DYNAMIC SPANNING IN THE CONSUMPTION-BASED CAPITAL ASSET PRICING MODEL

Transcription

1 DYNAMIC SPANNING IN THE CONSUMPTION-BASED CAPITAL ASSET PRICING MODEL PETER OVE CHRISTENSEN, SVEND ERIK GRAVERSEN, AND KRISTIAN R. MILTERSEN Absrac. Under he assumpions of he Consumpion-based Capial Asse Pricing Model (CCAPM), Pareo opimal consumpion allocaions are characerized by each agen s consumpion process being adaped o he filraion generaed by he aggregae consumpion process of he economy. The wealh processes of he agens, however, are adaped o he finer filraion generaed by aggregae consumpion and he condiional disribuion of fuure aggregae consumpion. Therefore, in order o achieve Pareo opimal consumpion allocaions, a sufficienly varied se of asses mus exis such ha any wealh process adaped o his finer filraion can be implemened by dynamically rading in ha se of asses. We provide sufficien condiions for he exisence of such a se of asses based on dynamically rading coningen claims on aggregae consumpion. In addiion, we give sufficien condiions for he exisence of equilibria in a dynamically effecively complee marke in which agens are only able o rade in coningen claims on aggregae consumpion, he marke porfolio of firms, and a (numeraire) zero-coupon bond. We demonsrae he role of shor- and long-erm coningen claims on aggregae consumpion for he implemenaion of Pareo opimal allocaions in he presence of shor- and long-erm risks. In addiion, in he presence of personal risks, we demonsrae he role of insurance conracs. 1. Inroducion The prime equilibrium model of asse pricing is he Consumpion-based Capial Asse Pricing Model (CCAPM) firs formulaed in coninuous ime by Breeden (1979). Duffie and Zame (1989) provide condiions on he primiives of a coninuous-ime economy under which equilibria obeying he CCAPM exis. The basic assumpions are homogeneous beliefs, ime-addiive preferences, and dynamically complee markes. The assumpion of dynamically complee markes ensures ha consumpion allocaions are Pareo opimal. 1 A basic characerisic of such allocaions is ha any agen s equilibrium consumpion a any dae is measurable wih respec o he aggregae consumpion a ha dae. Based on his observaion, we define a concep called dynamically effecively complee markes. I has he propery ha any consumpion plan, which is measurable wih respec o aggregae consumpion, can be implemened. This is accomplished by ensuring ha porfolio sraegies exis which span informaion ses disinguished by eiher differences in curren aggregae consumpion or differences in condiional probabiliy disribuions for fuure aggregae consumpion. 2 Given sandard assumpions, we show ha his ype of marke is sufficien o ensure he exisence of an equilibrium wih Pareo opimal consumpion allocaions, despie Dae: Augus This version: April 28, Mahemaics Subjec Classificaion. G13. Key words and phrases. Consumpion-based capial asse pricing model, CCAPM, dynamic rading. The paper was presened a Aarhus Universiy, Århus, Denmark, a he Conference on Mahemaical Finance I, Aarhus Universiy, Århus, Denmark, a Universiy of Briish Columbia, Vancouver, Canada, and a he 13 h AFFI Conference in Geneva, Swizerland. We are graeful o Jesper Andreasen, Ben Jesper Chrisensen, Anders Damgaard, Darrell Duffie, David Heah, and oher seminar paricipans for valuable commens. All hree auhors graefully acknowledge financial suppor of he Danish Naural and Social Science Research Councils. Besides, he hird auhor graefully acknowledges financial suppor of Danske Bank. Documen ypese in LATEX. 1 There is also a version of he CCAPM for incomplee markes, bu he issue of exisence of equilibria is unresolved in his seing (cf. Duffie (1996)). 2 This presumes ha he individual agens consumpion endowmens are measurable wih respec o aggregae consumpion. We consider he case of general consumpion endowmens in Secion 7. 1

2 DYNAMIC SPANNING IN THE CONSUMPTION-BASED CAPITAL ASSET PRICING MODEL 2 he fac ha agens may no be able o implemen any financially feasible consumpion plan as hey are in a dynamically complee marke. In addiion, we idenify relaionships beween he ypes of securiies ha consiue a dynamically effecively complee marke and he characerisics of he process for he condiional probabiliy disribuions for fuure aggregae consumpion and he agens consumpion endowmen processes. In paricular, we invesigae he role of shor- and long-erm coningen conracs on aggregae consumpion as well as insurance conracs ha faciliae an efficien sharing of personal risks. The CCAPM is as he CAPM and he APT based on he gains of diversificaion. Tha is, he impac of evens ha influence he reurns on individual asses bu no he level of aggregae consumpion is opimally eliminaed by agens hrough diversificaion. A dynamically complee marke requires securiies ha enables agens o implemen any sae-coningen consumpion plan, boh in erms of diversifiable risk and non-diversifiable risk. Recognizing ha agens eliminae diversifiable risk, he securiies on he marke only have o provide insurance agains non-diversifiable risk as depiced by changes in curren aggregae consumpion and beliefs abou fuure aggregae consumpion. Hence, if diversifiable risks exis, he number of securiies required o ensure Pareo opimaliy may be dramaically reduced compared o he number of securiies required o ensure a (fully) dynamically complee marke. These poins are illusraed in he following using he simple even ree informaion srucure shown in Figure 1. There are wo consumpion daes, 1and 2, and eigh saes, ω Ω revealed a dae 2 and four signals, σ Σ revealed a dae 1. σ 1 ω 1 ω 2 σ 2 ω 3 ω 4 σ 3 ω 5 ω 6 σ 4 ω 7 ω Figure 1. Even ree informaion srucure. A complee marke requires welve primiive securiies, one for each signal a dae 1andonefor each sae a dae 2. In ha marke, all rading occurs a dae wih no subsequen rading occurring a dae 1, cf. Rubinsein (1975). However, wih dynamic rading only four sufficienly disinc long-lived securiies are required o ensure Pareo opimal consumpion allocaions. This is due o he fac ha if, a each dae, agens can span wealh vecors wih he same dimension as he cardinaliy of he informaion se for he following dae, i.e., he marke is dynamically complee, hen any wealh process can be implemened by dynamic rading. Moreover, he minimum number of securiies o consiue a dynamically complee marke is equal o he maximum cardinaliy of informaion ses

