( )vB\u03c1 )(cS\u03c1 )(cpS c v Responding interval Proposing interval Figure 2.2: Proposing and responding strategies in a non-full-trade equilibrium with sepa- rated supports 21 Chapter 2. Dynamic Matching and Two-sided Private Information Bargaining Also define the responding interval as [\u03c1S (0) , \u03c1B (1)], and the proposing interval as [pB (v), pS (c\u0304)]. We see from Lemma 2(a) that [pB (v) , pS (c\u0304)] \u2282 [\u03c1S (0) , \u03c1B (1)]. In words, the proposing interval is contained in the responding interval. Since buyers and sellers always leave the market in pairs, the entry flows must also be balanced in steady state, i.e. b[1 \u2212 F (v)] = sG (c\u0304).21 Then it is clear that the marginal entering types v and c\u0304 must be on different sides of the Walrasian price p\u2217. Then from Lemma 2(a), it is not hard to prove that Walrasian price p\u2217 must fall within the proposing interval [pB (v) , pS (c\u0304)]. Lemma 3 In any nontrivial steady-state equilibrium, p\u2217 \u2208 [pB (v) , pS (c\u0304)] \u2282 [\u03c1S (0) , \u03c1B (1)]. Proof. The second inclusion is straight implication of Lemma 2(a). To see the first inclusion, simply notice that pB (v) \u2264 min {c\u0304, v} \u2264 p\u2217 \u2264 max {c\u0304, v} \u2264 pS (c\u0304) . The first and last inequalities are from Lemma 2(a). The other two inequalities in the middle are due to the facts that b[1\u2212 F (v)] = sG (c\u0304) and b[1\u2212 F (p\u2217)] = sG (p\u2217). 2.5 Full-trade equilibria Although the previous section provides a series of results that characterize equilibrium patterns, in general there is no analytic solution for a nontrivial steady-state equilibrium. However, we have more to say about the qualitative properties of equilibria. There are two qualitatively different possibilities that could happen in equilibrium. First, it may happen that in an equilibrium every meeting results in a trade. In contrast, it can be that not every meeting results in a trade. We call these two types of equilibria full-trade equilibria and non-full-trade equilibria respectively. 21 It can be formally derived from (2.6) and (2.7). 22 Chapter 2. Dynamic Matching and Two-sided Private Information Bargaining Definition 2 A nontrivial steady-state equilibrium is called a full-trade equilibrium if in this equilibrium every meeting results in a trade. A nontrivial steady-state equilibrium is called a non-full-trade equilibrium if it is not a full-trade equilibrium. The above definition is sensible for any bargaining mechanism, and for either private information or full information bargaining. In the current context, Lemma 2(a) implies that full-trade equilibria must have the following properties: (i) the supports for active buyers\u2019 types and active sellers\u2019 types are separate, i.e. v > c\u0304; (ii) the lowest buyers\u2019 offer pB (v) is exactly at the level acceptable to all active sellers, i.e. pB (v) = c\u0304; and (iii) the highest sellers\u2019 offer pS (c\u0304) is exactly at the level acceptable to all active buyers, i.e. pS (c\u0304) = v. It is easy to see that the converse is also true. (Clearly, a full-trade equilibrium must be with separated supports; or equivalently an equilibrium with overlapping supports must be non- full-trade.) Thus we could alternatively define a full-trade equilibrium to be a nontrivial steady-state equilibrium with pB (v) = c\u0304 and pS (c\u0304) = v. Non-full-trade equilibria are illustrated in Figure 2.1 and Figure 2.2 in the previous section. Figure 2.3 illustrates the qualitative features of strategies played in a full-trade equilibrium. In particular, the proposing strategies must be flat and the dynamic type functions must be linear.22 We are interested in full-trade equilibria for several reasons. First, our uniqueness and existence results are closely related to full-trade equilibria. Second, our discussions on bargaining efficiency and the effect of information structure on entry in the next chapter will also be intimately related to full-trade equilibria. Third, full-trade equilibria admit a very simple characterization, which we present now.23 In full-trade equilibria (if any), the marginal type equations in Lemma 2(c) take the 22 If a full-trade equilibrium and a non-full-trade equilibrium coexist (whether they can coexist is an open question), our results do not imply the full-trade equilibrium Pareto dominates the non-full-trade one. Indeed, the non-full-trade equilibrium could have more entry (i.e. lower v and higher c\u0304) than the full-trade one, so that the marginal entrants strictly prefer the non-full-trade equilibrium. 23As a matter of fact, the analysis of Mortensen and Wright (2002) is based only on full-trade equilibria (although they do not use this term). 23 Chapter 2. Dynamic Matching and Two-sided Private Information Bargaining ,c v 0 1 1 )(vpB ( )vB\u03c1 )(cS\u03c1 )(cpS vc Figure 2.3: Proposing and responding strategies in a full-trade equilibrium form \u03b1B(\u03b6)\u03b2B (v \u2212 c\u0304) = \u03baB, (2.15) \u03b1S(\u03b6)\u03b2S (v \u2212 c\u0304) = \u03baS. (2.16) Noticing that \u03b1S(\u03b6)\/\u03b1B(\u03b6) = \u03b6, (2.15) and (2.16) can be easily solved for \u03b6 and v \u2212 c\u0304: \u03b6 = \u03b2B \u03b2S \u03baS \u03baB \u2261 \u03b60, (2.17) v \u2212 c\u0304 = K (\u03b60) , (2.18) where K (\u03b6) \u2261 \u03baB \u03b1B (\u03b6) + \u03baS \u03b1S (\u03b6) \u2200\u03b6. (2.19) In steady state, the inflow of active buyers must equal the inflow of active sellers: b[1\u2212 F (v)] = sG (c\u0304) . (2.20) Since v \u2212 c\u0304 is determined from (2.18), v and c\u0304 are uniquely pinned down by (2.20). It is clear that equations (2.18) and (2.20) have a solution for v < 1 and c\u0304 > 0 if and only if 24 Chapter 2. Dynamic Matching and Two-sided Private Information Bargaining K (\u03b60) < 1. Let us suppose K (\u03b60) < 1 and denote such a solution for (v, c\u0304) by (v0, c\u03040). That is to say, a full-trade equilibrium, if exists, must have its buyer-seller ratio and marginal entering types given by (\u03b60, v0, c\u03040). Other equilibrium objects are also easily obtained. In particular, \u03c7B(v) = \u23a7 \u23a8 \u23a9 1 if v \u2265 v0 0 otherwise , \u03c7S(c) = \u23a7 \u23a8 \u23a9 1 if c \u2264 c\u03040 0 otherwise , WB(v) = \u03c7B (v) \u03b1B(\u03b60) r + \u03b1B(\u03b60) (v \u2212 v0) WS(c) = \u03c7S(c) \u03b1S(\u03b60) r + \u03b1S(\u03b60) (c\u03040 \u2212 c) NB(v) = \u03c7B (v) b [F (v)\u2212 F (v0)] \u03b1B (\u03b60) NS(c) = [1\u2212 \u03c7S(c)] sG(c\u03040) \u03b1S (\u03b60) + \u03c7S(c) sG(c) \u03b1S (\u03b60) pB (v) = c\u03040 \u2200v \u2265 v0 pS (c) = v0 \u2200c \u2264 c\u03040. Throughout this dissertation, we identify an equilibrium with another one if they differs only in the proposing strategies of non-entrants and entry strategies of marginal entrants. Under this convention, there is at most one full-trade equilibrium. We call the above possible full-trade equilibrium the full-trade equilibrium candidate. We have seen that a unique full-trade equilibrium candidate exists if and only ifK (\u03b60) < 1.24 The function K (\u03b6), especially the value K (\u03b60), will play an important role in our analysis. It can be interpreted as the expected search costs incurred by a pair of buyer and seller when the buyer-seller ratio is \u03b6 and there is no discounting. In the full-trade equilibrium, this expected search cost, K (\u03b60), is equal to the entry gap v0 \u2212 c\u03040, as shown in (2.18). This value K (\u03b60) has yet an alternative interpretation. The following simple lemma, which will be used many times in our proofs, shows that K (\u03b60) can be interpreted 24Recall that for expositional simplicity we have assumed that the types are distributed on [0, 1]. If the support is [a1, a2], then the condition would read K (\u03b60) < a2 \u2212 a1. 25 Chapter 2. Dynamic Matching and Two-sided Private Information Bargaining 0 BB \u03b2\u03b6\u03b1 \u03ba )( B 0\u03b6 SS \u03b2\u03b6\u03b1 \u03ba )( S \u03b6 )( 0\u03b6K Figure 2.4: Interpretation of \u03b60 and K (\u03b60) either as a maximin or a minimax value of adjusted accumulated search costs until the next meeting. Lemma 4 For any matching function satisfying Assumption 2, we have K (\u03b60) = max\u03b6>0 min \u00bd \u03baB \u03b1B (\u03b6)\u03b2B , \u03baS \u03b1S (\u03b6)\u03b2S \u00be = min \u03b6>0 max \u00bd \u03baB \u03b1B (\u03b6)\u03b2B , \u03baS \u03b1S (\u03b6)\u03b2S \u00be = \u03baB \u03b1B (\u03b60)\u03b2B = \u03baS \u03b1S (\u03b60)\u03b2S . Proof. Consult Figure 2.4. Note that \u03b1B (\u03b6) is a nonincreasing function, while \u03b1S (\u03b6) is a nondecreasing function. The maximin and minimax values are realized at the intersection of the curves \u03baB \u03b1B (\u03b6)\u03b2B = \u03baS \u03b1S (\u03b6)\u03b2S which occurs if and only if \u03b6 = \u03b60. Corollary 1 For any matching function satisfying Assumption 2, the following statements are equivalent. (i) K (\u03b60) < 1. (ii) For some \u03b6 > 0, we have \u03b1B (\u03b6)\u03b2B > \u03baB and 26 Chapter 2. Dynamic Matching and Two-sided Private Information Bargaining \u03b1S (\u03b6)\u03b2S > \u03baS. (iii) For all \u03b6 > 0, we have \u03b1B (\u03b6)\u03b2B > \u03baB or \u03b1S (\u03b6)\u03b2S > \u03baS. (iv) \u03b1B (\u03b60)\u03b2B > \u03baB. (v) \u03b1S (\u03b60)\u03b2S > \u03baS.25 Even if K (\u03b60) < 1, so that a full-trade equilibrium candidate exists, this candidate may not constitute an equilibrium, since buyers may have an incentive to bid lower than c\u03040, and similarly sellers may have an incentive to bid above v0. Nevertheless, Theorem 1, which is the main result of this section, provides a necessary and sufficient condition under which such deviations are unprofitable and hence a full-trade equilibrium exists. Before stating this main result, we need to introduce the so-called virtual types of buyers and sellers. The buyers\u2019 and sellers\u2019 virtual type functions are respectively defined as: JB(v) \u2261 v \u2212 1\u2212 F (v) f(v) , JS(c) \u2261 c+ G(c) g(c) . It is well-known that the virtual type functions are nondecreasing for most usual probability distributions. We therefore take the monotonicity of JB and JS as a regularity condition. And this condition guarantees that only the first-order conditions for the proposers\u2019 prob- lems in (2.3) and (2.5) are sufficient for optimal proposing. Now we are ready to state the main theorem of this section. Theorem 1 (Existence of full-trade equilibrium) Assume the regularity condition that the virtual type functions JB and JS are nondecreasing. Then a (unique) full-trade equilib- rium exists if and only if (i) K (\u03b60) < 1 where \u03b60 \u2261 \u03b2B\u03baS\/\u03b2S\u03baB, and (ii) r \u2264 r\u2217 where r\u2217 is given by: r\u2217 \u2261 min \u00bd \u03baB\/\u03b2B max {c\u03040 \u2212 JB (v0) , 0} , \u03baS\/\u03b2S max {JS (c\u03040)\u2212 v0, 0} \u00be . (2.21) (If both denominators are 0, there is no upper bound so a full-trade equilibrium exists for all r. In this case we define r\u2217 =\u221e.) 25We will see in Section 2.7 that all these statements are equivalent to the existence of some nontrivial steady-state equilibrium. 27 Chapter 2. Dynamic Matching and Two-sided Private Information Bargaining Proof. We have already seen that a unique full-trade equilibrium candidate described before exists if and only if K (\u03b60) < 1. For proving existence of full-trade equilibrium, it suffices to verify that this candidate is really an equilibrium. Almost all equilibrium conditions are satisfied by construction, except that we need to verify pB (v) = c\u03040 \u2200v \u2265 v0 and pS (c) = v0 \u2200c \u2264 c\u03040 are buyers\u2019 and sellers\u2019 optimal proposing strategies. For notational simplicity, we omit the subscript \"0\". We focus on sellers\u2019 proposing problem in (2.5), which, according to our construction of the equilibrium candidate, can be rewritten as maxp\u2208[0,1] \u03c0\u0302S (c, p), where \u03c0\u0302S (c, p) = \u00b5 p\u2212 rc+ \u03b1S c\u0304 r + \u03b1S \u00b6Z 1 v I \u2219 p \u2264 rv + \u03b1Bv r + \u03b1B \u00b8 dF (v) 1\u2212 F (v) where I [\u00b7] is 1 if the condition inside the bracket holds, and is 0 otherwise. Notice that \u2202\u03c0\u0302S (c, p) \/\u2202p = 1 if p < v, so that any p < v is not optimal; moreover any p \u2265 r+\u03b1Bvr+\u03b1B is also not optimal because it implies \u03c0\u0302S (c, p) = 0. The partial derivative of \u03c0\u0302S w.r.t. p for any p \u2208 h v, r+\u03b1Bvr+\u03b1B i (it is right-hand derivative at the left boundary; and left-hand derivative at the right boundary) is: \u2202\u03c0\u0302S (c, p) \u2202p = \u2212 f \u00b3 (r+\u03b1B)p\u2212\u03b1Bv r \u00b4 1\u2212 F (v) r + \u03b1B r \u23a7 \u23a8 \u23a9 rJB \u00b3 (r+\u03b1B)p\u2212\u03b1Bv r \u00b4 + \u03b1Bv r + \u03b1B \u2212 rc+ \u03b1S c\u0304 r + \u03b1S \u23ab \u23ac \u23ad . For p = v being optimal for all c \u2264 c\u0304, a necessary condition is that \u2202\u03c0\u0302S (c\u0304, p) \/\u2202p \u2264 0 at p = v, because otherwise a type c\u0304 seller would deviate upward. This is also a sufficient condition because (i) \u2202\u03c0\u0302S (c, p) \/\u2202p is increasing in c, so that \u2202\u03c0\u0302S (c\u0304, p) \/\u2202p \u2264 0 implies \u2202\u03c0\u0302S (c, p) \/\u2202p \u2264 0 \u2200c \u2264 c\u0304; and (ii) due to the monotonicity of JB, \u2202\u03c0\u0302S (c, p) \/\u2202p \u2264 0 at p = v implies \u2202\u03c0\u0302S (c, p) \/\u2202p \u2264 0 at any p \u2208 h v, r+\u03b1Bvr+\u03b1B i . That is, we only need to verify rJB (v) + \u03b1Bv r + \u03b1B \u2212 c\u0304 \u2265 0. Similarly considering the buyers\u2019 proposing problem, we would see that pB (v) = c\u0304 \u2200v \u2265 v is optimal if and only if v \u2212 rJS (c\u0304) + \u03b1S c\u0304 r + \u03b1S \u2265 0. 28 Chapter 2. Dynamic Matching and Two-sided Private Information Bargaining Thus full-trade equilibrium exists if and only if both of these two inequalities hold, or equivalently, r \u2264 min \u00bd \u03b1B (\u03b6) (v \u2212 c\u0304) max {c\u0304\u2212 JB (v) , 0} , \u03b1S (\u03b6) (v \u2212 c\u0304) max {JS (c\u0304)\u2212 v, 0} \u00be . Finally, applying the full-trade equilibrium marginal type equations (2.15) and (2.16), we obtain the upper bound r\u2217 in (2.21). Corollary 2 Suppose the virtual type functions JB and JS are nondecreasing. Then, (a) In the region where r\u2217 < \u221e, if \u03baB and \u03baS increase, then r\u2217 increases, and vice versa. (b) Given any r > 0, there is a \u03ba\u0304 > 0 such that full-trade equilibrium does not exist whenever \u03baB, \u03baS < \u03ba\u0304. (c) Given any r > 0, a full-trade equilibrium exists when (\u03baB, \u03baS) is such that K(\u03b60) is less than but sufficiently close to 1. (d) Given any (\u03baB, \u03baS) such that K(\u03b60) < 1, a full-trade equilibrium exists when r is sufficiently close to 0. Proof. Consult Figure 2.4. The curve \u03baB\u03b1B(\u03b6)\u03b2B shifts up when \u03baB goes up. The curve \u03baS \u03b1S(\u03b6)\u03b2S shifts up when \u03baS goes up. Both of the two curves pointwise converge to 0 on {\u03b6 : \u03b6 > 0} as (\u03baB, \u03baS)\u2192 0. Obviously, K (\u03b60), as the height of the intersection, increases as \u03baB and \u03baS increase, and vice versa. An increase in K (\u03b60) in turn implies that v0 rises and c\u03040 drops, and and vice versa. Also, as (\u03baB, \u03baS)\u2192 0, we have K (\u03b60)\u2192 0, v0 \u2192 p\u2217 and c\u03040 \u2192 p\u2217. As K (\u03b60)\u2192 1 from below, we have v0 \u2192 1 and c\u03040 \u2192 0. From monotonicity of JB and JS , c\u03040 \u2212 JB (v0) and JS (c\u03040) \u2212 v0 drop as \u03baB and \u03baS increase. Then (a) follows. To prove (b), it suffices to prove r\u2217 \u2192 0 as (\u03baB, \u03baS) \u2192 0. Notice that c\u03040 \u2212 JB (v0) \u2265 c\u03040\u2212v0+(1\u2212v0)f\/f\u0304 and JS (c\u03040)\u2212v0 \u2265 c\u03040+ c\u03040g\/g\u0304\u2212v0. Therefore, as v0 \u2192 p\u2217 and c\u03040 \u2192 p\u2217, lim inf [c\u03040 \u2212 JB (v0)] \u2265 (1\u2212 p\u2217)f\/f\u0304 > 0 and lim inf [JS (c\u03040)\u2212 v0] \u2265 p\u2217g\/g\u0304 > 0. As a result, r\u2217 \u2192 0 as (\u03baB, \u03baS)\u2192 0, and (b) follows. To prove (c), notice that c\u03040 \u2212 JB (v0) \u2264 c\u03040 \u2212 v0 + (1 \u2212 v0)f\u0304\/f and JS (c\u03040) \u2212 v0 \u2264 c\u03040 + c\u03040g\u0304\/g \u2212 v0. Thus both of them are negative when v0 and c\u03040 are sufficiently close to 1 29 Chapter 2. Dynamic Matching and Two-sided Private Information Bargaining and 0 respectively. But v0 and c\u03040 can be made arbitrarily close to 1 and 0 respectively by letting K(\u03b60) be less than but close enough to 1. Hence r\u2217 = \u221e if K(\u03b60) is less than but close to 1. Then (c) follows. (d) is simply from r\u2217 > 0 for any \u03baB, \u03baS > 0 such that K(\u03b60) < 1. Remark 1 We need the monotonicities of virtual type functions JB and JS only in the proof of Theorem 1 and the proof of Corollary 2. Moreover, even if we do not assume these monotonicities, r \u2264 r\u2217 is still a necessary condition for the existence of full-trade equilibrium. 2.6 Uniqueness of equilibrium In this section we will show that a full-trade equilibrium is a unique equilibrium for small r. That is to say, there cannot be a non-full-trade equilibrium when r is small. The proof of this will utilize the following lemma. Lemma 5 In any nontrivial steady-state equilibrium, we have 1 > \u03c1B (1)\u2212 \u03c1S (0) > K (\u03b60) , (2.22) v \u2212 c\u0304 \u2264 K (\u03b60) . (2.23) Proof. Pick any nontrivial steady-state equilibrium. Lemma 1 implies WB (1) > 0 and WS (0) > 0. The first inequality in (2.22), which is equivalent to WB (1) +WS (0) > 0, follows. From the definition of \u03c0B and Lemma 2(a), we have \u03c0B (v) \u2264 v \u2212 pB (v) < \u03c1B (1)\u2212 \u03c1S (0). Then Lemma 2(c) implies \u03b1B (\u03b6)\u03b2B (\u03c1B (1)\u2212 \u03c1S (0)) > \u03baB. We can similarly prove \u03b1S (\u03b6)\u03b2S (\u03c1B (1)\u2212 \u03c1S (0)) > \u03baS . 30 Chapter 2. Dynamic Matching and Two-sided Private Information Bargaining It follows that \u03c1B (1)\u2212 \u03c1S (0) > max \u00bd \u03baB \u03b1B (\u03b6)\u03b2B , \u03baS \u03b1S (\u03b6)\u03b2S \u00be \u2265 K (\u03b60) . The last inequality is from Lemma 4. It proves the second inequality in (2.22). To prove (2.23), notice that \u03c0B (v) \u2265 v \u2212 c\u0304 because a type v buyer can always propose p = c\u0304 in his proposing problem (2.3) and this offer would be accepted with probability 1. Thus Lemma 2(c) implies \u03b1B (\u03b6)\u03b2B (v \u2212 c\u0304) \u2264 \u03baB. Similarly, we have \u03c0S (c\u0304) \u2265 v \u2212 c\u0304, so that \u03b1S (\u03b6)\u03b2S (v \u2212 c\u0304) \u2264 \u03baS . It follows that v \u2212 c\u0304 \u2264 min \u00bd \u03baB \u03b1B (\u03b6)\u03b2B , \u03baS \u03b1S (\u03b6)\u03b2S \u00be \u2264 K (\u03b60) . (2.24) The last inequality is again from Lemma 4. Lemma 5 is of interest on its own. Firstly, (2.22) implies an (nontrivial steady-state) equilibrium could exist only if K (\u03b60) < 1. Moreover, (2.22) can be written as WB (1) + WS (0) \u2208 (0, 1\u2212K (\u03b60)). It means in equilibrium the joint lifetime payoff of the best buyer- seller pair (i.e. type 1 buyer and type 0 seller) must be positive but smaller than their gains from trade, net of the expected accumulated search costs evaluated at \u03b6 = \u03b60. Roughly speaking, (2.23) says that in equilibrium the entry gap v\u2212 c\u0304 cannot be too large relative to the search costs, otherwise extramarginal traders would have strict incentives to enter. In order to prove non-full-trade equilibria cannot exist when r is close to 0, recall that a non-full-trade equilibrium could be either with overlapping supports (i.e. v \u2264 c\u0304), or with separated supports (i.e. v > c\u0304). Now we shall claim neither exists for small r. The following lemma implies that an equilibrium with overlapping supports cannot exist whenever r is lower than the search costs \u03baB and \u03baS . Lemma 6 In any nontrivial steady-state equilibrium, v \u2212 c\u0304 \u03c1B (1)\u2212 \u03c1S (0) > \u03ba\u2212 r r + \u03ba 31 Chapter 2. Dynamic Matching and Two-sided Private Information Bargaining where \u03ba \u2261 min {\u03baB, \u03baS}. Proof. Pick any nontrivial steady-state equilibrium. Step 1. Since qB is nondecreasing (from Lemma 1), qB(v) \u2265 qB (v) \u2265 \u03b2B Z {c:pB(v)\u2212c\u2265WS(c)} dNS (c) S for any v \u2265 v. Step 2. From Lemma 2(a) we have v > pB (v) \u2265 \u03c1S (0). Step 3. Combining the previous two steps and Lemma 2(c), we obtain \u03b1BqB(v) (v \u2212 \u03c1S (0)) \u2265 \u03baB \u2200v \u2265 v. From Lemma 1, we have \u03c10B (v) = 1\u2212W 0B (v) = r r + \u03b1BqB(v) \u2264 r r + \u03baB\/(v \u2212 \u03c1S (0)) . Hence \u03c1B (1)\u2212 v = Z 1 v \u03c10B(v)dv \u2264 r r + \u03baB\/(v \u2212 \u03c1S (0)) < r \u03baB\/(v \u2212 \u03c1S (0)) , \u03c1B (1)\u2212 v v \u2212 \u03c1S (0) < r \u03baB , \u03c1B (1)\u2212 v \u03c1B (1)\u2212 \u03c1S (0) = (\u03c1B (1)\u2212 v)\/(v \u2212 \u03c1S (0)) 1 + (\u03c1B (1)\u2212 v)\/(v \u2212 \u03c1S (0)) < r\/\u03baB 1 + (r\/\u03baB) = r r + \u03baB \u2264 r r + \u03ba , where \u03ba \u2261 min{\u03baB, \u03baS}. Step 4. Repeat the previous three steps with the roles of buyers and sellers interchanged, we can also get c\u0304\u2212 \u03c1S (0) \u03c1B (1)\u2212 \u03c1S (0) < r r + \u03ba . Sum these two inequalities up and rearrange terms. Then we get the desired inequality. Corollary 3 If r \u2264 \u03ba \u2261 min {\u03baB, \u03baS}, then a non-full-trade equilibrium with overlapping supports cannot exist (i.e. any nontrivial steady-state equilibrium has v > c\u0304). 32 Chapter 2. Dynamic Matching and Two-sided Private Information Bargaining Now we turn to the proof that a non-full-trade equilibrium with separated supports cannot exist. It is based on the following idea. As r \u2192 0, the dynamic types \u03c1B and \u03c1S , as functions of v and c, get flat, so that the support of dynamic types narrows down to a singleton. Consequently, a marginal entering trader who makes an interior offer in the support of his partner\u2019s dynamic types gains little relative to proposing at the boundary of the support (i.e. seller proposing v and buyer proposing c\u0304), but risks a substantially reduced probability of trading. We are able to show that bidding the endpoint of the support is the best response, so for small r it must be that pB (v) = c\u0304 and pS (c\u0304) = v. This leads to the following uniqueness result. Theorem 2 (Uniqueness of equilibrium) There is at most one nontrivial steady-state equilibrium, which is full-trade, if r \u2264 r where r is given by: r \u2261 \u03ba \u00b7 K (\u03b60)\u03c6 1 +K (\u03b60)\u03c6 , (2.25) where \u03ba \u2261 min {\u03baB, \u03baS} , \u03c6 \u2261 min \u00a9 bf, sg \u00aa M \u00a1 B\u0304, S\u0304 \u00a2 , B\u0304 \u2261 b \u03baB , S\u0304 \u2261 s \u03baS . Proof. We have seen in the text that there cannot be more than one full-trade equilib- rium. It suffices to prove that, if r is small, then in any (non-trivial steady-state) equilib- rium, pB (v) = c\u0304 and pS (c\u0304) = v. We will only consider r < \u03ba \u2261 min{\u03baB, \u03baS}, which through Lemma 3 implies v > c\u0304 in equilibrium. Now pick any equilibrium and focus on sellers. To prove pS (c\u0304) = v, it suffices to prove that p = v is the only maximizer of maxp\u2208[0,1] \u03c0\u0302S (c\u0304, p), where \u03c0\u0302S (c\u0304, p) = (p\u2212 c\u0304) Z 1 v I [p \u2264 \u03c1B (v)] dNB(v) B where I [\u00b7] is 1 if the condition inside the bracket holds, and is 0 otherwise. Since \u03c0\u0302S (c\u0304, p) is absolutely continuous in p, it is differentiable in p almost everywhere. Notice that \u2202\u03c0\u0302S (c\u0304, p) \/\u2202p is 1 if p < v, so that any p < v is never optimal. Proposing 33 Chapter 2. Dynamic Matching and Two-sided Private Information Bargaining p > \u03c1B (1), which implies \u03c0\u0302S (c\u0304, p) = 0, is also never optimal. If v < p < \u03c1B (1), whenever differentiable, we have \u2202\u03c0S (c\u0304, p) \u2202p = 1\u2212 \u0393B (p)\u2212 (p\u2212 c\u0304) \u03b3B (p) , (2.26) where \u0393B (p) \u2261 R 1 v I [p \u2264 \u03c1B (v)] dNB(v) B and \u03b3B (p) \u2261 \u03930B (p). Define \u03c6B (x) \u2261 N 0B (x) \/B. The function \u03c1B is strictly increasing (from Lemma 1 and r > 0), so that its inverse function \u03c1\u22121B is well-defined on the range of \u03c1B, and is also strictly increasing. Then \u03b3B (p) = \u03c6B \u00a1 \u03c1\u22121B (p) \u00a2 \u03c10B \u00a1 \u03c1\u22121B (p) \u00a2 \u2200p \u2208 [v, \u03c1B (1)] . We want to show that, when v < p < \u03c1B (1) the r.h.s. of (2.26) must be negative for all sufficiently small r > 0. Firstly, from r < \u03ba, Lemma 6 and 5, we obtain p\u2212 c\u0304 > v \u2212 c\u0304 \u2265 K (\u03b60) \u00b5 \u03ba\u2212 r r + \u03ba \u00b6 > 0. (2.27) Moreover, for all v \u2265 v, we have \u03c10B (v) = rr+\u03b1BqB(v) (from Lemma 1) and \u03b1BqB (v) \u2265 \u03baB (from Lemma 2(c)). Thus \u03c10B (v) \u2264 r\/ (r + \u03ba), and hence \u03b3B (p) \u2265 \u00b3 1 + \u03ba r \u00b4 \u03c6B \u00a1 \u03c1\u22121B (p) \u00a2 . (2.28) We now derive a lower bound on the market probability density of buyers\u2019 types \u03c6B. From the steady-state equation (2.6), we can deduce \u03c6B (v) = bf (v) M (B,S) qB (v) \u2265 bf M (B,S) \u2200v \u2265 v (2.29) and B = Z 1 v bf (v) dv \u03b1BqB (v) < b \u03baB \u2261 B\u0304. Similarly (2.7) implies S < s \u03baS \u2261 S\u0304. Since M (B,S) is nondecreasing in each of its arguments, M (B,S) \u2264M \u00a1 B\u0304, S\u0304 \u00a2 . Substitut- ing this bound into (2.29) we obtain \u03c6B (v) \u2265 bf M \u00a1 B\u0304, S\u0304 \u00a2 \u2261 \u03c6B \u2200v \u2265 v. (2.30) 34 Chapter 2. Dynamic Matching and Two-sided Private Information Bargaining Then apply (2.27), (2.28) and (2.30) to (2.26), and simplify, we find that for almost all p \u2208 [v, \u03c1B (1)], \u2202\u03c0S (c\u0304, p) \u2202p < 1\u2212K (\u03b60) \u00b3\u03ba r \u2212 1 \u00b4 \u03c6B. Similarly, we can consider a type v buyer\u2019s proposing problem and find that pB (v) \u2208 [\u03c1S (0) , c\u0304], and for almost all p \u2208 [\u03c1S (0) , c\u0304], we have \u2202\u03c0B (v, p) \u2202p > \u22121 +K (\u03b60) \u00b3\u03ba r \u2212 1 \u00b4 \u03c6S where \u03c0B (v, p) \u2261 (v \u2212 p) Z c\u0304 0 I[p \u2265 \u03c1S (c)] dNS(c) S , \u03c6S \u2261 sg M \u00a1 B\u0304, S\u0304 \u00a2 . Therefore, if 1 \u2212 K (\u03b60) \u00a1\u03ba r \u2212 1 \u00a2 \u03c6B \u2264 0 and \u22121 + K (\u03b60) \u00a1\u03ba r \u2212 1 \u00a2 \u03c6S \u2265 0, or equivalently r \u2264 r, then we have r < \u03ba, \u2202\u03c0S(c\u0304,p)\u2202p < 0 for almost every p \u2208 (v, \u03c1B (1)) and \u2202\u03c0B(v,p) \u2202p > 0 for almost every p \u2208 (\u03c1S (0) , c\u0304). Hence pS (c\u0304) = v and pB (v) = c\u0304. The following corollary provides the main properties of our uniqueness bound r and relates it to the other bounds, r\u2217 and min {\u03baB, \u03baS}, in Theorem 1 and Corollary 3. Corollary 4 We have (a) If \u03baB and \u03baS increase, then r increases, and vice versa; (b) 0 < r < min {\u03baB, \u03baS}; (c) r goes to 0 as \u03baB and \u03baS go to 0; and (d) if K (\u03b60) < 1 then r < r\u2217. Proof. (a)-(c) are obvious. Our derivation of r (in the proof of Theorem 2) shows that r \u2264 r is equivalent to K (\u03b60) \u00b3 \u03ba\u2212r r+\u03ba \u00b4 \u2212 1\u03c6 B r r+\u03ba \u2265 0 and K (\u03b60) \u00b3 \u03ba\u2212r r+\u03ba \u00b4 \u2212 1\u03c6 S r r+\u03ba \u2265 0, where \u03c6B and \u03c6S (defined in Theorem 2) are lower bounds of the market probability densities of buyers\u2019 and sellers\u2019 types in any equilibrium. On the other hand, r < r\u2217 is equivalent to K (\u03b60) \u2212 1\u03c6B0(v0) r r+\u03b1B(\u03b60) > 0 and K (\u03b60) \u2212 1\u03c6S0(c\u03040) r r+\u03b1S(\u03b60) > 0 where \u03c6B0 and \u03c6S0 are the market probability densities of buyers\u2019 and sellers\u2019 types in the full-trade equilibrium. Then 35 Chapter 2. Dynamic Matching and Two-sided Private Information Bargaining it is easy to verify that, given K (\u03b60) < 1 (so that \u03b1B (\u03b60) > \u03ba and \u03b1S (\u03b60) > \u03ba), r \u2264 r implies r < r\u2217. In other words, r < r\u2217 if K (\u03b60) < 1. Before closing this section, we give a simple example that visualizes the main results of this and the previous sections. In particular, whether in equilibrium every meeting results in a trade does not hinge on the level of frictions, but rather on the composition of different kinds of frictions (discount rate r and explicit costs \u03baB, \u03baS). More precisely, in the friction space of (r, \u03baB, \u03baS), any neighborhood of 0, no matter how small, must contain a region (where r is small relative to \u03baB, \u03baS) in which only full-trade equilibria exist, and also contain another region (where r is large relative to \u03baB, \u03baS) in which only non-full-trade equilibria exist. Example 1 Buyers and sellers are born at the same rate, i.e. b = s = 1\/2. The dis- tributions of buyers\u2019 valuations and sellers\u2019 costs are both uniform [0, 1], i.e. F (v) = v, G(c) = c. (It is easy to check that the monotonicity of the virtual type functions JB and JS is satisfied.) The bargaining power is evenly distributed, i.e. \u03b2B = \u03b2S = 1\/2. The matching function is given by M (B,S) = BS\/(B+S).26 One can check that the entry gap and marginal types in a full-trade equilibrium are given by v0 \u2212 c\u03040 = K(\u03b60) = 2(\u03baB + \u03baS), v0 = 1 2 + \u03baB + \u03baS c\u03040 = 1 2 \u2212 \u03baB \u2212 \u03baS . Also, r\u2217 can be calculated as r\u2217 = 4min{\u03baB, \u03baS} max {1\u2212 6(\u03baB + \u03baS), 0} . 