Setting the Upset Price in British Columbia Timber Auctions

Transcription

1 Settng the Upset Prce n Brtsh Columba Tmber Auctons Susan Athey, Peter Cramton, and Allan Ingraham Market Desgn Inc. and Crteron Auctons 2 September 2002 SUMMARY An mportant element of tmber auctons s the upset the mnmum acceptable prce, often called the reserve prce n other aucton envronments. The upset has three man purposes: () to guarantee substantal revenue n auctons where competton s weak but the upset s met, (2) to lmt the ncentve for and the mpact of collusve bddng, and (3) to provde useful nformaton to bdders. We analyze the determnaton of the upset n Brtsh Columba tmber auctons. Settng the upset too hgh results n unsold stands and produces an upward bas n prce f the compettve auctons are used to determne stumpage rates for non-auctoned tmber. Settng the upset too low wll reduce aucton revenue and can create downward bas when the aucton prces are used to calculate tmber prces for non-auctoned stands. It s therefore mportant to set the upset at or near the optmal level. We present the theory of upset prcng and then apply that theory to the data avalable from hstorcal tmber aucton sales n the BC Interor from 999 to We fnd that an upset of about 70 percent (a rollback of 30 percent) maxmzes aucton revenues f the Mnstry values tmber at about 52 to 56 percent of ts apprased value. Ths upset strkes the rght balance between enhanced revenues and unsold tmber stands. Gven ts mportance, ths upset calbraton should be refned as addtonal data becomes avalable to assure that the upset s not set too hgh or too low.. Susan Athey s Assocate Professor of Economcs at Stanford Unversty and a Prncpal of Market Desgn Inc. Her research focuses on aucton theory and the statstcal analyss of aucton data. She has publshed on a wde range of topcs n ndustral organzaton, ncludng market domnance and colluson. Peter Cramton s Professor of Economcs at the Unversty of Maryland and Presdent of Market Desgn Inc. He has advsed numerous governments on market desgn n energy, telecommuncatons, forestry, and the envronment. Hs research focuses on auctons, barganng, and market exchange. He has publshed many artcles on aucton theory and aucton practce n major journals. Allan Ingraham s Vce Presdent at Crteron Auctons. Hs areas of expertse are aucton desgn and strategy, detecton of bd rggng, ndustral organzaton, and econometrcs.

2 INTRODUCTION The Brtsh Columba Mnstry of Forests asked us to analyze the choce of the upset n Brtsh Columba tmber auctons. The upset, commonly called the reserve prce n other aucton envronments, 2 has three mportant functons: () t assures that aucton revenues wll be substantal even f competton s weak provded the upset s met, (2) t reduces the ncentve for, and mpact of, collusve bddng, and (3) t provdes useful prcng nformaton to potental bdders. Determnng a sutable upset s mportant n any aucton. But n the Brtsh Columba tmber auctons, the upset s mportance s magnfed, because an upset that s too hgh can upwardly bas the tmber prces pad for non-auctoned stands. In partcular, the prces for non-auctoned tmber are estmated usng the aucton prces from successful auctons. An aggressvely hgh upset wll lead to a sgnfcant percentage of unsuccessful auctons auctons wthout any bdders wllng to meet or exceed the upset. Those auctons are elmnated when estmatng the prces for non-auctoned tmber. However, those observatons occur when the market value s lower than expected, and therefore the average prce at successful auctons s greater than the average market value of all auctoned tmber. Such prcng creates an upward bas when successful aucton sales are used to prce non-auctoned tmber. Smlarly, settng an upset that s too low can downwardly bas the prce estmates for non-auctoned tmber. Theory predcts that bdders wll bd less aggressvely when the upset s too low. Furthermore, an nadequate upset creates greater scope for market manpulaton and collusve bddng. Gven the mportance of the upset n the B.C. tmber auctons, we recommend that the Mnstry of Forests recalbrate the upset perodcally. After the frst year of operaton of the program, the upset should be revewed. It also can be done when the prcng model s reestmated, whch lkely would occur every few years. At tmes of structural change, t may be necessary to temporarly adjust the upset on an emergency bass. If many stands are unsold the Mnstry should consder lowerng the upset. We begn by presentng the theory of upset prcng. We then apply the theory to the hstorcal tmber auctons n the Brtsh Columba Interor from 999 to Our analyss suggests that an upset of 70 percent (a rollback of 30 percent) maxmzes aucton revenues f the Mnstry values tmber at about 52 to 56 percent of ts apprased value. 2 THE THEORY OF UPSET PRICING Consder a seller auctonng off a sngle tem (a stand of tmber) to n rsk-neutral bdders, ndexed by =,,n. Each bdder,, has a prvately known value for the tem v. Nether the seller, nor the other bdders, know 's value; they know only that t s drawn from the probablty dstrbuton F wth postve densty f on support [0,h], ndependently from the other values v j, j. Ths model s called the symmetrc ndependent prvate values model, snce each bdder looks the same to the seller ex ante, and each bdder has a prvate value for the good, whch s drawn ndependently from other bdders values. The seller s value s denoted v 0. The seller receves a set of bds, and then chooses whether to accept them. In any equlbrum to the bddng game, the bddng strateges are strctly ncreasng. Thus, a - mappng between bds and values exsts. It s easer to descrbe the seller s problem n terms of the bdders values. So, we thnk of the seller as recevng a vector of bds b, nferrng what the correspondng values are v, and then determnng f one of the bdders receves the tem, or rather f the seller keeps t. That s, the seller chooses a ( v ) {0,}, where a ( v ) = ndcates that bdder s recevng the tem. If a ( v ) = 0 for all, the seller keeps the tem. 2. In ths paper we use the term upset to ndcate the mnmum bd prce that the Mnstry wll accept for a gven tmber stand. We use the term reservaton value to ndcate the Mnstry s estmate of a stand s value f that stand were ether resold at a later aucton or harvested by the Mnstry tself. 2

