Completing Markets in a One-Good, Pure Exchange Economy. Without State-Contingent Securities

Transcription

1 Compleing Markes in a One-Good, Pure Exchange Economy Wihou Sae-Coningen Securiies David M. Eagle Deparmen of Managemen, RVPT#3 College of Business Adminisraion Easern Washingon Universiy 668 N. Riverpoin Blvd., Suie A Spokane, Washingon USA Phone: ( Fax: ( deagle@ewu.edu 2004 by David Eagle. All righs reserved. Copyrigh will be ransferred o publishing journal when acceped. - -

2 Compleing Markes in a One-Good, Pure Exchange Economy Wihou Sae-Coningen Securiies ABSTRACT Pareo-efficien consumpion in a pure-exchange, one good economy varies over saes of naure wih respec o only wo facors: real aggregae supply and individual uiliy shocks. One s opimal conrac receips vary wih respec o only hese wo facors and he raio of one s endowmen o real aggregae supply. How one s Pareo-efficien consumpion varies wih real aggregae supply depends solely on how one s relaive risk aversion compares o he average. Complee markes can be approximaely achieved by four conracs dealing wih hese facors. This has implicaions concerning cenral banking, efficien insurance conrac design, and a possible new financial innovaion

3 Compleing Markes in a One-Good, Pure Exchange Economy Wihou Sae-Coningen Securiies Under fairly general assumpions, an Arrow-Debreu economy wih sae-coningen securiies resuls in a Pareo-efficien consumpion allocaion. However, for a variey of reasons, sae-coningen securiies are impracical in he real world. Many financial economiss make he presumpion ha insurance conracs and he financial innovaions of opions, fuures, swaps, and oher derivaives are helping o complee markes even hough hese conracs are no saeconingen securiies. This paper repors on a heoreical invesigaion ino wheher conracs or securiies oher han non-sae-coningen securiies can complee he markes in a pure-exchange economy wihou sorage. We find ha markes in such an economy can be approximaely compleed wih four differen ypes of conracs. These conracs are (i normal conracs, (ii endowmen-sharing conracs, (iii spending-sharing conracs, and (iv Real-Aggregae-Supply- Risk-Transfer (RASRT conracs. When Arrow (953 and Debreu (959 firs discussed economies wih sae-coningen securiies, hey noed ha complee markes would require n c T sae-coningen securiies where n is he number of saes, c is he number of commodiies ha exis, and T is he ime horizon of he economy. Arrow (953 did surmise ha, if he sae-coningen securiies were saed in erms of a numeraire, complee markes could be achieved wih n T such securiies. Laer, Radner (972 showed Arrow s surmise o be correc when markes were open in sequenial economies. Even so, for large T and large n, n T securiies will be impracically large. In paricular if we jus considered he characerisic of emperaure in 50,000 locaions hroughou he world and we considered only 0 emperaure ranges for each locaion, he - -

4 number of emperaure-locaion saes would be 0 50,000. If we would consider jus hree oher characerisics of each locaion wih each characerisic having 0 possible ranges, he number of possible saes would grow o 0 200,000. Ye, we have only begun o enumerae all he possible saes of naure. Oher characerisics would include diseases, erroris acs, volcanic aciviy, earhquakes, war, and much more. Enumeraing all saes of naure is clearly impracical. A furher problem is ha economic agens may no be able o conceive of all possible saes of naure. Even if hey could conceive of all possible saes of naure, he cos of wriing he legal documens o clearly define when a sae is considered o have occurred very well may be prohibiively expensive and oherwise impracical. Financial economiss presume ha insurance, fuures, opions, swaps, and oher derivaives are moving our economies owards having complee markes. However, his is a presumpion, no somehing ha has been logically proven heoreically. Answering his quesion for a compleely general economy is likely o be a very complex underaking. Insead, his paper focuses on pure-exchange economies wihou sorage. In such an economy, Eagle and Domian (2003 and 2004 show ha quasi-real bonds by hemselves complee markes as long as individuals have he same relaive risk aversion, no uiliy shocks occur, and individuals have raios of endowmen o real aggregae supply ha do no vary across saes of naure. This paper goes beyond he work of Eagle and Domian o discuss pure-exchange economies wihou sorage in general. In paricular he conclusions of his paper do apply o siuaions where individuals differ in heir relaive risk aversion, where heir raios of endowmen o real aggregae supply are sochasic, and where individual uiliy shocks occur. We define opimal conrac receips o be he real (inflaion-adjused amouns for each sae of naure ha an individual would receive (or pay if an economy wih complee sae

