A liation and Entry in First-Price Auctions with Heterogeneous Bidders

Transcription

1 A laton and Entry n Frst-Prce Auctons wth Heterogeneous Bdders Tong L y and Bngyu Zhang z Frst Verson: November 2008 Ths Verson: October 2009 Incomplete. Please do not quote Abstract In ths paper we study the tmber sales auctons n Oregon. We propsoe an entry and bddng model wthn the a lated prvate value (APV) framework and wth heterogeneous bdders, and establsh exstence of the entry equlbrum and exstence and unqueness of the bddng equlbrum wth the jont dstrbuton of prvate values belongng to the class of Archmedean copulas. We then estmate the resultng structural model, and nd that the haulng dstance plays a sgn cant role n bdders entry and bddng decsons. We quantfy the extent to whch the potental bdders prvate values and entry costs are a lated. The structural estmates are then used to conduct counterfactual analyses to address polcy related ssues. In partcular, we quantfy the e ects of reserve prce, a laton, and merger on the end aucton outcomes. We thank Luke Froeb, Ken Hendrcks, Al Hortacsu, Frank Wolak, and partcpants at the Second Conference on Auctons and Procurements n Penn State for comments and dscussons. L gratefully acknowledges support from the Natonal Scence Foundaton (SES ). y Department of Economcs, Vanderblt Unversty, VU Staton B #3589, Nashvlle, TN Phone: (65) , Fax: (65) , Emal: tong.l@vanderblt.edu z Department of Economcs, Natonal Unversty of Sngapore, Arts Lnk, Sngapore Phone: , Fax: , Emal: bngyu.zhang@nus.edu.sg

2 Introducton Auctons have long been used as a means for prce determnaton under a compettve settng and an ncomplete nformaton envronment. Aucton theory developed wthn the game-theoretc framework wth ncomplete nformaton (Harsany (967/968)) not only helps us understand how auctons work, but also o ers nsght n analyzng many other economc problems. A celebrated result n aucton theory s Vckrey s (96) revenue equvalence theorem, whch postulates that all the four aucton formats ( rst-prce sealed-bd, second-prce sealed bd, Englsh, and Dutch auctons) generate the same average revenue for the seller wth symmetrc, ndependent, and rskneutral bdders. The revenue equvalence theorem s a powerful result that o ers nsght nto how aucton mechansms work, and also rases mportant questons as to how ths powerful result can be a ected when the standard assumptons are relaxed. A large part of the aucton theory has focused on answerng these questons. Mlgrom and Weber (982) gve revenue rankng wth symmetrc and a lated bdders n whch the Englsh aucton generates hghest revenue among the four formats and the second-prce aucton ranks next; they also establsh that wth symmetrc, a lated, and rsk-averse bdders who have constant absolute rsk averson, the Englsh aucton can generate at least as hgh revenue as the second-prce aucton. Myerson (98) derves the optmal auctons wth asymmetrc bdders, and Maskn and Rley (984) consder the case wth rsk-averse bdders. Levn and Smth (994) extend the revenue equvalence and rankng results from Vckrey (96) and Mlgrom and Weber (982) to the case wth symmetrc bdders (ndependent or a lated) usng mxed entry strateges. Usng tmber sale auctons organzed by the Oregon Department of Forestry (ODF), ths paper attempts to address a set of questons that nclude wth heterogeneous bdders and when entry s taken nto account, how the seller s revenue could change wth the extent to whch the bdders prvate values are a lated, and whether the reserve prce currently set by the ODF s optmal wth respect to maxmzng the seller s revenue/pro t. Moreover, merger and bdder coalton have been an mportant ssue to economsts nterested n competton polcy, yet no emprcal work has studed ths ssue takng nto account endogenous partcpaton from potental bdders. That we consder heterogeneous bdders s motvated by the evdence from the prevous work studyng the tmber auctons n Oregon (e.g. Brannman and Froeb (2000) usng data consstng of oral auctons, and L and Zhang (2008) usng the same data used n ths paper comprsng rst-prce sealed-bd auctons), that haulng dstance plays an mportant role n bdders bddng (Brannman and Froeb (2000)) and entry (L and Zhang (2008)) decsons. Ths means that bdders are asymmetrc and heterogeneous. Furthermore, L and Zhang (2008) nd a small but strongly sgn cant level of a laton among potental bdders prvate nformaton (ether prvate values or entry costs). Lastly, recent emprcal work n auctons n general and n tmber auctons n partcular (e.g. Athey, Levn and Sera (2004), Bajar and Hortacsu (2003), Kransnokutskaya and Sem (2006), L and Zheng (2007, 2009)) has demonstrated that bdders partcpaton and entry decson s an ntegrated part of the decson makng process that has to be taken nto account when studyng auctons. In vew of these, n ths

