Endogenous Uncertainty in a General Equilibrium Model with Price Contingent Contracts * Received: March 14, 1995; revised version: November 9, 1995

Transcription

1 Endogenous Uncerany n a General Equlbrum Model wh Prce Conngen Conracs * 2 Mordeca urz and Ho-Mou Wu Deparmen of Economcs, Serra Sree a Galvez, Sanford Unversy, Sanford, CA USA 2 Deparmen of Economcs, Naonal Tawan Unversy and Insue of Economcs, Academa Snca, 2 Hsu Chow Road, Tape 0020, TAIWAN Receved: March 4, 995; revsed verson: November 9, 995 Summary. Ths paper vews uncerany and economc flucuaons as beng prmarly endogenous and nernally propagaed phenomena. The mos mporan Endogenous Uncerany examned n hs paper s prce uncerany whch arses when agens do no have srucural nowledge and are compelled o mae decsons on he bass of her belefs. We assume ha agens adop Raonal Belefs as n urz [994a]. The radng of endogenous uncerany s accomplshed by usng Prce Conngen Conracs (PCC) raher han he Arrow-Debreu sae conngen conracs. The paper provdes a full consrucon of he "prce sae space" whch requres he expanson of he exogenous sae space o nclude he "sae of belefs." Ths consrucon s cenral o he analyss of equlbrum wh endogenous uncerany and he paper provdes an exsence heorem for a Raonal Belef Equlbrum wh PCC. I shows how he PCC complees he mares for radng endogenous uncerany and lead o an allocaon whch s Pareo opmal. The paper also demonsraes ha endogenous uncerany s genercally presen n hs new equlbrum. I. On he Naure of Uncerany The sandard componen of he heory of an ndvdual decson problem n an unceran envronmen s he specfcaon of he "world" whch s wha he ndvdual s unceran abou. The "sae of he world" s hen a complee descrpon of he world such ha once s revealed o he * Ths research was suppored n par by he Fondazone En Enrco Mae of Mlan, Ialy, and by he Naonal Scence Councl of Tawan. The auhors han Carsen. Nelsen for valuable suggesons.

2 3 ndvdual, no uncerany remans. The "sae" may be an enrely subjecve objec: may no be observable by ohers or comparable o he saes of oher agens. There s no requremen ha a decson maer be able o communcae hs sae o oher agens or ha hs sae can even be comprehended by ohers. We sress hese deals snce he heorec framewor of he ndvdual decson problem became he bass for he reamen of uncerany n general equlbrum analyss. The drasc concepual leap ha was aen by Arrow [953], Debreu [959] and Arrow and Debreu [954] was he assumpon ha he sae space was common o all agens. In conras o he ndvdual problem where he sae of he world s merely a subjecve descrpon of ndvdual uncerany, n general equlbrum heory descrbes commodes, denfes mares and s a bass for wrng conracs and specfyng propery rghs. In an Arrow-Debreu economy all mares for he exogenously specfed sae conngen clams requre, for her vably, an emprcally coheren descrpon of all saes. Moreover, n he formulaon of an equlbrum of a sequence economy where secures replace mares for sae conngen clams, Arrow [953] and Radner [972] elevae he concep of he sae o even more crucal role. In he formulaon of he equlbrum hey adop he raonal expecaons hypohess (whch has also been called he condonal perfec foresgh hypohess) where agens are assumed o now a each dae he map beween fuure realzed saes and fuure equlbrum commody and asse prces. Thus, n such an equlbrum wh secures he realzed sae mus resolve all ndvdual uncerany ncludng he uncerany of fuure prces. I s wdely recognzed ha hs exogenous and objecve concep of he sae whch s 3 See Savage [954], page 9. 2

3 4 common o all agens, has emprcal conen only for nsurance mares. An exogenous sae whch s observable and common o all agens canno resolve mos unceranes. On he oher hand, a sae whch expresses all ndvdual unceranes consss mosly of unobservable and ncomparable componens. Arrow [953] hmself explcly recognzed hs when explanng ha mares for exogenously specfed sae-conngen commody clams do no exs and herefore we mus consder secures as he man vehcles for reallocang socal uncerany. However, for he exogenous sae o be a useful ool for he prcng of secures agens need o now he maps from saes a fuure daes o prces n he fuure and s enrely unrealsc o assume ha agens can fnd ou wha hs sequence of maps s. I s hen clear ha he consrucon of an exogenous sae, common o all agens, serves as a convenen mahemacal devce whch enables an Arrow-Debreu heory o formally ncorporae he complex phenomenon of uncerany by merely relabelng commodes. As a useful mahemacal devce, he exogenous sae space ogeher wh he raonal expecaons hypohess n he sense of Arrow [953] have enabled exremely mporan developmens n he felds of general equlbrum heory and fnance. Wh he vew of exendng hs heory furher we noe ha he argumens agans hs approach are boh emprcal as well as heorecal. On he emprcal sde we now ha mares for sae conngen conracs rarely exs and 4 The Arrow-Debreu sae space has been useful only for nsurance mares. However, n hs case Malnvaud [972][973] showed ha snce hese mares handle only ndvdual, dosyncrac rss, he dversfcaon acheved hrough such mares mply ha hese ndvdual rss are of lle relevance o general equlbrum consderaons. The man allocaon problem whch general equlbrum heory mus address s hen he allocaon of socal rss; hese rss are represened mosly by he flucuaons of economc varables over me. 3

4 he exogenous "sae" s hardly descrbable. On he oher hand, an exensve array of prce conngen conracs (o be called "PCC" n hs paper) are raded by agens for hedgng rss. Such conracs are obvously raded daly n large volume on major fnancal mares across he world. Moreover, such nsrumens play an essenal role n he ordnary conduc of busness. A few examples wll llusrae he pon. Real esae developers and naural resource companes use opons and oher PCC as hedgng devces n her plannng of major projecs. Purchasers of large scale plan and equpmen such as arcrafs and gas ppelnes use PCC as a normal devce for handlng he rs of flucuang demand. Even owners of spor clubs use opons n any labor conrac o assure hemselves he connuy of servce whou prce rs. The use of PCC o rade prce uncerany s he consequence of he fac ha agens do no now he Arrow [953] maps from exogenous saes o fuure prces. In such a framewor conceps of "complee" and "ncomplee" mares s devod of emprcal conen snce here s no emprcal way of deermnng f a mare s complee or no. In fac, he presumpon should be ha mares are always "complee" snce under he assumpons of Arrow [953] agens now he map from saes o prces and consequenly can always add enough "dervave" secures o complee he mares. Ths observaon s due o Ross [976]. Also, when he mare srucure s complee "dervave" secures are "redundan" (see Haansson [978]). All he above are crcally cenral ssues o a general equlbrum heory wh secures. They are, however, of mnor mporance relave o our major heorecal argumen agans he Arrow [953] and Arrow-Debreu [954] sae space and raonal expecaons wh secures. The cenral ssue a sae s he naure of uncerany n economc sysems. The Arrow [953] and Arrow-Debreu [954] formalsm whch was adoped by all subsequen developmens, vews all uncerany n he economy as beng generaed by forces whch are exernal o he economc 4