3 DYNAMIC SPANNING IN THE CONSUMPTION-BASED CAPITAL ASSET PRICING MODEL 3 beween any wo daes, cf., e.g., Kreps (1979) for discree ime and finie sae space seings and Duffie and Huang (1985) for he exension o coninuous-ime models. The basic assumpions of he consumpion based capial asse pricing model are ime-addiive and sae-independen preferences and homogeneous beliefs. Wih hese addiional assumpions i is wellknown ha Pareo opimal consumpion plans are measurable wih respec o aggregae consumpion (AC-measurable), cf., e.g., Breeden and Lizenberger (1978). Therefore, only AC-measurable consumpion plans have o be implemenable in order o achieve Pareo opimaliy. As a consequence, much fewer securiies may be required o ensure Pareo opimaliy han o ensure a (dynamically) complee marke. Breeden and Lizenberger (1978) demonsrae in a discree ime model wih a finie sae space and no endowmen risks ha Pareo opimaliy can be achieved if agens can rade in a complee se of coningen claims on aggregae consumpion, i.e., if here exiss a primiive claim on aggregae consumpion for each aggregae consumpion level a each dae. Consequenly, all rading occurs a he iniial rading dae. As i is he case for he sandard Arrow-Debreu model, he Breeden-Lizenberger concep of an effecively complee marke does no generalize o a coninuous-ime model wih coninuous sae space, since his would require an infinie number of primiive claims on aggregae consumpion. However, in he same spiri as in Duffie and Huang (1985) we show ha coninuous rading of long-lived coningen claims on aggregae consumpion can subsiue he need for an infinie number of primiive securiies. In our discree ime and finie sae space seing, as depiced in Figure 1, we illusrae in he following example he concep of a dynamically effecively complee marke in which any consumpion plan, measurable wih respec o aggregae consumpion, can be achieved hrough dynamic rading. In he main par of he paper we formalize his concep o he coninuous-ime seing of he CCAPM. Agens consumpion possibiliies are deermined by he reurn from heir porfolio sraegies and heir consumpion endowmens. However, suppose for simpliciy ha agens have no consumpion endowmens such ha agens consumpion processes are equal o he dividend processes of heir porfolio sraegies. 3 Hence, in a dynamically effecively complee marke, i is possible for agens o choose a porfolio sraegy wih any financially feasible AC-measurable dividend process. However, in general, he value process of any such porfolio sraegy and, herefore, he agen s wealh process, will no be AC-measurable. This can be illusraed by considering, in Figure 2, he same even ree as in Figure 1 augmened wih he level of aggregae consumpion, γ, for each signal a each dae. Noe, in paricular, ha a number of signals have idenical aggregae consumpion levels, i.e., he risk in asse reurns associaed wih hese signals is diversifiable. Consider a given porfolio sraegy wih payoffs ha are measurable wih respec o aggregae consumpion, γ, a each dae. Hence, he dividends a dae 1 of his porfolio sraegy depend only on wheher signal σ 1 is obained or no, whereas he dae 2 dividends only depend on wheher an even or an odd sae occurs. Hence, no maer which signal occurs a dae 1, he porfolio chosen a dae 1 has he same payoff vecors a dae 2. However, he ex-dividend values a dae 1 of hose porfolios will be differen for he four signals due o differences in he sae-coningen prices. These prices are he producs of condiional probabiliies for he fuure saes and he marginal raes of subsiuion beween fuure sae-coningen consumpion and curren consumpion. The condiional probabiliies are idenical for he firs hree signals, bu he marginal raes of subsiuions are higher for σ 1 han for signals σ 2 and σ 3, since Pareo opimal consumpion plans, a dae 1, are sricly increasing in aggregae consumpion, cf., e.g., Breeden and Lizenberger (1978). Thus, 3 Dividends are here inerpreed as he ne-amoun wihdrawn from he porfolio a any poin in ime, i.e., he curren cum-dividend value of he porfolio acquired a he previous dae minus he ex-dividend value of he porfolio acquired a he curren dae.

4 DYNAMIC SPANNING IN THE CONSUMPTION-BASED CAPITAL ASSET PRICING MODEL 4 σ 1 γ 1 5 2/3 1/3 ω 1 ω 2 γ 2 5 γ 2 3 σ 2 γ 1 4 2/3 1/3 ω 3 ω 4 γ 2 5 γ 2 3 σ 3 γ 1 4 2/3 1/3 ω 5 ω 6 γ 2 5 γ 2 3 σ 4 γ 1 4 1/2 1/2 ω 7 ω 8 γ 2 5 γ Figure 2. Even ree informaion srucure. he value of he porfolio acquired a dae 1ishigherforσ 1 han he value of he porfolios acquired for signals σ 2 and σ 3. For signal σ 4, he condiional probabiliies have shifed such ha he probabiliy of he low level of fuure aggregae consumpion has increased. The marginal raes of subsiuions, however, are idenical for σ 2, σ 3,andσ 4. Therefore, he sae-coningen price for he low sae increases whereas he saeconingen price for he high sae decreases. Since he porfolios acquired a dae 1 have idenical sae-coningen payoff vecors, he value of he porfolio acquired for σ 4 is, in general, differen from he value of he porfolios acquired for σ 2 and σ 3. 4 The dividends a dae 1 of he porfolio acquired a dae ishigherforσ 1 han for he oher hree signals since Pareo opimal individual consumpion plans are sricly increasing in aggregae consumpion. Since he ex-dividend value of he porfolio acquired a 1 is also higher for σ 1 han for σ 2 and σ 3, he cum-dividend value of he porfolio a dae 1musbehigherforσ 1 han for σ 2 and σ 3, which again, in general, is differen from he porfolio value for σ 4. However, he cum-dividend value as well as he ex-dividend value mus be idenical for σ 2 and σ 3. These wo signals are precisely characerized by he fac ha curren consumpion as well as he probabiliy disribuion for fuure aggregae (and individual) consumpion are idenical. Therefore, for he financial marke o be dynamically effecively complee i only has o span he hree informaion ses a dae 1 shown by he dashed circles and, hus, only hree sufficienly disinc long-lived securiies are required a dae o ensure Pareo opimaliy, whereas a dynamically complee marke would require four securiies. However, noe ha even hough here are only wo aggregae consumpion levels a 1 we need hree securiies o have a dynamically effecively complee marke due o he variaion in condiional probabiliies for fuure aggregae consumpion levels. This corresponds o wha Meron (1973) absracly erms changes in he invemen opporuniy se. In he following, we formalize he concep of a dynamically effecively complee marke in a coninuousime and coninuous sae-space economy. Tha is, wihou endowmen risks, a Pareo opimal consumpion allocaion can be ensured even if he financial marke only spans informaion ses disinguished 4 In he special case of logarihmic uiliy, here will be no difference in he values of he porfolio acquired a 1 due o he myopic behavior of his uiliy funcion.