3 paper we attempt to study the tmber auctons organzed by the ODF wthn a general framework n whch potental bdders are a lated and heterogeneous, and they make endogenous entry decsons before submttng bds. Aucton theory o ers lttle gudance n answerng these questons for auctons wth entry and asymmetrc potental bdders wth a lated prvate values. On the other hand, to gan nsght on these questons from an emprcal perspectve, one needs to observe two states of world, such as pre and post the change of the a laton level, or pre and post merger. Usually n aucton data, as s the case n our data, however, one cannot observe these two states of world. Therefore we adopt the structural approach n our emprcal analyss. We develop an entry and bddng model for asymmetrc bdders wthn the APV paradgm. Establshng exstence and unqueness of the entry and bddng equlbra wthn the APV model wth entry and wth heterogeneous bdders s a challengng problem. We extend the results by Lebrun (999, 2006) for the IPV case wth asymmetrc bdders and wthout entry to our case and establsh the exstence and unqueness of the bddng equlbrum and exstence of the entry equlbrum for a general class of jont (a lated) dstrbuton of prvate values. Ths can be vewed as a contrbuton of the paper to the lterature. Because of the general framework we adopt, the answers to the aforementoned questons of our nterest depend on the nteractons of a laton, entry, and asymmetry, as well as competton. As s well known, the optmal reserve prce n a symmetrc ndependent prvate value (IPV) model wthout entry does not depend on the number of potental bdders. Ths result can change f entry s ntroduced (see, e.g., Levn and Smth (994), Samuelson (985), L and Zheng (2007)), or f bdders have a lated prvate values (Levn and Smth (996), L, Perrgne, and Vuong (2003)). In our case, on the other hand, assessng the optmal reserve prce s complcated further by the APV framework wth entry and asymmetrc bdders. Therefore we can only address ths ssue through a counterfactual analyss usng the structural estmates. Furthermore, whle the e ect of the number of potental bdders on wnnng bds and seller s revenue s clear n an IPV model wth symmetrc bdders and wthout entry, t becomes less clear n a more general settng, such as the IPV model wth entry and symmetrc bdders (L and Zheng (2007, 2009)), and the APV model wthout entry (Pnkse and Tan (2005)). In partcular, L and Zheng (2007, 2009) show that n terms of the relatonshp between the number of potental bdders and the expected seller s revenue, n addton to the usual competton e ect, there s an opposte e ect due to the entry whch they term as the entry e ect. On the other hand, Pnkse and Tan (2005) postulate that n a condtonally ndependent prvate value model, a specal case of the APV paradgm, n addton to the competton e ect, there s an opposte e ect caused by a laton they term as the a laton e ect. Zhang (2008) shows that n the APV model wth entry and symmetrc bdders, these three e ects, namely, the competton e ect, the entry e ect, and the a laton e ect are at work. Whle we expect these three e ects to reman n the APV framework wth entry and asymmetrc bdders, t becomes challengng to pnpont them wth asymmetrc bdders. Snce the e ect of merger s closely related to how the seller s revenue changes wth the set of potental bdders,.e, 2

4 not only the number of potental bdders, but also the dentty of potental bdders when they are heterogeneous, and at the same tme, theory does not yeld good predctons, we rely on the structural analyss to gan nsght on ths ssue. We then develop a structural framework to estmate the entry and bddng model we propose. We use the estmated structural parameters to conduct counterfactual analyses of our nterest. We nd that for a representatve aucton, the optmal reserve prce should be much larger than the current one. In evaluatng the merger e ects we nd that (TO BE ADDED LATER) Asymmetry s an ndspensable element of the model gven the asymmetrc feature of the data. The analyss of the model, however, s complcated from both theoretcal and econometrc vewponts due to the ntroducton of asymmetry. Because of the complexty of the model, and n partcular, because that there s no closed form soluton for the bddng functon, we have to rely on some numercal approxmaton procedure. Moreover, whle the structural analyss of auctons wth asymmetrc bdders has focused on the case wth two types of bdders (Athey, Levn, and Sera (2004), Campo, Perrgne and Vuong (2003), and Kransnokutskaya and Sem (2006)), our model allows for all potental bdders to be d erent from each other, motvated by the fact that n our data, asymmetry s drven by the d erence among bdders haulng dstances. Ths paper makes contrbuton to the growng lterature of the structural analyss of aucton data snce Paarsch (992). Whle the structural approach has been extended to the APV paradgm by L, Perrgne and Vuong (2000, 2002), Campo, Perrgne and Vuong (2003), and L, Paarsch and Hubbard (2007), ths paper s the rst one n estmatng a structural model wthn the APV paradgm and takng nto account entry. On the other hand, whle the recent work has started to pay attenton to the problem of endogenous partcpaton and entry, all the work has focused on the IPV framework wth Bajar and Hortaçsu (2003) beng an excepton as they consder a common value (CV) model. In contrast, ths paper consders the entry problem wthn the APV paradgm, a more general framework. Our emprcal analyss of the tmber auctons and the resultng ndngs o er new nsght on tmber sale auctons and polcy related ssues. Whle most of the emprcal analyss of tmber sale auctons s based on the IPV model wthout entry (e.g. Paarsch (997), Baldwn, Marshall and Rchard (997), Hale (200), Hale and Tamer (2003), L and Perrgne (2003)) or the IPV model wth entry (Athey, Levn and Sera (2004), L and Zheng (2007)), ours s based on the APV model wth entry and heterogeneous bdders. As a result, our ndngs can be more robust, and also can be more useful for addressng the polcy-related ssues as our analyss takes nto account the a laton e ect, the entry e ect, and the asymmetry e ect. Moreover and probably more nterestngly, we study the merger e ect wthn the asymmetrc APV framework wth entry, and o er new nsght nto how merger as well as other ssues related to competton polcy can be a ected by complcatons arsng from a laton, entry, and asymmetry, and how they can be addressed wthn a un ed framework as adopted n ths paper. It s worth notng that to the best of our knowledge, Brannman and Froeb (2000), consderng oral tmber auctons wthn an IPV paradgm wthout entry, s the only paper assessng the merger e ect n auctons usng the structural approach. 3