5 sysem. In a emporal conex uncerany s mosly represened by he poenal flucuaons of economc varables. Hence, accordng o hs heory he flucuaons of asse prces, GNP or foregn exchange raes are all ulmaely explanable by exogenous facors such as he weaher, earhquaes, ec. Equally objeconable s he vew ha nohng ha agens do or hn has any mpac on he flucuaons of economc varables. I s our frm vew ha mos of he uncerany n an advanced ndusral socey arses from nernally propagaed flucuaons whch are generaed by he acons and belefs of he agens abou he naure of he mare and by her uncerany abou he acons of oher agens. Ths componen of uncerany was nroduced by urz [974a] who also nroduced he erm Endogenous Uncerany o descrbe. Such uncerany obvously canno arse n an Arrow [953] conex wh raonal expecaons. Our model recognzes he fac ha agens do no now he srucural relaonshps beween exogenous saes and opmal acons of oher agens or beween hese saes and prces. In such crcumsances agens are unceran abou fuure prces and urz [974a] proposed ha agens rade hs uncerany usng PCC such as opons. In such an economy PCC are no "redundan" n any sense: hey are he prmary vehcles o rade prce uncerany. These deas were furher developed by Svensson [98], Henroe [996] and urz [995]. Pvoal o hs approach s he exsence of dversy of belefs among he radng agens. The cenral ouloo of he endogenous uncerany approach s o vew uncerany and economc flucuaons as beng prmarly endogenous and nernally propagaed phenomena. I s herefore clear ha he developmen of hs approach mus be based on wo elemens. The frs s an negraon of a new heory of expecaons and belefs whch s compable wh he dversy of belefs among agens. The second s a comprehensve sudy of fnancal nsuons such as PCC 5

6 whch enable he radng of and reallocaon of endogenous uncerany. Our approach clearly recognzes he mporance of exogenous varables for equlbrum analyss. However, we defne he prce sae space o nclude he saes of belefs as well as oher endogenous varables such as prces or profs of frms. Such an economy conans he Arrow-Debreu economy as a specal case when here s no endogenous uncerany. In hs paper we wan o hghlgh he mporance of he "sae of belefs" and herefore wll exclude from he prce sae space oher endogenous varables. We pospone o laer papers he analyss of more complex prce sae spaces. Ths paper ams o develop an equlbrum heory wh endogenous uncerany n a one commody overlappng-generaons (OLG) conex and explore he role of prce-conngen conracs n he allocaon of rs. Our model has hree componens. Frs, agens are assumed o have Raonal Belefs (see urz [994a][994b][995][996]) whch nclude raonal expecaon as a specal case. The second componen of our model s he explc nroducon of a "prce sae space" and of prce conngen conracs. In conras wh he Arrow [953] and Radner [972] framewor where nowledge of he exogenous sae carres wh he complex nformaon needed o deermne prces, our prce sae space s eher a se of negers {, 2,..., M} n he case of fne prces or he un nerval n he case of connuum of prces. Thus, prces are hemselves he sae varables. On he oher hand, PCC are conracs ha specfy a delvery of commodes or secures a a fuure dae and such delveres are conngen on he prces whch wll be realzed a ha fuure dae. The hrd componen of our model s he sequenal srucure where mares reopen a each dae. Snce agens are assumed o hold Raonal Belefs he equlbrum concep employed here s a Raonal Belef Equlbrum nroduced by urz [994b][996]. In hese wo papers agens are nfnely lved bu are no allowed o rade PCC. The novely of he presen paper s ha here agens 6

7 can rade PCC and he framewor s an OLG model. Ths srucure wh a sngle consumable commody and a sngle producve acvy s a drasc smplfcaon. I does enable us, however, o hghlgh some of he essenal feaures of hs new approach o dynamc equlbrum analyss. II. The Model and he Raonal Belef Equlbrum (RBE) Concep (II.a) The Basc Model We use a sandard OLG model wh young agens n each generaon whch we denoe by =, 2,...,. There are also old agens n each generaon bu only he young receve an endowmen T, =, 2,... For each {T, =, 2,...} s a sochasc process whch wll be specfed below. Each young person s a replca of he old person who preceded hm where he erm "replca" refers o he ules and belefs. Ths s a model of "dynases" and he smplfyng assumpon made here s ha here s a fne number of such dynases. In addon o he mare for commodes raded n each perod, wo ypes of fnancal asses are ncluded. One s he common soc of a frm and a dae he supply (equals o ) of he soc s dsrbued among he old. Ths dsrbuon naes he fnancal secor and ulmaely ensures nergeneraonal effcency. The second asse class s a PCC whch enables an agen o conrac for he delvery of a un of he common soc a fuure daes conngen upon he prces whch preval a hese fuure daes raher han upon some absrac "saes" whch are realzed. The nfnely lved frm s assumed o be exremely smple: generaes exogenously a deermnsc sequence { R, =, 2,...} of dvdend paymens. Ths producon uses no resources and n mos of he developmen we shall assume ha R = R > 0. The PCC developed n Svensson [98], Henroe [996] and urz [995] enables agens o conrac for he fuure delvery of commodes and secures n a seng wh mulple commodes and secures. However, s 7

8 mmedae ha condons of "no arbrage" mply ha he only facor ha maers s he ably of agens o ransfer purchasng power across me raher han any specfc commodes or secures. We shall herefore specfy ha he PCC below perm an agen o conrac for he fuure delvery of he shares of he frm. Naurally, he prce of such nsurance could be so prohbve ha an agen may elec no o be fully nsured. We sress ha he only uncerany faced by he agens s he uncerany abou he prce of he soc and under our assumpons he mare s "complee" when hs erm means ha here s a feasble way of radng all prce uncerany n he second perod. In our model endowmen s random bu hs uncerany rases some neresng quesons. Snce hs uncerany exss before he agens are born and snce a "brh" hey are old wha T s, hs ex-pos varably of he endowmen s a useful ool of analyss. We could reallocae hs uncerany by assumng ha each agen s born wh a PCC and consequenly he uncerany would be shfed o he old agens. We see no advanage n such a devce snce boh observably and ncenves mae mpossble o drecly rade he uncerany of ndvdual endowmens and our concern n hs paper s he mare mechansm for allocaon of rs. The noaon we employ n hs paper s as follows: x - he consumpon of when young a ; 2 x - he consumpon of when old a +. Ths ndcaes ha was born a dae ; soc purchase of young agen a ; 2 0 -endowmen of he soc o an old agen a dae where 2 0 > 0 for all ; T - endowmen of when young a. Ths means ha =, 2,..., s among he young born a. Wrng T s unnecessary snce only he young receve an endowmen; p - he prce of he common soc a dae ; 8

