FRONTIERS AGBUS&MGR ECON
FRONTIERS AGBUS&MGR ECON AGEC 695
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Lecture Five Ximing Wu 4 Firstprice sealedbid and Dutch Auctions Recall that in these two auctions the optimal strategy is v N l f2 FV du 01 v FVUN1 41 Data generating process 0 Change of variables for PDF and CDF Suppose X is distributed according to FX with PDF fX Let X 9Y We have 0 If instead FY is known we have FX 56 FY 9 1CII 7 fY 91 M x g ltg 1ltw39 The second equality is by inverse function theorem Suppose f 1 is differen tiable at a f 1 Then 0 Let 3139 be the 2th bid in an auction We have vi 0 1 Thus F3 FV FV 0 1 8 7 fv018 M l a39 01 lt8 Note that we do not use 0 0 1 Why 0 From N 1 fv v f Fv WV 1cm a v FV WM We then have the joint density of the bids 31SN f 0 1 52 0 i015 i fS8 21 T H fv i018z i liAf 1gtfvlta1ltsagtj ti vuiM lduPyltvyv FV 018 N lovr 1gtif yltw t4du39 2 N H 2 Z 0 The winning bid in Dutch or firstprice sealedbid auction is the first order statistic with density jwaNmoWhoi Recall this density can be obtained by differentiating PrZZ g zfor a Thus the density function of the winning bid W equal to a Z is 7 fZ l01wl WWW 039 ia1ltwgti i NFV 01wN1fV 01010 N 1 fv lta1 M 1quot Fv ltugtN 1duFv W NFV 01 210 W 1 7 N 1 ff llwl FV QM 1 du 42 Identification and Estimation simplest case 421 Nonparametricidentification 0 Assume no reserve price and all bidders are risk neutral Also assume all bids are observed o Denote the population distribution of the observed bids by Fg s with PDF 243 the population distribution of valuation 1731 with PDF f3 1 0 From Chapter 2 we have 1 7 w v JV urge 11197 N 1f3U 0 Because in equilibrium 3 01 ii 1731 F3 0 1 F s 0 348 fl U WW v0v 0 v is F ltsgt W nfgrs 2 Thus f3 1 is identified from fg 8 Note that the optimal bidding strategy a does not appear in this identification equality 6 0 Assume only winning bids are observed or only the first order statistic of each auction is observed 0 Denote the population distribution of the first order statistic by F12 with PDF fSV Note that wamwgt mwmmWM o m m ddm n Dividing 3 by 4 WwiNWW w Plugging 5 into 2 AM N 1fii w Thus the distribution of WNW is identified through 111 Since the distribution of 141on is F3 N F3 is identified U zN w 422 Nonparametric estimation 0 Suppose we observe data from T auctions each with N bidders Denote the 2th bid from the tth auction by Sit o The CDF of the bids can be estimated by empirical CDF ECDF A 1 T 1 N 1733 N1S S 3 8 where 1 is the indicator function 0 The PDF of the bids can be estimated by a kernel density estimator KDE T N A 1 1 1 Sit 8 fsl8 f h gl hg 7 where I4 is a kernel smoothing function and hg is the bandwidth We will cover kernel density estimation in next lecture o The pseudovalues of the 2th bidder in the tth auction is estimated by F3 Sit N 1 f3 Sn Note that we use a hat above Wt since it is a random variable Which WtS t components in righthandside are random Which are not 0 The CDF and PDF of the valuation distribution are now estimated using the 9 pseudovalues 151 11 T N A A i 1 1 1 mt U fV i i T N hflt hg gt To avoid boundary bias one can estimate the PDF by 1 T 1 N 1 17 u A it hg gt1Sminhgltsitltsmaxhga 251 11 9 where 3min and SmX are the minimum and maximum of the observed bids 0 Suppose only winning bids are observed eg Dutch auction We first esti mate the CDF and PDF of the winning bids by o The pseudovalue of the highest bidder in the tth auction is estimated by A i N FW Wt WW W N 1fw Wt The CDF and PDF of the valuation of the highest bidder are estimated by HIH T Fm quizW9 151 A 1 T 1 VN Z fZ Zh K 1 Wmin hg S Wt S Wmax hg7 251 9 l 9 where Wmm and VlmX are the minimum and maximum of observed winning bids 0 The CDF and PDF of the valuation distribution are estimated by 1 T 1W Fv v Fzv1N 1l71Nt 251 A Fzltvgt7i 1 zltvgt fv u using formula o Nonparametric methods have difficulties to control for covariate heterogeneity especially when sample size is small due to the curse of dimensionality Lecture Three Ximing Wu 3 Vickrey and English Auctions 31 Nonparametric identification