5 Ths paper s organzed as follows. Secton 2 descrbes the data we analyze n the paper. In Secton 3 we propose the asymmetrc APV model wth entry. Secton 4 s devoted to the structural analyss of the data, and Secton 5 conducts a set of counterfactual analyses studyng the e ects of reserve prces, a laton levels, and mergers. Secton 6 concludes. 2 Data The data we study n ths paper are from the tmber auctons organzed by the ODF between January 2002 and June Before an aucton s advertsed, the ODF cruses the selected tract of tmber and obtans nformaton of the tract, such as the composton of the speces, the qualty grade of the tmber and so on. Based on the nformaton t obtans, the ODF sets ts apprased prce for the tract, whch serves also as the reserve prce. After the cruse, a detaled bd notce s usually released 4-6 weeks pror to the sale date, whch provdes nformaton about the aucton, ncludng the date and locaton of the sale, speces volume, qualty grade of the tmber, apprased prce as well as other related nformaton. Potental bdders acqure ther own nformaton or prvate values through d erent ways and decde whether and how much to bd. Bds are submtted n sealed envelopes that are opened at a bd openng sesson at the ODF dstrct o ce o erng the sale. The sale s awarded to the bdder wth the hghest bd. All the sales are therefore rst prce sealed bd scale auctons. The orgnal data contan 45 sales n total. Among them, some sales have more than one bd speces, whch are deleted from our sample because of the skewed bddng ssue dscussed n Athey and Levn (200). We focus on the sales n whch Douglas- r s the only bd speces and drop the sales wth other than Douglas- r as bd speces, because Douglas- r s a majorty speces. Consderng the tme that our estmaton program takes, we focus on the auctons wth at most 8 potental bdders. The resultng nal sample has 203 sales and 074 observed bds. For each sale, we drectly observe some sale-spec c varables ncludng the locaton and the regon of the sale, apprased prce, apprased volume measured n thousand board feet or MBF, length of the contract, and dameter at breast heght (DBH) as well. Notng that the bd speces s often a combnaton of a mxture of several grades of qualty, we use number, 2,, up to 8 to denote the letter-grades used by ODF so that the nal grade of a sale s the weghted average of grades wth volumes of grades as the weght. In addton to sale-spec c varables, as shown n Brannman and Froeb (2000) and L and Zhang (2008), respectvely, haulng dstance s an mportant bdder-spec c varable that a ects bdders bddng and entry decsons. However, haulng dstance s not observed drectly. We use the haulng dstance varable constructed n L and Zhang (2008) who transfer the locaton of a tract nto lattude and longtude through the Oregon Lattude and Longtude Locator 2 and nd the dstances between the tract and the mlls of rms by usng Google Map. The key nformaton related to endogenous entry s the denttes of potental bdders, whch 2 It s avalable at 4

6 are not observed. Unlke some procurement auctons, where nformaton on bdders who have requested bddng proposal s avalable and can be used as a proxy for potental bdders (L and Zheng (2009)), we do not have such nformaton n our case, as s usual for tmber sale auctons. Therefore we follow Athey, Levn, and Sera (2004) and L and Zheng (2007) to construct potental bdders. Spec cally, we rst dvde all sales n the orgnal data set nto 46 groups each of whch contans all sales held n the same dstrct n the same quarter of the same year. The potental bdders of a sale are then all bdders who submt at least one bd n the sales of the group that the sale belongs to. In other words, all auctons n the same group have the same set of potental bdders. Note that n constructng the potental bdders we use the orgnal data set ncludng all auctons removed from our nal sample. Summary statstcs of the data are gven n Table. Notably, the entry proporton, whch s calculated as the rato of the number of actual bdders and the number of potental bdders, s about 0.66 on average, meanng that whle there s strong evdence of entry pattern from the potental bdders, on average more than half of the potental bdders would partcpate n the aucton. 3 The Model In ths secton we propose a theoretcal two-stage model to characterze the tmber sales, extendng the models n Athey, Levn, and Sera (2004) and Krasnokutskaya and Sem (2006) wth two groups of bdders wthn an IPV paradgm to the APV paradgm that allows potental bdders to be d erent from each other. Spec cally, motvated by the ndng of Brannman and Froeb (2000) that the haulng dstance plays a sgn cant role n bdders bddng decson n oral tmber auctons n Oregon, and the ndng of L and Zhang (2008) usng the same data studed n ths paper that the haulng dstance s mportant n potental bdders entry decson and potental bdders are a lated through ther prvate nformaton (ether prvate values or entry costs), we consder a rst-prce sealed-bd aucton wthn the APV paradgm wth a publc reserve prce, endogenous entry, and asymmetrc bdders. In the model, a sngle object s auctoned o to N heterogenous and rsk-neutral potental bdders, who are a lated n ther prvate nformaton. Bdder has a prvate entry cost k ; ncludng the cost of obtanng prvate nformaton and bd preparaton, and does not obtan hs prvate value v untl he partcpates n the aucton. We allow both prvate values and entry costs to be a lated across bdders, that s V ; : : : ; V N and K ; : : : ; K N jontly follow a dstrbuton F (; : : : ; ) wth support [v = r; v] N, and a dstrbuton G (; : : : ; ) wth support k; k N, respectvely, where r s the publc reserve prce of the aucton. A laton s a termnology descrbng the postve dependence among random varables, whch was rst ntroduced nto the study of auctons by Mlgrom and Weber (982). It s equvalent to the concept called multvarate total postvty of order 2 (MTP 2 ) n the multvarate statstcs lterature. Followng Mlgrom and Weber (982), a laton has the followng formal de nton. De nton. Let z and z 0 be any two values of a vector of random varables Z R n wth a densty 5

7 f () : It s sad that all elements of Z are a lated f f (z _ z 0 ) f (z ^ z 0 ) f (z) f (z 0 ) ; where z _ z 0 denotes the component-wse maxmum of z and z 0 ; and z ^z 0 denotes the component-wse mnmum of z and z 0 : Intutvely, a laton means that large values for some of the components n Z make other components more lkely to be large than small. We also denote the margnal dstrbuton and densty of bdder s prvate value by F () and f () and margnal dstrbuton and densty of bdder s entry cost by G () and g (), respectvely, and assume that f () s contnuously d erentable and bounded away from zero on [v = r; v] : The subscrpt of dstrbuton functon mples that all potental bdders are of d erent types. Ths assumpton s motvated by the fact that heterogenety among bdders arses from d erent haulng dstances n our data. 3. Bddng Strategy Because the entry decson s based on the pre-entry expected pro t, whch depends on the bddng strategy of bdder, we rst descrbe the bddng strategy of bdder. We assume that bdder knows the number of the actual compettors n the bddng stage, 3 and thus bdder s bddng strategy s determned by the rst order condton of the followng maxmzaton problem, max (v b ) Pr (B j < b jv ; a ) ; where B j denotes the maxmum bd among other actual bdders and a 2 A = f(a ; : : : ; a N ) ja j = 0 or ; j = ; : : : N; j 6= g s one possblty of the 2 N combnatons of entry behavors of N other potental bdders. Denote the number of actual bdders of the combnaton a by n a : As usual we consder a contnuously d erentable and strctly ncreasng bddng strategy, b = s (v ) ; therefore the rst order condton s n X V F V jv sj jv (b ) ; j 6= jv + (v b ) j6= sj (b ) ; j 6= (b ) = 0; where F V jv denotes the jont dstrbuton of V j ; j 6= condtonal on V = v and s () s the nverse functon of the bddng functon of bdder : A set of equaton () for = ; : : : ; n form a system of d erental equatons characterzng the equlbrum bds for all n actual bdders. We denote the post-entry pro t of bdder by (v ja ) : 3 When the lower support of prvate value s below the reserve prce, bdder only knows the actve bdders who partcpate n the aucton but not actual bdders who submt bds. In our case, the number of actve bdders s equal to the number of actual bdders, snce the lower support of prvate value s assumed to be just the reserve prce. 6