9 c p - he prce of he consumpon good a dae ; f (p ) - he condonal densy of when young represenng hs belef a dae regardng + he dsrbuon of p a dae +. The subscrp denoes he me dependency and + suppresses he condonng on I - he hsory up o. $(p ) - he prce a dae of one un of a PCC for delvery of un of he common soc + a dae + conngen on p. + z (p ) - he amoun of PCC purchased a dae by agen, each for delvery of one un of + he soc conngen on p upon p + Max 4 (x, 2 m,z ) 0 p c x +. The owner of such a PCC receves a dae +, conngen, boh he soc as well as he dvdends on he soc a dae +. Ths assumpon s merely a convenon whch we follow for analycal smplcy. We assume ha he young are beng nformed of he realzaon of her own endowmens and consequenly he opmzaon problem of agen when young s as follows: () subjec o (2a) (2b) 4 u (x,x 2 % (p )) f % (p )dp % % % p 2 % $ m (p % )z (p )dp ' % % pc T 0 p c % x 2 % (p % ) ' 2 (p % % pc % R % ) % z (p % )(p % % pc % R % ). The budge equaons (2a)-(2b) are homogenous of degree zero n prces and because of prce normalzaon, he uncerany abou p s all he uncerany an agen faces. For he problem above + o mae sense, he funconal expressng he value of he amoun z of he PCC purchased by mus be well defned and hs leads o complex echncal dffcules (see Svensson [98], Henroe [996] and urz [995]). In all of our analyss below we wor wh he case n whch boh $(@) and p ae fne number of values. In fac, an mporan concluson of hs paper s ha by nroducng 9

10 he condons of raonaly of belefs, he prce sae space could be made o conss of only a fne number of elemens hus dspensng wh mos of he echncal problems menoned above. Also, n he case when {R, =, 2,...} s a random process, ssues of ncenves mae mpossble o rade a dae he uncerany of R. Tha s, snce n he real economy managemen decsons + nduce he sochasc process {R, =, 2,...} such ncenves would be affeced by he ably of managers o be on he oucome of her own decsons. The ably o rade common socs has some elemen of hese same ncenve problems, bu o a much lesser exen and even hen subsanal publc regulaons have been nsued o preven he dsorons of such ncenve effecs. As a modelng sraegy we assume ha all PCC can be raded bu do no perm he radng of conracs whch are conngen on he profs of he frm. Our prevously specfed assumpon of R = R s hen jusfed by he desre for smplfcaon. I also ensures ha each generaon wll have a complee se of mares o rade uncerany. Quesons of arbrage free prcng are mporan. To see wha resrcons hey mpose noe ha f an agen purchases un of he common soc a he prce p hen he receves he amoun of (p + R) uns of consumpon n perod +. On he oher hand, suppose ha + p c % he buys he consan PCC conrac - as a funcon of fuure prces - z (p ) ' %. He hen receves un of he common soc for sure. Snce by our convenon he also receves he dvdend nex perod, hs conrac generaes a ( + ) he same value as he ownershp of he common soc. The cos of hs compose PCC s $ m (p % )dp %. These consderaons lead us c o nroduce he followng: A prce sysem (p, p, $) s sad o be arbrage free f for all 4 0 (3) 4 p ' $ m (p % )dp %. 0 0

11 In he developmen below we requre ha he prce vecors (p,p c, $ ) sasfy (3). (II.b) Raonal Belef Equlbra of he OLG Model wh Prce-Conngen Conracs Our developmen here follows urz [994a] and [994b]. Raonal Belef Equlbrum (RBE) requres mare clearance. Thus, we say ha mares clear a all daes f, for all hsores (4a) (4b) j 2 ' ',2,... ' j z (@) ' 0 ' ',2,... If follows from (2a) - (2b) and (4a) - (4b) ha when mares clear hen (5a) (5b) p c x % p ' p c T ',2,... p c x 2 ' p % p c R ',2,... 2 where x, x and T are he aggregaes defned by x (6a) - (6b) ' j x =, 2 ' (6c) T ' j T. ' The non-arbrage condon (3) and a naural normalzaon are boh used n he selecon of an c approprae prce space S v for v = (p, p ). We need such a space o be a compac subse of a complee and separable merc space and hen denoes he Borel F-feld of he space n queson. (S 4 v, Û(S4 v )) s a measurable space where Û(@) 2 To defne a Raonal Belef Equlbrum le (x, x, 2, z, =, 2,...) be a sequence of opmal decson funcons whch are maps from hsores o acons for =, 2,...,. These funcons nduce a mare clearng process of prces (p,p c ), ', 2,... over he space (S 4 v, Û (S4 v ), A) and an assocaed sequence of funcons ( $, =, 2,...). One of he objecves

12 of hs paper s o show how he raonaly of belef condons enable us o wor wh a space wh fne number of elemens. Noe he crucal observaon ha he mare clearng probably A 2 was nduced by he belefs (Q, Q,..., Q ) of he dynases (represened earler by he 2 condonal denses (f, f,..., f )). In a RBE we have he dual propery ha A s nduced by (Q 2, Q,..., Q ) and each one of he Q s a Raonal Belef relave o A (as n urz [994a][994b]). Ths movaes our basc concep Defnon : A Raonal Belef Equlbrum wh Prce Conngen Conracs s a sequence of 2 decson funcons {(x, x, 2, z, =, 2,..., ) =, 2,...}, a sochasc process of prces c {(p (S 4 v, Û, p ) =, 2,...} on (S4 v ), A), a sequence of funcons {$ (p +), =, 2,...} 2 and a se of probably belefs (Q, Q,..., Q ) such ha 2 c () (x, x, 2, z ) s opmal relave o Q and {p, p, $ (p ) }, =, 2, () The mares clear a all daes and for all hsores. () Q s a Raonal Belef relave o A for =, 2,...,. The defnon of RBE does no address drecly he ssue of mulple equlbra. eep n mnd ha we are modelng he economy as a dynamcal sysem n whch nfne random draws are assocaed wh defnve sequences of realzed economc allocaons. Ths means ha f a any dae he economy can have mulple mare clearng oucomes, hen as par of he dynamcs posulaed here s a procedure for selecng a parcular one of hem whch, n urn, generaes he daa observed n he economy. Ths, ndrecly, addresses also he ssue of sunspo equlbra. Such equlbra requre a devce for alernang random selecons from among mulple equlbra of some underlyng economy over me. If such an equlbrum s o be realzed hen hs selecon mus be par of he descrpon of he dynamcal sysem. Moreover, a formal coordnaon among agens s feasble only f one of he observable exogenous varables provdes he needed sgnal for jon acon and hen we mus nerpre he flucuaons of he economy whch are due o he commonly observed 2