5 coningen securiy markes exised. We find ha he combinaion of four ypes of non-saeconingen securiies can approximae hese opimal conrac receips. Imporan o his finding are he following realizaions repored in his paper:. Pareo-efficien consumpion only varies wih respec o wo facors: (i real aggregae supply and (ii individual uiliy shocks. 2. Opimal conrac receips vary wih respec o only hree facors: (i real aggregae supply, (ii individual uiliy shocks, and (ii changes in he raio of endowmen o real aggregae supply. 3. When here are no individual uiliy shocks, he proporionaliy of an individual s Pareo-efficien consumpion o real aggregae supply depends only on how he individual s relaive risk aversion compares o average relaive risk aversion. Also, his paper derives he precise relaionship beween relaive risk aversion and how an individual s Pareo-efficien consumpion changes when real aggregae supply changes. All of hese resuls are imporan o help us undersand why he four ypes of conracs discussed in his paper will approximaely lead o complee markes. Secion II below reviews he sandard Arrow-Debreu pure exchange economy and presens and proves he consumpion-aggregae-supply-invariance propery, which is he propery upon which much of his paper s resuls res. This secion also proves and derives oher imporan resuls including he very imporan relaionship beween relaive risk aversion and how individuals Pareo-efficien consumpion varies wih real aggregae supply. Secion III hen discusses how normal conracs, endowmen-sharing conracs, and spending-sharing conracs complee he markes when all consumers have he same relaive risk aversion. Secion IV discusses how RASRT conracs could be designed o help consumers ransfer real-aggregaesupply risk among each oher if heir relaive risk aversions differ. Secion V discusses he pricing of RASRT conracs. Secion VI hen presens an example of how consumers could use all four conracs simulaneously o approximaely replicae heir opimal conrac receips

6 Secion VII summarizes his papers resuls and discusses how hese resuls have implicaions o real world economies. II. Arrow-Debreu Pure Exchange Economy wih Sae-Coningen Securiies This secion reviews a sandard Arrow-Debreu pure exchange economy wihou sorage consising of one nonsorable consumpion good. Assume each consumer j s ime-separable uiliy funcion is: U T n ( c + π su js ( c js = s= β ( where c is j s consumpion a ime 0, c js is j s consumpion in sae s a ime, β is he ime discoun facor, and π is he probabiliy of sae s occurring a ime. The funcions U c s ( and U c are coninuous, wice differeniable, sricly concave, and sricly increasing. To js ( js rule ou corner soluions, assume limu ( c = limu ( c = +. The ime frame for he s c 0 c 0 subscrip is deermined by he subscrip nex o he s subscrip. For example, he s in c js refers o one of he possible saes ha can occur a ime. A ime 0, consumers can buy or sell sae-coningen securiies. These sae-coningen securiies are prepaid securiies where he buyer pays he seller he price of he securiy a ime 0. Le x js represen individual j s demand a ime 0 for he sae-coningen securiy ha delivers one consumpion good a ime iff sae s occurs a ime. Define js Ω s so ha he price of his securiy equals P π Ω 0 s s. Wih i so defined, s Ω represens he real pricing kernel. Each consumer j chooses x js for all s and o maximize ( subjec o: P c T S + Pπ Ω x = P y (2 0 0 s s js 0 = s= - 4 -