3 Usng ths formulaton, one can show that the seller s profts equal () n E a ( v ) v v v,.., vn = 0 Fv ( ) f( v ) The dervaton of ths result s beyond the scope of ths paper, but t s standard n the lterature; see, Myerson (98) or Bulow and Roberts (989). Hence, to maxmze profts, the seller allocates the tem to the bdder that has the hghest margnal revenue: Fv ( ) (2) MR( v) = v v0. f ( v ) Margnal revenue s less than the gans from trade (the gap between the value of the tem to the buyer and the seller). The reason s that the bdder s able to capture some profts as a result of the bdder s prvate nformaton. For example, n a frst-prce sealed-bd aucton, the bdder shades ts bd below ts true value n order to maxmze ts profts. Bd shadng allows the bdder to extract a porton of the gans from trade. By settng an upset prce above v 0, the seller elmnates bd shadng by a bdder wth a value equal to the upset. By mplcaton, the seller also reduces the bd shadng of bdders wth values above the upset, snce these bdders are competng to wn the tem from other bdders that are bddng more aggressvely as a result of the upset. The ncentve to bd hgh ncreases wth a reducton n the lkelhood that a low bd wns the aucton. By refusng to sell the tem even at bds above the seller s value, the seller makes t less temptng to bd low, and bd shadng s reduced. It s standard to assume that MR(v ) s ncreasng n v. Ths s called the regular case, and holds for many common probablty dstrbutons (unform, exponental, etc.). In the regular case, the seller maxmzes profts by sellng to the bdder wth the hghest margnal revenue, provded the margnal revenue s non-negatve. Hence, the upset prce s set at r such that MR(r) = 0; n other words, r s the value that solves Fr ( ) (3) r v0 =. f () r Clearly, t wll not be worthwhle for any bdder wth values below r to bd at or above the upset, whereas all bdders wth values greater than r gan some surplus by bddng above the upset. A bdder wth value r smply places a bd equal to r. In the regular case, snce MR(v ) s ncreasng n v, f the seller fnds t optmal to award the tem to anyone, the seller awards t to the bdder wth the hghest value v max = max {v,,v n }. Symmetry mples ths result. The seller assgns the good effcently to maxmze the gans from trade subject to the constrant that the seller never sell at a prce below r. In the benchmark model, the optmal upset prce does not depend on the number of bdders. Ths result s a good frst-approxmaton even when we relax one or more of the assumptons of the benchmark model. The purpose of the upset s to guarantee substantal revenues even when competton s weak. Indeed, the optmal upset s the same as the optmal take-t-or-leave-t offer that the seller would make to a sngle buyer wth a value v drawn from the dstrbuton F. To get a sense of how the optmal upset vares wth the dstrbuton F, consder the example where v 0 = 0 and v s dstrbuted accordng to the dstrbuton Fv ( ) = v α, on [0,]. For α =, values are unformly 3

4 dstrbuted, MR(v ) = 2v, and the optmal upset s r = ½. Notce that although the upset does not n change wth n, the probablty of trade, Fr () = n, s very much dependent on the number of bdders. 3 2 As the seller gets more confdent that the expected value s near one, the upset ncreases. However, the probablty of trade also ncreases, snce the no-trade opton becomes less attractve to the seller as the expected gans from trade ncrease. Fgure and Table, below, llustrate these two ponts. Fgure dsplays the optmal upset and the dstrbuton functon F for α =, 2, 4, 8, and 6. Table dsplays the expected hghest value, the optmal upset, and the probablty of trade wth n = 4, for α =, 2, 4, 8, and 6. FIGURE. VALUE DISTRIBUTION FUNCTIONS AND THE CORRESPONDING OPTIMAL UPSET The fve horzontal lnes n Fgure represent the optmal upset prces that correspond to the fve upward slopng lnes, whch are the value dstrbuton functons, F(v). The red upward slopng lne s the dstrbuton functon when α =, and therefore F(v) = v. The upward slopng purple lne corresponds to F(v) = v 6. The horzontal purple lne at an approxmate value of.84 represents the optmal upset for the value dstrbuton F(v) = v 6. Because the dfference between the dstrbuton functon evaluated at any two ponts along the x-axs represents the probablty that a bdder draws a value between those two x- axs ponts, the movement from the red upward slopng lne to the purple upward slopng lne ndcates a that bdders are more lkely to have hgher values. Put dfferently, when the dstrbuton functon s F(v) = v, the probablty of a bdder havng a value between.95 and.9 s F(.95) - F(.9) =.05. If the dstrbuton functon s F(v) = v 6,however, the smlar calculaton becomes (.95) 6 (.9) 6 =.255. There s a 5 percent chance of a bdder havng value between.9 and.95 f the dstrbuton functon s F(v) = v, but there s a 25.5 percent chance that a bdder has a value between.9 and.95 f the dstrbuton s F(v) = v 6. The upset ncreases as the dstrbuton puts more weght on hgher values. Now, consder Table below, whch dsplays the expected hghest value, the optmal upset, and the probablty of trade wth n = 4, for α =, 2, 4, 8, and The probablty of trade s the Prob{at least one bdder has value above r} = Prob{all bdders have value below r}. Thus, the probablty of trade s as follows: Prob{ trade r = } = F( r) = n. n 2 2 4