5 DYNAMIC SPANNING IN THE CONSUMPTION-BASED CAPITAL ASSET PRICING MODEL 5 by eiher differences in curren aggregae consumpion or differences in condiional probabiliy disribuions for fuure aggregae consumpion. If invesors have consumpion endowmen risks, hen addiional insurance conracs or financial claims may be needed in order o hedge hese personal risks. In Secion 2, we ouline he basic coninuous-ime model. In Secion 3 and Secion 4, we characerize he wealh processes of he agens and we characerize which and how many securiies he agens need in order o implemen he Pareo opimal consumpion plans. Moreover, we expand he exisence resuls in Secion 5, and in Secion 6 we show ha shor erm uncerainy is insured by shor erm coningen claims and long erm uncerainy is insured by long erm coningen claims. Finally, we exend he characerizaion of he wealh processes of he agens o include personal risk in Secion 7. All proofs are defered o he appendix. 2. The Coninuous-Time Economic Model The coninuous-ime and coninuous sae-space seing of he CCAPM follows ha of Duffie and Huang (1985), Duffie (1986), and Duffie and Zame (1989). The primiives are a complee probabiliy space (Ω, F,P), a finie ime horizon T : [,T, and a filraion F : {F } T generaed by a K- dimensional Wiener process, W, compleed in he usual way. Tha is, F fulfills he usual condiions. The consumpion space, L, 5 is he vecor space of square-inegrable previsible real-valued sochasic processes represening he agens consumpion a each poin in ime of he single commodiy. Each of he agens i Iin he finie se I is represened by (U i, ĉ i, ˆθ i ), where U i is a von Neumann-Morgensern uiliy funcion on L + represening agen i s preferences, ĉ i L + is agen i s consumpion endowmen process, and ˆθ i is an F -measurable sochasic variable represening agen i s endowed porfolio of real asses. Agens can inves in a finie se of asses, A, conaining boh he se of real asses, J,andheseof financial asses, N. Tha is, A J Nand, moreover, by consrucion J N. Hence, here is a finie se J of firms each represened by a real dividend process δ j L, j J. 6 For laer use, define he cumulaive dividend processes of he real asses as D j : δs j ds, j J. In addiion, here is a finie se N of financial asses in zero ne supply wih cumulaive dividend processes D n, n N, also measured in unis of consumpion, wih D denoing he lump-sum dividends paid a ime. To simplify he noaion, we sack he cumulaive dividend processes of real and financial asses in he same vecor, D, ofdimension A. To separae asse dividends for a se of asses we use projecions. E.g., o separae ou financial asse dividends, we define he projecion of D, denoed D N, also of dimension A such ha coordinae a is (D N ) a D n, if a N, :, if a N. Hence, D D J + D N. We will use his noaion for asse prices, porfolios, gain processes, ec. as well. The enire economy is hus described by a collecion E ( (Ω, F, F,P), {(U i, ĉ i, ˆθ i )} i I,D ). 5 Formally, L is defined as in Duffie and Zame (1989, Foonoe 4, p. 1283). Tha is, L L 2 (Ω [,T, P,ν), where P is he predicable σ-field on Ω [,T, (ha is, he σ-field generaed by he lef-coninuous adaped processes) and ν is he produc measure of P and he Lebesgue measure on [,T resriced o P. 6 By real, we mean ha he dividend process pays ou in unis of consumpion.

6 DYNAMIC SPANNING IN THE CONSUMPTION-BASED CAPITAL ASSET PRICING MODEL 6 Agens ake an A -dimensional ex-dividend asse price process, S, for graned. In addiion o having he dividends and asse prices denominaed in he consumpion numeraire (denoed real dividends and real prices), we also use nominal dividends and nominal asse prices, denoed D and S, respecively. Assume ha one of he financial asses, n N, is a zero-coupon bond paying one consumpion uni a mauriy dae T. Nominal dividends and asse prices are denominaed in unis of he price of his (numeraire) asse, S n. To ease he noaion denoe he nominal price of one consumpion uni as π, i.e., π 1 S n wih π T 1. Hence, he nominal asse price process and dividend process are defined by S : π S, D : π D, d D π dd. We assume ha he (nominal) gain process, defined by G S + D, isaniô process, d G µ G d + σ G dw, where µ G is an A -dimensional previsible sochasic process and σ G is an A K-dimensional previsible sochasic process such ha T E σ G σ G ds <, where denoes ranspose. This allows us o define cumulaive gains from rade for an A -dimensional previsible porfolio process, θ, as 7 θ s d G s, if θ fulfills he following inegrabiliy condiions T E θ (σ G σ G θ )d < (, and T E θ µ G d <. A budge-feasible plan for agen i is a consumpion process c L + and a porfolio process θ such ha, for any dae T, θ ( S + D ) θ s d G s + π s (ĉ i s c s)ds + ˆθ i S J, and such ha θ T D T. 8 A budge-feasible plan (c, θ) for agen i is opimal for agen i if here is no oher budge feasible plan ( c, θ) such ha U i ( c) >U i (c). An equilibrium for he economy E is a collecion ( ) S, {(c i,θ i )} i I such ha, given he asse price process S, he budge-feasible plan (c i,θ i )is opimal for each agen i I, and such ha markes clear: c i ĉ, i I θ i,a 1, i I a J, 7 x y denoes he sandard Euclidean inner produc beween x and y in R A. 8 Equaliies of random variables are o be undersood as almos surely ideniies.

