An Empirical Equilibrium Model of a Decentralized Asset Market

Transcription

1 An Empirica Equiibrium Mode of a Decentraized Asset Market Aessandro Gavazza June, 212 Abstract I estimate a search-and-bargaining mode of a decentraized market to quantify the effects of trading frictions on asset aocations and asset prices, and to quantify the effects of intermediaries that faciitate trade. Using business-aircraft data, I find that, reative to the Warasian benchmark, 2.3 percent of the assets are misaocated, and prices are 6.61-percent ower. Deaers pay an important roe in reducing frictions: In a market with no deaers, 32.3 percent of the assets woud be misaocated, and prices woud decrease by 5.25 percent. I am gratefu to Ricardo Lagos, Jonathan Levin, Gianuca Vioante, Pierre-Oivier Wei and many seminar audiences for usefu comments and discussions. Leonard N. Stern Schoo of Business, New York University. 44 West 4th Street, New York, NY 112. Teephone: (212) Fax: (212) Emai: agavazza@stern.nyu.edu. 1

2 1 Introduction How arge are trading frictions? How do they affect the aocations and prices of assets? What is the roe of intermediaries in reducing trading frictions? This paper estimates a structura mode of a decentraized market to provide quantitative answers to these questions. Many assets trade in decentraized markets. Cassic exampes are financia assets such as bonds and derivatives, consumer durabe goods such as cars and houses, and firms capita assets such as pants and equipment. The fundamenta characteristics of decentraized markets are that agents must search for trading partners and that, once a buyer and a seer meet, they must bargain to determine a price. Moreover, in response to trading frictions, amost a decentraized markets have intermediaries. Indeed, starting with Demsetz (1968), trading frictions have been used to expain the existence and behavior of intermediaries. The key roe of intermediaries in such markets is to reduce frictions, thereby faciitating trade. In this paper, I ay out a mode of trading in decentraized markets with two-sided search and biatera bargaining. The mode formaizes the effects of trading frictions on asset aocations and prices, and the effects of intermediaries in aeviating these frictions. I then quantify the roe of frictions and of deaers, estimating this mode using data on business jet aircraft. The theoretica framework combines eements from Rubinstein and Woinsky (1987) and Duffie, Gâreanu and Pedersen (25), extending them to capture key features of rea asset markets, such as the depreciation of assets. A fow of agents enters the market in every period, seeking to acquire an aircraft. They contact seers at a rate that depends on the mass of aircraft for sae and on traders search abiity, and they contact deaers at another rate that refects deaers inventories and search abiity. Deaers hod inventories because they do not meet buyers and seers simutaneousy (Grossman and Mier, 1988), and they extract surpus by shortening the time that buyers and seers have to wait in order to trade (Rubinstein and Woinsky, 1987). When agents meet or meet deaers, they bargain over the terms of trade. Gains from trade arise from heterogeneous vauations of hoding the aircraft. For exampe, an aircraft owner, usuay a corporation, wishes to se because its vauation for the aircraft has dropped, perhaps because the firm s profitabiity has decined. Equiibrium aocations and prices depend in an intuitive way on agents and deaers search abiities, bargaining powers and vauations. I estimate the mode by using data on the secondary market of business jet aircraft a typica decentraized market. The data are we-suited to studying the effects of search 2

3 frictions and the roe of intermediaries. In particuar, they report the number of aircraft for sae and number of aircraft transactions in each month, and their ratio is informative on the magnitude of trading deays. Simiary, the data report deaers inventories and deaers transactions, and their ratio is informative on the roe of deaers in reducing deays. In addition, the data report two series of prices: retai prices between fina users of the aircraft; and whoesae prices between aircraft owners (as seers) and deaers (as buyers). Their differences are usefu in understanding how much deaers are abe to command by suppying immediacy of trade (and, thus, seers are wiing to forego). The estimation reveas that trading deays are non-trivia: On average, aircraft stay on the market approximatey five months before a seer is abe to finaize a sae. The quantitative importance of these deays depends on how frequenty agents seek to trade, determined by a drop in their vauations; this happens, on average, every four and a haf years. Moreover, the estimation impies that the deaers enjoy strong bargaining powers, capturing amost the entire surpus of transactions. I use the estimated parameters to simuate two counterfactua scenarios. In the first one, I quantify the roe of trading frictions on asset aocations and prices by computing a Warasian market equiibrium. The estimates impy that trading frictions generate moderate inefficiencies. Compared to the Warasian benchmark, aircraft prices are 6.61-percent ower, and 2.3 percent of a business aircraft are misaocated: 11.6 percent are on the market for sae, whereas, for 8.7 percent of a aircraft, the trading frictions are arger than seers expected gross gains from trade; thus, their ow-vauation owners prefer to keep them rather than put them on the market for sae. Two forces affect aocations and prices in the estimated mode reative to this Warasian benchmark: search costs and trading deays. The parameter estimates impy that trading deays quantitativey account for amost a differences, and the effect of search costs on aocations and prices is sma. In the second counterfactua, I quantify the effect of deaers on the decentraized market equiibrium, computing a decentraized market with no deaers. The estimates impy that deaers pay an important roe in reducing frictions: In a market without them, 32.3 percent of the assets woud be misaocated, and prices woud decrease by 5.25 percent. Thus, the effect on aocations is arger than the effect on prices. The reason is that seers outside options improve reative to buyers, thus partiay offsetting the effects of higher search costs and sower trade on asset prices. Therefore, an interesting concusion of this counterfactua anaysis is that it may be difficut to infer excusivey from the magnitude of changes in asset prices the magnitude of changes in trading frictions (or, more generay, changes in trading 3

