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Optimal Price Ceilings in a Common Value Auction

                                            Archishman Chakraborty
                                   Baruch College, City University of New York



                                                     Abstract
       A simple common value auction is considered where it is optimal to set a ceiling price in
       addition to a reserve price. The ceiling price prevents the better informed bidder from
       outbidding the less informed bidders. This guarantees participation from the less informed
       bidders raising the seller's revenues. The seller is better off by selling the good in an auction
       with a price ceiling compared to selling the good at a fixed price.




Thanks are due to Faruk Gul.
Citation: Chakraborty, Archishman, (2002) "Optimal Price Ceilings in a Common Value Auction." Economics Bulletin, Vol. 3,
No. 7 pp. 1−7
Submitted: March 11, 2002. Accepted: April 18, 2002.
URL: http://www.economicsbulletin.com/2002/volume3/EB−02C70005A.pdf
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Optimalprice

  • 1. Optimal Price Ceilings in a Common Value Auction Archishman Chakraborty Baruch College, City University of New York Abstract A simple common value auction is considered where it is optimal to set a ceiling price in addition to a reserve price. The ceiling price prevents the better informed bidder from outbidding the less informed bidders. This guarantees participation from the less informed bidders raising the seller's revenues. The seller is better off by selling the good in an auction with a price ceiling compared to selling the good at a fixed price. Thanks are due to Faruk Gul. Citation: Chakraborty, Archishman, (2002) "Optimal Price Ceilings in a Common Value Auction." Economics Bulletin, Vol. 3, No. 7 pp. 1−7 Submitted: March 11, 2002. Accepted: April 18, 2002. URL: http://www.economicsbulletin.com/2002/volume3/EB−02C70005A.pdf