7 c = y + x (3 js js js where (3 applies for all s=,2,,s for all =,2,,T where S is he finie number of saes of naure a ime. The marke clearing condiions are ha c j 0 = Y0, c js = Y s, and n j= n j= n x js j= = 0 for all saes s a ime and for =,2, T, where he aggregae supply of he consumpion good is represened by Y 0 a ime 0 and Y s in sae s a ime respecively. Consumer j s opimizaion problem is saisfied when which implies ha U ( c β π su js ( c js P 0 = P π Ω 0 s s for all s=,2,,s and for all =,2,,T, Ω s β U js ( c js = U ( c (4 The lef side of (4 is he real pricing kernel and he righ side is he ineremporal marginal rae of subsiuion. Some lieraure misakenly defines he pricing kernel as he ineremporal marginal rae of subsiuion (See, for example, Campbell, Lo, and MacKinlay, 997, p The equaliy beween he real pricing kernel and he ineremporal marginal rae of subsiuion shown in (4 is an equilibrium condiion no a definiion. Since his is a sandard one-good Arrow-Debreu pure-exchange economy wih well behaved uiliy funcions, a unique compeiive equilibrium exiss and ha compeiive equilibrium is Pareo efficien. Also, he following propery holds: Consumpion-Aggregae-Supply Invariance Propery: Le and 2 represen any wo differen saes of naure. If real aggregae supply and each consumer s uiliy funcion U js (. is he same in boh saes of naure (i.e., here are no uiliy shocks, hen every individual s consumpion will be he same in boh saes of naure

8 Proof by conradicion. Assume here is some consumpion allocaion in a compeiive equilibrium where for some saes and 2, each consumer s uiliy funcion is he same for boh saes and 2, Y =Y 2, and here are wo individuals j and k such ha c and ck ck 2 c < j j2 >. Since his is an Arrow-Debreu compeiive equilibrium, he consumpion ~ c + c ~ c + c. allocaion mus be Pareo efficien. Define ( c j 2 j j 2 and ( ck 2 k k 2 Define a new consumpion allocaion where for all consumers, for all saes of naure, and for all ime periods, he new consumpion equals he old consumpion excep ha j s consumpion in saes and 2 are boh c~ j and k s consumpion in saes and 2 are boh c~ k. The new consumpion allocaion is obviously feasible since he original allocaion was feasible. Because boh j and k are sricly risk averse, hey are boh beer off wih his new consumpion allocaion. However, ha conradics he saemen ha he original consumpion allocaion is Pareo efficien. We, herefore, conclude ha he consumpion allocaion mus be he same as long as neiher aggregae oupu nor he form of he uiliy funcions changes. Q.E.D. The consumpion-aggregae-supply-invariance propery is he foundaion for his paper. Before we discuss implicaions of his very imporan propery, le us firs disinguish beween aggregae uiliy shocks and individual uiliy shocks. Define aggregae uiliy shocks o be shocks o everyone s uiliy so ha individuals Pareo-efficien consumpion do no change as a resul. Individual uiliy shocks, on he oher hand, do affec no only he individual s Pareo-efficien consumpion bu also oher individuals Pareo-efficien consumpion. Wih his disincion made, we are now ready o discuss wo very imporan corollaries of he consumpion-aggregae-supply-invariance propery: Corollary : Le j represen any paricular individual in a pure-exchange economy wihou sorage. Individual j s Pareo-efficien consumpion a ime varies across saes of naure a ime only if changes occur in one of wo and only wo facors: These facors are (i real aggregae supply a ime, and (ii individual uiliy shocks a ime, eiher o individual j or someone else. The consumpion-aggregae-supply-invariance propery assumes no uiliy shocks and saes ha as long as real aggregae supply remains he same, an individual s Pareo-efficien - 6 -