26Gale (1987) assumes this matching function, although the matching function is not explicitly stated there. In his model each trader in the market is randomly matched for each period with another trader, either a buyer or a seller. So, for a buyer, the probability per period of being matched with a seller is S\/(B+S). Similarly a seller is matched with a buyer with probability B\/(B+S). With continua of buyers and sellers in the market, the total mass of matches made per period is BS\/(B + S). 36 Chapter 2. Dynamic Matching and Two-sided Private Information Bargaining 0 B\u03ba S\u03ba \u03ba r 2 1 6 1 0 4 1 12 1 2 1 6 1 \u2217r \u03ba r Figure 2.5: Different patterns of equilibria in different regions of friction space Therefore, a full-trade equilibrium exists for all discount rate r if 16 \u2264 \u03baB + \u03baS < 1 2 , shown in the left panel of Figure 2.5. If \u03baB +\u03baS < 16 , full-trade equilibrium may or may not exist, depending on whether r is sufficiently small. Now let us assume \u03baB = \u03baS = \u03ba, then we have r\u2217 = 4\u03ba max {1\u2212 12\u03ba, 0} , r \u2261 8\u03ba 3 1 + 8\u03ba2 . The shaded area in the right panel of Figure 2.5 shows the values of r and \u03ba for which a full-trade equilibrium exists. Under the dashed ray \u03ba, a non-full-trade equilibrium with overlapping supports cannot exist. Under the dashed curve r, a unique equilibrium, which is full-trade, exists. 2.7 Necessary and sufficient condition for existence In this section we prove that the condition K (\u03b60) < 1 alone is a necessary and sufficient for the existence of a (full-trade or non-full-trade) nontrivial steady-state equilibrium. 37 Chapter 2. Dynamic Matching and Two-sided Private Information Bargaining Theorem 3 (General existence) At least one nontrivial steady-state equilibrium exists if and only if K (\u03b60) < 1. Taken together with Corollary 2(b), Theorem 3 implies that a non-full-trade equilibrium exists if the search costs are sufficiently small relative to the discount rate. Corollary 5 (Existence of a non-full-trade equilibrium) Given any r > 0, there is some \u03ba\u0304 > 0 such that a non-full-trade equilibrium exists whenever \u03baB, \u03baS < \u03ba\u0304. It is relatively easy to see that the condition K (\u03b60) < 1, a necessary condition for the existence of a full-trade equilibrium, is also necessary for the existence of any nontrivial equilibrium of our model. Indeed, it is already proved by (2.22) in Lemma 5. Perhaps surprisingly, the condition K (\u03b60) < 1 is also sufficient for the existence of a nontrivial equilibrium of our model. It might be natural to guess that a nontrivial equilibrium exists if and only if the expected search cost incurred by a buyer-seller pair (i.e. K(\u03b6)) is smaller than the maximum gains from trade, which is 1. However, this alone does not give us a meaningful condition for existence. It is because the buyer-seller ratio \u03b6 in equilibrium (if any) is endogenous, and the set {K(\u03b6) : \u03b6 > 0} is unbounded since lim\u03b6\u21920K(\u03b6) = lim\u03b6\u2192\u221eK(\u03b6) =\u221e. However, Theorem 3 tells us that in order to know whether a friction profile is compatible with a nontrivial market, it suffices to check only the expected search costs in the full-trade equilibrium candidate, although the true equilibrium (if any) might be non-full-trade. This result can be informally understood as follows. The market might have to break down because the expected search cost K(\u03b6) is too high that it does not pay for traders to enter. So a case where K(\u03b6) is very small is an \"inframarginal situation\". What matters to the existence condition is the \"marginal situation\" where K(\u03b6) is close to 1. If we insert the full-trade equilibrium buyer-seller ratio \u03b60 into K(\u03b6) and then consider the marginal situation, Corollary 2(c) asserts that a full-trade equilibrium does exist, which in turn validates \u03b60 in the first place. 38 Chapter 2. Dynamic Matching and Two-sided Private Information Bargaining The rest of this section is devoted to the main elements of the formal proof of Theorem 3. Additional details are provided in the Appendix. As usual, we want to construct a mapping T such that its fixed point characterizes an (nontrivial steady-state) equilibrium, and prove that T has a fixed point. The mapping T is informally described as follows. Start with a pair of value functions (WB,WS) and a pair of distribution functions (NB,NS), we construct best-response proposing strategies (pB, pS) and entry strategies (\u03c7B, \u03c7S). Then from those strategies and the original functions (WB,WS, NB,NS), we define a new pair of value functions (W \u2217B,W \u2217 S) through the Bellman equations, and a new pair of distribu- tion functions (N\u2217B, N \u2217 S) through the steady-state equations. Thus a fixed point of T (i.e. (WB,WS , NB, NS) = (W \u2217B,W \u2217 S , N \u2217 B, N \u2217 S)) characterizes an equilibrium. We will apply the Schauder fixed point theorem: if D is a nonempty compact convex subset of a Banach space and T is a continuous function from D to D, then T has a fixed point. In order to make this theorem applicable, certain difficulties need to be overcome. The main difficulty is that as we apply the mapping T , we need to preserve positive entry. To deal with this difficulty, we first prove existence of what we call an \u03b5-equilibrium, which is an actual equilibrium of the \u03b5-model described below. The \u03b5-model differs from our original model in three ways. First, we add a subsidy that ensures that all buyers with type v \u2265 1\u2212\u03b5 and all sellers with type c \u2264 \u03b5 enter. Every new- born trader is qualified to receive a flow of subsidy for her market participation, provided that (i) her type satisfies v \u2265 1\u2212 \u03b5 or c \u2264 \u03b5, and (ii) she would choose not to participate if no subsidy were available. Further, the flow rate of the subsidy for a qualified trader is the least amount sufficient to make this trader participate, i.e. the flow subsidies are infimum subsidies to attain WB (v) \u2265 0 and WS (c) \u2265 0 for v \u2208 [1\u2212 \u03b5, 1] and c \u2208 [0, \u03b5]. Because any subsidized traders are simply indifferent between entering or staying out, the Bellman equations for (WB,WS) and optimality conditions for (pB, pS) do not need to be changed. Although we now have a positive lower bound for the inflows of traders, we may not have a positive lower bound for the mass of traders in the market because the out- flow rate (i.e. \u03b1B(\u03b6)qB(v) or \u03b1S(\u03b6)qS(c)) could be potentially very large. To overcome 39 Chapter 2. Dynamic Matching and Two-sided Private Information Bargaining this difficulty, we impose the second modification, which ensures that the arrival rates \u03b1B(\u03b6) and \u03b1S(\u03b6) are bounded from above by some \u03b1\u0304. We modify the matching function M(B,S) as min {M(B,S), B\u03b1\u0304, S\u03b1\u0304}. Notice that this modified one inherits all the proper- ties of a matching function. But under the modified matching function we make sure that \u03b1B(\u03b6), \u03b1S(\u03b6) \u2264 \u03b1\u0304. While the first two modifications are made to make the mass of traders bounded from below, we also want it to be bounded from above, because our domain D needs to be compact. It suffices to have a lower bound for the outflow rate (\u03b1B(\u03b6)qB(v) or \u03b1S(\u03b6)qS(c)). For a type who chooses to enter without subsidy, there is naturally an upper bound for its mass because her expected trading surplus must be larger than her search cost. More precisely, for a participating v-buyer who is not subsidized, \u03b1B(\u03b6)qB(v) \u2265 \u03baB. However, a subsidized buyer could have \u03b1B(\u03b6)qB(v) < \u03baB. Our third modification is to disqualify subsidized traders in a way that ensures the outflow rates of subsidized types are at least \u03baB or \u03baS. The disqualification process is a Poisson process, with the rate equal to the minimum that makes the outflow rate at least \u03baB or \u03baS . For example, a currently qualified v-buyer with \u03b1B(\u03b6)qB(v) < \u03baB will be disqualified and exit immediately at a Poisson rate \u03baB \u2212 \u03b1B(\u03b6)qB(v); while a currently qualified v-buyer with \u03b1B(\u03b6)qB(v) \u2265 \u03baB will not be disqualified. Notice that for any v-buyer, either subsidized or not, the gross outflow rate must bemax {\u03b1B(\u03b6)qB(v), \u03baB}. Therefore, in the steady-state equations (2.6) and (2.7) that define N\u2217B and N \u2217 S we now use max {\u03b1B(\u03b6)qB(x), \u03baB} and max {\u03b1S(\u03b6)qS(x), \u03baS} instead of \u03b1B(\u03b6)qB(x) and \u03b1S(\u03b6)qS(x). It completes the descriptions of our \u03b5-model. We will show that our \u03b5-model has at least one equilibrium, which we shall call an \u03b5- equilibrium (Proposition 1). Next, we will prove that if \u03b5 > 0 is sufficiently small and \u03b1\u0304 sufficiently large, then an \u03b5-equilibrium is an equilibrium of our original model (Proposition 1). The main ideas of the proof are illustrated graphically in Figure 2.6. First, as in Lemma 5, we show that in any \u03b5-equilibrium, we must have v \u2212 c\u0304 \u2264 K (\u03b60). Second, we show that the trading flows are almost balanced, the discrepancy bounded in absolute value by (a multiple of) \u03b5. Imposing these constraints on the set of values 40 Chapter 2. Dynamic Matching and Two-sided Private Information Bargaining c ( )0\u03b6Kcv =\u2212 v ( )0\u03b6K 0)()](1[ =\u2212\u2212 cGvFb \u03b5acGvFb =\u2212\u2212 )()](1[ \u03b5acGvFb \u2212=\u2212\u2212 )()](1[ },max{ where gsfba \u2261 A B 1 Figure 2.6: Illustration of the idea behind the existence proof (c\u0304, v), we obtain the set of feasible values given by the shaded area in Figure 2.6. As the graph makes clear, the shaded area collapses to the curvilinear segment AB. Consequently, as \u03b5 gets arbitrarily small, the minimum c\u0304 in the shaded area is arbitrarily close to the horizontal coordinate of point A, and the maximum feasible v is arbitrarily close to the vertical coordinate of A. It follows that for small enough \u03b5 > 0, the constraints c\u0304 \u2265 \u03b5 and v \u2264 1 \u2212 \u03b5 become non-binding. In other words, our subsidy policy does not have a bite because no entrant is subsidized. It further implies the marginal entrants must be able to recover their search costs, and hence \u03b6 is bounded away from 0 and \u221e. Thus as long as \u03b1\u0304 are chosen to be large enough, our modification of the matching function does not have a bite. It follows that if \u03b5 > 0 is small and \u03b1\u0304 large, then an \u03b5-equilibrium is an equilibrium of our original model. The following is our formal treatments. We first define an appropriate domain D\u03b5, and then a mapping T\u03b5 : D\u03b5 \u2192 D\u03b5. Definition 3 Fix \u03b1\u0304 > max {\u03baB, \u03baS} and \u03b5 \u2208 (0, \u03b5\u0304], where \u03b5\u0304 \u2261 min \u00bd 1, f\u0304 \u03b1\u0304 \u03baBf , g\u0304\u03b1\u0304 \u03baSg \u00be . Let C[0, 1] be the Banach space of real continuous bounded functions defined on [0, 1], en- 41 Chapter 2. Dynamic Matching and Two-sided Private Information Bargaining dowed with the supremum norm. Then we define D\u03b5 \u2282 (C[0, 1])4 as the set of all tuples (WB,WS , NB, NS) such that (i) WB, NB and NS are nondecreasing, while WS is nonincreasing, (ii)WB,WS, NB and NS have Lipschitz constants no greater than \u03b1\u0304\/ (r + \u03b1\u0304), \u03b1\u0304\/ (r + \u03b1\u0304), bf\u0304\/\u03baB and sg\u0304\/\u03baS respectively, and (iii) WB (0) =WS (1) = NB (0) = NS (0) = 0 and NB(1) \u2265 \u03b5bf\/\u03b1\u0304, NS(1) \u2265 \u03b5sg\/\u03b1\u0304. Lemma 7 D\u03b5 is nonempty, convex and compact for any \u03b1\u0304 > max {\u03baB, \u03baS} and any \u03b5 \u2208 (0, \u03b5\u0304]. Proof. In Appendix A. Definition 4 Fix \u03b1\u0304 > max {\u03baB, \u03baS} and \u03b5 \u2208 (0, \u03b5\u0304]. Define a mapping T\u03b5 : D\u03b5 \u2192 D\u03b5 as follows. For any (WB,WS , NB, NS) \u2208 D\u03b5, define B \u2261 NB (1), S \u2261 NS (1), \u03b1B \u2261 min {M(B,S), B\u03b1\u0304, S\u03b1\u0304} \/B and \u03b1S \u2261 min {M(B,S), B\u03b1\u0304, S\u03b1\u0304} \/S. Then construct pB, pS, W \u2217B, W \u2217 S , \u03c7B, \u03c7S, N \u2217 B, N \u2217 S by pB (v) \u2261 max \u23a7 \u23aa\u23a8 \u23aa\u23a9 arg max p\u2208[0,1] Z {c:p\u2212c\u2265WS(c)} (v \u2212 p\u2212WB(v)) dNS(c) S \u23ab \u23aa\u23ac \u23aa\u23ad (2.31) pS (c) \u2261 min \u23a7 \u23aa\u23a8 \u23aa\u23a9 arg max p\u2208[0,1] Z {v:v\u2212p\u2265WB(v)} (p\u2212 c\u2212WS (c)) dNB(v) B \u23ab \u23aa\u23ac \u23aa\u23ad . (2.32) W \u2217B (v) \u2261 max \u03c7\u2208{0,1} \u03c7 \u00b7 { \u03b1B r + \u03b1B [\u03b2B\u03c0B (v) +\u03b2S Z {c:v\u2212pS(c)\u2265WB(v)} (v \u2212 pS(c)\u2212WB(v)) dNS(c) S ] \u2212 \u03baB r + \u03b1B }+ \u03b1B r + \u03b1B WB(v) (2.33) W \u2217S(c) \u2261 max \u03c7\u2208{0,1} \u03c7 \u00b7 { \u03b1S r + \u03b1S [\u03b2S\u03c0S (c) +\u03b2B Z {v:pB(v)\u2212c\u2265WS(c)} (pB (v)\u2212 c\u2212WS(c)) dNB(v) B ] \u2212 \u03baS r + \u03b1S }+ \u03b1S r + \u03b1S WS(c). (2.34) 42 Chapter 2. Dynamic Matching and Two-sided Private Information Bargaining \u03c7B (v) and \u03c7S (c) are defined as the maximizers in (2.33) and (2.34) respectively; wherever multiple maximizers exist, we pick 1. N\u2217B (v) \u2261 Z v 0 \u03c7\u2217B (x) b max {\u03b1BqB (x) , \u03baB}dF (x) (2.35) where \u03c7\u2217B (v) is 1 if \u03c7B (v) = 1 or v \u2265 1\u2212 \u03b5, and is 0 otherwise. N\u2217S (c) \u2261 Z c 0 \u03c7\u2217S (x) s max {\u03b1SqS (x) , \u03baS}dG (x) (2.36) where \u03c7\u2217S (c) is 1 if \u03c7S (c) = 1 or c \u2264 \u03b5, and is 0 otherwise. Now T\u03b5(WB,WS ,NB, NS) is defined by the constructed (W \u2217B,W \u2217 S , N \u2217 B, N \u2217 S). In Appendix A we show that our definition of T\u03b5 is legitimate, i.e. it is well-defined and T\u03b5 (D\u03b5) \u2282 D\u03b5. Lemma 8 The mapping T\u03b5 : D\u03b5 \u2192 D\u03b5 is continuous for any \u03b1\u0304 > max {\u03baB, \u03baS} and any \u03b5 \u2208 (0, \u03b5\u0304]. Proof. In Appendix A. Lemma 9 Fix any \u03b1\u0304 > max {\u03baB, \u03baS} and any \u03b5 \u2208 (0, \u03b5\u0304]. There exists some E \u2208 D\u03b5 such that T\u03b5(E) = E. (That is, there exists an \u03b5-equilibrium). Proof. As claimed before, D\u03b5 is a nonempty, convex and compact set in a Banach space (C[0, 1])4 and the mapping T\u03b5 is continuous. Then we obtain our result by applying the Schauder Fixed Point Theorem (which is stated before). Proposition 1 Suppose K(\u03b60) < 1. Then if \u03b5 > 0 is small enough and \u03b1\u0304 large enough, any fixed point of T\u03b5 characterizes a nontrivial steady-state equilibrium. (That is, if \u03b5 > 0 is small and \u03b1\u0304 large, any \u03b5-equilibrium is in fact an equilibrium of our original model.) Proof. Suppose E = (WB,WS , NB, NS) \u2208 D\u03b5 is a fixed point of T\u03b5. Then E, together with the constructed objects through the transformation from E to T\u03b5 (E), constitutes what 43 Chapter 2. Dynamic Matching and Two-sided Private Information Bargaining we call an \u03b5-equilibrium. Moreover, an \u03b5-equilibrium satisfies all the equilibrium conditions in Definition 1 if one can verify that (i) v\u2217 \u2261 inf {v : \u03c7\u2217B (v) = 1} < 1\u2212 \u03b5, and c\u0304\u2217 \u2261 sup {c : \u03c7\u2217S (c) = 1} > \u03b5; (ii) \u03b1BqB (v) \u2265 \u03baB if \u03c7\u2217B (v) = 1, and \u03b1SqS (c) \u2265 \u03baS if \u03c7\u2217S (c) = 1; and (iii) \u03b1B, \u03b1S < \u03b1\u0304. The following steps 1-6 will show that, for any (r, \u03b1\u0304)\u00c0 (0,max {\u03baB, \u03baS}), any \u03b5 \u2208 (0, \u03b5\u0304], and any associated fixed point of T\u03b5, a bunch of equilibrium properties hold. Then steps 7 and 8 will show that (i)-(iii) also hold if \u03b5 > 0 is small enough and \u03b1\u0304 large enough. Step 1. E \u2208 D\u03b5 implies v \u2212WB (v) and c +WS (c) are strictly increasing. Thus, from (2.31) and (2.32), we have pB (v) \u2264 c\u0304\u2217 +WS (c\u0304\u2217) and pS (c) \u2265 v\u2217 \u2212WB (v\u2217). Step 2. The expression inside the curly bracket in (2.33) can be written as \u03b1B r + \u03b1B \u2219 \u03b2B\u03c0B (v) + \u03b2S Z max {v \u2212 pS (c)\u2212WB (v) , 0} dNS (c)S \u2212 \u03baB \u03b1B \u00b8 , (2.37) which is continuous in v. Then by definition of v\u2217, \u03c7 = 0 is a maximizer in (2.31) when v = v\u2217. In other words, (2.37) is non-positive when v = v\u2217. Now evaluate (2.33) at v = v\u2217. From the above result and that W \u2217B = WB, we have WB (v \u2217) = \u03b1Br+\u03b1BWB (v \u2217), or WB (v\u2217) = 0. Step 3. The fact that (2.37) is non-positive when v = v\u2217 can be simplified as \u03b2B\u03c0B (v\u2217) \u2264 \u03baB \u03b1B because v\u2217 \u2212WB (v\u2217) = v\u2217 is no greater than pS (c), due to step 1. The logic in this and the previous step can be applied to the sellers\u2019 side. Thus we also have WS (c\u0304\u2217) = 0 and \u03b2S\u03c0S (c\u0304\u2217) \u2264 \u03baS\u03b1S . Step 4. Notice that \u03c0B (v\u2217) \u2265 v\u2217 \u2212 c\u0304\u2217 since the choice variable p in the definition (2.3) of \u03c0B can be taken as c\u0304\u2217. Similarly \u03c0S (c\u0304\u2217) \u2265 v\u2217 \u2212 c\u0304\u2217. Then step 3 implies v\u2217 \u2212 c\u0304\u2217 \u2264 min \u00bd \u03baB \u03b1B\u03b2B , \u03baS \u03b1S\u03b2S \u00be \u2264 K (\u03b60) . (2.38) Step 5. The expression inside the curly bracket in (2.33), which can be written as (2.37), is increasing in v. Hence \u03c7B and \u03c7\u2217B are increasing. Therefore, if v \u2265 v \u2261 inf {v : \u03c7B (v) = 1}, then (2.37) is non-negative, which implies \u03b1BqB (v) \u2265 \u03baB. Similarly, \u03c7S and \u03c7\u2217S are decreasing, and for any c \u2264 c\u0304 \u2261 sup {c : \u03c7S (c) = 1}, we have \u03b1SqS (c) \u2265 \u03baS . 44 Chapter 2. Dynamic Matching and Two-sided Private Information Bargaining Step 6. Equation (2.35), N\u2217B = NB, and step 5 imply b [1\u2212 F (v\u2217)]\u2212 Z 1 v\u2217 \u03b1BqB (v) dNB (v) = Z 1 v\u2217 max {0, \u03baB \u2212 \u03b1BqB (v)} dNB (v) . (2.39) The r.h.s. of (2.39) is clearly non-negative. Moreover, it is also no greater than bf\u0304\u03b5. To see this, consider two (exhaustive) cases: v\u2217 = v and v\u2217 < v. First consider the case that v\u2217 = v. From step 5 the r.h.s. of (2.39) is 0. Then consider the case that v\u2217 < v. Due to the definition of v\u2217 and v, we have v\u2217 = 1 \u2212 \u03b5. The r.h.s. of (2.39) is no greater than bf\u0304\u03b5 because dNB (v) \u2264 bf\u0304\u03baB . Similar logic can be applied to the sellers\u2019 side. Therefore we obtain 0 \u2264 b [1\u2212 F (v\u2217)]\u2212 Z 1 v\u2217 \u03b1BqB (v) dNB (v) \u2264 bf\u0304\u03b5 0 \u2264 sG (c\u0304\u2217)\u2212 Z c\u0304\u2217 0 \u03b1SqS (c) dNS (c) \u2264 sg\u0304\u03b5. On the other hand, by definition of \u03b1B, qB, \u03b1S, qS , we haveZ 1 v\u2217 \u03b1BqB(v)dNB(v) = Z c\u0304\u2217 0 \u03b1SqS(c)dNS(c). Therefore, |b[1\u2212 F (v\u2217)]\u2212 sG(c\u0304\u2217)| \u2264 max\u00a9bf\u0304 , sg\u0304\u00aa \u00b7 \u03b5. (2.40) Step 7. The previous six steps work with a particular fixed point of T\u03b5 given (\u03b5, \u03b1\u0304). In this and the next step, we let (\u03b5, \u03b1\u0304) \u2192 (0,\u221e) and consider an associated sequence of fixed points. Along any subsequence, c\u0304\u2217 cannot approach to 0 because otherwise (2.40) implies v\u2217 \u2192 1 and hence v\u2217\u2212 c\u0304\u2217 \u2192 1, violating (2.38) and K(\u03b60) < 1. Similarly, v\u2217 cannot approach to 1 along any subsequence. Therefore, in the tail of the sequence, we have c\u0304\u2217 > \u03b5 and v\u2217 < 1\u2212 \u03b5, i.e. (i) holds. Notice that (i) implies v\u2217 = v and c\u0304\u2217 = c\u0304. Thus step 5 implies (ii) also holds in the tail. Step 8. From steps 5 and 7, we have \u03b1B (\u03b6) \u2265 \u03baB and \u03b1S (\u03b6) \u2265 \u03baS in the tail as (\u03b5, \u03b1\u0304)\u2192 (0,\u221e). Thus \u03b6 \u2261 B\/S is bounded away from 0 and \u221e. It follows that, in the tail, \u03b1B < \u03b1\u0304 and \u03b1S < \u03b1\u0304, i.e. (iii) holds. Proof of Theorem 3. The necessity of the condition K (\u03b60) < 1 has been proved by (2.22) in Lemma 5. The sufficiency is implied by Lemma 9 and Proposition 1. 45 Chapter 2. Dynamic Matching and Two-sided Private Information Bargaining 2.8 Concluding remarks We have analyzed a steady-state search-theoretic model with two-sided private information bargaining. Although the model is not complicated, analyzing the equilibrium is a highly nontrivial job. It is because traders\u2019 best-response bargaining behaviors in general depend on the buyers\u2019 and sellers\u2019 distributions in a nontrivial manner. And these distributions in turn depend on the traders\u2019 bargaining behaviors through steady-state equations. Moreover the existence of equilibrium is also elusive. It is because a trivial no-entry equilibrium always exists, so that we cannot solely apply a fixed-point argument to prove existence of some nontrivial equilibrium. In spite of these difficulties, we are able to provide quite a few results. We have provided a necessary and sufficient condition K (\u03b60) < 1 under which some nontrivial steady-state equilibrium exists. Not surprisingly, in equilibrium the market must breakdown (i.e. nobody enters) if search costs are too large. Besides, the qualitative pattern of equilibrium (whether every meeting results in a trade) mainly depends not on the level of frictions, but the relativity of the two kinds of frictions (time discounting and explicit search costs). This result can only be obtained in a model with the coexistence of two kinds of frictions. Before closing this chapter, we make a few remarks on the existence conditionK (\u03b60) < 1. First, although the trivial no-entry equilibrium always exists because of the coordination problem, it is appealing to assume that the trivial equilibrium will not be selected as long as a nontrivial one exists. After all, every nontrivial equilibrium Pareto dominates the trivial one. With this assumption, we predict that the market will open if and only if K (\u03b60) < 1. Second, the discount rate r does not enter into the condition K (\u03b60) < 1. One way to understand it is that in steady state, when a trader is going to decide whether to enter or not, he just need to compare the expected gain from search for a period of very short length dt and the corresponding search costs incurred for the same period. Since this dt can be arbitrarily small, the discount rate has to be irrelevant for this entry decision. Third, the conditionK (\u03b60) < 1 depends on the distribution of bargaining power between 46 Chapter 2. Dynamic Matching and Two-sided Private Information Bargaining buyers and sellers. In particular, if the relative bargaining power of sellers \u03b2S is close to 0 or 1, the market must breakdown. The intuition is that if \u03b2S is close to zero, sellers do not have enough incentive to participate. If \u03b2S is close to one, buyers do not have enough incentive to participate. The openness of the market requires some balance between the interests of the two sides. 47 Chapter 3 Role of Information Structure in Dynamic Matching Markets 3.1 Introduction This chapter studies how the information structure at the bargaining stages affects the equilibrium outcome of a dynamic matching market.27 We analyze and compare two search- theoretic dynamic matching and bargaining games. They are called the private information model and the full information model. The former is the one we have seen in Chapter 2; the latter is the one in Mortensen and Wright (2002). In both models, searching for a trading partner takes time and other resources (e.g. money, effort). Thus there are two kinds of search frictions, one parameterized by a discount rate and one parameterized by explicit search costs. The two models are identical except for only one aspect: in the private information model, when a buyer and a seller meet each other they bargain without knowing each other\u2019s characteristics; while in the full information model they observe each other\u2019s characteristics once they meet. We show that the private information and full information models have some similarities. They have the same necessary and sufficient condition for the existence of a nontrivial steady-state equilibrium (Theorem 4); or putting it another way, information structure has no impact on whether the market would open or breakdown. Moreover, in both models, whether there exists a full-trade equilibrium (i.e. in which every meeting results in a trade) 27The chapter includes materials in my manuscript \"Bilateral Matching and Bargaining with Private Information\", which is joint with my thesis co-supervisor Artyom Shneyerov. 