Timing: ex ante, interim, ex post. Definition. This is a draft; me with comments, typos, clarifications, etc.

Transcription

1 Ths s a draft; emal me wth comments, typos, clarfcatons, etc. Tmng: ex ante, nterm, ex post In secton, we unntentonally ran nto the concepts of ex ante, nterm, and ex post expectatons. Whle these may not formally be of much value, the ntuton they generate s qute useful n consderng the nuances between varous aucton optmzaton strateges (those of the seller and of the buyers). To ths end, we ll state some smple defntons below mostly taken from MWG (p89) and then use ntuton to flesh out the meanng behnd the math. These terms are ntended wth respect to the type space underlyng the game. Defnton Let represent the type of agent and represent the types of all other agents. Agent s ex post expected utlty s Her nterm expected utlty s Her ex ante 1 expected utlty s u (σ ( ), σ ( )) E [u (σ ( ), σ ( ))] E [u (σ ( ), σ ( ))] Note that the term ex post expected utlty s a lttle napproprate: there s no expectaton wth respect to types! It s, of course, possble that based on the constructon of the game there are stll some elements of chance once types are revealed, but we ll table that for now. The ntuton underlyng these concepts s not partcularly dffcult: ex post expected utlty s taken once all types are known n the mechansm; nterm expected utlty s taken once my type s known to me; ex ante utlty s taken before I even know my type. In ths sense, t s natural to thnk about these expectatons as ssues of tmng. We can support ths wth a throwaway story: an agent s drvng around LA and sees a sgn for a yard sale; she s bored so she goes to check t out. She gets out of her car and starts perusng the avalable goodes, and to her surprse dscovers a mnt-condton, orgnal-release LP of Purple Ran. Excted, she approaches the homeowner and haggles for the tem. Of course the homeowner dd not ntend to put such a valuable tem up for sale and mmedately takes t nsde. Our agent does not receve the tem. Here s how these tmng concepts play out n the above story: Before the agent has been to the yard sale, she s evaluatng her ex ante expected utlty (of the yard sale; the LP here s a red herrng). When we dscuss partcpaton constrants, we are n general dscussng ex ante expected utlty ncentves; here, the expected utlty from checkng out the yard sale was presumably greater than the expected utlty from more-mmedately gettng where she was gong, so she checked out the yard sale. Once the agent has valued the yard sale but before she knows how others value the yard sale here, she s dscovered somethng worth buyng but doesn t know how much t mght be sold for she s evaluatng her nterm expected utlty. That s, she knows that the yard sale holds somethng good, but she does not know yet whether or not she can acqure t as ths s a functon of the types of others (n partcular, the type of the seller). Aprl 27, 211 1

2 After the agent has haggled wth the seller over the tem and lost, she s evaluatng her ex post expected utlty (here, we make the not-qute-correct presumpton that hagglng reveals types). Once t s known that she does not value the LP as much as the owner, her now-known value of the yard sale drops to. We could of course have set ths up as an aucton proper but ths would not have the stayng power of When Doves Cry. How does ths te to our optmal aucton developments? We ve been dscussng the wn probablty functon w (); ths functon s dfferent from Pr( wns) n an mportant way: the wn probablty functon w ( ) s evaluated once types are known. That s, n a drect-revelaton mechansm the seller should, at some pont, know the types of the potental buyers. He s then free to allocate the tem (or not) as he sees ft. So w () s s probablty of wnnng the tem once all types are known; how ths functon s nondegenerate becomes clearer once we ve solved through a few questons. Snce t s evaluated wth full nformaton of types, t s an ex post concept. A partcular buyer does not know the other buyer s valuatons, but knows her own. We express her value functon as V ( ) = w () ( r ()) df ( ) = E [w () ( r ())] That s, her value s evaluated as an expectaton over the types of others; ncentve compatblty through frst-order condtons and the envelope theorem s constructed to support ths. Therefore the buyer s problem s evaluated at the nterm stage 2. The seller must construct a mechansm wthout any knowledge of types. Thus when the seller attempts to maxmze expected revenue over all possble realzatons, E [R()] he s optmzng the mechansm ex ante. It may seem odd that he s ex ante-optmzng over ex post actons (the wn probablty w ( )); but once he knows types, he s free to allocate the good accordng to w ( ) however he sees ft. Ths optmzaton over actons serves to cause agents to behave n a manner whch s consstently optmal for the seller. Optmal aucton subsdes Ths dscusson wll [unntentonally] roughly parallel Essental Mcroeconomcs exercse Hopefully the fact that the two are slght varants of one another wll make the underlyng concepts abundantly clear. Suppose we have an aucton wth two buyers, {1, 2}, whose values are dstrbuted accordng to F ( ) on [, 1]. Value functons, as usual, are gven by where r( ) s the expected payment of type. V ( ) = Pr( wns) r( ) Prevously (.e., earler n the quarter), we had assumed Pr( wns) = F ( ); that s, the probablty that type wns s essentally the probablty that she has a hgher type than her opponent. In optmal auctons, we make ths probablty more generc; the probablty that wns s now expressed as w (), a functon of both agents types. It s not mmedate why ths buys the seller power n optmzaton, but that should be made clear. Note that ths transformaton has the effect of makng the r( ) notaton potentally nsuffcent; we then change expected payment to r (), so that payments may now depend on the type of the other player. The agent s value functon then becomes V ( ) = E [w () r ()] 2 It s also evaluated ex ante, when the buyer s decdng whether or not to partcpate; assumng partcpaton constrants hold, the buyer s problem s fully n the nterm stage. Aprl 27, 211 2