9 consumpion will remain he same. Therefore, i immediaely follows ha Pareo-efficien consumpion only varies wih changes in real aggregae supply or uiliy shocks. However, by definiion, aggregae uiliy shocks do no affec Pareo-efficien consumpion. Hence, Pareoefficien consumpion varies only wih changes in real aggregae supply and individual uiliy shocks. Corollary 2: Individual j s opimal conrac receips a ime vary across saes of naure a ime only if changes occur in one of hree and only hree facors: (i real aggregae supply a ime, (ii individual uiliy shocks a ime, eiher o individual j or someone else, and (iii he raio of j s endowmen o real aggregae supply a ime. Remember how j s consumpion relaes o j s endowmen and j s holding of he relevan sae-coningen securiy. Equaion (3 saes ha c = y + x. Where individual j chooses js js js c js and x js opimally, x js will represen j s opimal conrac receips in sae s a ime. Therefore, he opimal conrac receips equal he difference beween j s Pareo-efficien consumpion and j s endowmen. By corollary above, if real aggregae supply does no change and no individual uiliy shocks occur, hen he Pareo-efficien consumpion does no change. Therefore, he only oher reason by which he opimal conrac receips will change will be if j s endowmen changes. Clearly, if real aggregae supply does no change, hen he raio of j s endowmen o real aggregae supply will change iff j s endowmen changes. Therefore, j s opimal conrac receips can only vary if real aggregae supply changes, individual uiliy shocks occur, or j s raio of endowmen o real aggregae supply changes. (We sae corollary 2 in erms of j s raio of endowmen o real aggregae supply raher han jus j s endowmen because i fis beer wih subsequen resuls concerning he relaionship beween relaive risk aversion and he proporionaliy of Pareo-efficien consumpion o real aggregae supply

10 Anoher imporan propery of he pure exchange economy wihou sorage is he relaionship beween relaive-risk aversion, Pareo-efficien consumpion, and real aggregae supply when here are no individual uiliy shocks. When no individual uiliy shocks occur, individual j s Pareo-efficien consumpion is a funcion solely of real aggregae supply. Define he implici funcion c ~ ( o be how he Pareo-Efficien consumpion by individual j a ime j Y depends on aggregae supply. Noe ha c~ ( is a reduced form; i is no he srucural j Y consumpion funcion. To help us avoid his confusion, we refer o Y as real aggregae supply a ime, no income. Define ~ j a j ( Y U ~ j j ( c~ j ( Y c ( Y U, which is he funcion of how he coefficien of absolue ( risk aversion varies wih real aggregae supply. Define ρ ( Y c ~ ( Y a~ ( Y ~ j j j, which is he funcion of how he relaive risk coefficien varies wih real aggregae supply. Also, define ρ m ( Y ( Y j= ~ρ j dc~ dy j, which is he weighed average of he relaive risk coefficiens using he derivaives of ~ ( c as he weighs. Finally, define α ( Y j Y relaive risk coefficien compares o he average relaive risk coefficien. ~ ρ ( Y ( Y ~ j j, which is how j s ρ Since equaion (4 is rue for all j, U ( ~ ( ~ j c j U c = U ( c U,0 ( c,0 (5 There is no jus one Pareo-efficien consumpion allocaion, bu raher a coninuum of such allocaions, each corresponding o a paricular allocaion of endowmens across saes. We can hink abou his Pareo-efficien consumpion allocaion as he one ha corresponds o he exising allocaion of endowmens

11 j ( c ~ for j=2..m. Toally differeniaing (5 wih respec o Y gives U ( c U j dc j U ( c dc ~ dy = U Dividing he lef and righ sides of his by he lef and righs sides of (5 respecively gives: U j ( c ~ j dc~ j U ( c ~ dc~ U ( c ~ = dy U ( c ~ (6 dy j j Subsiuing ~ j a j ( Y U ~ j j and rearranging slighly gives: dc~ j dy a~ dc~ = a~ j dy ( c~ j ( Y c ( Y U ino (6 and muliplying boh sides by a minus sign ( By summing boh sides of (7 over all consumers, we ge:,0 ~ ( c,0 ~ dy (7. m j= dc~ j dy dc~ dy = a~ m j= a~ j (8 By equilibrium in he marke for he consumpion good a ime, c j = Y, which also m dc implies ha dy j= j =. Therefore, solving (8 for imply ha he following is rue for all j. ~ dc dy gives dc~ dy = m j= m j= a ~ a~ j. This and (7 dc ~ dy j = m a ~ s= j a ~ s This resul was firs derived by Wilson (968, see his heorem 5. (9 Nex, we need o deermine he value of ρ ( Y. The following sars ou wih he definiion of ρ ( Y, hen subsiues in he definiion of j ( Y ρ ~ and he resul in (9: - 9 -