4 mechanisms and institutions) on market inefficiencies. This paper makes two main contributions. First, it provides a framework suited to empiricay anayzing decentraized asset markets. Search modes have proved usefu in understanding key features of abor markets, and, more recenty, researchers have started to appy search modes to financia markets. To my knowedge, this paper is the first to estimate a biatera search mode that investigates the microstructure of the market of a capita asset/durabe good, quantifying the effects of trading deays and of intermediaries that faciitate exchange. Second, the empirica findings suggest that, even within a wedefined asset cass such as business aircraft, trading frictions are a non-trivia impediment to the efficient aocation of assets and have significant effects on asset prices. Thus, the paper innovates on recent works that study the process of asset reaocations (Ramey and Shapiro, 1998, 21; Maksimovic and Phiips, 21; Schingemann et a., 22; Eisfedt and Rampini, 26; Gavazza, 211a and 211b. For an empirica mode of business transfers, see, aso, Homes and Schmitz, 1995) by quantifying its inefficiencies. This is a necessary step to understanding how asset markets work and, therefore, to the design of any poicy that affects them. The paper proceeds as foows. Section 2 reviews the iterature. Section 3 presents some institutiona detais on the business-aircraft market. Section 4 introduces the data. Section 5 presents the theoretica mode, and Section 6 estimates it. Section 7 performs counterfactua anaysis. Section 8 concudes. The appendices present the anaytica soution of the mode with the assumption of no depreciation and the estimates of the mode with different assumptions on the matching function, respectivey. 2 Reated Literature This paper contributes to the important iterature that anayzes decentraized markets. The theoretica iterature is vast. The most cosey reated papers examine biatera search markets, in which both buyers and seers search for a trading counterpart, and prices are determined through biatera bargaining (Rubinstein and Woinsky, 1985 and 1987; Gae, 1987; Mortensen and Wright, 22; Duffie, Gâreanu and Pedersen, 25 and 27; Miao, 26). The main focus of these theoretica papers is to investigate whether the equiibrium converges to the competitive outcome as frictions vanish. To my knowedge, this paper is 4

5 the first to estimate a biatera search mode of a rea asset market to quantify the effects of trading frictions i.e., the focus of the theoretica iterature. The paper further contributes to the iterature on intermediaries. Severa papers investigate the roe of brokers/deaers in financia markets and their inventory-management poicies; for a survey, see O Hara (1995). Spuber (1999) presents a thorough anaysis of intermediaries between customers and suppiers. Wei (27) presents a search-and-bargaining mode to understand how intermediaries provide iquidity by accumuating inventories when seing pressure is arge, and then dispose of those inventories after that seing pressure has subsided. Recent empirica anaysis of non-financia intermediaries incude Ha and Rust (2), who focus on the inventory investment of a singe stee whoesaer, and Gavazza (211a), who focuses on the roe of essors in reaocating commercia aircraft. To my knowedge, this paper presents the first structura empirica anaysis of the roe of intermediaries in a search-and-bargaining framework. This paper is aso reated to the empirica iterature on the structura estimation of search modes. Most appications focus on abor markets. Eckstein and Wopin (199) pioneered this iterature by estimating the mode of Abrecht and Axe (1984). Eckstein and van der Berg (26) provide a thorough survey of this iterature. One key difference between the current paper and previous research is that this paper seeks to understand how search frictions affect the eve of asset prices and of asset aocations, whie most papers that structuray estimate abor-market search modes focus on how search frictions affect wage dispersion across workers (notabe exceptions are Gautier and Teuings, 26 and 21). Search modes have aso been appied to housing markets, and Carrio (212) is the cosest empirica paper. Finay, this paper is reated to a few that investigate aircraft transactions. Using data on commercia-aircraft transactions, Puvino (1998, 1999) finds that airines under financia pressure se aircraft at a 14-percent discount, and that distressed airines experience higher rates of asset saes than non-distressed airines do. Using data on business jets simiar to the data that I use in this paper, Giigan (24) finds empirica evidence consistent with the idea that easing ameiorates the consequences of information asymmetries about the quaity of used aircraft. Gavazza (211b) empiricay investigates whether trading frictions vary with the size of the asset market in commercia-aircraft markets. However, none of these papers quantifies the importance of trading frictions by estimating a structura mode. 5

6 3 Business-Aircraft Markets For severa reasons, the business-aircraft market is an interesting context in which to investigate the effects of search frictions and the roe of intermediaries. First, used business aircraft trade in decentraized markets, organized around privateynegotiated transactions. Amost a buyers (and seers) are weathy individuas or corporations, that use the aircraft to fy their executives. To initiate a transaction, a prospective seer must contact mutipe potentia buyers or se its aircraft to a deaer. For buyers, comparing two simiar aircraft is costy since aircraft saes invove the materia inspection of the aircraft, which coud be in two different ocations. Thus, aircraft markets share many features with other over-the-counter markets for financia assets (mortgage-backed securities, corporate bonds, bank oans, derivatives, etc.) and for rea assets (rea estate), in which trading invoves materia and opportunity costs (Duffie, Gâreanu and Pedersen, 25 and 27). Moreover, compared to financia markets and other equipment markets, businessaircraft markets are thin : Sighty more than 17, business jet aircraft were operated wordwide in December 28. In thin markets, the search costs to find high-vaue buyers are usuay arge (Ramey and Shapiro, 21; Gavazza, 211b). Second, intermediaries pay an important roe in mediating transactions. Most intermediaries operate as brokers who match buyers and seers. Some arger intermediaries operate aso as deaers, acquiring aircraft for inventories. Finay, business aircraft are registered goods with a major ife events (date of first fight, maintenance, scrappage, etc.) recorded, so detaied data are avaiabe. In the next section, I describe them. 4 Data Patterns in the data suggest that trading deays are an important feature of aircraft markets. Moreover, the avaiabe data dictate some of the modeing choices of this paper. Hence, I describe the data before presenting the mode. This description aso introduces some of the identification issues that I discuss in more detai in Section 6. 6