48 Chapter 3. Role of Information Structure in Dynamic Matching Markets mainly depends on the relative magnitudes of the two kinds of frictions (Corollary 2 and Corollary 6). Furthermore, the two models become completely identical if the discount rate is zero (Proposition 2); or in other words, information structure has no impact if agents are perfectly patient. Information structure also makes qualitative differences. Due to private information the bargaining between a buyer and a seller might breakdown even when the gain from trade is larger than the total value of their outside options. We show that whether this bargaining inefficiency occurs in equilibrium again depends on the relative magnitudes of the two kinds of frictions (Proposition 3). Besides, private information has an entry-deterring effect, so that typically less potential traders enter in the private information model (Proposition 4). Why this is so can be understood through the following logic. Suppose we start with a nontrivial steady-state equilibrium under full information. In this equilibrium there are marginal entrants: the lowest-value active buyers and the highest-cost active sellers. They are indifferent between entering or not. Now let us think about how the entry incentives of the marginal entrants would change if we make the bargaining to be under private information. Recall that the bargaining protocol in our market is the so-called random-proposer protocol. Notice the followings: First, when being a responder, a buyer with the lowest value in the market (or a seller with the highest cost in the market) would never receive an offer that makes him better off on top of his outside option. This is true no matter information is full or private. Second, when being a proposer, not knowing the responder\u2019s type would make a marginal entrant lose some information rent. Summing up these two concerns, the marginal entrants would expect less participating gains on average, if information is switched to be private. They, originally indifferent between entering or not, would become non-participants. Hence less potential traders enter in the private information model. Because entry decisions have externalities through the matching process, this entry- deterring effect could either improve or deteriorate the aggregate social welfare. We also provide and interpret sufficient conditions under which this entry effect improves or deteri- 49 Chapter 3. Role of Information Structure in Dynamic Matching Markets orates social welfare (Theorem 6). The rest of this chapter is organized as follows. Section 3.2 reviews the private infor- mation model. Section 3.3 presents the full information model, and the associated results. Section 3.4 solves the no-discounting case for both models. Section 3.5 analyzes the con- cept of bargaining efficiency and its relation with full-trade equilibria. Section 3.6 studies how the information structure affects social welfare through its impact on entry decisions. Section 3.7 concludes. Appendix B contains the calculations for Section 3.6. 3.2 Private information model Our model of dynamic matching market with private information bargaining, or private information model for short, is the one we use in Chapter 2. To make this chapter somehow self-contained and at the same time avoid too much repetition, let us for now only briefly review the model, recall the notations, write down the equations that define our equilibrium concept, and present our central results in Chapter 2. This is a continuous time, steady state model of a decentralized market with continua of risk-neutral traders (buyers and sellers). Different buyers (with unit demand) have different valuations v \u2208 [0, 1] for an indivisible good, and different sellers (with unit supply) have different costs c \u2208 [0, 1] for the good. Traders in the market are randomly matched pairwise at the aggregate flow rateM(B,S), which depends on the mass of buyers B and the mass of sellers S currently in the market. Once a buyer and a seller meet, they bargain following the random-proposer protocol: with probability \u03b2B \u2208 (0, 1) the buyer makes a take-it-or-leave- it offer to the seller, and with probability \u03b2S \u2261 1\u2212 \u03b2B the seller makes a take-it-or-leave-it offer. The traders leave once they successfully trade. New potential buyers are born at the rate b and sellers at the rate s. We normalize the aggregate born rate to be 1, i.e. b+s = 1. Once a potential buyer (seller) is born, his valuation (cost) is drawn i.i.d. from the c.d.f. F (v) (G(c)). The market is continuously replenished with new-born buyers and sellers who voluntarily choose to enter. We study the steady-state perfect Bayesian equilibria with positive entry, so called nontrivial steady-state equilibria. There are (except the asymmetric 50 Chapter 3. Role of Information Structure in Dynamic Matching Markets information) two kinds of frictions: time discounting at rate r > 0 and explicit search costs at rates \u03baB > 0 for buyers and \u03baS > 0 for sellers. The matching function M exhibits constant returns to scale. (For other assumptions we make on the functions F , G and M , see Assumptions 1 and 2 in Section 2.2.) Definition 5 Under the private information model, a nontrivial steady-state equilibrium is a pair of value functions WB,WS : [0, 1] \u2192 R+, a pair of entry strategies \u03c7B, \u03c7S : [0, 1] \u2192 {0, 1}, a pair of proposing strategies pB, pS : [0, 1] \u2192 [0, 1], and a pair of distribution functions NB, NS : [0, 1]\u2192 R+ such that B \u2261 NB (1) > 0, S \u2261 NS (1) > 0, rWB(v) = max \u03c7\u2208{0,1} \u03c7 \u00b7 {\u03b1B(\u03b6)[\u03b2B\u03c0B (v) (3.1) +\u03b2S Z {c:v\u2212pS(c)\u2265WB(v)} (v \u2212 pS(c)\u2212WB(v)) dNS(c) S ]\u2212 \u03baB} rWS(c) = max \u03c7\u2208{0,1} \u03c7 \u00b7 {\u03b1S(\u03b6)[\u03b2S\u03c0S (c) + (3.2) \u03b2B Z {v:pB(v)\u2212c\u2265WS(c)} (pB (v)\u2212 c\u2212WS(c)) dNB(v) B ]\u2212 \u03baS} b\u03c7B (v) dF (v) = \u03b1B(\u03b6)qB (v) dNB(v) (3.3) s\u03c7S (c) dG(c) = \u03b1S(\u03b6)qS (c) dNS(c) (3.4) where \u03b6 \u2261 B\/S, \u03b1B(\u03b6) \u2261M(1, 1\/\u03b6), \u03b1S(\u03b6) \u2261M(\u03b6, 1), \u03c0B (v) \u2261 max p\u2208[0,1] \u23a7 \u23aa\u23a8 \u23aa\u23a9 Z {c:p\u2212c\u2265WS(c)} (v \u2212 p\u2212WB(v)) dNS(c) S \u23ab \u23aa\u23ac \u23aa\u23ad (3.5) \u03c0S (c) \u2261 max p\u2208[0,1] \u23a7 \u23aa\u23a8 \u23aa\u23a9 Z {v:v\u2212p\u2265WB(v)} (p\u2212 c\u2212WS (c)) dNB(v) B \u23ab \u23aa\u23ac \u23aa\u23ad (3.6) qB (v) \u2261 \u03b2B Z {c:pB(v)\u2212c\u2265WS(c)} dNS(c) S + \u03b2S Z {c:v\u2212pS(c)\u2265WB(v)} dNS(c) S 51 Chapter 3. Role of Information Structure in Dynamic Matching Markets qS (c) \u2261 \u03b2S Z {v:v\u2212pS(c)\u2265WB(v)} dNB(v) B + \u03b2B Z {v:pB(v)\u2212c\u2265WS(c)} dNB (v) B , and \u03c7B, \u03c7S , pB, pS solve the optimization problems in (3.1), (3.2) (3.5), and (3.6) respec- tively. The equilibrium objects have the following interpretations: \u2022 WB(v),WS(c): buyers\u2019 and sellers\u2019 continuation payoffs when unmatched, \u2022 \u03c7B(v), \u03c7S(c): buyers\u2019 and sellers\u2019 entry strategies (1 represents \"enter\" and 0 repre- sents \"not enter\"), \u2022 pB(v), pS(c): buyers\u2019 and sellers\u2019 proposing strategies, i.e. what trading prices they propose, \u2022 NB(v), NS(c): buyers\u2019 and sellers\u2019 steady-state distributions of types in the market, \u2022 B,S: buyers\u2019 and sellers\u2019 steady-state masses in the market, \u2022 \u03b6: steady-state buyer-seller ratio (or market tightness), \u2022 \u03b1B (\u03b6) , \u03b1S (\u03b6): buyers\u2019 and sellers\u2019 Poisson arrival rates of being matched, \u2022 \u03c0B (v) , \u03c0S (c): buyer\u2019s and sellers\u2019 capital gains when they become a proposer, and \u2022 qB (v) , qS (c): buyer\u2019s and sellers\u2019 trading probabilities in a given meeting. Equations (3.1) and (3.2) are buyers\u2019 and sellers\u2019 Bellman equations. Equations (3.3) and (3.4) are the steady-state equations: the inflow rate of the traders of each type is equal to the outflow rate due to trading. The buyer\u2019s and sellers\u2019 responding strategies are also captured in the above equilibrium definition: A type v buyer accepts a price offer p if and only if p \u2264 v \u2212WB (v); a type c seller accepts a price offer p if and only if p \u2265 c+WS (c). Thus buyers\u2019 and sellers\u2019 reservation prices, also called dynamic types, are given by \u03c1B (v) \u2261 v \u2212WB (v) , (3.7) 52 Chapter 3. Role of Information Structure in Dynamic Matching Markets \u03c1S (c) \u2261 c+WS (c) . (3.8) In general there is no analytic solution for the system of equations (3.1) through (3.4). However, we know from Theorem 3 (in Chapter 2) that our private information model has at least one nontrivial steady-state equilibrium if and only if K (\u03b60) < 1 where \u03b60 \u2261 \u03b2B \u03b2S \u03baS \u03baB , (3.9) K (\u03b6) \u2261 \u03baB \u03b1B (\u03b6) + \u03baS \u03b1S (\u03b6) \u2200\u03b6. (3.10) By Theorem 2 and Corollary 4, if the discount rate r is small relative to the search costs \u03baB and \u03baS , then the (nontrivial steady-state) equilibrium is unique and has the property that every meeting results in a trade. We call this kind of equilibria full-trade equilibria. By Theorem 1 and Corollary 2 (see also Remark 1), if the discount rate is large relative to the search costs, then in equilibrium some meetings do not result in a trade. We call this kind of equilibria non-full-trade equilibria. In particular, whether there exists a nontrivial equilibrium depends on the search costs (\u03baB, \u03baS) (but not on the discount rate r) and the distribution of bargaining power (\u03b2B, \u03b2S). Whether in equilibrium every meeting results in a trade depends on the relative magnitudes of r and (\u03baB, \u03baS). 3.3 Full information (Mortensen-Wright) model 3.3.1 Model Our model of dynamic matching market with full information bargaining, or full information model for short, is the one in Mortensen and Wright (2002). Mortensen and Wright (2002) consider a model that differs from our private information model only in one respect: they assume full information bargaining, i.e. bargainers know each other\u2019s type once they meet. Consequently, proposers hold their partners to their reservation values (i.e., to their dynamic types), and the proposing strategies depend on both the proposer\u2019s and the responder\u2019s type. In other words, for a meeting between a type v buyer and a type c seller, if the 53 Chapter 3. Role of Information Structure in Dynamic Matching Markets buyer proposes, he will propose the offer pB (v, c) = \u03c1S (c) if v \u2212 \u03c1S (c) \u2265 WB(v), while the offer can be defined as any price less than \u03c1S (c) if v \u2212 \u03c1S (c) < WB(v) (such a price will be rejected by the seller). Similarly, if the seller proposes, she will propose the offer pS (v, c) = \u03c1B (v) if \u03c1B (v)\u2212 c \u2265WS(c). In the context of full information bargaining, the random-proposer protocol is equivalent to the generalized Nash bargaining solution. To see this, notice that under random-proposer protocol, a meeting between a type v buyer and a type c seller results in a trade if and only if \u03c1S(c) \u2264 \u03c1B(v). And conditional on trade, the expected trading price p(v, c) is the weighted average of the seller\u2019s offer \u03c1B(v) and the buyer\u2019s offer \u03c1S(c): p(v, c) = \u03b2S\u03c1B(v) + \u03b2B\u03c1S(c). (3.11) Now consider the generalized Nash bargaining with the buyer\u2019s relative bargaining power being \u03b2B \u2208 (0, 1), and the seller\u2019s relative bargaining power being \u03b2S \u2261 1\u2212 \u03b2B. The joint matching surplus to be shared is v\u2212 c\u2212WB(v)\u2212WS(c) and the threat points of the buyer and the seller is WB(v) and WS(c) respectively. Therefore, a trade occurs if and only if v \u2212 c\u2212WB(v)\u2212WS(c) \u2265 0 or equivalently \u03c1S(c) \u2264 \u03c1B(v). Conditional on that, the trading price p(v, c) is determined by p(v, c) \u2208 argmax p [v \u2212 p\u2212WB(v)]\u03b2B [p\u2212 c\u2212WS(c)]\u03b2S , for which the solution is exactly (3.11). Thus, no matter we use random-proposer protocol or generalized Nash bargaining, the buyer\u2019s and seller\u2019s capital gains from the meeting are given respectively by v \u2212 p (v, c)\u2212WB (v) = \u03b2B \u00b7 (\u03c1B (v)\u2212 \u03c1S (c)) , p (v, c)\u2212 c\u2212WS (c) = \u03b2S \u00b7 (\u03c1B (v)\u2212 \u03c1S (c)) . In this regard, the random-proposer bargaining is an extension of Nash bargaining into the environment of private information. 54 Chapter 3. Role of Information Structure in Dynamic Matching Markets Here we define nontrivial steady-state equilibria for the full information model in a way parallel to Definition 5. Definition 6 Under the full information model, a nontrivial steady-state equilibrium is a pair of value functions WB,WS : [0, 1] \u2192 R+, a pair of entry strategies \u03c7B, \u03c7S : [0, 1] \u2192 {0, 1}, and a pair of distribution functions NB, NS : [0, 1]\u2192 R+ such that B \u2261 NB (1) > 0, S \u2261 NS (1) > 0, rWB(v) = max \u03c7\u2208{0,1} \u03c7 \u00b7 {\u03b1B(\u03b6)\u03b2B Z {c:\u03c1B(v)\u2265\u03c1S(c)} (\u03c1B(v)\u2212 \u03c1S (c)) dNS(c) S \u2212 \u03baB} (3.12) rWS(c) = max \u03c7\u2208{0,1} \u03c7 \u00b7 {\u03b1S(\u03b6)\u03b2S Z {v:\u03c1B(v)\u2265\u03c1S(c)} (\u03c1B(v)\u2212 \u03c1S (c)) dNB(v) B \u2212 \u03baS} (3.13) b\u03c7B (v) dF (v) = \u03b1B(\u03b6)qB (v) dNB(v) s\u03c7S (c) dG(c) = \u03b1S(\u03b6)qS (c) dNS(c) where \u03b6 \u2261 B\/S, \u03b1B(\u03b6) \u2261M(1, 1\/\u03b6), \u03b1S(\u03b6) \u2261M(\u03b6, 1), \u03c1B (v) \u2261 v \u2212WB (v) \u03c1S (c) \u2261 c+WS (c) qB (v) \u2261 Z {c:\u03c1B(v)\u2265\u03c1S(c)} dNS(c) S (3.14) qS (c) \u2261 Z {v:\u03c1B(v)\u2265\u03c1S(c)} dNB (v) B (3.15) and \u03c7B, \u03c7S solve the optimization problems in (3.12) and (3.13) respectively. The interpretations for the equilibrium conditions and equilibrium objects in Definition 6 are the same as in the previous section. 55 Chapter 3. Role of Information Structure in Dynamic Matching Markets 3.3.2 Basic equilibrium properties The analysis of Mortensen and Wright (2002) is based only on full-trade equilibria (although they do not use this term). That would not be enough for the purposes of this and the next Chapter. Now let us provide some lemmas for nontrivial steady-state equilibria in general. Our methodology here is similar to the one in Section 2.4. Lemma 10 Under full information, in any nontrivial steady-state equilibrium, there are marginal entering types v, c\u0304 \u2208 (0, 1) such that the supports of NB and NS are [v, 1] and [0, c\u0304] respectively. Marginal entrants (i.e. type v buyers and type c\u0304 sellers) are indifferent between entering or not, while the entry preferences of all others are strict. {v : \u03c7B (v) = 1} is either [v, 1] or (v, 1]. {c : \u03c7S (c) = 1} is either [0, c\u0304] or [0, c\u0304). WB is absolutely continuous, convex, nondecreasing on [0, 1], strictly increasing on [v, 1], with WB (v) = 0; whenever differentiable, W 0B(v) = \u03c7B (v) \u03b1B (\u03b6)\u03b2BqB(v) r + \u03b1B (\u03b6)\u03b2BqB(v) . (3.16) WS is absolutely continuous, convex, nonincreasing on [0, 1], strictly decreasing on [0, c\u0304], with WS (c\u0304) = 0; whenever differentiable, W 0S(c) = \u2212\u03c7S (c) \u03b1S (\u03b6)\u03b2SqS(c) r + \u03b1S (\u03b6)\u03b2SqS(c) . (3.17) The trading probability qB is strictly positive and nondecreasing on [v, 1], while qS is strictly positive and nonincreasing on [0, c\u0304]. Proof. We prove the results for buyers only. We use an argument parallel to that for Lemma 1. For any v, v\u0302 \u2208 [0, 1], define \u03a0B (v, v\u0302) \u2261 Z {c:\u03c1B(v\u0302)\u2265\u03c1S(c)} (v \u2212 \u03c1S (c)) dNS(c) S . The buyers\u2019 Bellman equation (3.12) implies for any v, v\u0302 \u2208 [0, 1] and any \u03c7 \u2208 {0, 1}, rWB(v) \u2265 \u03c7 \u00b7 {\u03b1B\u03b2B [\u03a0B (v, v\u0302)\u2212 qB (v\u0302)WB (v)]\u2212 \u03baB} 56 Chapter 3. Role of Information Structure in Dynamic Matching Markets or equivalently WB (v) \u2265 \u03c7 \u00b7 uB (v, v\u0302) where uB (v, v\u0302) \u2261 \u03b1B\u03b2B\u03a0B (v, v\u0302)\u2212 \u03baB r + \u03b1B\u03b2BqB (v\u0302) . And the inequality becomes equality if v\u0302 = v and \u03c7 = \u03c7B (v). Let UB (v)\u2261maxv\u0302\u2208[0,1] uB (v, v\u0302). We then have WB (v) = \u03c7B (v)uB (v, v) = \u03c7B (v)UB (v) = max {UB (v) , 0}. For any v\u0302, uB (v, v\u0302) is affine and nondecreasing in v. Milgrom and Segal (2002) Envelope Theorem im- plies UB (v) is absolutely continuous, convex, nondecreasing, and with slope \u03b1B\u03b2BqB (v) \/(r+ \u03b1B\u03b2BqB (v)) whenever differentiable. The same properties are inherited by WB (v), except that its slope becomes \u03c7B (v)\u03b1B\u03b2BqB (v) \/(r + \u03b1B\u03b2BqB (v)). Obviously UB (0) < 0. Let v \u2261 sup {v \u2208 [0, 1] : UB (v) < 0}. By continuity of UB, we have v > 0 and UB (v) \u2264 0. But UB (v) < 0 is impossible in nontrivial equilibrium because it implies \u03c7B (v) = 0 \u2200v \u2208 [0, 1] and hence B = 0. Thus UB (v) = WB (v) = 0. By monotonicity of UB, for all v < v, we have UB (v) < 0 and hence \u03c7B (v) = WB (v) = 0. Moreover, qB (v) > 0 for all v \u2265 v. It is because qB (v) \u2265 \u03a0B (v, v), and for all v \u2265 v, the fact UB (v) \u2265 0 implies \u03b1B\u03b2B\u03a0B (v, v) \u2265 \u03baB > 0. It furthermore implies U 0B (v+) \u2265 \u03b1B\u03b2BqB (v+) \/(r + \u03b1B\u03b2BqB (v+)) > 0. Thus for all v > v, we have UB (v) > 0 and hence \u03c7B (v) = 1 and WB (v) = UB (v). From the buyers\u2019 steady-state equation, [v, 1] is the support of NB. Since the inflow distribution F does not have atom point, neither does NB. Hence B > 0 implies v < 1. Finally, the convexity of UB implies that qB is nondecreasing on [v, 1]. The thresholds v and c\u0304 in Lemma 10 are called marginal entering types. Those buyers with type v and those sellers with type c\u0304 are called marginal entrants. Since the flow and stock masses of marginal entrants (who are indifferent between entering or not) is zero anyway, we will without loss of generality assume throughout they enter, i.e. \u03c7B (v) = \u03c7S (c\u0304) = 1. Comparing Lemma 10 above with Lemma 1 in Chapter 2, we see that they looks almost the same, except that the trading probabilities qB and qS in Lemma 1 are replaced by \u03b2BqB 57 Chapter 3. Role of Information Structure in Dynamic Matching Markets and \u03b2SqS respectively in Lemma 10. To see the intuition, recall that under full information bargaining a buyer can gain from a meeting (on top of his outside option) only when he proposes, the probability of which is \u03b2B. In other words, he will be indifferent between accepting or rejecting an offer whenever he is a responder. Therefore, keeping \u03b1B and qB unchanged, we can evaluate the buyer\u2019s lifetime payoff WB as if he will reject any offer. If so, his counterfactual trading probability becomes \u03b2BqB instead of qB. A similar logic applies to sellers. As a direct implication, keeping \u03b1B and qB unchanged, private information bargaining makes the slopes of lifetime payoffs WB(v) and WS(c) steeper. It should not be surprising because it is well-known that information rents are monotone in types. As another direct implication, again keeping \u03b1B and qB unchanged, the slopes of dynamic types \u03c1B(v) and \u03c1S(c) become flatter: \u03c10B(v) = r r + \u03b1B (\u03b6)\u03b2BqB (v) > 0 a.e. v \u2208 [v, 1] (3.18) \u03c10S(c) = r r + \u03b1S (\u03b6)\u03b2SqS (v) > 0 a.e. c \u2208 [0, c\u0304] . (3.19) The following lemma provides the indifference conditions for the marginal entrants. Lemma 11 Under full information, in any nontrivial steady-state equilibrium, \u03c1B(v) = v and \u03c1S(c\u0304) = c\u0304. Moreover, \u03b1B (\u03b6)\u03b2B Z max{v \u2212 \u03c1S(c), 0} dNS(c) S = \u03baB (3.20) \u03b1S (\u03b6)\u03b2S Z max{\u03c1B(v)\u2212 c\u0304, 0} dNB(v) B = \u03baS. (3.21) Proof. From Lemma 10 we have WB(v) =WS(c\u0304) = 0, hence \u03c1B(v) = v and \u03c1S(c\u0304) = c\u0304. Evaluate (3.12) and (3.13) at v = v and c = c\u0304, we get the results. Since the buyers\u2019 and sellers\u2019 reservation prices \u03c1B and \u03c1S (also called dynamic types) are increasing, the buyers\u2019 lowest and highest reservation prices are v and \u03c1B (1). The sellers\u2019 lowest and highest reservation prices are \u03c1S (0) and c\u0304. 58 Chapter 3. Role of Information Structure in Dynamic Matching Markets The following lemma shows that \u03c1S (0) < v, otherwise type v buyers prefer not to enter as they cannot recover the search costs. Similarly c\u0304 < \u03c1B (1), otherwise type c\u0304 sellers prefer not to enter as they cannot recover the search costs. Lemma 12 Under full information, in any nontrivial steady-state equilibrium, \u03c1S (0) < v and c\u0304 < \u03c1B (1). Proof. We prove \u03c1S (0) < v first. Suppose v \u2264 \u03c1S (0). Then the left-hand side of (3.20) is 0, while the right-hand side is strictly positive, a contradiction. To prove c\u0304 < \u03c1B (1), simply apply (3.21) instead of (3.20). As in Chapter 2, define the Walrasian price p\u2217 as the price that clears the flow demand and flow supply: b [1\u2212 F (p\u2217)] = sG(p\u2217). Since buyers and sellers always leave the market in pairs, the entry flows of buyers and sellers must be balanced in steady state, i.e. b [1\u2212 F (v)] = sG(c\u0304).28 Therefore the marginal entering types v and c\u0304 must lie on different sides of the Walrasian price p\u2217. Although both v \u2264 p\u2217 \u2264 c\u0304 and c\u0304 < p\u2217 < v are possible, the comparisons between \u03c1B (1), \u03c1S (0) and p\u2217 are, as under private information, unambiguous. Lemma 13 Under full information, in any nontrivial steady-state equilibrium, \u03c1S (0) < p\u2217 < \u03c1B (1). Proof. We prove \u03c1S (0) < p\u2217 only. The other part is completely parallel. Suppose \u03c1S (0) \u2265 p\u2217. Then clearly c\u0304 \u2265 p\u2217. Moreover, from Lemma 12 we have v > \u03c1S (0) \u2265 p\u2217. But then b [1\u2212 F (v)] < b [1\u2212 F (p\u2217)] = sG(p\u2217) \u2264 sG(c\u0304), a contradiction. The following lemma is parallel to Lemma 5 in Chapter 2. 28 It can be formally derived from steady-state equations (3.3) and (3.4). 59 Chapter 3. Role of Information Structure in Dynamic Matching Markets Lemma 14 Under full information, in any nontrivial steady-state equilibrium, we have 1 > \u03c1B (1)\u2212 \u03c1S (0) > K (\u03b60) , (3.22) v \u2212 c\u0304 < K (\u03b60) . (3.23) Proof. Pick any nontrivial steady-state equilibrium. Lemma 10 implies WB (1) > 0 and WS (0) > 0. The first inequality in (3.22), which is equivalent to WB (1) +WS (0) > 0, follows. Condition (3.20) implies \u03b1B (\u03b6)\u03b2B (\u03c1B (1)\u2212 \u03c1S (0)) > \u03baB, because (i) \u03c1B (1) > v and (ii) \u03c1S(c) > \u03c1S (0) for any c on [0, c\u0304] (which is the support of NS). Similarly (3.21) implies \u03b1S (\u03b6)\u03b2S (\u03c1B (1)\u2212 \u03c1S (0)) > \u03baS , so that \u03c1B (1)\u2212 \u03c1S (0) > max \u00bd \u03baB \u03b1B (\u03b6)\u03b2B , \u03baS \u03b1S (\u03b6)\u03b2S \u00be \u2265 K (\u03b60) . The last inequality is from Lemma 4 in Chapter 2. This proves (3.22). We turn to prove (3.23). Notice that (3.20) implies \u03b1B (\u03b6)\u03b2Bmax {v \u2212 c\u0304, 0} < \u03baB, because \u03c1S(c) < c\u0304 for any c on [0, c\u0304). Similarly, (3.21) implies \u03b1S (\u03b6)\u03b2Smax {v \u2212 c\u0304, 0} < \u03baS , from which it follows that max {v \u2212 c\u0304, 0} < min \u00bd \u03baB \u03b1B (\u03b6)\u03b2B , \u03baS \u03b1S (\u03b6)\u03b2S \u00be \u2264 K (\u03b60) . (3.24) The last inequality is again from Lemma 4 in Chapter 2. This proves (3.23). 60 Chapter 3. Role of Information Structure in Dynamic Matching Markets 3.3.3 Necessary and sufficient condition for existence Mortensen and Wright (2002) do not provide a necessary and sufficient condition under which a nontrivial steady-state equilibrium exists. We can fill this gap by applying the technique we developed in Chapter 2. Indeed, our general existence proof for private in- formation model (see Section 2.7 and Appendix A) adapts to full information model with minor changes. In particular the necessary and sufficient condition for the existence is the same as before, which is K (\u03b60) < 1.29 Having developed the results in the previous subsection, it is now easy to see the necessity of K (\u03b60) < 1. Indeed, if there exists some nontrivial steady-state equilibrium, then Lemma 14 implies the condition K (\u03b60) < 1. For the sufficiency part, the proof for the full information model is strictly easier than that for the private information model (which is provided in Section 2.7 and Appendix A) because we do not have to consider proposing strategies in our construction mapping T , whose fixed point characterizes an equilibrium. The essential changes involved are to modify Definition 4 of T\u03b5 by (i) deleting the proposers\u2019 problems (2.31) and (2.32), (ii) replacing the expressions inside the square brackets in (2.33) and (2.34) by \u03b2B Z {c:\u03c1B(v)\u2265\u03c1S(c)} (\u03c1B(v)\u2212 \u03c1S (c)) dNS(c) S and \u03b2S Z {v:\u03c1B(v)\u2265\u03c1S(c)} (\u03c1B(v)\u2212 \u03c1S (c)) dNB(v) B respectively, and (iii) redefining qB and qS according to (3.14) and (3.15). Theorem 4 Given the parameters (b, s, F,G,M, \u03b2B, \u03b2S , r, \u03baB, \u03baS), a nontrivial steady-state equilibrium exists in the full information model if and only if a nontrivial steady-state equi- librium exists in the private information model. More precisely, for either the private in- 29The value \u03b60 \u2261 \u03b2B\u03baS \u03b2S\u03baB in the full information model should not be interpreted as the buyer-seller ratio in full-trade equilibrium. Nevertheless, it can be, like in the private information model, interpreted as the equilibrium buyer-seller ratio when r = 0. 61 Chapter 3. Role of Information Structure in Dynamic Matching Markets formation or the full information model, a necessary and sufficient condition for existence of a nontrivial steady-state equilibrium is K (\u03b60) < 1, where \u03b60 and the function K are defined by (3.9) and (3.10). For the intuition of the existence conditionK(\u03b60) < 1, see Section 2.7. Here let us discuss the intuition of the invariance of this condition across different information structures. It suffices to consider the \"marginal situation\" where the search costs are such that K (\u03b60) is smaller than but very close to 1. Then only those potential buyers with valuations very close to 1 and those potential sellers with costs very close to 0 would enter. That is to say, all buyers (sellers) in the market are virtually homogeneous in their valuations (costs). It is no wonder that the information structure at the bargaining stages does not alter the existence condition in this situation. 