8 3.2 Entry Decson In the ntal partcpaton stage, each potental bdder only knows hs own entry cost, jont dstrbutons of entry costs and prvate values. Therefore the entry decson of bdder s determned by hs pre-entry expected pro t from partcpaton, : Spec cally, he partcpates n the aucton only f hs entry cost s less then : Let p denote the entry probablty of bdder ; respectvely. The ex ante expected pro t s gven by = X a 2A Z v v (v ja ) df (v ) Pr (a ja = ) ; (2) where Pr (a ja = ) s a functon of p ; = ; : : : ; N; whch can be denoted by Pr (a ; p ; : : : ; p N ja = ) : As a result, the pre-entry expected pro t s the sum of 2 N products of the post-entry pro ts and correspondng probabltes wth the unknown prvate value ntegrated out. On the other hand, the ex ante probablty of entry s gven by p = Pr (K < ) = G ( ) : Note that although the number of potental bdders does not drectly a ect the bddng strategy n the bddng stage, t a ects the number and the denttes of actual bdders, whch n turn have mpact on the bddng strategy. 3.3 Characterzaton of the Equlbrum Exstence and unqueness of the Bayesan Nash equlbrum wth asymmetrc bdders has been a challengng problem studed n the recent aucton theory lterature. See, e.g. Lebrun (999, 2006) and Maskn and Rley (2000, 2003) wthn the IPV framework, Lzzer and Persco (2000) wthn the APV framework and two types of bdders. The analyss of our model s further complcated by the ntroducton of a laton and entry, as well as that we allow all potental bdders to be d erent from each other. To address the ssue of exstence and unqueness n our case, we look at the case where the jont dstrbuton of bdders prvate values s characterzed by the famly of Archmedean copulas. For the copula concept and the characterzaton of the Archmedean copulas, see Nelsen (999). Copula can provde a exble way of modelng jont dependence of multvarate varables usng the margnal dstrbutons. Spec cally, by Sklar s theorem (Sklar (973)), for a jont dstrbuton F (x ; : : : ; x N ) ; there s a unque copula C; such that C (F (x ) ; : : : ; F N (x N )) = F (x ; : : : ; x N ) : For the Archmedean copulas, the copula C can be expressed as C (u ; : : : ; u n ) = [ ] ( (u ) + + (u n )) ; where s a generator of the copula and s a decreasng and convex functon, and [ ] denotes the pseudonverse of 4 : The famly of Archmedean copulas nclude a wde range of copulas. For example, the generators (u) = q (u q ) ; (u) = ( ln (u)) q ; and (u) = ln correspond to the exp(qu) exp(q) 4 s a decreasng convex functon from [0; ] to (0; ] wth () = 0: [ ] s de ned as [ ] (u) = (u) ; 0 u (0) ; 0; (0) u : 7

9 wdely used Clayton copula, Gumbel copula, and Frank copula, respectvely. Snce we consder a d erentable bddng strategy, we have to con ne ourself to the strct generator, that s [ ] = : Snce C (F (x ) ; : : : ; F N (x N )) = F X jx (x ; : : : ; x N ) (e.g. L, Paarsch, and Hubbard (2007)), the rst order condton () determnng the equlbrum bds can be wrtten as follows ds (b) = db 2 0 P k F k s k (b) n a 0 F s (b) f s (b) 00 P k F k s k (b) 4 X k6= 3 n a 2 k (b) b 5 s : (b) b Note that wth the copula spec caton for the jont entry cost dstrbuton, the entry probabltes n (2) can be expressed n terms of the jont entry cost dstrbuton. s (3) For example, Pr (a ; p ; : : : ; p N ja = ) ; for the case that gven the partcpaton of bdder ; bdder up to bdder as where partcpate n the aucton whle bdder + up to bdder N do not, can be expressed Pr (a = a = ; a + = a N = 0ja = ) (4) = Pr (a = a = ; a + = a N = 0) Pr (a = ) Pr (a = a = ; a + = a N = 0) X = C (p ; : : : ; p ; ; : : : ; ; q k ) C (p ; : : : ; p ; p j ; ; : : : ; ; q k ) +jn + ( ) N C (p ; : : : ; p N ; q k ) ; and Pr (a = ) = C (; : : : ; ; p ; ; : : : ; ; q k ) : Equlbrum of the model conssts of two parts, entry equlbrum and bddng equlbrum. Based on the choce of Archmedean copulas for the jont dstrbuton of prvate values, the exstence of the equlbrum s guaranteed. Moreover, wth some addtonal condtons, the bddng equlbrum s unque. The next proposton descrbes the equlbrum formally. Proposton (Characterzaton of Equlbrum). Assume (a) the margnal dstrbuton of entry cost of bdder, G s contnuous over k; k for all ; (b) margnal dstrbuton of prvate value of bdder s d erentable over (v; v] wth a dervatve f locally bounded away from zero over ths nterval for all ; (c) jont dstrbuton of prvate values follows an Archmedean Copula.. Bddng Equlbrum In the bddng equlbrum, bdder adopts a contnuously d erentable and strctly ncreasng bddng functon b = s (v) over (v; v]: The nverse functons of s for all ; s ; : : : ; s n are the soluton of the system of d erental equatons (3) wth boundary 8

