Identification of Time and Risk Preferences in Buy Price Auctions

Transcription

1 Identification of Time and Risk Pefeences in Buy Pice Auctions Daniel Ackebeg Univesity of Michigan Keisuke Hiano Univesity of Aizona Quazi Shahia San Diego State Univesity Apil 27, 2011 Abstact Buy pice auctions mege a posted pice option with a standad bidding mechanisms, and have been used by vaious online auction sites including ebay and GMAC. A buye in a buy pice auction can accept the buy pice to win with cetainty and end the auction ealy. Intuitively, the buy pice option may be appealing to biddes who ae isk avese o impatient to obtain the good, and a numbe of authos have examined how such mechanisms can incease the selle s expected evenue ove standad auctions. We show that data fom buy pice auction can be used to identify biddes isk avesion and time pefeences. We develop a pivate value model of bidde behavio in a buy pice auction with a tempoay buy pice. In ou setup, biddes aive stochastically ove time, and the auction poceeds as a second-pice sealed bid auction afte the buy pice disappeas. Upon aival, a bidde in ou model is allowed to act immediately (i.e. accept the buy pice if it is still available, o place a bid) o wait and act late. Allowing fo geneal foms of isk avesion and impatience, we fist chaacteize equilibia in cutoff stategies and descibe the condition unde which all symmetic pue-stategy subgamepefect Bayesian Nash equilibia ae in cutoff stategies. Given sufficient exogenous vaiation in auction chaacteistics such as eseve and buy pices and in auction lengths, we then show that the aival ate, valuation distibution, utility function, and time-discounting function in ou model ae all nonpaametically identified. We also develop extensions of the identification esults fo settings in which the vaiation in auction chaacteistics is moe limited. We thank Matin Dufwenbeg, Jin Hahn, Phil Haile, Geg Lewis, Isabelle Peigne, Benad Salanié, Quang Vuong, John Woodes, Robet Zeithamme, and semina paticipants at U. of Aizona, UC Ivine, UCLA, Canegie Mellon U., Columbia U., Bown U., Havad U., U. of Michigan, U. of South Floida, Yale U., and the Confeence on Identification of Demand at Bown Univesity fo comments and suggestions. Special thanks ae due to Robet Ackebeg fo paticulaly helpful discussions. All eos ae ou own. 1

2 1 Intoduction This pape studies identification of bidde pefeences in single unit buy pice (BP) auctions. BP auctions mege a posted pice selling envionment with an auction envionment, and have been used by ebay (in thei Buy-it-Now auctions), GMAC, and othe oganizations as an altenative to standad fist o second pice auctions. We show that data fom BP auctions can be paticulaly infomative about isk avesion and time pefeences among potential biddes, in a way that standad auctions ae not. As a esult, it is possible to ecove bidde pefeences fom widely available obsevational data, o cay out expeiments to obtain appopiate data. Thee is a lage theoetical liteatue that shows how BP auctions can incease expected evenue ove standad auctions; see Budish and Takeyama (2001), Mathews (2004), Mathews and Katzman (2006), Hidvégi, Wang and Whinston (2006), Gallien and Gupta (2007), Wang, Montgomey, and Sinivasan (2008), and Reynolds and Woodes (2009). BP auctions allow some o all potential biddes to puchase the item immediately at a posted buy pice. If this does not happen, a standad auction is held. Intuitively, the buy pice option may be appealing to biddes who ae isk avese o impatient to obtain the good. The existing theoetical models typically assume isk avesion, impatience, o both, on the pat of biddes, and show that BP mechanisms can incease expected evenue to the selle. Since biddes decisions in BP auctions depend on thei isk avesion and thei impatience, one might conjectue that obseved data fom BP auctions could be infomative about isk avesion and impatience. Many of the existing theoetical models of BP auctions abstact significantly fom the specific mechanisms used in pactice. Fo example, some of the models ae puely static, wheeas in pactice many BP auctions have two phases, a buy pice phase and a bidding phase, with paticula ules about when the bidding phase stats and how long it lasts. The models of Mathews (2004) and Gallien and Gupta (2007) do featue sequential aival of biddes and an auction fomat closely modeled on ebay auctions, but impose specific paametic foms on time discounting o iskavesion. We fist develop a theoetical model fo a BP auction which captues some key dynamic featues of eal-wold BP auctions and allows fo geneal foms of both isk avesion and impatience, but leads to a tactable equilibium and elatively staightfowad identification esults. In ou model, biddes have independent pivate values and aive accoding to a time-vaying Poisson pocess. Any potential bidde who aives in the buy pice phase can puchase the good at the buy pice (theeby winning the item and ending the auction), can bid (theeby initiating the bidding phase), o can wait. Potential biddes who aive duing the bidding phase (o biddes who aived duing the buy pice phase and have waited) can place bids. The bidding phase lasts fo a fixed amount of time and is modeled as a second-pice sealed bid auction. 1 We descibe conditions unde 1 Ou fixed length bidding phase diffes fom the setup of Mathews (2004) and Gallien and Gupta (2007), who assume a fixed oveall length of the auction (simila to ebay). The eason we conside this altenative is because it makes ou basic identification esults most staigtfowad. In Section 5, we extend ou identification esults to ebay-style models. 2