7 DYNAMIC SPANNING IN THE CONSUMPTION-BASED CAPITAL ASSET PRICING MODEL 7 and θ i,a, a N, where he exogenously given aggregae consumpion process, ĉ, is defined as 9 ĉ : ĉ i + δ j. i I j J i I We make he following hree assumpions similar o Duffie and Zame (1989, (A.1) (A.3), pp ) wih slighly more resricive regulariy condiions. Assumpion 2.1. Agen i s uiliy funcion, U i, is represened by a smooh coninuously differeniable funcion, u i : R + T R of he form, T U i (c) E u i (c,)d, where, for each T, he funcion u i (,):R + R is sricly concave, increasing, and hree imes coninuously differeniable on (, ), wih firs derivaive denoed u ic (,) saisfying lim k u ic (k, ) +. Assumpion 2.2. The aggregae consumpion process, ĉ,isaniôprocess, uniformly bounded away from zero, represened as (1) dĉ µ d + σ dw, such ha E T µ 2 d <, E T σ 2 d <, µ and σ have coninuous sample pahs. Moreover, we assume ha, for all, a regular condiional disribuion of {ĉ r } r (,T given F (denoed P (ĉ F )) exiss such ha he condiional marginal disribuion for aggregae consumpion, P (ĉ r F ), is absoluely coninuous for all r and wih r. Assumpion 2.3. A (numeraire) zero-coupon bond wih mauriy T exiss in addiion o a subse of securiies L Afor which he accumulaed dividend process, D L,isanIô process represened as wih E T σd σ D dd L µ D d + σ D dw, d < and E T µd d <, such ha he vecor maringale defined by M L σd s dw s forms a maringale generaor for F (under he measure P ). Tha is, for a given onedimensional F-maringale ˆM, anf-previsible process ψ exiss such ha ˆM ˆM + ψ s dm L s. From Duffie and Zame (1989) 1 and Duffie (1988) 11 we collec he following resuls. 9 Even hough we have inroduced firms, he economy is an exchange economy since producion decisions are exogenously given. 1 Duffie and Zame (1989, Theorem 1, p and Corollary, p. 1289) sill holds even hough we have added addiional srucure o he model. The real dividends from he firms do no change anyhing excep giving he represenaive agen addiional endowmens. Moreover, in Assumpions 2.1 and 2.2 we included addiional assumpions on he primiives of he model, bu no on he equilibrium ha we wan o find. Finally, we have relaxed he spanning assumpion in Assumpion 2.3 such ha we allow for a subse of he asses o span he uncerainy generaed by he filraion F. 11 Under slighly less resricive regulariy condiions, Duffie (1988) proves Equaion (3) as Exercise The resul follows from he firs order condiions for he mahemaical program defining he uiliy funcion for he represenaive agen. In addiion, cf. Huang (1987, Proposiion 3.3, p. 125).

8 DYNAMIC SPANNING IN THE CONSUMPTION-BASED CAPITAL ASSET PRICING MODEL 8 Proposiion 2.4. An equilibrium wih a represenaive agen (u λ, ĉ ) exiss such ha, for any ime, he equilibrium price of a cumulaive dividend process,, is S 1 T u λc (ĉ,)e u λc (ĉ s,s)d (2) s F, [,T), for such ha E 2 T <, where T denoes he oal variaion of he process in he ime inerval [,T. Furhermore, he equilibrium consumpion of agen i, a dae, c i, is measurable wih respec o aggregae consumpion a dae. Tha is, measurable non-negaive funcions k i : R + T R, fori I, exis such ha (3) c i k i (ĉ,). Afer collecing hese resuls on he exisence and characerizaion of equilibria in dynamically complee markes, we urn o he implemenaion of Pareo opimal consumpion plans. 3. Characerizaion of he Wealh Processes of he Agens As we have demonsraed in he discree-ime, finie sae-space example in he inroducion, aggregae consumpion affecs individual consumpion (see also Equaion (3) in Proposiion 2.4) and condiional probabiliy disribuions for fuure aggregae consumpion affec implici sae prices and, hus, he curren value of he invesors fuure consumpion sream. Similarly, in order o characerize he relevan informaion for he agens implemenaion of heir Pareo opimal consumpion plans in he coninuousime and coninuous sae-space economy, he following σ-fields defines his informaion G : σ(ĉ ), G : G s, s H : σ ( ĉ,p(ĉ F ) ), H : s H s, where ĉ is a shorhand noaion for {ĉ r } r (,T. Assume ha he σ-fields are compleed by adjoining all P -null-ses. Furhermore, we denoe he filraions {G } T resp. {H } T as G resp. H. The filraion, G keeps rack of pas values of aggregae consumpion, whereas H keeps rack of boh pas values of aggregae consumpion and relaed condiional probabiliies given he invesors informaion F. Noe ha, since aggregae consumpion is exogenously given, boh he filraions G and H are exogenously given. The following example illusraes he key differences beween he filraions F, H, andg. Example 3.1. Le Y be a sae process such ha (ĉ,y) is a Markov process governed by he following sochasic differenial equaion, ( ) ( ) dĉ µ(ĉ dy,y )d + σ(ĉ,y dw 1 ), dw 2 where σ is a non-singular marix funcion suiably chosen o mee he condiions of Assumpion 2.2. Clearly, G σ(ĉ s : s ) and H σ(ĉ s,y s : s ) (up o P -null-ses). The σ-fields G and H will, herefore, in general, no coincide if ĉ is no Markov by iself. Hence, he filraion G is a proper sub-filraion of H, in general. Moreover, he filraion H can easily be a proper sub-filraion of F. To give an example of his relaed o he economy of his paper consider he case where agens have no consumpion endowmens and he

9 DYNAMIC SPANNING IN THE CONSUMPTION-BASED CAPITAL ASSET PRICING MODEL 9 firms marke shares of aggregae dividends are affeced by he Wiener process W (W 3,..., W K ) wih K>2. Assume ha he marke share of firm j, a j, defined from he relaion δ j ajĉ,isaniôprocess on he form da j µ aj d + σ aj dw. Alhough he process for aggregae consumpion is only affeced by he firs wo coordinaes of he Wiener process, individual firm dividends (and also prices of he individual asses) are affeced by all he coordinaes of he Wiener process. The agens, however, would diversify heir porfolio holdings of he real asses such ha hey are no gambling on he oucome of which of he individual firms ha pay high and which pay low dividends. This is so because he uncerainy relaed o which of he firms ha pay high and which pay low dividends is independen on he uncerainy relaed o he level of aggregae consumpion. Therefore, he agens would no be worse off if hey could only inves in a muual fund holding all he real asses, which has a G-adaped dividend process. However, in order o implemen a Pareo-opimal allocaion he invesors may in addiion o he fund of real asses need a second asse o dynamically hedge he uncerainy associaed wih he wo sae variables. Tha is, in general, hey need hree asses, wo asses wih gain processes ha span he filraion H and one numeraire securiy. As indicaed in he inroducion, curren aggregae consumpion and he condiional probabiliy disribuion for fuure aggregae consumpion and, hus, he filraion H, play a key role in describing he sae prices in he economy. We sar by showing ha he price densiy process peraining o aggregae consumpion, p r (c) is adaped o he filraion H. Lemma 3.2. The implici price densiy process, {p r(c)} [,r, peraining o aggregae consumpion of level c, a a given fuure dae, r, is adaped o he filraion H. Moreover, in he following lemma we derive he coninuous-ime version of he basic discree ime and finie sae space valuaion equaion in Breeden and Lizenberger (1978, Theorem 2). Lemma 3.3. The equilibrium price of an absoluely coninuous cumulaive dividend process, { } : δ r dr saisfying he inegrabiliy condiion of Proposiion 2.4, is T (4) S R p r (c)e[δ r ĉ r c, F dc dr. Using Lemma 3.2 and Lemma 3.3 he price process of a securiy will no be H-adaped unless is dividend process is G-adaped, in general. Moreover, even if he dividend process is G-adaped he price process will no be G-adaped, because he implici price densiy processs is only H-adaped. The key characerisic of valuaion equaion (4) is ha risk adjusmens only perain o variaions in aggregae consumpion whereas here is no risk adjusmen for diversifiable risk no affecing aggregae consumpion. Tha is, he agens preferences wih respec o ineremporal consumpion and risk are capured by he implici price densiies peraining o aggregae consumpion. This can also be seen from he measure ransformaion o he risk adjused probabiliy measure where he gain processes are maringales. The following lemma demonsraes ha his ransformaion is H-adaped., T