5 TABLE. OPTIMAL AUCTION CHARACTERISTICS WITH 4 BIDDERS α Expected Hgh Value Optmal Upset Probablty of Trade The expected hgh value 4 ncreases wth α, because hgher values are more lkely when α ncreases. The probablty of trade also ncreases wth α. In the model above, the optmal upset does not depend on the number of bdders. Ths result requres that the bdders are not colludng. However, f the bdders are colludng to keep the aucton prce low, then the optmal upset ncreases wth the number of bdders. In the extreme case, the n bdders are colludng perfectly, wth all bds at or just above the upset, provded one of the bdders has a value above the upset. Ths stuaton s equvalent to a seller sellng to a sngle bdder wth valuaton drawn from the dstrbuton F n. The optmal upset n ths case solves n Fr ( ) (4) r v0 = n nf() r f () r Table 2 shows the expected hgh value, the optmal upset when v 0 = 0, and the probablty of trade for the unform dstrbuton on [0,] by the number of bdders from to 6. TABLE 2. OPTIMAL AUCTION WITH COLLUDING BIDDERS Number of Bdders, n Expected Hgh Value Optmal Upset Probablty of Trade When bdders collude, the optmal upset ncreases wth the number of bdders. Because the expected hgh value ncreases wth the number of bdders, the gans from trade wll also rse wth the number of bdders. However, the cartel wll capture all ncremental gans from trade f the upset remans constant when the cartel adds new members. To capture a share of the ncremental gans, the seller must ncrease the upset prce. 5 Snce we beleve that Brtsh Columba has establshed the necessary laws and enforcement to dscourage colluson, we focus on the non-collusve case n the next secton. The underlyng theory and a dscusson of the estmaton technque s presented n Appendx B. Appendx A presents an example showng how an optmal upset s determned from sample data. 4. The expected hgh value s nothng more than the expectaton of the hghest order statstc. 5. The reader may notce that the optmal upset prces n Table and 2 are the same. Ths s smply a mathematcal concdence of ths partcular example. The fact that the expected hgh value and the probablty of trade are equal n Table 2 s also an artfact of ths partcular example. 5

6 3 APPLYING THE THEORY TO ESTIMATE THE OPTIMAL UPSET IN BRITISH COLUMBIA The analyss above ndcates that the optmal upset prce depends crtcally on the shape of the dstrbuton of bdder valuatons. A growng lterature n appled econometrcs s devoted to estmatng such value dstrbutons. Ths research uses aucton theory and aucton data (bds, the number of bdders, etc.) to estmate the underlyng dstrbuton from whch bdders draw ther values. Once the value dstrbuton s estmated, the optmal upset prce can be determned usng the theory from the benchmark model. Below, we dscuss the varables a researcher would requre to perform ths estmaton. We then outlne the estmaton, and present our results. Our estmaton s based on auctons conducted under the Small Busness Forest Enterprses Program (SBFEP) n the BC Interor from 999 to The BC tmber auctons are frst-prce sealed-bd auctons. 6 The Mnstry currently records the wnnng bd, rather than all the bds. The Mnstry also records the number of bdders at aucton. In our analyss, we assume that the bdders are symmetrc n the sense that each bdder s value s drawn from the same probablty dstrbuton. Because the dataset represents loggers only, ths assumpton s reasonable. In the future, however, both loggers and tenure holders may bd, n whch case the Mnstry should consder applyng an asymmetrc bdder assumpton (covered n Athey and Hale (2002) for the case when only the wnnng bd s observed). We also assume that condtonal on the apprased value of the tract, bdders' valuatons are ndependent of one another. Ths assumpton s commonly made n the lterature, and s reasonable f the apprased value ncorporates most of the mportant characterstcs of the tract that affect harvestng costs and the value of the tmber. The non-parametrc estmaton for a frst prce aucton wth symmetrc bdders and ndependent prvate values s studed n Guerre, Perrgne, and Vuong (GPV) (995) and GPV (2000). GPV (995) provde dstnct estmaton procedures for two alternatve scenaros: () an explct upset prce already exsts n the market and (2) only the transacton prce s observed. GPV (2000) shows the asymptotc propertes of the non-parametrc estmator when all the bds are observed. Here, we combne the two procedures n the 995 paper to correct for the fact that the SBFEP auctons already have an upset and only the wnnng bd s observed. 7 The nonparametrc procedure s mult-step. Frst, we estmate the number of potental bdders. The number of potental bdders s equal to the number of bdders that would have placed a bd n the absence of an upset prce. However, because some bdders have values below the upset, only a subset of the potental bdders actually places a bd n the aucton. Followng GPV (995), as a proxy for the number of potental bdders, we use the maxmum of the observed number of bdders n a gven group (.e. tmber areas, whch are pooled dstrcts wth smlar characterstcs). Because the bd functon depends on the number of potental bdders, the estmaton procedure s performed separately for auctons wth dfferent numbers of potental bdders. We assume that the values, call them ν are generated multplcatvely through the apprasal, α, as follows: v = αξ, 6. See Paarsch (997) for an analyss of BC s earler oral auctons. 7. In the future, we recommend that the Mnstry should collect data about all bds, rather than just the wnnng bd, because the estmaton procedure wll be far smpler and more powerful. However, to remove any ncentves bdders mght have to place low-ball bds that have lttle chance of success, n the hope of manpulatng future reserve prce calculatons, t may be prudent to use only the top two or three n the estmaton. Athey and Hale (2002) show how such data can be ncorporated. 6