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  • 4. nqrzv Zh zloo dvvxph wkdw lv glvwulexwhg xqlirupo| lq dfc o dqg wkdw iru 5 ic 2j wkh vljqdov r 5 iMc uj zlwk hdr ' M m o ' Ixuwkhu/ frqglwlrqdo rq / wkh vljqdov r dqg r2 duh lqghshqghqw1 Wkh vhoohu krogv d vhfrqg sulfh dxfwlrq1 Kh fdq fkrrvh d uhvhuyh sulfh R 5 dfc o dqg wkh o kljkhvw elgghu zlqv wkh remhfw dqg sd|v wkh pd{lpxp ri wkh vhfrqg kljkhvw elg dqg R Zh o doorz wkh vhoohu wr dovr fkrrvh d fhlolqj sulfh R 5 dfc oc zlwk R R Wkdw lv/ wkh vhoohu rqo| S S o dffhswv elgv deryh R dqg ehorz R o S Jlyhq d fhlolqj sulfh/ wlhv zloo qrw eh }hur suredelolw| hyhqwv lq wklv dxfwlrq/ hvshfldoo| ehfdxvh vrph elgghuv pd| elg dw wkh fhlolqj sulfh1 Wkurxjkrxw zh zloo dvvxph wkdw wlhv duh eurnhq xqlirupo| zkhq ? elgv wlh/ hdfk elgghu jhwv wkh jrrg zlwk suredelolw|  ? Qrwlfh wkdw fkrrvlqj R S ' R R lv htxlydohqw wr vhoolqj wkh remhfw dw wkh {hg sulfh R o ' Krzhyhu/ lw zloo eh vkrzq wkdw d fhlolqj sulfh ri R  dqg d uhvhuyh sulfh R : f/ zlwk S o R : R lv rswlpdo iru vhoohu uhyhqxhv1 S o Jlyhq d fkrlfh ri d fhlolqj sulfh dqg d uhvhuyh sulfh/ zh zloo orrn dw Ed|hvldq Qdvk htxloleuld ri wkh uhvxowlqj dxfwlrq1 Ohw K Er eh wkh elg ri elgghu 5 ic 2j dqg K
  • 5. E wkh elgglqj vwudwhj| ri elgghu 6 Zh zloo uhvwulfw dwwhqwlrq wr v|pphwulf htxloleuld zlwk uhvshfw wr elgghuv  dqg 2 zkhuh KEr ' K2Er ' KEr iru hdfk r1 514 Vhfrqg Sulfh Dxfwlrq= Qr Fhlolqj Sulfh Zh uvw frqvlghu d vhfrqg sulfh dxfwlrq zlwk qr fhlolqj sulfh RS '  Fodlp 4 Vxssrvh R '  Lq dq| v|pphwulf htxloleulxp zlwk uhvshfw wr elgghuv 4 dqg 5/ S K Er ' f iru doo r 5 iMc uj dqg elgghu 6 elgv zkhqhyhu R Wkh rswlpdo uhvhuyh sulfh o lv R ' fD iru dq h{shfwhg uhyhqxh ri f2D o Surri1 Iru dq| R 5 dfc oc lw fdq hdvlo| eh fkhfnhg wkdw d zhdno| grplqdqw vwudwhj| iru o elgghu 6 lv wr elg zkhqhyhu R o Ixuwkhu/ lq dq| v|pphwulf htxloleulxp/ zh pxvw kdyh wkdw elgghuv 4 dqg 5 gr qrw elg/ l1h1/ hdKEr : fo ' f Iru vxssrvh hdKEr : fo : f dqg ohw 4@ KEr ' K : f Zkhq elgghu  r elgv K dqg zlqv wkh dxfwlrq wkhq K Li K2 wkhq elgghu  sd|v dqg pdnhv }hur surwv1 Krzhyhu li K2 elgghu  sd|v K2 dqg pdnhv qhjdwlyh surwv1 Vlqfh K2 lv vwulfwo| srvlwlyh zlwk vwulfwo| srvlwlyh suredelolw|/ elgghu 4 pdnhv qhjdwlyh h{shfwhg surwv iurp elgglqj K dqg vr lv ehwwhu r qrw elgglqj1 5
  • 6. Wkxv/ lq dq| v|pphwulf htxloleulxp zlwk uhvshfw wr elgghuv 4 dqg 5/ rqo| elgghu 6 elgv d srvlwlyh dprxqw zlwk srvlwlyh suredelolw| dqg dozd|v sd|v wkh sulfh R Wkh vhoohu fkrrvhv o wkh uhvhuyh sulfh wr vroyh 4@ R E R o o Wkxv wkh rswlpdo uhvhuyh sulfh lv Ro ' fD iru dq h{shfwhg uhyhqxh htxdo wr f2D W 515 Fhlolqjv ) Iorruv D fhlolqj sulfh zloo uhgxfh wkh