3 which all symmetic, pue-stategy, subgame-pefect Bayesian Nash equilibia (BNE) of this game ae in cutoff stategies, whee a potential bidde aiving in the buy pice phase accepts the posted pice if he valuation is sufficiently high. Having chaacteized equilibium stategies fo potential biddes in the auction, we conside identification. In ou model, biddes ae heteogeneous in thei valuations, but have common utility and time-discounting functions. This allows biddes to be isk avese, impatient, o both. Ou setup imposes some estictions on the natue of bidde pefeences, but unde these assumptions, we show that the aival ate function, the distibution of valuations, the utility function, and the time-discounting function ae nonpaametically identified unde an assumption of exogenous vaiation in the auction setup (e.g. eseve and buy pices), and some suppot conditions. The assumption that eseve and buy pices vay exogeneously is somewhat stong, but povides a natual stating point fo identification analysis and could be elaxed in vaious ways. Ou esults could also be used by selles (o economists) who wish to expeiment with eseve and buy pices in ode to lean about the pefeences of buyes. We also show that the model is oveidentified, in the sense that it imposes testable estictions on the distibution of obseved data. In addition, if the suppot conditions ae not fully met, then cetain local vesions of the stuctual objects ae identified. Although ou model captues many featues of eal-wold BP auctions, it does diffe in some details fom the BP auctions used by both ebay and those used by GMAC. We show that unde some additional assumptions, pimaily used to guaantee that biddes use cutoff stategies immediately upon aival, ou identification aguments can be extended to these two cases. The extended identification esults fo ebay-style Buy-it-Now auctions ae being used in ongoing empiical wok (Ackebeg, Hiano, and Shahia, 2006). Ou findings contibute to the liteatue on identification of auction models, and moe geneally to the liteatue on ecoveing isk avesion and othe featues of pefeences fom evealed behavio. Beginning with Guee, Peigne, and Vuong (2000), Li, Peigne, and Vuong (2002), and Athey and Haile (2002), a lage liteatue has emeged exploing identification in vaious auction fomats. 2 If biddes ae isk avese, identification becomes much moe challenging in these fomats; see, e.g. Campo, Guee, Peigne, and Vuong (2010), Bajai and Hotacsu (2005), Campo (2006), Peigne and Vuong (2007), Lu and Peigne (2008), Athey and Haile (2007), and Guee, Peigne, and Vuong (2009). To ou knowledge, ou pape is the fist to examine identification of bidde pefeences in BP auctions, and ou identification esults indicate that these auctions can povide consideable infomation on bidde isk pefeences. Moeove, ou esults show that bidde popensities to accept buy pices can be used to infe both thei isk avesion and time pefeences. Some of ou identification aguments may also be useful moe geneally in the context 2 See Haile and Tame (2003), Bajai and Hotacsu (2003), Song (2004), Adams (2004, 2007, 2009), Gonzales, Haske, and Sickles (2004), Canals-Ceda and Peacy (2008), Nekipelov (2008), Zeithamme and Adams (2009), and Backus and Lewis (2009), among othes. Athey and Haile (2007) povides a suvey of identification esults fo auction models. 3

4 of the liteatue on identification of isk pefeences when thee is unobseved heteogeneity (see Chiappoi, Gandhi, Salanie, and Salanie (2009) and the efeences thee). 2 Model and Equilibium We stat with a simple continuous time, independent pivate value (IPV) BP auction. 3 The auction stats at time 0, has a eseve pice (minimum bid) [0, ), and a buy pice p [, ). At times t > 0, potential biddes aive at the auction accoding to a Poisson pocess with ate λ(t). These potential biddes have pivate valuations v dawn independently fom the distibution F V (v). Thee ae two phases in the auction. At any time t in the the fist phase (the buy pice phase), any potential bidde who has peviously aived at the auction can take one of the following actions: 1. Immediately puchase the object at p (Accept the BP). In this case, the auction ends. 2. Submit a sealed bid b > fo the object (Reject the BP). In this case, the buy pice phase ends, and the auction immediately entes the second phase (the bidding phase). The bidding phase lasts fo fixed length τ > 0. Duing the bidding phase, potential biddes no longe have the option to puchase the object immediately at p. Othe potential biddes who eithe have aleady aived, o who aive duing the bidding phase, can also submit a sealed bid b > fo the object. These sealed bids can be placed at any time duing the bidding phase. At the end of the bidding phase, the auction ends and the object is awaded to the bidde who has placed the highest sealed bid. The winning pice is the maximum of eithe the eseve pice o the highest sealed bid of the othe biddes. We assume that biddes do not diectly obseve the actions o aivals of othe biddes. Howeve, we assume that any bidde who is pesent at the auction at t knows whethe the auction is cuently in the buy pice phase o the bidding phase, and if the latte, that the bidde knows at what point in time the auction enteed the bidding phase. In the teminology of Gallien and Gupta (2007), ou auction featues a tempoay buyout option, which disappeas once the buy pice is ejected and the bidding phase begins. 4 Howeve, as long as no potential bidde accepts o ejects the BP, the auction continues indefinitely. Thee ae a numbe of possible vaiations on this mechanism. Fo example, we could conside a design whee if by time T τ, no bidde has accepted o ejected the BP, the auction automatically entes the bidding phase. Altenatively, we could fix the oveall length of the auction at T (unless the BP is accepted). In this case, the length of the bidding phase is T t, whee t is the point in the auction at which the BP is ejected. This coesponds to the setup of ebay s Buy-it-Now auctions and 3 See Shahia (2008) fo a model of BP auctions with common values. 4 In contast, unde Gallien and Gupta s pemanent buyout option scheme, the option to puchase the good at pice p emains fo the entie duation of the auction. This was used by Yahoo! on thei now defunct auction site. 4

5 the models of Mathews (2004) and Gallien and Gupta (2007). We could also conside altenative foms fo the bidding phase, fo example by explicitly modeling ebay s poxy bidding system. We begin with ou stylized setup because it simplifies the equilibium analysis and leads moe diectly to identification esults. In Section 5, we conside identification unde some of these altenative BP auction designs and ae able to extend ou esults unde additional assumptions. Conside a bidde who aives at time t with a valuation v fo the object. We assume that if this bidde wins the object at time t and pays pice p, she obtains payoff δ(t t)u(v p ), whee U( ) is a utility function and δ( ) is a function captuing impatience, i.e. the idea that a bidde would pefe to win the object ealie. If the bidde does not obtain the object, she obtains utility 0. λ( ), F V ( ), U( ) and δ( ) ae pimitives of ou model; the vaiables (p,, τ) chaacteize the auction setup. We make the following assumptions on these pimitives. Assumption 1 The model pimitives {λ( ), F V ( ), U( ), δ( )} satisfy: 1. {λ( ), F V ( ), U( ), δ( )} ae common knowledge to all potential biddes; 2. λ( ) is twice continuously diffeentiable and satisfies 0 < λ(t) < fo all t 0; 3. F V ( ) is twice continuously diffeentiable on [0, ). F V (0) = 0, F V (v) > 0 fo all v > 0, F V (v)dv = 1; 0 4. U ( ) is twice continuously diffeentiable; 5. U ( ) > ɛ fo some ɛ > 0; 6. B < U ( ) 0 fo some 0 < B < (weak isk-avesion); 7. δ ( ) < 0 (stict impatience), δ( ) > 0; 8. U(0) = 0, U (0) = 1, δ(0) = 1 (nomalizations). We additionally make the following assumption about bidde behavio. Assumption 2 Biddes do not play weakly dominated stategies. The bidding phase is essentially a second-pice sealed bid auction, which geneally have unusual equilibia in weakly dominated stategies (Milgom (1981), Plum (1992), Blume and Heidhues (2004)). Assumption 2 is a simple way to ule these unusual equilibia out, and it ensues that we 5

