Problem Set 3: Model Solutions

Transcription

1 Ecoomc 73 Adaced Mcroecoomc Problem et 3: Model oluto. Coder a -bdder aucto wth aluato deedetly ad detcally dtrbuted accordg to F( ) o uort [,]. Let the hghet bdder ay the rce ( - k)b f + kb to the eller, where k [,], b f = the (frt) hghet of the bd, ad b = the ecod-hghet of the bd. h a k+ t rce aucto: f k = we hae a frt-rce aucto, ad f k = we hae a ecod-rce aucto. he utlty fucto for bdder the u (b,b,, ) [( k)b k max b ] f b max b, otherwe. o f the - other bdder lay the trategy b( ), ' bet reoe ole max ( k)b kb(y)}dg(y) b b (b ) where y max ad G(y) = F (y) the dtrbuto of the hghet of - radom arable deedetly dtrbuted accordg to F(). he frt-order codto L b (b ) N M O Q P d (- k)dg(y) b (b ) { [( k)b kb(b (b ))]}g[b (b )] db where g(y) = ( - )F -2 (y)f(y) the dety of G ad f the dety of F. Ug the fact that he frt-order-codto become d db b (b ) = b [b ( b )], ( [ ( f[b (b )] k)b b b )] = ( b )( ) = F[b ( b )]. For the ecod-rce aucto (k = ), the aboe mle b = o matter what trategy b() the other are layg. I fact, t a domat trategy for each layer to bd b =. 2. he ayoff fucto for th game are u (b,b,, ) b - f b m b, - m b otherwe. If the - other are layg b() the ' bet reoe ge alue ole

2 where b b ( b ) Z Y ( b ) max (b - ) dh(z) + - b(z)dh(z) b he frt-order codto H(z) = Pr{m < z} = - Pr{m Z Y d dh(z) b ( b ) z} = -[- F(z)]. db b b b h b b b b b h b ( ) ( ) ( ( )) ( ( )) ( ( b )), where h(z) = H'(z). By the uual ere of te, th codto become F I HG K J h( ) b ( ) H( ) b( ) ( ) f( ) F( ) Ug a tegratg factor a aboe yeld b( ) [ F(z)] dz [ F( )] F HG U V W, b( ) I K J. where the cotat the defte tegral are determed the ame way a aboe. For the uform cae, F(z) = z, o we get b( ) + ( ). 3. We characterze aget tratege term of qualty q roduced at a ge kll leel, rather tha the effort e exerted. ecallg that q = e, ad aget are rk-eutral, we ca wrte the exected utlty of aget a u (q, ) = Pr[q > q ] q (a) h a ubcae of art (b) ad (c); whe = 2, the equlbrum trategy we foud there recely q() = 2 /2, whch quadratc kll a ecfed. (b) We how that q mootoe odecreag. Fx two kll leel, for aget, ad let q, q be the roduced qualte at thee kll leel, reectely. ce aget mut (weakly) refer reortg q to reortg q at kll leel, we hae Pr[q > q ] q Pr[q > q ] q mlarly, ce q (weakly) referred to q at, we hae Pr[q > q ] q Pr[q > q ] q UVW 2

3 Combg thee ge u that q q (Pr[q > q ] Pr[q > q ]) q q or, equaletly, ( )(q q ), ad o q mut be mootoe. (c) Coder ome aget, ad aume all aget emloy the equlbrum trategy q(). At kll leel, roducg qualty q ge exected utlty u (q, ) = Pr[q > q( ) ] q ce we kow q mootoe, ad kll leel are dtrbuted d uformly o [,], u (q, ) = (q (q )) q akg the F.O.C. wth reect to q ge du dq = ( )(q (q )) 2 q (q (q )) = ce q the equlbrum trategy, we make the ubttuto q (q ) = aboe ad rearrage term to get q () = ( ) ce I mle that t a bet reoe to roduce qualty q = at kll leel =, we coclude that the ymmetrc equlbrum trategy q() = ( ) d = ( ) (d) Mootocty of q mle that the maxmum-qualty ubmo wll come from the mot klled artcat. ce kll leel are dtrbuted d uformly o [,], the dtrbuto of the max kll leel wll be F max () =. hu, the exected qualty of the wg ubmo E [max q( )] = q( )df max ( ) = ( ) d = 2 ( ). O the other had, each aget wll exed (exected) effort E [ q() ] = ( ) d = ( ). ce there are artcat we ca ee that, exectato, the total effort exeded by all artcat wll be twce the tem qualty receed by the comay. 4. ecall that both buyer ad eller hae alue draw d from U[,]. Alo, both the buyer ad eller are treated a bdder the acedg aucto; oly the aymet fucto dtguh them. (a) Otmalty for the buyer mmedate, ce by drog out before the rce reache b the buyer mly forgoe the oblty of ome roftable trade ad by cotug after the rce exceed b the buyer' ayoff at mot ad may be egate. Ge th trategy of the buyer, the eller' roblem to determe a dro out rce to 3

4 max ( )d ( )( ) whch ha a uque maxmum at = /2 +/2. Note that th aucto rocedure reult the ame outcome a a ecod-rce ealed bd aucto, a doe a extee-form whch the eller make a gle take-t-or-leae-t offer. (b) he cotructo of the eller-otmal ex ate effcet mecham mlar to the cotructo of the mecham that maxmze the exected ga from trade Myero ad atterthwate [983]. Our roblem to max U()d.t. (IC) ad (I).,x By alyg Lemma the ote to tate U() term of P() ad the tatg (IC) ad (I) a (*) the Corollary, are roblem become (aumg U() = ):.t. (*) max Z Y P(t)dtd [2b - 2](, b)ddb Z Y. ecallg the defto of P(), our obecte t Z Y Z Y Z Y Z Y Z Y Z Y Z Y (t, b)dbdtd (t, b)ddtdb (, b)ddb where the frt equalty follow from chagg the order of tegrato, ad the ecod from erformg the tegrato (ad reamg t a ). he Lagraga for the otmzato Z Y Z Y L(, ) [ + (2b - 2)](, b)ddb. We wh to fd ad to maxmze L(,) uch that the cotrat (*) bdg. For a fxed, L(,) maxmzed (otwe) by,b) = f b 2 ( f b 2 where = - /(2). he otmal foud by determg the alue of for whch the cotrat (*), 4

5 bdg. mle calculu reeal that the otmal = /2 (ote that > a requred), ce (*) the reduce to [2b - 2]dbd Z Y. /2+/2 Hece, the ex ate effcet mecham,b) = f b 2 2 * ( f b 2 2. But th mly the outcome of the aucto (a). 5