vhyhuh zlqqhu*v fxuvh sureohp vhhq lq wkh v|pphwulf htxl0 oleulxp deryh/ dv orqj dv elgghuv vrphwlphv klw wkh fhlolqj1 Wklv lv ehfdxvh/ frqglwlrqdo rq zlqqlqj dw wkh fhlolqj sulfh/ wkhuh lv srvlwlyh suredelolw| wkdw wkh ydoxh ri wkh jrrg lv dovr deryh wkh fhlolqj sulfh1 Wkh qh{w fodlp vkrzv wkh elgglqj ehkdylru lq dq| v|pphwulf htxloleulxp/ jlyhq d fhlolqj sulfh R  S Fodlp 5 Vxssrvh f R R  Lq dq| v|pphwulf htxloleulxp zlwk uhvshfw wr elgghuv 4 o S dqg 5/ elgghu 5 ic 2j hlwkhu elgv R ru grhv qrw elg +elgv f, Elgghu 6 elgv 4?ER c zkhq S S R o Surri1 Lw lv vwudljkwiruzdug wr fkhfn wkdw lw lv d zhdno| grplqdqw vwudwhj| iru elgghu 6 wr ehkdyh dv vshflhg/ jlyhq wkdw kh fdqqrw elg deryh R S Vxssrvh elgghuv 5 ic 2j elg lq wkh lqwhuydo dR c R zlwk vwulfwo| srvlwlyh suredelolw|/ o S lq vrph v|pphwulf htxloleulxp1 Iru dq| vxfk elg/ li elgghu zlqv wkh dxfwlrq/ wkhq lv qrw juhdwhu wkdq klv elg1 Elgghu hduqv }hur surwv li elgghu
  • 7. lv wkh vhfrqg kljkhvw elgghu zlwk R dqg hduqv vwulfwo| qhjdwlyh surwv li R ru li elgghu 5 ic 2jc ' / lv o o 9 wkh vhfrqg kljkhvw elgghu1 Vlqfh wkh odvw srvvlelolw| kdv vwulfwo| srvlwlyh suredelolw| lq rxu v|pphwulf htxloleulxp/ lw lv ehwwhu iru elgghu wr elg f lqvwhdg ri elgglqj dq| dprxqw lq wkh lqwhuydo dRo c RS 1 Lq frqwudvw wr d elg vwulfwo| ohvv wkdq R c li elgghu 5 ic 2j zlqv dw d elg htxdo wr R c S S pljkw eh vwulfwo| juhdwhu wkdq wkh sulfh sdlg zlwk vwulfwo| srvlwlyh suredelolw|1 Jlyhq d Zkhq sK wkhuh duh dovr dv|pphwulf htxloleuld ri wkh irup e +v, @ e # +3 4` iru doo v e# +v, @ 3 iru @ 3/ doo v= Elgghu 6 elgv y= Wkh pd{lpxp uhyhqxh htxloleuld ri wklv irup kdv e @ 4 iru uhyhqxh ri = Krzhyhu vxfk # htxloleuld gr qrw vxuylyh lwhudwhg ghohwlrq ri +zhdno|, grplqdwhg vwudwhjlhv1 Iru dq| uhvhuyh sulfh sK A 3 elgghuv 4 dqg 5 zloo qrw elg dq| srvlwlyh dprxqw lq dq| htxloleulxp/ dv wkh| pdnh vwulfwo| qhjdwlyh surw zkhq wkh| elg pruh wkdq sK dqg y ? sK 1 6
  • 8. fkrlfh ri d fhlolqj sulfh dqg d rru sulfh/ wkhuh duh wkxv wzr srvvleoh fdvhv wr frqvlghu1 Lq r dqg r2 wkh uvw fdvh/ elgghuv 4 dqg 5 elg dw wkh fhlolqj sulfh uhjdugohvv ri wkhlu vljqdov Lq wkh vhfrqg fdvh/ elgghuv 4 dqg 5 elg dw wkh fhlolqj rqo| zkhq wkhlu vljqdo htxdov M Zh frqvlghu hdfk ri wkhvh wzr fdvhv lq vxffhvvlrq1 D Orz Fhlolqj +ru Il{hg, Sulfh Frqvlghu d fhlolqj sulfh RS vxfk wkdw/ lq htxloleulxp/ elgghuv 4 dqg 5 elg wkh sulfh RS uhjdugohvv ri wkhlu vljqdo1 Wkh remhfw lv vrog zlwk suredelolw| 4 dw d sulfh htxdo wr R Wkxv/ vhwwlqj vxfk d fhlolqj sulfh lv htxlydohqw wr vhwwlqj d {hg S sulfh Ohw wkh ghqvlw| ri jlyhq r eh ghqrwhg e| s Emr G ; ? s Emr = 2E li r ' u +4, 2 li r ' M Jlyhq wkh ehkdylru ri elgghu 6 dqg elgghu 5 ic 2j/ 9' / zkhq elgghu 5 ic 2j elgv R S kh zlqv zlwk suredelolw| R dqg rwkhuzlvh/ uhjdugohvv ri wkh vljqdo ri elgghu 
  • 9. zkhq S  2 1 Dqg kh dozd|v sd|v wkh sulfh R zkhq kh zlqv1 Wkxv/ wkh h{shfwhg surw iru elgghu S 5 ic 2j iurp elgglqj R jlyhq d vljqdo rc lv htxdo wr S  ] ]   E R s E mr_ n E R s Emr_ R5 S +5, S 2 f
  • 10. R5 Elgghu zrxog eh zloolqj wr elg dw R dv orqj dv kh hduqv qrqqhjdwlyh surwv1 S Wkh rswlpdo vxfk sulfh vkrxog eh vhw vxfk wkdw elgghu 5 ic 2j pdnhv }hur h{shfwhg suri0 lwv iurp elgglqj R zkhq r ' u Xvlqj +4, lq +5, iru r ' u/ dqg shuiruplqj wkh lqwhjudwlrqv S zh rewdlq wkdw R pxvw vroyh= S 
  • 11. R
  • 12. S S 2 R2 n H  R   S S S 'f Wkh vroxwlrq lv R S ' f2be2H Xvlqj wklv ydoxh ri R r ' Mc zh rewdlq wkdw wkh S lq +5, iru h{shfwhg surw iru elgghu 5 ic 2j iurp elgglqj R c jlyhq d vljqdo r ' Mc lv htxdo wr S f22./ vr wkdw elgghu 5 ic 2j lv dovr zloolqj wr elg R zkhq r ' M Vlqfh wkh jrrg lv S vrog zlwk suredelolw|  iru vxfk d fhlolqj +{hg, sulfh/ wkh h{shfwhg uhyhqxh iru wkh vhoohu lv htxdo wr f2be2H1 Wkxv/ wkh vhoohu lv ehwwhu r iurp fkdujlqj wklv {hg sulfh frpsduhg wr wkh dxfwlrq deryh1 7
  • 13. D Kljk Fhlolqj Zh qrz wxuq wr wkh vhfrqg fdvh1 Vxssrvh wkdw wkh vhoohu vhwv R dqg S R vxfk wkdw/ lq htxloleulxp/ elgghuv 4 dqg 5 gr qrw elg zkhq wkhlu vljqdo htxdov uc dqg o elg R zkhq wkhlu vljqdo htxdov M 1 Dv xvxdo/ elgghu 6 elgv R li R dqg elgv zkhq S S S 5 dR c R dqg grhv qrw elg rwkhuzlvh1 o S Elgglqj R lv rswlpdo iru elgghuv 4 dqg 5 jlyhq r li wkh| hduq qrqqhjdwlyh h{shfwhg S surwv iurp grlqj vr1 Ohw Z Ec r ( R c R eh wkh h{shfwhg surw iru elgghu 5 ic 2j iurp S o S 9 elgglqj R c iru hdfk dqg elgghu ' *v vljqdo r c jlyhq wkdw doo wkh rwkhu elgghuv duh elgglqj dv deryh1 Wkh wdeoh ehorz ghvfulehv Z Ec r ( R c R iru glhuhqw ydoxhv ri dqg r = S o r 'M r 'u R S E R *
