8 Jun ANSWERS TO EVEN NUMBERED EXERCISES IN CHAPTER SECTION : INCENTIVE COMPATABILITY Exrcis - Educaional Signaling A yp consulan has a marginal produc of m( ) = whr Θ = {,, 3} Typs ar uniformly disribud Th bs ousid opporuniy is (, r ) = (, ) Th cos of q accumulaing ducaional crdnial q is C(, q) = so ha h payoff o a yp consulan, if sh rcivs a paymn r for hr srvics is q u(, q, r) = r (a) Wha is h bs sparaing PBE of h gam (b) Wha is h bs pooling quilibrium in which h lowr wo yps ar poold (c) Wha is h bs pooling quilibrium wih all hr yps poold? (d) Compar h quilibrium payoffs () Characriz h s of PBE in which h lows wo yps ar poold ANSWER (a) In a PBE h lows yp dos no signal Thus ( z, r) = (, ) = (,) Th local upward consrain is binding In a sparaing quilibrium r = = Thrfor (, ) (, ) = u z r = u z r = z and so ( z, r ) = (,) Th local upward consrain is 3 binding for yp hrfor = u( z, r) = u( z3, r3) = 3 z3 and so ( z, r 3) = (5,3) 4 4 (b) In a PBE h bs quilibrium wih h lows yps poold has a zro ducaion rquirmn Th avrag produciviy is hr for ( z, r) = ( z, r) = (,) Th local upward consrain is binding for yp hrfor = u ( z, r ) = u ( z, r ) = 3 z and so ( z, r 3) = (6,3) 3 3 4 3 (c) Wih all hr yps poold h quilibrium wag is (d) Th lows yp gains from pooling and h largr h pool h br Typ is wors off in h parial signaling quilibrium and bs of in h pool Typ 3 is bs off in h Answrs o Chapr 9 pag
8 Jun sparaing quilibrium Thus among h s of PBE all hr quilibria ar Paro fficin () L ( z, ) b h parial pooling oucom for h lows yps For any ducaion lvl h minimum wag is a PBE is an his is suppord by h blif ha i is h lows yp choosing ha ducaion lvl Thus u ( z, ) = z and so z [, ] Th local upward consrain mus b binding for yp Thrfor z = u ( z, ) = u ( z, r ) = 3 z and so z3 = 6 z 3 3 4 3 SECTION : INFORMATION REVELATION, ADVERSE SELECTION AND SIGNALLING Exrcis -: Can local incniv consrains hold for fficin signals? Th uiliy of yp is u(, q, r) = r q / Thr ar T yps and < < T If i is blivd ha h yp snding h signal q is, h scond sag rspons funcion is R(, q) = q (a) Show ha h fficin acion for yp, rspons R and h rsuling uiliy (, q ( )) q ( ) = Hnc solv for h fficin U ( ) (b) Suppos ha + = a Show ha h local upward consrain u(, q ( + ), r ( + )) U ( ), is saisfid if and only if 3 4 a a 3 (c) Confirm ha his consrain holds if a > and is violad for all a (, ] Rmark: This illusras a vry gnral poin Unlss ach yp is sufficinly diffrn, i is only possibl o spara by ngaging in socially xcssiv signaling ANSWER Answrs o Chapr 9 pag
8 Jun (a) In an fficin signaling quilibrium ach yp s signal is rvaling hnc a yp agn s payoff is u(, q, R(, q)) = q q/ This is maximizd a q ( ) = Th oucom for yp is hrfor consrain for yp, u(, z, r) u(, z +, r+ ), can b rwrin as follows 4 4 3 3 + + + 3 ( q, r) ( q, q( )) (, ) Dfin a so ha + = a Thn h local upward consrain is Dfin 4 3 3 a ( a + ) ha ( ) a 4 3 = a + No ha () h = and = = Th local upward ( ) = ( 3) Thus ( ) h a a a ha is 3 dcrasing ovr [, ] and incrasing for all highr a No also ha h () = Thrfor 3 ha ( ) is posiiv for all a and is ngaiv if a (, ] Exrcis -4: Choic of signals wih coninuously disribud yps Coninu wih h modl of h prvious xrcis bu now yps ar coninuously disribud on h inrval [,] Th signaling cos funcion is C(, q, ) = q/ Th marginal produc of a yp consulan is m( ) = (Appaling o h abov limiing argumn, h lows signal, x( ), is zro (a) Solv for (b) Wri down h firs ordr condiion for incniv compaibiliy Ingra and hnc show ha h quilibrium cos of signaling for yp is C whr τ = /, a + + (, ) = ( τ ) Answrs o Chapr 9 pag 3