6 get unique equilbium play in the bidding phase whee biddes follow the weakly dominant stategy of submitting bids equal to thei valuations (o not submitting a bid if v < ). With these assumptions, we can state the following Poposition 1 Unde Assumptions 1 and 2, any symmetic, pue stategy, pefect Bayesian-Nash equilibium (BNE) of this auction game has the following popeties: 1. Potential biddes with v < neve take any action; 2. Potential biddes with v > who aive duing the buy pice phase immediately eithe a) Accept the BP, i.e. puchase the good at p; o b) Reject the BP by placing a sealed bid equal to v; 3. Potential biddes with v > who aive duing the bidding phase place a sealed bid equal to v at some point befoe the end of the auction. Poof: Appendix A. Pat 2 of the Poposition states that in equilibium, potential biddes with v > aiving duing the buy pice phase do not wait they eithe accept o eject the BP immediately. The incentives to act immediately in equilibium aise fom two souces. Fist, waiting delays the time at which the bidde may potentially win the item, geneating lowe utility due to impatience. Second, waiting engendes moe competition fom othe potential biddes fo the object. Fo example, delaying accepting the BP incus the isk that anothe potential bidde will ente and accept the BP fist. Delaying ejecting the BP lengthens the time until the end of the auction (since the bidding phase has fixed length τ), inceasing the expected numbe of competitos that the bidde will face in the sealed-bid auction. Pat 3 (combined with Pat 2) of the Poposition implies that we get the well known 2nd-pice sealed bid auction outcome fo any auction that entes the bidding phase. Specifically, the bidde with the highest valuation wins the object at the valuation of the second highest bidde, o when thee ae no othe bids placed. 2.1 Optimal BP Decision and Conditions fo a Cutoff Equilibium Poposition 1 does not fully chaacteize the BNE, as it does not specify whethe a potential bidde aiving duing the buy pice phase (with v > ) will accept o eject the BP. We now chaacteize this decision. Conside such a bidde who aives at t. If the bidde accepts the BP option immediately, she will obtain payoff U A (v, p) := U(v p). 6

7 If the bidde ejects the BP and places a sealed bid at time t, then she will win the object if she has placed the highest sealed bid by time t + τ, and pay a pice equal to the valuation of the next highest bidde (if thee is anothe bid), o equal to if thee ae no othe bids placed. Let γ = t+τ t λ(s)ds, so the numbe of othe biddes who aive afte t is Poisson(γ). (Note that γ is a function of t and τ, but we suppess this in the notation.) Then the bidde s expected utility fom ejecting the BP is { } U R (v,, τ, t) := δ(τ) e γ γ n e γ U(v ) + FV n (v)e n [U(v max{, Y } Y v], n! n=1 whee FV n(v) = [F V (v)] n and E n is the expectation when Y has CDF F n. In this fomulation, n epesents the numbe of othe biddes that aive afte the BP is ejected, and Y epesents the maximum of these othe biddes valuations. Note that U R (v,, τ, t) does not depend on p, because Poposition 1 implies that any bidde aiving pio to t with v > would have aleady eithe accepted o ejected the BP. Hence, a bidde who aives while the BP is still available knows, in equilibium, that no pio aiving bidde has v >. The following poposition povides a simple expession fo U R (v,, τ, t) which we will use extensively in the sequel. Poposition 2 U R (v,, τ, t) = δ(τ) α(, τ, t)u(v ) + whee α(, τ, t) = exp(γf V () γ), h(y, τ, t) = exp(γf V (y) γ)γf V (y), and α(, τ, t), h(y, τ, t) satisfy: v U(v y)h(y, τ, t)dy, α(, τ, t) + h(y, τ, t)dy = 1, and α(, τ, t) = h(, τ, t). Poof: Appendix A. 7

8 Fom the pespective of a bidde ejecting the BP at t, α(, τ, t) is the pobability that no othe bidde will aive duing the bidding phase with a valuation geate than, and h(y, τ, t) is the density of the maximum of the valuations of biddes who aive duing the bidding phase. With expessions fo U A (v, p) and U R (v,, τ, t) in hand, we can now chaacteize the choice of whethe to accept o eject the BP. In paticula, we examine conditions unde which this decision depends on a bidde s valuation v in a paticulaly simple way: the bidde accepts the BP option if he valuation is above a cutoff value and ejects the BP (i.e. initiates bidding) othewise. We call this a cutoff stategy. Given Poposition 1, biddes with v > who aive duing the buy pice phase immediately eithe accept o eject the BP. Clealy, in any BNE, the bidde must accept the BP if and only if U A (v, p) U R (v,, τ, t) o U(v p) δ(τ) α(, τ, t)u(v ) + v U(v y)h(y, τ, t)dy. Define M(v,, τ, t) = U 1 δ(τ) α(, τ, t)u(v ) + v U(v y)h(y, τ, t)dy,, v, t, τ. M(v,, τ, t) is the cetainty equivalent of the andom outcome obtained by ejecting the BP option. Whethe o not the BP decision follows a cutoff stategy depends on how the cetainty equivalent vaies with v. Conside the following assumption: Assumption 3 Fo some ɛ > 0, M v (v,, τ, t) < 1 ɛ,, v, t, τ. This is a sufficient condition fo equilibium BP decisions to follow cutoff stategies. 5 Poposition 3 Unde Assumptions 1, 2, and 3, in any symmetic, pue stategy, pefect BNE, thee exists a finite valued cutoff function c(p,, τ, t), implicitly defined by the equation U(c(p,, τ, t) p) = δ(τ) α(, τ, t)u(c(p,, τ, t) ) + c(p,,τ,t) U(c(p,, τ, t) y)h(y, τ, t)dy, (1) 5 A necessay condition fo the accept/eject decision to follow a cutoff stategy fo any p,, t, and τ is that M v(v,, τ, t) 1. 8