10 condtons (5) and (6) : for some. s (v) = v (5) s () = v: (6). Unqueness of Bddng Equlbrum Moreover, f F (v) > 0 and 0 (u) 00 (u) n u, then the bddng equlbrum s unque. s decreasng. Entry Equlbrum In the entry equlbrum, bdder chooses to partcpate n the aucton f hs entry cost s less than the threshold (p) and stay out otherwse, where p = (p ; : : : p N ) and p s the entry probablty of bdder and s determned by p = G ( (p)) : (7) As s seen here, the exstence of the entry equlbrum s equvalent to the exstence of the entry probablty p ; gven by the equaton (7) : Snce s contnuous n p and thus G s contnuous over [0; ] ; there exsts a soluton p of equaton (7) ; accordng to Kakutan s xed pont theorem (Kakutan (94)). To show the unqueness of the bddng equlbrum s to show that there s a unque such that s () = v: Then startng from ; accordng to Lpschtz unqueness theorem, s s unque over (v; ]: Note that a Clayton copula sats es the condton for unqueness that 0 (u) 00 (u) s decreasng n u: The formal proofs are provded n Appendx A. 4 The Structural Analyss We estmate the model proposed n the last secton usng the tmber sales data. Our objectve s to recover the underlyng jont dstrbutons of prvate values and entry costs usng observed bds and the number of actual bdders. The structural nference n our case s complcated because of the generalty of our model that accounts for a laton, asymmetry, and entry. Our approach crcumvents the complcatons arsng from the estmaton of our model and makes the structural nference tractable. Frst, to model the a laton n a exble way, we adopt the copula approach n modelng the jont dstrbuton of prvate values and the jont dstrbuton of entry costs. 5 Second, snce we allow bdders to be asymmetrc, the system of d erental equatons consstng of equaton (3) that characterzes bdders Bayesan Nash equlbrum strateges does not yeld closed-form solutons. To address ths problem we adopt a numercal method based on Marshall, Meurer, Rchard, and Stromqust (994) and Gayle (2004). Thrd, because of the varous covarates we try to control for and the relatvely small sze of the data set, the nonparametrc method does not work well here. Therefore, we adopt a fully parametrc approach. 5 L, Paarsch, and Hubbard (2007) use the copula approach to model a laton wthn the symmetrc APV framework wthout entry and propose a semparametrc estmaton method. 9

11 4. Spec catons We adopt the Clayton copula to model the jont dstrbutons of both prvate values and entry costs. Wth the generator of Clayton copula gven above, the jont dstrbuton of prvate value s spec ed as F (v ; : : : v n ) = P F (v ) qv n + =q v ; and the jont dstrton of entry costs s spec ed as G (k ; : : : ; k n ) = P G (v ) q k n + =q k ; where q v and q k are dependence parameters and F and G are the margnal dstrbutons of prvate value and entry cost, whch are spec ed as truncated exponental dstrbutons gven as follows, F V` (vjx` ; ) = G K` (kjx` ; ) = exp k` exp k` k k` k k` exp k` exp k` k k` k` k exp v` exp v` v v` v v` exp v` exp v` v v` v` v for bdder of the `-th aucton, ` = ; : : : ; L; where L s the number of auctons, v` and k` are the prvate value and entry cost means and equal exp (x` ) and exp (x` ), respectvely, and x` s a vector of covarates that are aucton spec c or bdder spec c, and n our case ncludes varables such as haulng dstance, volume, duraton, grade, and DBH. 6 In practce, v s equal to the reserve prce of `-th aucton, v s equal to $500/MBF, the lower bound of entry cost s equal to zero and the upper bound k s $940/MBF, an arbtrarly large number. 7 We then model the jont dstrbutons of prvate values and entry costs n aucton ` as Clayton copula wth d erent dependence parameters q v and q k : The use of the Clayton copula o ers several advantages. Frst, t guarantees the exstence and unqueness of the equlbrum as dscussed n Secton 3.3. Second, t preserves the same dependence structure when the number of potental bdders changes. Thrd, t s relatvely easy to draw dependent data from the Clayton copula, as t has a closed form that can be used to draw data recursvely. Lastly, snce q s the only parameter that measures the dependence, we can easly evaluate the mpact of the dependence level on the end outcomes of an aucton by changng the value of q: Note that n these spec catons, the asymmetry across potental bdders s captured by the ncluson of the haulng dstance varable n x`, whle both and are kept constant across d erent bdders. Ths enables us to estmate a relatvely parsmonous structural model and at the same tme control for the asymmetry. 4.2 Estmaton Method Because of the complexty of our structural model, we employ the ndrect nference method to estmate the model. Intally proposed n the nonlnear tme seres context by Smth (993) and developed further by Goureroux, Monfort, and Renault (993) and Gallant and Tauchen (996), the ndrect nference method s smulaton based and obtans the estmates of parameters by mnmzng a measure of dstance between the estmates for the auxlary parameters of an auxlary model usng the orgnal data and smulated data. More spec cally, let denote the vector of parameters of 6 Here we do not ntroduce unobserved aucton heterogenety nto the model, as L and Zhang (2008) show that t does not have a sgn cant e ect n bdders entry behavors. 7 Here we need an upper bound for the prvate value to make the algorthem of ndng the equlbrum bds possble. See Appendx B for the detal of the algorthem. ; 0