8 Jun (c) Confirm ha + C (, ) + τ ( ) = Hnc show ha C (,) < C (,) for all + C (, ) + τ > (d) Show ha C (, ) < C (, ) for all > and > ANSWER (a) Appaling o h argumns of h prvious xrcis r = m( ) hnc a = r and so = r / a (b) W hrfor look for a sparaing quilibrium for all yps on h inrval r = L q(, ) b h quilibrium signal Typ is h lows yp so a [,] [,] q(, ) = If yp chooss q (, ), h quilibrium signal of yp, sh is assumd o b yp and so hr payoff is u(,,, x) = a C(,, q (, )) = a q (, )/ For ruh-lling o b a bs rspons, his mus ak on is maximum a q u(,,, q( )) = a (, )/ = a = q Hnc (, ) = a Ingraing, + + ( ) Sinc q(, ) =, q(, ) = a + Finally, h quilibrium signaling cos is + q(, ) = a + k + FOC = + + + τ C (, ) = q(, )/ = a( ) = a( ) ( + ) + (c) Appaling o (b) whr τ = / + C (, ) + τ = ( + ) C (, ) + τ Hnc 3 C (,) τ + τ + τ τ = ( ) = ( ) = ( + ) C (,) 3 τ 3 + τ 3 + τ Answrs o Chapr 9 pag 4
8 Jun I is radily confirmd ha his funcion is incrasing for τ [,] and quals a τ = Thus for all > (and hnc τ < ), C (,) < C (,) + C (, ) + τ + (d) Dfin r( τ ) = = ( ) No ha r() = < and appaling + C (, ) + τ + o l Hôpial s Rul r () = To prov h rsul w show ha for any ˆ τ such ha r( ˆ τ ) =, h slop r ( ˆ τ ) > For hn r( τ ) mus b sricly grar han for all τ > ˆ τ Bu his is impossibl sinc w hav shown ha r () = Taking h logarihm of r( τ ) and diffrniaing, + r ( τ) ( + ) τ ( + ) τ ( + ) τ + + τ = = ( τ ( ) ) + + + + r( τ) τ τ τ + + τ τ r( τ ) = ( + ) τ ( ) + τ Suppos r( τ ) = Thn τ r ( τ) = ( + ) τ [ ] + τ For all τ < h righ hand sid is sricly posiiv For τ =, h numraor and dnominaor of h brackd xprssion ar boh zro An appal o l Hôpial s Rul sablishs ha r () = > Exrcis -6: Produciv Signal A workr of yp has a marginal produc of m(, q) q / = if h achivs ducaion lvl q His cos of ducaion is C(, q) = q/ Typs ar coninuously disribud on h inrval [,4] Thr is no ousid opporuniy (a) Wih full informaion show ha yp will choos 4 q ( ) = and ha his wag will b 3 m(, q ( )) = (b) Wih asymmric informaion, xnd h argumn abov o show ha h quilibrium wag funcion rq ( ) mus saisfy h following ordinary diffrnial quaion dr () rq 4 / q dq = Answrs o Chapr 9 pag 5
8 Jun (c) Solv for h quilibrium lvl of ducaion q( ) and h wag funcion rq ( ) ANSWER (a) Wih full informaion, yp chooss q o maximiz / q U(, q, m(, q)) = q Diffrniaing by q yilds h FOC U / = q = z 4 Solving, q ( ) = Thn yp has 3 m(, q ( )) = (b) In a signaling quilibrium yp chooss a signal q( ) and arns a wag Rq ( ( )) For incniv compaibiliy his mus b prfrrd o any ohr pair ( q ( ), Rq ( ( )) Tha is, q () u(, ) = U(, q( ), R( q( ))) = R( q( )) aks on is maximum a = Hnc ( R ( q()) ) q () = a = FOC Also, in quilibrium ach yp arns his marginal produc Thus Rq ( ( )) m(, q( )) q( ) / = = Subsiuing for from h FOC and rarranging, dr R( q) / q dq = Ingraing and noing ha h lows yp has a marginal produc of zro, i follows ha Bu Rq ( ) 8 = q 3 3/ Rq ( ( )) ( ) / /4 = ( 3) q / = q Thn 8 = and so 9 4 q( ) = Invring, 4 3/ 4 3 q q Thn R( q) = ( q) q = ( ) q / / 3/4 3 Answrs o Chapr 9 pag 6