9 such that a potential bidde who aives at t duing the buy pice phase with v > immediately accepts the BP if v > c(p,, τ, t), immediately ejects the BP if v < c(p,, τ, t), and is indiffeent between immediately accepting and immediately ejecting the BP if v = c(p,, τ, t). The cutoff function c(p,, τ, t) satisfies: 1. c p (p,, τ, t) > 0, c (p,, τ, t) < 0, c τ (p,, τ, t) < 0; 2. c(,, τ, t) =, c(p,, τ, t) > p when p >. Poof: Appendix A. Assumption 3 is a high level assumption. In geneal, whethe o not it holds will depend on the foms of U( ), α(, τ, t), h(y, τ, t), and δ(τ). Some intuition can be obtained in the the case whee the bidde is isk neutal, i.e. U(x) = x. Then we have M(v,, τ, t) = δ(τ) α(, τ, t)(v ) + M v (v,, τ, t) = δ(τ) α(, τ, t) + v v (v y)h(y, τ, t)dy, h(y, τ, t)dy < δ(τ) < 1, t, τ, v, since α(, τ, t) + v h(y, τ, t)dy < 1, δ(0) = 1, and δ (τ) < 0. Hence unde isk neutality, equilibia always involve cutoff stategies, egadless of the foms of α(, τ, t), h(y, τ, t), and δ(τ). The intuition is faily clea in the isk neutal case. When a bidde s valuation v inceases fom v to v + 1, U A (v, p) inceases by 1. On the othe hand, U R (v,, τ, t) inceases by less than 1 because of discounting, and because some of the utility gains fom the valuation incease ae lost to competing biddes with valuations between v and v + 1. Since the utility fom accepting the BP option inceases in v faste than the utility fom ejecting the BP option, optimization implies a cutoff ule whee biddes with high valuations accept the BP, and biddes with low valuations eject the BP. It is possible to obtain moe pimitive conditions ensuing an equilibium in cutoff stategies. Fo example, Appendix B shows that U (x) 0 is a sufficient condition fo Assumption 3 to hold fo any pimitives {λ( ), F V ( ), δ( )} satisfying ou conditions. Howeve, given a paticula {λ( ), F V ( ), δ( )}, thee will geneally be utility functions that do not satisfy U (x) 0 but do satisfy Assumption 3. Regading the popeties of c(p,, τ, t), it is intuitive that when the BP p inceases, the cutoff inceases, making a bidde less likely to accept the BP. When the eseve pice inceases, the cutoff deceases because the expected utility fom ejecting the BP deceases. When τ inceases, the expected utility fom ejecting the BP deceases, and the cutoff deceases. Thee ae two 9

10 easons fo this. Fist, inceasing τ inceases the expected numbe of competitos enteing in the bidding phase, loweing the expected utility fom ejecting the buy pice. Second, the expected utility fom ejecting the BP deceases as τ inceases due to impatience. Popety 2 simply states that when the BP exactly equals the eseve pice, all enteing biddes with v > will accept the BP. As p inceases above, c(p,, τ, t) also inceases, and is stictly above p. Lastly, note that c t (p,, τ, t) may be positive o negative (o 0), depending on how the Poisson ate λ(t) vaies acoss t. 2.2 Invese cutoff function p(c,, τ, t) Given that the cutoff function c(p,, τ, t) is stictly inceasing in p, we can invet it to obtain an invese cutoff function p(c,, τ, t). The invese cutoff function tells us, fo a given, τ, and t, what the BP would have to be fo a bidde with valuation c to be indiffeent between accepting and ejecting the BP. Following equation (1), the invese cutoff function solves U(c p(c,, τ, t)) = δ(τ) α(, τ, t)u(c ) + U(c y)h(y, τ, t)dy. (2) Unlike the cutoff function, we can explicitly solve out fo the invese cutoff function as a function of model pimitives, i.e. p(c,, τ, t) = c U 1 δ(τ) α(, τ, t)u(c ) + U(c y)h(y, τ, t)dy. (3) The invese cutoff function and this altenative epesentation of the indiffeence condition will be useful in the identification aguments below. The following popeties will also be helpful: Poposition 4 The invese cutoff function p(c,, τ, t) satisfies the following popeties: 1. 0 < p c (c,, τ, t) < 1, p (c,, τ, t) > 0, p τ (c,, τ, t) 0; 2. p(c,, τ, t) c, p(c,, τ, t) = c iff c = ; 3. p cc (c,, τ, t), p (c,, τ, t), and p c (c,, τ, t) exist and ae bounded away fom and ; 4. p c (z, z, τ, t) = 1 δ(τ)α(z, τ, t), p (z, z, τ, t) = δ(τ)α(z, τ, t), p cc (z, z, τ, t) = U (0) δ(τ)α(z, τ, t) (1 δ(τ)α(z, τ, t)) δ(τ)h(z, τ, t), p (z, z, τ, t) = U (0) δ(τ)α(z, τ, t) (1 δ(τ)α(z, τ, t)) + δ(τ)α (z, τ, t), p c (z, z, τ, t) = U (0)δ(τ)α(z, τ, t) (1 δ(τ)α(z, τ, t)). 10

11 Poof: Appendix A. Popety 4 in this poposition concens the behavio of the invese cutoff function when the buy pice (and thus the cutoff) equals the eseve pice (in which case all aiving biddes with v > accept the BP). Since we assume that p (and thus c ), the deivatives when c = should be intepeted as one-sided deivatives. 3 Identification We now conside identification of the stuctual demand paametes {F V ( ), λ( ), U( ), δ( )} of this model. Heuistically, we suppose we have many independent obsevations of auctions with the same {F V ( ), λ( ), U( ), δ( )}, and exogenous vaiation in the eseve pice (), buy pice (p), and bidding phase length (τ). Fomally, we define andom vaiables R, P, Υ, whose ealizations ae, p, τ. Let F,p,τ denote thei joint distibution. Conditional on R =, P = p, Υ = τ, we have a distibution fo the auction outcomes detemined by the (fixed) stuctual demand paametes, and the auction mechanism and equilibium solution descibed in Section 2. Given knowledge of the joint distibution of R, P, Υ and the auction outcomes, we want to ecove the stuctual demand paametes. The assumption that vaiation in the eseve pice, buy pice, and length is exogenous may be stong in some situations. Even if we view the identification analysis as conditional on auctionlevel covaiates, the vaiation in, p, and τ could aise fom unobseved (to the econometician) diffeences acoss auctions that selles take into account when choosing the auction featues (Kasnokutskaya (2004), Aske (2008), Robets (2009), Descaolis (2009)). It may be possible to elax this exogeneity assumption using instumental vaiables techniques, but we do not pusue this in the cuent pape. Howeve, the assumption may be cedible in some makets whee one believes the majoity of vaiation in auction setup is due to selle chaacteistics athe than unobseved demand factos, o in expeiments with tue andomized vaiation (in the lab, field, o elsewhee). Ou single-unit theoetical model in Section 2 also embodies an assumption that thee ae no othe auctions, eithe simultaneously, o in the futue. This is often assumed in the auction liteatue (with some notable exceptions, e.g. Pesendofe and Jofe-Benet (2003), Zeithamme (2006, 2007, 2009), Nekipelov (2008), Zeithamme and Adams (2009), and Backus and Lewis (2009)), but it may be questionable fo some poduct categoies on ebay and simila makets. 3.1 Obsevational Model Now we specify the auction outcomes that ae obseved. Let T 1 be the time of the fist action (eithe accepting o ejecting the BP) taken by any bidde in the auction. Let B = 0 indicate that the fist acting bidde ejected the BP, and let B = 1 indicate that the fist acting bidde 11