7 where ξ s an d error term. Hence, we take the log of both sdes to get the standard addtve model: log( v ) = log( α) + ε, where ε = log( ξ). As we show below, a non-stochastc component n the values wll carry through drectly to the bds. Therefore, f one were to subtract the log of the apprasal from the log of the bds, one would have then constructed d bds. Thus, our non-parametrc analyss s performed on the percent varaton of the bds away from the apprased stand value. Smlarly, our value estmates are really estmates of the varaton of the correspondng values away from the apprased stand value. To smplfy language below, we wll refer to these values as bd dsturbances and value dsturbances. A bd dsturbance of -.25 ndcates that the wnnng bd was 25 percent below the apprased value. An estmated value dsturbance of. translates to an estmated value that was 0 percent above the apprasal. We now proceed wth an overvew of the estmaton method. The estmaton method proceeds by usng a kernel to estmate the dstrbuton of wnnng bd dsturbances. The theory of equlbrum bddng specfes that each bdder must respond optmally to the antcpated dstrbuton of opponent bd dsturbances. In an ndependent prvate values aucton, the dstrbuton of dsturbances conveys enough nformaton for the researcher to calculate the dstrbuton of bds that any gven bdder expects to face at aucton. Ths dstrbuton dffers from the dstrbuton of wnnng bds, snce a bdder s own bds enter nto the latter dstrbuton. Then, one can determne for each bd dsturbance, the value dsturbance and correspondng value that would make ts bd optmal. Formally, suppose that there s no upset prce, so that the number of potental bdders s known and fxed at N, and suppose that the dstrbuton of wnnng bds s G () ( b ). Then, the dstrbuton of a typcal () bdder s bds s Gb ( ) = ( G ( b)) N. Antcpatng ths dstrbuton of the maxmum rval bd, a bdder wth value v solves: max ( v )( ( )) N b b G b, snce the bdder wll wn wth bd b when all N opponents bd less than b. Takng the frst-order condton, the bd b correspondng to value v must solve or, rearrangng, 2 ( )( )( ( )) N N v b N G b g( b) ( G( b)) 0 =, (5) Gb ( ) v= b+. ( N ) g( b) We use ths relatonshp to nfer what the dstrbuton of values must be. In partcular, we use a kernel regresson to estmate the rght-hand sde of (5), usng the observed bds. Then, for every observed bd b, we substtute that b nto our estmate of the rght-hand sde of (5), and nfer what the value v must have been to satsfy the equaton. Gven ths constructed dataset of nferred or pseudo values, we can then use a kernel densty estmator to estmate the probablty densty of the value dstrbuton. In turn, we ntegrate the densty to obtan an estmate of the dstrbuton of values. In Appendx B, we outlne the methodology n more detal, and show how t extends to the case where there s an upset prce. We also dscuss how we subtract the non-stochastc porton of the bds (assumed to be the apprased stand value) and run the estmaton on the resultng bd dsturbances. 7

8 3. Results from nonparametrc estmaton for the Interor To account for regonal characterstcs n the auctons, we run the nonparametrc estmaton by geographc areas. The areas are Forest Dstrcts whose bd data we pooled together because common bdders tend to compete wthn the combned area. We lst these areas, and the assocated Forest Dstrcts n Table 3. 8 TABLE 3. ESTIMATION AREAS AND ASSOCIATED INTERIOR DISTRICTS Bddng Area Dstrcts Comprsng the Area Observatons Fort Nelson, Fort St. John, Dawson Creek 59 2 MacKenze, Prnce George 30 3 Prnce George, Robson Valley, Quesnel, Ft. St. James, Vanderhoof 96 4 Bulkley-Casser, Kspox, Kalum 6 5 Quesnel, Horsefly, Wllams Lake, 00 Mle House, Chlcotn, 78 Vanderhoof 6 Lllooet, Kamloops, Clearwater, Merrtt 53 7 Salmon Arm, Vernon, Pentcton, Boundary 27 8 Columba, Invermere, Cranbrook 9 9 Arrow, Boudary, Kootenay Lake 33 0 Lakes, Maurce, Vanderhoof 4 Certan bddng areas contan the same Forest Dstrct. For example, Prnce George Dstrct s ncluded n both Area 2 and Area 3. The overlap between areas wll not affect our estmaton wthn a sngle area, because we perform the estmaton separately for each area. We frst estmate the potental number of bdders n the area as the maxmum number of bdders that bd for any sngle aucton n that area. We then assume a probablty of any bdder comng to the aucton at an upset rate of 70 percent. Ideally, ths probablty should be calculated from the data, but gven current sze lmtatons ths proves dffcult. Thus, we proceed by varyng ths probablty and recalculatng our results, notng f substantal changes occur. After a kernel densty estmaton of the observed bds, we then retreve the pseudo values that s, the estmates of the underlyng values. These steps are also done ndvdually, by area. As an example, the bd functon that we estmated for Area 3 (Prnce George, Robson Valley, Quesnel, Ft. St. James, and Vanderhoof dstrcts) s shown n Fgure Gurerre, Perrgne, and Vuong (2000), show that sample szes of 200 generate hghly accurate results. Our estmaton contans sample szes much less than 200. In the future, more data should be collected to ncrease the sample sze, and mprove the accuracy of the estmates. 8

9 FIGURE 2. ESTIMATED BID FUNCTION FOR PRINCE GEORGE, ROBSON VALLEY, QUESNEL, FT. ST. JAMES, AND VANDERHOOF DISTRICTS Estmated Bd Dsturbance Estmated Value Dsturbance In Fgure 2, we see that the bd dsturbances the varaton of the log bds away from the log of the apprased stand values n Area 3 vared from between 33 percent below the apprasal and 8 percent above the apprasal. 9 Gven ths varaton, we estmate that the underlyng values vared from between 9 percent below the apprasal and 86 percent above the apprasal. 0 An mportant specfcaton test for the estmaton procedure s to verfy that the nverse bd functon s monotonc. Only a monotonc bd functon s consstent wth theory. (Note that we plot the nverse bd functon wth values on the x-axs and bds on the y-axs, so that t can be nterpreted as a bd functon; a non-monotoncty n the nverse bd functon corresponds to a backward bend n the curve.) Above we see that the estmated bd functon s ncreasng, as requred. We can therefore proceed to the next step, more confdent n our results. Gven the estmated nverse bd functons, such as those dsplayed n Fgure 2 above, we then compute the pseudo-value for each bd. The pseudo-values, and correspondng percent varatons away 9. Because we express the dsturbances n log form, the percentages we descrbe dffer from the percentages the mnstry uses. In partcular, a log bd dsturbance that s 33 percent below the apprasal represents a bd that exceeds the 30 percent rollback used for the upset. The upset s determned as the product of 0.7 and the apprased value. The log of 0.7 s Thus, a log bd dsturbance that s 33 percentage ponts below the apprasal corresponds to a bd that exceeds the upset. 0. Our estmate of the number of potental bdders n area 3 s nne. On average, over four bdders submtted bds n ths area. 9