and estimation 0 Assumptions IPVP no reserve price note that risk attitude does not matter here 0 Denote the bid and valuation for bidder 2 by Bi and W with corresponding CDF F2 and F3 BFV gtFgv F3v 1 Thus the distribution of bids identifies the distribution of valuations 0 Suppose there are T identical auctions each with the same number of bidders N lfall bids in these auctions are observed F3 is estimated by empirical CDF A 1 T N FMhWZZMBusv 151 2391 where 1 is the indicator function and Bit the bid of bidder 2 in auction 75 oWhat if only winning bids the second highest bid are observed Denote W V2N the CDF of W is see Appendix A2 for details 0 Fvlw F VwNN 1 uN21 udu 0 0 N l 0 N NltN1gtFvgt1 y s0 nglw Wi Show that ltp F3 is strictly increasing in F3 Hence F12 identi fies F3 0 Estimation 1 Fwy Zf11wt lt v 2 Fv v ltp 1 FW 1 solved numerically o Inferences under standard regularity conditions FW 1 FSV i N OVW F3 1 where W 173 W FSVW 1 Fvov 10 s0 PM W 1 s0 WW WH By the delta method 13mm F3 20 1gt N 0 VV F3 1 N where VW WW W N 4 VVFvUN N2N12F 9v2 1 1119 4 The derivation of W F3 1 N is left as an exercise The delta method if VT XT 6 i N o a then VT 9 0m g 6 i N o 9 a 02 provided that g is differentiable at 1 8 and g 8 y 0 32 Covariates o ldentical auctions are rare Researchers typically have to use data from auc tions differing in many respects and therefore need to control for differences across auctions 0 Suppose each auction can be characterized by a set of attributes Z If Z is discrete one can in principle estimate a conditional value distribution FVIZ for each auction type39 Not practical because of curse of dimensionality for multivariate Z 0 Similar difficulties with continuous Z 33 Singleindex model semiparametric estimation 0 Assuming F19Z Ulzl Fviz UlzTVO gt FllZ wlzl Fwiz wlzT70 The dimension of condition variables is reduced to one 0 Estimation 1 Estimate 70 gt 3 2 Estimate FWIZ wlzT l gt sz wlzT 3 Calculate FVIZ vlzT l from sz wlzT o Singleindex model implies that the conditional mean of W given Z is only a function of zTVO or 5 u vao Since is unknown 70 is identified only up to some location and scale normalization To see this define uquot u It follows that 7 27 Thus for identification z typically does not include the constant term and certain normalization is imposed on 70 For instance let 70 71 7 llvoll 23112 1 331 Densityweighted derivative estimator Powell Stock and Stoker 1989 See textbook p56 59 for details 332 Maximum rankcorrelation estimator Han 1987 and Sherman 1993 See textbook p5961 for details Lecture One Ximing Wu 1 Introduction 0 Auction format the rules governing how the potential bidders must behave in an auction 7 how bids must be tendered who wins the auction 7 what the winner pays 0 Information asymmetry the sellers know little concerning the valuation of potential buyers potential buyers have no incentive to tell the seller their valuations o The role of auction is to get the potential buyers to reveal their valuations 2 Overview of auction theory 21 Auction formats and rules 211 Oral ascending price English auction 0 the most frequently used format 0 Auction begins at some low price The bidders raise their bids during the auction Auction stops when no bidders are willing to raise the bid further 212 First price sealed bid format 0 Participants submit bids in a sealed envelope At some prespecified time all envelopes are opened The participant with the highest bid wins the auction and pays what he bid 0 Procurement low price sealed bid auction consider this bidding in terms of negative price to pay it is also a first price auction 213 Oral descending price Dutch auction 0 The auction starts at a very high price and drops continuously until a bidder signals his is willing to pay the current price 0 Frequently used to sell flowers or fish at auctions 214 Second price sealed bid Vickrey format 0 Same as first price sealed bid auction except that winner pays second highest bid 0 Named after William Vickrey who first analyzed this format in Vickrey 1961 o Desirable to economists