3.3.4 Full-trade equilibrium In the context of full information bargaining, a nontrivial steady-state equilibrium is full- trade (i.e. every meeting results in a trade) if and only if v \u2265 c\u0304. (That is, the dichotomy of full-trade\/non-full-trade and the dichotomy of separated\/overlapping supports we introduce in Chapter 2 are the same thing in the context of full information.) We will characterize the full-trade equilibria under full information later (see Section 3.6). For now, let us present the results on the existence and uniqueness of full-trade equilibrium, which are due to Mortensen and Wright (2002). We need some definitions in order to state the following theorem. Let \u03c8B : (0,\u221e)\u2192 (0,\u221e] and \u03c8S : (0,\u221e)\u2192 (0,\u221e] be \u03c8B(\u03b6) \u2261 \u03b2S\u03baB\u03b6 \u03b2Bmax nR p\u2217 0 (p \u2217 \u2212 c) dG(c)G(p\u2217) \u2212 \u03baB \u03b1B(\u03b6)\u03b2B , 0 o , \u03c8S(\u03b6) \u2261 \u03b2B\u03baS \u03b2S\u03b6max nR 1 p\u2217(v \u2212 p\u2217) dF (v) 1\u2212F (p\u2217) \u2212 \u03baS \u03b1S(\u03b6)\u03b2S , 0 o . 62 Chapter 3. Role of Information Structure in Dynamic Matching Markets (These functions take the value \u221e whenever their defining expressions have a denominator 0.) Now let r\u0302 \u2208 (0,\u221e] be the unique value such that r\u0302 = \u03c8B(\u03b6\u0302) = \u03c8S(\u03b6\u0302) for some \u03b6\u0302 > 0.30 Theorem 5 (Mortensen and Wright, 2002) Under full information, a (unique) full- trade equilibrium exists if and only if K(\u03b60) < 1 and r \u2264 r\u0302. Moreover, if r is sufficiently close to 0, then non-full-trade equilibrium does not exist, implying uniqueness of equilibrium. From the above definition of r\u0302 we can prove the following results (for full information model), which is parallel to Corollary 2 (for private information model). Corollary 6 (a) In the region where r\u0302 <\u221e, if \u03baB and \u03baS increase, then r\u0302 increases, and vice versa. (b) Given any r > 0, there is a \u03ba\u0304 > 0 such that full-trade equilibrium in the full information model does not exist whenever \u03baB, \u03baS < \u03ba\u0304. (c) Given any r > 0, a full-trade equilibrium in the full information model exists when (\u03baB, \u03baS) is such that K(\u03b60) is less than but sufficiently close to 1. (d) Given any (\u03baB, \u03baS) such that K(\u03b60) < 1, a full-trade equilibrium exists when r is sufficiently close to 0. Proof. For any \u03b6 > 0 such that \u03c8B(\u03b6) is finite, \u03c8B(\u03b6) is strictly increasing in \u03baB. Similarly, for any \u03b6 > 0 such that \u03c8S(\u03b6) is finite, \u03c8S(\u03b6) is strictly increasing in \u03baS. Hence (a) holds. For any \u03b6 > 0, we have \u03c8B(\u03b6) \u2192 0 as \u03baB \u2192 0; and \u03c8S(\u03b6) \u2192 0 as \u03baS \u2192 0. Therefore r\u0302 \u2192 0 as (\u03baB, \u03baS)\u2192 0, and (b) follows. It follows from Lemma 4 that, for any \u03b6 > 0, we must have either \u03baB\u03b1B(\u03b6)\u03b2B \u2265 K(\u03b60) or \u03baS \u03b1S(\u03b6)\u03b2S \u2265 K(\u03b60). Also notice that both R p\u2217 0 (p \u2217\u2212c) dG(c)G(p\u2217) and R 1 p\u2217(v\u2212p\u2217) dF (v) 1\u2212F (p\u2217) are constants strictly smaller 1. Then according to the definitions of \u03c8B(\u00b7) and \u03c8S(\u00b7), it is impossible to 30 It is easy to see that \u03c8B and \u03c8S are continuous. Moreover, \u03c8B is nondecreasing and \u03c8S is nonincreasing, and lim \u03b6\u21920 \u03c8B(\u03b6) = lim\u03b6\u2192\u221e \u03c8S(\u03b6) = 0. Therefore r\u0302 is well-defined. 63 Chapter 3. Role of Information Structure in Dynamic Matching Markets keep both \u03c8B(\u03b6) and \u03c8S(\u03b6) finite if we let K(\u03b60) go to 1 from below. Therefore r\u0302 = \u221e when K(\u03b60) is less than but sufficiently close to 1. Hence (c) follows. (d) is simply from r\u0302 > 0 for any \u03baB, \u03baS > 0 such that K(\u03b60) < 1. Usually a full-trade equilibrium is easier to exist under full information bargaining. Example 2 Take the parameters as in Example 1, i.e. b = s = 1\/2, F (v) = v, G(c) = c, \u03b2B = \u03b2S = 1\/2, and M (B,S) = BS\/(B + S). Also take \u03baB = \u03baS = \u03ba. The r\u0302 in Theorem 5 (the threshold of r below which a full-trade equilibrium exists under full information) is r\u0302 = 4\u03ba max {1\u2212 16\u03ba, 0} , and the r\u2217 in Theorem 1 (the threshold of r below which a full-trade equilibrium exists under private information) is r\u2217 = 4\u03ba max {1\u2212 12\u03ba, 0} . Obviously r\u0302 \u2265 r\u2217 and it is strict unless r\u0302 = r\u2217 =\u221e. In other words, a full-trade equilibrium is strictly easier to exist in full information model than in private information model. 3.4 No-discounting case The previous two sections describe the private information model and the full information model respectively. The equilibrium properties of the two models exhibit some similarities. They have the same necessary and sufficient condition for the existence of some (nontrivial steady-state) equilibrium. In either model, keeping other parameters unchanged, any equi- librium must be full-trade if the discount rate r is sufficiently close to 0; and any equilibrium must be non-full-trade if r is sufficiently large. In this section, we completely solve the equilibria for both of the two models for the case where there is no time discounting. Formally, we extend our private information model to allow r = 0, and define an associated nontrivial steady-state equilibrium as a tuple (WB,WS , \u03c7B, \u03c7S , pB, pS, NB, NS) such that (i) it satisfies the conditions in Definition 5 evaluated at r = 0, and (ii) it is the limit of some private information equilibrium sequence 64 Chapter 3. Role of Information Structure in Dynamic Matching Markets as r \u2192 0 from above. Similarly, we extend our full information model to allow r = 0, and define an associated nontrivial steady-state equilibrium as a tuple (WB,WS , \u03c7B, \u03c7S ,NB, NS) such that (i) it satisfies the conditions in Definition 6 evaluated at r = 0, and (ii) it is the limit of some full information equilibrium sequence as r \u2192 0 from above.31 Both models are greatly simplified in the no-discounting case. Furthermore, the no- discounting case provides a benchmark in which the information structure at the bargaining stages plays no role. Indeed, if r = 0, the two models are equivalent, in the sense that any equilibrium of the full information model must be an equilibrium of the private informa- tion model, and conversely any equilibrium of the private information model must be an equilibrium of the full information model. To see this, consider the full information model with r > 0 and let r \u2192 0. From (3.18) and (3.19), in the limit we have v = \u03c1B (1) and \u03c1S (0) = c\u0304. In words, all buyers (sellers) in the market have the same dynamic valuation (dynamic cost). Participating traders are homogeneous in their dynamic types although they are heterogeneous in their original types. It follows that a trader within a meeting does not really need to observe his partner\u2019s types, because all that matter for bargaining are the dynamic types rather than the original types. Therefore, any equilibrium under no discounting would still be an equilibrium when we switch the information structure into the private one. Conversely, consider the private information model with r > 0 and let r \u2192 0. From (2.12) and (2.13), in the limit we have v = \u03c1B (1) and \u03c1S (0) = c\u0304. Clearly a seller (buyer) whenever being a proposer would have no choice but propose the trading price v (c\u0304), and this offer would be accepted. Letting the proposer know the responder\u2019s type does not have a bite, because the proposer already knows the responder\u2019s dynamic type, which is all he needs to make the decision of proposing. Thus, this equilibrium would still be an equilibrium when we switch into the full information model. We now solve the no-discounting case analytically.32 We have already claimed that v = \u03c1B (1) and \u03c1S (0) = c\u0304 in any of the two models. Then both Lemma 2(c) (for private 31We will see (ii) actually implies (i) in either models. 32Mortensen and Wright (2002) have already solved it for their full information model. 65 Chapter 3. Role of Information Structure in Dynamic Matching Markets information) and Lemma 11 (for full information) are (in the limit) reduced to: \u03b1B(\u03b6)\u03b2Bmax {v \u2212 c\u0304, 0} = \u03baB, (3.25) \u03b1S(\u03b6)\u03b2Smax {v \u2212 c\u0304, 0} = \u03baS . (3.26) Equation (3.25) simply means that the type v buyers are indifferent between entering or not: the left-hand side is the expected gain from participating in the market per unit time, and the right-hand side is the corresponding search cost. Similarly equation (3.26) is the indifference condition for marginal sellers. Noticing that \u03b1S(\u03b6)\/\u03b1B(\u03b6) = \u03b6, equations (3.25) and (3.26) uniquely pin down the buyer-seller ratio \u03b6 and entry gap v \u2212 c\u0304: \u03b6 = \u03b60, (3.27) v \u2212 c\u0304 = K (\u03b60) > 0, (3.28) where \u03b60 and K (\u00b7) are defined by (3.9) and (3.10). In steady state, the incoming flow of active buyers must equal the incoming flow of active sellers. Thus in either model we have the following inflow balance equation: b[1\u2212 F (v)] = sG (c\u0304) . (3.29) Given that K (\u03b60) < 1, equations (3.28) and (3.29) have a unique solution for (v, c\u0304), which is denoted as (v0, c\u03040) (see Figure 3.1). Hence, when r \u2192 0 and K (\u03b60) < 1, the equilibrium buyer-seller ratio and marginal types are unique and given by (\u03b60, v0, c\u03040). Other endogenous variables are easily obtained. In particular, the equilibrium is full- trade, so that qB(v) = qS(c) = 1 for any v \u2208 [v0, 1] and c \u2208 [0, c\u03040]. As a result, the aggregate inflow-outflow balance equations become b [1\u2212 F (v0)] = B\u03b1B(\u03b60), sG(c\u03040) = S\u03b1S(\u03b60), which pin down the steady-state masses of buyers and sellers in the market: B = b [1\u2212 F (v0)] \u03b1B(\u03b60) , S = sG(c\u03040) \u03b1S(\u03b60) . 66 Chapter 3. Role of Information Structure in Dynamic Matching Markets Furthermore, the market distributions of types must be proportional to the corresponding distributions of inflow types: NB(v) = B \u00b7 F (v)\u2212 F (v0) 1\u2212 F (v0) , NS(c) = S \u00b7 G(c)G(c\u03040) . From either Lemma 1 (for private information) or Lemma 10 (for full information), we have W 0B(v) = \u2212W 0S(c) = 1 for any v \u2208 [v0, 1] and c \u2208 [0, c\u03040]. Therefore the equilibrium lifetime payoffs are WB(v) = max {v \u2212 v0, 0} , WS(c) = max {c\u03040 \u2212 c, 0} . It is easy to verify that these equilibrium objects indeed satisfy Definition 5 and Definition 6. The above analysis yields the following proposition. Proposition 2 If r = 0, the private information model and the full information model are equivalent, in the sense that the two models have the same set of equilibria.33 In fact, this set is either empty (if K (\u03b60) \u2265 1) or a singleton (K (\u03b60) < 1).34 As in Mortensen and Wright (2002), we generally define the welfare measure W as the aggregate lifetime payoffs of a cohort: W \u2261 bW eaB + sW eaS (3.30) 33Formally, this statement is not completely rigorous, because by Definition 5 an equilibrium of the private information model is a tuple (WB ,WS , \u03c7B, \u03c7S , pB, pS , NB ,NS), while by Definition 6 an equilibrium of the full information model is a collection (WB ,WS , \u03c7B , \u03c7S ,NB,NS); and the proposing strategies (pB, pS) in the private information model are functions with one argument, while in the full information model they are functions with two arguments (own type and partner\u2019s type). Evidently these can be taken care of, but it would not be interesting at all and we do not bother to do so. 34This is under the convention that we identify an equilibrium with another one if they differs only in the proposing strategies of non-entrants and entry strategies of marginal entrants. 67 Chapter 3. Role of Information Structure in Dynamic Matching Markets where W eaB (W ea S ) is a buyer\u2019s (seller\u2019s) ex-ante utility, i.e. W eaB \u2261 Z WB(v)dF (v), (3.31) W eaS \u2261 Z WS(c)dG(c). (3.32) The welfare measure W is also interpreted as the ex-ante utility of an agent before knowing whether he is a buyer or a seller and what his valuation\/cost is. (Recall the normalization that b+ s = 1 so that the measure of a cohort is 1.) For the no-discounting case, the welfare measure W can be written as W = b Z 1 v0 (v \u2212 v0) dF (v) + s Z c\u03040 0 (c\u03040 \u2212 c) dG(c) = Z 1 v0 b [1\u2212 F (v)] dv + Z c\u03040 0 sG(c)dc. Figure 3.1 illustrates the equilibrium for the no-discounting case. The black area represents the welfare W . The Walrasian price p\u2217 must be bracketed by the marginal types v = v0 and c\u0304 = c\u03040. Intuitively, it is analogous to the standard demand-supply analysis, with a transaction cost K (\u03b60) that must be incurred for each transaction. 3.5 Full-trade equilibria and bargaining efficiency The previous section shows that private information in bargaining has no effect in the special case where there is no time discounting. However, when the discount rate is strictly positive, private information will have impacts. This section turns to the question of whether the private information affects the efficiency with respect to bargaining. Let us start with a definition. Definition 7 In either the full information or the private information model, a nontrivial steady-state equilibrium is said to be bargaining-efficient if in this equilibrium the bargaining outcome of every meeting is always ex-post efficient, in the sense that every buyer-seller meeting (on the equilibrium path) results in a trade if and only if the matching surplus is 68 Chapter 3. Role of Information Structure in Dynamic Matching Markets 0 )](1[ vFb \u2212 0vv = )(csG 0cc = )( 0\u03b6K 1 (Flow demand) (Flow supply) *p inflow, transaction flow cv, ),( SBM Figure 3.1: Equilibrium when discount rate is zero non-negative (i.e. v\u2212 c \u2265WB(v) +WS(c)), or equivalently the buyer\u2019s dynamic value is at least as high as the seller\u2019s dynamic cost (i.e. \u03c1B(v) \u2265 \u03c1S(c)). A nontrivial steady-state equilibrium is said to be bargaining-inefficient if it is not bargaining-efficient. Clearly, in the full information model, any nontrivial steady-state equilibrium is bargaining- efficient. In the private information model, it is not hard to see that any nontrivial steady-state equilibrium is bargaining-efficient if and only if it is full-trade. Suppose an equilibrium is full-trade, then the entry gap v \u2212 c\u0304 must be strictly positive, so that every meeting (on the equilibrium path) must have positive matching surplus. Thus the equilibrium is also bargaining-efficient. Now suppose an equilibrium is non-full-trade, then either (i) the marginal buyer\u2019s offer pB (v) will not be accepted with probability 1, or (ii) the marginal seller\u2019s offer pS (c\u0304) will not be accepted with probability 1. To be concrete, let us say (i) is the case. Then there must be sellers with \u03c1S (c) \u2208 (pB (v) , v). But then the marginal buyers, 69 Chapter 3. Role of Information Structure in Dynamic Matching Markets when they propose, would not trade with those sellers, although the matching surplus is positive. Therefore the equilibrium is also bargaining-inefficient. Proposition 3 Under full information, any nontrivial steady-state equilibrium is bargaining- efficient. Under private information, any nontrivial steady-state equilibrium is bargaining- efficient if and only if it is full-trade. We therefore, in the context of private information, only need to recall our results in Chapter 2 on the full-trade equilibrium. By Theorem 2 and Corollary 4, if the discount rate r is small relative to the search costs \u03baB and \u03baS , then in equilibrium bargaining efficiency is attained. In contrast, by Theorem 1 and Corollary 2 (see also Remark 1), if the discount rate is large relative to the search costs, then in equilibrium some meetings do not result in a trade although there is a positive matching surplus. Before closing this section, we want to point out that even when private information does not result in bargaining inefficiency in equilibrium, the equilibrium welfare level would still be altered. It is because the private information affects the way bargainers split the matching surplus. This redistribution effect would in turn have impacts on potential traders\u2019 entry decisions and hence aggregate welfare. We will discuss it in the next section. 3.6 Entry effect of private information If the frictions (r, \u03baB, \u03baS) are zero and the agents behave as if in the Walrasian equilibrium (i.e. type v buyers enter if and only v \u2265 p\u2217, type c sellers enter if and only if c \u2264 p\u2217, and every participating trader proposes p\u2217 whenever she is a proposer, and takes p\u2217 as his reservation price whenever she is a responder), then the welfare measure of our models (either full information or private information) would be at the Walrasian level, denoted as 70 Chapter 3. Role of Information Structure in Dynamic Matching Markets W \u2217: W \u2217 \u2261 b Z 1 p\u2217 (v \u2212 p\u2217)dF (v) + s Z p\u2217 0 (p\u2217 \u2212 c)dG(c) = b Z 1 p\u2217 vdF (v) + s Z p\u2217 0 cdG(c) = Z 1 p\u2217 bF (v)dv + Z p\u2217 0 sG(c)dc. Our search models (either full information or private information) necessarily have lower welfare than the Walrasian level. There are three sources of welfare loss. The first source of welfare loss is the direct effect of frictions (r, \u03baB, \u03baS): search takes time (and traders discount), and search costs have to be paid. The second one is the entry effect: the entry (or search) of a trader induces positive externality to the opposite side of the market (so called \"thick market effect\") and negative externality to the same side (so called \"congestion effect\"). The entry of buyers and sellers could be either too much or too little relative to the constrained optimal level. The third effect is bargaining inefficiency, which occurs only in non-full-trade equilibria of the private information model. In the next Chapter, we will see \"convergence results\" that imply as the frictions vanish, the last two \"behavioral effects\" also vanish.35 However, in this Chapter we are interested in a market with positive frictions, which is why search theory was developed in the first place. We have seen in the previous section that bargaining inefficiency may or may not exist under private information, by and large depending on the composition of frictions. As for entry effect, it is well understood since Diamond (1981) and Mortensen (1982) that, even under full information, search equilibria are generally not constrained efficient due to entry externalities (or search externalities). However, because this strand of literature typically assumes full information bargaining, it does not tell us the interaction between the entry effect and the information structure. In this section, we shall see private information in bargaining affects welfare even when bargaining inefficiency does not arise. Basically, even when the private information does not reduce the pie of matching surplus, it redistributes 35Various versions of this claim for other comparable models can be found in Gale (1987), Mortensen and Wright (2002), Satterthwaite and Shneyerov (2007), Atakan (2008) and Lauermann (2008), etc. 71 Chapter 3. Role of Information Structure in Dynamic Matching Markets the surplus and hence affects the incentives of entry. As a result, private information alters the level of welfare through the channel of entry externalities. To elaborate the above point, let us compare the full-trade equilibria of the two models. To ensure both models have some full-trade equilibrium, we assume K (\u03b60) < 1 and 0 \u2264 r \u2264 min {r\u2217, r\u0302}, where r\u2217 is given by Theorem 1 and r\u0302 is given by Theorem 5. Also, we shall use subscript \"p\" to denote private information (e.g. \u03b6p) and use subscript \"f\" to denote full information (e.g. \u03b6f ). Section 2.5 shows that, in the context of private information, the unique full-trade equi- librium can be characterized by the following three simple equations with three unknowns\u00a1 \u03b6p, vp, c\u0304p \u00a2 :36 \u03b1B(\u03b6p)\u03b2B(vp \u2212 c\u0304p) = \u03baB, (3.33) \u03b1S(\u03b6p)\u03b2S(vp \u2212 c\u0304p) = \u03baS, (3.34) b[1\u2212 F (vp)] = sG (c\u0304p) . (3.35) Equations (3.33) and (3.34) are marginal entering buyers\u2019 and sellers\u2019 indifference conditions between entering or not. Equation (3.35) is inflow balance equation, which must hold in steady state. We now turn to the full information full-trade equilibrium.37 Since the equilibrium is full-trade, we have vf \u2265 c\u0304f , and NBf (v) Bf = F (v)\u2212 F \u00a1 vf \u00a2 1\u2212 F \u00a1 vf \u00a2 , NSf (c) Sf = G (c) G (c\u0304f ) . Also, (3.18) and (3.19) gives the dynamic types: \u03c1Bf (v) = rv + \u03b1B \u00a1 \u03b6f \u00a2 \u03b2Bvf r + \u03b1B \u00a1 \u03b6f \u00a2 \u03b2B , \u03c1Sf (c) = rc+ \u03b1S \u00a1 \u03b6f \u00a2 \u03b2S c\u0304f r + \u03b1S \u00a1 \u03b6f \u00a2 \u03b2S . 36Equations (3.33) through (3.35) are the same as equations (3.25), (3.26) and (3.29) for the no discounting case. It is simply because in the private information model, the full-trade equilibrium \u0003 \u03b6p, vp, c\u0304p \u0004 does not vary with r. 37This is already characterized by Mortensen and Wright (2002). 72 Chapter 3. Role of Information Structure in Dynamic Matching Markets Substituting these into the marginal type equations (3.20) and (3.21) in Lemma 11, we obtain \u03b1B \u00a1 \u03b6f \u00a2 \u03b2B Z c\u0304f 0 \u00a3 vf \u2212 \u03c1Sf (c) \u00a4 dG (c) G (c\u0304f ) = \u03baB, (3.36) \u03b1S \u00a1 \u03b6f \u00a2 \u03b2S Z 1 vf \u00a3 \u03c1Bf (v)\u2212 c\u0304f \u00a4 dF (v) 1\u2212 F \u00a1 vf \u00a2 = \u03baS . (3.37) The inflow balance equation still holds here: b \u00a3 1\u2212 F (vf ) \u00a4 = sG(c\u0304f ). (3.38) Equations (3.36)\u2212 (3.38) uniquely pin down \u00a1 \u03b6f , vf , c\u0304f \u00a2 for all r \u2264 r\u0302. We are now ready to claim that private information in bargaining deters entry, at least within the full-trade class of equilibria. Proposition 4 Fix the parameters (b, s, F,G,M, \u03b2B, \u03b2S, r, \u03baB, \u03baS) such that r > 0 and both the full information model and the private information model have a full-trade equilibrium (i.e. K (\u03b60) < 1 and 0 < r \u2264 min {r\u2217, r\u0302}). Comparing the two full-trade equilibria, we must have vp > vf and c\u0304p < c\u0304f . Proof. From the inflow balance equations (3.35) and (3.38), the two inequalities vp > vf and c\u0304p < c\u0304f are equivalent. Therefore it suffices to prove vp\u2212 c\u0304p > vf \u2212 c\u0304f . We will consider two cases: \u03b6p \u2265 \u03b6f and \u03b6p < \u03b6f . Suppose \u03b6p \u2265 \u03b6f first. Then \u03b1B(\u03b6p) \u2264 \u03b1B(\u03b6f ) since \u03b1B is nonincreasing. Now the buyers\u2019 marginal equations (3.33) and (3.36) imply \u03b1B(\u03b6p)(vp \u2212 c\u0304p) = \u03b1B(\u03b6f ) \u2219 vf \u2212 Z c\u0304f 0 \u03c1Sf (c) dG(c) G(c\u0304f ) \u00b8 vp \u2212 c\u0304p \u2265 vf \u2212 Z c\u0304f 0 \u03c1Sf (c) dG(c) G(c\u0304f ) . Moreover, r > 0 implies \u03c1Sf (0) < c\u0304f and henceZ c\u0304f 0 \u03c1Sf (c) dG(c) G(c\u0304f ) < c\u0304f . Combining the above results, we have vp \u2212 c\u0304p > vf \u2212 c\u0304f , as desired. 73 Chapter 3. Role of Information Structure in Dynamic Matching Markets Now suppose \u03b6p < \u03b6f . Applying a symmetric logic to the sellers\u2019 marginal equations (3.34) and (3.37), we can easily obtain vp \u2212 c\u0304p > vf \u2212 c\u0304f again. To see the intuition of Proposition 4, we need to compare the entry incentives of the two models. In the full information model, a bargainer extracts the full surplus of matching if he proposes, but gets none if he responds. In contrast, if types are private information, then typically some information rent is redistributed from the proposer of a meeting to the responder. Now recall that the proposer of a meeting is randomly chosen and notice that the responder\u2019s expected information rent is higher if his type is better (i.e. higher value or lower cost). As a result, the redistribution of rent from proposers to responders would be translated into a redistribution from inefficient (i.e. low value or high cost) traders to efficient (i.e. high value or low cost) traders in the market. Therefore, if the information structure is switched from full information into private information, inframarginal entrants have higher incentive to enter while marginal entrants have lower incentive to enter. Now it is clear that the private information model tends to induce less entry, because what matters to the equilibrium amount of entry is the entry incentive of marginal entrants. Proposition 4 simply says that this is unambiguously true when r > 0 and we are comparing two full-trade equilibria. Put it more concretely. Suppose, as a thought experiment, that the marginal types v and c\u0304 are the same across the two models. Notice that a marginal entering buyer as a responder gets zero rent from bargaining anyway. But as a proposer, a marginal entering buyer under full information extracts the full rent from the seller he meets; while he is (in a full-trade equilibrium) only able to extract the rent of the most inefficient type c\u0304 of sellers under private information. If r = 0, it makes no difference because the distribution of sellers\u2019 dynamic types collapses to a single point. But if r is positive, that marginal entering buyer tends to be worse off under private information than under full information. This tendency might be reversed only when the buyer-seller ratio is lower (so that the buyers\u2019 arrival rate of being matched is higher) under private information, i.e. \u03b6p < \u03b6f . Similarly, if r > 0, a marginal seller is worse off under private information than under full information, 74 Chapter 3. Role of Information Structure in Dynamic Matching Markets unless \u03b6p > \u03b6f . Since \u03b6p < \u03b6f and \u03b6p > \u03b6f cannot hold together, either more buyers or more sellers must be attracted to enter. Finally, the entry must be more for both the buyers\u2019 side and the sellers\u2019 side, because the inflows of the two sides have to balance in steady state. Then why is the result in Proposition 4 only for full-trade equilibria? It is because in non-full-trade equilibria of the private information model the marginal entrants\u2019 propos- ing gains also depend on the steady-state distributions of dynamic types in the market. These distributions and the traders\u2019 bargaining behaviors affect each other in a highly non- trivial way. Thus it is conceivable that, from a marginal buyer\u2019s (seller\u2019s) standpoint, the distribution of sellers\u2019 (buyers\u2019) dynamic types in the private information model is much more favorable than the full information counterpart. And this effect might dominate the aforementioned information rent effect. Our next goal is to evaluate the impact of private information on the equilibrium buyer- seller ratio \u03b6 and level of welfare W . To proceed, in the rest of this section we shall focus on cases where r is positive but sufficiently close to 0. In such cases, both models have a unique equilibrium, which is full-trade. Doing this has several advantages, both methodologically and technically. First, we can annihilate bargaining inefficiency so that the entry effect is isolated out. Second, by virtue of uniqueness we do not need to worry about the selection of equilibria. Third, the relatively simple structures of full-trade equilibria in both models make it feasible to compare the levels of welfare under private and full information. Fourth, by virtue of equivalence between the two models in the no-discounting case, studying sufficiently small discounting case only amounts to working out the \"first- order effects\" of r around the r = 0 case. Yet the main insight gained from our analysis should also enlighten our understanding of the main driving forces in the general case. From now on, we shall think of the equilibrium objects as functions of r. For example, we shall write \u03b6p(r), \u03b6f (r), WBp(v; r), WSf (c; r) etc., although the dependency on r might be suppressed for notational simplicity. 