13 sunspo varable as exogenously caused. In an RBE where an exogenous sunspo sgnal no presen, s possble ha he agens form belefs whch wll vary over me and would, n a sponaneous way, be perfecly coordnaed. In hs narrow sense, a sunspo equlbrum can be realzed as an RBE bu s a mos unlely equlbrum. Gven he defnon of an RBE he res of he paper s organzed as follows. Secon III wors ou a smple example whch demonsraes how he prce sae space of an RBE s consruced and why he raonaly condons allow us o wor wh a fne sae space. Our use of he raonaly condons n he consrucon of he prce sae space may be conrased wh he reamen n Svensson [98], Henroe [996] and urz [993] who do no use any raonaly condons and end up needng o wor wh a space of he order of he connuum. The secon also provdes a defnon of he mporan concep of Endogenous Uncerany. Secon IV provdes a proof of he exsence of an RBE wh PCC for he famly of SIDS processes developed by Nelsen [994]. The proof demonsraes how he PCC "complee" he mare for radng endogenous uncerany for every generaon once he endowmen of he young s nown. Secon V dscusses he Pareo opmaly properes of he equlbrum and he role of he PCC n he opmaly properes. III. Endogenous Uncerany and Raonal Belefs The sudy of RBE n full generaly enals complex echncal dffcules bu he properes of he sae space and endogenous uncerany become very clear even n smple models. Consequenly, hs secon wll be devoed o explore a specal case where four smplfcaons are made: () R = R > 0 for all, () = 2, 2 H H 2H L L 2L () (T, T ) can ae only wo values T = (T, T ) and T = (T, T ), (v) u (x,x 2 % (p % )) ' u (x ) % u 2 (x 2 % (p % )), (v) In order o avod ssues of prce normalzaon we use he erm "prce" n hs secon o 3

14 c mean he par ( p, p ). However, young agens form belefs only abou he relave prce c c p /p and hus he erm "prce" n hs secon means "relave prce p /p ". (III.a) Raonal Belefs and Prce Saes 2 We hn of {T = (T, T ), =, 2,...} as a sochasc process defned on an exogenous sae space S whch s he sae space for exogenous varables. In he presen case, hs space s T very smple (7) S = {H, L}. T The rue process s assumed o be non-saonary and s consruced n he followng way. Selec a paron {D, V} of he posve negers such ha on any nfne se of daes {, +,...} he fracon of members of D n he se s, say, B. A smple mechansm o selec he ses D and V D s o oss, a each dae, a con wh probably of H beng B. If H s realzed you declare he D dae +, D, f no +, V. We selec he value of B = 2/5 and hs s acheved by an..d. D con-ossng process wh a probably of beng 2/5. Thus we have B D = 2/5, B V = 3/5. 2 Fnally, we selec {(T, T ), =, 2,...} o be an ndependen sequence of random varables wh wo denses and such ha g T g 2 T (8) P(T,T2 ) ' g 2 T (T, T2 ) f %, V. g T (T, T2 ) f %, D For example, we selec g T (T, T2 ) ' (TH, T 2H ) ' 3 4 g 2 T (T, T2 ) ' (TH, T 2H ) ' 3. Denoe by A T he probably of he sochasc process {T, =, 2,...} gven {D, V}. Ths non- saonary process s an example of an SIDS process suded by Nelsen [994],[996]. I follows from Nelsen [996] ha s sable and has a saonary measure m defned by he..d. process T 4

15 3 2 5 % 3 5 ' 8%2 ' H 2H where he probably of (T, T ) s ½. Ths s so snce We sress s non-saonary characer n comparson wh (S 4 T, Û (S4 T ), m T ) by nong ha one mus nerpre he rue process as a composon of a selecon rule of daes n D or V ogeher wh he probables whch apply n D or n V. Thus, f you were o be a on he oucome a + hen he nowledge of A wll lead you o be dfferenly a dfferen daes. Ths T s no so under m T where he process {T, =, 2,...} s vewed as an..d. process and hence saonary. Neher one of he wo agens now he rue sochasc process of he endowmens. Moreover, despe he smplcy of he process n he model, ams o represen he process of exogenous varably due o echnology, nvenons, clmae changes, dscovery of naural resources, ec. Mos of hese are neher observable nor fully undersood and he model smplfcaon of all hese down o a bnary process wh wo dfferen probables s purely a maer of llusraon. The naural urge s o explo he model's smplcy o exrac valuable nformaon. From our perspecve hs s no consrucve snce we need o hn of he process {T, =, 2,...} as represenng he complexy of he exogenous envronmen of modern socey. We hus propose o hn of he aggregae endowmen T as unobservable. Agens observe her own endowmens and mare prces and need o form belefs abou fuure prces. We assume ha agen ype beleves ha hs envronmen s saonary. The saonary measure represenng hs belef wll be derved below. Agen ype 2 beleves ha hs economc envronmen consss of wo dfferen regmes of prce dsrbuons. Such a belef srucure means ha here are wo dfferen saes of belef (denoed 2 by and 2) a whch he probables adoped by agen ype 2 are f and f. Combned wh he wo exogenous saes, here are four possble prces whch may be realzed. Insead of hnng abou he values of prces we focus on he space on whch prces are laer defned by an equlbrum process. We hus see ha we have here four "prce saes": wo are nduced by exogenous facors 2 and wo are nduced endogenously by he belef of agen 2. Ths means ha f and f are defned 2 on a sae space of dmenson four. Denoe he wo probables (f, f ) by 5

16 a b (9) a 2 a 3 f ', f2 '. b 3 a 4 b 4 Noe ha a any momen of me agen 2 beleves ha any of he four prces may be realzed b 2 excep ha he does no beleve ha he dsrbuon s saonary. To formulae hs belef on nfne sequences of prces he selecs frs a rule of paronng he negers no {D, V } n a manner smlar o he rule of selecng {D, V} so ha on any nfne se {, +, + 2,...} he proporons are B D ' 3, B V ' 2 3. He hen selecs a dae he probables represenng hs belef abou prces a dae + accordng o he followng rule: (0) a selec f 'f f %,D a selec f 'f 2 f %,V. We denoe by Q he Raonal Belef specfed above on he space sequences. (S 4 p, Û (S4 p )) of nfne prce In summary, he four prce saes of he model may be hough of n he followng manner H Prce sae : (T = T, f = f ) () H 2 Prce sae 2: (T = T, f = f ) L Prce sae 3: (T = T, f = f ) L 2 Prce sae 4: (T = T, f = f ). We sress he fac ha one may only hn abou he saes n hs manner for wo reasons. Frs, H neher T nor f are observable and from he pon of vew of he agens here are smply four prces ha may be observed. Second, he nerpreaon of he "prce sae" depends upon he srucure of nformaon gven o he agens. For hs reason we defne he Prce Sae Space by (N) S = {, 2, 3, 4} p and he sochasc process {p, =, 2,...} of equlbrum prces s an nfne sequence of random 6