3 To construct an optmal aucton, we are concerned wth maxmzng the seller s revenue. The seller receves r () from each of {1, 2}; however, the seller s unaware of agents types that s why an aucton mechansm s beng run n the frst place! so the evaluaton of revenue must take place as an ex ante expectaton. The seller s expected revenue s then E [R()] = E [r 1 () + r 2 ()] = E [w 1 () 1 V 1 ( 1 ) + w 2 () 2 V 2 ( 2 )] At ths pont, we need to obtan an expresson for E [V ( )]. From the ntegral form, we have E [V ( )] = V ( )df ( ) = V ( )(1 F ( )) 1 + = V ( )(1 F ( ))d ( ) 1 F = V ( ) ( ) f( )d f( ) [ ( )] 1 F = E V ( ) ( ) f( ) Notce that we have appled here the fact that V () = wthout much fanfare; that the low type receves payoff s not the pont of ths exercse, and follows from standard arguments. By standard envelope theorem arguments, we have V ( ) = E [w ()] Then the expectaton derved above becomes ( )] 1 F ( ) E [V ( )] = E [E [w ()] f( ) [ ( )]] 1 F ( ) = E [E w () f( ) ( )] 1 F ( ) = E [w () f( ) From ths, we obtan an expresson for the expected revenue of the seller, ( ( )) ( ( ))] 1 F1 ( 1 ) 1 F2 ( 2 ) E [R()] = E [w 1 () 1 + w 2 () 2 f 1 ( 1 ) f 2 ( 2 ) Transformng ths nto Rley s J ( ) functon, ths may be expressed succnctly as ] E [R()] = E [ 2 =1 w ()J ( ) The summaton notaton here s of course a lttle overkll, but helps to demonstrate just how cleanly ths should generalze to the case of many agents. How does ths te to maxmzaton of seller revenue? The above becomes, n ntegral form, E [R()] = 2 w ()J ( )df 2 ( 2 )df 1 ( 1 ) =1 Presumably, the denotaton optmal here refers to the fact that, n the real world, the seller s generally the mechansm desgner (or, n the case of thngs lke ebay, at least the mechansm selector). In ths sense, f we are to act lke reasonable economsts t behooves us to thnk lke the desgner should n practce; our concern for optmzaton s then the seller s revenue. Aprl 27, 211