7 4.1 Data Sources I combine two distinct datasets. The first is an extensive database that tracks the history of business-aircraft transactions. The second reports the average vaues of severa aircraft modes, simiar to Bue Book prices. I now describe each dataset in more detai. Aircraft Transactions This database is compied by AMSTAT, a producer of aviationmarket information systems. 1 It provides summary reports that track business-aircraft market transactions. For each month from January 199 to December 28 and for each mode (e.g., Cessna Citation V or VI), the dataset reports information on the active feet (e.g., the number of active aircraft; number of new deiveries); information on aircraft for sae (e.g., the number of aircraft for sae, by owners and by deaers; the average vintage); and information on competed transactions (e.g., the tota number of transactions; the number of retai-to-retai, retai-to-deaer, deaer-to-deaer and deaer-to-retai transactions). The dataset aso reports detaied characteristics of each aircraft mode (e.g., average number of seats, maximum range, fue consumption). I restrict the anaysis to business jets, thus excuding turbo-propeers. Aircraft Prices I obtained business-aircraft prices from the Aircraft Buebook Historica Vaue Reference. 2 This dataset is an unbaanced pane, reporting quartery historic vaues of different vintages for the most popuar business-aircraft modes during the period Two series are reported: average retai prices and average whoesae prices. Average retai prices report prices between fina users of the aircraft, and average whoesae prices report average transaction prices between an aircraft owner (as a seer) and a deaer (as a buyer). A prices are based on the company s experience in consuting, appraisa and feet evauation. A vaues are in U.S. doars, and I have defated them using the GDP Impicit Price Defator, with 25 as the base year. It is important to note that the construction of the Aircraft Buebook Historica Vaue Reference impies that the retai and whoesae price series are free of severa biases. First, whoesae prices refer to saes to deaers thus before deaers coud make any improvement to the aircraft. Second, the database reports historica retai and whoesae prices, even for 1 stat.htm.the website mentions that: AMSTAT s customers are aircraft professionas, whose primary business is seing, buying, easing and/or financing business aircraft, as we as providers of reated services and equipment. 2 The dataset is avaiabe at The website describes the dataset as: The Aircraft Buebook Historica Vaue Reference is specificay designed for ease companies, bankers, aircraft deaers, or anyone who needs to know the pricing history of an individua aircraft. 7

8 Tabe 1: Summary statistics Pane A: Aircraft Transactions Obs. Mean St. Dev. Median Modes of Aircraft 26, Active Aircraft 26, Aircraft for Sae 26, by Owners 26, by Deaers 26, Average Age, Aircraft for Sae 23, Retai-to-Retai Transactions 26, Retai-to-Deaer Transactions 26, Deaer-to-Deaer Transactions 26, Deaer-to-Retai Transactions 26, Tota Number of Transactions 26, Pane B: Aircraft Prices Modes of Aircraft 31, Retai Price (in $1,) 31, 524 7, 67 8, 534 4, 343 Whoesae Price (in $1,) 31, 524 6, 731 7, 555 3, 849 Age (in Years) 31, Notes This tabe provides summary statistics of the variabes used in the empirica anaysis. Pane A presents summary statistics for the Aircraft Transactions dataset. Each observation represents a mode-month pair. Pane B presents summary statistics for the Bue Book prices dataset. Each observation represents a mode-vintage-quarter tupe. Aircraft prices are in thousands of U.S. doars and have been defated using the GDP Impicit Price Defator, with 25 as the base year. those mode-vintage pairs for which ony one unit (i.e., one seria number) exists, suggesting that the price series are not affected by seers seection based on aircraft quaity (observabe or unobservabe to both trading parties) or on their vauations 4.2 Data Description Tabe 1 provides summary statistics of the main variabes used in the empirica anaysis. Pane A refers to the Aircraft Transactions Dataset, a sampe containing 161 modes (the definition of a mode is quite fine in this dataset), comprising a tota of 26,237 aircraft mode-month observations. 3 For each mode, the stock of Active Aircraft equas approximatey 9 units, of which approximatey ten are on sae in a given month. The number of transactions is sma reative to the number of aircraft on sae: On average, there are 3 The quantitative anaysis in Section 6 does not expoit the cross-sectiona differences across modes. 8

9 Fraction Age Fig. 1: Histogram of the average ages across modes of aircraft on the market in January Retai-to-Retai Transactions and.83 Deaer-to-Retai Transactions per mode-month pair (I incude Lease transactions in Deaer-to-Retai Transactions). There are aso a few Deaer-to-Deaer Transactions.28 per month, on average suggesting that deaers smooth their inventories by trading with other intermediaries. The Tota Number of Transactions, defined as the sum of Retai-to-Retai Transactions and Deaer-to-Retai Transactions, indicates that, on aggregate, approximatey 1.5 aircraft per mode trades in a given month and that a deaer is the seer to the fina retai user in approximatey 6 percent of these transactions. The average age of aircraft for sae is years, with a arge dispersion across observations (mode-month pairs). 4 This dispersion indicates that aircraft of many vintages are on the market simutaneousy. For exampe, Figure 1 pots the histogram of the average ages across modes of aircraft on the market in January 21, showing that the youngest vintages are frequenty for sae. This pattern is in stark contrast with the evidence from other asset markets in which repacement purchases are the main motives for trade, such as, for exampe, the car market. In particuar, in the car market, the fraction of cars sod is owest for the youngest vintages (Stoyarov, 22; Gavazza, Lizzeri and Rokestkiy, 212). The reason is that, because of transaction costs, few househods se their cars one year after purchasing them, so the resae rates are increasing when cars are reativey new. Hence, Figure 1 suggests that repacement is not the first-order motive for trade. 4 The number of observations of Average Age, Aircraft for Sae is ower than a other variabes because there are no aircraft for sae in some mode-month pairs. 9