12 accepted the BP. The paametes of the model {F V ( ), λ( ), U( ), δ( )} detemine a joint distibution fo (T 1, B) given (P = p, R =, Υ = τ). Let F 1 ( p,, τ) denote the conditional distibution of T 1 given (P = p, R =, Υ = τ), and let P(B = 1 p,, τ, t 1 ) denote the conditional pobability of the BP option being accepted given (P = p, R =, Υ = τ, T 1 = t 1, ). Ou basic identification esults will only equie that the outcomes T 1 and B ae obseved (along with the exogenous vaiables (P = p, R =, Υ = τ)). This will be enough to identify {F V ( ), λ( ), U( ), δ( )}, using the implications of Popositions 1 and 3. In pinciple, we might obseve othe outcome vaiables, fo example the final pice in the auction, o the sealed bids placed by paticipants, o the poxy bids in ebay auctions. 6 these othe outcome vaiables. In Sections 4 and 5 we examine the identifying powe of some of To deive the simplest vesion of ou identification esults, we make the following assumption on the suppot of (R, P, Υ): Assumption 4 The maginal distibution of R has suppot [0, ) and the conditional distibution of P given R = has suppot [, ). The conditional distibution of Υ given (R =, P = p) has suppot [0, ). This suppot condition is elaxed in Section Identification of λ( ) and F V ( ) We begin by examining identification of the aival ate and valuation distibution. Ou aguments fo identification of these two objects ae simila to Canals-Ceda and Peacy (2008), who conside identification in ebay auctions without buy pices (and without impatience o isk avesion) Recall that potential biddes aive accoding to a Poisson pocess with aival ate λ(t), and by Poposition 1, if no othe action has yet been taken, the aiving bidde takes an action if he valuation V. Hence the time of the fist obseved action T 1 in an auction, given (P = p, R =, Υ = τ), has conditional hazad ate θ(t 1 p,, τ) = λ(t 1 )(1 F V ()), (4) To sepaately identify λ( ) and F V ( ), note that when = 0, F V () = 0, so λ(t 1 ) = θ(t 1 p, 0, τ). Since the Poisson intensity is bounded, T 1 P, R = 0 has full suppot [0, ), so conditioning on R = 0 identifies λ( ) on [0, ). Given that λ( ) has been identified, we can identify F V ( ) by using equation (4) fo values of othe than 0. This identifies F V ( ) on [0, ). 6 In some cases, these othe vaiables may be had to intepet, e.g. ebay s poxy bids. 12

13 Since λ( ) and F V ( ) ae identified, it follows that α(, τ, t 1 ) and h(y, τ, t 1 ) ae identified ove thei full suppots, since we can fom γ = t 1 +τ t 1 λ(s)ds fo any t 1 and then diectly constuct α(, τ, t 1 ) = exp(γf V () γ), h(y, τ, t 1 ) = exp(γf V (y) γ)γf V (y). 3.3 Identification of c(p,, τ, t 1 ) and p(c,, τ, t 1 ) To identify the cutoff function and its invese, we use the obseved distibution of B given (P = p, R =, Υ = τ, T 1 = t 1 ). Fo a bidde who takes an action at time t 1, the distibution of he valuation V is F V tuncated below at. By Poposition 3, this bidde will accept the buy pice if V c(p,, τ, t 1 ). Theefoe, P(B = 1 p,, τ, t 1 ) = 1 F V (c(p,, τ, t 1 )). 1 F V () Given knowledge of F V and the conditional pobability of B = 1, we can invet to obtain c(p,, τ, t 1 ) = F 1 V (1 (1 F V ()) P(B = 1 p,, τ, t 1 )). This identifies the cutoff function on the joint suppot of (P, R, Υ, T 1 ). Having identified c(p,, τ, t 1 ), we can then invet it to obtain the invese cutoff function p(c,, τ, t 1 ) and the fist and second deivatives of p(c,, τ, t 1 ) with espect to c and. 3.4 Identification of U( ) We now conside identification of the utility function. By definition, the cutoff function c(p,, τ, t 1 ) gives the valuation of the bidde who is indiffeent between accepting o ejecting the BP given (p,, τ, t 1 ). Similaly, the invese cutoff function p(c,, τ, t) gives the BP p that would make a bidde with valuation c indiffeent between accepting o ejecting the BP. As noted in Section 2, this indiffeence condition can be witten as ( U(c p(c,, τ, t 1 )) = δ(τ) α(, τ, t 1 )U(c ) + ) U(c y)h(y, τ, t 1 )dy. (5) This integal equation will hold fo all t 1 [0, ), [p, ), c [, ), and τ (0, ). To identify the utility function U( ), we need to show that given knowledge of α(, τ, t 1 ), h(y, τ, t 1 ) and p(c,, τ, t 1 ), thee is a unique utility function U ( ) that satisfies this integal equation. Equation (5) is an integal equation with a vanishing delay tem. 7 In geneal, such equations do 7 The delay tem is p(c,, τ, t) in U(c p(c,, τ, t)). The delay is called vanishing because (c p(c,, τ, t)) 0 as c 0. 13

14 not have simple solutions. Howeve, the delay tem can be eliminated by diffeentiating the integal equation with espect to c: ( U (c p(c,, τ, t 1 )) (1 p c (c,, τ, t 1 )) = δ(τ) α(, τ, t 1 )U (c ) + and with espect to : ) U (c y)h(y, τ, t 1 )dy ; ( ( )) α(, τ, U (c p(c,, τ, t 1 )) ( p (c,, τ, t 1 )) = δ(τ) α(, τ, t 1 )U t1 ) (c ) + U(c ) h(, τ, t 1 ) ; (7) U (c p(c,, τ, t 1 )) ( p (c,, τ, t 1 )) = δ(τ)α(, τ, t 1 )U (c ); (8) whee the second line follows because α(,τ,t 1) = h(, τ, t 1 ). Unde ou assumptions, each side of (8) is bounded away fom 0. Dividing (6) by (8) cancels the delay tem (and eliminates the impatience tem), and diffeentiating the esulting equation with espect to esults in an odinay fist ode linea diffeential equation in U ( ), i.e. U (c ) = (Φ (c,, τ, t 1 ) + h(, τ, t 1 )) U (c ), (9) Φ(c,, τ, t 1 ) (6) whee [ ] (1 pc (c,, τ, t 1 )) Φ(c,, τ, t 1 ) = α(, τ, t 1 ) 1. p (c,, τ, t 1 ) Since h(, τ, t 1 ) and all the components of Φ(c,, τ, t 1 ) have aleady been shown to be identified, we only need to conside whethe thee is a unique solution U ( ) to this diffeential equation (with U(0) = 0 and U (0) = 1). Poposition 5 Unde Assumptions 1, 3, and 4, thee is a unique U ( ) on suppot [0, ) satisfying (9). Hence, U ( ) is identified on suppot [0, ). Poof: Appendix A. The intuition hee is that fist ode linea diffeential equations like (9) typically have a unique solution given an initial condition (which in ou case is U (0) = 1). Note that (9) holds fo any values of (c(p,, τ, t 1 ),, τ, t 1 ). So, fo example, one can fix (, τ, t 1 ) and use vaiation in p acoss its suppot (i.e. in c(p,, τ, t 1 )) to tace out U ( ). Since this can be done at any (, τ, t 1 ), this will geneate oveidentifying estictions, which will be discussed in the next section. Equation (9) can be ewitten as U (c ) U (c ) = (Φ (c,, τ, t 1 ) + h(, τ, t 1 )). (10) Φ(c,, τ, t 1 ) 14