theoretically rarely used in practice 215 Additional rules Reserve price a minimum price that must be bid A reserve price can be publicly announced or remains secret until a winner is declared Bid increment a minimum amount needed to raise a bid 0 Multi unit auction bidder might need to specify a price for each identical unit or a price schedule 3 o Multi object auction the objects are not identical 22 Information structure 221 Independent private values paradigm IPVP 0 Auction plays an allocational role in IPVP 0 Suppose there are N potential bidders each getting an iid draw from the value distribution FV 1 In this sense the bidders are assumed to be symmetric ex ante o FV v PrV S 1 By convention upper cases are used for random variables and lower cases for realizations of random variables Define the survivor function SV 1 Pr V gt v 1 FV 1 It gives the proportion of the population whose valuation is higher than 1 and thus demand the object at price 1 Then the expected demand curve is given by NSV v 222 Common value paradigm CVP o The value of the object is the same to all potential bidders 0 Example In oil exploration a number of firms have seismic information concerning the likelihood of finding oil on a particular tract Their information thus estimated value of the tract might be different But conditional of having found oil the value of the oil is the same to all potential bidders o The distribution of information plays a central role in CVP 223 Affiliated value paradigm AVP o A middle ground between IPVP and CVP o Often unidentified in econometric sense 224 Asymmetries o In IPVP this often means bidders valuations are drawn from two or more different valuation distributions 23 Bidder preferences 0 Bidders are often assumed to be risk neutral in theoretical work Denote a bidder s valuation and chosen strategy by v and 5 respectively His expected utility from participating at the auction is EU v 5 Pr strategy 5 wins the auction 0 Risk averse bidder CARA UY 1 exp aY oz 2 0 gt U YUY oz A practical difficulty of CARA is that it does not easily nest the risk neutral case via a parameter restriction on oz 7 CRRA U0 0517775 2 0777 Z 1 gt YU YUY 77 177 1 Clearly it reduces to the risk neutral case when 77 1 24 Purposeful bidding and game equilibrium 241 Vickrey auctions o The simplest case 0 Assuming IPVP ii risk neutral bidder iii no reserve price or zero reserve price Denote the value and bid of the ith bidder by V2 and Bi 3 5 Vi V2 6 where B is the bidder s strategy In this case the dominant strategy is to bid the true value Here is why 7 Suppose B2 and Bj are the two highest bids 7 a Suppose 1 bids Vl 6161 gt 0 gtllt If Bj lt Vi it does not matter if1 bids Vl or Vl a he pays Bj anyway gtllt If Bj gt Vl 61 again it does not matter to 1 but matters to j gtllt If Vl lt Bj lt Vl 611 has to pay Bj resulting a net loss Hence bidding V2 is at least as good as bidding V2 a i b How about bidding Vl a where a gt 0 gtllt If Bj gt Vi it does not matter to 1 whether1 bids Vl or Vl a it matters to j though gtllt If Bj lt Vl 61 again it does not matter to 1 since he pays Bj anyway gtllt If Vl a lt Bj lt Vi it matters since if1 bids Vl a he loses the auction but he wins it with bidding Vi Hence bidding V2 is at least as good as bidding Vl a Combining a and b 1 will not bid above Vi nor below Vi hence he bids Vi o Bidder with the highest value wins the auction The winning bid W is the second order statistic from a sample of size N Given a sample 1117 UN the corresponding order statistics are defined by rearranging 7 the sample such that U1 2 mm 2 Z U7 where WIN is called the 1th order statistic of the sample 242 English auctions o The dominant strategy is similar to that of the Vickrey auction 0 Consider the clock model by Milgrom and Weber 1982 The clock is set initially at some reserve price or zero in the absence of a reserve price The price is allowed to rise continuously A bidder leaves the auction once the current price passes his valuation The auction stops at the point when a bidder bidders leaves such that only one bidder