12 nterest, be the parameters of the auxlary model, b T and b (p) ST () be the estmates of the auxlary model usng the orgnal data and the p-th smulated data out of P sets of smulated data from the model gven a spec c, respectvely. Then the estmator of, denoted by b ST ; s de ned as b ST = arg mn 2 4b T P PX p= b (p) 3 ST () 5 0 " b T P PX P = b (p) ST () # ; (8) where s a symmetrc sem-postve de nte matrx. Therefore to mplement the ndrect nference method, we have to draw data from the model for a gven ; whch nvolves calculatng the equlbrum bds and the thresholds of the entry costs. Bascally we use numercal approxmaton method smlar as the ones n Marshall, Meurer, Rchard, and Stromqust (994) and Gayle (2004) to nd the equlbrum bds and teraton to nd the equlbrum entry probabltes, both of whch are llustrated n detal n Appendx B. The auxlary model, whch s usually smpler than the orgnal model and easer to estmate as well, plays an mportant role n the ndrect nference method. In ths paper, followng the dea n L (2005) we employ a relatvely smple and easy-to-estmate auxlary model to make the mplementaton tractable and the nference feasble. Spec cally, snce we use both bds and the number of actual bdders n the estmaton of entry and bddng model, our auxlary model ncludes two separate regressons: a lnear regresson of the observed bds and a Posson regresson of the number of actual bdders, whch are descrbed as follows b` = 0 + HX X h` h + h= Pr (n` = k) = exp ( `) k` k! HX Xh`2h h= ; ` = 20 + HX X h` 2h + h= HX Xh`mh m + " ; h= HX Xh`22h h= HX h= X m h`2mh; where b` s the average bd of aucton `; and X h`, h = ; :::H, denote the vector of aucton-spec c covarates of aucton ` and the average of bdder-spec c covarates, and H s the number of such covarates, whch s 6 n our case. m = 2 makes our model over-dent ed. An ssue arsng from the mplementaton of the ndrect nference method s the dscontnuty of the objectve functon of equaton (8) because of the dscrete dependent varable (the number of actual bdders) n the auxlary model that makes gradent-based optmzaton algorthm nvald. We address ths ssue by usng smplex, a nongradent-based algorthm. Alternatvely, one can follow Keane and Smth (993) to smooth the objectve functon usng a logstc kernel. 4.3 Estmaton Results Table 2 reports the estmaton results. For the (margnal) prvate value dstrbuton, all the estmated parameters have the expected sgns. Of partcular nterest s the parameter of the haulng dstance varable, whch s used to control for heterogenety across bdders. Its estmate s negatve,

13 meanng that the longer the haulng dstance s, the less s the prvate value mean. Furthermore, the average margnal e ect of the haulng dstance varable s about -0.09, meanng that one mle ncrease n the dstance would reduce the prvate value mean by $0.09/MBF whle everythng else s xed. Another parameter of partcular nterest s the dependence parameter q v n prvate values, whch turns out to be relatvely small (q v = 0:04) but sgn cant. To get some dea of how large the dependence s wth q v = 0:04; we use a measure called Kendall s (Nelsen (999)), whch s used to measure the concordance of two random varables. Concordance s not really the same concept as a laton, but measures the postve dependence n a smlar way. Kendall s s de ned as the probablty of concordance mnus the probablty of dscordance: X;Y = Pr [(X X 2 ) (Y Y 2 ) > 0] Pr [(X X 2 ) (Y Y 2 ) < 0] : For the Clayton copula, = q=(q + 2) = 0:047: Therefore q v = 0:04 mples that the event of any two bdders prvate values beng concordant s about 4.7% more lkely than the event of beng dscordant. Two ponts n the estmates n the dstrbuton of entry costs are worth notng. Frst, the haulng dstance varable s postve n the entry cost dstrbuton and ts margnal e ect s.34. Second, the dependence level among the entry costs s , mplyng a Kendall s of 0.2 n the two bdders case. Ths ndcates that the a laton among the entry behavors s manly drven by the a laton among the entry costs. 4.4 Model Ft TO BE ADDED LATER 5 Counterfactual Analyses Wth the estmated structural parameters we can now answer the questons put forward n the ntroducton secton emprcally. We focus on both end outcomes, namely, the number of actual bdders, and wnnng bds (or seller s unt revenue). We conduct counterfactual analyses on the 99th aucton of our data. We use ths aucton as a representatve aucton, as the values of covarates of ths aucton are close to the average values of all auctons n our data set. In partcular, the number of potental bdders n ths aucton s 7, about the same as the average number of potental bdders n the orgnal data. In dong so we assume that the estmates derved from a subset of the data t the whole data. The seller s expected unt revenue s gven as follows E (w) = E (wjw > 0) Pr (w > 0) + E (wjw < 0) Pr (w < 0) = E (wjw > 0) Pr (w > 0) + v 0 Pr (w < 0) ; where w denotes the wnnng bd and v 0 s the valuaton of the tmber to the seller, and the second 2