15 This implies that to identify the Aow-Patt measue of isk avesion at a cetain point, one only needs to compute the values of Φ(c,, τ, t 1 ), Φ (c,, τ, t 1 ), and h(, τ, t 1 ) at that point. On a elated note, we investigate how sensitive ou identification esults ae to the suppot condition (Assumption 4) in Section Identification of δ( ) Lastly, conside identification of the impatience function. Manipulating the indiffeence condition (5) gives us U(c p(c,, τ, t 1 )) δ(τ) = ( ). (11) α(, τ, t 1 )U(c ) + U(c y)h(y, τ, t 1 )dy Since all of the tems on the ight-hand-side have aleady been shown to be identified, it is clea that δ( ) is identified. Since τ vaies ove suppot (0, ) and we have nomalized δ(0) = 1, δ( ) is identified on [0, ). 4 Extensions 4.1 Relaxing Suppot Conditions Assumption 4 equies the exogenous andom vaiables (R, P, Υ) to have lage suppots, which may be unealistic in many applications. We now examine how estictions on the suppots affect ou identification esults. Fist conside a situation whee all the auctions in the data set have a fixed bidding phase length τ 0. Assumption 5 The maginal distibution of R has suppot [0, ) and the conditional distibution of P given R = has suppot [, ). The conditional distibution of Υ given (R, P ) has P (Υ = τ 0 R, P ) = 1 almost suely. Poposition 6 Unde Assumptions 1, 3, and 5, {F V ( ), λ( ), U( )} ae identified ove thei full suppots, and δ(τ) is identified at τ 0. Intuitively, if the bidding phase has fixed length τ 0, we can only identify δ( ) at that point. Howeve, the othe stuctual functions ae identified ove thei entie suppots by ou pevious aguments. In pactice, it may also be difficult to estimate λ(t) fo lage t, because this would equie obseving many auctions that last until time t without an action being taken. To captue this situation, we suppose that obsevations on auctions ae tuncated at some time T : Assumption 6 Thee is some T > τ 0 such that we only obseve (T 1, B) when T 1 < T. 15

16 In this case, even though we can only identify the Poisson pocess pio to time T, we can still identify F V ( ) and U( ). Specifically, the following esult follows fom the aguments in Section 3. Poposition 7 Unde Assumptions 1, 3, 5, and 6, {F V ( ), U( )} ae identified ove thei full suppots, λ( ) is identified on suppot [0, T ), and δ(τ) is identified at τ 0. Next, we conside esticting the suppot of R to the bounded set [, ]. Assumption 7 The maginal distibution of R has suppot [, ] and the conditional distibution of P given R = has suppot [, ). The conditional distibution of Υ given (R, P ) has P (Υ = τ 0 R, P ) = 1 almost suely. Thee is some T > τ 0 such that we only obseve (T 1, B) when T 1 < T. Poposition 8 Unde Assumptions 1, 3, and 7, λ(t)(1 F V ()) is identified on [, ] and t [0, T ), δ(τ) is identified at τ 0, and U( )is identified on [0, ]. Poof: Appendix A. In this case, since the eseve pice neve goes below, we cannot distinguish between a nonaival and an aival of a bidde with valuation below. As a consequence, λ(t) and F V () ae not sepaately identified. Howeve, ove limited suppots, we can identify the composite function λ(t)(1 F V ()), and though this α(, τ, t 1 ), h(y, τ, t 1 ) and c(p,, τ, t 1 ). A slight modification of ou oiginal identification agument leads to the final esult. The suppot on which U ( ) is identified depends on the ange of the eseve pice in the suppot of the data. Lastly, we futhe estict the suppot of the buy pice P. Assumption 8 The maginal distibution of R has suppot [, ] and the conditional distibution of P given R = has suppot containing [p 0 ɛ, p 0 +ɛ] fo some ɛ > 0. The conditional distibution of Υ given (R, P ) has P (Υ = τ 0 R, P ) = 1 almost suely. Thee is some T > τ 0 such that we only obseve (T 1, B) when T 1 < T. Thee exists (, ) and t < T τ 0 such that c(p 0,, τ 0, t ) (, ). Poposition 9 Unde Assumptions 1, 3, and 8, λ(t)(1 F V ()) is identified on [, ], δ(τ) is identified at τ 0, and U ( ) U ( ) is identified at the point c(p 0,, τ 0, t ). Poof: Appendix A. The last condition of Assumption 8 equies the obseved buy pice p 0 being low enough such that the cutoff at this buy pice is within the ange [, ] (fo some and t ). This is needed to identify 16