10 DYNAMIC SPANNING IN THE CONSUMPTION-BASED CAPITAL ASSET PRICING MODEL 1 Lemma 3.4. Le q u λc (ĉ,) and le be an absoluely coninuous cumulaive dividend process, { } : δ r dr saisfying he inegrabiliy condiion of Proposiion 2.4. An equivalen measure Q exiss such ha he equilibrium price of he cumulaive dividend process,, is T S S n E Q 1 d Ss n s F. The Radon-Nikodym derivaive, dq dp F s, can be expressed as (5) N s : 1 E[q T Sn s q s. The real price of he numeraire securiy, S n, and he Radon-Nikodym derivaive, N, arebohh-adaped. In nominal erms, he equilibrium price of he nominal cumulaive dividend process,, can be deermined as, T S E Q [ T F, and he corresponding gain process, G,isan(F,Q)-maringale. Using he valuaion resuls in he previous lemmas, he equilibrium wealh processes of he agens can be characerized by inerpreing he equilibrium wealh of an agen as he curren value of he agen s fuure equilibrium consumpion process. 12 I follows from Proposiion 2.4 ha he agen s fuure equilibrium consumpion process is G-adaped. Hence, using Lemma 3.2 and Lemma 3.3 he curren value of his consumpion process is H-adaped. Proposiion 3.5. The equilibrium wealh process of agen i, w i, is adaped o he filraion H, for all i I. Observe ha, in general, he wealh process will no be adaped o he filraion G generaed by he aggregae consumpion process. Tha is, even hough curren and pas aggregae consumpion are idenical for wo evens, he equilibrium wealh may differ for hose evens since he implici price densiy may differ. This poin is also illusraed in our simple discree-ime and discree sae-space example presened in he inroducion of he paper. However, in single-period models, he wealh process is adaped o he filraion G, since in hose models wealh equals consumpion, cf., e.g., Demange and Laroque (1995). This resul can be re-esablished in a muli-period seing if he aggregae consumpion processismarkovashefollowingresulshows. Corollary 3.6. Suppose ĉ is Markov. Then he opimal wealh process of agen i, w i, is adaped o he filraion G, for all i I. Having demonsraed ha he agens equilibrium wealh processes are H-adaped, we urn o he quesion of characerizing he ype of asses ha he agens need in order o implemen Pareo opimal consumpion plans. 12 Noe, ha he filraion H is precisely consruced as he minimal filraion ha makes he equilibrium wealh process for all he agens adaped o ha filraion. In a no-arbirage, wo-dae consumpion framework, Chamberlain (1988) consrucs a minimal Wiener filraion ha makes he agens final-dae consumpion measurable wih respec o he finaldae σ-algebra of ha filraion. This implies ha he wealh processes are adaped o he Wiener filraion. In general, his Wiener filraion is a finer filraion han H.

11 DYNAMIC SPANNING IN THE CONSUMPTION-BASED CAPITAL ASSET PRICING MODEL Implemenaion of Pareo Opimal Consumpion Plans In his secion we show, given a spanning condiion o be made precise, ha he agens are able o implemen heir Pareo opimal consumpion plans by rading coningen claims on aggregae consumpion. However, even hough he equilibrium wealh processes are H-adaped, he rading sraegies ha implemen hese wealh processes are no necessarily H-adaped. We begin wih a definiion of maringale generaors for sub-filraions of F ha recognizes his. 13 Definiion 4.1. Le L : {L } T be a filraion such ha L F, for all T, andlem be an n-dimensional (F,Q)-maringale. M is an (F,Q)-maringale generaor for L if, for any given onedimensional L-adaped (F,Q)-maringale N, ann-dimensional F-previsible process ψ exiss such ha N N + ψ s dm s, (, Le M N, be a subse of he financial asses consising of coningen claims on aggregae consumpion. Tha is, he corresponding cumulaive dividend processes, D M,areH-adaped. Noe by Lemma 3.4 ha G M is an (F,Q)-maringale denoing he gain process of hese coningen claims (augmened wih zeros on he coordinaes ouside he se M). Proposiion 4.2. Le an equilibrium, ( ) S, {(c i,θ i )} i I, be given. If GM forms an (F,Q)-maringale generaor for H and he agens consumpion endowmen processes, ĉ i, are adaped o he filraion G, for all i I, hen he equilibrium consumpion plan can be implemened by rading only in he se of financial asses M, he marke porfolio of real asses, and he numeraire zero-coupon bond. The proposiion demonsraes ha financial asses coningen on aggregae consumpion can play a key role in implemening Pareo opimal consumpion plans when he agens consumpion endowmen processes, ĉ i, are adaped o he filraion G. 14 When here are personal risks in he agens consumpion endowmens more securiies (or insurance conracs) may be needed o faciliae an efficien insurance agains hose risks. We reurn o ha issue in Secion 7. We focus on he role of coningen claims on aggregae consumpion for spanning purposes because a maringale generaor for H mus be adaped o H. 15 Noe by Proposiion 2.4, ha he gain process of a coningen claim wih cumulaive dividend process adaped o G, whichweermasimple coningen claim on aggregae consumpion, ish-adaped. This resul is formalized in he following proposiion. Proposiion 4.3. Le an equilibrium, ( ) S, {(c i,θ i )} i I be given. The gain process of a simple coningen claim on aggregae consumpion is H-adaped. Hence, hese simple coningen claims on aggregae consumpion are naural candidaes for forming a maringale generaor. Non-simple coningen claims on aggregae consumpion may or may no have Z 13 Noe ha for any F-previsible process ψ, i will no, in general, be he case ha he maringale generaed as ψ s dm s (, is L-adaped. 14 In a N-dimensional Markov seing, Meron (1973) demonsraes a similar muual fund separaion resul in which any budge-feasible consumpion process can be obained by coninuous rading in N +2 porfolios including one hedge porfolio for each of he N sae variables. In a similar vein as in Example 3.1 suppose aggregae consumpion is a Markov process in iself and i is one of Meron s sae variables. Then, Proposiion 4.2 shows ha only hree muual funds are needed in order o implemen a Pareo opimal allocaion whereas Meron uses N + 2. However, asse prices depend on all sae variables. 15 Noe ha he proposiion gives a separaion resul and is no a saemen on he minimal number of financial asse needed in order o implemen a Pareo opimal consumpion allocaion. For example, real asses migh also be used as par of he maringale generaor. In paricular, he marke porfolio of real asses can be used, since is gain process is H-adaped given he assumpion on he consumpion endowmens.

