(A1) and bˆ represents the expected price conditional on b. being one of the winning bids. Rewrite expression (A1) as follows:.
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1 Aendx: Proo o Prooston : Bdder s (lnear) exected utlty rom dng s: S Q U (A) where Q reresents the roalty that she wns the aucton wth and reresents the exected rce condtonal on eng one o the wnnng ds Rewrte exresson (A) as ollows: S Q S Q U (A2) Q Q ote that the second term o exresson (A2) s ndeendent o Thereore choosng to maxmze (A2) s dentcal to choosng to maxmze the ollowng exresson: U* * (A3) S Q where * xresson (A3) s o the same orm o the rolem aced y a dder Q n a second-rce aucton; wth roalty dder wns a unt o the good worth * and ays a rce equal to In ths more amlar settng the ash equlrum ddng strategy has each dder ddng her alue Here the result ollows: S Q UH * (A4) Q Proo o Prooston 3: Assume that dder elees all other dders are usng the ncreasng d uncton l or l where the other suscrts hae een suressed Choosng to maxmze S Q S Q F U F U U yelds the ollowng rst order Q Q condton: ' U U ' F ' U U (A5) S Q where Q Susttutng or ' and or the exresson reduces to ' U ' F U U U ' ' (A6) Rearrangng exresson (A6) yelds: U U U F U (A7)
2 ote that exressons (A5) (A6) and (A7) characterze the rst order condton or the equlrum d uncton under the dscrmnate hyrd regardless o the dder s rsk reerences ow assumng rsk neutralty and rearrangng terms we hae F (A8) DH DH ecause U Rewrte (A8) as d F d DH Because DH (A9) the soluton to the aoe s gen y DH x x dx (A) F In order to conrm that the soluton to (A) s the ash equlrum we must conrm that the yelds: d uncton s ndeed ncreasng n Solng exresson (A8) or DH DH DH F (A) DH x xdx F whch s oste roded A comarson o the equlrum d unctons or the dscrmnate hyrd and the dscrmnate aucton when the lottery s asent (Harrs and Ra s [98] equaton ()) conrms ths nequalty Followng Harrs and Ra [98] the equlrum d uncton (assumng rsk neutralty) or dder acng a dscrmnate aucton denoted D s equal to D x xdx (A2) F Susttutng D nto the exresson or DH yelds DH D (A3) whch suggests that the resence o the lottery n the hyrd mechansm causes rsk neutral S Q dders to shade ther ds y the roalty o losng the lottery Harrs Q and Ra [98] show that D Comnng the results we hae the ollowng nequalty: S Q S Q DH D (A4) Q Q Proo o Prooston 5: where The rst art o the roo roceeds y showng mn DH A DH DH A DH
3 For the remander o the roo all suscrts are suressed By strct concaty o U U ( ) U ( ) U( ) whch mles U ( ) U ( ) U( ) ecause U( ) Multlyng the nequalty y - and addng and sutractng rom the rght hand sde yelds: U ( ) U ( ) U( ) Rearrangng we hae: U ( ) ( ) U( ) U ( ) U( ) (A5) By concaty o U and U we know that U ( ) ( ) U( ) When = U () U( ) y strct concaty o U Ths term aroaches zero as ncreases and s equal to zero at some * For * U ( ) U( ) Suose DH A DH then where s dened aoe s such that U ( ) U( ) Wth the ollowng holds: U ( ) U ( ) ( ) U( ) U( ) U ( ) ( ) U( ) U ( ) U( ) Thereore U ( ) U ( ) ( ) U( ) U( ) or such that DH A DH
4 On the other hand suose A DH then s such that DH U ( ) U( ) whch comned wth the act that U ( ) ( ) U( ) ges us the ollowng: U ( ) U ( ) ( ) U( ) U( ) or such that DH A DH Thereore U ( ) U ( ) ( ) U( ) U( ) or such that DH A DH ote that the nequalty holds y strct concaty or such that DH A DH Rearrangng terms mles U ( ) U () U( ) or (A6) By roertes o densty and dstruton unctons ( ) F ( ) In turn U U DH U DH A F F So or mn DH A DH we hae DH A DH ow we show that DH A DH or all such that DH A DH By contnuty there exsts such that DH A DH Recall the denton o mn DH A DH From aoe we hae DH A DH so that DH A DH or n a neghorhood o ut then DH A DH or all Otherwse DH A would cross whch contradcts the denton o Thereore or all ut Suose not Then there exsts one DH A DH DH at some
5 DH A DH so DH A DH and roes DH A DH or all Proo o Prooston 6: n a neghorhood o zero Ths contradcts or all DH A DH The equalty ollows drectly rom Proostons and 3 and rom Harrs and Ra s [98] Theorem 6 ( ) To roe the nequalty consder the exected reenue or the dscrmnate hyrd wth rsk-aerse dders: RDH A DH A Q h DH A Q DH A DH Because d and y Prooston 2 DH A DH R R DH DH A we hae Addtonal reenue hyothess: S Q Prooston A: RUH Q RU RU Proo: xected reenue n the unorm rce hyrd wth rsk neutral dders s gen y: SQ RUH Q Q Q (A7) SQ R Q U where RU Q Q S Q ollows rom y equaton 22 o Cox Smth and Walker [985] The nequalty Q SQ Prooston A2: R R R DH Q D D Proo: xected reenue n the dscrmnate hyrd wth rsk neutral dders s gen y: RDH DH Q Q DH h where h s the densty uncton o the th order statstc n a samle sze o Susttutng or DH d usng equaton (9) n the text yelds (A8)
6 SQ RDH Q D h Q SQ Q D h Q SQ R Q D where the last equalty ollows rom equaton (4) o Harrs and Ra [98] Prooston A3: R R R UH UH A U d d (A9) Proo: Proe the second nequalty rst xected reenue n the unorm rce hyrd wth rsk aerse dders s gen y ~ RUH A Q UH A ~ where UH A reresents the rst reected d n the unorm rce hyrd wth rsk aerse dders or the Q th d xected reenue n the unorm rce aucton s gen y ~ RU Q Q By Prooston 2 UH A Q whch mles RUH A RU A smlar logc roes the rst nequalty xamle wth heterogeneous rsk reerences: CRRA utlty and the unorm rce hyrd mechansm The equlrum d uncton or the unorm rce hyrd wth CRRA utlty and heterogeneous S Q r rsk reerences s gen y Q where and r denote dder s alue and coecent o relate rsk aerson resectely Consder two dders wth r r 2 and r2 r Assume r and so that oth dders are rsk aerse ut dder s relately more so An ecent mechansm would guarantee that n equlrum dder 2 (dder ) outds dder (dder 2) roded ( ) We roceed y solng or the alue o or whch 2 Let r S Q r where r Q Consder the ollowng three ranges o ossle alues or : and For alues o less than zero or greater than the dder wth the hgher alue sumts the hgher d and the unorm rce hyrd s ecent Howeer when 2 ut 2 ; under the unorm rce hyrd dder s more lkely to wn a unt o the good een though dder 2 has a hgher alue or the good Thereore the unorm rce hyrd s not n general ecent under heterogeneous CRRA rsk reerences
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