remains in the auction 0 For those nonwinners their strategy is essentially bidding their true value 0 The bidder with the highest value wins the auction The winning bid is again the second order statistic Hence English auctions are sometimes referred to as second price auctions 0 Unlike the Vickrey auction the valuation of the winner is not revealed in English auctions We only know that his valuation is higher than the winning bid the second order statistic o The winner s strategy is seemineg different lntuitively the difference is due to the ex post asymmetry among the bidders that the bidder with the highest valuation does not need to reveal his valuation 243 Dutch and First price seal bid auctions 0 Further assume that the value distribution is defined on 75 and has a continuous density function fV v The value distribution and number of potential bidders N are common knowledge 0 Under the above assumptions the Dutch and first price seal bid auctions are strategically equivalent as the bidder needs to decide how high he is willing to bid 0 Consider a representative bidder bidder 1 with value 111 who bids 51 Under risk neutrality his expected profit is 111 51 Pr wins1 where Pr wins1 Pr 32 S 51 Q SN 3 51 N HPr S S 51 by independence i2 Assume a symmetric Bayes Nash equilibrium Suppose that the N 1 opponents use a common strategy 6r V 7 which is increasing and differentiable in V We then have by monotonicity Pr Sl S 51 Pr CAT l S d 151 FV 5771091 7 which implies Prwins1 FV 771 51W71 by iid assumption It follows that the expected profit is 111 51 FV amp151W 1gt Note that 51 is the only control value for bidder 1 Taking first order condition with respect to 51 yields FV amp7181N 1 111 51 N 1 FV amp151W 2gt fV 5H 51 612 8551 0 2 Symmetry among bidders implies that bidder 1 also uses the strategy 6r such that 81 CAT U1 By implicit function theorem dCAT7181 1 181 CAT U1 Plugging 3 and 4 into 2 and rearranging yields the solution for all feasible values of 111 N fV v v N vaV v FV U FV U Qgt 6 v The solution of this differential equation assuming the boundary condition a y y in the absence of a reserve price takes the form f FV uW71du 01 v W 5 FV v i 0 Now let s take a closer look at the equilibrium strategy which depends on v N and FV below is left as an exercise 7 01 lt v the bidders are deceptive Confirm that do 1 dv gt 0 the higher is the value the higher the bid The auction is efficient in 11 the sense that the bidder with the highest value wins the auction hint one needs to use the Leibniz formula lf Ft fig ftxdx for all t where a b and f are differentiable functions of t then b F t f lt22 b a b t f lt22 a a a t 16 lt22 m czar do vgN dN gt 0 The higher is the number of potential bidders the higher the bid 0 In English auction only the winner bids below his valuation But in Dutch and first price sealed bid auctions the dominant strategy is that every bidder bids below his valuation The difference is because in English auction there is an asymmetry in that all bidders but the winner reveal their valuations by leaving the auction 0 Unlike at English and Vickrey auctions bidders risk attitude matters here Using CRRA specification we obtain f F V WW4 du 0vnNv W Risk averse bidders behave as if they were competing with more opponents than in the risk neutral case Since do vgN dN gt 0 it follows immediately that 80 mmV 877 gt 0 25 A binding reserve price 0 In Vickrey and English auctions the bidders do not change their strategy in the presence of a binding reserve price 7 But the number of bidders will be different because only those with value higher than 7 participate Denote the actual number of participants by N It follows a binomial distribution given by N TL mm FvrN 1 Fvrnn 01N check the definition of binomial distribution 0 Assuming the bidders are unaware of the actual number of bidders in a first price sealed bid auction a binding reserve price does not change bidder strategy except that y is replaced by r such that f Fv WW4 du v FV mm71 r lt v 6 7 01 What is the effect of r on the bidder strategy 0 Note that