17 varables on S. p The assumpon of non-srucural nowledge by he agens s cenral o our wor. I means ha agens now ha n equlbrum here are only four possble prces whou he nowledge of he srucure whch nduces hese saes. We use hs smple srucure because s analycally convenen. However, agens are requred o dsregard her ndvdual mpac on he srucure. More specfcally, agen ype 2, whose belef nduces some of he varably of prces, s assumed o be "compeve" or "small" and s herefore specfcally prohbed from nowng how hs own belef srucure conrbues o he naure of uncerany of fuure prces. In a model where he n endowmen T may ae any random value n a subse of ß, where he dvdend process R may + ae any value n a compac nerval and where he number of agens s very large, he assumpon ha each agen s "compeve" s hen naurally made and would hardly be quesoned. We now urn o he specfcaon of he resrcons of raonaly on he belefs of a ype 2 B D ' 2 agen. Ths agen selecs f = f wh frequency when +, D and f = f wh 3 B V ' 2 frequency when +, V. Ths selecon s done ndependenly of he realzaon n 3 H L 3 {T, T } whch are seleced wh probables of when, D and when, V. Ths 4 3 ndependence nduces wo ses of condons on he saonary vecor µ = (µ, µ 2, µ 3, µ 4 ) of prce probables. From he pon of vew of ype 2 agens, we mus have B D f % B V f 2 ' µ and consequenly a a 2 % a 3 a b b 2 b 3 b 4 ' µ µ 2 µ 3 µ 4. On he oher hand, consder each of he µ. Gven he ndependen selecon of D and D follows ha on he evens DD = {, D, +, D }, DV = {, D, +, V }, VD = {, V, +, D } and VV = {, V, +, V } we have he frequences B DD ' 2 5, B DV ' 4 5, B VD ' 3 5, B VV '

18 H To calculae he µ consder, for example, µ. I s generaed when T = T and f = f. The even f = f s realzed only on daes n DD and VD. Gven ha, D and +, D he probably H of T = T s and when (, V, +, D ) hs probably s Ths leads o µ = and 6 he calculaons of he res of hem follows: µ ' 3 4 B DD % 3 B VD ' 6 µ 2 ' 3 4 B DV % 3 B VV ' 3 µ 3 ' 4 B DD % 2 3 B VD ' 6 µ 4 ' 4 B DV % 2 3 B VV ' 3. 2 These calculaons mply ha he raonaly resrcons on he belefs (f, f ) of agen ype 2 are (2a) (2b) (2c) (2d) 3 a % 2 3 b ' 6 3 a 2 % 2 3 b 2 ' 3 3 a 3 % 2 3 b 3 ' 6 3 a 4 % 2 3 b 4 ' 3. Added o (2a) - (2d) are he naural resrcons on probables whch are (2e) a + a 2 + a 3 + a 4 = (2f) b + b 2 + b 3 + b 4 = We hus have 5 ndependen equaons wh 8 unnowns mplyng 3 degrees of freedom whch ndcae he sze or he dmenson of ndeermnacy leadng o mulple RBE. Alhough we assumed ha agens canno observe he aggregae endowmen relaxng hs assumpon would have made no dfference o he calculaons of he saonary measure. The reason s ha an nspecon of () reveals ha when +, D he agen beleves (a ) ha he even {p H or p 2 } occurs wh probably (a + a 2 ) and hs s also he probably of he even {T = T }. When +, V he beleves (a ) ha he even {p or p 2 } occurs wh probably (b + b 2). Snce B D ' 3 and B V ' 2 3 hs nowledge requres ha 8

19 (3) 3 (a % a 2 ) % 2 3 (b % b 2 ) ' 2 The on he rgh hand sde of (3) comes abou from he saonary measure m T of he 2 endowmen process. However, (3) adds no resrcon snce s mpled by (2a) - (2f). An example of a Raonal Belef of agen ype 2 s herefore f ' , f 2 ' where B D f % B V f 2 ' µ. Now denoe by he space of nfne sequences s of members of S p ; by Û(S p ) he Borel F-feld of subses of S p and by (S p, Û(S p), A) he equlbrum probably space. A s he rue probably of prce sae sequences S 4 p s p 0 S 4 p nduced by A n T 4 4 (8) and by he selecon rules specfed n (0). The saonary measure on (S, Û(S ) ) s denoed by m and s defned by he..d. process wh densy µ a each dae. We presen n he able below an example of he hree probably measures of our example. A s he rue probably; Q 2 = m s he belef of agen ype and Q = Q s he belef of agen ype 2. p p p Probables of p a + (, + ), D D D V V D V V A Q=m Q A Q=m Q A Q=m Q A Q= Q m Prce Sae 3/4 /6 /4 0 /6 /8 /3 /6 /4 0 /6 /8 Prce Sae 2 0 /3 /4 3/4 /3 3/8 0 /3 /4 /3 /3 3/8 Prce Sae 3 /4 /6 /4 0 /6 /8 2/3 /6 /4 0 /6 /8 Prce Sae 4 0 /3 /4 /4 /3 3/8 0 /3 /4 2/3 /3 3/8 9

20 We now urn o he defnon of he crucal concep of "Endogenous Uncerany." As s clear from he descrpon of he sae spaces S T and S p we have ha (4) S = {, 2, 3, 4} and S = {H, L}. p T However, hnng of H and L as subses of he prce sae space follows from () ha H = {, 2} and L = {3, 4}. I s ypcal ha he collecon of all ses n he exogenous sae space s a non-rval paron of he prce sae space. The erm "non-rval" means ha a leas one member of S conans more han T one member of S. p Now suppose ha agens had full srucural nowledge. They would hen now ha a daes n D he densy of T s g T bu T ' T L or T ' T H. In V he densy s g 2 T bu T ' T L or T ' T H. Hence he four prce saes under full srucural nowledge are Prce Sae : Prce Sae 2: Prce Sae 3: Prce Sae 4: (T ' T H,g T ' g T ) (T ' T H,g T ' g 2 T ) (T ' T L,g T ' g T ) (T ' T L,g T ' g 2 T ) In conras, prce varaons n an RBE occur only as a resul of varaons n he realzed value of exogenous varables and varaons n he "sae of belefs" of he agens. Wh hs n mnd we defne s(t, f) o be he prce saes n an RBE where T s he exogenous sae and f s he vecor of one perod probably belefs of he agens. However, s mporan o see ha he mere fac ha prces may ae more values han s warraned by he number of dfferen values aen by he exogenous varables s only a necessary condon for he presence of Endogenous Uncerany. Ths s so snce n he example here may be rue ha p = p and p = p. In hs case he varaons n he belefs of agens ype 2 have no mpac on acual varably of equlbrum prces. 20