4 The Mrrlees trck tells us that maxmzaton on the left-hand sde may be attaned by pontwse-maxmzaton of the ntegrand on the rght-hand sde. So the frm s actually solvng (for each = ( 1, 2 ) ) max w () =1 2 w ()J ( ) But ths maxmzaton s lnear! Snce w ( ) s constraned to be a vald probablty dstrbuton (possbly ncludng the opportunty for no allocaton), t s mmedate that w () = χ (, J ( ) > J ( ) J ( ) > ) That s, the object s allocated to the bdder wth the hghest J ( ), provded ths value s postve. If no buyers have postve J ( ), the object remans wth the seller. At ths pont, t s reasonable to ask why we are even botherng to construct optmal auctons, gven the revenue equvalence theorem. But we need to be careful when dscussng ths result: revenue equvalent only holds for ex post effcent outcomes, where the hghest-typed bdder wns the tem for sure. Here, we are causng a hgher-typed buyer to lose some fracton of the tme n order to extract more expected revenue from hm. Thus the antecedents of revenue equvalence do not hold, and optmzaton s not hamstrung by ths mathematcal necessty. It s ths neffcency of outcomes that leads to the noton of subsdy. Of course, we are not n ths case actually provdng a monetary subsdy to the less-powerful bdder, but n tltng the odds n ther favor we are offerng an mplct subsdy to ther actons. Whle t s unclear that ths causes the weaker bdder to bd more strongly, ntutvely t makes sense that t should cause the stronger bdder to bd even hgher than before. Ths can be seen by solvng through the partcular bd functons (usng explct dstrbutons, f necessary), but the algebra s nasty and not partcularly nformatve. Ironng In problems of consumer demand from a monopolst, we have seen over the past few weeks that monotoncty of the quantty functon q( ) s necessary to ensure ncentve compatblty. In Rley s lecture notes, we saw an example of when, followng the standard method of solvng for q( ), the functon s nonmonotonc. We ll solve through that example here and dscuss the technque for fxng the nonmonotoncty problem. Suppose we are gven a standard monopolst setup: there s a frm whch produces quantty q at cost c(q), and a buyer wth value dstrbuted wth CDF F ( ) on [α, β]. The agent s value s as usual, V () = B(q(); ) r() For ncentve compatblty, we have Ex ante utlty s V () = B (q(); ) E [V ()] = β α V ()F ()d Aprl 27, 211 4

5 Integratng by parts, we fnd E [V ()] = V (α) + = V (α) + β α β α β V ()(1 F ())d B (q(); ) (1 F ()) d ( 1 F () = V (α) + B (q(); ) α F [ ( () )] 1 F () = V (α) + E B (q(); ) F () We can obtan an expresson for the expected revenue of the seller, E [r()] = = = = β α β α β α β α r()df () (B (q(); ) V ()) df () B (q(); ) df () E [V ()] ( B (q(); ) B (q(); ) ) df () ( )) 1 F () F df () () Now, the frm would lke to maxmze expected revenue. Assumng the functons nvolved are well-behaved, we can pursue a pontwse maxmzaton strategy; that s, for each we maxmze ( ) 1 F () R(q; ) B (q; ) B (q; ) F () accordng to the suppled quantty q. Recall that B (q; ) = q p(t; )dt If we are free to swtch the order of the dervatves, we can obtan from frst-order condtons ( ) 1 F () q R(q; ) = p(q; ) p (q; ) F () To proceed any further, t wll serve us well to specfy the problem more completely. Suppose types are dstrbuted n Θ = [, 2], { F () = 4 f < otherwse We assume a lnear cost functon, c(q) = kq, and lnear prce demand, p(q; ) = q (that s, B(q; ) = q q2 2 ). From standard analyss, we have that at the optmum margnal revenue should equal margnal cost. Then q s defned by ( ) 1 F () q = k F () In a more-closed form, ths gves q = { 2 k 4 f [, 1) 2 k 2 f [1, 2] Aprl 27, 211 5

6 Of course, we stll need to do some edge correctng to make sure the quantty suppled s never negatve, but ths s a nonssue 4. Notce that ths q s nonmonotonc (check the jump at 1). Intutvely, ths volates ncentve compatblty: to mantan contnuty of the value functon, the frm wll need to offer two prces for the same quantty. (we have seen ths before: that q must be monotone) How then can we transform q to respect ncentve compatblty? Usng Rley logc, we were able to see that t only makes sense to ntroduce a flat regon nto the q functon; that s, devaton to a nonconstant quantty wll cause the frm to sacrfce more revenue relatve to the optmum than t must otherwse. The queston then becomes, where should we locate ths flat regon? One key prncple s that ths regon must be anchored at both ends to the exstng q functon. That s, let [, ] represent the regon whch has been flattened; to properly specfy that a regon s flat, we need q() = q(). Let s assume that ths relatonshp can be modeled as = m() for some m( ). We clam that the best the seller can do s mnmze lost proft (relatve to the opmum obtaned from q), and ths s farly ntutve: f we need a new devce to be ncentve compatble but we are solvng n the context of the seller s optmum, we should mnmze devatons from the unconstraned optmzaton problem. Snce q wll reman unchanged outsde of the range [, ], we can constran all of our reasonng to ths partcular range. At the optmum, the seller s proft from buyers n ths range s m() (R (q(); ) c (q())) df () Snce the quantty suppled n the flattened, ncentve compatble mechansm s anchored to q(), we can represent the seller s proft from flattenng as m() (R (q () ; ) c (q ())) df () The optmum level of proft must always le [weakly] above the flattened level of proft, so mnmzaton as stated s equvalent to 5 mn m() (R (q(); ) c (q())) df () m() (R (q () ; ) c (q ())) df () To locate the mnmum, we take frst-order condtons wth respect to and obtan = (R (q (m ()) ; m ()) c (q (m ()))) F (m ()) m () (R (q () ; ) c (q ())) F ()... ((R (q () ; m ()) c (q ())) F (m ()) m () (R (q () ; ) c (q ())) F ())... m() (R q (q () ; ) c (q ())) q () df () From Rley logc, we were able to see that q() = q() = q(m()). It follows that the leadng four terms n the above expresson cancel, and we are left wth m() (R q (q () ; ) c (q ())) q () df () = 4 It was mentoned n secton that the easest way to get around a large part of ths s smply to assume k =, all producton s free of charge. Our analyss wll hold up wth or wthout ths assumpton, but f you re graphng along at home k = wll make thngs reasonably nce. 5 In secton, we ether reversed the order of subtracton, or napproprately used mn when we should have used max (dependng on how you want to look at t). Snce we ddn t bother checkng second-order condtons none of our subsequent math was affected by ths mstake. Aprl 27, 211 6