10 Pane B provides summary statistics for the Aircraft Prices dataset. This sampe contains 72 modes (indicating that the definition of a mode is coarser than in the Aircraft Transactions Dataset), comprising a tota of 31,524 aircraft mode-vintage-quarter observations. The average Retai Price of an aircraft in the sampe is 7.6 miion (year 25) doars, and the average Whoesae Price is 6.7 miion (year 25) doars. Moreover, there is substantia variation in both prices: The standard deviation of retai prices is 8.5 miion doars and of whoesae prices is 7.5 miion doars. The two datasets provide a rich description of the business-aircraft market and are wesuited to investigating the importance of frictions and the roe of deaers. Specificay, three key patterns suggest that trading deays may be non-trivia and that deaers reduce them. First, the difference between the number of Aircraft for Sae (on average, 11 aircraft per mode-month) and the Tota Number of Transactions (on average, 1.5 per modemonth) means that aircraft stay on the market for severa months before seing, indicating that trading deays are substantia. Second, the ratio between Retai-to-Deaer Transactions and Aircraft for Sae by Deaers is higher than the ratio between Tota Number of Transactions and Aircraft for Sae, suggesting that deaers are faster than owners at turning aircraft over. 5 Third, the difference between the Retai Price and the Whoesae Price is quite arge (on average, 13 percent), corroborating that frictions are reevant in this market and that deaers are abe to command a substantia markup by suppying immediacy of trade. With a their advantages, however, the datasets pose some chaenges. In my view, the main imitation is that both datasets provide aggregate statistics of the market for different modes. This imitation impies that a mode with rich heterogeneity of agents and deaers, whie theoreticay feasibe, woud not be identified with these aggregated data. For exampe, the aircraft-price dataset does not aow the identification of rich heterogeneity in asset vauations, which woud be possibe if transaction prices were avaiabe. Therefore, the mode admits a parsimonious binary distribution of vauations, high and ow. In Section 8, I discuss the impications of this imited heterogeneity for the interpretation of the empirica resuts. Simiary, the aircraft-transaction dataset reports ony aggregate deaers inventories, imiting the possibiity of identifying heterogeneity across deaers. Finay, an additiona imitation is that the aircraft-transaction dataset does not report whether retai buyers or 5 These differentia trading patterns are difficut to expain through a mode of thin markets i.e., markets with a very sma number of buyers and seers wishing to trade at non-perfecty synchronized times. The differentia patterns are aso comparabe between more- and ess-popuar aircraft. 1

11 seers hired a broker to search for trading counterparts, athough these intermediaries are popuar in business-aircraft markets. 5 Mode In this section, I ay out a mode of a decentraized market with two-sided search to theoreticay investigate the effects of search frictions on asset aocations and prices. The mode combines eements from Rubinstein and Woinsky (1987) and Duffie, Gâreanu and Pedersen (25), extending them to capture key features of rea asset markets, such as the depreciation of assets. I mode frictions of reaocating assets expicity. In particuar, each agent contacts another agent randomy, and this is costy for two reasons: 1) There is an expicit cost c s of searching; and 2) there is a time cost in that a agents discount future vaues by the discount rate ρ >. 5.1 Assumptions Time is continuous and the horizon infinite. A mass of risk-neutra agents enters the economy at every instant. A entrants have an exogenous parameter z = z h > that measures their vauation for an aircraft. This vauation parameter is a Markov chain, switching from z h to z = with intensity λ; z is an absorbing state. The vauationprocesses of any two agents are independent. Hence, the mass of high-vauations agents is equa to λ. There is a constant fow x of new aircraft entering the economy in every instant. Aircraft depreciate continuousy over time and are exogenousy scrapped when they reach age T, but coud be endogenousy scrapped earier. Thus, the endogenous tota mass A of aircraft is, at most, equa to xt. I assume that > xt, impying that > A, which means that λ λ the margina owner in a Warasian market is a high-vauation agent. An aircraft of age a generates an instantaneous fow of utiity equa to π(z,a) to its owner with vauation z. The function π(z,a) satisfies the foowing: 1) It is increasing in the vauation z, π(z h,a) > π(z,a); 2) it is decreasing in the age a of the asset, π(z,a) a compementarity between the vauation z and the age a of the asset, π(z h,a) a < ; and 3) it exhibits negative < π(z,a) a. Each agent can own either zero or one aircraft. 6 Agents can trade aircraft: A given agent 6 Hence, I do not consider quantity decisions, as do Duffie, Gâreanu and Pedersen (25 and 27), Miao (26), Vayanos and Wang (27), Vayanos and Wei (28), and Wei (27 and 28). See Lagos and 11