10 from the apprasals, can then be used to estmate the densty and dstrbuton functons of those pseudovalues. A kernel densty estmator and numercal ntegraton are used for ths procedure. Because we only observe the wnnng bds, we plot the densty and dstrbuton of the hghest ordered value (among all of the bdders). We then transform these denstes and dstrbutons so that they correspond to the value dstrbuton of a typcal bdder, rather than the dstrbuton for the hghest-value bdder. Fnally, the resultng dstrbutons can be used to determne the optmal upset. Fgures 3, 4, and 5 contan graphs of these estmates. Fgure 3 contans the estmated densty of the hghest pseudo value n Area 3, whle Fgure 4 contans the correspondng dstrbuton estmate. Then, n Fgure 5, we present the estmated dstrbuton of the underlyng value dsturbances. FIGURE 3. ESTIMATED DENSITY OF VALUE DISTURBANCES FOR PRINCE GEORGE, ROBSON VALLEY, QUESNEL, FT. ST. JAMES, AND VANDERHOOF f () (e) e 0

11 FIGURE 4. ESTIMATED DISTRIBUTION OF THE HIGHEST VALUE DISTURBANCES FOR PRINCE GEORGE, ROBSON VALLEY, QUESNEL, FT. ST. JAMES, AND VANDERHOOF F () (e) e

12 FIGURE 5. ESTIMATED DISTRIBUTION OF THE VALUE DISTURBANCES FOR PRINCE GEORGE, ROBSON VALLEY, QUESNEL, FT. ST. JAMES, AND VANDERHOOF F(e) e In Fgures 3 and 4, we see that the kernel process estmates a smooth densty and dstrbuton functon for Area 3. However, lumps n the underlyng densty do occur n some areas. The estmaton s more challengng n those areas. For example, the bd densty n Area 5 was bmodal even at hgh bandwdths, makng t dffcult to extract nformaton for ths Area. 3.2 Usng the estmates to determne the optmalty of the upset rule Recall from above that the margnal revenue of the bdders s Fv ( ) MR( v) = v v0, f ( v ) when values are expressed n levels. However, we chose to estmate the logs of the bds rather than the bds themselves, and therefore must account for ths transformaton n the margnal revenue equaton. Let H be the probablty dstrbuton for the log of the bdder values. Then, H(log( v )) MR(log( v)) = log( v) log( v0 ). h(log( v )) 2

13 Recall that ε s the stochastc component of the log values, so that ε = log( v α). Let Φ be the dstrbuton functon of ε and let φ be the densty. Then, we have Φ(log( v α)) MR(log( v)) = log( v) log( v0 ). φ(log( v α)) We now take ρ to be the upset rate (for the Mnstry, ths s.7), so that the mnmum bd accepted wll be α ρ. Then, we must set an optmal upset that solves Φ(log( ρ)) (6) log( ρ) = log( v0 ) log( α) φ(log( ρ)) The optmal upset depends crtcally on the Mnstry s reservaton value v 0 for the stand, expressed as a percentage of apprased value. Snce t s mpossble to estmate the optmal upset rate ρ wthout knowng the reservaton value v 0, we nstead fx ρ and estmate the reservaton value that would yeld an optmal upset rate of ρ. Also, snce the actual upset rate was 70 percent, whch truncated the bd dsturbances that we observe at 70 percent, we can only perform the estmaton for values of ρ greater than 70 percent. We therefore consder an upset rate of 79 percent and ask the queston, What Mnstry reservaton value, v 0, would make such an upset rate optmal? Usng 20 percent as the probablty of drawng a value dsturbance less than the reserve rate, we fnd that an upset rate of 79 percent concdes wth a reservaton stand value that s between 55.3 percent and 56.9 percent of the apprased stand value. Put dfferently, the Mnstry would need to value the tmber for ts own purposes at between 55.3 percent and 56.9 percent of the apprased value for an upset rate of 79 percent to be optmal. These estmates are precse; the bootstrap standard errors are small, and the bootstrap confdence ntervals ndcate that the Mnstry s true reservaton value s between 52 and 62 percent of the apprased value wth 95 percent probablty. 2 We preset these results n Table 4 for the areas where we are able to estmate the underlyng value dstrbuton wth a comfortable degree of precson.. 79 percent s the smallest upset for whch the left-hand sde of equaton (6) allows us to estmate the mplct reservaton value for Areas, 2, 3, 4, 6, and For a dscusson of the methodology used to obtan the bootstrap standard errors and confdence ntervals, see Secton 2 of the Appendx A. 3

14 Area TABLE 4. INFERRED MINISTRY RESERVATION VALUES, ASSUMING AN OPTIMAL UPSET OF 79 PERCENT Reservaton Value as Percent of Apprasal, Assumng Optmal 79% Upset Bootstrap Standard Error Bootstrap 95% Confdence Interval (Percentle Method) Average Apprased Value ($) Inferred Mnstry Reserve for Optmal 79% Upset ($) (53.4, 60.) (52.8, 6.6) (54.8, 58.) (52.8, 58.5) (53.0, 59.9) (53.8, 60.4) Results n Table 4 ndcate that the Mnstry s upset rule sgnfes a hgher opportunty cost of auctonng tmber than the smple costs of reforestaton, and land management. The Mnstry s nternalzng ts alternatve of lettng the tmber at aucton. In partcular, t could cut the tmber tself and sell the end product, or t could hold the tmber for sale at a later aucton. The Mnstry would then make more money through ths polcy than t would by acceptng the statutory mnmum for ts tmber. To better gauge the optmal upset, one must better understand the Mnstry s true reservaton value relatve to ts prvate-sector apprasal. However, we cauton that for low Mnstry reservaton values, we wll be unable to calculate the optmal upset rule, because data n the relatve porton of the value dstrbuton wll be unavalable. That s, we cannot nfer the shape of the value dstrbuton n regons of the dstrbuton that are censored by the upset prce. Extrapolatng from Table 4, we run a lnear regresson on our optmal upset results n the range.74 to.99, and then nfer back to obtan the value of v 0 that mples an optmal upset of 70 percent. 3 In partcular, we apply a quadratc specfcaton to the data, and estmate that equaton usng least squares. We obtan the regresson coeffcents, and then predct what the reservaton rate would be at an optmal upset of 70 percent f had we been able to observe those data. Table 5 contans these results. 3. The regresson range s dfferent for each Area, but each range s 20 percentage ponts wde. For example, we could have determned the reservaton rate for an optmal upset of.75 n Area. Thus, we use the range.75 to.95 for the lnear regresson, and predct the optmal upset at 70 percent. Lkewse, we could only determne the reservaton rate for an optmal upset of 79 percent n Area 4 because of data restrctons. The regresson sample n Area 4 therefore ranges from upset rates between.79 and.99. 4