7 Margnal cost prcng tells us c (q()) = R q (q(); ), so the above becomes m() (R q (q () ; ) R q (q () ; )) q () df () = Here, t becomes panful to proceed wthout substtutng n for the exstng problem specfcaton 6. Note frst that we can explctly calculate m(), 2 k 4 = 2 k 2 = + 1 Further, we have a clear boundary pont delmtng where the ntegrand changes sgns; n partcular, when q() < q() we should have R q (q(); ) > c(q()) (q() should ncrease to the maxmum) and when q() > q() we should have R q (q(); ) < c(q()). Snce we have a sngle pont of nonmonotoncty (n some sense), we can restate the problem as (R q (q () ; ) R q (q () ; )) q () df () = (R q (q () ; ) R q (q () ; )) q () df () On the nteror of both of these ranges, q ( ) = 2 s well-defned. In the left-hand range, df () = 4 the rght-hand range, df () = 1 4. So we may smplfy further and n (R q (q () ; ) R q (q () ; )) d = (R q (q () ; ) R q (q () ; )) d Substtutng n for known equatons above, we have (for a partcular γ dependng on the sde of the equaton n dscusson) ( ( )) 4 R q (q () ; ) R q (q () ; ) = q () (γ ) q () ( ) 4 = γ Then our overall optmum smplfes qute ncely to Solvng through, ( ) = d = = = = = ( ( + 1 ( ( + 1 = = d ) ) ) ( + 1 ) ( =1 ) Indeed, t seems Myerson s specfcaton leaves roughly ths form as the fnal soluton. ( + 1 ) ) 2 ( + 1 ) Aprl 27, 211 7

8 Then we have a full characterzaton of quanttes n the optmal ncentve compatble mechansm, [ ] 2 k 4 k+4 f 6, 7 { 6 max } [ 7 k, f ] 6, 9 6 q() = [ 9 max{2 k 2, } f ] 6, 2 otherwse Ostroy topcs At ths pont, we have not yet covered much economcs proper n Ostroy s class; ths wll change n the comng weeks. Stll, we should address some of the topcs that he wll be coverng and engage n our usual comp revew. It s necessary to formally ntroduce two new concepts to cover the selected queston ths week, but do not use ths as the go-to reference: Ostroy wll cover ths far more thoroughly than we can hope to here, and always take hs word over the notes. Defnton A prce-takng equlbrum for a gven vector of value functons v conssts of a normalzed prce vector (p, 1) and an allocaton (z, m ) for each I such that (z, m ) d(v, p) and markets clear, z =, m = That s, a prce-takng equlbrum n the quaslnear model s precsely the dea of equlbrum that we are accustomed to: agents utlty-maxmze gven prces, and markets clear. Ths defnton ponts to one of the key features of the quaslnear model: all endowments may be normalzed to and encoded n the value functon (that s, rather than havng a utlty functon and an endowment floatng around for each agent, we alter the reference pont for utlty and place all nformaton nto the value functon) and budgets dsappear. Market clearng s then fully descrbed by all nterestng quanttes summng to, or that one agent s ncreased consumpton relatve to hs endowment must be balanced by another agent s lowered consumpton relatve to hers. Defnton Gven an economy E = I, {X }, { }, {e } (where ths formulaton accommodates the quaslnear case), the k-replcate E k s defned by E k = k I h, {X h }, { h }, {e h } h=1 where I h = I, X h = X, h =, and e h = e. That s, E k s the orgnal economy E replcated k tmes, wth k copes of each agent and her attached preferences, endowments, and consumpton opportuntes. In Ostroy s class, replcates wll eventually be used to make the case for competton; as more agents are ntroduced to the system, sets whch were once nonconvex become approxmately convex on an approprate Aprl 27, 211 8