75 Chapter 3. Role of Information Structure in Dynamic Matching Markets The welfare measure (defined by (3.30) in general) in the full information model is: Wf (r) = b Z WBf (v; r)dF (v) + s Z WSf (c; r)dG(c) (3.39) = \u03b1B \u00a1 \u03b6f \u00a2 \u03b2B r + \u03b1B \u00a1 \u03b6f \u00a2 \u03b2B b Z 1 vf \u00a1 v \u2212 vf \u00a2 dF (v) + \u03b1S \u00a1 \u03b6f \u00a2 \u03b2S r + \u03b1S \u00a1 \u03b6f \u00a2 \u03b2S s Z c\u0304f 0 (c\u0304f \u2212 c) dG (c) . And, the welfare measure in the private information model is: Wp (r) = b Z WBp(v; r)dF (v) + s Z WSp(c; r)dG(c) (3.40) = \u03b1B (\u03b60) r + \u03b1B (\u03b60) bW eaB0 + \u03b1S (\u03b60) r + \u03b1S (\u03b60) sW eaS0, where W eaB0 (W ea S0) is a buyer\u2019s (seller\u2019s) ex-ante utility in the no-discounting case, i.e. W eaB0 \u2261 Z 1 v0 (v \u2212 v0) dF (v) , W eaS0 \u2261 Z c\u03040 0 (c\u03040 \u2212 c) dG (c) . (3.41) It is clear from (3.33)\u2212 (3.35) that, under private information and small r, the equilib- rium buyer-seller ratio \u03b6p and the marginal entering types vp and c\u0304p do not change when r varies. Thus \u03b6p, vp and c\u0304p are simply at the levels of the no-discounting case. Mathemat- ically, \u03b6p(r) = \u03b60, vp(r) = v0 and c\u0304p(r) = c\u03040 for all r sufficiently close to 0. Furthermore, as we have claimed in Section 3.4, when r = 0, the two models are equivalent. Indeed, it is easy to verify that \u03b6f (0) = \u03b60, vf (0) = v0, c\u0304f (0) = c\u03040 and Wp (0) = Wf (0). By virtue of these, the comparison between the two models for sufficiently small r > 0 amounts only to working out the derivatives \u03b6 0f (0), W 0p (0) and W 0f (0). Proposition 5 For all sufficiently small r > 0, the private information model has higher (resp. lower) buyer-seller ratio compared to the full information model if sW eaS0 \u03baS \u2212 bW ea B0 \u03baB is negative (resp. positive). 76 Chapter 3. Role of Information Structure in Dynamic Matching Markets Proof. It is shown in Appendix B that the sign of \u03b6 0f (0) is the same as that of the expression sW ea S0 \u03baS \u2212 bW ea B0 \u03baB . This, together with \u03b6f (0) = \u03b6p(r) for sufficiently small r, implies the result. Since sW ea S0 \u03baS \u2212 bW ea B0 \u03baB could be positive or negative, Proposition 5 implies that the pri- vate information model may have higher or lower buyer-seller ratio compared to the full information model. Under private information, the slope of the welfare W 0p (r) evaluated at r = 0 is W 0p (0) = \u2212 bW eaB0 \u03b1B (\u03b60) \u2212 sW ea S0 \u03b1S (\u03b60) . (3.42) This is simply the direct effect of discounting. In particular, the effect of discounting on buyers\u2019 (resp. sellers\u2019) welfare is proportional to their expected searching time 1\/\u03b1B (resp. 1\/\u03b1S). Under full information, in contrast, the slope of the welfare W 0f (r) evaluated at r = 0, as shown in Appendix B, is W 0f (0) = \u2212 bW eaB0 \u03b1B (\u03b60) \u2212 sW ea S0 \u03b1S (\u03b60) \u2212 sG (c\u03040)K 0 (\u03b60) \u03b6 0f (0) . (3.43) Other than the direct effect, the increase in r away from 0, by inducing additional entry, could increase or decrease the buyer-seller ratio \u03b6f , which in turn affects the expected searching time 1\/\u03b1B and 1\/\u03b1S. Thus the indirect effect on the total accumulated search costs incurred by a cohort is the last term in (3.43). In Appendix B, we also show that the difference of the two slopes can be written as W 0p (0)\u2212W 0f (0) = sG (c\u03040)K 0 (\u03b60) \u03b6 0f (0) = K (\u03b60) [\u03c3S (\u03b60)\u2212 \u03b2S ] \u00b5 sW eaS0 \u03baS \u2212 bW ea B0 \u03baB \u00b6 (3.44) where \u03c3S (\u03b6) \u2261 1 \u2212 \u03b6m0 (\u03b6) \/m (\u03b6) is the elasticity of the matching function with respect to the mass of sellers (i.e. \u03c3S (\u03b6) = SM2 (B,S) \/M (B,S)). We thus have the following theorem. 77 Chapter 3. Role of Information Structure in Dynamic Matching Markets Theorem 6 For all sufficiently small r > 0, the private information welfare Wp (r) is higher (resp. lower) than the full information welfare Wf (r), if [\u03c3S (\u03b60)\u2212 \u03b2S ] \u00b5 sW eaS0 \u03baS \u2212 bW ea B0 \u03baB \u00b6 is positive (resp. negative). It is easy to see that the difference W 0p (0)\u2212W 0f (0) may be either positive or negative, depending on the elasticity of the matching function, the search costs, the new-born rates and the new-born distributions. For example, if the new-born rates are equal (i.e. b = s), the new-born distributions F and G are flips of each other (i.e. 1 \u2212 F (x) = G(1 \u2212 x) for all x \u2208 [0, 1], so that W eaS0 = W eaB0) and the matching function is Cobb-Douglas \u221a BS (so that \u03c3S = 1\/2), then the sign of W 0p (0) \u2212W 0f (0) is the same as \u00a1 1 2 \u2212 \u03b2S \u00a2 (\u03baB \u2212 \u03baS). In other words, when the discount rate is positive but small, the private information welfare is higher (resp. lower) than the full information welfare if the side with greater bargaining power incurs higher (resp. lower) search costs. The intuition behind Theorem 6 is the following. Basically, the first factor \u03c3S (\u03b60) \u2212 \u03b2S summarizes entry externalities, while the second factor sW eaS0 \u03baS \u2212 bW ea B0 \u03baB represents how information structure affects the equilibrium buyer-seller ratio. The product of the two hence represents the interaction between entry externalities and information structure. To get more insight, recall that traders\u2019 entry imposes positive externality to the opposite side of the market and negative externality to the same side. In case of constant returns to scale matching technology and zero discount rate, Mortensen and Wright (2002) show that the positive and negative externalities completely cancel out only when the Hosios (1990) condition holds, i.e. \u03c3S = \u03b2S. If, for example, the elasticity of matching function with respect to the mass of sellers, \u03c3S, is larger than sellers\u2019 bargaining weight \u03b2S , then the equilibrium buyer-seller ratio is higher than the constrained optimal level, hence decreasing \u03b6 would be welfare enhancing. On the other hand, Proposition 5 implies that for small positive r, if sW ea S0 \u03baS > bW ea B0 \u03baB , then the private information model has smaller \u03b6. Therefore, the private information model could have better welfare performance if both \u03c3S > \u03b2S and sW eaS0 \u03baS > bW ea B0 \u03baB hold. 78 Chapter 3. Role of Information Structure in Dynamic Matching Markets 3.7 Concluding remarks Until recently, the literature of search models and dynamic matching and bargaining games usually adopts (generalized) Nash bargaining solution, which inevitably requires that the bargainers know each other\u2019s type during the bargaining. This might not be an appealing assumption for many applications. In order to understand the impact of releasing this common assumption, we have an- alyzed and compared two models of dynamic matching markets: the private information model and the full information model. The two models differ in only one aspect: whether the bargainers observe each other\u2019s type during the bargaining. There are two kinds of frictions: discount rate and search costs. If the discount rate is zero, private information bargaining has no impact at all. More generally, the smaller the discount rate relative to the search costs, the more alike the two models are. The bargaining efficiency, an equilibrium property of the full information model, is maintained under private information bargaining as long as the discount rate is small enough relative to the search costs. Furthermore, private information bargaining does not affect when the market would breakdown and when open. The private information model induces less potential traders to enter, at least when the discount rate is small. Intimately relating to Hosios (1990) condition, the impact of private information on social welfare could be either positive or negative. Before closing this chapter, we note that the dynamic structure of our models is im- portant to obtain our results. An easy way to see this is to notice that the discount rate plays a crucial role in our analyses and results; and the discount rate can play a role only in dynamic models. Indeed, the uniqueness of full-trade equilibria hinges on small discount- ing. Our results on entry effect hinges on the uniqueness and simple characterization of full-trade equilibria. 79 Chapter 4 Rate of Convergence towards Perfect Competition 4.1 Introduction This chapter continues our study of decentralized dynamic matching markets.38 The previ- ous two chapters analyze the markets with non-vanishing frictions. This chapter, in contrast, studies the convergence properties of dynamic matching markets, as search frictions vanish. Our baseline model is the one in Chapter 2. In particular, the buyers and sellers partici- pating in the market are matched pairwise; and every pair of buyer and seller bargains over the trading outcome under the so-called random-proposer bargaining protocol, and under two-sided private information. In order to reduce repetition, this chapter is not prepared to be self-contained. Readers should have read either Chapter 2 or Chapter 3 before reading this chapter. Our basic results are as follows. As frictions vanish, we not only show that the equilib- rium price range collapses to the Walrasian (or market-clearing) price, but also show that the rate of convergence is linear, i.e. of the same order as frictions (Theorem 7 and Corollary 7). Furthermore, under random-proposer bargaining, equilibrium welfare also converges to the first best Walrasian level at the linear rate, which is shown to be the fastest possible rate among all bargaining mechanisms (Theorem 9, Theorem 10 and Corollary 11). 38The chapter significantly includes the materials in my manuscript \"The Rate of Convergence to Perfect Competition of a Simple Matching and Bargaining Mechanism\", which is joint with my thesis co-supervisor Artyom Shneyerov. 80 Chapter 4. Rate of Convergence towards Perfect Competition We also provide two robustness checks for our basic results. The first one is to assume full information bargaining, as in Mortensen and Wright (2002), rather than private infor- mation bargaining. We show that our basic results are robust to this switch of information structure (Theorem 8 and Corollary 10). The second robustness check is to assume bilat- eral double auction bargaining, as first introduced by Chatterjee and Samuelson (1983), instead of random-proposer take-it-or-leave-it bargaining. We show that our basic results are not robust to this switch of bargaining protocol. More precisely, along some sequences of nontrivial steady-state equilibria under double auction bargaining, the equilibrium price range does not collapse to the Walrasian price, and the equilibrium welfare level does not converge to the Walrasian welfare level. These results suggest that information structure at the bargaining stages does not affect asymptotic efficiency, but bargaining protocol might. To understand why the random-proposer bargaining has robust convergence property but the double auction bargaining does not, first notice the well known fact that double auction bargaining generates plethora of equilibria. As it turns out, although there are sequences that are convergent to perfect competition, we are also able to select sequences of progressively inefficient equilibria that keep far from it, no matter how small the frictions are. This inefficiency along our non-convergent sequences is due to a positive entry gap v\u2212 c\u0304 that is bounded away from zero even when frictions vanish. (Recall that v denotes the lowest valuation of those buyers who enter; and c\u0304 denotes the highest cost of sellers who enter.) It is because under double auction, the bargaining power of any bargainer is not guaranteed. One can construct a double auction full-trade equilibrium with large entry gap by giving one side of the market, say sellers, large bargaining power. It makes the buyers reluctant to enter. But it does not mean the sellers have strong incentive to enter. It is because in steady state, more entry of sellers makes the stock of sellers accumulate, so that the steady-state buyer-seller ratio is so low, cancelling the incentive of entry brought by the high bargaining power. This is the intuition behind Theorem 11. In short, under double auction, the potentially unbalanced distribution of bargaining 81 Chapter 4. Rate of Convergence towards Perfect Competition power between buyers and sellers can seriously deter entry of both of the two sides, leading to large inefficiency for arbitrarily small frictions. Random-proposer bargaining, on the other hand, distributes the bargaining power between buyers and sellers rather evenly, which guarantees that both sides have the right entry incentives in the limit. We also discuss another class of progressively inefficient equilibria under double auction, which are so-called two-step equilibria. In such equilibria, there is again positive entry gap that is bounded away from zero, but there is no unbalancedness between buyers and sellers. Instead, we play with some kind of unbalancedness between good traders (i.e. high valuation buyers and low cost sellers) and bad traders (Theorem 13). The structure of this chapter is as follows. Section 4.2 borrows the framework in Chapter 2, which assumes private information random-proposer bargaining, as our baseline model. Section 4.3 derives for our baseline model the rate of convergence of equilibrium price range to the Walrasian price as frictions are removed. This section also shows that the rate of convergence remains unchanged if we assume full information bargaining instead. Section 4.4 gives the rate of convergence of equilibrium welfare to the Walrasian welfare level. This section also proves that this rate is the fastest possible rate among all bargaining mechanisms, either under private or full information. Section 4.5 presents and proves our results for the double auction bargaining. Section 4.6 concludes. 4.2 The baseline model We take the dynamic matching and bargaining game we study in Chapter 2 as the baseline model of this Chapter. In particular, the market we study is decentralized; searching for a trading partner is costly; the trading decisions and trading prices are determined by bilateral bargaining under two-sided incomplete information. Our equilibrium concept is the one we call nontrivial steady-state equilibrium. Readers can consult Chapter 2 for the details and Section 3.2 for a brief review. Recall that the flow rate of pairwise matching generated in the market is given by a matching function M(B,S), where B and S are the masses of active buyers and active 82 Chapter 4. Rate of Convergence towards Perfect Competition sellers currently in the market. Since we want to study the convergence properties of our model as search frictions vanish, let us embed a shifter \u03c4 in the matching function, and write M(B,S; \u03c4) instead of M(B,S). This shifter \u03c4 is chosen to be inversely proportional to the rate of matching. That is, M(B,S; \u03c4) \u2261 M\u0303(B,S) \u03c4 for some function M\u0303 : R2+ \u2192 R+ that satisfies Assumption 2 (see p.12). Therefore \u03c4 is analogous to the time length between matches in discrete time models, e.g. Satterthwaite and Shneyerov (2007). The parameter \u03c4 plays a crucial role throughout this chapter, because we are interested in the asymptotic properties of our model as \u03c4 \u2192 0.39 Let us say a bit more about \u03c4 . Note that \u03c4 is proportional to an active trader\u2019s expected waiting time until his next meeting. To see this, notice that, given steady-state active trader masses B and S, trad- ing opportunities for a buyer arrive at the Poisson rate M(B,S; \u03c4)\/B or equivalently M\u0303(B,S)\/\u03c4B. Therefore the expected waiting time is \u03c4 \u00b7 B\/M\u0303(B,S). Similarly, the ex- pected waiting time for the seller is \u03c4 \u00b7 S\/M\u0303(B,S). The inverse of \u03c4 can also be interpreted as the level of competition intensity that is analogous to the number of traders in the centralized double auction literature. To see why, recall that in a centralized market, traders are competing intratemporally with all other traders on the same side. In contrast, in the dynamic matching environment here, traders, owing to the matching frictions, are not directly competing with all other traders on the same side whenever they bargain with their partners. But they do intertemporally compete with others in the sense that their partners have the option to search another to trade with. Since 1\/\u03c4 is proportional to arrival rates, it measures the local market size that reflects the intensity of this intertemporal competition. Since the buyers\u2019 and sellers\u2019 arrival rates of being matched, denoted as \u03b1B and \u03b1S in Chapter 2, directly depend on \u03c4 , let us write \u03b1B(\u03b6, \u03c4) and \u03b1S(\u03b6, \u03c4 ) instead of \u03b1B(\u03b6) and 39All of our results hold equally well if we fix \u03c4 and let the discount rate r, search costs \u03baB and \u03baS tend to 0 proportionally, instead of letting \u03c4 \u2192 0. 83 Chapter 4. Rate of Convergence towards Perfect Competition \u03b1S(\u03b6). (Recall that \u03b6 \u2261 B\/S.) More precisely, \u03b1B(\u03b6, \u03c4 ) \u2261 M\u0303(\u03b6, 1) \u03c4\u03b6 , \u03b1S(\u03b6, \u03c4) \u2261 M\u0303(\u03b6, 1) \u03c4 . The function K (see p.24) also directly depends on \u03c4 , so we write K(\u03b6, \u03c4) rather than K(\u03b6). Note that, given any \u03b6, K(\u03b6, \u03c4) is proportional to \u03c4 . Specifically, K(\u03b6, \u03c4) \u2261 \u03baB \u03b1B (\u03b6, \u03c4) + \u03baS \u03b1S (\u03b6, \u03c4) = \u03c4 \u00b7K (\u03b6, 1) . All other notations are left unchanged. 4.3 Rate of convergence of trading prices In (nontrivial steady-state) equilibrium, trading prices are different across transactions, simply because buyers and sellers in the market are heterogeneous, and the matching and the selection of proposer are random. Recall that, in our terminology and notations in Chapter 2, the price offer proposed by the proposer (either buyer or seller) of a meeting must fall within what we call the proposing interval [pB (v) , pS (c\u0304)]; while the reservation price (or dynamic type) of the responder must fall within what we call the responding interval [\u03c1S (0) , \u03c1B (1)]. (Of course, these intervals implicitly depend on \u03c4 , and on which equilibrium is prevailing.) Besides, the Walrasian price (or market-clearing price) p\u2217 is the price that clears the flow demand b [1\u2212 F (\u00b7)] and flow supply sG (\u00b7): b[1\u2212 F (p\u2217)] = sG (p\u2217) . The purpose of this section is to prove that both the proposing interval [pB (v) , pS (c\u0304)] and the responding interval [\u03c1S (0) , \u03c1B (1)] collapse to Walrasian price p\u2217 as \u03c4 \u2192 0, and furthermore to show the speed of it. 84 Chapter 4. Rate of Convergence towards Perfect Competition At this point it is helpful to recall Lemma 3, which asserts that in any nontrivial steady- state equilibrium, p\u2217 \u2208 [pB (v) , pS (c\u0304)] \u2282 [\u03c1S (0) , \u03c1B (1)] . Hence it suffices to consider convergence of the length \u03c1B (1)\u2212 \u03c1S (0). Indeed, we will show that \u03c1B (1)\u2212 \u03c1S (0) is O (\u03c4). In other words, as \u03c4 \u2192 0, the length \u03c1B (1)\u2212 \u03c1S (0) converges to 0 at the linear rate. 4.3.1 Convergence of full-trade equilibria Before proving our general rate of convergence theorem, we show how the linear rate is obtained when we restrict our attention to full-trade equilibria. This can be done in a simple manner because a full-trade equilibrium admits a simple characterization. Recall that, in a full-trade equilibrium (if it exists), the buyer-seller ratio \u03b6, and the marginal entering types v and c\u0304 are uniquely determined by the following three equations: \u03b6 = \u03b2B \u03b2S \u03baS \u03baB \u2261 \u03b60, v \u2212 c\u0304 = K (\u03b60, \u03c4) , b[1\u2212 F (v)] = sG(c\u0304). It follows that the entry gap v\u2212 c\u0304 converges to 0 at the linear rate in \u03c4 . Recall Lemma 1 (see p.16). Since qB (v) = qS (c) = 1 and \u03b6 = \u03b60 in the full-trade equilibrium, the slopes of responding strategies also converge to 0 linearly in \u03c4 . Consequently, \u03c1B (1) \u2212 \u03c1S (0) converges at that rate as well. 4.3.2 General convergence theorem Proving that all equilibria (i.e. also non-full-trade) converge at the linear rate in \u03c4 is much harder. However, our result is neat. 85 Chapter 4. Rate of Convergence towards Perfect Competition Theorem 7 (Rate of convergence for trading prices) Fix \u03c4 > 0. In any nontrivial steady-state equilibrium, we have K (\u03b60, \u03c4) \u2264 pS (c\u0304)\u2212 pB (v) \u2264 \u03c1B (1)\u2212 \u03c1S (0) \u2264 K (\u03b60, \u03c4) \u00b5 1 + 2r \u03ba \u00b63 , where \u03b60 \u2261 \u03b2B\u03baS\/\u03b2S\u03baB and \u03ba \u2261 min{\u03baB, \u03baS}. We will prove Theorem 7 in the next subsection. Notice that both the upper and lower bounds in the theorem are proportional to \u03c4 . We thus conclude that the proposing interval and responding interval collapse at the linear rate as \u03c4 \u2192 0. Moreover, the upper bound provided in Theorem 7 converges to the lower bound as r gets small relative to \u03ba \u2261 min{\u03baB, \u03baS}. It indicates that our bounds are tight at least when the discount rate is small relative to the search costs. As a corollary of Theorem 7, traders\u2019 proposing and responding strategies must converge to the Walrasian price at no-slower-than-linear convergence rate. Corollary 7 Fix (r, \u03baB, \u03baS) \u00c0 0. For any sequence of nontrivial steady-state equilibria parametrized by \u03c4 such that \u03c4 \u2192 0, the proposing interval [pB\u03c4 (v) , pS\u03c4 (c\u0304)] and respond- ing interval [\u03c1S\u03c4 (0) , \u03c1B\u03c4 (1)] collapse to the Walrasian price {p\u2217} at no-slower-than-linear convergence rate. More precisely, max {|pB\u03c4 (v)\u2212 p\u2217| , |pS\u03c4 (c\u0304)\u2212 p\u2217| , |\u03c1S\u03c4 (0)\u2212 p\u2217| , |\u03c1B\u03c4 (1)\u2212 p\u2217|} \u2264 K (\u03b60, \u03c4) \u00b5 1 + 2r \u03ba \u00b63 . Before turning to the proof of Theorem 7, we make two remarks. Remark 2 In the previous two chapters and Mortensen and Wright (2002), frictions are represented by the discount rate and search costs. Our result can equally well be interpreted that way: fix the matching function and let the discount rate and search costs be \u03c4 \u00b7(r, \u03baB, \u03baS), then the equilibrium responding interval and proposing interval would collapse at linear rate as \u03c4 \u2192 0. Indeed, the upper and lower bounds in Theorem 7 do not change if we replace \u03c4 by 1 and then (r, \u03baB, \u03baS) by \u03c4 \u00b7 (r, \u03baB, \u03baS) (note that K(\u03b60, 1) will also be replaced by \u03c4 \u00b7K(\u03b60, 1)). 86 Chapter 4. Rate of Convergence towards Perfect Competition Remark 3 We interpret Theorem 7 as a rate of convergence result because our interest of this chapter is the convergence of decentralized market towards perfect competition. But it is clear that Theorem 7 is much more than merely an asymptotic result. More precisely, Theorem 7 provides upper and lower bounds of the lengths pS (c\u0304)\u2212pB (v) and \u03c1B (1)\u2212\u03c1S (0) (and hence the deviation of trading prices from the Walrasian price) for any parameter profile and any nontrivial steady-state equilibrium. In other words, it is also a result for the world of non-vanishing frictions. In this regard, it is complementary to our results in the previous two chapters. The above two remarks can be made for all the results throughout this and the next sections. 4.3.3 Proof of Theorem 7 We are now ready to prove Theorem 7. The following formal proof will be followed by some intuition behind it. Proof of Theorem 7. Step 1 : We claim that (a): v \u2212 \u03c1S (0) \u03c1B (1)\u2212 \u03c1S (0) \u2265 \u03baB r + \u03baB (b): \u03c1B (1)\u2212 c\u0304 \u03c1B (1)\u2212 \u03c1S (0) \u2265 \u03baS r + \u03baS . We provide the proof for part (a) only. The proof for part (b) is the flip of that for part (a). The buyers\u2019 marginal type equation in Lemma 2(c) (see p.18) can be written as \u03b1B (\u03b6)\u03b2B\u0393S(pB (v)) [v \u2212 pB (v)] = \u03baB where \u0393S (p) \u2261 R {c:p\u2212c\u2265WS(c)} dNS(c) S . Notice that qB(v) \u2265 \u03b2B\u0393S(pB (v)) > 0 whenever v \u2208 [v, 1], and that v \u2212 \u03c1S (0) \u2265 v \u2212 pB (v) > 0, we have \u03b1BqB(v)(v \u2212 \u03c1S (0)) \u2265 \u03baB whenever v \u2208 [v, 1]. Then for almost all v \u2208 [v, 1], \u03c10B(v) = r r + \u03b1BqB(v) \u2264 r \u03baB\/(v \u2212 \u03c1S (0)) . Hence \u03c1B (1)\u2212 v = Z 1 v \u03c10B(v)dv \u2264 r \u03baB\/(v \u2212 \u03c1S (0)) , 87 Chapter 4. Rate of Convergence towards Perfect Competition \u03c1B (1)\u2212 v v \u2212 \u03c1S (0) \u2264 r \u03baB , v \u2212 \u03c1S (0) \u03c1B (1)\u2212 \u03c1S (0) = 1 1 + (\u03c1B (1)\u2212 v)\/(v \u2212 \u03c1S (0)) \u2265 1 1 + r\u03baB = \u03baB r + \u03baB . Step 2 : We claim that (a): min{v, c\u0304}\u2212 \u03c1S (0) \u2264 4r (r + \u03baB) \u03b1S\u03b2S\u03baB (b): \u03c1B (1)\u2212max{v, c\u0304} \u2264 4r(r + \u03baS) \u03b1B\u03b2B\u03baS . Again by symmetry, we only provide a proof for (a). Recall that pS (c) solves the sellers\u2019 proposing problem in (2.5). In other words, pS(c) \u2208 argmaxp\u2208[0,1] [1\u2212 \u0393B(p)] [p \u2212 \u03c1S(c)] where \u0393B(p) \u2261 R {v:v\u2212p\u2265WB(v)} dNB(v) B . Let y \u2261 min{v, c\u0304}\u2212 \u03c1S (0). Consider a type c seller with \u03c1S(c) \u2264 \u03c1S (0) + y\/2, then [1\u2212 \u0393B(pS(c))] [pS(c)\u2212 \u03c1S(c)] \u2265 [1\u2212 \u0393B(v)] [v \u2212 \u03c1S(c)] \u2265 v \u2212 \u00b3 \u03c1S (0) + y 2 \u00b4 \u2265 v \u2212 \u03c1S (0) 2 . Consequently, such a seller\u2019s probability of trade in a given meeting, qS (c), is bounded from below by qS (c) \u2265 \u03b2S[1\u2212 \u0393B(pS(c))] \u2265 \u03b2S pS(c)\u2212 \u03c1S(c) v \u2212 \u03c1S (0) 2 \u2265 \u03b2S 2 v \u2212 \u03c1S (0) \u03c1B (1)\u2212 \u03c1S (0) \u2265 \u03b2S\u03baB 2 (r + \u03baB) . The last inequality is from step 1(a). Then from (2.13) in Lemma 1 (see p.16), \u03c10S (c) = r r + \u03b1SqS (c) \u2264 r \u03b1S\u03b2S\u03baB\/2r (r + \u03baB) = 2r (r + \u03baB) \u03b1S\u03b2S\u03baB . Now we can see that y 2 = Z {c:\u03c1S(c)\u2208[\u03c1S(0),\u03c1S(0)+ y2 ]} \u03c10S (c) dc \u2264 2r (r + \u03baB) \u03b1S\u03b2S\u03baB , which is the same as (a). 