17 the cutoff function. Howeve, povided this holds, we only need a small amount of vaiation in the buy pice to identify the Aow-Patt measue of isk avesion at a paticula c(p 0,, τ 0, t ). If thee is a set of points (, t ) such that (, ), t < T τ 0, and c(p 0,, τ 0, t ) (, ), then the Aow-Patt measue of isk avesion will be identified at all the coesponding values of c(p 0,, τ 0, t ). In summay, we can elax ou oiginal suppot conditions in vaious ways and still obtain local identification of the stuctual objects. Howeve, even Assumption 8 makes a significant joint suppot condition on and p. Intuitively, to identify U ( ) and δ ( ) locally, we need vaiation in the eseve pice and we need that thee ae buy pices low enough that the equilibium cutoff is sometimes in this ange. 4.2 Testable Restictions Ou model has a numbe of testable estictions on the obseved data (o functions of the obseved data). These follow fom the discussion in Sections 2 and 3 and include: 1. The conditional hazad θ(t 1 p,, τ) does not depend on p o τ; 2. The atio θ(t 1 p,,τ) θ(t 1 p,0,τ) does not depend on p,τ, o t 1; 3. p(c,, τ, t 1 ) c, 0 < p c (c,, τ, t 1 ) < 1, p (c,, τ, t 1 ) > 0, p τ (c,, τ, t 1 ) > 0, p c (z, z, τ, t 1 ) = 1 α(z, τ, t 1 ), p (z, z, τ, t 1 ) = α(z, τ, t 1 ); 4. The coefficient in the diffeential equation (9), i.e. (Φ (c,, τ, t 1 ) + h(, τ, t 1 )), Φ(c,, τ, t 1 ) does not depend on τ and t 1. The coefficient only depends on c and though the diffeence c. 8 These estictions could be tested, and also imply that moe flexible vesions of the model can be identified. Fo example, estictions 1 and 2 imply that one can identify a model whee the distibution of a bidde s valuation depends on thei time of aival, i.e. F V (v, t). Restiction 4 implies we could identify an extended model whee a bidde s utility function depends on thei time of aival, i.e. δ(τ)u (v p, t). 9 8 The fact that (9) implies Ideally, we would like to find necessay and sufficient conditions U (c ) U (c ) = (Φ(c,, τ, t1) + h(, τ, t1)) Φ(c,, τ, t 1) geneates these estictions. 9 One could potentially investigate whethe even moe geneal models ae identified o patially identified, e.g. the utility function U(τ, v p, t) o U (τ, v, p, t), o models whee biddes ae heteogeneous in thei isk attitudes o thei impatience athe than in thei valuations. Even moe challenging would be models whee bidde heteogeneity cannot be summaized by a scala. 17

18 on θ(t 1 p,, τ) and P(B = 1 p,, τ, t 1 ) (which can be estimated diectly fom data) fo these to be ationalized by ou model {F V ( ), λ( ), U( ), δ( )} (see, e.g. Ayal, Peigne, and Vuong (2009)). Howeve, this appeas to be quite complicated in ou context. Fo example, the invese cutoff function (3) implies that the elationship between p(c,, τ, t 1 ) (and thus P(B = 1 p,, τ, t 1 )) at two values of t 1 depends in a complicated way on the aival pocess between those two points in time (as an integand though h(, τ, t 1 ) and then though the invese utility function). 4.3 Additional Data on Final Pices Ou basic identification agument only uses data on T 1, the time of the fist obseved aival, and B, the indicato fo whethe the BP option was accepted o ejected. This data identifies the aival ate λ(t), the valuation distibution F V (v), and the invese cutoff equation p(c,, τ, t 1 ). Using λ(t) and F V (v), we can identify the functions α(, τ, t 1 ) and h(y, τ, t 1 ) in ou integal equation (5). Given knowlege of α(, τ, t 1 ), h(y, τ, t 1 ), and p(c,, τ, t 1 ), we then showed identification of the utility components U ( ) and δ ( ). This subsection consides an altenative appoach to identifying α(, τ, t 1 ) and h(y, τ, t 1 ). Recall that α(, τ, t 1 ) is the pobability that, given ejection of the BP at t 1, no othe bidde with valuation geate than aives duing the bidding phase. h(y, τ, t 1 ) is the density of the maximum valuation of all futue biddes aiving duing the bidding phase (o the eseve pice). 10 Instead of fist ecoveing λ(t) and F V (v), we can use the final outcomes fo auctions that ente the bidding phase to lean α(, τ, t 1 ) and h(y, τ, t 1 ). This appoach will be paticulaly useful fo ou extension to ebay Buy-It-Now auctions consideed below. 11 We make the following assumption about winning pices. Assumption 9 If an auction entes the bidding phase, the final pice in the auction is the 2nd highest valuation of all biddes who aived at the auction. Note that Assumption 9 will hold in equilibium if the bidding phase consists of eithe a 2nd picesealed bid auction, a button auction, o an ebay style poxy-bidding auction (up to bid incements). Conside an auction with setup (p,, τ). Suppose that at T 1 = t 1, an aiving bidde ejects the BP option (B = 0). This bidde has value V distibuted accoding to F V tuncated between and c(p,, τ, t 1 ). Let Ỹ equal the highest valuation among biddes aiving afte time t 1, o equal R if no futhe biddes aive to the auction with valuations geate than. Unde ou assumptions, Ỹ is conditionally independent of V, i.e. Ỹ V P = p, R =, Υ = τ, T 1 = t 1, B = In equilbium, no bidde who aived pio to t 1 has valuation >. 11 The appoach detailed in this section is elated to a lage liteatue on estimation methods fo dynamic models initiated by Hotz and Mille (1993). 18

19 Suppose we obseve the andom vaiable W, an indicato that the bidde who ejected the BP option ended up winning the auction. Note that W = 1(Ỹ < V ), since the bidde who ejected the BP only wins if he valuation is highe than the biddes enteing duing the bidding phase. Suppose that conditional on W = 1, we obseve the final pice in the auction, Z. Unde Assumption 9, Z = Ỹ if W = 1. In othe wods, if the bidde who ejected the BP option wins the auction, then the final pice Z is equal to the highest valuation of biddes aiving afte time t 1 (o if thee ae no such biddes with valuations geate than ). Since W is obseved and Z is obseved (given W = 1), we can identify P (W = 1 p,, τ, t 1, B = 0), and p Z (z W = 1, p,, τ, t 1, B = 0). The fist tem is the pobability that the BP ejecto wins the auction. the distibution of the final pice given that the BP ejecto wins the auction. The second tem is (In geneal the conditional distibution of Z will have point mass at, so we intepet conditional densities as being with espect to the sum of Lebesgue measue and counting measue at.) By Bayes Theoem, we can wite p Z (z W = 1,, p, τ, t 1, B = 0) = p y(z, p, τ, t 1, B = 0)P (W = 1 z,, p, τ, t 1, B = 0) P (W = 1, p, τ, t 1, B = 0) = p y(z, p, τ, t 1, B = 0) (F V (c(p,, τ, t 1 )) F V (z)), P (W = 1, p, τ, t 1, B = 0) whee p y indicates the conditional density of Ỹ. Assuming we have aleady identified F V (v) as in Section 3, we can use this equation to ecove p y (z p,, τ, t 1, B = 0). h(y, τ, t 1 ), since α(, τ, t 1 ) = P (Ỹ =, p, τ, t 1, B = 0), This identifies α(, τ, t 1 ) and and h(y, τ, t 1 ) = p y (y, p, τ, t 1, B = 0) fo < y < c(p,, τ, t 1 ). Assuming we have also identified c(p,, τ, t 1 ) 12, we can now use the integal equation (5) to identify U ( ) and δ ( ) (note that Equation (5) only depends on h(y, τ, t 1 ) on the suppot c(p,, τ, t 1 ) > y > ). This appoach allows us to identify U ( ) and δ ( ) without equiing that λ ( ) be sepaately identified at late points in the auction. Note that this additional data povides additional testable estictions of the model. Fo example, p y (y, p, τ, t 1, B = 0) should not depend on p. 12 Note that one does not need to know λ ( ) at late points in the auction to identify F V ( ) and c(p,, τ, t 1) (fo low t 1), using the aguments of Section 3. 19