9 scale. Ths leads to nce propertes for optmzaton and equlbrum-attanment. The ratonale behnd replcatng an exstng economy rather than ntroducng new agents and creatng a new one s that n replcaton, we have a set of propertes whch transfer from the orgnal economy to ts replcates (that s, we know somethng about the underlyng agents n a proof-frendly way). Essental Mcroeconomcs, exercse 12.- Two bdders have values that are contnuously dstrbuted on [, 1]. The CDF for bdder s F (). The hazard rate s hgher for the second bdder; that s, f 1 () 1 F 1 () < f 2() 1 F 2 () (a) Show that F 1 () < F 2 (), (, 1), so that buyer 1 s the strong bdder. Soluton: note that at =, the hazard rate nequalty gves us f 1 () < f 2 (). Thus wthn a neghborhood of, for ε > suffcently small we should have F 1 (ε) < F 2 (ε). Suppose there exsts some such that F 1 () F 2 (). By contnuty of the underlyng dstrbutons, there must exst some such that F 1 ( ) = F 2 ( ); let be such that = mn { : F 1 () = F 2 ()} By contnuty, we know that ths mnmum s well defned. Knowng that F 1 (ε) < F 2 (ε), we may assume that < (that s, we are analyzng the frst crossng of F 2 by F 1 ). Agan followng from the hazard rate nequalty, we see that f 1 ( ) < f 2 ( ); then for δ > suffcently small, a Taylor expanson around wll gve us F 1 ( δ) > F 2 ( δ). But f ths s the case, contnuty tells us that there must exst some < δ such that F 1 ( ) = F 2 ( ), a contradcton of our defnton of. Hence there cannot be such that F 1 () F 2 (), and so F 1 () < F 2 () for all (, 1). (b) Show that the buyer s payoff from any sellng scheme can be expressed smply as a functon of the allocaton rule w ( 1, 2 ), {1, 2}, where these are the probabltes that the tem s assgned to bdder. Soluton: we assume that the queston ntends, any ncentve compatble sellng scheme. 7 We have, as usual V ( ) = E [w (, ) ( b( ))] Followng ncentve compatblty, we take frst-order condtons to obtan Knowng that V () =, we can then see V ( ) = E [w (, )] V ( ) = (c) Show that the expected seller revenue s E [w (, )] d [w 1 ()J 1 ( 1 ) + w 2 ()J 2 ( 2 )] df 1 ( 1 )df 2 ( 2 ), J (v) v 1 F (v) f (v) 7 Although, n a way, an ncentve ncompatble sellng scheme does not make much sense. Aprl 27, 211 9

10 Soluton: from the seller s perspectve, we have [ E [V ( )] = E E [w (, ) ( b( ))] ] = E [w (, ) ] E [w (, )b( )] = E [w (, ) ] E [r ()] Thus we have E [r ()] = E [w (, ) ] E [V ( )] Followng our standard trcks, we see E [V ( )] = V ( ) df ( ) = V ( ) (1 F ( )) 1 = + = = E [w (, )] (1 F ( ))d ( 1 F ( ) E [w (, )] F ( ) ( )]] 1 F ( ) F ( ) = E [ E [w (, ) V ( ) (1 F ( ))d ) df ( ) We then see [ ( )]] 1 F ( ) E [r ()] = E [w (, ) ] E E [w (, ) F ( ) ( ( ))] 1 F ( ) = E [w (, ) F ( ) = E [w (, )J ( )] The seller s expected revenue s the sum of the expected revenue from each of the two bdders, E [R()] = E [w 1 ( 1, 2 )J 1 ( 1 )] + E [w 2 ( 1, 2 )J 2 ( 2 )] = E [w 1 ()J 1 ( 1 ) + w 2 ()J 2 ( 2 )] Expressng ths expectaton n ntegral form gves the desred result, E [R()] = [w 1 ()J 1 ( 1 ) + w 2 ()J 2 ( 2 )] df 2 ( 2 )df 1 ( 1 ) (d) Hence comment on whether or not the optmal sellng scheme s symmetrc, and whether the playng feld should be tlted n favor of the weak player. Soluton: usng the Mrrlees trck, maxmzaton of expected revenue s dentcal to pontwse maxmzaton of the ntegrand. That s, for each = ( 1, 2 ) the seller should maxmze the quantty w 1 ()J 1 ( 1 ) + w 2 ()J 2 ( 2 ) The quantty whch the seller s optmzng s the vector of wn probabltes, w(). We know that the wn probabltes are constraned so that w 1 () + w 2 () 1 (that s, they represent vald probabltes and the good does not necessarly need to be allocated). Under ths constrant, the frm s optmum decson s ntutve: f both J 1 ( 1 ) and J 2 ( 2 ) are less than, let w() = ; otherwse, set w () = 1 f J ( ) > J ( ). We gnore the possblty of tes, as they are zero-probablty events and wll not enter nto agent decsons. Aprl 27, 211 1