15 TABLE 5. LEAST SQUARES PREDICTION OF THE MINISTRY S RESERVATION VALUE GIVEN AN OPTIMAL UPSET OF 70 PERCENT Area Reserve Value as Percent of Apprasal, Assumng Optmal 70% Upset Bootstrap Standard Error Bootstrap 95% Confdence Interval (Percentle Method) (44.4, 66.5) (47., 59.7) (52., 57.9) (45.0, 57.7) (42.8, 65.3) As Table 5 ndcates, we estmate that an optmal upset of 70 percent mples that the Mnstry s reservaton value s between 5.8 percent and 55.8 percent of the apprasal. 4 These estmates are less precse than the estmates presented n Table 4. The bootstrap standard errors are at least twce as large, and the bootstrap 95-percent confdence ntervals n some cases exceed 20 percentage ponts. The lower amount of precson stems from the out-of-sample predcton of the left-hand sde of equaton 6. The natural varaton n the shape of ths expresson, nduced by re-samplng, s magnfed by our lnear regresson model. 5 Put dfferently, small changes n the estmated quadratc equaton are magnfed when we predct far away from the observed data. Fgure 6 graphcally llustrates ths pont. 4. We cannot estmate the reserve value for Area 0 at an upset of 70 percent because, even after extrapolaton, the left-hand sde of equaton 6 s not ncreasng monotoncally n r. The lowest upset for whch we can estmate the reserve value s 78 percent, whch yelds an mplct reserve value of 56.4 percent of the apprasal. 5. For a dscusson of bootstrap termnology, see Secton 2 of the Appendx A. 5

16 Φ(log( ρ)) log( ρ) φ(log( ρ)) FIGURE 6. MAGNIFICATION OF ERROR FROM OUT-OF-SAMPLE PREDICTION OF THE RESERVATION RATE A B Value Dsturbance Fgure 6 presents two (hypothetcal) quadratc lnes from bootstrap samples. Both these lnes (called A and B) were estmated usng data samples from pseudo value dsturbances between 0.79 and The two regresson lnes closely match one another for value dsturbances n that range, but when those equatons are used to predct reservaton rates for a value dsturbance of 0.70, the dfference between the two regresson lnes ncreases. Thus, larger varaton n the estmate of the reservaton value from replcaton to replcaton occurs when those data are used to make predctons for out-of-sample reservaton values. The ncreased varaton n the estmated reservaton rate wll ncrease the standard errors of that pont estmate, and therefore the confdence ntervals surroundng that pont estmate wll rse. Thus, the level of precson of the estmated reservaton rate decreases when that rate s estmated from out-of-sample predctons. 3.3 Potental revenue loss f the upset s set ncorrectly Usng the above methodology, we can estmate the lost revenue, n percentage terms, from settng an upset of 80 percent, 85 percent, or 90 percent when the optmal upset s n fact 79 percent, gven the Mnstry s reservaton value (opportunty cost of auctonng the tmber). To perform ths calculaton, frst assume that the optmal upset s 79 percent and consult equaton 6 to fnd the opportunty cost, v 0, that would command such an upset rate. We then use an upset of 80 percent and the v 0 mpled by the optmal 79 percent rule to determne the revenue that the Mnstry would experence for the average stand from equaton 2. After calculatng the revenue from equaton 2, we can take the dfference, n percentage terms, to fnd the rate of lost revenue from the sub-optmal upset. Below, we show the rate of revenue loss by settng an upset of between 80 and 90 percent when then optmal upset s 79 percent. 6

17 TABLE 6. RATE OF REVENUE LOSS WHEN THE UPSET IS CHOSEN INCORRECTLY Percent Reducton n Percent Reducton n Percent Reducton n Revenue at 80% Upset, Revenue at 85% Upset, Revenue at 90% Upset, Area Assumng Optmal 79% Assumng Optmal 79% Assumng Optmal 79% Upset Upset Upset When the upset s off by percent, the estmated revenue loss s between 0.2 percent and 0.8 percent. Ths revenue loss s neglgble. When the upset s off by percent, however, we predct revenue losses that are more substantal. The smallest revenue loss we predct s 7.2 percent, whle the largest s 0.2 percent. Ths estmaton shows the mportance of understandng the opportunty cost of auctonng the tmber n each dstrct. The hgher s the value that the Mnstry places on that tmber, the hgher should be the upset prce, but note that for reasons we expand on below, the tmber should not be auctoned at upset prces below the Mnstry s value of the tmber. Fnally, we cannot drectly predct the revenue loss f the upset s currently too hgh, because we cannot drectly retreve the bd dstrbuton below the 79 percent upset. However, usng least squares to predct the revenue loss when the optmal upset s 70 percent but the upset s ncorrectly set at 75 percent, we estmate that the Mnstry wll loose between 0 percent and 3.0 percent of revenues dependng on the area n queston. 6 If the optmal upset s 70 percent, but the upset s set at 85 percent, we estmate that revenue losses wll vary between 5.4 percent and. percent. 7 4 THE OPPORTUNITY COST OF AUCTIONING TIMBER, AND CHANGING THE UPSET RATE The opportunty cost of auctonng tmber s a key parameter when settng an upset rate to maxmze the aucton proceeds. For ths reason the Mnstry should obtan a reasonable estmate of ts value of the tmber relatve to the prvate sector apprasal, and n all lkelhood should not consder changng the upset rate untl an accurate estmate of ths value s determned. Once the Mnstry s opportunty cost of auctonng tmber s estmated, better data s collected (as we descrbed above), and the estmaton s refned, changng the upset may be n order. We dscuss some examples below, where rasng or lowerng the upset may be approprate. 6. Ths range excludes Area 0 because, as mentoned earler, we cannot calculate the reserve value for upsets of less than 78 percent. 7. Ths range also excludes Area 0. At an upset of 90 percent, the margnal revenue loss n area 0 s 7.5 percent, assumng the optmal upset s 78 percent. 7