88 Chapter 4. Rate of Convergence towards Perfect Competition Step 3 : Let \u03ba be min{\u03baB, \u03baS}. We claim that \u03c1B (1)\u2212 \u03c1S (0) \u2264 min \u00bd \u03baS \u03b1S\u03b2S , \u03baB \u03b1B\u03b2B \u00be \u00b7 \u00b3 1 + r \u03ba \u00b4\u00b5 1 + 2r \u03ba \u00b62 . To prove it, first notice that from step 2(a) and inequality (2.24) (see p.31), we have v \u2212 \u03c1S (0) = min{v, c\u0304}\u2212 \u03c1S (0) +max {v \u2212 c\u0304, 0} \u2264 4r (r + \u03baB) \u03b1S\u03b2S\u03baB + \u03baS \u03b1S\u03b2S . Then from step 1(a), \u03c1B (1)\u2212 \u03c1S (0) \u2264 r + \u03baB \u03baB (v \u2212 \u03c1S (0)) \u2264 r + \u03baB \u03b1S\u03b2S\u03baB \u2219 4r(r + \u03baB) \u03baB + \u03baS \u00b8 = \u03baS \u03b1S\u03b2S \u00b5 1 + r \u03baB \u00b6 \u2219 1 + 4r \u03baS \u00b5 1 + r \u03baB \u00b6\u00b8 \u2264 \u03baS \u03b1S\u03b2S \u00b3 1 + r \u03ba \u00b4 \u2219 1 + 4r \u03ba \u00b3 1 + r \u03ba \u00b4\u00b8 = \u03baS \u03b1S\u03b2S \u00b3 1 + r \u03ba \u00b4\u00b5 1 + 2r \u03ba \u00b62 . Similarly, from step 2(b), inequality (2.24) and step 1(b), \u03c1B (1)\u2212 \u03c1S (0) \u2264 \u03baB \u03b1B\u03b2B \u00b3 1 + r \u03ba \u00b4\u00b5 1 + 2r \u03ba \u00b62 . We get our claim by combining the above two upper bounds of \u03c1B (1)\u2212 \u03c1S (0). Step 4 : We claim that \u03c1B (1)\u2212 \u03c1S (0) \u2265 pS (c\u0304)\u2212 pB (v) \u2265 max \u00bd \u03baS \u03b1S\u03b2S , \u03baB \u03b1B\u03b2B \u00be . To prove it, simply observe that Lemma 2(a,c) (see p.18) implies \u03baB \u2264 \u03b1B\u03b2B(v \u2212 pB (v)) \u2264 \u03b1B\u03b2B(pS (c\u0304)\u2212 pB (v)), \u03baS \u2264 \u03b1S\u03b2S(pS (c\u0304)\u2212 c\u0304) \u2264 \u03b1S\u03b2S(pS (c\u0304)\u2212 pB (v)), and pS (c\u0304)\u2212 pB (v) \u2264 \u03c1B (1)\u2212 \u03c1S (0). Step 5 : Combine steps 3 and 4, we get max \u00bd \u03baS \u03b1S\u03b2S , \u03baB \u03b1B\u03b2B \u00be \u2264 pS (c\u0304)\u2212 pB (v) \u2264 \u03c1B (1)\u2212 \u03c1S (0) \u2264 min \u00bd \u03baS \u03b1S\u03b2S , \u03baB \u03b1B\u03b2B \u00be \u00b7 \u00b3 1 + r \u03ba \u00b4\u00b5 1 + 2r \u03ba \u00b62 \u2264 min \u00bd \u03baS \u03b1S\u03b2S , \u03baB \u03b1B\u03b2B \u00be \u00b7 \u00b5 1 + 2r \u03ba \u00b63 . (4.1) 89 Chapter 4. Rate of Convergence towards Perfect Competition From Lemma 4 (see p.26) we have min \u00bd \u03baS \u03b1S\u03b2S , \u03baB \u03b1B\u03b2B \u00be \u2264 K (\u03b60, \u03c4) \u2264 max \u00bd \u03baS \u03b1S\u03b2S , \u03baB \u03b1B\u03b2B \u00be . Combine the above two results, we obtain the theorem. As a by-product of the above proof, we also obtain upper and lower bounds for the equilibrium buyer-seller ratio \u03b6. These bounds do not depend on \u03c4 , which implies that \u03b6 is O (1) as \u03c4 \u2192 0. Corollary 8 In any nontrivial steady-state equilibrium, we have \u03b60 \u00b7 \u00b5 1 + 2r \u03ba \u00b6\u22123 \u2264 \u03b6 \u2264 \u03b60 \u00b7 \u00b5 1 + 2r \u03ba \u00b63 where \u03b60 \u2261 \u03b2B\u03baS\/\u03b2S\u03baB and \u03ba \u2261 min{\u03baB, \u03baS}. Proof. From (4.1) we have \u03baS \u03b1S\u03b2S \u2264 \u03baB \u03b1B\u03b2B \u00b5 1 + 2r \u03ba \u00b63 and \u03baB \u03b1B\u03b2B \u2264 \u03baS \u03b1S\u03b2S \u00b5 1 + 2r \u03ba \u00b63 . Recall that \u03b1S\/\u03b1B = \u03b6. Then we get the result by simple rearranging of terms. We now turn to the intuition behind the proof of Theorem 7. The main parts of the above proof are step 2 through step 4. Steps 2 and 3 derive an upper bound (proportional to \u03c4) for the length of responding interval \u03c1B (1)\u2212\u03c1S (0), while step 4 derives a lower bound for the length of proposing interval pS (c\u0304)\u2212 pB (v). The lower bound part is relatively easy. We have seen in Lemma 2(a) (on p.18) that the marginal entering types v and c\u0304 must fall within the proposing interval [pB (v) , pS (c\u0304)] in equilibrium. If the length pS (c\u0304)\u2212 pB (v) is too small, the marginal entrants would not be able to recover the search costs they incur. (The accumulated search cost is O(\u03c4).) Therefore pS (c\u0304) \u2212 pB (v) is bounded below by \u03c4 multiplied by some constant. The upper bound part is subtler. Following the logic we use to show the linear rate of convergence for full-trade equilibria, we want to show the slopes of responding strategies 90 Chapter 4. Rate of Convergence towards Perfect Competition \u03c1B(v) and \u03c1S(c) are O(\u03c4). Look at sellers for example, from (2.13) (on p.18), \u03c10S(c) is indeed O(\u03c4) for those c such that the probability of trade qS(c) is bounded away from 0. Such a boundedness of qS(c) in turn can be obtained for low cost sellers (\u03c1S(c) \u2264 \u03c1S (0) + y\/2 in step 2) since those sellers, with substantial profitability, would never prefer to make an offer that is accepted with a too low probability. Therefore, \u03c10S(c) is O(\u03c4) for a subset of active sellers. Moreover, our choice of the subset allows us to extend the result to bound min {v, c\u0304}\u2212\u03c1S (0); and then the statement claimed in step 1 further extends the boundedness to the whole length of responding interval \u03c1B (1)\u2212 \u03c1S (0). 4.3.4 Full information model This subsection deviates from our baseline model by considering full-information bargaining as introduced in Section 3.3. We will see that the rate of convergence remains unchanged in this full information model. Furthermore, the proof of the rate of convergence for the full information model is similar to (actually a bit easier than) that for our baseline (private information) model, although it seems not convenient to unify the two proofs. Recall that in the full information model we study in Chapter 3, we assume that, as in our baseline model, traders bargain using the random-proposer protocol with buyers\u2019 bargaining weight \u03b2B \u2208 (0, 1) and sellers\u2019 bargaining weight \u03b2S \u2261 1\u2212 \u03b2B. All trading prices must fall within the interval [\u03c1S (0) , \u03c1B (1)], where \u03c1S (0) is the lowest dynamic type of active sellers and \u03c1B (1) is the highest dynamic type of active buyers. In this full information context, one could also equivalently assume that the bargaining outcome of a matched pair is given by the generalized Nash bargaining solution with buyer\u2019s and seller\u2019s relative bargaining powers being \u03b2B and \u03b2S . Then the following theorem shows that, as \u03c4 \u2192 0, the length \u03c1B (1)\u2212 \u03c1S (0) converges to 0 at the linear rate in \u03c4 .40 40Like in Theorem 7, the upper bound provided in Theorem 8 converges to the lower bound as r gets small relative to \u03ba \u2261 min{\u03baB , \u03baS}. It indicates that our bounds are tight at least when the discount rate is small relative to the search costs. 91 Chapter 4. Rate of Convergence towards Perfect Competition Theorem 8 Under full information bargaining, in any nontrivial steady-state equilibrium, we have K (\u03b60, \u03c4) \u2264 \u03c1B (1)\u2212 \u03c1S (0) \u2264 K (\u03b60, \u03c4) \u00b3 1 + r \u03ba \u00b42 , where \u03b60 \u2261 \u03b2B\u03baS\/\u03b2S\u03baB and \u03ba \u2261 min{\u03baB, \u03baS}. Proof. Step 1 : We claim that (a): v \u2212 \u03c1S (0) \u03c1B (1)\u2212 \u03c1S (0) \u2265 \u03baB r + \u03baB and (b): \u03c1B (1)\u2212 c\u0304 \u03c1B (1)\u2212 \u03c1S (0) \u2265 \u03baS r + \u03baS . We provide the proof for part (a) only. The proof for part (b) is the flip of that for part (a). Applying (3.20) (which is on p.58), we have \u03baB \u03b1B\u03b2B = Z v\u2265\u03c1S(c) [v \u2212 \u03c1S(c)] dNS(c) S \u2264 Z v\u2265\u03c1S(c) [v \u2212 \u03c1S (0)] dNS(c) S = qB(v)(v \u2212 \u03c1S (0)). Thus for any v \u2265 v, we have \u03b1B\u03b2BqB(v) \u2265 \u03baB\/(v \u2212 \u03c1S (0)). Then for almost all v \u2208 [v, 1], \u03c10B(v) = r r + \u03b1B\u03b2BqB(v) \u2264 r \u03baB\/(v \u2212 \u03c1S (0)) . Hence \u03c1B (1)\u2212 v = Z 1 v \u03c10B(v)dv \u2264 r \u03baB\/(v \u2212 \u03c1S (0)) , \u03c1B (1)\u2212 v v \u2212 \u03c1S (0) \u2264 r \u03baB , v \u2212 \u03c1S (0) \u03c1B (1)\u2212 \u03c1S (0) = 1 1 + (\u03c1B (1)\u2212 v)\/(v \u2212 \u03c1S (0)) \u2265 1 1 + r\u03baB = \u03baB r + \u03baB . Step 2 : We claim that (a): min{v, c\u0304}\u2212 \u03c1S (0) \u2264 r \u03b1S\u03b2S and (b): \u03c1B (1)\u2212max{v, c\u0304} \u2264 r \u03b1B\u03b2B . Again by symmetry, we only provide a proof for (a). It is clear that qS (c) = 1 if \u03c1S(c) \u2264 min{v, c\u0304}. Thus, min{v, c\u0304}\u2212 \u03c1S (0) = Z \u03c1S(c)\u2264min{v,c\u0304} \u03c10S(c)dc \u2264 r r + \u03b1S\u03b2S \u2264 r \u03b1S\u03b2S . 92 Chapter 4. Rate of Convergence towards Perfect Competition Step 3 : We claim that \u03c1B (1)\u2212 \u03c1S (0) \u2264 min \u00bd \u03baS \u03b1S\u03b2S , \u03baB \u03b1B\u03b2B \u00be\u00b5 1 + r \u03baB \u00b6\u00b5 1 + r \u03baS \u00b6 . To prove it, first notice that from step 2(a) and inequality (3.24) (see p.60), we have v \u2212 \u03c1S (0) = min{v, c\u0304}\u2212 \u03c1S (0) + max {v \u2212 c\u0304, 0} \u2264 r \u03b1S\u03b2S + \u03baS \u03b1S\u03b2S = \u03baS \u03b1S\u03b2S \u00b5 1 + r \u03baS \u00b6 . Then from step 1(a), \u03c1B (1)\u2212 \u03c1S (0) \u2264 r + \u03baB \u03baB (v \u2212 \u03c1S (0)) \u2264 \u03baS \u03b1S\u03b2S \u00b5 1 + r \u03baB \u00b6\u00b5 1 + r \u03baS \u00b6 . Similarly, from step 2(b), inequality (3.24), and step 1(b), we have \u03c1B (1)\u2212 \u03c1S (0) \u2264 \u03baB \u03b1B\u03b2B \u00b5 1 + r \u03baB \u00b6\u00b5 1 + r \u03baS \u00b6 . Step 4 : We claim that \u03c1B (1)\u2212 \u03c1S (0) \u2265 max \u00bd \u03baS \u03b1S\u03b2S , \u03baB \u03b1B\u03b2B \u00be . To prove it, observe that Lemma 11 (on p.58), together with Lemma 12 (p.59), implies \u03baB \u2264 \u03b1B\u03b2B(\u03c1B (1)\u2212 \u03c1S (0)) \u03baS \u2264 \u03b1S\u03b2S(\u03c1B (1)\u2212 \u03c1S (0)). Step 5 : Combine steps 3 and 4, we get max \u00bd \u03baS \u03b1S\u03b2S , \u03baB \u03b1B\u03b2B \u00be \u2264 \u03c1B (1)\u2212 \u03c1S (0) \u2264 min \u00bd \u03baS \u03b1S\u03b2S , \u03baB \u03b1B\u03b2B \u00be\u00b5 1 + r \u03baB \u00b6\u00b5 1 + r \u03baS \u00b6 \u2264 min \u00bd \u03baS \u03b1S\u03b2S , \u03baB \u03b1B\u03b2B \u00be\u00b3 1 + r \u03ba \u00b42 . (4.2) From Lemma 4 (see p.26) we have min \u00bd \u03baS \u03b1S\u03b2S , \u03baB \u03b1B\u03b2B \u00be \u2264 K (\u03b60, \u03c4) \u2264 max \u00bd \u03baS \u03b1S\u03b2S , \u03baB \u03b1B\u03b2B \u00be . 93 Chapter 4. Rate of Convergence towards Perfect Competition Combine the above two results, we obtain the theorem. As a by-product of the above proof, we also obtain upper and lower bounds for the equilibrium buyer-seller ratio \u03b6. These bounds do not depend on \u03c4 , which implies that \u03b6 is O (1) as \u03c4 \u2192 0. Corollary 9 Under full information bargaining, in any nontrivial steady-state equilibrium, we have \u03b60 \u00b7 \u00b3 1 + r \u03ba \u00b4\u22122 \u2264 \u03b6 \u2264 \u03b60 \u00b7 \u00b3 1 + r \u03ba \u00b42 where \u03b60 \u2261 \u03b2B\u03baS\/\u03b2S\u03baB and \u03ba \u2261 min{\u03baB, \u03baS}. Proof. From (4.2) we have \u03baS \u03b1S\u03b2S \u2264 \u03baB \u03b1B\u03b2B \u00b3 1 + r \u03ba \u00b42 and \u03baB \u03b1B\u03b2B \u2264 \u03baS \u03b1S\u03b2S \u00b3 1 + r \u03ba \u00b42 . Recall that \u03b1S\/\u03b1B = \u03b6. Then we get the result by simple rearranging of terms. Corollary 10 Under full information bargaining, for any sequence of nontrivial steady- state equilibria parametrized by \u03c4 such that \u03c4 \u2192 0, the proposing interval [\u03c1S\u03c4 (0) , \u03c1B\u03c4 (1)] collapses to the Walrasian price {p\u2217} at no-slower-than-linear convergence rate. More pre- cisely, max {|\u03c1S\u03c4 (0)\u2212 p\u2217| , |\u03c1B\u03c4 (1)\u2212 p\u2217|} < K (\u03b60, \u03c4) \u00b3 1 + r \u03ba \u00b42 . Proof. Recall from Lemma 13 (on p.59) that \u03c1S (0) < p\u2217 < \u03c1B (1). Then the result is a straight implication of Theorem 8. 4.4 Rate of convergence of welfare In this section, we turn to the rate of convergence of welfare. We will consider both the private information model and the full information model in a unified way. 94 Chapter 4. Rate of Convergence towards Perfect Competition Recall that the lifetime payoff of a particular new-born type v buyer (type c seller) is denoted as WB(v) (WS(c)). To provide a benchmark for our results, we define their Walrasian counterparts in the usual manner, as W \u2217B (v) \u2261 max{v \u2212 p\u2217, 0}, W \u2217S (c) \u2261 max{p\u2217 \u2212 c, 0}. Recall that we generally define (on p.67) the measure of aggregate welfare W as the aggregate lifetime payoffs of a cohort: W \u2261 bW eaB + sW eaS (4.3) where W eaB (W ea S ) is a buyer\u2019s (seller\u2019s) ex-ante utility, i.e. W eaB \u2261 Z WB(v)dF (v), W eaS \u2261 Z WS(c)dG(c). The Walrasian counterpart of W is: W \u2217 \u2261 b Z 1 p\u2217 (v \u2212 p\u2217) dF (v) + s Z p\u2217 0 (p\u2217 \u2212 c) dG (c) . The following lemma shows that, in either the private or full information model, every trader\u2019s interim lifetime utility converges no slower than the length of responding interval \u03c1B (1)\u2212 \u03c1S (0). Lemma 15 In either the private or full information model, and in any nontrivial steady- state equilibrium, we have |W \u2217B(v)\u2212WB(v)| \u2264 \u03c1B (1) \u2212 \u03c1S (0) and |W \u2217S(c)\u2212WS(c)| \u2264 \u03c1B (1)\u2212 \u03c1S (0), for any v, c \u2208 [0, 1]. Proof. We will only prove the result for buyers. That for sellers can be proved by a symmetric argument. Recall that if v \u2265 v then WB(v) = v \u2212 \u03c1B(v); and if v < v then 95 Chapter 4. Rate of Convergence towards Perfect Competition WB(v) = 0. Consequently, W \u2217B(v)\u2212WB(v) = max{v \u2212 p\u2217, 0}\u2212WB(v) = \u23a7 \u23aa\u23aa\u23aa\u23aa\u23a8 \u23aa\u23aa\u23aa\u23a9 \u03c1B(v)\u2212 p\u2217 if v \u2265 p\u2217 and v \u2265 v \u03c1B(v)\u2212 v if v < p\u2217 and v \u2265 v v \u2212 p\u2217 if v \u2265 p\u2217 and v < v 0 if v < p\u2217 and v < v . In any of the four cases, we must have |W \u2217B(v)\u2212WB(v)| < \u03c1B (1) \u2212 \u03c1S (0). It is obvious for the fourth case. For the other three cases, recall Lemma 2 and Lemma 3 in Section 2.4, Lemma 12 and 13 in Subsection 3.3.2, and the monotonicity of \u03c1B in both models. We can see that (i) p\u2217 \u2208 [\u03c1S (0) , \u03c1B (1)], (ii) \u03c1B(v) \u2208 [\u03c1S (0) , \u03c1B (1)] under the conditions of the first and second cases, and (iii) v \u2208 [\u03c1S (0) , \u03c1B (1)] under the conditions of the second or third cases. Combine Lemma 15, Theorem 7 and Theorem 8, we obtain the following rate of con- vergence theorem for interim lifetime utilities. Theorem 9 (Rate of convergence for interim lifetime utilities) Fix (r, \u03baB, \u03baS)\u00c0 0. Then the interim lifetime utilities WB\u03c4 (v), WS\u03c4 (c) converge to their Walrasian counter- parts W \u2217B (v) and W \u2217 S (c) at least as fast as linear rate, as \u03c4 \u2192 0. More precisely, for all v, c \u2208 [0, 1], we have max {|W \u2217B(v)\u2212WB\u03c4 (v)| , |W \u2217S(c)\u2212WS\u03c4 (c)|} \u2264 K (\u03b60, \u03c4) \u00b5 1 + 2r \u03ba \u00b63 for both the private information model and the full information model. Remark 4 In Theorem 9, absolute values for both W \u2217B(v)\u2212WB\u03c4 (v) and W \u2217S(c)\u2212WS\u03c4 (c) are needed because they are not guaranteed to be positive. Indeed, if v\u03c4 < p\u2217, then buyers with type v \u2208 (v\u03c4 , p\u2217] would have strictly positive utilities in equilibrium but have 0 Wal- rasian utilities. Furthermore, we have not precluded the possibility that some interim utility converges at a faster-than-linear rate. It is because we do not have a positive lower bound for 1\u03c4 |W \u2217B(v)\u2212WB\u03c4 (v)| and 1\u03c4 |W \u2217S(c)\u2212WS\u03c4 (c)|. Indeed, for some types v, we could have W \u2217B(v) =WB\u03c4 (v) = 0. 96 Chapter 4. Rate of Convergence towards Perfect Competition Our baseline model assumes a random-proposer take-it-or-leave-it bargaining game. But the treatment can be straightforwardly extended to any bargaining protocol as long as traders\u2019 types are private information and attention is still restricted to steady state. In particular Lemma 1 (on p.16) holds for the double auction bargaining protocol as well, although, as shown later, our convergence results fail for arbitrary protocol. We now show that no bargaining mechanism can generate the (steady-state) aggregate welfare W converging at a faster than linear rate in \u03c4 , regardless of whether information is full or private. Any bargaining game played in each meeting results in a trading probability q (v, c) \u2208 [0, 1] and expected payment t (v, c) from the buyer to the seller, as functions of traders\u2019 types. In steady-state equilibrium, the bargaining outcomes q and t are unchanged over time. Then, contingent on entry, buyers\u2019 and sellers\u2019 lifetime payoff WB and WS are given by the following Bellman equations: rWB(v) = \u03b1B(\u03b6)[qB(v)v \u2212 tB(v)\u2212 qB (v)WB (v)]\u2212 \u03baB rWS(c) = \u03b1S(\u03b6)[tS (c)\u2212 qS (c) c\u2212 qS (c)WS (c)]\u2212 \u03baS where qB(v) \u2261 Z q(v, c) dNS(c) S , qS(c) \u2261 Z q(v, c) dNB(v) B , tB(v) \u2261 Z t(v, c) dNS(c) S , tS(c) \u2261 Z t(v, c) dNB(v) B . The functions qB (v) and qB (c) are, as before, the trading probabilities in a given meeting conditional only on traders\u2019 own types; tB (v), tS (c) are the expected payments conditional only on own types. Given the bargaining mechanism, the entry of traders is voluntary. We assume the entry of every trader is a one-time decision: once being born, every trader can choose either to stay away from the market forever (in which case his payoff is 0), or to stay in the market until he trades successfully. In steady state this restriction is not binding. Let \u03c7B(v) and \u03c7S(c) be the buyers\u2019 and sellers\u2019 entry probabilities respectively. From the above Bellman equations, we can write WB(v) = \u03c7B (v) \u00b7 \u03b1B(\u03b6)[qB(v)v \u2212 tB(v)]\u2212 \u03baB r + \u03b1B(\u03b6)qB (v) (4.4) 97 Chapter 4. Rate of Convergence towards Perfect Competition WS(c) = \u03c7S(c) \u00b7 \u03b1S(\u03b6)[tS (c)\u2212 qS (c) c]\u2212 \u03baS r + \u03b1S(\u03b6)qS (c) . (4.5) Individual rationality requires that WB(v) \u2265 0 and WS(c) \u2265 0 for all v, c \u2208 [0, 1]. Equiva- lently, individual rationality requires \u03b1B[qB(v)v \u2212 tB(v)] \u2265 \u03baB if \u03c7B(v) > 0, \u03b1S[tS(c)\u2212 qS(c)c] \u2265 \u03baS if \u03c7S(c) > 0. (4.6) The steady-state equations for market distributionsNB andNS are maintained as before. We now prove that no individually rational bargaining mechanism can have a faster- than-linear rate of convergence for the steady-state welfare W , by establishing an explicit lower bound on W \u2217 \u2212W . Theorem 10 For any individually rational bargaining protocol, in steady-state equilibrium we have W \u2217 \u2212W \u2265 \u03bc \u00b7min \u03b6>0 K (\u03b6, \u03c4) , (4.7) where \u03bc is the equilibrium mass of buyers (or sellers) who enter the market per unit time. Proof. Rewrite the Walrasian welfare level W \u2217: W \u2217 = b Z 1 p\u2217 (v \u2212 p\u2217)dF (v) + s Z p\u2217 0 (p\u2217 \u2212 c)dG(c) = max \u03c7B ,\u03c7S \u23a7 \u23aa\u23aa\u23a8 \u23aa\u23aa\u23a9 b R \u03c7B(v)vdF (v)\u2212 s R \u03c7S(c)cdG(c) s.t. b R \u03c7B(v)dF (v) = s R \u03c7S(c)dG(c), 0 \u2264 \u03c7B(v) \u2264 1, 0 \u2264 \u03c7S(c) \u2264 1 \u23ab \u23aa\u23aa\u23ac \u23aa\u23aa\u23ad . (4.8) On the other hand, the equilibrium welfare levelW for any individually rational bargaining mechanism can be bounded as follows. For any active buyer type v (i.e. \u03c7B(v) 6= 0), individual rationality requires \u03b1B[qB(v)v \u2212 tB(v)] \u2265 \u03baB. Hence, from (4.4) we have WB (v) \u2264 \u03c7B(v) \u00b7 \u03b1B[qB(v)v \u2212 tB(v)]\u2212 \u03baB \u03b1BqB(v) = \u03c7B(v) \u00b7 \u2219 v \u2212 \u03baB \u03b1BqB(v) \u2212 tB(v) qB(v) \u00b8 . 98 Chapter 4. Rate of Convergence towards Perfect Competition Similarly for sellers: WS (c) \u2264 \u03c7S(c) \u00b7 \u2219 \u2212c\u2212 \u03baS \u03b1SqS(c) + tS(c) qS(c) \u00b8 . Substituting these bounds into the definition (4.3), W \u2264 b Z \u03c7B(v)vdF (v)\u2212 s Z \u03c7S(c)cdG(c) (4.9) \u2212b Z \u03c7B(v) \u03baB \u03b1B dF (v)\u2212 s Z \u03c7S(c) \u03baS \u03b1S dG(c) \u2212b Z \u03c7B(v) tB(v) qB(v) dF (v) + s Z \u03c7S(c) tS(c) qS(c) dG(c). (In the second line, we have used qB(v) \u2264 1 and qS(c) \u2264 1.) In view of (4.8), the terms in the first line of the right hand side do not exceed the Walrasian surplus W \u2217. Also, since the steady-state condition implies that b\u03c7B(v)dF (v) = \u03b1BqB(v)dNB(v) for buyers and s\u03c7S(c)dG(c) = \u03b1SqS(c)dNS(c) for sellers, and the transfers are balanced,Z tB(v) dNB(v) B = Z tS(c) dNS(c) S , the last line in (4.9) is 0. Taking all these into account, we have W \u2264W \u2217 \u2212 b\u03baB \u03b1B Z \u03c7B(v)dF (v)\u2212 s \u03baS \u03b1S Z \u03c7S(c)dG(c) and therefore W \u2217 \u2212W \u2265 \u00b5 \u03baB \u03b1B + \u03baS \u03b1S \u00b6 \u03bc, where \u03bc \u2261 b Z \u03c7B(v)dF (v) = s Z \u03c7S(c)dG(c) is the equilibrium mass of buyers (or sellers) who enter the market per unit time. Further- more, \u03baB \u03b1B(\u03b6, \u03c4) + \u03baS \u03b1S(\u03b6, \u03c4 ) = K (\u03b6, \u03c4) \u2265 min \u03b6>0 K (\u03b6, \u03c4) . The inequality (4.7) follows. As \u03c4 \u2192 0, we must have \u03bc\u03c4 \u2192 sG (p\u2217) whenever W\u03c4 \u2192 W \u2217. We therefore have the following corollary. 99 Chapter 4. Rate of Convergence towards Perfect Competition Corollary 11 No individually rational bargaining mechanism can attain a faster-than- linear convergence rate for the traders\u2019 aggregate welfare level W\u03c4 as \u03c4 \u2192 0. Remark 5 Since Theorem 10 does not require incentive compatibility, it in particular im- plies that even with full information, as in Mortensen and Wright (2002), convergence cannot be faster than linear. Remark 6 Theorem 9 and Theorem 10 together imply that the traders\u2019 aggregate welfare level W\u03c4 , in either the private or full information model, converges to W \u2217 at exactly linear rate as \u03c4 tends to 0. It follows that the random-proposer bargaining mechanism attains the fastest possible convergence towards the first best (i.e. Walrasian welfare level), independent of the infor- mation structure of bargaining. The intuition for why no other bargaining mechanism can attain a faster rate for welfare is that matching delays will still be present regardless of the efficiency of bargaining. Even if only the buyers with v \u2265 p\u2217 and sellers with c \u2264 p\u2217 enter and always trade to full efficiency, there still will be welfare loss at rate \u03c4 because of costly search (and discounting), since the expected time between matches is proportional to \u03c4 . This might motivate one to separate the welfare loss into the loss directly due to delay and search costs, and the loss indirectly due to equilibrium behaviors. More precisely, let us explicitly think of buyers\u2019 lifetime utility as functions of \u03c3B \u2261 (\u03c7B, qB, tB) and (r, \u03baB): W\u0302B (v;\u03c3B; r, \u03baB) \u2261 \u03c7B(v) \u00b7 \u03b1B [qB(v)v \u2212 tB(v)]\u2212 \u03baB r + \u03b1BqB(v) . And similarly for sellers, W\u0302S (c;\u03c3S ; r, \u03baS) \u2261 \u03c7S(c) \u00b7 \u03b1S [tS(c)\u2212 qS(c)c]\u2212 \u03baS r + \u03b1SqS(c) . Obviously W\u0302B (\u00b7;\u03c3B; r, \u03baB) and W\u0302S (\u00b7;\u03c3S; r, \u03baS) become the Walrasian counterparts W \u2217B and W \u2217S when (i) (r, \u03baB, \u03baS) = 0, and (ii) \u03c3B and \u03c3S are at their Walrasian values, i.e. \u03c7B(v) = I [v \u2265 p\u2217] , \u03c7S(c) = I [c \u2264 p\u2217] 100 Chapter 4. Rate of Convergence towards Perfect Competition where I [\u00b7] is 1 if the condition inside the bracket holds, and is 0 otherwise; and for all v \u2265 p\u2217 and all c \u2264 p\u2217, qB(v) = qS(c) = 1, tB(v) = tS(c) = p\u2217. Then we can define the welfare loss indirectly due to equilibrium behaviors as b Z h W \u2217B(v)\u2212 W\u0302B (v;\u03c3B; 0, 0) i dF (v) + s Z h W \u2217S(c)\u2212 W\u0302S (c;\u03c3S ; 0, 0) i dG(c), and define the welfare loss directly due to delay and search costs as b Z h W\u0302B (v;\u03c3B; 0, 0)\u2212 W\u0302B (v;\u03c3B; r, \u03baB) i dF (v) +s Z h W\u0302S (c;\u03c3S; 0, 0)\u2212 W\u0302S (c;\u03c3S; r, \u03baS) i dG(c). The driving force of Theorem 10 is that the direct part is O (\u03c4), for any bargaining protocol. However the indirect part could vanish at a faster-than-linear rate. To see this, notice that this indirect part can be simplified as b Z [I (v \u2265 p\u2217)\u2212 \u03c7B(v)] vdF (v)\u2212 s Z [I (c \u2264 p\u2217)\u2212 \u03c7S(c)] cdG(c). Under random-proposer bargaining (or any other protocol with private information), the entry strategies must be cutoff strategies, i.e. \u03c7B(v) = I [v \u2265 v] and \u03c7S(c) = I [c \u2264 c\u0304]. Hence the indirect loss is simply the familiar deadweight loss triangle: b Z v p\u2217 vdF (v)\u2212 s Z p\u2217 c\u0304 cdG(c). It is easy to see that in our baseline model (i.e. random-proposer bargaining with private information), along a sequence of full-trade equilibria with \u03c4 \u2192 0, this indirect welfare loss vanishes at quadratic rate in \u03c4 , because the entry gap v \u2212 c\u0304 vanishes at linear rate.41 4.5 Results for k-double auction We have seen that the rate of convergence results for our baseline model are robust to other information structure of bargaining. In this section, we are interested in whether our 41 It is not hard to verify that this is also true for our full information model. 