11 What s key here s the second element of the optmum decson. Suppose that the two players draw the same type, 1 = 2 ; for sake of argument, assume both are so that J ( ) >. From the hazard rate nequalty, we see 1 = 2 ( ) ( ) 1 F1 ( 1 ) 1 F2 ( 2 ) = 1 < 2 f 1 ( 1 ) f 2 ( 2 ) J 1 ( 1 ) < J 2 ( 2 ) Then even though both bdders have the same valuaton, bdder 2 should be allocated the object! As long as the hazard rate s smooth, there wll be a neghborhood where 1 > 2 and yet bdder 2 s stll allocated the object. That s, at the optmum the odds are tlted n bdder 2 s favor. Ths s done to encourage bdder 1 to bd hgher than he would otherwse, n an ncentve-compatble way; t so happens that the [opportunty] losses from allocatng the good neffcently to bdder 2 are more than offset by the gans from bdder 1 ncreasng hs bd strategy. Essental Mcroeconomcs, exercse There are two buyers, and values are ndependent. Each buyer has a value of 1 wth probablty 1 p and a value of 2 wth probablty p. (a) Show that the equlbrum strategy n the sealed, hgh-bd aucton s for a low-value buyer to bd hs value and for a hgh-value buyer to bd a mxed strategy b wth support [1, 1 + p]. Soluton: the ratonale behnd the low-value buyer bddng hs value s gone over thoroughly n the week 1 secton notes. Intutvely, the low-value buyer wll never bd beyond hs valuaton due to negatve expected utlty; f we assume monotoncty of the bd functons, t follows that he s unwllng to bd below hs valuaton n equlbrum, due to the postve probablty assocated wth tebreakng. The ratonale behnd assumng a nce, well-behaved bddng CDF for the hgh-value buyer s also fully covered n the week 1 secton notes, so here we wll merely solve the ndfference condtons to demonstrate the desred clam. By lmtng ndfference, we know that the hgh-value buyer s expected utlty (n the lmt; recall the bd 1 + ε argument) from bddng 1 s (1 p)(2 1) = 1 p At the upper bound b of her mxture, she wns wth probablty 1; therefore by ndfference we have (1 p) = (2 b) = b = 1 + p It follows that the hgh-value bdder s randomzng over support [1, 1 + p]. (b) Confrm that the payoff to each buyer type s the same n the sealed, hgh-bd and open, ascendng-bd aucton. Soluton: from above, we know that n the frst-prce aucton (sealed, hgh-bd) expected payoffs are V fp 1 =, V fp 2 = 1 p In the second-prce aucton (open, ascendng-bd) each buyer bds hs type. We know V sp 1 = Aprl 27,