8 Jun Exrcis -8: Equilibrium whn highr yps hav br ousid opporuniis A yp workr has a valu g( ) = and a signaling cos C(, q) = q/ Typs ar coninuously disribud on [,] Th ousid payoff o yp is w ( ) = γ +, whr γ < (a) Show ha wih full informaion, ach yp γ /( ) < = is br off aking his ousid opporuniy (b) Suppos ha wih asymmric informaion, yp ˆ is indiffrn bwn choosing h signal q( ˆ ) > and saying ou whil all highr yps signal For ˆ, dfin V( ) = u(, q( ), r( )) = q( )/ dv q( ) Show ha = Hnc show ha h quilibrium payoff funcion V ( ) mus d dv saisfy h diffrnial quaion + V ( ) = Ingra and so sablish ha d k V(, ) = + k (c) Dpic h 45 lin and h curv V(, k) for k > Show ha a h inrscion of his curv and 45 lin h slop of h payoff funcion is zro (d) Draw in h ousid opporuniy lin and xplain why h Paro prfrrd signaling funcion mus b angnial o h ousid opporuniy lin () Solv for h minimum yp ha signals, ˆ, and h minimum signal q( ˆ ) ANSWER (a) Wih full informaion, yp can arn in h indusry or r ( ) = + ousid Thus h is br off ousid if + >, ha is < = Sinc yps ar disribud on h inrval [,], all ar ousid unlss + < (b) Arguing as in prvious xrciss, if ( q( ), r( q( )) is h signaling quilibrium q () oucom for, hn u(, ) = r( q( )) aks on is maximum a = Dfin U( ) = u(, ) Appaling o h Envlop Thorm, Answrs o Chapr 9 pag 7
8 Jun du q( ) q( ) = Also U( ) = Eliminaing q( ) yilds h ordinary diffrnial d quaion du U( ) d + = Solving, k U(, ) = + k (c) Two mmbrs of h s of soluions for k ar dpicd blow U U = U(, ) k k = + U (,) = ˆ Fig 5-: Soluions o h Diffrnial Equaion (d) Sinc U(, k) is a sricly incrasing funcion of k, a highr valu of k is sricly prfrrd by hos who signal Sinc h lows yp who signals mus b indiffrn bwn signaling and accping his ousid payoff, U( ˆ, k) = + ˆ Thus h Paro prfrrd payoff funcion mus ouch h ousid opporuniy lin This is h dashd lin in h figur abov ˆ () A h poin of angncy, ˆ ˆ q U(, k) = = + ˆ Also h slops ar qual, hus ˆ du ˆ qˆ = ( ) = ˆ d From h scond condiion, ˆq ˆ = Subsiuing his ino h firs condiion, ˆ = /( ) Sinc [,] no yp signals unlss /( ) <, ha is + < Answrs o Chapr 9 pag 8
8 Jun SECTION 3: MECHANISM DESIGN Exrcis 3-: Opimal slling schm A sor ownr has singl uni for sal (an aniqu) A consumr walks ino h sor H knows ha his valuaion of h im is V = [, ] Howvr h sor ownr knows only h disribuion of possibl valuaions, ha is, h pdf f ( ) (a) L q( ) b h quilibrium probabiliy ha a rad will b mad o a yp buyr, l r( ) b his yps quilibrium xpcd paymn and l U ( ) b h xpcd buyr payoff Show ha h quilibrium marginal informaional rn is U ( ) = q( ) Hnc show ha h xpcd rvnu of h sllr is R = q( ) J( ) f( ) d U( ) whr J( ) = ( F( ))/ f( ) (b) Is i ncssarily h cas ha q( ) is an incrasing funcion? (c) Suppos ha J ( ) is a sricly incrasing funcion Explain why hr xiss som (, ) such ha J ( ) > if and only of >, < (d) Hnc show ha h opimal schm is o s q( ) =, () Wha is a simpl implmnaion of his opimal schm? (f) Suppos ha J ( ) changs sign a, and Explain why, for an opimal schm q( ) =, < and q( ) =, q Hnc R = J( ) q( ) f( ) d + J( ) f( ) d Argu ha if q( ) is sricly incrasing on som sub-inrval of [, ] i is no rvnu maximizing Thus q( ) = qˆ in [, ] (g) Hnc show ha ihr q ˆ = or q ˆ = is opimal and again discuss h implmnaion of h opimal schm ANSWER Answrs o Chapr 9 pag 9
8 Jun If a yp cusomr claims o b yp x his payoff is u(, x) = q( x) r( x) This mus hav a a maximum x U( ) = q( ) r( ) = hus h quilibrium payoff is Appaling o h Envlop Thorm, U ( ) = u(, x) = q( ) Th xpcd payoff of h buyr is x= U = U( ) f( ) d = U( )( F( ) ) U ( )( F( ) ) d = U() q( )( F( )) d Th oal surplus gnrad is q( ) f( ) d Th sllrs gain (hr rvnu) is h oal surplus lss h cusomr s xpcd payoff, ha is R = q( ) f( ) d U Th claim hn follows by subsiuion (b) Sinc h singl crossing propry holds, a ncssary condiion for a mchanism o b incniv compaibl is ha q( ) mus b incrasing (c) Sinc F() =, J() < Sinc F( ) =, J( ) > Thn, sinc J ( ) is sricly incrasing hr is a uniqu (, ) a which J ( ) changs sign (d) Poin-wis opimizaion of R = q( ) J( ) f( ) d U( ) yilds q ( ) =, < and q ( ), = Sinc q ( ) is incrasing, i is incniv compaibl Th lows yp has h ousid opion of a zro payoff so U () = () Th sllr announcs a fixd pric p = Thn h im is radd if and only if p = Answrs o Chapr 9 pag