101 Chapter 4. Rate of Convergence towards Perfect Competition results are robust to other bargaining protocol. Although we have not been able to prove a general theorem in this direction, we have a theorem showing that another well-studied trading mechanism, the double auction, does not have robust convergence properties. In other words, some sequences of equilibria do not converge to perfect competition. Recall the rules of the bilateral k-double auction introduced by Chatterjee and Samuel- son (1983): once a meeting occurs, the buyer and the seller simultaneously and indepen- dently submit a bid price pB and an ask price pS respectively, and then trade occurs if and only if the buyer\u2019s bid is at least as high as the seller\u2019s ask, at the weighted average price (1\u2212 k)pS + kpB, where k \u2208 (0, 1). As in the baseline model, we assume that the buyer and the seller do not observe each other\u2019s type during the bargaining. We maintain the notation as before up to a bit reinterpretations. The functions pB(v) and pS(c) are now the strategies of submitting bids and asks respectively. There is no responding strategy under double auction, but \u03c1B(v) and \u03c1S(c) are still buyers\u2019 and sellers\u2019 reservation prices, and also called dynamic types. The definition for nontrivial steady-state equilibria can be obtained as a straightforward revision from the baseline (random-proposer) case. Furthermore, Lemma 1 (on p.16) still holds here. The proof goes through almost word-by-word, with the trading probability function replaced with qB(v) \u2261 Z pS(c)\u2264pB(v) dNS(c) S and the expected payment function replaced with tB(v) \u2261 Z pS(c)\u2264pB(v) [kpB (v) + (1\u2212 k)pS (c)] dNS(c) S . In this k-double auction model, as in the baseline model, a nontrivial steady-state equi- librium could be either full-trade or non-full-trade. We claim that the full-trade class of double auction equilibria includes equilibria that are very inefficient, even with arbitrarily small frictions. (But at the same time, this class also includes equilibria that converge to perfect competition.) The set of full-trade equilibria is even easier to characterize for the double auction model. In particular, from Lemma 1, the sets of active buyers\u2019 and sellers\u2019 types are still 102 Chapter 4. Rate of Convergence towards Perfect Competition intervals [v, 1] and [0, c\u0304] for some marginal types v and c\u0304; and we also have \u03c1B(v) < v and \u03c1S(c) > c for all v > v and all c < c\u0304. Since all active traders\u2019 trading probabilities are strictly positive, they must in equilibrium submit serious bids\/asks, and therefore, we must have pB(v) \u2264 \u03c1B(v) < v and pS(c) \u2265 \u03c1S(c) > c for all v > v and all c < c\u0304. Now it is clear that for an equilibrium to be full-trade, we must have c\u0304 \u2264 v, and all traders submit a common bid\/ask p and hence every matched pair trades at the price p. Furthermore, the marginal entrants (i.e. type v buyers and type c\u0304 sellers) have to recover their search costs, thus in any full-trade equilibrium we have c\u0304 < p < v for some p \u2208 (0, 1). Any full-trade equilibrium for the double auction model must satisfy the following mar- ginal type equations and inflow balance equation: \u03b1B (\u03b6, \u03c4) (v \u2212 p) = \u03baB, (4.10) \u03b1S (\u03b6, \u03c4) (p\u2212 c\u0304) = \u03baS , (4.11) b[1\u2212 F (v)] = sG (c\u0304) . (4.12) Unlike in the baseline model, it is easy to see that the converse is also true, i.e. any quadruple {p, \u03b6, v, c\u0304} satisfying (4.10), (4.11), (4.12) and K (\u03b6, \u03c4) < 1 must characterize a full-trade equilibrium. In particular, any trader\u2019s best-response bid\/ask strategy is p, given that all other active traders submit p.42 From equations (4.10) and (4.11), it follows that the entry gap is v \u2212 c\u0304 = K (\u03b6, \u03c4) . (4.13) The next proposition shows that v \u2212 c\u0304 can be arbitrarily close to 1 for all \u03c4 (such that an equilibrium exists), so that the equilibrium outcomes can be arbitrarily far from efficiency even with small frictions. The set of equilibrium entry gaps converges to the full-range (0, 1) as frictions disappear, so the set of full-trade equilibria ranges from the perfectly competitive one to the almost perfectly inefficient ones. Moreover, the set of equilibrium prices also converges to the full-range (0, 1) as frictions disappear. Thus indeterminacy 42Clearly, equations (4.10)-(4.12) still characterize a full-trade equilibrium even if we assume full informa- tion double auction bargaining. 103 Chapter 4. Rate of Convergence towards Perfect Competition ( )\u03c4\u03b6 ,K \u03b6 \u03b6 \u03b6 ( )csG )](1[ vFb \u2212 ( )csG c v 0\u03b61\u03b6 cv , 1 Figure 4.1: Construction of a double auction full-trade equilibrium grows rather than vanishes with competition, contrary to the results in the static double auction literature. Theorem 11 Under double auction, a full-trade equilibrium exists if and only if min \u03b6>0 K (\u03b6, \u03c4 ) < 1. (4.14) The set of equilibrium values of v\u2212c\u0304 in full-trade equilibria is an interval [min\u03b6>0K (\u03b6, \u03c4) , 1). As \u03c4 \u2192 0, this set and the set of equilibrium prices converge to (0, 1). In particular, there ex- ist sequences of full-trade equilibria that converges to perfect competition, and also sequences that do not converge. Proof. The proof follows the graphical argument shown in Figure 4.1. Given \u03c4 , the right panel shows the marginal types v and c\u0304 in a steady-state equilibrium. The left panel shows the supportable values of buyer-seller ratio \u03b6 and \u03b6\u0304 that correspond to the given entry gap v \u2212 c\u0304 < 1. (In general, there can be one, two or more such values.) Fix any \u03c4 > 0. Our assumption M (0, S; \u03c4) = M (B, 0; \u03c4) = 0 implies \u03b1B (\u221e, \u03c4) = 104 Chapter 4. Rate of Convergence towards Perfect Competition \u03b1S (0, \u03c4) = 0. It in turn implies lim \u03b6\u21920 K (\u03b6, \u03c4) = lim \u03b6\u2192\u221e K (\u03b6, \u03c4 ) =\u221e, (4.15) as depicted in the left panel. Given that (4.15) holds, a solution \u03b6 to the equation K (\u03b6, \u03c4) = v \u2212 c\u0304 exists if and only if v \u2212 c\u0304 \u2208 [min\u03b6>0K (\u03b6, \u03c4) , 1). Since lim\u03c4\u21920K (\u03b6, \u03c4) = 0 for any \u03b6 > 0, we also must have min\u03b6>0K (\u03b6, \u03c4) \u2192 0 as \u03c4 \u2192 0. It proves that the set of supportable values of entry gap v \u2212 c\u0304 converges to the interval (0, 1). Now fix any \u03c4 such that min\u03b6>0K (\u03b6, \u03c4) < 1. Consider the longest interval [\u03b60, \u03b61] such that K (\u03b60, \u03c4) = K (\u03b61, \u03c4) = 1 and K (\u03b6, \u03c4) < 1 for \u03b6 \u2208 (\u03b60, \u03b61). For any \u03b6 \u2208 (\u03b60, \u03b61), v and c\u0304 can be found uniquely from (4.13) and (4.12) (graphically shown in Figure 4.1). Denote v\u03c4 (\u03b6) and c\u0304\u03c4 (\u03b6) as the results. The equilibrium price p can also be found uniquely from equation (4.10) or equation (4.11): p\u03c4 (\u03b6) \u2261 c\u0304\u03c4 (\u03b6) + \u03baS \u03b1S (\u03b6, \u03c4) (4.16) ( = v\u03c4 (\u03b6)\u2212 \u03baB \u03b1B (\u03b6, \u03c4) ). (4.17) This formally defines a continuous mapping p\u03c4 (\u00b7) of [\u03b60, \u03b61] into R+. Consequently, its image is a closed interval that contains the points p (\u03b60) and p (\u03b61); and the set of supportable equilibrium price contains this interval. The definitions of \u03b60 and \u03b61 imply that \u03b60 \u2192 0 and \u03b61 \u2192\u221e as \u03c4 \u2192 0. Now c\u0304\u03c4 (\u03b61) = 0 for all \u03c4 and \u03b1S (\u03b61, \u03c4)\u2192\u221e as \u03c4 \u2192 0, therefore (4.16) implies that lim\u03c4\u21920 p\u03c4 (\u03b61) = 0. Similarly, v\u03c4 (\u03b60) = 1 for all \u03c4 and \u03b1B (\u03b60, \u03c4) \u2192 \u221e as \u03c4 \u2192 0, so that (4.17) implies that lim\u03c4\u21920 p\u03c4 (\u03b60) = 1. It proves that the set of supportable equilibrium price converges to (0, 1). It is not hard to see that the condition min\u03b6>0K (\u03b6, \u03c4) < 1 is also necessary for any nontrivial steady-state equilibrium to exist. We thus have the following theorem. Theorem 12 Under double auction, there exists a nontrivial steady-state equilibrium (ei- ther full-trade or non-full-trade) if and only if min \u03b6>0 K (\u03b6, \u03c4 ) < 1. 105 Chapter 4. Rate of Convergence towards Perfect Competition Proof. Having Theorem 11, it now suffices to claim the necessity of \u03baB\/\u03b1B(\u03b6, \u03c4) + \u03baS\/\u03b1S(\u03b6, \u03c4) < 1 for a nontrivial equilibrium to exist. Recall the notation for a general bargaining game introduced in Section 4.4. Individual rationality (4.6) implies \u03baB \u03b1B(\u03b6, \u03c4) \u2264 Z Z [q(v, c)v \u2212 t(v, c)]dNB(v) B dNS(c) S , \u03baS \u03b1S(\u03b6, \u03c4) \u2264 Z Z [t(v, c)\u2212 q(v, c)c]dNB(v) B dNS(c) S , and hence \u03baB \u03b1B(\u03b6, \u03c4) + \u03baS \u03b1S(\u03b6, \u03c4) \u2264 Z Z (v \u2212 c)dNB(v) B dNS(c) S < 1. Remark 7 Compare the necessary and sufficient conditions for equilibrium existence under double auction (given by Theorem 12) and under random-proposer bargaining (given by Theorem 3 on p.37). The condition under double auction is weaker than the one under the baseline (random-proposer bargaining) model, which is K (\u03b60, \u03c4) < 1. In this sense, the market is easier to open under double auction. Theorem 11 shows that the set of double-auction equilibria, even if we restrict attention to the full-trade ones, is very large. For more intuition, rewrite the first two marginal type equations of the double-auction full-trade equilibrium in a parallel way to the baseline model: \u03b1B (\u03b6, \u03c4)\u03b2DAB (v \u2212 c\u0304) = \u03baB, \u03b1S (\u03b6, \u03c4)\u03b2DAS (v \u2212 c\u0304) = \u03baS , where \u03b2DAB \u2261 1\u2212 \u03b2DAS , \u03b2DAS \u2261 p\u2212 c\u0304 v \u2212 c\u0304 . We may call \u03b2DAB and \u03b2 DA S the buyers\u2019 and sellers\u2019 relative bargaining powers under double- auction full-trade equilibrium. These equations are the same as the marginal type equations (2.15) and (2.16) (on p.24) that characterize a full-trade equilibrium in our baseline model, with the only difference that the exogenous bargaining power \u03b2S is now replaced with the 106 Chapter 4. Rate of Convergence towards Perfect Competition endogenous bargaining power \u03b2DAS . (The remaining inflow balance equation is the same in both models.) If \u03b2DAS = \u03b2S, the equilibria in both models have the same marginal types v and c\u0304, and once these are solved for, the price p is uniquely determined from the equation \u03b2DAS = \u03b2S, or equivalently p = c\u0304 + (v \u2212 c\u0304)\u03b2S . In other words, to any \u03b2S \u2208 (0, 1) there corresponds a double-auction full-trade equilibrium with \u03b2DAS = \u03b2S and the same marginal types v and c\u0304 as in the random-proposer full-trade equilibrium candidate. The above discussion has the following two implications. First, since \u03b2DAS can be ar- bitrary, in the double auction model, there is a great multiplicity of equilibria.43 Second, since we know that full-trade equilibria of the baseline model converge in terms of welfare level at the linear rate, it follows immediately that there is a sequence of double-auction equilibria that also converges at the linear rate to perfect competition. We state this finding as a corollary. Corollary 12 As \u03c4 \u2192 0, there are double-auction full-trade equilibria that converge, in terms of welfare level, at the linear rate in \u03c4 . Remark 8 The logic of Theorem 12 and Corollary 12 has nothing to do with the assumption of private information bargaining. They hold equally well if every pair of buyer and seller bids and asks knowing each other\u2019s type, because all they need to know is only the equilibrium price p. The above discussion explains why double auction full-trade equilibria can have non- Walrasian limit while it cannot be the case in our baseline model. But Figure 4.1 and the logic in the proof of Theorem 11 also make it clear that for the double auction full-trade equilibria to be non-convergent to the Walrasian outcome, we have to let the bargaining power of one side vanish (i.e. either v\u2212p\u2192 0 or p\u2212 c\u0304\u2192 0 as \u03c4 \u2192 0) and also let the market become extremely unbalanced (i.e. either \u03b6 \u2192 0 or \u03b6 \u2192 \u221e as \u03c4 \u2192 0). One might wonder if all equilibria (e.g. non-full-trade) will converge to the Walrasian outcome if we preclude 43The nature of indeterminacy here is analogous to that in the Nash demand game. As is well-known, the outcome of double auction is highly indeterminate even when information is complete. 107 Chapter 4. Rate of Convergence towards Perfect Competition ,c v 0 1 1 )(cpS p p c\u0302 *p c v\u0302v )(vpB *p Figure 4.2: A two-step equilibrium under double auction that class of equilibria (which is perhaps a natural restriction on equilibrium selection). It turns out that this is not so, as we show next. We construct a non-full-trade equilibrium of the following nature (see Figure 4.2). There are two seller cutoff types c\u0302 \u2208 (0, 1) and c\u0304 \u2208 (0, 1) with c\u0302 < c\u0304, and two buyer cutoff types v\u0302 \u2208 (0, 1) and v \u2208 (0, 1) with v\u0302 > v. The sellers with c \u2208 [0, c\u0302) enter and submit pS(c) = p, where p is some constant strictly below the Walrasian price p\u2217. The sellers with c \u2208 [c\u0302, c\u0304] enter and submit pS(c) = p\u0304, where p\u0304 > p\u2217. The sellers with c \u2208 (c\u0304, 1] do not enter. Similarly, the buyers with v \u2208 (v\u0302, 1] enter and submit p\u0304, the buyers with v \u2208 [v, v\u0302] enter and submit p, and the buyers with v \u2208 [0, v) do not enter. We call the equilibria of this kind two-step equilibria. The following theorem gives our non-convergence result for the two-step (non-full-trade) equilibria.44 Theorem 13 For any constant a \u2208 (0, 1), there exist r0 > 0, \u03c40 > 0 and W\u0304 < W \u2217 such that for all r \u2208 (0, r0) and \u03c4 \u2208 (0, \u03c40), there exists a two-step equilibrium in which the price 44As a by-product, we also prove the existence of non-full-trade equilibrium for small \u03c4 and r. 108 Chapter 4. Rate of Convergence towards Perfect Competition spread is larger than a, i.e. p\u0304\u2212 p > a, and the welfare level is lower than W\u0304 , i.e. W < W\u0304 . Proof. We derive a system of equations characterizing the set of two-step equilibria. But before doing so, it is convenient to introduce some notations. In a two-price equilibrium, the buyers with v > v\u0302 who submit the high bid price p\u0304, trade with any seller they meet. Buyers with v \u2208 [v, v\u0302], who submit the low bid price p, trade only with those sellers with c < c\u0302, who submit p; their probability of trading is equal to NS(c\u0302)\/S. Similarly sellers with c < c\u0302 trade with any buyer they meet, and sellers with c \u2208 [c\u0302, c\u0304] trade only with those buyers with v > v\u0302; their probability of trading is equal to 1\u2212NB(v\u0302)\/B. In our constructed equilibria NS(c\u0302)\/S and 1\u2212NB(v\u0302)\/B will converge to 0 as \u03c4 goes to 0, so it is convenient to divide them by \u03c4 : \u03bbB \u2261 1 \u03c4 \u2219 1\u2212 NB(v\u0302) B \u00b8 , \u03bbS \u2261 1 \u03c4 NS(c\u0302) S . Since type v buyers and type c\u0304 sellers are indifferent between entering or not, we have \u03b1B\u03c4\u03bbS(v \u2212 p) = \u03baB (4.18) \u03b1S\u03c4\u03bbB(p\u0304\u2212 c\u0304) = \u03baS . (4.19) Since type v\u0302 buyers are indifferent between biding p or p\u0304, and type c\u0302 sellers are indifferent between asking p or p\u0304, we have \u03c4\u03bbS[\u03c1B(v\u0302)\u2212 p] = \u03c4\u03bbS \u00a9 \u03c1B(v\u0302)\u2212 [(1\u2212 k)p+ kp\u0304] \u00aa + (1\u2212 \u03c4\u03bbS) [\u03c1B(v\u0302)\u2212 p\u0304] (4.20) \u03c4\u03bbB[p\u0304\u2212 \u03c1S(c\u0302)] = \u03c4\u03bbB \u00a9 [(1\u2212 k)p+ kp\u0304]\u2212 \u03c1S(c\u0302) \u00aa + (1\u2212 \u03c4\u03bbB) [p\u2212 \u03c1S(c\u0302)]. (4.21) Since Lemma 1 still hold here, we have W\u0302B = (v\u0302 \u2212 v) m\u0303 (\u03b6)\u03bbS \u03b6r + m\u0303 (\u03b6)\u03bbS (4.22) W\u0302S = (c\u0304\u2212 c\u0302) m\u0303 (\u03b6)\u03bbB r + m\u0303 (\u03b6)\u03bbB . (4.23) where we denoted m\u0303(\u03b6) \u2261 M\u0303(\u03b6, 1), W\u0302B \u2261WB (v\u0302) and W\u0302S \u2261WS (c\u0302). 109 Chapter 4. Rate of Convergence towards Perfect Competition To complete the description of the two-step equilibrium, the indifference conditions are supplemented with steady-state inflow balance conditions for each interval of types. Here, it suffices to require that the total inflows into the intervals [v, 1] and [0, c\u0304] are balanced with outflows, b [1\u2212 F (v)] = Sm\u0303(\u03b6) [\u03bbS + \u03bbB (1\u2212 \u03c4\u03bbS)] , (4.24) sG(c\u0304) = Sm\u0303(\u03b6) [\u03bbB + \u03bbS(1\u2212 \u03c4\u03bbB)] (4.25) and that the inflows into the intervals v \u2208 [v\u0302, 1] and [0, c\u0302] are also balanced with outflows, b[1\u2212 F (v\u0302)] = Sm\u0303(\u03b6)\u03bbB, (4.26) sG(c\u0302) = Sm\u0303(\u03b6)\u03bbS . (4.27) (Observe that the matching rate is Sm\u0303 (\u03b6) \/\u03c4 for both buyers and sellers, and that \u03c4 cancels out.) We also define the price spread, a0 \u2261 p\u0304\u2212 p. Then equations (4.18) through (4.27) form a 10-equation system with 12 endogenous variables {p, a0, \u03b6, v, c\u0304, v\u0302, c\u0302, \u03bbB, \u03bbS , S, W\u0302B, W\u0302S}. This system does characterize an equilib- rium. Indeed, one can easily see that buyers with v \u2208 (v\u0302, 1] strictly prefer to bid p\u0304, buyers with v \u2208 (v, v\u0302) strictly prefer to bid p, and buyers with v \u2208 [0, v) strictly prefer not to enter. Similar remark applies for sellers. Since we have two degrees of freedom, we can fix some \u03b6 > 0 and a0 \u2208 (a, 1) and then let equations (4.18) - (4.27) determine {p, v, c\u0304, v\u0302, c\u0302, \u03bbB, \u03bbS, S, W\u0302B, W\u0302S}. We claim that solution exists for small enough \u03c4 and r. To see this, one can check that when \u03c4 = r = 0, we have a (unique) solution with p implicitly determined by b[1\u2212F (p+ a0)] = sG(p), and all other variables given by c\u0304 = p, v = p\u0304 = p+ a0, \u03bbB = \u03baS m\u0303(\u03b6)a0 , \u03bbS = \u03baB\u03b6 m\u0303(\u03b6)a0 , S = sG(p)a0 \u03baB\u03b6 + \u03baS , 1\u2212 F (v\u0302) = [1\u2212 F (p\u0304)]\u03baS \u03baB\u03b6 + \u03baS , G(c\u0302) = G(p)\u03baB\u03b6 \u03baB\u03b6 + \u03baS , 110 Chapter 4. Rate of Convergence towards Perfect Competition W\u0302B = v\u0302 \u2212 p\u0304, W\u0302S = p\u2212 c\u0302. One can also check that the Jacobian evaluated at \u03c4 = r = 0 is not zero.45 Therefore the Implicit Function Theorem applies. Because p\u0304 \u2212 p \u2261 a0 > a, there exists a two-step equilibrium with p\u0304 \u2212 p > a when \u03c4 and r are small enough. Moreover, since v \u2192 p\u0304 and c\u0304 \u2192 p as (\u03c4 , r) \u2192 (0, 0), the spread v \u2212 c\u0304 is also bounded below by a. It follows that the associated welfare W is bounded away from the Walrasian welfare W \u2217. Unlike Theorem 11, the construction in the proof of Theorem 13 treats buyers and sellers symmetrically. In particular, \u03b6 could be fixed at any value. Then why does the double auction mechanism has non-Walrasian limit equilibria while the random-proposer mechanism does not?46 One can verify that the dynamic types do collapse to singletons even in the two-step non-convergent equilibria. Thus to fix the idea, let us simply suppose the discount rate r is 0 so that the ultimate trading probabilities are 1 and therefore the dynamic types are constant and equal to \u03c1S = c\u0304 \u2192 p and \u03c1B = v \u2192 p\u0304. Also suppose \u03c4 is very small. Then all buyers have dynamic types approximately p\u0304 and all sellers have dynamic types approximately p. Unlike under random-proposer bargaining, the dynamic types are no longer the acceptance levels. Effectively the bids\/asks also play this role. A seller submitting an ask lower than the dynamic types of all buyers does not guarantee herself a successful trade. To guarantee a trade, she has to ask lower than all buyers\u2019 bids. Consider a seller with c < c\u0302. This seller\u2019s equilibrium ask price is p. She realizes fully that the buyer\u2019s dynamic willingness-to-pay is always p\u0304 approximately, and would like to demand that much if acceptance is guaranteed, as it would be under the random-proposer bargaining. However, demanding that much under the double auction protocol runs into the risk of being countered with the buyer\u2019s bid of p, resulting in no trade. In our equilibrium with \u03c4 small, most of the active buyers bid p. Weighing these trade-offs carefully, the seller 45The Mathematica R\u00b0 notebook that contains the evaluation of the Jacobian is available at http:\/\/grad.econ.ubc.ca\/adamwong. 46Note that the non-convergent sequence constructed in the proof of Theorem 13 does not converge to the trivial no-trade equilibrium. Indeed, as revealed in the proof, when r is small and \u03c4 tends to 0, both the entry levels and steady-state stocks of buyers and sellers do not go to 0. 111 Chapter 4. Rate of Convergence towards Perfect Competition decides to submit p rather than p\u0304. Similar logic applies to the buyers. Now consider a seller with c = p + \u03b5 where \u03b5 > 0 is small. Although her type (or dynamic type) is significantly lower than buyers\u2019 dynamic types, which is p\u0304 approximately, she chooses not to enter even when the expected search costs incurred to obtain a meeting is very small as \u03c4 becomes very small. It is again because most of the active buyers bid p, making her prospect of trade meager. Similar logic applies to the buyers. Finally, to complete our logic, we explain why the fraction of active buyers bidding p is very high relative to the fraction bidding p\u0304. It is because, in our equilibrium, buyers bidding p can only trade with those sellers asking p, which makes their outflow rate tiny. On the other hand, buyers bidding p\u0304 trade in any meeting. Thus in steady state, the buyers who bid p accumulate and dominate the buyers\u2019 side of the market. Similar logic applies to the sellers. These arguments together explain why marginal traders do not enter to quest the significant size of the unexploited surplus v \u2212 c\u0304, keeping a positive gap between v and c\u0304. The rules of the double auction do not provide a tight connection between the dynamic types and actual acceptance levels as would be the case under the random-proposer bar- gaining. Here, a bid\/ask is both an offer and an acceptance level. On the other hand, under random-proposer bargaining, proposing strategies and responding strategies are separate decisions because traders are clear about who is proposer and who is responder. Ex-post, the bargaining power is given to one party, and thus well-defined. Therefore the responder is always held to her acceptance level, which creates strong incentive to enter. Ex-ante, both parties could have the full bargaining power. Therefore the incentives to enter are evenly distributed over both sides of the market, driving the marginal entering types close to each other and to the Walrasian price, and leading to rapid convergence. 4.6 Concluding remarks This chapter studies the equilibrium convergence properties of a decentralized dynamic matching and bargaining market, as search frictions vanish. The literature on dynamic matching and bargaining games has concentrated on whether the game-theoretic equilib- 112 Chapter 4. Rate of Convergence towards Perfect Competition rium outcome converges to the perfect competition. Although other papers (as reviewed in Chapter 1) have shown convergence in the contexts of similar models (for the sake of providing foundation of Walrasian equilibrium), this chapter has fundamental contributions on top of the literature. First, we not only prove the convergence, but also derive the rate of convergence, for our baseline model in which the decentralized bargaining is under two-sided private information and the random-proposer take-it-or-leave-it protocol. Second, we show that the market with such a simple bargaining protocol has the property that equilibrium welfare converges to the Walrasian (first best) welfare at the fastest possible rate among all bargaining protocols. Although we have not been able to characterize the most efficient bargaining mechanism for our decentralized market, our result can be interpreted to be an asymptotic efficiency result. Third, we show that the information structure of bargaining does not alter the convergence and its speed, but the convergence might fail if we assume another bargaining protocol, double auction. It suggests that information structure is not essential to the asymptotic efficiency of a dynamic matching and bargaining market, but the bargaining protocol might. Before closing this chapter, we make two remarks. Our first remark is a caveat on our non-convergence results for the double auction model. Under double auction, there is a great deal of multiplicity of equilibria, and some sequences of equilibria do converge to perfect competition. Also, the non-convergent equilibria we have constructed might be rather special. Our possible approach to address this is to impose additional assumptions on equilibrium selection (e.g. continuity of strategies and boundedness of the ratio of buyers to sellers) with the purpose of proving convergence. Secondly, as we point out in Remark 2, we can think of frictions as the \"cost of delay\", i.e. the discount rate r and the search costs \u03baB and \u03baS , as we did in the previous two chapters. Then Theorem 7 (for private information bargaining) and Theorem 8 (for full information bargaining) imply that market equilibria converge to perfect competition as the friction profile (r, \u03baB, \u03baS) tends to zero proportionally. But what if (r, \u03baB, \u03baS) tends to zero non-proportionally? 113 Chapter 4. Rate of Convergence towards Perfect Competition It might be natural that the search costs (\u03baB, \u03baS) would vanish slower than the discount rate r. Let us discuss in the language of a discrete time model, so that the matching among the market participants occurs once per period; and the discount rate and search costs are measured per period. Then as the period length is shortened (in other words matches are made more frequently), the discount rate per period would decrease at the same rate as the period length. But the search costs per period might decrease at a slower rate, reflecting that making matches more frequently is costly. It is easy to see from our theorems that, as friction profile (r, \u03baB, \u03baS) tends to 0, conver- gence (for the baseline model) holds as long as the search costs (\u03baB, \u03baS) vanish not faster than the discount rate r. As a matter of fact, convergence holds even when the vanish- ing of (\u03baB, \u03baS) is mildly faster. To be more concrete, let us say \u03baB = \u03baS = r\u03b8 for some \u03b8 > 0. Then Theorem 7 implies that, for private information bargaining, convergence holds if \u03b8 < 32 ; while Theorem 8 implies that, for full information bargaining, convergence holds if \u03b8 < 2. Finally, what if the vanishing of (\u03baB, \u03baS) is much faster? Is there a \"uniform convergence\" result? This is still an open question. 114 Chapter 5 Conclusion 5.1 Summary This dissertation studies a decentralized market with frictions (e.g. labor market, housing market). In the market, which we call a dynamic matching market, there are a large number of traders and the trading decisions and prices are determined by countless bilateral negotiations. More precisely, we model our market as a steady-state dynamic matching and bargaining game. The bargaining games are always bilateral, i.e. between a buyer and a seller; and each bargainer holds private information about his own willingness-to-pay or cost of providing the good. The main purpose of Chapter 2 is to prove the existence of equilibrium for our baseline model, and to understand the equilibrium patterns and properties, under different combi- nations of frictions. While the results in this chapter are interesting on their own right, they are also the foundation of the analyses of Chapter 3 and Chapter 4. Chapter 3 studies the role of private information bargaining in our baseline model. Our approach is to compare the equilibrium predictions of our baseline model (in which bar- gaining is under private information) with those of the full information bargaining version of the same model (i.e. Mortensen-Wright model).47 We find both qualitative similarities and differences between them. In particular, the two models have completely the same predictions if agents are perfectly patient. Besides, if agents are impatient, private infor- mation bargaining has an entry-deterring effect. In other words, typically less potential 47Part of this chapter\u2019s contribution is that we have derived new results (most importantly the general condition for equilibrium existence) for Mortensen-Wright model. It is done by applying the techniques we developed in Chapter 2. 