12 wishing to trade (either a buyer or a seer) pays a fow cost c s to search for a counterparty. 7 Whie searching, he makes contact with other agents pairwise independenty at Poisson arriva times with intensity γ >. Thus, given that matches are determined at random, the arriva rateγ s foraseer (therateat which hemeets buyers) isγ s = γ b, and thearriva rate γ b (a) for a buyer (the rate at which he meets seers of an age-a aircraft) is γ b (a) = γ s (a), where b and s (a) are the endogenous equiibrium masses of buyers and seers of an age-a aircraft, respectivey. 8 In addition, there is an endogenous mass d of independent used-aircraft deaers that meets traders aso through a search process. Each deaer has a fow cost equa to k and has, at most, one unit of inventory. An agent wishing to trade meets deaers pairwise independenty at Poisson arriva times with intensity γ >. Thus, a buyer meets a deaer with a vintage-a aircraft to se at rate γ bd (a) = γ do (a), and a deaer with an aircraft to se meets a buyer at rate α ds = γ b, where do (a) is the endogenous mass of deaers with a vintage-a aircraft to se. The rates γ bd (a) and α ds represent the sum of the intensity of buyers search for deaers and deaers search for buyers. A seer meets a deaer wiing to buy an aircraft at rate γ sd = γ dn, and a deaer wiing to buy an aircraft meets a seer of an aicraft of age a at rate α db (a) = γ s (a), where dn is the endogenous mass of deaers with an aircraft to se. There is free entry into the deaers market. Once a buyer and a seer meet, or they meet a deaer, parties negotiate a price to trade. I assume that a buyer and a seer negotiate a price according to generaized Nash bargaining, where θ s [,1] denotes the bargaining power of the seer. Simiary, when an agent meets a deaer, they negotiate a price, and θ d [,1] denotes the bargaining power of the deaer. 9 Rocheteau (29) for a mode that considers quantity decisions. 7 The main roe of the search cost c s is to avoid swaps of assets. For exampe, a ow-vauations agent with a young aircraft and a high-vauation agent with an od aircraft may want to swap their assets. However, a positive fow cost of search c s eiminates these trades with sma gains from trade. 8 Thus, the matching function exhibits increasing returns to scae. Appendix A reports on the estimates of the mode and on the counterfactua anayses under the assumption that the matching function exhibits constant returns to scae. 9 The mode assumes symmetry of information about the quaity of the asset. Severa institutiona features of aircraft markets support this assumption. First, the aviation authorities often reguate aircraft maintenance. Second, maintenance records are frequenty avaiabe, and a parties can observe the entire history of owners of each aircraft. Finay, a transactions invove a thorough materia inspection of the aircraft. 12

13 5.2 Soution There are four types of agents in the mode economy and two types of deaers: high- and ow- vauation owners and non-owners, and deaers with and without an aircraft for sae. I denote their types ho, o, hn, n, do, dn, respectivey. The owner of an age-a aircraft with vauation z can keep operating it, put it up for sae, or scrap it. In the first case, he enjoys the fow profit π(z,a). In the second case, he meets potentia trading partners at rate γ s and a deaer at rate γ sd. In the ast case, he becomes an agent with no aircraft. Intuitivey, owners prefer to se their assets when their vauations are ow and the assets are reativey young since the compementarity between vauation z and age a of the fow utiity π(z,a) means that the gains from trade between buyers and seers are arger for younger aircraft. Moreover, high-vauation owners scrap their od aircraft, seeking to purchase newer ones. Simiary, an agent with no aircraft can meet active seers of age-a aircraft at rate γ b (a) and a deaer at rate γ bd (a), or he can exit the market. Non-owners choose to search when their vauation is high, and they purchase ony reativey young assets. Deaers without inventories operate simiary to buyers, and deaers with inventories operate simiary to seers, with the key difference that deaers opportunity costs differ from buyers and seers and, thus, their choice of which assets (i.e., vintages) to purchase differs. I now formay derive the vaue functions for a agents of the economy and the transaction price at which trade occurs. These vaue functions aow us to pin down the equiibrium conditions and derive the endogenous distribution of owners vauations. This distribution describes in an intuitive way how frictions generate aocative inefficiencies and affect aircraft prices Agents Vaue Functions Let U ho (a) be the vaue function of an agent with vauation z h who owns an aircraft of age a and is not seeking it to se it. U ho (a) satisfies: ρu ho (a) = π(z h,a)+λ(v o (a) U ho (a))+u ho (a). (1) Equation (1) has the usua interpretation of an asset-pricing equation. An agent with vauation z h enjoys the fow utiity π(z h,a) from an aircraft of age a. At any date one possibe event, at most, might happen to him: At rate λ, his vauation drops to z, in 13

14 which case he chooses between continuing to operate the aircraft (enjoying the vaue U o (a)) and activey seeking to se it (enjoying vaue S o (a)). Thus, the agent obtains a vaue V o (a) = max{u o (a),s o (a)} and a capita oss equa to V o (a) U ho (a). Moreover, the aircraft depreciates continuousy, so that the agent has a capita oss equa to U ho (a). Simiary, the vaue function S ho (a) of an agent with vauation z h who owns an aircraft of age a and is activey seeking to se it satisfies: ρs ho (a) = π(z h,a) c s +λ(v o (a) S ho (a))+γ s max{p(a)+v hn S ho (a),}+ γ sd max{p B (a)+v hn S ho (a),}+s ho(a). (2) An agent with vauation z h enjoys the fow utiity π(z h,a) from an aircraft of age a, but he pays the fow cost c s whie he activey seeks to se it. At any date, one of three possibe events, at most, might happen to him: 1) At rate λ, his vauation drops to z. In this case, the agent chooses between keeping the aircraft (enjoying the vaue U o (a)) and activey seeking to se it (enjoying vaue S o (a)). Hence, the agent obtains a capita oss equa to V o (a) S ho (a). 2) At rate γ s, the agent meets a potentia buyer and chooses between trading the aircraft or keeping it. If he trades it at price p(a), he becomes a high-vauation non-owner with vaue V hn, thus obtaining a capita gain equa to p(a)+v hn S ho (a). If he keeps it, he obtains a capita gain of zero. 3) At rate γ sd, he meets a deaer and chooses between trading the aircraft or keeping it. If he trades it at price p B (a), he then chooses between activey searching for another aircraft or not, thus obtaining a capita gain equa to p B (a)+v hn S ho (a). If he keeps it, he obtains a capita gain of zero. Moreover, the aircraft depreciates continuousy, so that the agent has a capita oss equa to S ho (a). The vaue functions U o (a) and S o (a) of an agent with vauation z who owns an aircraft of age a satisfy the foowing Beman equations, respectivey: ρu o (a) = π(z,a)+u o (a), (3) ρs o (a) = π(z,a) c s +γ s max{p(a)+v n S o (a),}+ γ sd max{p B (a)+v n S o (a),}+s o (a). (4) The interpretation of equations (3) and (4) is now straightforward. An agent with vauation z enjoys the fow utiity π(z,a) from an aircraft of age a. If he does not seek to se the aircraft, the ony event that affects his utiity is the depreciation of the aircraft, generating 14