8 Jun (f) Poin-wis opimizaion on [, ] yilds [, ] yilds Hnc q ( ) = and poin-wis opimizaion on q ( ) = Sinc q lis on h inrval [,], his is incniv compaibl R = J( ) q( ) f( ) d + J( ) f( ) d Suppos q( ˆ ) = q For all [, ], J ( ) hus i is opimal o mak q( ) as larg as possibl Bu q( ) q( ˆ ) = q(sinc q( ) mus b incrasing) Thus q ( ) = qon ˆ [, ] A similar argumn sablishs ha q ( ) = qon [, ] ˆ (g) Subsiuing ino h xprssion for xpcd rvnu R = J( ) q ( ) f( ) d + J( ) f( ) d = qˆ J( ) f( ) d + J( ) f( ) d This is a linar funcion of ˆq so ihr q ˆ = or q ˆ = mus b a soluion Thus again h opimal soluion is a sp funcion whr h sp is ihr a or Implmnaion is hrfor xacly as in h monoonic cas 3-4: Pric Discriminaion wih a fixd supply Th dmand pric funcions of yp is p(, q) = q, whr is coninuously disribud on [,] Th fracion of cusomrs whos yps ar no grar han is F( ) Th aggrga supply of h commodiy is Q I is illgal for h monopoly sllr o pric discrimina dircly Insad h mus offr a schdul of slling plans ( qr, ), ha is, q unis for a oal cos of R (a) L V ( ) b h payoff of yp in h profi maximizing schm and l q( ) b dv h allocaion rul undr his schm Show ha q( ) d = Hnc obain an xprssion for h oal profi R (b) Hnc show ha oal rvnu is Answrs o Chapr 9 pag
8 Jun R = B(, q( )) f( ) d U = ( ) q( ) d (c) Th aggrga supply is fixd so ha h monopoly dsigns an incniv compaibl schm o solv Max{ R q( ) f ( ) d Q} Explain why h Lagrang muliplir of q( ) q his consraind opimizaion problm is h marginal rvnu of h monopoly (d) Solv for h opimal allocaion rul as a funcion of h shadow pric () Ingra o obain an xprssion for ha oal allocaion and hnc solv for h shadow pric λ ( Q) = MR( Q) as a funcion of h aggrga supply Hnc xplain how o solv for h profi maximizing slling schm for any cos funcion CQ ( ) ANSWER (a) L ( q( ), R( )) Θ b an IC allocaion and paymn Dfin V = Max u q x R x = q x q x R x ( ) { (, ( ), ( ) ( ) ( ) ( )} x Appaling o h Envlop Thorm, du q( ) d = Ingraing ovr yps, h oal consumr payoff is Also Thrfor dv U = V( ) f( ) d = ( F( )) d = q( )( ) d d V = q q R ( ) ( ) ( ) ( ) R = B(, q( )) f( ) d U = ( ) q( ) d Answrs o Chapr 9 pag
8 Jun (c) Th allocaion mus saisfy h aggrga supply consrain q( ) d Q Th sllr hn solvs h following opimizaion problm subjc o h IC consrain ha h allocaion rul { q( )} q( ) Θ Max{ R Q q( ) d } (d) Forming h Lagrangian, mus b non-dcrasing L = (( ) q( ) q( ) d + λ( Q q( ) d) L R b h soluion o h maximizaion problm Appaling o h Envlop Thorm dr MR( Q) = dq = λ No nx ha h Lagrangian can b rwrin as follows L = λ q q d λq (( ) ( ) ( ) ) Poin-wis maximizaion yilds h following soluion q( ) = Max(, λ) Sinc his is non-dcrasing i is incniv compaibl () Subsiuing ino h consrain, ( ( + λ))d = Q ( + λ ) Ingraing h lf hand sid, Thn ( ( + λ)) = ( ( λ)) = Q MR / = λ = Q Answrs o Chapr 9 pag 3
8 Jun Thus for any oal cos funcion TC( Q) w can firs fix h oal supply Q and solv for marginal rvnu and so obain h profi maximizing oal rvnu TR( Q ) Thn w can choos Q o maximiz TR( Q) TC( Q) Answrs o Chapr 9 pag 4