115 Chapter 5. Conclusion traders enter in the private information model. We also show when the private information bargaining would generate a higher level of social welfare. Unlike most works in the literature on DMBG, Chapter 2 and Chapter 3 focus on \"out-of-the-limit\" results (i.e. the frictions are fixed rather than vanishing). They are particularly of interest when we are concerned with those markets with significant frictions (e.g. labor market, housing market), rather than concerned with providing a foundation for the Walrasian equilibrium. The concern of Chapter 4 is convergence. However it is different than the literature in that this chapter does not merely provide a foundation of Walrasian equilibrium based on the convergence of a DMBG, but also shows how fast the equilibrium outcome converges to the Walrasian first best outcome. In other words, this chapter studies the \"asymptotic effi- ciency\", in terms of the rate of convergence, of dynamic matching and bargaining markets. Our results suggest that whether there is private information in bargaining does not affect the asymptotic efficiency, but the choice of bargaining protocol could have a significant effect. 5.2 Discussions Here let us discuss which underlying assumptions we have made are crucial, and which are not. Some of the following discussions are based on conjectures. 5.2.1 Continuous time, continuous types First of all, our assumption that time is continuous does not matter. All of our results hold under the discrete time version of our model, with only minor modifications. Our assumption that types are continuous (together with strictly positive densities) should not matter in any significant way. However, if types are discrete, we have to allow mixed (or asymmetric) strategies in order to have nontrivial equilibrium. For example, the marginal entrants must be allowed to enter probabilistically (or asymmetrically). The proposing 116 Chapter 5. Conclusion strategies would probably have to be mixed (with nondegenerate support) as well.48 5.2.2 Symmetric pure strategies Although we implicitly assume that traders use symmetric pure strategies, this is merely for simplicity of exposition. At a cost in notation we could define trader-specific and mixed strategies and then prove that they must be (essentially) symmetric and pure. To see this intuitively, recall that the matching in the market is anonymous and random. Even if different traders follow distinct strategies, every buyer with the same type v would still face the same market environment. (This is strictly true because we assume a continuum of traders.) Therefore, for a given value v, every buyer will have the identical continuation payoff, implying essentially identical responding and entry strategies. Moreover, every buyer has identical best-response correspondence for proposing strategy. Lemma 2(b) still holds so that every selection from this correspondence is nondecreasing. Consequently, the best- response is single-valued apart from a measure zero set of values where jumps could occur. But because the set is of measure zero, the selection\/mixing over that set has no consequence for the maximization problems of the other traders. The same logic applies to sellers. 5.2.3 Random-proposer bargaining If the bargaining games proceed under full information, then assuming our random-proposer bargaining protocol is equivalent to assuming the generalized Nash bargaining solution (see Subsection 3.3.1 for more details). While the Nash bargaining solution is so standard in the context of full information bargaining, there is no standard modeling method for a bilateral bargaining with two-sided private information. The tractability of our model relies on the assumption of random-proposer bargain- ing even under private information. Under this bargaining protocol, the signaling issue is assumed away, because the proposers directly make take-it-or-leave price offers so that re- sponders do not need to know their proposers\u2019 types. Also, this bargaining protocol ensures 48Gale (1987) proves convergence in a model with discrete type setting. 117 Chapter 5. Conclusion the bargaining games are one-shot. We justify our assumption of random-proposer bargaining under private information as follows. First, it is a natural generalization of the Nash bargaining solution to a private information setting. Second, it is used in some of the recent labor search literature, e.g. Kennan (2007). In addition, it is actually much less restrictive than it looks. We can allow the proposers to propose a general mechanism (which is an informed principal mechanism design problem), and shows that in equilibrium the proposers would still make take-it-or- leave-it price offers, as in Atakan (2008).49 5.2.4 Choice of friction space Recall that our notion of frictions includes two things: time discount rate r and search costs (\u03baB, \u03baS).50 For our analyses to be interesting, we have to include both of them. If search costs are positive and there is no time discounting, as we have seen in Section 3.4, the private information in bargaining plays no role at all. Equilibrium existence and convergence can all be proved in a very simple manner. On the other hand, if search costs are zero, it is impossible to have a nontrivial steady- state equilibrium, given that entry is endogenous. The reason is that, if search costs are zero and there exists a nontrivial steady-state equilibrium, the marginal entrants (who are indifferent between entering or not) must have zero probability of trade. But then these marginal entrants would accumulate and eventually clog the matching process.51,52 Assuming an exogenous death rate (or exit rate) \u03b4 as in Satterthwaite and Shneyerov (2008) can restore the nontrivial steady-state equilibrium. What if we take (r, \u03b4) or (r, \u03b4, \u03baB, \u03baS) as our notion of frictions? This is an open question. 49Atakan (2008) does that by extending the results of Riley and Zeckhauser (1983) and Yilankaya (1999). His logic can be applied here as well. 50The parameter \u03c4 in Chapter 4 can be interpreted as a common multiplier of the discount rate and the search costs. 51The argument here is rather loose, but it can be made rigorous. 52 In Gale (1987), this problem is resolved by adding an entry fee. 118 Chapter 5. Conclusion 5.2.5 Constant-returns-to-scale matching function We assume that the matching function exhibits constant returns to scale. I conjecture that our main results would not be changed qualitatively if the matching function exhibits decreasing returns instead. What if the matching function exhibits increasing returns? Then things could be different. It is well-known that it is easy to have multiplicity of equilibria under increasing returns. Hence at least the uniqueness of full-trade equilibrium would not hold any more. Our convergence results should also have to be modified. I conjecture that as frictions vanish, some sequence of nontrivial steady-state equilibria still converges to perfect competition, but some other sequence converges to the trivial (i.e. no- entry) equilibrium, since now the trivial equilibrium becomes \"stable\". Besides, our proof of equilibrium existence does rely on constant returns. It is not clear how the necessary and sufficient condition for the existence of a nontrivial steady-state equilibrium would change if we release the assumption of constant returns. 5.2.6 Continuum of traders We assume the market has continua of buyers and sellers. It is a common assumption in the literature, and it is technically crucial to our analysis. If the number of traders in the market is finite (of course, it is endogenous, so we need to assume the number of traders born within any finite length of time being finite), then the number and distribution of traders in the market cannot stay at some steady-state value. They have to follow some stochastic process because the matching is random and the law of large number does not apply. The equilibrium analysis would become much less tractable, but I conjecture that the equilibrium (defined appropriately) of such a \"finite market\" converges to the equilibrium of the \"corresponding continuum market\", at least in some sense. 119 Chapter 5. Conclusion 5.3 Further research The previous section has pointed out some unanswered questions that are left for future research. This section suggests several more. First, our \u03b5-equilibrium technique (see Section 2.7) is seemingly applicable to prove existence of nontrivial equilibrium for other dynamic matching and bargaining games with heterogeneous types and free entry. For example, Satterthwaite and Shneyerov (2008) have been unable to prove existence of equilibrium, unless a distribution of new-born types is assumed to be concave. But, as they points out, \"concavity is not an economically plausible assumption to impose on type distributions\". Besides, the existence theorem in Satterthwaite and Shneyerov (2007) requires sufficiently small discount rate relative to the search costs (together with sufficiently small search costs); and it is only for full-trade equilibria. As another example, the existence theorem in Atakan (2008) requires what he calls Free First Draw for Low Cost Sellers, which is an artificial assumption. To sum up, all these papers have gaps in the equilibrium existence, and I expect our \u03b5-equilibrium technique is useful to fill those gaps. Another line of related research could be introducing competitive search (or directed search), like in Moen (1997). In particular, we could ask: would competitive search make the convergence faster? If the discount rate is zero, the competitive search version of our model (which is analyzed in Mortensen and Wright (2002)) is equivalent to the random-proposer model with a specific bargaining weight. If the discount rate is positive, different buyers and different sellers would choose to enter different submarkets. One might conjecture that even when the discount rate is positive, our rate of convergence results for the random-proposer model maintain in the competitive search model. Another further research could be on general bargaining mechanism. We have touched on that in Section 4.4, but there are still unanswered questions. In particular, we have not solved the socially optimal bargaining mechanism when both the discount rate and the search costs are positive.53 Moreover, what kind of mechanisms ensure convergence, and at 53Mortensen and Wright (2002) solve it for the no-discounting case. 120 Chapter 5. Conclusion what rate? This is also interesting to explore. 121 Bibliography Apostol, T. (1974): Mathematical analysis. Addison Wesley. Atakan, A. 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(1999): \u201cA Note on the Seller\u2019s Optimal Mechanism in Bilateral Trade with Two-Sided Incomplete Information,\u201d Journal of Economic Theory, 87(1), 267\u2014271. 125 Appendix A Additional Details for Existence of Nontrivial Steady-state Equilibrium This appendix provides the additional details for the proof of Theorem 3, which asserts that: In the private information model, at least one nontrivial steady-state equilibrium exists if and only if K (\u03b60) < 1. (For the adaptations needed for the full information model, see subsection 3.3.3.) We first claim that our definition of mapping T\u03b5 is legitimate. Definition 4 of T\u03b5 is legitimate. Fix \u03b1\u0304 > max {\u03baB, \u03baS} and \u03b5 \u2208 (0, \u03b5\u0304]. We need to claim that T\u03b5 is well-defined and its range, as stated in the definition, is contained in its domain D\u03b5. The restrictions we impose on D\u03b5 are important to claim that. Pick any E \u2261 (WB,WS, NB,NS) \u2208 D\u03b5. Firstly, by construction B > 0 and S > 0, so that \u03b1B and \u03b1S are well-defined. Second, NB (v) and \u03c1B (v) \u2261 v \u2212WB (v) are continuous in v; NS (c) and \u03c1S (c) \u2261 c + WS (c) are continuous in c. Third, \u03c1B and \u03c1S are strictly increasing (since E \u2208 D\u03b5 and r > 0). It follows that the objective functions in (2.31) and (2.32) are continuous in p. Therefore the argmax correspondences in (2.31) and (2.32) are nonempty- valued and compact-valued. Thus pB and pS are well-defined. Now it is obvious that all other constructed objects, in particular W \u2217B,W \u2217 S , N \u2217 B, N \u2217 S, are well-defined. It remains to verify that (W \u2217B,W \u2217 S , N \u2217 B, N \u2217 S) \u2208 D\u03b5. First, by our construction W \u2217B, W \u2217S , N\u2217B, N \u2217 S are absolutely continuous; and whenever differentiable, W \u22170B (v) = \u03c7B (v) \u03b1B r + \u03b1B qB (v) \u00a3 1\u2212W 0B (v) \u00a4 + \u03b1B r + \u03b1B W 0B(v), 126 Appendix A. Additional Details for Existence of Nontrivial Steady-state Equilibrium W \u22170S (c) = \u2212\u03c7S (c) \u03b1S r + \u03b1S qS (c) \u00a3 1 +W 0S (c) \u00a4 + \u03b1S r + \u03b1S W 0S(c), N\u22170B (v) \u2261 \u03c7\u2217B (v) bf (v) max {\u03b1BqB (v) , \u03baB} , N \u22170 S (c) \u2261 \u03c7\u2217S (c) sg (c) max {\u03b1SqS (c) , \u03baS} . From these derivatives we see (W \u2217B,W \u2217 S , N \u2217 B, N \u2217 S) satisfies the conditions (i) and (ii) in Definition 4. Second, it is easy to verify that (W \u2217B,W \u2217 S , N \u2217 B, N \u2217 S) also satisfies the condition (iii) in Definition 4. Therefore (W \u2217B,W \u2217 S , N \u2217 B, N \u2217 S) \u2208 D\u03b5. We conclude that Definition 4 of T\u03b5 is legitimate. Next, we prove D\u03b5 is nonempty, convex and compact (i.e. Lemma 7). Proof of Lemma 7. Obviously, D\u03b5 is convex and closed. To see D\u03b5 is nonempty, let WB (v) =WS (c) = 0 for all v, c, and NB (v) = bf\u0304v\/\u03baB, NS = sg\u0304c\/\u03baS. Since \u03b5 \u2264 \u03b5\u0304, we have NB(1) \u2265 \u03b5bf\/\u03b1\u0304 and NS(1) \u2265 \u03b5sg\/\u03b1\u0304. All other restrictions of D\u03b5 are obviously satisfied, thus D\u03b5 is nonempty. To see the compactness, notice that D\u03b5 is a uniformly bounded family of functions on a compact set [0, 1], and is also an equicontinuous family of functions because the Lipschitz constant for every function in D\u03b5 is at most max \u00a9 1, bf\u0304\/\u03baB, sg\u0304\/\u03baS \u00aa . By Ascoli-Arzela Theorem (see e.g. Royden (1988) p.169), D\u03b5 is compact. It remains to prove the continuity of T\u03b5 (i.e. Lemma 8). It requires the following lemma. Lemma 16 Let {\u03a6n} be a sequence of continuous c.d.f.\u2019s with supports contained in [0, 1] and {\u03c8n} a sequence of real functions on [0, 1]. Suppose (i) {\u03a6n} is uniformly convergent to some c.d.f. \u03a6; (ii) {\u03c8n} is convergent to some real function \u03c8 almost everywhere on [0, 1]; and (iii) the absolute values and total variations of {\u03c8n} and \u03c8 are bounded by some constant C. Then \u03c8n is Riemann integrable with respect to \u03a6n for each n; and \u03c8 is Riemann inte- grable with respect to \u03a6. Moreover, lim n\u2192\u221e Z 1 0 \u03c8n (x) d\u03a6n (x) = Z 1 0 \u03c8 (x) d\u03a6 (x) . Proof. For each n, since \u03c8n is of bounded variation and \u03a6n is continuous, hence \u03c8n is Riemann integrable with respect to \u03a6n (see e.g. Apostol (1974) p.159 Theorem 7.27 and 127 Appendix A. Additional Details for Existence of Nontrivial Steady-state Equilibrium p.144 Theorem 7.6). Similarly, \u03c8 is of bounded variation and \u03a6 (as the uniform limit of a sequence of continuous functions) is continuous, hence \u03c8 is Riemann integrable with respect to \u03a6. Moreover,\u00af\u0304\u0304\u0304Z 1 0 \u03c8nd\u03a6n \u2212 Z 1 0 \u03c8d\u03a6 \u00af\u0304\u0304\u0304 \u2264 \u00af\u0304\u0304\u0304Z 1 0 \u03c8nd\u03a6n \u2212 Z 1 0 \u03c8nd\u03a6 \u00af\u0304\u0304\u0304 + \u00af\u0304\u0304\u0304Z 1 0 \u03c8nd\u03a6\u2212 Z 1 0 \u03c8d\u03a6 \u00af\u0304\u0304\u0304 . The first part of the right-hand side can be written, through integration by parts for Riemann-Stieltjes integrals (see e.g. Apostol (1974) p.144 Theorem 7.6), as \u00af\u0304R [\u03a6\u2212 \u03a6n] d\u03c8n \u00af\u0304 and hence is bounded by C \u00b7 supx\u2208[0,1] |\u03a6 (x)\u2212 \u03a6n (x)|, which converges to 0 as n\u2192\u221e, due to the uniform convergence of {\u03a6n}. The second part also converges to 0 as n\u2192\u221e, due to Lebesgue\u2019s dominated convergence theorem (see e.g. Apostol (1974) p.270 Theorem 10.27). Proof of Lemma 8. Fix (r, \u03b1\u0304) \u00c0 (0,max {\u03baB, \u03baS}) and \u03b5 \u2208 (0, \u03b5\u0304]. We write the constructed objects in Definition 4 as functions of E \u2261 (WB,WS , NB, NS) explicitly, e.g. B (E), \u03b1B (E), pB (v,E),WB (v,E), NB (v,E) etc. We need to show that: for any sequence {En} on D\u03b5, En \u2192 E implies T\u03b5 (En)\u2192 T\u03b5 (E). (Recall that we use the uniform metric on D\u03b5.) Step 1. Obviously B (E), S (E), \u03b1B (E) and \u03b1S (E) are continuous in E. Step 2. It is easy to see that: I [p \u2265 c+WS (c)] (where I [\u00b7] is 1 if the condition inside the bracket holds, and 0 otherwise), as a function of (c, p, E), is continuous on {(c, p,E) : p 6= c+WS (c)}. Similarly, I [p \u2264 v \u2212WB (v)], as a function of (v, p, E), is con- tinuous on {(v, p, E) : p 6= v \u2212WB (v)}. Step 3. \u03c0\u0302B (v, p, E) \u2261 [v \u2212 p\u2212WB (v)] R 1 0 I [p \u2265 c+WS (c)] dNS(c) S(E) is continuous in (v, p, E). To see this, let (vn, pn, En)\u2192 (v, p, E). Then firstly vn\u2212pn\u2212WBn (vn)\u2192 v\u2212p\u2212WB (v) (note that the convergence WBn \u2192WB is uniform); secondly from step 2, I [pn \u2265 c+WSn (c)]\u2192 I [p \u2265 c+WS (c)] except at the c such that p = c+WS (c) (note that there is at most one such c since r > 0 and E \u2208 D\u03b5 imply c +WS (c) is strictly increasing). Applying Lemma 16, we obtain \u03c0\u0302B (vn, pn, En) \u2192 \u03c0\u0302B (v, p, E). Thus \u03c0\u0302B (v, p,E) is continuous. Similarly, \u03c0\u0302S (c, p, E) \u2261 [p\u2212 c\u2212WS (c)] R 1 0 I [p \u2264 v \u2212WB (v)] dNB(v) B(E) is continuous in (c, p, E). Step 4. From step 3 and Berge\u2019s maximum theorem, \u03c0B (v,E) (which is equal to 128 Appendix A. Additional Details for Existence of Nontrivial Steady-state Equilibrium maxp\u2208[0,1] \u03c0\u0302B (v, p, E)) is continuous in (v,E), and PB (v,E) \u2261 argmaxp\u2208[0,1] \u03c0\u0302B (v, p, E) is nonempty-valued, compact-valued, and upper-hemicontinuous in (v,E). Analogous results can be proved for \u03c0S (c,E) and PS (c, E) \u2261 argmaxp\u2208[0,1] \u03c0\u0302S (c, p, E). Step 5. pB (v,E) is continuous on {(v,E) : PB (v,E) is a singleton}. To see this, let (vn, En) \u2192 (v,E) and let pB (vn, En) \u2192 p. Then from step 4, p \u2208 PB (v,E). Thus, if p 6= pB (v,E) then PB (v,E) is not a singleton. Moreover, pB (v,E) is continuous on {(v,E) : v \u2212WB (v) > WS (0)}. Analogous result can be proved for pS . Step 6. Let E \u2208 D\u03b5 and En \u2192 E. Then pB (v,En) \u2192 pB (v,E) a.e. v \u2208 [0, 1]. To see this, firstly consider those v with v \u2212WB (v) < WS (0). Then it is easy to see that \u03c0B (v,En) = 0 = \u03c0B (v,E) and PB (v,En) = [0,WSn (0)] = PB (v,E). Thus pB (v,En) \u2192 WS (0) = pB (v,E). Now consider those v with v \u2212 WB (v) > WS (0). By a standard revealed preference argument, any selection of PB (\u00b7, E) |{v:v\u2212WB(v)>WS(0)} is nondecreasing. It follows that, for all but countably many v\u2019s in {v : v \u2212WB (v) > WS (0)}, PB (v,E) is a singleton. Then pB (v,En) \u2192 pB (v,E) a.e. from step 5. Analogous result can be proved for pS. Step 7. Let E \u2208 D\u03b5 and En \u2192 E. Then, from steps 1, 2, 4, 6, and Lemma 16, W \u2217B (v,En)\u2192W \u2217B (v,E) \u2200v and W \u2217S (c, En)\u2192W \u2217S (c, E) \u2200c. Step 8. It is easy to see that \u03c7B (v,E) is continuous on {(v,E) : \u03b1B (E)\u03a0B (v,E) 6= \u03baB}, where \u03a0B (v,E) is the expression inside the square bracket in (2.33). Furthermore, given E, there is at most one v such that \u03b1B (E)\u03a0B (v,E) = \u03baB. To see this, notice that \u03b1B (E)\u03a0B (v,E) is nondecreasing in v, and if \u03b1B (E)\u03a0B (v,E) = \u03baB then \u03b1B (E) qB (v,E) \u2265 \u03baB and hence \u2202\u2202v [\u03b1B (E)\u03a0B (v,E)] = \u03b1B (E) qB (v,E) [1\u2212W 0B (v)] \u2265 \u03baBr r+\u03b1B(E) > 0. As a result, given any E \u2208 D\u03b5, if En \u2192 E then \u03c7B (v,En) \u2192 \u03c7B (v,E) a.e. v \u2208 [0, 1]. Obviously \u03c7\u2217B has the same property, and analogous results can be proved for \u03c7S and \u03c7 \u2217 S . Step 9. Let E \u2208 D\u03b5 and En \u2192 E. Then, from steps 1, 2, 6, and Lemma 16, qB (v,En)\u2192 qB (v,E) a.e. v \u2208 [0, 1], and qS (c, En) \u2192 qS (c,E) a.e. c \u2208 [0, 1]. This together with step 8 implies that N\u2217B (v,En) \u2192 N\u2217B (v,E) \u2200v and N\u2217S (c, En) \u2192 N\u2217S (c, E) \u2200c, again due to Lemma 16. 129 Appendix A. Additional Details for Existence of Nontrivial Steady-state Equilibrium Step 10. Let E \u2208 D\u03b5 and En \u2192 E. From steps 7 and 9, W \u2217B (\u00b7, En), W \u2217S (\u00b7, En), N\u2217B (\u00b7, En) and N\u2217S (\u00b7, En) converge pointwise toW \u2217B (\u00b7, E),W \u2217S (\u00b7, E), N\u2217B (\u00b7, E) and N\u2217S (\u00b7, E) respectively. Moreover, the pointwise convergence is equivalent to uniform convergence, because each of those function sequences form an equicontinuous family of functions with a compact domain [0, 1] (see e.g. Royden (1988) p.168). We therefore conclude that T\u03b5 (En)\u2192 T\u03b5 (E). 130 Appendix B Calculations for Section 3.6 The goal of this Appendix is to derive the slopes \u03b6 0f (0) and W 0f (0) in Section 3.6. As a by-product, we also show that v0f (0) < 0 and c\u0304 0 f (0) > 0. First of all, divide the buyers\u2019 marginal type equation (3.36) through by \u03b1B \u00a1 \u03b6f \u00a2 , apply integration by parts to the integral in left-hand side, differentiate through at r = 0, and rearrange: d dr \" \u03b2B Z c\u0304f 0 \" vf \u2212 rc+ \u03b1S \u00a1 \u03b6f \u00a2 \u03b2S c\u0304f r + \u03b1S \u00a1 \u03b6f \u00a2 \u03b2S # dG (c) G (c\u0304f ) # r=0 = d dr \" \u03baB \u03b1B \u00a1 \u03b6f \u00a2# r=0 \u03b2B \u00b7 d dr \" vf \u2212 c\u0304f + r r + \u03b1S \u00a1 \u03b6f \u00a2 \u03b2S Z c\u0304f 0 G (c) G (c\u0304f ) dc # r=0 = \u03baB\u03b7B(\u03b60)\u03b6 0 f (0) \u03b2B \u2219 v0f (0)\u2212 c\u03040f (0) + W eaS0 \u03b1S (\u03b60)\u03b2SG (c\u03040) \u00b8 = \u03baB\u03b7B(\u03b60)\u03b6 0 f (0) (B.1) where \u03b7B(\u03b60) \u2261 d d\u03b6 \u2219 1 \u03b1B (\u03b6) \u00b8 \u03b6=\u03b60 = \u2212 \u03b1 0 B (\u03b60) [\u03b1B (\u03b60)] 2 > 0. Work with the sellers\u2019 marginal type equation (3.37) in the same fashion, we have \u03b2S \u2219 v0f (0)\u2212 c\u03040f (0) + W eaB0 \u03b1B (\u03b60)\u03b2B [1\u2212 F (v0)] \u00b8 = \u2212\u03baS\u03b7S(\u03b60)\u03b6 0f (0) (B.2) where \u03b7S(\u03b60) \u2261 \u2212 d d\u03b6 \u2219 1 \u03b1S (\u03b6) \u00b8 \u03b6=\u03b60 = \u03b10S (\u03b60) \u03b1S (\u03b60) 2 > 0. Equations (B.1) and (B.2) can be solved for c\u03040f (0) \u2212 v0f (0) and \u03b6 0f (0). After some rewriting from the characterizing equations of (\u03b60, v0, c\u03040), we get \u03b6 0f (0) = K (\u03b60) sG (c\u03040) \u2219 \u03baS\u03b7S(\u03b60) \u03b2S + \u03baB\u03b7B(\u03b60) \u03b2B \u00b8\u22121\u00b5sW eaS0 \u03baS \u2212 bW ea B0 \u03baB \u00b6 , (B.3) 131 Appendix B. Calculations for Section 3.6 c\u03040f (0)\u2212 v0f (0) = K (\u03b60) sG (c\u03040) \u2219 \u03baS\u03b7S(\u03b60) \u03b2S + \u03baB\u03b7B(\u03b60) \u03b2B \u00b8\u22121 \u00b7 \u2219 \u03baS\u03b7S(\u03b60) \u03b2S sW eaS0 \u03baS + \u03baB\u03b7B(\u03b60) \u03b2B bW eaB0 \u03baB \u00b8 . Notice that \u03baB\u03b7B(\u03b60) \u03b2B \u2219 \u03baS\u03b7S(\u03b60) \u03b2S + \u03baB\u03b7B(\u03b60) \u03b2B \u00b8\u22121 = 1\u2212 \u03b60m 0 (\u03b60) m (\u03b60) \u2261 \u03c3S(\u03b60) > 0 and \u03baS\u03b7S(\u03b60) \u03b2S \u2219 \u03baS\u03b7S(\u03b60) \u03b2S + \u03baB\u03b7B(\u03b60) \u03b2B \u00b8\u22121 = \u03b60m0 (\u03b60) m (\u03b60) \u2261 \u03c3B(\u03b60) > 0. Then c\u03040f (0)\u2212 v0f (0) can be further simplified: c\u03040f (0)\u2212 v0f (0) = K (\u03b60) sG (c\u03040) \u2219 \u03c3B(\u03b60) sW eaS0 \u03baS + \u03c3S(\u03b60) bW eaB0 \u03baB \u00b8 > 0. (B.4) Now (B.3) gives the result for \u03b6 0f (0), while (B.4) and the flow balance equation (3.38) imply that v0f (0) < 0 and c\u0304 0 f (0) > 0. Next, by direct calculation, the private information slope of welfare W 0p (0) is what we state in (3.42). The full information slope of welfare W 0f (0) is W 0f (0) = \u2212 1 \u03b2B bW eaB0 \u03b1B (\u03b60) \u2212 1 \u03b2S sW eaS0 \u03b1S (\u03b60) + sG (c\u03040) [c\u03040f (0)\u2212 v0f (0)]. (B.5) Sum (B.1) and (B.2), and insert the resulting c\u03040f (0)\u2212 v0f (0) into (B.5), and cancel terms, we obtain: W 0f (0) = \u2212 bW eaB0 \u03b1B (\u03b60) \u2212 sW ea S0 \u03b1S (\u03b60) \u2212 sG (c\u03040) [\u03baB\u03b7B (\u03b60)\u2212 \u03baS\u03b7S (\u03b60)] \u03b6 0f (0) = W 0p (0)\u2212 sG (c\u03040)K 0 (\u03b60) \u03b6 0f (0) which gives (3.43). To obtain (3.44), simply substitute (B.4) into (B.5) and rewrite. 132","type":"literal","lang":"en"}],"http:\/\/www.europeana.eu\/schemas\/edm\/hasType":[{"value":"Thesis\/Dissertation","type":"literal","lang":"en"}],"http:\/\/vivoweb.org\/ontology\/core#dateIssued":[{"value":"2009-11","type":"literal","lang":"en"}],"http:\/\/www.europeana.eu\/schemas\/edm\/isShownAt":[{"value":"10.14288\/1.0067729","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/language":[{"value":"eng","type":"literal","lang":"en"}],"https:\/\/open.library.ubc.ca\/terms#degreeDiscipline":[{"value":"Economics","type":"literal","lang":"en"}],"http:\/\/www.europeana.eu\/schemas\/edm\/provider":[{"value":"Vancouver : University of British Columbia Library","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/publisher":[{"value":"University of British Columbia","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/rights":[{"value":"Attribution-NonCommercial-NoDerivatives 4.0 International","type":"literal","lang":"en"}],"https:\/\/open.library.ubc.ca\/terms#rightsURI":[{"value":"http:\/\/creativecommons.org\/licenses\/by-nc-nd\/4.0\/","type":"literal","lang":"en"}],"https:\/\/open.library.ubc.ca\/terms#scholarLevel":[{"value":"Graduate","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/title":[{"value":"Essays on dynamic matching markets","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/type":[{"value":"Text","type":"literal","lang":"en"}],"https:\/\/open.library.ubc.ca\/terms#identifierURI":[{"value":"http:\/\/hdl.handle.net\/2429\/13397","type":"literal","lang":"en"}]}}