15 a capita oss equa to U o (a). If he seeks to se the aircraft, his fow utiity is reduced by the search cost c s. Then, at any date, he meets a potentia buyer at rate γ s, in which case he chooses between seing the aircraft at price p(a) thus obtaining a capita gain equa to p(a) + V n S o (a) or keeping it thus obtaining a capita gain of zero. Simiary, he meets a deaer at rate γ sd, in which case he chooses between trading the aircraft at price p B (a) thus, obtaining a capita gain equa to p B (a) + V n S o (a) or keeping it thus obtaining a capita gain of zero. Moreover, the aircraft depreciates continuousy, so that the agent has a capita oss equa to S o (a). The vaue functions of high- and ow-vauation agents with no aircraft satisfy: ρs hn = c s +λ(v n S hn )+ γ b (a)max{v ho (a) p(a) S hn,}da+ (5) γ bd (a)max{v ho (a) p A (a) S hn,}da, ρu hn =, (6) ρs n = c s + γ b (a)max{v o (a) p(a) S n,}da+ (7) γ bd (a)max{v o (a) p A (a) S n,}da, ρu n =. (8) Equation (5) says that a high-vauation agent with no aircraft who is paying the search cost c s has a capita oss equa to V n S hn when, at rate λ, his vauation drops from high to ow; has a capita gain equa to max{v ho (a) p(a) S hn,} when, at rate γ b (a), he meets a seer of an aircraft of age a; and has a capita gain equa to max{v ho (a) p A (a) S hn,} when, at rate γ bd (a), he meets a deaer seing an aircraft of age a, where V ho (a) = max{u ho (a),s hn }. Since the hn-agent does not know the age of the aircraft that the counterparty wi have, he takes expectation over the possibe capita gains that arise from the different vintages. Simiary, equation (7) says that a ow-vauation agent with no aircraft who is paying the search cost c s to search for a counterparty has an expected capita gain equa to max{v o (a) p(a) S n,} when he meets a potentia seer of an aircraft of age a, and an expected capita gain equa to max{max{v o (a),w o (a)} p A (a) W n,} when he meets a deaer. Equations (6) and (8) say that agents without aircraft who are not searching have a zero vaue. 15

16 5.2.2 Deaers Vaue Functions As in Rubinstein and Woinsky (1987), deaers can extract surpus by shortening the time that buyers and seers have to wait in order to trade. They are capacity-constrained and cannot store more than one aircraft. Thus, the vaue functions J do (a) and J dn of deaers with and without an aircraft for sae, respectivey, satisfy: ρj do (a) = max{ k +α ds (p A (a)+j dn J do (a))+j do (a),ρj dn}, (9) ρj dn = k + α db (a)max{j do (a) p B (a) J dn,}da. (1) Equation (9) says that a deaer who owns an aircraft of age a chooses between two aternatives: 1) He can activey seek to se it, meeting a potentia buyer at rate α ds. In this case, the deaer trades the aircraft at a negotiated price p A (a), obtaining a capita gain equa to p A (a)+j dn J do (a). Moreover, the deaer pays the fow cost k and its aircraft depreciates continuousy, so the deaer has a capita oss equa to J do (a); and 2) he can scrap the aircraft, thereby becoming a deaer without aircraft for sae. Simiary, equation (1) says that a deaer without aircraft pays the fow cost k whie he activey seeks to purchase one. At rate α db (a), this deaer meets a potentia seer of an aircraft of age a, in which case the deaer decides whether or not to trade, thus enjoying a capita gain equa to max{j do (a) p B (a) J dn,}. Since the dn-deaer does not know the age of the aircraft that the counterparty wi have, he takes expectation over the possibe capita gains that arise from the different vintages. Deaers free entry requires that J dn = i.e., deaers expected capita gain is exacty equa to their fixed operating cost Prices When a buyer and a seer meet and agree to trade, the negotiated price p(a) = (1 θ s )(S o (a) V n )+θ s (U ho (a) S hn ) (11) is the soution to the foowing symmetric-information bargaining probem: max[u ho (a) p(a) S hn ] 1 θs [p(a)+v n S o (a)] θs p(a) subject to: U ho (a) p(a) S hn and p(a)+v n S o (a). 16

17 Simiary, the ask and bid prices p A (a) and p B (a) satisfy: p A (a) = (1 θ d )(J do (a) J dn )+θ d (U ho (a) S hn ), (12) p B (a) = (1 θ d )(J do (a) J dn )+θ d (S o (a) V n ). (13) Agents and Deaers Poicies We can simpify the vaue functions of a types of agents {ho,o,hn,n} recognizing that gains from trade may arise ony when high-vauation non-owners meet ow-vauation owners. Moreover, since the fow payoff π(z, a) exhibits compementarity between agents vauation z and the age a of the asset, the gains from trade are arger for younger assets. In turn, this impies that, if the assets are sufficienty heterogeneous between them (i.e., either δ 2 or T is arge), not a assets trade in equiibrium. More specificay, since U ho (a) is intuitivey decreasing in a, there exists a cutoff age a h such thatho-agentsscraptheir assetswhen itreaches agea h i.e., a h satisfies U ho(a h ) = S hn. Moreover, this same cutoff age a h determines whether or not hn-agents purchase an age-a asset. To understand this, note that hn-agents purchase an age-a asset if their gains from trade arepositive: U ho (a) p(a) S hn, or p(a) U ho (a) S hn. Since the negotiated prices (11) are ower than the prices that seers woud command if they captured a surpus (i.e., θ s = 1), then the condition p(a) U ho (a) S hn aways hods, as ong as o-owners of these assets are wiing to se them (see next paragraph). An identica argument impies that hn-agents purchase assets of age a a h from deaers as ong as deaers have inventories of these assets. The poicy of o-owners satisfies a cutoff rue, as we. Specificay, since the fow utiity π(z,a)isdecreasingintheageaoftheasset, whereasthefowsearchcostc s isconstant, there exists a cutoff age a such that o-agents se their aircraft if it is younger than a, but keep it ifitisoder i.e., U o (a) > S o (a)ifa < a, U o(a o ) = S o(a o )andu o(a) < S o (a)ifa > a. Equiibrium requires that o-agents se assets that hn-agents are wiing to purchase thus, a a h. Moreover, since o-agents enjoy positive utiity from these od assets, and this utiity is greater than the opportunity cost of searching for a younger aircraft, they keep these od assets unti they reach the scrappage age T. 17