14 equalty follows the assumpton that f the tmber s not sold successfully then the seller gets hs own value. In the followng analyses we assume v 0 = 0, thus the expected revenue s equvalent to the expected pro t. 5. E ects of Reserve Prce and Dependence Level Intutvely the e ect of the reserve prce can be seen from two aspects. On one hand, a hgher reserve prce s assocated wth a lower ex ante expected pro t,.e., a lower cut-o entry cost accordng to equaton (2) as t narrows the ntegraton range, and thus fewer partcpatng bdders and lower probablty of beng sold, whch may lower bds n our APV model wth asymmetrc bdders. On the other hand, a hgher reserve prce rases the lowest acceptable bds and of course makes bdders bd hgher. Our counterfactuals shown n Fgure con rm such tradeo. The three panels n Fgure show how the reserve prce a ects the number of actual bdders, the probablty of beng sold, and the seller s revenue. The number of actual bdders s decreasng n the reserve prce as s shown n the rst panel. The average number of partcpatng bdders drops dramatcally from 3.96 to.04 when the reserve prce s rased from $293.42/MBF to $880/MBF. The probablty of beng sold s negatvely related to the reserve prce. The change n the wnnng bd s the nal result of all e ects assocated wth change n the reserve prce. As s seen n the last panel, the optmal reserve prce s around $498/MBF, whch s almost twce as large as the current reserve prce. Ths mples that when the reserve prce s below $498/MBF the postve e ect on the wnnng bd outweghts the negatve e ect assocated wth the lower probablty of beng sold. The APV model we estmated also enables us to quantfy the e ects of the dependence level among bdders. To ths end, we change the values of the dependence parameters of both prvate values and entry costs whle keepng other parameters xed. We are able to conduct such analyss as the change of the dependence parameter does not a ect the margnal dstrbutons of prvate values and entry costs. As n the analyss of the e ects of the reserve prce, we are nterested n three e ects of the dependence parameters. Results are provded n Fgure 2 and Fgure 3. As s seen n the two gures, the dependence level does not a ect the probablty of beng sold, whch remans at about 98 percent. In Fgure 2, t can be seen that the number of partcpatng bdders, the probablty of beng sold and the seller s revenue are all slghtly negatvely related wth the dependence level of prvate values. Snce the dependence level of prvate values does not a ect bdders entry costs, t must be true that f bdders become more correlated wth each other, ther pre-entry expected pro ts become less, whch leads to a decrease n the number of partcpatng bdders. The drop n the probablty of beng sold s a drect result of the decrease n the number of partcpatng bdders. Also, As can be seen n Fgure 2, the seller s revenue drops as the dependence level of prvate values become hgher. The e ects of the dependence level of entry costs are d erent than the dependence level of prvate values. The rst panel n Fgure 3 presents a postve relatonshp between the dependence level of entry costs and the number of partcpatng bdders at least for the dependence levels that are less than 0.20 (kendall s ). It s ntutve because the entry process s manly governed by the 3

15 entry cost. When the entry costs become more a lated, more bdders wll choose the same entry decson, whch s to partcpate n the aucton n ths case. Ths s also the dea of the a laton test n L and Zhang (2008). Note that the relatonshp between the dependence level of entry costs has the same pattern as the relatonshp between the dependence level of entry costs and the number of partcpatng bdders, whch can be seen from the comparson of the rst panel and the last panel of Fgure 3. Ths s consstent wth what we found n Fgure E ect of Bddng Coalton or Merger TO BE ADDED LATER 6 Concluson TO BE ADDED LATER 4

16 Appendx A: Proof of Proposton The proof of Proposton adapts Lebrun (999, 2006). We rst need the followng lemma. Lemma. Consder a contnuously d erentable and strctly ncreasng bddng strategy. Assume 0 (u) s decreasng n u: If e > and es 00 (u) (b) and s (b) for all are two solutons of the system of d erental equatons (3) wth boundary condton (6) over (e; e] and (; ], respectvely, then the nverse bddng functons satsfy the followng condton: es (b) < s (b) for all b n (max (; e) ; ]; where > v: Proof. Snce we know that s s strctly ncreasng over (; ]; we have es () < s () = v: De ne g n [max (; e) ; ] as follows: g = nf b 2 [max (; e) ; ] es b 0 < s b 0 ; for all and all b 0 2 (b; ] : We want to prove that g = max (; e) : Accordng to the de nton of g; > g: Suppose that g > max (; e) : By contnuty, there exsts such that es (g) = s (g) : From the de nton of g; we also have es j (g) sj (g) for all j: Moreover, there exsts j 6= such that es j (g) < sj (g) ; because all the solutons conncde at the pont g and therefore conncde n (g; ] due to the fact that the rght hand sde of equaton (3) s locally Lpschtz at b = g, whch contradcts the fact that at pont es () < s () : From equaton (3), we know ds (b) =db s a strctly decreasng functon of sj (b) ; for all j 6= ; snce 0 (u) s decreasng n u: Consequently, des 00 (u) (g) =db > ds (g) =db: Therefore there exsts > 0 such that es (b) > s (b) ; for all b n (g; g + ) : Ths contradcts the de nton of g: Proof of Proposton Proof. Frst we prove the rst part of the proposton by showng that there exst an, such that s () = v: () Bddng Equlbrum Let ; n denote bdders who have the hghest bds, denoted by 0 ; at the upper bound of prvate value v and j; j n denote bdders who has the second hghest bd, denoted by ; at the upper bound of prvate value v. So 0 : For bdder ; we know that It s obvous that Pr (B < 0 jv) = v 0 Pr B < 0 jv (v ) Pr (B < jv) : Pr (B < jv) = Pr (b j < ; b k < ; k 6= ; jjv = v) = Pr (b k < ; k 6= ; jjv = v) ; snce b j s not larger than : Pr (B j < jv) = Pr (b < ; b k < ; k 6= ; jjv j = v) : 5

17 Snce the jont dstrbuton of prvate values follows Archmedean copulas, we have 0 Pr (B < jv) = X k6=;j F k s k () + F j s j () 0 = X F k s k () + () + () A 0 () k6=;j and Pr (B j < jv) = 0 X F k s k () + (F j (v)) + F s k6=;j 0 = X F k s k () + () + F s k6=;j + (F (v)) A 0 (F (v)) () A 0 (F j (v)) () A 0 () If F s () < ; then F s () > () and Pr (B < jv) > Pr (B j < jv) snce 0 () < 0 and 0 (x) s ncreasng n x. Therefore v 0 Pr B j < 0 jv > (v ) Pr (B j < jv) snce Pr (B j < 0 jv) = : But ths s mpossble because the optmal bd of bdder j at v s ; therefore we have F s () = and 0 = : () Unqueness of Bddng Equlbrum Suppose that there exst two equlbra and thus two d erent values and e such that the respectve solutons s (b) and es (b) are also solutons of the system ofd erental equatons for all. Wthout loss of generalty, we can assume that < e: The value of ln Pr v j < sj (b ) ; j 6= jv at b = s strctly larger than the value of ln Pr v j < es j (b ) ; j 6= jv at the same pont. We have shown that es (b) < s (b) for all b n (v; ]. When b converges to v; s (v) = v: On the other hand, the rst order condton can be wrtten as follows Therefore d ln(pr(v j<s j (b);j6=jv )) d ln Pr v j < sj db (b ) ; j 6= jv db < d ln(pr(v j<es j (b);j6=jv )) db = s : (b ) b : Therefore, the d erence between these two logarthms ncreases as b decreases towards v: On the other hand, ln (Pr (v j < v; j 6= jv )) s a nte value snce F j (v) > 0: Therefore for two solutons, ln Pr v j < sj (b ) ; j 6= jv cannot both coverge to the same nte value as b decreases towards v: Therefore and e conncde and the equlbrum s unque. () Entry Equlbrum The entry probablty p s determned by equaton (7) : Let p = (p ; : : : ; p n ) 2 [0; ] n and 6