12 DYNAMIC SPANNING IN THE CONSUMPTION-BASED CAPITAL ASSET PRICING MODEL 12 H-adaped gain processes. In coninuaion of Example 3.1, he following example illusraes ha if aggregae consumpion and he condiional probabiliy disribuion for fuure aggregae consumpion is a Markov process as in Example 3.1, hen non-simple coningen claims on aggregae consumpion will also have H-adaped gain processes. Example 4.4. Le he siuaion be as in Example 3.1. Le δ be he rae of dividends on a coningen claim wih an H-adaped absoluely coninuous cumulaive dividend process. Hence, i follows from Proposiion 2.4 ha he real price of he coningen claim is given by S 1 T u λc (ĉ,)e u λc (ĉ s,s)δ sds F 1 T u λc (ĉ,) E u λc (ĉ s,s)δ s ds H, where he las equaliy follows from (ĉ,y) being a Markov process. Since all erms on he righ hand side are H-adaped, he price of he claim is H-adaped. I hen follows from Lemma 3.4 ha he nominal price and he gain process are boh H-adaped. However, claims wih dividend processes no adaped o H will rarely have gain processes adaped o H. Examples of simple coningen claims on aggregae consumpion are forward conracs and European opions wrien direcly on he level of aggregae consumpion, whereas examples of non-simple coningen claims are fuures, American opions, and opions wrien on a forward conrac on he level of aggregae consumpion. 5. Exisence of Equilibria in Dynamically Effecively Complee Markes In obaining he resul of Proposiion 4.2 i has been assumed ha he marke is dynamically complee, i.e., securiies forming a maringale generaor for F exis, cf., Assumpion 2.3. However, i is a direc consequence of Proposiion 4.2 ha he equilibrium can be susained in an oherwise idenical (dynamically incomplee) economy in which he coningen claims on aggregae consumpion forming a maringale generaor for H and he numeraire zero-coupon bond are he only financial asses raded. We erm his economy dynamically effecively complee. In fac, his saemen can be exended o prove he exisence of equilibria for dynamically effecively complee markes as opposed o he exisence of equilibria for dynamically complee markes shown in Duffie and Zame (1989). Proposiion 5.1. Le an economy E ( (Ω, F, F,P), {(U i, ĉ i, ˆψ i )} i I,D ) saisfying Assumpions be given. Suppose H is generaed by some linear funcion of he Wiener process, i.e., H σ({aw s } s [, ), where A is a consan marix, and ha he agens consumpion endowmen processes, ĉ i, are adaped o he filraion G, for all i I. Financial asses coningen on aggregae consumpion can be consruced such ha an equilibrium exiss in he economy augmened wih hese asses and he numeraire zero-coupon bond wih mauriy dae T. The proposiion demonsraes ha any economy saisfying Assumpions wih H generaed by some linear funcion of Wiener processes can be augmened wih financial asses coningen on aggregae consumpion such ha an equilibrium exiss for he augmened economy. However, he choice of dividend processes for he financial asses ha ensure ha heir gain processes form a maringale generaor for H depends on he equilibrium prices in he exended economy. If, insead, we assume ha he cumulaive dividend processes for he financial asses as opposed o he gain processes are Iô processes and form a maringale generaor for H, hen he necessary financial asses can be specified exogenously independenly

13 DYNAMIC SPANNING IN THE CONSUMPTION-BASED CAPITAL ASSET PRICING MODEL 13 of equilibrium prices as in Assumpion 2.3. However, his would exclude he spanning role of opionlike coningen claims on aggregae consumpion which have non-coninuous dividend processes. The corresponding gain processes of hese claims, on he oher hand, are ypically assumed o be Iô processes and, herefore, i is naural o sae he spanning condiion in erms of he gain processes forming a maringale generaor for H. 6. Shor- and Long-erm Coningen Claims Proposiion 5.1 provides sufficien condiions for he exisence of equilibria in dynamically effecively complee markes in erms of exisence of a se of absrac coningen claims on aggregae consumpion. The following proposiions provide relaionships beween he process of condiional probabiliy disribuions for fuure aggregae consumpion and characerisics of he simple coningen claims on aggregae consumpion which are useful o implemen Pareo opimal consumpion plans. Proposiion 6.1. Suppose he condiional probabiliy disribuion for aggregae consumpion afer dae 1, P (ĉ 1 F ),ish -measurable for all [, 1 and here is a se L of long-erm simple coningen claims on aggregae consumpion such ha D L DL,, [, 1 and ha σ(ds L DL 1 : s> 1 ) σ(ĉ 1 ). Then he se of gain processes of he asses in he se L, { { Gl } [, 1} l L, has a Q-maringale mulipliciy of one in he ime inerval [, 1. The key assumpion in he proposiion is ha no new informaion is revealed in he ime-period from o 1 abou aggregae consumpion possibiliies afer In ha case, long-erm simple coningen claims on aggregae consumpion are no useful in he implemenaion of shor-erm Pareo opimal consumpion plans. In order o implemen shor-erm Pareo opimal consumpion plans, he agens mus be able o rade in a sufficienly varied se of shor-erm simple coningen claims on aggregae consumpion (or non-simple coningen claims on aggregae consumpion). Proposiion 6.2. Suppose he process of aggregae consumpion before dae 1, ĉ 1,isH -measurable, for a given < 1,andhereisaseL of shor-erm simple coningen claims on aggregae consumpion such ha D L DL,, > 1. Then he se of gain processes of he asses in he se L, { { G l } } [, 1 of one in he ime inerval [, 1. l L, has a Q-maringale mulipliciy The key assumpion in his proposiion is ha no new informaion is revealed abou shor-erm aggregae consumpion possibiliies. The informaion revealed perains only o long-erm aggregae consumpion possibiliies. Consequenly, only long-erm coningen claims on aggregae consumpion are useful in order o implemen Pareo opimal consumpion plans. The las wo proposiions demonsrae ha here is a close connecion beween he mauriy srucure of useful simple coningen claims on aggregae consumpion and he informaion ha is revealed abou fuure aggregae consumpion. An efficien sharing of shor-erm risk in aggregae consumpion is faciliaed by rading shor-erm simple coningen claims, whereas an efficien sharing of long-erm risk in aggregae consumpion requires rading in long-erm simple coningen claims (or shor-erm non-simple coningen claims on aggregae consumpion). 16 Noe ha his implies, e.g., ha he erm srucure of forward raes is deerminisic for mauriies laer ha 1.