18 Thus, the agents vaue functions are: V ho (a) = V o (a) = { Uho (a) for a < a h, V hn = S hn, V n =. S hn for a a h, S o (a) for a < a, U o (a) for a a < T, V n for a = T, Simiar arguments appy to deaers, as we. Their vaue function J do (a) is intuitivey decreasing in a because younger aircraft se at greater margins, as the gains from trade are arger. Thus, since deaers fixed cost k is independent of the age of the asset, there exist two cutoff ages a dn and a do such that deaers do not purchase aircraft oder than a dn and scrap aircraft when they reach age a do. Since the asset purchase price is sunk at the time of scrapping it, but not at the time of buying it, a dn < a do. Moreover, equiibrium requires that deaers se assets that buyers are wiing to purchase. Hence, a do a h Distribution of Agents Let i (a) be the masses of owners of age-a aircraft whose state is i {ho,o,do}, and et i be the masses of non-owners whose state is i {hn,n,dn}. The masses of owners evove over time according to the foowing system of differentia equations: ho (a) = γ b (a) hn +γ bd (a) hn λ ho (a) for a < a h, (14) o (a) = λ1(a < a h ) ho(a) γ s 1(a < a h )1(a < a ) o(a) (15) γ sd 1(a < a dn)1(a < a ) o (a) for a < T, do (a) = α db (a)1(a a dn ) dn α ds do (a) for a < a do, (16) withinitiaconditions ho () = xand o () = do () =,andterminaconditions ho (a) = for a h < a < T and do(a) = for a do < a < T. The notation 1(Y) represents an indicator function equa to one if the event Y is true, and zero otherwise. The intuition for these equations is simpe. Specificay, equation(14) states that the mass of high-vauation agents with an age-a asset is the resut of fows of three sets of agents: 1) 18

19 the infow of high-vauation non-owners that found a seer of an age-a aircraft the term γ b (a) hn ; 2) the infow of high-vauation non-owners that found a deaer seing an age-a aircraft the term γ bd (a) hn ; and 3) the outfow of high-vauation age-a aircraft owners whose vauation just dropped the term λ ho (a). Moreover, equation (14) incorporates the equiibrium outcomes that ow-vauation owners of age-a aircraft are wiing to se it rather than keep it (i.e., γ b (a) > ) and deaers have inventories of age-a aircraft (i.e., γ bd (a) > ) ony if high-vauation non-owners are wiing to buy them (i.e., a < a h ). Equation (15) states that the mass of ow-vauation agents with an age-a asset is the resut of fows of three sets of agents: 1) the infow of high-vauation agents whose vauation just dropped. Since high-vauation agents scrap their assets when they reach age a h, this infow appies ony to owners of assets younger than a h. Hence, the term λ1(a < a h ) ho(a); 2) the outfow of owvauation aircraft owners who prefer to se their aircraft rather than keep it (i.e., an aircraft of age a < a ) and found a buyer wiing to purchase it (i.e., an aircraft of age a < a h ). Hence the term γ s 1(a < a h )1(a < a ) o(a); and 3) the outfow of ow-vauation aircraft owners that prefer to se their aircraft rather than keep it (i.e., an aircraft of age a < a ) and that found a deaer wiing to purchase it (i.e., an aircraft of age a < a dn ). Hence the term γ sd 1(a < a dn )1(a < a ) o(a). The intuition for equation (16) is simiar. Moreover, in a sma time interva of ength ǫ, up to terms in o(ǫ), the masses of highvauation non-owners and deaers without inventories evove from time t to time t + ǫ according to: hn = ( x)+ ho (a ho ) λ dn hn hn γ b (a)da hn γ bd (a)da, do dn = α ds do (a)da dn whereas, in each instant, the mass of n-agents equas: α db (a)1(a a dn )da+ do(a dn ), n = λ hn +γ s a o (a)1(a a h)da+γ sd a o (a)1(a a dn)da+ o (T). Steady state imposes the foowing constraints on the evoution of these masses: 1) The aggregate masses of owners T i(a)da for i {ho,o,do} and the aggregate masses hn and dn are constant over time; 2) the mass n of exiters equas the mass of new entrants; 3) the tota mass of agents with high vauation hn + T ho(a)da equas ; 4) the tota λ mass of deaers d equas T do(a)da+ dn ; 5) the aggregate masses of assets sod and pur- 19