12 ρ m ( Y ( Y j= ~ρ j dc dy j = m j= c~ j a~ j a~ m k = j a~ k = m j= m k= c~ j a~ k However, he sum of consumpion across all consumers in his pure exchange economy equals aggregae supply for ha period. Therefore, ~ ρ j wih c~ Y ~ ( Y j = α j m gives us: ρ = m (0 c~ j= Y a ~ j ~ j ( From he definiion of ~ ρ Y α j, we can wrie ~ ρ = ~ j α j ρ and hen replace ρ Y c ~ a~ j j and j= a~ j a ~ j ( Y ρ wih (0 o ge (. Using (9, we can rewrie his as: c~ ( Y c~ j a~ j = ~ α j. Dividing boh sides by Y m j j= Y a ~ j a~ gives c ~ j dc~ j = ~ α j. Dividing boh sides by ~ α j Y dy j i j i = ~ ( α j ( Yi Yi Equaion ( is he very imporan relaionship beween how consumpion changes as real aggregae supply changes and how j s relaive risk aversion compares o average relaive risk aversion. The derivaive of j s Pareo-efficien consumpion wih respec o real aggregae supply equals a muliplier imes he proporion of one s consumpion o real aggregae supply. The muliplier is inversely relaed o how one s relaive risk aversion compares o he average relaive risk aversion. For example, if aggregae supply decreases by %, hen ( says ha he Pareo-efficien consumpion will decrease by half a percen for someone who has wice he - 0 -

13 average relaive risk aversion, whereas i will decrease by 2% for someone having half he average relaive risk aversion. By decreasing heir consumpion more han proporionaely, he lower (relaively risk-averse consumers are enabling he higher risk-averse consumers o reduce heir consumpion less han proporionaely. In essence, he lower risk-averse consumers are agreeing o absorb more of he risk concerning real aggregae supply so ha higher risk-averse consumers can absorb less risk. 2 The RASRT conracs discussed laer in his paper are designed o ry o mee he need o ransfer he risk relaed o possible changes in real aggregae supply. III. Compleing Markes When All Have The Same Relaive Risk Aversion In he previous secion, we showed ha in a pure exchange economy wihou sorage, a consumer j s opimal conrac receips vary across saes of naure only when (i real aggregae supply changes, (2 individual j s raio of endowmen o real aggregae supply changes, and (3 individual uiliy shocks occur, eiher o individual j or someone else. We also derived equaion ( which saes he precise relaionship of how relaive risk aversion solely deermines how Pareo-efficien consumpion and hence he opimal conrac receips vary wih real aggregae supply. These resuls from he previous secion are imporan because if a cerain se of conracs were o be able o replicae he opimal conrac receips, he real paymens on hese conracs would need o respond o individual uiliy shocks, sochasic changes in endowmen raios, and sochasic changes in real aggregae supply. Also, hese conracs would need o enable consumers o ransfer he real-aggregae-supply risk from he more relaively-risk-averse consumers o he less relaively-risk-averse consumers. 2 The relaionship in ( is relaed o one derived by Viard (993, alhough he assumed ha all income was derived from pas invesmens in risky asses whereas we assume all income comes from endowmens. - -