21 In hs case he four prce saes are reduced o wo and here s no endogenous uncerany: all he sochasc varably of prces s enrely caused by sochasc varaons n he exogenous varables. To formally defne he concep of Endogenous Uncerany we need o ae no accoun he varably of g, he rue probables of he exogenous varables. Noe frs ha hese probables T are nown n a raonal expecaons equlbrum bu no n an RBE. As a resul, prces n a raonal expecaons equlbrum vary when g vares bu hs s no he case n an RBE. To enable a T dsncon beween hese wo equlbrum conceps we defne he par (g, s) as he exended sae T of he economy. Alhough he frs componen does no nfluence prces n an RBE he consrucon s useful n clarfyng he followng concep: Defnon 2: Endogenous Uncerany s sad o be presen n an RBE f (a) (b) he exogenous sae space s a non-rval paron of he prce sae space; 2 2 here exs wo exended saes (g, s ) and (g, s ) sasfyng s = (T, f ) and s = (T, f ), g = g and f ú f such ha p(s ) ú p(s ). T T T T Endogenous Uncerany hus arses when he belefs of agens nfluence equlbrum prces when he rue probably of he exogenous varables remans he same. Ths allows for he possbly ha an RBE s a raonal expecaons equlbrum and he defnon spulaes ha Endogenous Uncerany canno be presen n a raonal expecaons equlbrum. Tha s, suppose ha a wo dfferen daes he exogenous varables have he same realzed T bu dfferen rue probables of fuure values of he exogenous varables. Under raonal expecaons agens now ha he rue probables are dfferen and hence prces wll be dfferen bu hs varably, accordng o our defnon, does no consue endogenous uncerany. (III.b) When he Uly Funcon s Logarhmc Endogenous Uncerany s No Presen The frs order condons requre ha he decson funcons (x,x 2 %,2,z ) depend upon 2

22 all varables (p, p, $( ), T ). In wrng x we suppress hs dependence bu n wrng x we explcly recognze he dependence of x + on he realzed prce sae j a +. Supposng ha p ( (j)' p (j) u (x, y) = log x + log y, le, we have p c (j) (j) (5a) x ' 8 pc (5b) 4 j j ' f x 2 % (j) (p ( % (j))( p % (j) % pc % (j)r ) ' 8 p c % (j) p (5c) f x 2 % (j) (p ( % (j))(p % (j)% pc % (j)r) ' 8 $ (p ( % (j))pc % (j) j',2,3,4. Equaons (5a) - (5b) mply ha (6) x ' 2 T x 2 ' 2 T % R p ' 2 T. I hen follows from (6) ha no Endogenous Uncerany can be presen n hs RBE snce he demand of he young s ndependen of her prce expecaons. Equaons (5a) - (5c) also mply ha he consumpon of he old n prce sae j sasfes (7) p c % (j)x2 % (p % (j)) ' 2 T f (p ( % (j)) (p (j) % $ (p ( % (j)) % pc % (j)r ). Equaon (7) shows ha alhough here s no Endogenous Uncerany n he model, he belefs of he agens nfluence he allocaon. Addng (7) over reveals he equlbrum funcon $, (8) 2 $ (p ( % (j)) ' j ' T ( p % (j) % p c 2R% T p c % (j) % (j)r )f (p ( % (j)). I s seen n (9) ha he prce of a PCC s exacly wha one can call he mare belef: s he weghed average of he probables of he agens when her relave endowmens provde he weghs. We can hen conclude as follows: 22

23 Proposon : When u (x, y) = log x + log y, all, here s no Endogenous Uncerany n any RBE. (III.c) Endogenous Uncerany s Generally Presen n a RBE The logarhmc uly funcon s borderlne case when fuure perceved nvesmen opporunes have no effec on curren consumpon and consequenly prce expecaons have no effec on curren consumpon. For any oher uly funcon he dversy of belefs n an RBE gves rse o Endogenous Uncerany. To llusrae, consder an alernave case where for all (9) u (x, y) ' Ax & 2 x 2 % By & 2 y 2 (x # A, y # B). The frs order condons of he ndvdual opmzaon are (20a) A & x ' 8 pc (20b) 4 j (B & x 2 % (j)f (p ( % (j)) ( p % (j)% pc % (j)r ) ' 8 j ' p c % (j) p (20c) (B & x 2 % (j))f (p ( % (j))(p % (j) % pc % (j)r)' 8 $ (p ( % (j))pc % (j). In hs case he expecaons of he fuure are of cenral mporance. Noe ha due o he properes of SIDS processes, condonal and uncondonal expecaons are he same. Hence, condon (20c) ogeher wh he no-arbrage condon (3) mply ha f Q, =, 2 are he belefs of he wo ypes of agens hen 2A & x 2 ' j E Q (B & x 2 % )(p % % pc % R p c (2) ). ' p c % p c c Snce by (5a) p x = p T - p follows ha (22) p p c ' T & 2A % pc p 2 j E Q ' (B & x 2 % )(p % % pc % R ). p c % To see how endogenous uncerany s refleced n (22) noe ha p s dfferen when 23

24 H L T = T or T = T. However, s also dfferen dependng upon wheher +, D or +, V snce he prce expecaons of agen ype 2 are dfferen n hese wo cases. The prce of a PCC sll reflecs he "mare belef" bu here he weghs are dfferen. Ths follows from he fac ha (20a) and (20c) mply ha (23) 2 $ (p ( % (j)) ' j ' B& x 2 % (j) A & x f (p ( % (j))( p % (j) % pc % (j)r p c % (j) ). Ths example shows ha resource allocaons of an agen wh non - logarhmc uly funcon s sensve o hs belefs and n equlbrum wll, generally, ranslae no endogenous uncerany. IV. Raonal Belef Equlbrum wh PCC and Endogenous Uncerany Ths secon demonsraes he exsence of an RBE for he economy of Secons II and III. An exsence heorem requres no only a proof of he exsence of mare clearng prces bu also a demonsraon ha agens hold raonal belefs and ha equlbrum quanes and prces consue a sable dynamcal sysem. Ancpang he sably requremen we carry ou he analyss n wo sages. In he frs sage we consruc he prce sae space by selecng he probably of he endowmen process and he probably belefs of he agens o be jonly SIDS measures snce follows from Nelsen [996] ha hs nduces sable equlbrum sysem. Such an SIDS sysem generalzes he example of Secon III. In he second sage we prove he exsence of equlbrum 5 prces and quanes for he specfed sysems. (IV.a) The Srucure of Uncerany and Belefs We reurn o he model of Secon II. The process {T ' (T,..., T ), ',2,...} s a 5 Ths approach was proposed by Nelsen [994], [996] who proves exsence n economes where PCC are no raded. 24