18 G p = (G (p) ; : : : ; G n n (p)) : Snce s (v) and G s contnuous, the pre-entry expected proft and G s contnuous n p: So G p : [0; ] n! [0; ] n and s contnuous n p: A xed pont of p follows Kakutan s xed pont theorem (Kakutan (94)). Appendx B: Solvng Equlbrum Bds and Entry Probabltes Equlbrum Bds Note that wth the choce of Clayton copula, the rst order condton gven n equaton (3) can be wrtten as follows, ( + q) (v b) X j6= df q j s j db (b) De ne F q s (b) = l (b) ; then v = F l q = q nx k= F q k s k (b) n + (b) ; and F.O.C. becomes! : ( + q) F l q (b) X b lj 0 (b) = j6= q! nx l k (b) n + k= Rewrtng all terms n the equaton as polynomals F l F l q l (b) = l 0 (b) = q l (b) = q (b) (b) = b = F (x) = x q = X a ;j (b b 0 ) j ; j=0 X (j + ) a ;j+ (b b 0 ) j ; j=0 X g ;j (b b 0 ) j ; j=0 X p ;j (b b 0 ) j ; j=0 X ep ;j (b b 0 ) j ; j=0 X d ;j (x x 0 ) j ; j=0 X c ;j (x x 0 ) j ; j=0 where ep ;0 = p ;0 b 0 ; ep ; = p ; ; and ep ;j = p ;j for j > : Computaton of p ;j ; g ;j : followng the lemma n Appendx C n Marshall, Meurer, Rchard, 7

19 and Stromqust (994), we have p ;J = ;r;j = g ;J = ' ;r;j = JX d ;r ;r;j z J ; p ;0 = F q l (b 0 ) r= JXr+ s= (0a) g ;s ;r ;J s ; ;0;0 = ; (0b) JX c ;r ' ;r;j ; r= JXr+ s= (0c) a ;s ' ;r ;J s ; ' ;0;0 = : (0d) Computaton of a ;j : from the FOC, we have 0 X ( + ep ;j (b b 0 ) j A X X (s + ) a j;s+ (b j=0 j6= s=0 0 0 b 0 ) s = q X X ( + ep ;j (b b 0 ) j A (s + X a j;s+ A (b j=0 s=0 j6= 0 b 0 ) s = q X sx X ( + q) (s + ep ;s A (b b 0 ) s = q s=0 r=0 j6= a j;s+ r! nx X a k;s (b b 0 ) s n + k= s=0 X s=0 k= X s=0 k=!! nx a k;s (b b 0 ) s n +!! nx a k;s (b b 0 ) s n + Comparng the coe cents of (b b 0 ) s we have 0 sx X ( + q) (s + ep ;s r=0 j6= a j;s+ r A = q X ( + q) p ;0 a j; = q j6=! nx a k;s ; for s > 0 (a) k=! nx a k;0 n + ; for s = 0: (b) k= Algorthm:. d ;j ; c ;j for j = ; : : : ; J; can be computed by Taylor expanson. In practce, J = 3: 2. Decde a ;0 ; ep ;0 ; ;0;0 ;and ' ;0;0 by the boundary condtons. 3. Calculate ep ; from equatons (0) gven a ;0 ; ep ;0 ; ;0;0 ;and ' ;0;0 : 4. Calculate a ; from equatons () gven ep ; : 5. Repeat step 3 and 4 untl a ;j ; j = ; : : : ; J are calculated. 8

20 Now we have found the coe cents of the Taylor expanson of the nverse bddng functon up to the J-th order, so we are able to nd the equlbrum bd for a gven prvate value for bdder through the obtaned Taylor expanson at a approprate pont. One ssue regardng the algorthm s the boundary condtons. From the Proposton we know that there are two boundary condtons assocated wth the equlbrum. Note that t cannot be used here although the boundary condton at the lower bound of bds s known to us, snce t causes the problem of sngularty. Therefore we have to use the condton at the upper bound, whch s unfortunately unknown to us. To address ths problem we follow the method descrbed n Marshall, Meurer, Rchard, and Stromqust (994) and Gayle (2004) to nd the common rst. Roughly, t s to nd an whch generates the best equlbrum bds at pont v accordng to the algorthm descrbed above. Please refer to the two papers for detals. Equlbrum Entry Probabltes As s seen from the Proposton and equaton (7) the equlbrum entry probabltes are determned by a xed pont problem. We solve t through teraton, descrbed n detal as follows,. Gven an ntal guess of p old ; = ; : : : ; N; we calculate the probabltes of all possble entry behavor occurrng accordng to equaton (4) : 2. Gven the calculated R v v (v ja ) df (v ) for all possble entry behavor n equaton (2) and assocated probabltes gven n step, we calculate the post-entry expected pro t ; = ; : : : ; N: 3. Calculate new entry probabltes accordng to p new = G ( ) ; = ; : : : ; N: 4. If the d erence of p new p new let p old and p old s smaller than a gven small postve number, "; then ; = ; : : : ; N are the equlbrum entry probabltes and the teraton stops; otherwse, = p new and go to step. 9

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