14 DYNAMIC SPANNING IN THE CONSUMPTION-BASED CAPITAL ASSET PRICING MODEL Personal Risk In Proposiion 4.2 we showed ha agens can implemen Pareo opimal consumpion plans by resricing heir rading sraegies o he marke porfolio of real asses, coningen claims on aggregae consumpion, and he numeraire zero-coupon bond. This resul was shown under he assumpion ha he agens consumpion endowmen processes conain no personal risks, i.e., ha he agens consumpion endowmen processes are adaped o he filraion generaed by aggregae consumpion, G. When his is no he case, more securiies (or insurance conracs) are needed o faciliae an efficien insurance agains personal risks. One simple way of implemening he agens opimal consumpion plan in he case where he agens are exposed o personal risks is o inroduce an insurance fund. This insurance fund offers conracs wih he following dividend processes ɛ i { E[ĉ i G ĉ i } T, for all i I. Clearly, conrac i is explicily ailored o agen i in he sense ha conrac i is designed o insure precisely agains agen i s personal risk. The value of conrac i a dae is T R p r(c)e [ E[ĉ i r G r ĉ i r ĉ r c, F dc dr, for i I. Noe ha he value of he conrac is zero a dae zero, such ha i has no budge consequences a dae zero for he agen o buy his insurance conrac. Moreover, noe ha if agen i buys insurance conrac i a dae zero, he agen s insured endowmen process is {E[ĉ i G } T, whichisg-adaped. Hence, afer having inroduced he insurance fund, we are back in he case of Proposiion 4.2, in which he individual agens personal endowmen processes are G-adaped. The insurance fund is funded by issuing a real asse wih he dividend process i I ɛi. The porfolio consising of he real asses including he asse issued by he insurance fund forms he marke porfolio. The dividend process of he marke porfolio is { ĉ i + δ j E[ĉ i G } T { ĉ E[ĉ i G } T. i I j J i I i I Hence, he dividend process of he marke porfolio is G-adaped. The following resul now follows almos immediaely from Proposiion 4.2. Proposiion 7.1. Le a subse of financial asses, M N, and an equilibrium for a dynamically complee marke, ( ) S, {(c i,θ i )} i I, be given. Denoe he corresponding gain processes for M by GM. If G M forms an (F,Q)-maringale generaor for H and he agens have access o he insurance conracs wih dividend processes {ɛ i } i I, hen he equilibrium consumpion plan can be implemened by buying he individually ailored insurance conracs and dynamically rading only in he se of financial asses M, he marke porfolio of real asses including he asse issued by he insurance fund, and he numeraire zero-coupon bond. Noe ha he agens buy-and-hold he individually ailored insurance conracs, since he insurance conracs insure he consumpion endowmens for he whole lifeime of he agens. If hese individually ailored long-erm insurance conracs are no available, he Pareo opimal consumpion allocaions are implemenable if he se of financial asses and shor-erm insurance conracs is rich enough for he agens o hedge heir personal risks hrough shor-erm insurance conracs and dynamically rading of

15 DYNAMIC SPANNING IN THE CONSUMPTION-BASED CAPITAL ASSET PRICING MODEL 15 financial asses. In order o invesigae his issue, define he following σ-fields, P : σ (ĉ,p(ĉ F ), ĉ i, { ) {E[ĉ i r ĉ r c, F } c C r }r (,T : i I, ĉ C, P : P s, s and he filraion, P : {P } T. The equivalen proposiion o Proposiion 4.2 now akes he following form. Proposiion 7.2. Le a subse of financial asses, M N, and an equilibrium for a dynamically complee marke, ( ) S, {(c i,θ i )} i I, be given. Denoe he corresponding gain processes for M by GM.If G M forms an (F,Q)-maringale generaor for P, hen he equilibrium consumpion plan can be implemened by rading only in he se of financial asses M, he marke porfolio of real asses, and he numeraire zero-coupon bond. The key issue in implemening Pareo opimal consumpion plans wih personal risks is ha agens hrough insurance conracs and dynamic rading are able o insure hemselves agains no only variaions in curren consumpion endowmens bu also in variaions of he value process of heir fuure consumpion endowmen process. However, noe ha all new informaion abou he agens fuure consumpion endowmens does no have o be capured by he maringale generaor only informaion relevan for he value process of fuure consumpion endowmens. Irrespecively of equilibrium prices, his is assured by he assumpion ha he informaion generaed by each individual agen s fuure expeced consumpion endowmens condiional on levels of aggregae consumpion is capured by he maringale generaor. Hence, for example, new informaion abou condiional variances of he agens fuure consumpion endowmens has no impac on he value process. Appendix A. Proofs The following equivalen characerizaion of he filraion H is useful in several of following proofs: Lemma A.1. The σ-field H can, up o P -null-ses, be wrien as σ ( ĉ s,e[g(ĉ s ) F s:g C b (C s, R),s ), where C s denoes all fuure possible aggregae consumpion pahs, given he informaion a dae s, and C b (C s, R) denoes he se of bounded coninuous funcions mapping from C s o R. ProofofLemma3.2. Le Fr (c) denoe he value, a dae, of a claim paying ou one uni of consumpion if, and only if, he aggregae consumpion level is less han or equal o c a dae r. By Proposiion 2.4 Fr(c) E[u λc(ĉ r,r)1 {ĉ r c} F u λc (ĉ.,) Fr (c) ish -measurable by Lemma A.1. Moreover, Fr ( ) is non-decreasing and righ coninuous. By he Radon-Nikodym Theorem and he assumpion (cf. Assumpion 2.2) ha he condiional marginal disribuion for aggregae consumpion, P (ĉ r F ), is absoluely coninuous, a non-negaive, H -measurable funcion, p r( ), exiss such ha (6) F r(c) c p r(u)du.