25 2 sochasc process on he exogenous sae space S T where (T, T,..., T ) s he endowmen vecor of he young agens. Ths defnes a dynamcal sysem (S 4 T, Û(S 4 T ), A T,T). The assumpon whch we mae here and whch wll hen lead o he posulaed SIDS sysem s: Assumpon : The process {T,', 2,...}, T, ß, aes only a fne number of values n he se F T ' {T, T 2,..., T N 0 } of N elemens. Moreover, A s an SIDS probably measure under 0 T whch {T,', 2,...} s an ndependen sequence of random varables. Assumpon mples ha we can defne he exogenous sae space S by he coordnaes. Hence T (24) S T ' {,2,...,N 0 }. We nex nroduce he belefs of he agens whch are probables Q, =, 2,..., on a measurable space (S 4 p, Û(S 4 p )) To do ha we nroduce where S - he prce sae space - s a Borel subse ye o be defned. p c Assumpon 2: The belefs Q, =, 2,..., specfy he process {(p, p ), =, 2,...} o be a sequence of ndependen random varables wh dae probably on (S p, Û(S p )) denoed by f. Ths probably s seleced from a fne se of such probables members. F ' {f,f 2,...,f N } wh N The mporan mplcaon of Assumpons and 2 s ha we can derve from condons (4a) - (4b) an equlbrum map whch aes he form (25) (p c,p, $ (@)) ' (T M(,f,f 2,...,f ). Inspecon of (25) reveals ha under Assumpons and 2 he maxmal number of prces ha can be observed s M ' N 0 N N 2... N. Ths leads o he concluson ha he prce sae space, whch s he doman of he prce process, can be defned by (26) S = {, 2,..., M}. p Gven he fne sae spaces of he endowmen process and he belefs of he agens we wan o 25

26 specfy he selecon process { (T,f,f 2,...,f ), =, 2,...} o be jonly sable. Formally, noe ha n vew of he equlbrum map (25) follows ha we can hn of he exogenous sae space S T as a paron of he prce sae space S p as explaned n (4) and hs mples ha we can hn of he probables A T and Q as measures on he same space. Wh hs n mnd we specfy (see Nelsen[996] Proposon 6): 2 Assumpon 3: The probably measures (A T, Q, Q,...,Q ) are jonly SIDS. We remar ha Assumpons and 3 are essenally assumpons abou he sochasc process {(T,f,f 2,...,f ), =, 2,...} whch s called a "generang process" and s he drvng mechansm of an SIDS process (see Nelsen [996] Secons 4-5 ). The jon SIDS propery of he endowmen and he belefs of he agens s cenral o our exsence argumen snce a proof of he exsence of an RBE requres a demonsraon ha all mare clearng varables consue a sable process and he belefs of he agens are raonal wh respec o he probably of he equlbrum process. An SIDS sysem s "self referenal" or "closed" n he precse sense ha f Assumpons, 2, 3 are sasfed hen equlbrum prces are also SIDS and for each exogenous process here exs SIDS belefs whch are raonal wh respec o he resulng equlbrum. Moreover, for each dynamcal sysem (S 4 T, Û(S 4 T ), A T,T) he equlbrum 2 dynamcs of prces and he raonaly condons on he belefs (Q, Q,..., Q ) are all specfed n erms of he dynamcal sysem (S 4 p, Û(S 4 p ), A,T) of prce saes whou specfyng he numercal values of equlbrum prces. Also, he saonary measure of he equlbrum sochasc process of prces s ndependen of he parcular sequence (T,f,f 2,...,f ) whch s realzed; depends only on he generang process self. Ths self-referenal propery, whch s a subsue for a fxed pon argumen, has been exended by urz and Schneder [996] o subclasses of Marov processes. To complee he developmen we need o ensure ha we have a conssen prce sae space n he sense ha for each endowmen process here s a map (25), an SIDS equlbrum prce process and SIDS belefs Q for all such ha he belefs are raonal wh respec o he equlbrum 26

27 dynamcs of prces and he belefs nduce he equlbrum SIDS of prces. We hen have he resul: Lemma :(Nelsen [996]) For any endowmen dynamcs (S 4 T, Û(S4 T ),A T,T) sasfyng Assumpon here exss a class of generang processes { (T,f,...,f ), =, 2,...}such ha under he map (25) a conssen prce sae space s nduced n he sense ha Assumpons 2 and 3 are sasfed and. he mpled equlbrum dynamcs of prces (S 4 p, Û(S 4 p ), A,T) s a non-saonary SIDS, 2. he mpled SIDS belefs (Q,..., Q ) of he agens are raonal wh respec o A. Lemma complees he frs sage of he exsence argumen. Wha s lef o show s he exsence of mare clearng prces whch are compable wh he sochasc srucure posulaed. Before movng on o hs problem we need o clarfy he ndexng of he M prce saes. A member of hs collecon s denfed by a selecon (27) (T 0,f,f 2 2,..., f ), 0{,2,...,N } ' 0,, 2,...,. These members of he sae space are denfed by he permuaons (,,,..., ) of selecons 0 2 from he ses {,2,...,N } ' 0,,2,...,. I s convenen o order hese permuaons and map hem one-for-one no he se of he M negers {, 2,..., M}. We hus specfy by (28) ( 0 (), (), 2 (),..., ()) ', 2,..., M a rule o map each permuaon no an neger n {,..., M}. We hen replace (27) wh (29) (T 0 (),f (),f2 2 (),...,f ()) for ',2,...,M. The map (29) esablshes he correspondence beween each prce sae and he confguraon of endowmen and belefs whch defnes ha sae. c I s evden ha each prce vecor (p,p, $ ) n sae has M + 2 coordnaes. Alernavely, c c 2 we can hn of he prce vecor a as conssng of a par ( (p, p ), $ ) where (p, p ), ß and + $ (p c ( ) s a funcon specfyng, for each %,p % ), he cos a of a clam on one share a + conngen on he specfed prces realzed a +. Snce here are M possble values whch 27

28 (p c %,p % ) can ae, $ can ae M dfferen values. Bu snce here are M dfferen possble prces (and prce saes) here mus also be M dfferen funcons ($, $,..., $ ) ha may be 2 M realzed as an equlbrum schedule a any dae. Snce n he model a hand a young person s unceran, a each dae, only abou he prces probably belef vecor s M. f j (p c %,p % ) a dae +, he dmenson of each (IV.b) Exsence of an RBE wh a complee se of PCC (IV.b.) Demand Correspondences and he Inerpreaon of Prce Saes Alhough he equlbrum values of each one of he prce vecors ( p c p c, p, $ j ) has no been esablshed as ye, he mplcaon of Lemma s ha we mus hn of a "prce sae" as exacly such a vecor. In oher words, n an RBE an agen does no hn of some absrac and unobservable "sae" and hen consders equlbrum prces o be measurable funcons on hs sae space. Insead, he hns of vecors of prces as saes over whch he places hs probably belefs. Equally mporan, he PCC conracs used o rade uncerany are no conngen upon an exogenously specfed saes bu raher, on specfc and observable prce vecors (, p ) ha may be realzed nex perod. In he presen paper, where we assume a complee se of PCC a each dae, he wo-sage procedure of our proof has he mplcaon ha sandard echnques of Arrow-Debreu heory can be used o prove he exsence of an RBE despe he new fnancal srucure whch we posulae. We reurn o he problem () - (2) of agen assumng R = R n order o resae n erms of he prce saes (p c,p ), ',,2,...,M and he assocaed prces of he PCC. Sarng wh hs prce sysem, s now an M M marx [$ ] and he arbrage free condon (3) s now j wren n he followng form (30) p ' j M j ' $ j ',2,...M. The budge equaons (2a) - (2b) for =,..., M are now wren n he form (3a) (3b) p c x % p 2 M % ' j ' $ j z j ' p c T, p c j x 2 j ' (2 % z j )(p j % p c j R), j ',..., M. 28