12 We may compute the expected payoff of the hgh-value buyer as V sp 2 = (2 1)(1 p) + 1 (2 2)p 2 = 1 p Thus we have revenue equvalence between the two aucton mechansms. (c) For an optmal aucton, argue that the local downward constrant must be bndng. Characterze an optmal drect revelaton scheme. Soluton: suppose that the local downward constrant s not bndng. Then the hgh-value bdder strctly prefers bddng truthfully to representng as the low-value type. But, as usual, ths means that we may ncreasng the hgh-value s expected payment slghtly wthout affectng ncentve compatblty. Snce ths acton leaves the seller better off, a mechansm n whch the local downward constrant does not bnd cannot be optmal. From ths constructon, we can see that we need the hgh-value buyer to be strctly ndfferent between reportng a hgh type and reportng a low type. Necessarly as n part (a) the low-value buyer wll pay 1 n an optmal, ncentve compatble mechansm. To construct ths as an equlbrum, we assume that a hgh-value buyer wll truthfully report; then another hgh-value buyer s expected payoff from reportng as the low type s V 21 = 1 (1 p)(2 1) 2 The other hgh-value buyer s expected payoff from reportng as the hgh type, lettng r 2 be hs expected payment provsonal on wnnng, s V 22 = ((1 p) + 12 ) p (2 r 2 ) Equatng the two valuatons as we must to support bndng local downward constrants, we have 1 (1 p) = (1 12 ) 2 p (2 r 2 ) 2 2p 4 2p = 2 r 2 r 2 = 6 2p 4 2p = p 2 p Then n an optmal mechansm, the low-value buyer pays 1 f she receves the good whle the hgh-value buyer expects to pay 6 2p 4 2p f he receves the good. Outcomes follow effcency crtera, wth one of the hgh-value buyers recevng the tem wth probablty 1 f one exsts n the mechansm. (d) Show that t s optmal to sell usng a sealed second-prce aucton where buyers are only able to make one of two possble bds. Soluton: from the above argument, we may construct a sealed second-prce aucton n two possble bds. It s evdent that one bd should be 1. The other, b 2, we solve usng the prevous expected payment equaton. Usng Bayes rule, we see that the hgh-value buyer expects to pay the low-value buyer s bd wth probablty 2 2p 2 p, condtonal on wnnng; he expects to pay the hgh-value buyer s Aprl 27,

13 bd wth probablty p 2 p. Applyng ths to the equaton above, we have ( ) ( ) 2 2p p + b 2 = p 2 p 2 p 2 p (2 2p) + pb 2 = p pb 2 = 1 + p b 2 = 1 + p p So we may construct a second-prce aucton n two possble bds, b 1 = 1 and b 2 = 1+p p, whch s optmal and ncentve compatble. Notce that, perhaps perversely, f the probablty of meetng another hgh bdder s suffcently low, b 2 s qute large (and well above 2, the bdder s valuaton); the trck here s that we are supportng ndfference between bds, not guaranteeng that the buyer s necessarly happy wth the outcome ex post. For pedagogy s sake, we ll ensure that ndfference s acheved. The expected utlty to the hgh-value buyer from bddng b 1 s V 21 = 1 1 p (1 p)(2 1) = 2 2 Hs expected utlty from bddng b 2 s V 22 = (1 p)(2 1) + 1 ( 2 p p ) p = (1 p) + 1 (p 1) 2 = 1 p 2 V 22 = V 21 Thus the bndng downward constrant s verfed. 21 Sprng comp, queston 6 Suppose (z ) s a feasble allocaton, z =, for the quaslnear model v = (v 1,..., v N ), where each v s merely contnuous. In the followng questons, determne f the statement s true or false. If true, demonstrate; f false, provde a counterexample. (a) If there exsts a postve nteger k and a feasble allocaton (zh k ) for the k-replca of v such that v (z h ) > k v (z ) then (z ) cannot be a prce-takng equlbrum for v. Soluton: true 8. h Suppose that (z ) s a prce-takng equlbrum supported by prces p; then each consumer s solvng z v (p). It follows that f we extend ths allocaton to the k-replcate, each agent s stll solvng 8 Ths s the opposte of what I clamed n secton. Aprl 27, 211 1