14 In his secion, we show how consumers can use (i normal conracs, (ii endowmensharing conracs, and (iii spending-sharing conracs o replicae heir opimal conrac receips when everyone has he same relaive risk aversion. The secion following his one hen discusses he RASRT conrac which would be needed when consumers have he same relaive risk aversion. We define normal conracs o be conracs whose real paymens are proporional o real aggregae supply. When he cenral bank of an economy keeps nominal aggregae demand from varying across saes of naure, nominal conracs behave as normal conracs. To see his, hink abou he equaion of exchange as MV=N=PY where M is he money supply, V is he income velociy 3 of money, N is nominal aggregae demand, P is he price level, and Y is real aggregae supply. If X is a cash flow a some fuure ime, hen he real value of his cash flow will be X / P. By he equaion of exchange, P N / Y =. Therefore, he real value of X will be X Y / N, which shows ha as long as N does no vary across saes of naure, he real value of X will be proporional o real aggregae supply. A cenral bank ha is rying o keep nominal aggregae demand from varying across saes of naure is following wha is currenly called nominal-income argeing. 4 If a cenral bank does no arge nominal income or nominal aggregae demand or if he cenral bank is unable o make nominal aggregae demand invarian o changes in saes of naure, nominal conracs will no behave as normal conracs. Quasi-real indexing as proposed by Eagle and Domian (995 is a way o make conracs behave as normal conracs even when 3 The erm income velociy of money is unforunaely he sandard erminology here. The erm income usually refers o aggregae supply no aggregae demand. However, he money supply imes velociy equals nominal aggregae demand, which equals nominal aggregae supply only in equilibrium. 4 A more effecive sraegy o ge nominal aggregae demand o be unaffeced by differen saes of naure would be o following nominal aggregae demand argeing insead of nominal income argeing. Once again, nominal income is associaed wih nominal aggregae supply no nominal aggregae demand. Therefore, if disequilibrium does occur, nominal aggregae demand argeing could be more effecive han nominal income argeing

15 he cenral bank does no pursue nominal income argeing or nominal aggregae demand argeing. A quasi-real-indexed conrac would ake a paymen X a ime and muliply i by N / N 0. Therefore, he nominal paymen would be X N / N 0 and he real paymen would be X N P N 0. Since Y = N / P by he equaion of exchange, he real paymen would be X Y N 0, which is proporional o real aggregae supply. Eagle and Domian (2003 and 2004 assume a pure exchange economy wihou sorage bu wih consumers having he same relaive risk aversion and he raios of endowmen o real aggregae supply ha do no vary across saes of naure. They show ha quasi-real bonds by hemselves complee he markes under hese assumpions. In oher words, under hese assumpions, normal conracs by hemselves enable consumers o replicae heir opimal conrac receips. Eagle and Domian (2003 and 2004 also assume no uiliy shocks. When individual uiliy shocks do occur or when consumers raios of endowmen o real aggregae supply are sochasic, or when consumers have differing relaive risk aversion, normal conracs will no longer enable consumers o replicae heir opimal conrac receips. This secion exends Eagle s and Domian s analysis o show ha endowmen-sharing conracs can handle sochasic raios of consumers endowmen o real aggregae supply, and spending-sharing conracs can handle individual spending shocks. The secion following his one hen shows ha RASRT conracs can approximaely handle he ransfer of real-aggregae-supply risk need when consumers have differen relaive risk aversion. To exemplify how normal conracs can replicae he opimal conrac receips when all consumers have he same relaive risk aversion and raios of endowmen o real aggregae supply ha do no vary across saes of naure and no individual uiliy shocks occur, assume a wo

16 period, pure exchange economy wihou sorage. Assume wo individuals exis, named A and B, where each consumer j maximizes his/her logarihmic uiliy funcion n 0 ln( c j 0 + βπ sε s ln( c js s= ε subjec o (2 and (3 where ε 0 and ε s represen aggregae uiliy shocks a ime 0 and in sae s a ime respecively. Assume only wo consumers exis, labeled consumer A and consumer B. A ime 0 assume real aggregae supply is 00, wih consumer being endowed wih 20 consumpion unis and he res going o consumer 2. A ime assume here are five saes of naure where real aggregae supply equals 30, 60, 90, 20, and 50 wih each sae equally likely. Assume he raio of consumer s endowmen o real aggregae supply is 40% in a ime, wih he res of real aggregae supply going o consumer 2. Table shows equaions for he consumpion demands and he real pricing kernels as well as he values for he relevan variables, including he demands for he sae-coningen securiies Equaions for Logarihmic Example: y + E[ Ω s y js] β ξ c =, y + E[ Ω s y ] s js c js =, E[ ξ s ] Ω ξ E[ ξ ] + β s 0 s + β ξ 0 ξ 0 Ω s β ξ s = ξ 0 Y 0 Y s Example Value Resuls: consumer A consumer B real agg. endowm consumpion x s endowm consumpion x 2s supply real ime ime : pricing sae kernel prob Table : Equaions and Values for Logarihmic Example - 4 -