29 We observe now ha he assumed complee PCC srucure perms hedgng, a dae, of all perod + rss whch are compable wh a sngle budge consran. Ths maes feasble o selec rsless consumpon sreams. To show ha all pars of perods and + consumpon are feasble f hey sasfy a sngle ncome consran when young, subsue (30) no (3a) and se p c j $^ j ' $ j ( p j % p c j R ) o oban one neremporal budge consran for he wo vecors x x 2 ' (x 2 2 of consumpon and,..., x for each =,..., M (32) p c x p c M ) M % ' $^ j x 2 j j ' ' p c T. Hence we have n (32) a sngle budge consran relave o whch s feasble o choose a each sae a rsless (.e. consan ) consumpon sream. The budge equaons (3a)-(3b) do no apply o he old members a dae who rade he endowmen 2 > 0, allocaed nally o her "dynasy", n a compeve mare. If he economy s n sae j hen he budge equaon of such an old agen s (38) j y j ' 2 (p j % p c j R). In (33) we use he symbol y j o denfy he consumpon of he old a dae and sae j. Equaon (33) reveals ha alhough he young a any dae can hn of (32) as her effecve consran hs s no he case for he old a dae. Ths s a drec consequence of he OLG srucure of he model whch we need o eep n mnd n esablshng he exsence of an RBE. We now nroduce he followng assumpon. Assumpon 4: For each, he uly funcon u s connuous, quasconcave and srcly monoonc. The uly funcon n () can hen be wren for a gven belef f ' (f j ) as (N) U (x,x 2 j M ) ' ' j ' u (x,x 2 j )f j, ',..., M. 29

30 I s clear from (3a)-(3b) ha f (x *, 2 *, z *) s an opmal allocaon of a young agen M n sae where we use he noaon z = (z,...,z M ), ß, hen any oher porfolo (2, z ) whch sasfy he condon 2 * + z j * = 2 + z j s also an opmal. Ths ndeermnacy s ypcal for fnancal models and we handle by smply eepng he soc ownershp fxed a he nal level (34) 2 ( ' 2 0 >0 for ', 2,..., M, ', 2,...,. Condon (34) means ha only he vecors z are needed o be chosen and hs deermnes he M (M+) porfolo. We use he noaon q = (q,...,q ), ß where q ' (p c, $,..., $ M ), ßM %, and M gven he arbrage free condon (30) hese vecors specfy he prce sysem of he economy. We also use he noaon $ = [$ ] for he marx of PCC prces and $ = ($,...,$ ) for he PCC prce j M vecors n sae. The arbrage free condon s hen p ' 6 for all where 6 ' (,...,). The vecors (x,z ), ß M % denoes he vecor of choces of young raders n sae whle y, ß M denoes he consumpon vecor of dae old n sae. The arbrage free condon and he homogeney propery n (3a)-(3b) gve us one degree of freedom o normalze each one of he prce vecor q for =,..., M. We hen employ he sandard smplex used n Arrow-Debreu ype proofs M (35) ) ' q, ß M % * p c % % ' $ j ' and hence q, = = ) )... ) (M mes). j ' The budge correspondence of young raders n sae s hen wren, for =, 2,..., as (35a) (q,t ) ' (x B,z ), ß % ßM * p c x M % ' $ j (2 % z j ) # p c T ; x2 j $0 n (36b) q,=. j ' I s mporan o noe ha he budge correspondence (35a) depends upon he enre se of prces 30

31 q snce n (3b) he agen needs o ensure non-negave consumpon n all saes. Tha s, a each prce sae he agen plans hs old age consumpon a all possble prce saes j and hence depends upon he enre prce vecor p = (p, p,..., p ). By he arbrage free condons (30) he 2 M vecor p s a lnear funcon of all he elemens n $. The budge correspondence of old raders =, 2,..., a dae n sae s hen (35b) (q, 2 ) ' y, ß % p c y # 2 (p % p c R), p ' 6,q,). B y The followng s hen sandard: Lemma 2: The budge se correspondences of he young and he old are non-empy and for each q hey are convex and compac valued, and connuous on he neror of =. In he followng proof we encouner he usual problem where demand correspondences are no defned on he boundary when some prces equal o 0. We denoe by Ú he real economy wh n whch we wor. We now nroduce a sequence of economes Ú where for each n he economy s bounded n a cube nw. The se W s a compac cube cenered on he zero vecor and all he orgnal budge ses are hen nerseced wh nw o creae new budge ses whch are hen compac subses of nw even when some prces equal 0. These budge correspondences are non-empy, convex and compac valued, and connuous a all prce vecors n =. A consrucon of he n economes Ú requres complex addonal noaon. Snce hs s a sandard procedure we shall avod such added noaon (for deals on hs procedure see urz [974b, secons 6-7]). Thus, when n we say below ha "a varable aes he value +n n Ú " we mean ha s on he boundary of he n resrced budge se of he agen n Ú. Turnng o demand correspondences, for =,..., and =,...,M he noaon used for he young s x,n 0, z j,n j, j',..., M, n ' (n 0, n,..., n M ), n (q)' (n,..., n M ). For old agens we use he noaon y,n y (q, 2 ), n y ' (n 2,..., n2 M ). Now, defne he demand correspondences 3

32 (36a) n (q, T ) ' (x,z ), ß % ß M * (x, z ) maxmzes ( ) )on B (q, T ) q, n =. (36b) n y (q, 2 ) ' y, ß % y s maxmal on B y (q,2 ) q, n). I hen follows from he heorem of he maxmum and from Lemma 2 ha y Lemma 3: The demand correspondences (n (q), n (q)) for =, 2,..., are non empy, convex and compac valued, and upper hemconnuous on n=. In each of he unformly bounded n y economes Ú, he vecor of demand correspondences (n (q), n (q)) s non-empy, convex and compac valued, and upper hemconnuous on he enre prce space =. (IV.b.2) Exsence Proof In he OLG economy a hand he mare clearng condons (5a)-(5b) spulae ha no maer wha he sae a dae - s, n an RBE he aggregae consumpon of he young and he aggregae consumpon of he old a dae has o add o he oal supply. Tha s, (37) % ' x 2 ' T % R for, j =,..., M. ' x ' ' Lemma 4: For all saes and j, n equlbrum j ' x 2 j ' ' ' y / y. ' 2 Proof: A dae he requremen of maeral balance specfes ha x + y = T + R. However, he demand of he young a any dae depends only upon he sae a ha dae and hence hs condon holds for all daes. Comparng wh (37) we can conclude ha n equlbrum a all daes. ' x 2 j ' ' y þ holds By Lemma 4 we rewre (37) o requre (38) % ' y ' T % R, for all =, 2,...,M. ' x ' ' Nex, he fnancal mares mus clear and snce all PCC are n zero ne supply we requre ha 32