in English and Dutch auctions the bidding strategy will be different from Vickrey and first price sealed bid auction respectively if there is only one bidder in the auction Bidding behavior depends on observed competition However in Dutch auction a reserve price does not change the decision problem To see this note that we can replace FV v in 5 by CV 11 FV v 1 FV 7 S v and N by n 13 the actual number of bidders which yields f FV 107171 du mil 7 lt v FV v 7 011 26 Expected revenues and optimal auctions 261 Revenue equivalence proposition REP o Riley and Samuelson 1981 show that under IPVP and risk neutrality all four formats generate the same average revenue for the seller They are efficient in the sense that the potential bidder with the highest valuation wins the auction 0 Assuming symmetric IPVP and risk neutrality Riley and Samuelson 1981 focus on a representative bidder bidder 1 Let bidx be the bidding strategy whose argument is the revealed information from the bidder The expected gain to bidder 1 is H mm 111 x Pr winning Expected payment Under risk neutrality one can separate the probability of winning from the expected payment because the utility to bidder 1 is linear in the payoffs Clearly the expected payment is a function of all bids Payment bid 7 bid v2 7 7 bid UN 7 where x is the value reported by bidder 1 Since 1127 7UN are unknown to bidder 1 he uses expected payment in his decision P E Payment bid 7 bid 7 7 bid Since PrbidV lt bidx PrVj lt x by the monotonicity of bid function the probability of bidder 1 winning the auction is Pr v1 lt m y 1 FvN71 Hence bidder 1 s expected profit is 11057111 11va ml1 P 05 Taking first order condition and substituting x by 111 under truth telling we obtain P 111 v1 1Fvv1N 2 fV v1 If v1 r the expected payment P r rFV m 1 Why single out this point So the expected payment for bidder 1 is i N71 v1 P 111 i rFV r P u du rFV MA 1 7 1va 11N 1 FvuN 1du 7 U udFV WA 1 Thus the expected revenue to the seller is N times the expectation of P 111 N qu UV1 FV 1 1 dv dFV u N Mfr10 FV u 1 FvuN 1du Lastly since the derivation does not depend on the auction format the result holds for all four formats discussed above under maintained assumptions o Krishna 2002 shows that one can interpret the expected revenue as the expected value of the second order statistic of valuations from a sample of size N Intuitively the expected revenue or payment in a second price sealed bid auction is the expected second order statistics By REP this applies to other formats too Lecture Seven Ximing Wu 5 MultiUnit Auctions Multiunit auctions refer to the auctions where multiple units of identical objects are auctioned The essential difference between a singleobject auction and a multiunit auction resides not in that there are multiple units in the latter but that the object is divisible Bidders in multiunit auction submit a demand function or price schedule which specifies the price they are willing to pay for each unit they demand How different is it from the purchase decision one has to make say at the grocery 51 Weber 1983 s Classification System 511 Simultaneousdependent auctions o Bidders are required to take a single action that determines both the allocation of the units and the payments to the seller 0 The highest bidders win the available units There are two different pricing rules 1 Uniform price Each bidder who wins some units pays the same price 2 Discriminatory price Each winner pays what he bid for the the unit he has won payasbid auctions 512 Simultaneousindependent auctions The sale of one unit does not depend on the outcome of other sales 513 Sequential auctions 0 One unit or lot containing several units of the same item is sold at a time 0 Additional rules Eg the following two might motivate riskaverse bidders The winner ofthe current sale in the auction often has the option to choose how many of the remaining units to take at the current price The selling price of a previous unit is important in determining the reserve price for subsequent units 52 Pricing Rules Recall that in singleobject