  • 14. E R *2 S S +6, R : R E R *2 f S o S R : o E R *2 R S o Ohw wkh mrlqw ghqvlw| ri dqg r jlyhq r eh ghqrwhg e| s Ec r mr1 Wkhq ; ? s Ec r mM = 2E li r ' u +7, 22 li r ' M dqg ; ? s Ec r mu = 2E li r ' u 2 +8, 2E li r ' M Ohw Z ER c R eh wkh h{shfwhg surw iurp elgglqj R jlyhq d vljqdo r 5 iMc uj/ dqg jlyhq R r S o S S dqg R =o r S o ] [ Z ER cR f ZEc r ( RSc Ro s Ec r mr_ +9, r Wkh vshflhg ehkdylru deryh lv dq htxloleulxp li wkh h{suhvvlrq lq +9, lv qrqqhjdwlyh zkhq r ' M lv qrqsrvlwlyh zkhq r ' u RS dqg hlwkhu r ru r2 htxdov M Wkh sulfh rewdlqhg e| wkh vhoohu lv wkh fhlolqj sulfh li ru li RS dqg erwk r dqg r2 htxdo M Li 5 dRo c RS dqg h{dfwo| rqh ri r dqg r2 htxdov Mc wkhq wkh sulfh rewdlqhg lv Wkh sulfh lv htxdo wr Ro rwkhuzlvh/ dv orqj dv wkh jrrg lv vrog1 Wkh jrrg lv qrw vrog li Ro dqg erwk r dqg r2 htxdo u Wkxv wkh h{shfwhg uhyhqxh iru wkh vhoohu lv= 8
  • 15. ERS cRo ' ]R R o n 2Ro E _ ] R SR 2 E f n 22 E n Ro E 2 _ S 2 ]R R S n E o  n RS S E 2 n 2RS E n Ro E 2_ +:, Zh orrn iru wkh fhlolqj dqg rru sulfhv wkdw vroyh= 4@ RS cRo ERS c Ro v1w1 +l, ZM ERS c Ro fc +;, +ll, Z u ERS c Ro f +lll, Ro RS Zh vroyh wkh sureohp e| ljqrulqj wkh odvw wzr frqvwudlqwv +ll, dqg +lll, dqg wkhq fkhfnlqj wkdw wkh| duh vdwlvhg dw wkh rswlpxp1 Shuiruplqj wkh lqwhjudwlrqv lq +9, dqg +:,/ wkh uhod{hg sureohp lv 4@ RS cRo SRS  R
  • 16. n 2 RS  Re n e R
  • 17. R2 n  Ro  e
  • 18. S
  • 19. 2 o
  • 20. o o
  • 21. v1w1 +S, eRe eRS n SR
  • 22. n SRe 2R
  • 23. n b f S S o o Wkh frqvwudlqw lq +S, pxvw elqg1 Frpsxwdwlrqv |lhog wkdw wkh rswlpdo sulfhv duh RS ' WW f.DHc Ro ' f2
  • 24. iru dq h{shfwhg uhyhqxh ri fe
  • 25. .2b Dw wkh rswlpxp/ Z u ERS c Ro ' ffH..2 WW vr wkdw doo wkh rplwwhg frqvwudlqwv duh vdwlvhg1 Wkxv/ fkrrvlqj d fhlolqj sulfh vwulfwo| ohvv wkdq 4 dqg vwulfwo| juhdwhu wkdq d srvlwlyh uhvhuyh sulfh lv rswlpdo iru wkh vhoohu1 Frqvlghu qrz wkh vroxwlrq wr +;, zlwk wkh dgglwlrqdo frqvwudlqw wkdw wkh vhoohu fkdujhv qr uhvhuyh sulfh vr wkdw Ro ' f Wkh fruuhvsrqglqj rswlpdo fhlolqj sulfh lv rewdlqhg e| vroylqj ZM ERSc f ' f iru RS 1 Wkh vroxwlrq lv RS ' f.2.be. iru dq h{shfwhg uhyhqxh ri fef
  • 26. D2 W Vwduwlqj iurp wklv srlqw/ li wkh vhoohu udlvhv wkh uhvhuyh sulfh iurp f/ kh lqfuhdvhv wkh zlqqhu*v fxuvh sureohp iru elgghuv  dqg 2 iurp elgglqj RS Wklv lv ehfdxvh zkhq r ' u dqg Ro wkh| kdyh wr qrz sd| Ro zkhq wkh| zlq/ lqvwhdg ri / dqg vr hduq qhjdwlyh surwv1 Wkxv/ wr jhw elgghuv 5 ic 2j wr sduwlflsdwh/ wkh vhoohu kdv wr orzhu wkh fhlolqj sulfh dv kh udlvhv wkh uhvhuyh sulfh1 Dw wkh rwkhu h{wuhph/ frqvlghu wkh fdvh zkhuh wkh vhoohu lv frqvwudlqhg wr fkdujh d {hg sulfh ri RS ' Ro ' R Vroylqj ZM ERc R ' fc zh rewdlq wkdw wkh vroxwlrq lv R W ' fD.SDSc iru 9