18 When consderng the approprate opportunty cost of auctonng tmber, the Mnstry must frst realze that ts opportunty cost of auctonng the tmber should be less than ts apprasal of that tmber s value. In partcular, the apprasal process s the Mnstry s attempt to estmate the value of the tmber stand to the prvate sector. However, the Mnstry presumably cannot harvest the tmber tself more effcently than the prvate sector, so the value of the stand arses from the value of retanng the tmber, ether ndefntely or for future resale. When a tract does not sell at aucton, the Mnstry acqures certan nformaton. If bdders values for a partcular tract are forever constant, and f the bdders are bddng under the equlbrum descrbed above, then the Mnstry learns that the value of the tract s less than the upset t set. In other words, the expected resale value of tracts that do not sell at a gven upset cannot be greater than the upset tself. Alternatvely, f some components of bdder values are transent (for example, bdders values for a partcular tract depend on ther current access to other, smlar tracts), then a tract that fals to sell today mght sell for a future prce that s greater than today s upset prce. Overall, the expected future resale value of a tract that fals to sell today depends on the correlaton between today s values and tomorrow s values by frms. Such a correlaton cannot be easly estmated from the exstng data, because a large sample of tracts that faled to sell ntally and were offered agan later would be requred. One factor that could lead to correlaton n values over tme (condtonal on the apprased value) s that bdders may measure certan features of tracts that they consder to be mportant n that tract s valuaton. The Mnstry, however, may be unable to nclude all of these characterstcs n ts apprasal. If bdder values are correlated over tme, lttle s ganed by settng a hgh upset prce and contnually reofferng a tract n the hopes t wll sell at that hgh prce. Further, holdng multple auctons for a sngle tract utlzes resources from both the Mnstry and the bdders that nvestgate the auctons. The ssues assocated wth the dynamcs of optmal upset prcng have receved lttle attenton n the economcs lterature to date, and a full analyss of them s beyond the scope of ths paper. Such an analyss would requre and analyss of the extent to whch a bdder values ts opton of bddng on a stand n the future f that stand were to go unsold today. Between January, 999 and December 3, 2000 the Mnstry auctoned 476 stands. Of those 476 stands, 5 stands or 0.7 percent receved no bds, and 37 stands or 7.8 percent receved only one bd. Also, 72 stands or 36. percent receved wnnng bds above the Mnstry s apprased value. These statstcs ndcate that ncreasng the upset rate should be done wth cauton, because a sgnfcant ncrease n the number of unsold stands could result. Wth fewer stands sold at aucton, the prcng nformaton from those margnal stands to the Mnstry s prcng equaton would dmnsh. Now consder the decson to change the upset rate. The precedng analyss suggests a procedure for optmally settng the upset that can be mplemented by the BC government once better data s avalable. The frst step s to estmate the optmal upset and ts assocated confdence nterval. The government can begn by calculatng v o relatve to ts apprasal of the tmber s worth on a dstrct-by-dstrct, or area-byarea bass. It can then employ the estmaton procedure used n ths paper to fnd the optmal upset equaton and use the estmated reservaton value, v o, to fnd the optmal upset va ths equaton. Fnally, the government can use the bootstrap to fnd a 95 percent confdence nterval for ths estmate. The second step s to determne whether the prevalng upset s suffcently close to the optmal upset n each area. The prevalng upset mght not be consdered optmal f () t does not le wthn the 95 percent confdence nterval of the estmated optmal upset n numerous areas and (2) t sgnfcantly reduces revenue relatve to the optmal upset n those areas. If both of these condtons are met, the Mnstry should consder changng the upset rate. If only condton () s met, then the economc benefts of changng the upset may be relatvely small, and the government should serously consder whether the benefts of adjustment outwegh the costs. 8

19 Also, we must consder that a sngle upset rate s set throughout the provnce. Thus, we must understand the economc sgnfcance that a provncal upset rate may have on ndvdual dstrcts or areas. Partcular attenton should be pad to the effects of adjustng the upset on total revenues. For example, settng the upset optmally for a group of dstrcts representng 20 percent of revenues could ncrease revenues 0 percent n these dstrcts whle reducng revenues by 5 percent n all other dstrcts. Ths partcular change n upset polcy, whle optmal for some dstrcts, actually reduces total revenues. 5 CONCLUSION The upset prce plays a crtcal role n the BC tmber auctons. It serves to enhance revenues n stuatons where competton s weak, to lmt the ncentve for collusve bddng, and to provde nformaton to bdders. Our analyss of the data suggests that an upset of about 70 percent (a rollback of 30 percent) maxmzes aucton revenues when the Mnstry values tmber at approxmately 52 percent to 56 percent of ts apprased value. In arrvng at ths number, the analyss makes the followng assumptons: The bdders do not collude. The bdders values are drawn ndependently from the same probablty dstrbuton n a gven area. The bdders bd to maxmze ther expected profts from the aucton. The number of potental bdders does not depend on the upset. These assumptons are standard n the lterature and are a good frst approxmaton. Stll, gven the mportance of settng an approprate upset, we urge the BC government to refne ts estmate of the upset as better data becomes avalable. An mproved dataset would nclude all the observed bds, not just the wnnng bd, and a better understandng of the Mnstry s opportunty cost of auctonng the tmber. We have assumed that the BC government s objectve s to maxmze aucton revenues. If nstead, the goal s effcency (maxmzaton of the gans from trade) and colluson s not a problem, then the upset should be set at the government s reservaton value for the tmber. However, n stuatons where the reservaton value s low, then settng the upset at the reservaton value may create too strong an ncentve for the bdders to collude, n whch case rasng the upset above the reservaton value s justfed on both revenue and effcency grounds. On the other hand, settng a lower upset may ncrease bdder partcpaton n whch case settng a lower upset may ncrease both revenues and effcency (see Bulow and Klemperer 996). 9