20 This agument only uses the final pice conditional on the BP ejecto winning the auction. In pinciple, thee is moe infomation that could be used fo identifying stuctual objects and geneating testable estictions of ou model. Fo example, we could use the final pice egadless of who wins the auction. If we also obseve all the bids in the bidding phase, and these bids epesent biddes valuations, then we could use this infomation as well. 13 If we also obseve clicksteam data measuing when a use fist visited a paticula auction, this could povide an altenative souce of identification fo aival ates. 5 Applications 5.1 ebay s Buy-it-Now Auctions ebay s popula Buy-it-Now auctions featue a buy pice option that disappeas as soon as any bidde places a bid, which ou oiginal model captues. Howeve, in ebay s BP auction fomat, thee is a fixed length fo the oveall auction. Since ebay s auctions end at some fixed time T, the bidding phase has length T t 1, not a fixed length τ as we assumed in ou model. We call ou BP auction model a Fixed τ BP auction, wheeas ebay s Buy-it-Now auction is a Fixed T BP auction. In a Fixed T auction, the envionment is defined by (p,, T ). The cutoff function fo a bidde aiving at t 1 is then c(p,, T, t 1 ), depending on p,, and the oveall length of the auction T. Thee ae a numbe of papes that have focused on empiically studying ebay (o elated) Buy-it-Now auctions, including Wan, Teo, and Zhu (2003), Chan, Kadiyali, and Pak (2007), and Andeson, Fiedman, Milam, and Singh (2008). Ackebeg, Hiano, and Shahia (2006) estimate a paametic stuctual model elated to the non-paametic model we descibe hee. Geneally speaking, identification of paametes in a Fixed T auction is simila to that in a Fixed τ auction. Vaiation in eseve pices acoss auctions can tace out aival ates and valuations, vaiation in p identifies the cutoff function c(p,, T, t 1 ), and an integal equation simila to (5), i.e. U(c p(c,, T, t 1 )) = δ(t t 1 ) α(, T, t 1 )U(c ) + identifies U( ) and δ( ). U(c y)h(y, T, t 1 )dy, (12) Howeve, the Fixed T BP auction has a complication that does not aise in ou oiginal model. Conside a potential bidde aiving at some t, with a valuation geate than, but less than c(p,, T, t). In ou oiginal model, such a bidde has an incentive to immediately eject the BP. This is because immediately ejecting ends the auction soone, and minimizes the expected amount of competition that bidde faces in the bidding phase. In contast, in a Fixed T model, this bidde 13 Depending on the context, bids may only povide bounds on valuations (Haile and Tame (2003), Zeithamme and Adams (2009)). 20

21 does not have a stict incentive to immediately eject the BP, since immediately ejecting does not end the auction soone o limit competition. 14 This can lead to multiple equilibia, because biddes ae indiffeent between ejecting the BP immediately, o waiting. 15 Then we may not be able to identify F V (), since we ae not cetain that obseved actions ae being taken by biddes who have just aived. 16 We can esolve this poblem by eithe modifying the model, o using a solution concept that ensues that biddes who eject the BP act immediately. Gallien and Gupta (2007) discuss esticting attention to tembling hand pefect BNE, whee the tembles involve a bidde accidentally accepting the BP. This esults in an equilibium whee biddes who want to eject the BP do so immediately. The same is tue if one adds a small monitoing cost to the model that is incued when one waits to eject the BP. 17 In both cases, ou identification aguments in Sections 2 and 3 can be applied to Fixed T auctions. Because we need to ensue that cetain subsets of the biddes act immediately upon enteing the auction, 18 we ae also elying heavily on the assumption that auctions ae isolated. If the same (o a simila) good wee potentially available in othe ebay auctions, then biddes might have an incentive to wait befoe taking an action. This suggests that it might be valuable to investigate how ou identification aguments extend to moe dynamic settings (e.g. Jofe-Bonet and Pesendofe (2003), Zeithamme (2006), Backus and Lewis (2009)). 19 While ou extension to ebay s Buy-it-Now auctions equie some futhe stong assumptions about behavio of biddes, ou appoach to identification has a numbe of attactive featues. Fist, ecall that in Fixed τ auctions, if τ is fixed in the data at τ 0, one can identify δ( ) only at τ 0. In a Fixed T auction, even if T is fixed in the data at T 0, one can identify the impatience function δ( ) at moe than one point (due to vaiation in t 1 - see Equation (12)). Second, note that ou identification aguments do not equie data on ebay poxy bids. As noted by Zeithamme and Adams (2009) among othes, these bids may be had to intepet on ebay. We investigate many of these and 14 Moeove, thee is no eason to act immediately to pevent anothe bidde fom enteing and accepting the BP. This is because in a model whee biddes ae only heteogeneous in thei valuations, this othe bidde would always win the auction phase. 15 In the Fixed T envionment, biddes with valuations above the cutoff c(p,, T, t 1) do have an incentive to accept the BP immediately, as they do not want to lose the item to anothe aiving bidde. 16 This point illustates an inteesting theoetical advantage of Fixed τ BP auctions vs Fixed T BP auctions. Specifically, identification is easie in Fixed τ auctions because of the stonge incentives fo biddes to act immediately. It would be inteesting to compae expected evenue acoss the two types of BP auctions, though that seems beyond the scope of the cuent pape. 17 Anothe petubation is anothe model in Gallien and Gupta (2007), whee thee is assumed to be a point mass of despeate biddes who ae vey impatient and accept the BP immediately if they aive. This also ceates incentives fo nomal bidde to eject the BP immediately. Howeve, with this model, one would want to explicitly conside identification of the point mass of despeate biddes, and check whethe this affects identification of the othe model components. The same issue aises in a petubation whee one adds a small utility benefit of paticipating in the bidding phase. 18 We could extend the agument to allow fo an exogenous andom delay befoe acting. 19 In the ebay context it may also be had to identify the aival pocess late in the auction, because it equies data on auctions in which no action is taken until close to the end of the auction. Hence the altenative appoach descibed in Section 4.3 to identify α(, T, t 1) and h(y, T, t 1) using final pices might be paticulaly useful hee. 21