20 chased by deaers equa the aggregate masses of assets purchased from and sod to deaers: do hn γ bd (a)da = α do ds do (a)da and dn dn α db (a)da = γ dn sd o (a)da; and 6) deaers aggregateinventoriesareconstantovertime: do hn γ bd (a)da = α do ds do (a)da = dn dn α db (a)da = γ dn sd o (a)da. In addition, equiibrium requires that the tota mass of owners of age-a aircraft ho (a) + o (a) + do (a) equas the mass x of aircraft for a min{a do,a h }, and ho(a)+ o (a)+ do (a) < x for a > min{a do,a h }. A these steady-state equaities and the equiibrium condition ho (a)+ o (a)+ do (a) x aow us to sove for the endogenous masses as a function of the exogenous parameters x,t, and λ, and the exogenous parameters of the matching functions. Lettingγ increase, im γ + o (a)convergesto: Whenfrictionsvanish, noow-vauation agentownsanaircraft. Simiary, im γ + do (a)convergesto.thus, themasses T o(a)da and T do(a)da i.e., the masses of ow-vauation agents and deaers with inventories are measures of assets inefficienty aocated in the economy. Based on the parameters estimated from the data, in Section 7.1, I compute these measures to quantify the effects of trading frictions on asset aocations. 6 Quantitative Anaysis The mode does not admit an anaytic soution for the asset aocations and asset prices as a function of a the mode primitives, except in the specia case when assets are homogeneous i.e., δ 2 =. AppendixBreportsthisspecia case. Hence, thegoaofthequantitative anaysis is to choose the parameters that best match moments of the data with the corresponding moments computed from the mode s numerica soution. Since δ 2 is estimated to be sma (approximatey equa to.2), the anaytic soution of the mode under the assumption that δ 2 =, presented in Appendix B, represents a good approximation of the genera mode, thus making the intuition for the identification of the parameters more transparent. 6.1 Estimation and Identification I estimate the mode using the data on business aircraft described in Section 4, assuming that they are generated from the mode s steady state. I set the unit of time to be one quarter. Unfortunatey, the data ack some detaied information to identify a parameters. Therefore, I have to fix some vaues. Specificay, the discount rate ρ is traditionay difficut to 2

21 identify, and I fix it to ρ =.15. Moreover, I fix the usefu ifetime of an aircraft to be equa to T = 16 quarters. Furthermore, I use a directory of aircraft deaers to estimate tota deaers capacity to be equa to d = 1. Moreover, I assume that aircraft owners fow payoff equas: π(z,a) = z ( δ +δ 1 e δ 2a ), (17) further imposing that δ 1 = 1, as this parameter is not separatey identified from the baseine vauation z δ (this is just a normaization). I estimate the vector of other parameters ψ = {λ,γ s,γ sd,α ds,,z h,z,δ,δ 2,c s,k} using a minimum-distance estimator that matches key moments of the data with the corresponding moments of the mode. More precisey, for any vaue of these parameters, I sove the mode described in Section 5 to find agents and deaers poicy functions, and agents and deaers distributions that are consistent with each other. Based on the mode s soution, I cacuate the vector m(ψ) composed by the foowing moments: 1. The fraction of aircraft for sae, which, in the mode, equas: o (a)da+ do do (a)da. (18) A 2. The fraction of aircraft for sae by deaers (i.e., aggregate deaers inventories), which equas: do do (a)da. (19) A 3. The fraction of retai-to-retai transactions to tota aircraft, which equas: γ s o (a)da. (2) A 4. The fraction of deaer-to-retai transactions to tota aircraft, which equas: α do ds do (a)da. (21) A 21

22 5. The average age of aircraft for sae, which equas: a o (a)da+ do a do (a)da o (a)da+ do do (a)da. 6. The tota mass of assets A, which equas: T ( ho (a)+ o (a)+ do (a))da. 7. The mass of deaers d, which equas: dn + do do (a)da. 8. For each age a [,12], the price p(a), which equas: p(a) = (1 θ s )(S o (a) V n )+θ s (U ho (a) S hn ). (22) 9. For each age a [,12], the bid price p B (a), which equas: p B (a) = (1 θ d )(J do (a) J dn )+θ d (S o (a) V n ). (23) 1. Deaers free-entry conditions: J dn =. The minimum-distance estimator minimizes the criterion function (m(ψ) m S ) Ω(m(ψ) m S ), where m S is the vector of sampe moments corresponding to the vector m(ψ) of moments, and Ω is a symmetric, positive-definite weighting matrix. In practice, I choose Ω to be a diagona matrix with diagona eements that are proportiona to the inverse of m S m S to make the scae comparabe across moments. Moreover, since I am matching the moments p(a) and p B (a) for every age a [,12], I weight each of these price moments by to make their aggregate weight in the criterion function comparabe to the weight of each of the non-price moments. 22

23 Athough the mode is highy noninear, so that (amost) a parameters affect a outcomes, the identification of some parameters reies on some key moments in the data. Specificay, equations (18)-(21), aong with the differentia equations (14)-(16) that define the functions ho (a), o (a) and do (a), identify the parameters λ, γ s, γ sd and α ds. Tabe 1 reports that the fraction of aircraft for sae is non-trivia, thus indicating that trading deays are reevant. Hence, the mode can fit this key feature of the data very we. Simiary, Tabe 1 reports that deaers are faster than seers at trading aircraft, and the mode captures this difference through the trading parameters γ s, γ sd and α ds. From the estimates of the parameters λ, γ s, γ sd, α ds, together with A and d, we can recover, γ and γ from the arriva rates (24)-(29) and from the steady-state condition: T λ = hn + ho (a)da. For any vaue of the bargaining parameters θ s and θ d, from the price equations (22) and (23), we can recover the vaues of the parameters z h, z, c s, δ and δ 2. The price differences across vintages identify the parameter δ 2, whereas the eve of prices identifies the parameters z h, z, c s, δ. Moreover, the separate identification of z h, z, and c s, δ reies on the assumptions that the vauations z h and z interact with the age of the aircraft in the fow utiity (17), whereas the search cost c s and δ do not. Finay, I can recover k from deaers free-entry condition J dn =. The bargaining parameters θ s and θ d are not point-identified. The reason is that, conditiona on the trading frictions, prices depend on the gains from trade(i.e., agents vauations) and how these gains are spit between the parties (i.e., agents bargaining parameters). Thus, the prices aone do not separatey identify agents vauations and bargaining parameters. Therefore, I estimate an interva for their vaues by imposing that the estimated parameters satisfy some restrictions dictated by the mode. Specificay, I require the foowing 23