20 APPENDIX A. AN EXAMPLE OF THE DETERMINATION OF THE OPTIMAL UPSET Suppose that values, v, are a random sample drawn from the densty functon f(v) =, on the unt nterval, wth correspondng dstrbuton functon F(v) = v. We conduct 200 sealed-bd auctons wthout an upset prce, and four bdders submt bds n every aucton (n each aucton, each bdder receves a new, ndependent draw from the dstrbuton). Also, assume that we only record the wnnng bd n each aucton. Thus, our dataset ncludes 200 wnnng bds for frst-prce sealed bd auctons where four bdders competed. Below, we go through a nonparametrc analyss of optmal upset estmaton for the full sample, and then use bootstrappng on 500 samples of 200 observatons (drawn from the orgnal sample, wth replacement) to estmate the standard error of the optmal upset. THE NONPARAMETRIC ESTIMATION () From the statstcs lterature, the dstrbuton of the hghest (.e. wnnng) value, v, has the dstrbuton functon F () () v = F() v N () 4, where N s the number of bdders. Thus, F () v = v n ths example. We therefore sample 00 wnnng values from ths dstrbuton functon, and calculate the correspondng wnnng bds accordng to the optmal bddng rule: v N b = s( v, N) = v F ( x) dx N F ( v ). Substtutng Fv () = v, N = 4, and v = 0 nto the optmal v bd equaton, we can smplfy the equlbrum bd strategy to: (A) b 3v = s( v,4) = 4 Thus, we transform our values nto bds accordng to equaton A. As researchers, we pretend that the only observables we have are the wnnng bds that we generated above, and the number of bdders n each aucton, whch s four. 20

21 We are now ready to perform the steps of the nonparametrc estmaton to determne the optmal upset. The frst step of ths technque s to run a kernel densty estmate on the observed bds to obtan an estmate of g () ( b ), the densty of the wnnng bds. Ths estmaton yelds the densty functon depcted n Fgure A, and numercal ntegraton of the estmated densty yelds the wnnng bd dstrbuton shown n Fgure A FIGURE A. KERNEL DENSITY ESTIMATES OF THE WINNING BID DENSITY g () (b) b 2

22 FIGURE A2. ESTIMATE OF THE WINNING BID DISTRIBUTION G () (b) b The kernel densty estmate of g () ( b) seems reasonable, as most of ts mass occurs for hgher bds, as we would expect. The estmated wnnng bd dstrbuton also appears realstc. It has the proper shape that s, t entrely les below the 45-degree lne. After obtanng estmates of the densty and dstrbuton of the wnnng bds, we now apply the nverse bd functon to obtan estmates of the correspondng values called pseudo values n the lterature. Ths step s done by applyng equaton (A2) 8 to the observed bds, denstes, and dstrbutons. (A2) vˆ () () ˆ () () ( b ) () () ˆ N G = b + N g ( b ) In equaton A2, G ˆ () and gˆ() are the dstrbuton and densty estmates, respectvely, from the kernel procedure. We now have estmated our bd functon, because we have both actual bds and estmates of the values that generated those bds. Fgure A3 shows the estmated bd functon, and how t compares to the true bd functon. 8. Ths equaton s gven on page 27 of ELLV (995). It can be derved drectly from the bdder s maxmzaton problem, as descrbed n the text. 22

23 FIGURE A3. ESTIMATED AND ACTUAL INVERSE BID FUNCTIONS.4.2 v () Value Est. Value b () In Fgure A3, we note that our estmaton of the true bd functon s precse, wth the excepton of the upper tal. That we have dffculty estmatng the tal s lttle surprse gven the dscusson above. Nevertheless, the kernel procedure performs well n ths example, as we would hope. After we have calculated a pseudo-value for each bd n the dataset, we then use kernel estmaton once agan to obtan an estmate of the densty of those pseudo-values. The dstrbuton can be constructed usng numercal ntegraton. As a fnal step before determnng the optmal upset, we must () transfer our estmates of the dstrbuton and densty of the hghest value, ˆ () F () and f ˆ () respectvely, nto estmates of the populaton dstrbuton and densty namely, F ˆ () and f ˆ(). Ths transformaton s performed accordng to equatons A3 and A4, below. Equaton A3 s smply the nverse () 5 of the dstrbuton of the hghest ordered value n auctons of fve bdders: F () = F(), and equaton A4 s calculated by dfferentatng equaton A3 wth respect to v. (A3) (A4) ˆ ˆ () F() = F () 4 ˆ f() = fˆ () Fˆ () 4 3 () () 4 Fnally, we are ready to estmate the optmal upset. Recall from equaton 3, that wth v 0 = 0, the optmal upset solves the followng equaton: 23

24 (A5) Fr ˆ ( ) r = 0 fˆ( r ) In our dataset, we generate a functon dentcal to that n equaton (A5) usng the estmates of the densty and dstrbuton functons, f ˆ( r ) and Fr ˆ () respectvely. Fgure A4 graphs ths functon on the doman where t s monotone ncreasng. FIGURE A4. OPTIMAL UPSET EQUATION (-F(r))/f(r) - r r The upset prce at whch the optmal upset equaton holds that s, where the lne ntersects the horzontal axs s the optmal upset prce. In ths example, the optmal upset prce s approxmately 0.443, whch s close to the true optmal upset prce of To better determne the precson of our estmated optmal upset prce s to the true optmal upset prce, we need to calculate the standard error of the estmate. 2 ESTIMATING THE STANDARD ERROR OF THE OPTIMAL UPSET THROUGH BOOTSTRAPPING Bootstrappng s a technque that, roughly speakng, samples many tmes from the orgnal sample to determne the varance of a consstent estmator. 9 Intutvely, t works by calculatng how the estmates change when the sample changes. Bootstrappng s lkely the smplest method for calculatng the standard error of the estmated optmal upset prce n ths applcaton. 9. For a more thorough explanaton of bootstrappng, see Efron and Tbshran (993) or Greene (997). 24