14 z h v (p). Hence (z h) s a prce-takng equlbrum n the k-replcate, supported by prces p. Socal welfare at ths allocaton s v (z h ) = k v (z ) h A prce-takng equlbrum must be effcent 9. If there exsts some feasble allocaton (zh k ) such that h v (zh k ) > k v (z ), the allocaton (z h ) s not solvng the maxmzaton problem necessary for effcency. Hence (z h ) cannot be a prce-takng equlbrum, and so (z ) s not a prce-takng equlbrum; the clam n queston s true. (b) If for all postve ntegers k and all feasble allocatons (zh k ), h {1,..., k} for the k-replca of v, v (z h ) k v (z ) then (z ) s a prce-takng equlbrum for v. Soluton: true. h Ths s a verson of the Second Welfare Theorem. That s, we can see that, snce z h = z s a feasble allocaton n every k replca t must be that (z ) maxmzes socal welfare n every k-replca; hence (z ) s effcent, snce effcency s equvalent to maxmzaton of socal utlty n the quaslnear model. From the Second Welfare Theorem and the prncple of no wealth effects, t follows that (z ) s a prce-takng equlbrum. Suppose that (x ) s a feasble allocaton for the ordnal preferences model of an exchange economy E = {(X ), ( ), (ω )}, where X = R l + and each s merely contnuous. (c) If there exsts a postve nteger k and a feasble allocaton (x k h ) for the k-replca of E such that x k h x,, h {1,..., k} then (x ) cannot be a prce-takng equlbrm for E. Soluton: ths depends on the defnton of the demand functon d(, p) n the ordnal preferences model of an exchange economy wthout quaslnear utlty 1. In the quaslnear model, there s no free dsposal (we must le on the fronter of our budget set); we wll presume the same thng here. In ths case, the clam s demonstrably false, but n the case wth free dsposal, the clam s true. Suppose there are 2 commodtes, l = 2, and that there s a sngle agent wth preferences representable by { { } } u (z; κ) = max mn 1 z y y N 2 κ, Intutvely, for κ suffcently small ths preference relaton has the agent ndfferent between most possble allocatons, wth steep utlty cones around natural coordnates. In a sense, the agent wants to consume only whole amounts of goods and hates fractons; these preferences are contnuous snce all upper and lower contour sets are closed for any κ >. Suppose the agent s endowment s ω = ( 1 2, 1 2 ). 9 Ths appears n the class notes, but the ntuton sn t so bad n the dfferentable case wth nonsataton: f we are not maxmzng, we can make a dfferental change n allocatons to mprove socal welfare. But ths mples that agents margns cannot equal the same prce level, hence they are not utlty-maxmzng. Then the allocaton s not an equlbrum. In partcular, n the quaslnear framework preferences are necessarly locally nonsatated snce more of the money commodty s always a good thng; ths s suffcent for the Frst Welfare Theorem. 1 For whch I cannot fnd a good defnton n Ostroy s notes. Aprl 27,

15 In the 1-replcate, we can support x = ω as a prce-takng equlbrum [roughly] wth prces wth a non-natural rato, κ suffcently small. So let p = (2, ); then the agent s budget fronter s z 2 = 5 4z 1 6 Importantly, the ntercepts are not natural numbers, so there must exst some κ > whch yelds only utlty along the budget fronter; so gven the prce vector p = (2, ), the agent s maxmzng utlty by consumng hs endowment. Now consder the 2-replcate. By tradng wth hmself (between replcas), the allocaton x 1 = (1, ), x 2 = (, 1) may be attaned (that s, ths allocaton s feasble). By constructon, x h x for all h. Thus the premse of the queston holds, but x may be supported as a prce-takng equlbrum. Hence the statement s false. How does the statement change wth free dsposal? Suppose there s some allocaton x k h such that x k h x for all, h. Assume that x s a prce-takng equlbrum supported by prces p; we clam that at least one agent h s consumng somewhere whch was ntally feasble accordng to a budget constrant wth free dsposal. Suppose otherwse; then p x k h > p x. It then follows that p x k h > k p x h ( ) p > kp p h ( k x k h e ) > kp But ths s a contradcton! So we must have at least one agent wth p x k h < p x. But snce we have assumed x k h x, ths cannot hold wth the agent utlty-maxmzng n a world wth free dsposal. It follows that f x s a prce-takng equlbrum, there exsts no k-replcate wth feasble x k h such that x k h x for all and all h. Then the suggested clam s true. (d) Suppose that for every k and every feasble allocaton (x k h ) for the k-replca of E, x k h x,, h {1,..., k} = x k h x Can you conclude that (x ) s a prce-takng equlbrum for E? Soluton: no (or, false ). Suppose l = 2, and that preferences are representable by the utlty functon u (x) = x 1 + x 2 There s one agent, and hs endowment s (e 1, e 2 ) = (, 1). In the unque feasble allocaton n the 1-replca, x = (, 1). In all k-replcas of ths economy, [x k h ] 1 = by market clearng. Then f x k h x for all h, we must have x k h = x ; hence x k h x for all h. Thus the assumptons stated n the clam above hold. However, x = (, 1) s not a prce-takng equlbrum. For (, 1) to possbly be supported as optmal consumpton subject to prces, we must have p 2 > otherwse the demand for good 2 s nfnte. Then snce the agent s provded wth a postve budget and the Inada condtons hold at (the x e Aprl 27,

16 endowment of good 1), the agent wll have strctly postve demand for good 1 for any fnte prce level p 1. Fnte prces are an mplct requrement for equlbrum (the budget constrant s ll-defned f the agent has unts of good 1 but ts prce s nfnte), so ths cannot support a prce-takng equlbrum. Snce the market clearng constrants tell us that [x ] 1 =, t follows that x s not a prce-takng equlbrum. Aprl 27,