16 DYNAMIC SPANNING IN THE CONSUMPTION-BASED CAPITAL ASSET PRICING MODEL 16 ProofofLemma3.3. By Proposiion 2.4 S 1 T u λc (ĉ,) E u λc (ĉ r,r)d r F T [ uλc (ĉ r E,r)δ r u λc (ĉ,) F dr T [ uλc (ĉ E r,r)e[δ r F G r u λc (ĉ F dr.,) Using he condiional densiy derived in Equaion (6) and he relaion beween condiional probabiliies and condiional probabiliies given a fixed aggregae consumpion level, c, from Hoffmann- Jørgensen (1994, pp ) yields Equaion (4). ProofofLemma3.4. In order o show ha N, defined in Equaion (5), is a Radon-Nikodym derivaive, we mus show ha N is a non-negaive (F,P)-maringale wih E[N T 1. Noe ha from Proposiion 2.4 he real price of he numeraire securiy is given by (7) S n 1 q E[q T F. E[q T is well-defined since q T is bounded. This follows from he assumpion ha ĉ is uniformly bounded away from zero and ha u λc (,T) is decreasing and sricly posiive. Muliplying boh sides of Equaion (7) wih q yields ha N is a non-negaive (F,P)-maringale. Moreover, i follows from Equaion (7), Lemma A.1, and he definiion of N ha boh S n and N are H-adaped. From he expression of he Radon-Nikodym derivaive i follows ha 1 T E q s d s F q S T E T E E Q T q s q d s N s S n N Ss n S n Ss n T S n E Q F d s F d s F 1 S n s d s F. The nominal price formula follows direcly from subsiuion of he definiions of nominal erms, i.e., T E Q d s F S E Q [ T F. Since G S + i follows ha G is an (F,Q)-maringale. Proof of Proposiion 3.5. Define he cumulaive consumpion process, C i, of agen i by C i : c i sds.

17 DYNAMIC SPANNING IN THE CONSUMPTION-BASED CAPITAL ASSET PRICING MODEL 17 Using Lemma 3.3 and Proposiion 2.4, agen i s wealh a ime can be wrien as T w i S Ci R p r(c)e[c i r ĉ r c, F dc dr (8) T R p r (c)ki (c, r)dc dr. Lemma 3.2 hen gives ha w i, is adaped o he filraion H. Proof of Corollary 3.6. If ĉ is Markov hen he process {P (ĉ r F s)} s [,r) {P (ĉ r ĉ s )} s [,r) is adaped o he filraion G, for all r T. Thais,H G, for all T. Hence he wo filraions G and H are idenical. Proof of Proposiion 4.2. The proof goes in wo seps: (i) Allocae rading sraegies o each agen which generae ha agen s equilibrium consumpion process and which clears markes. (ii) Show ha no agen has any incenive o deviae from he allocaed rading sraegy. Sep i: Choose an agen, i I. Define I : I\{i }. For each agen, i I,len i c i ĉ i be he neconsumpion requiremen from he agen s rading sraegy. Since c i is adaped o G by Proposiion 2.4, and ĉ i is adaped o G by assumpion, n i is adaped o G. Denoe agen i s porfolio wealh process as X i : θ i (S + D). Clearly he porfolio wealh is going o finance all fuure ne-consumpion requiremens. Hence, using Proposiion 2.4, X i is given by X i 1 T u λc (ĉ,) E u λc (ĉ r,r)ni r dr F (9) 1 T u λc (ĉ E[u λc (ĉ,) r,r)n i r F dr. Since n i is adaped o he filraion G and he σ-field H includes condiional probabiliies of G-adaped processes i follows ha E[u λc (ĉ r,r)ni r F ish -measurable. Hence, by Equaion (9), X is adaped o he filraion H. Define he cumulaive ne consumpion requiremens as N i : n i s ds. Since N i is H-adaped, i hen follows from Lemma 3.4 ha (1) M i : X i + Ñ i EQ [Ñ i T F is an H-adaped (F,Q)-maringale. Hence, an F-previsible rading sraegy ψ i exiss such ha (11) M i X i + ψs i d G M s, since by assumpion G M forms an (F,Q)-maringale generaor for H. Wihou loss of generaliy we can choose ψt i. Recall ha ψi is augmened wih zeros ouside he se of coordinaes M. 17 ψ i is agen i s new rading sraegy in he financial asses coningen on aggregae consumpion. The only hing missing is o specify he amoun raded in he numeraire securiy such ha X i ψ i ( S + D) ψ i ( S M + D M )+ψ in. 17 In paricular, (ψ i ) J, since agen i is now only rading in he se of financial asses, N M.

18 DYNAMIC SPANNING IN THE CONSUMPTION-BASED CAPITAL ASSET PRICING MODEL 18 Consequenly, ψ in mus be deermined as ψ in : X ψ i ( S M + D M ), T. I follows ha he new rading sraegy, ψ i, is budge feasible since by Equaion (1) and (11) X i ψs i d G M s Ñ i + X i ψ i s d G s Ñ i + X i, ψ i s d G s π s n i s ds + ˆθ i S J. Tha ψi s d G M s ψi s d G s follows since G n is idenically one by consrucion of he numeraire securiy. Thus, we have shown ha he rading sraegy ψ i implemens he equilibrium consumpion process, c i, of agen i I I\{i }. The rading sraegies of agen i is such ha all asse markes clear: ψ J i 1 i I ψ J i 1, ψ N i i I ψ N i. Sep ii: By consrucion, agens i I have no incenive o deviae from heir allocaed rading sraegies: hey precisely implemen heir equilibrium consumpion process, and ha process is weakly preferred o any oher budge feasible consumpion process. Similarly, we need o show ha agen i s rading sraegy implemens her equilibrium consumpion process. Bu his follows from he lineariy of sochasic inegrals and marke clearing of he equilibrium consumpion processes boh wih he old and he new rading sraegies. Proof of Proposiion 5.1. Firs, we show ha if he (exogenously given) filraion H is generaed by Wiener processes, hen coningen claims on aggregae consumpion exis forming a maringale generaor for H. Noe ha in Example 3.1, he filraion H is generaed by he Wiener process (W 1, W 2 ), such ha for his example (wo) coningen claims on aggregae consumpion exis which form a maringale generaor for H. Lemma A.2. Le an equilibrium, ( ) S, {(c i,θ i )} i I, for a dynamically complee marke be given. Suppose H is generaed by some linear funcion of he Wiener process, i.e., H σ({aw s } s [, ),wherea is a consan marix. Then financial asses coningen on aggregae consumpion can be consruced such ha heir gain processes form a maringale generaor for H. ProofofLemmaA.2. Define ˆN such ha N e ˆN 1 2 ˆN, where N is he Radon-Nikodym derivaive given in Lemma 3.4. Then M : AW ˆN,AW is an (F,Q)- maringale generaor for H by Iô s represenaion heorem. We show ha a se of coningen claims on aggregae consumpion can be consruced such ha he gain process of his se forms he maringale generaor M. Define a se of coningen claims on aggregae consumpion, Q, by he cumulaive nominal dividend process D Q such ha D T Q M T,