auctions bidder behaviors are determined by pricing rules The same principle applies to multiunit auctions Ausubel and Cramton 2002 pointed out uniform price is inefficient and leads to demand reduction For intuition see Figure 51 on page 186 521 Shading and demand reduction 0 Bidders tend to shade their bids and to reduce their demand for larger quanti ties at uniform price auctions If a bidder bids for a sufficiently large quantity there is a positive probability that his bid will be the winning bid at which he has to pay for all units he has won Thus he has an incentive to shade his bid in order to reduce the expected price on earlier units 522 An illustrative example 0 Consider a third price auction where there are two identical goods to be sold with only two bidders We use superscript to index bidder and subscript to index unit of the object Eg we denote the 2th bidders39 valuation for the jth unit by 0 Suppose bidder 1 demands just one unit with valuation V11 V1 Bidder 2 demands two units both with valuation V2 V1 and V2 are independent draws from the uniform distribution on 01 0 Each bidder submits two nonnegative bids The seller ranks them and awards the units to the submitters of the two highest bids The winning price is the thirdhighest bid for each unit won 0 As in Vickrey auction the dominant strategy is to bid the true valuation for 5 the first unit demanded o Bidder one bids zero for the second unit bidder two bids B3 which clearly is no greater than B why There are two likely outcomes B lt V1 Bidder 2 wins one unit and pays max BS 0 B3 2 V1 Bidder 2 wins both unit and pays V1 for each why 0 Bidder 239s expect profit H2 B3 V2 is B2 HNBavhVt BQO J e22 devl 0 VW u Vth Clearly it is maximized with B 0 which is independent of V2 0 The equilibrium outcome is each bidder wins one unit and pays the third 6 highest price zero This uniform price auction is obviously not efficient in what sense 523 A general model 0 Consider a divisible good normalized to have quantity one There are N bidders The seller39s valuation is zero 0 Bidder 2 can consume up to quantity C21 6 0 All where All E 01 Com petition for the good exists ie A1A2ANgt 1 0 Under IPVP bidder 2 gets his valuation Vi from a distribution defined on 01 Bidder i has a flat marginal value Vi for the good up to his capacity Al Consuming Q E 0 All and paying RT his payoff is 0 Auction mechanism Each bidder simultaneously and independently submits a demand curve ql pll t 0 1l gt lOaAll This demand curve expresses the quantity that bidder 2 is willing to purchase at price p This function is required to be weakly decreasing o The marketclearing price p0 is determined by the highest rejected bid N 190 inf 29 23 pll g 1 2391 Note the above price is defined as the infimum or minimum price at which 8 the aggregate demand is less than 1 Since the demand function is monotonic it is equivalent to the highest rejected price 0 Allocation If N qu 2907 V1 17 1 then each bidder 2 is assigned a quantity Ql ql p0 Vi If N qu 2907 V1 gt17 i1 then the aggregate demand curve is flat at 190 and some bidders39 demands at p0 will be rationed Lecture Two Ximing Wu 262 Optimal reserve price 0 The sellers can choose auction format and reserve price By REP auction format does not affect expected revenue How about reserve price 7 0 Assuming IPVP and risk neutral bidders The utility to seller is the sum of the expected revenue and the expected utility of retaining the object v0 11on m N f us u FV u 1 FvuN 1du Solving the first order condition with respect to r and replacing r with p yields 1 FV Pi 7 W 0 Suppose v0 6 01 and FV is the uniform distribution onl071 What is p What do we learn from this thought experiment Think in terms of the potential consequences of placing a reserve price too low or too high 263 Expected revenue and risk aversion 0 Under risk aversion REP doesn t hold 0 Consider Dutch and first price sealed bid auctions Let 6r and Er be the bid functions for risk neutral and risk averse bidders respectively The expected utility for a risk averse bidder is U 111 ar 