  • 28. bHbe Frpsdulqj zlwk wkh orz fhlolqj fdvh/ zh vhh wkdw wklv lv wkh rswlpdo fkrlfh ri d {hg sulfh iru wkh vhoohu1 Krzhyhu/ iru vxfk d {hg sulfh/ elgghuv kdyh wr sd| wkh {hg sulfh zkhqhyhu wkh| zlq1 Lq frqwudvw/ iru d fhlolqj sulfh ri RS ' f.2.be. W dqg d uhvhuyh sulfh ri f/ elgghuv sd| dqg hduq f surwv zkhq elgghu 6*v elg lv wkh vhfrqg kljkhvw elg1 Wklv orzhuv wkh zlqqhu*v fxuvh sureohp iru elgghuv 4 dqg 51 Kroglqj d vhfrqg sulfh dxfwlrq zlwk d fhlolqj sulfh ri f.2.be. dqg qr uhvhuyh sulfh |lhogv wkh vhoohu kljkhu uhyhqxhv wkdq vhoolqj dw d {hg sulfh ri fD.SDS1 Wkh wdeoh ehorz vxppdul}hv wkh uhvxowv1 Lw surylghv wkh udqnlqj ri vhoohu uhyhqxhv iurp vhwwlqj glhuhqw fhlolqj dqg rru sulfhv RS Ro Uhyhqxh 41 Rswlpdo Fhlolqj dqg Iorru f.DH f2
  • 29. fe
  • 30. .2b 51 Kljk Fhlolqj/ Qr Iorru f.2.be. f fef
  • 31. D2 61 Rswlpdo Il{hg Sulfh fD.SDS fD.SDS f
  • 32. bHbe 71 Orz Fhlolqj2Il{hg Sulfh f2be2H f2be2H f2be2H 81 Rqo| Iorru  fD f2D 6 Frqfoxvlrq Zkloh wkh h{dpsoh deryh lv dgplwwhgo| vlpsoh/ wkh lqwxlwlrq wkdw sulfh fhlolqjv udlvh vhoohu uhyhqxhv e| jhqhudwlqj sduwlflsdwlrq iurp ohvv lqiruphg elgghuv lv txlwh urexvw dqg vkrxog h{0 whqg wr ulfkhu hqylurqphqwv1 Iru h{dpsoh/ lw lv srvvleoh wr frqvwuxfw h{dpsohv zlwk pruh wkdq rqh ehwwhulqiruphg elgghu +lq dgglwlrq wr pxowlsoh ohvvlqiruphg elgghuv,/ zkhuh kroglqj d vhfrqg sulfh dxfwlrq zlwk d sulfh fhlolqj |lhogv kljkhu uhyhqxhv frpsduhg wr hlwkhu kroglqj d vhfrqg sulfh dxfwlrq zlwk d uhvhuyh sulfh ru iurp vhwwlqj d {hg sulfh dw zklfk wr vhoo wkh jrrg1 Wkxv wkh uhvxow deryh grhv qrw klqjh rq wkh h{lvwhqfh ri rqh ehwwhulqiruphg elgghu1 Lw lv dovr ri lqwhuhvw wr fkdudfwhul}h wkh qdwxuh ri wkh jdlqv iurp lpsrvlqj d sulfh fhlolqj dv d ixqfwlrq ri wkh uhodwlyh vl}hv ri wkh wzr fodvvhv ri elgghuv dqg wr lqyhvwljdwh wkh srvvlelolw| ri h{whqglqj wkh uhvxowv wr rwkhu dxfwlrq irupdwv1 Uhihuhqfhv ^4` Nohpshuhu/ S1 +4,/ Dxfwlrq Wkhru|= D Jxlgh wr wkh Olwhudwxuh/ Mrxuqdo ri Hfrqrplf Vxuyh|v/ 46 +6,/ Mxo|/ 55:0;91 :
  • 33. ^5` Plojurp S1 dqg U1 Zhehu +4;5,/ Wkh Wkhru| ri Dxfwlrqv dqg Frpshwlwlyh Elgglqj/ Hfrqrphwulfd/ 83 +8,/ Vhswhpehu/ 43;4551 ;