95 FV ml1 Its first order condition gives U011 5 111 N 1 fv v1 UU1 ampU1 Flt11 5011 lf UY Y the corresponding solution for a risk neutral bidder is N 1fvv1 CAT lv1 ET U1l FV v1 But for strictly risk averse bidder with U 0 0 we have WY WgtY This is left as an exercise It follows that for some 111 where 604 2 fr 111 7 we have N 1fvv1 7011 2 111 CAT 111 FV v1 CATU1 such that 6111 Z 67 for all vi 2 v1 Assume that y 0 and 60 6r0 0 it follows that 604 2 fr 111 for all positive 111 264 Optimal auctions o Myerson 1981 constructed the optimal selling mechanism under a set of rather general conditions See textbook p43 47 for details 27 Winner s Curse o Winner s curse can be a potential problem in CVP but not in IPVP o In CVP bidders base their bids on private information The winner is the bidder with the most optimistic 3 estimate Since the estimate is assumed to be unbiased on average the most optimistic estimate is an over estimate and thus the winner would pay more than the true value 0 Wilson 1977 showed that this argument does not necessarily hold Rational bidders anticipate this possibility and take it into account He showed that as the number of bidders got largethe winning bid converged almost surely to the true value of the object Lecture Six Ximing Wu 43 Binding Reserve Price 0 The number of actual bidders N1 is different from number of potential bidders M in auction if when there is a binding reserve price For simplicity assume M is a constant N for all auctions Also assume 7 is observed 0 Denote 1033 the truncated distribution of bids in the presence of a binding reserve price 7 Recall the optimal strategy of a bidder with valuation 7 is 00 7 We have F0 F0 verify Gg 7 0 and GE 3 1 and 0 1 x 958 alltvgt1 F30 To see this note that 91731 i 9173 0 1 93 i 93 We now have 1131 1 F3 7 G03 8 F3 7 0 v i 1 f3 v i 1 F3 7 9 8V 2 Recall from Chapter 2 fv lt2 i Fv ltvgt l Fv lt2 0 u Fv u n lt21 N 1 8 i1 F3ltrgti GgltsgtF3ltrgt N 1 l1 ltrgti ggltsgt 0 1 Gg 8 F3 7 8N 1lgltslll F3ltrlgltsl39 1 Thus the truncated distribution of F3 is identified by Gg note that distribu v0v tion of F3 1 with u lt 7 is not identified because no information is available for that region c When there is no binding reserve price 1 1738 2 N 1 rs lt8 l Relation 2 is actually a limiting case of For a reserve price to be non binding we need to have 7 y Under this condition we have G03 8 F30 8 9 8 fg 8 F3 r 0 2 follows immediately 0 In practice often N and F30 are not available To estimate them we assume 7 and N are constant across auctions A natural estimator for N is A T N 1n 1axZltt1 Let NT ZtT1 N15 we estimate F3 7 by A N FV1 AT N 0 Next we need to estimate Gg and 9 It is known that ggsaooassr where means approaching from aboveright lntuitively this is due to the jump39 in Gg at 7 To solve this technical difficulty we make the following transformation SrS 7 It follows that Ggr 3r Gg 7 33 9 8r 28mg 7 83 7 where 9 is bounded on its support When 8 7 3r O 7 83 7 92 03 gt 00 However sr gt 0 at a comparable rate thus 9 3r 229 7 33 is bounded when 8 7 We then have v33r 2s aw F80 1 N 1 gals 1 F3T9T8r c We can now estimate Ggr and 9 by 1 T 1 Nt G37 57 i 2 t 2 1 Shit 74 S 57 7 151 39 1 T Nt A 1 1 3711 57 gSrlt8r E E where Sn Sit 7quot o The pseudovalue is then computed as A G0 SM F0 1 WtSr7itr it Agrlt 70 Vfga A0 N 1 95 Smut 1 FV T 98 573115 The density of valuation conditional on V Z 7 is estimated by 1 T 1 Nt v A Vgt L fV39VZTM J Th ZNttzg lt hg 9 151 and its unconditional counterpart can be obtained as fVlU 1 Fvl fVV2r WW Z T 44 Riskaverse bidders oAssuming CRRA utility Uu nvln 77 Z 1 The BayesNash equilibrium bid function is N fJFS my du 01 1 F3 mmW1 It follows that Evaluating it at the upper support 17 yields HH 1 FM 770N 1 fglgo Given knowledge of 17 we identify the risk parameter 770 Given an estimate of 770 one can then estimate 0 Similarly when there is a binding reserve price 1 6 lt8 F3 r 77N 1 98 1 F3lt gtl9f lt8gt 39 The same strategy can be used to estimate 77 and the distribution 1731 for us qu
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