17 ( x js. Remember ha he opimal conrac receips equal he demands for consumpion unis 45 line he sae-coningen securiies. Figure shows he Pareo-efficien consumpion and endowmen curves for boh A and B as well as he opimal conrac receips for curve for B and a 45-degree line, which represens he c B y B y A c A opimal conrac real aggregae supply Figure : Complee Picure of Opimal Conrac in Logarihmic Example magniude of real aggregae supply mus be spli beween A and B. The Pareo-efficien resul is ha B exchanges some of his endowmen in period 0 for some of A s endowmen in period. As can be seen he opimal conrac receips are proporional o real aggregae supply. This resuls because he consumers raio of endowmen o real aggregae supply are consan and because he logarihmic uiliy funcion causes boh consumers o have a coefficien of relaive risk aversion of one (See equaion (. If he raios of each consumer s endowmen o real aggregae supply are sochasic, hen endowmen-sharing conracs as well as normal conracs would be needed o complee markes. The purpose of endowmen-sharing conracs is o handle sochasic variaions in he raio of one s endowmen o real aggregae supply. Le R j be he insurance company s conracual endowmen raio for individual j a ime. The insurance company should se R j o be individual j s implici average endowmen raio a ime, where we define he implici average endowmen raio o be he consan endowmen raio ha resuls in he same expeced presen value of j s endowmen a ime as j s acual endowmens. Using he real pricing kernel as he real sochasic discoun facor, he expeced presen value of individual j s endowmens a ime - 5 -

18 is E [ Ω y * j* ], which equals π sω ime herefore equals: 5 S s= s y js. Individual j s implici average endowmen raio a R j E[ Ω* y j* ] = (2 E[ Ω Y ] * * By enering ino an endowmen sharing conrac, he insurance company would make a y j paymen o individual j when < R j, and individual j would pay he insurance company when Y y j > R j. The size of he paymen would be Y e j R Y y (3 j j This paymen is in real erms. When e j is posiive hen he insurance company pays ha o individual j. A negaive value means individual j pays he insurance company. Noe ha if we sum (3 over all individuals, we ge zero: m j= e j m = R jy y j= m j= j = Y Y = 0 (Noe ha he R j values mus add up o one. This means ha if all individuals enered ino hese endowmen-sharing conracs, he insurance company would face no aggregae risk. All risk-averse consumers would choose o ener hese endowmen-sharing conracs. If more han one insurance company exiss, hen he insurance companies could reinsure hemselves in order 5 If R j wha he raio for saes s hen Solving for R j. If his proporionaliy exised, hen E[ Ω * y j* ] = E[ Ω* R jy* ] = R j E[ Ω* Y* ] could deermine he value of To deermine his equivalen gives he formula for his equivalen y = R Y, and E y ] = E[ Ω R Y ] = R E[ Ω Y ] js j s js [ * j* * j * j * * Ω. y = R Y, and j s. In realiy, his proporionaliy may no exis. Neverheless, we R j ha would resul in he same expeced presen value as he acual endowmens. R, we need o solve he equaion E y ] = R E[ Ω Y ] j R j gives (2. [ * j* j * * Ω for R j. This - 6 -

19 han none of hem face any risk. This absence of risk is consisen wih zero economic profi o he insurance company. Theoreically speaking all risk-averse individuals will choose o ener ino an endowmen-sharing conrac. However, if some individuals choose no o ener ino such conracs, he insurance company could srucure is paymens in he following manner so i sill faces no risk: e js = R j Y s k Z k Z y Y R ks s k y js (4 where Z is he se of all insured invesors. Summing (4 over all individuals in Z gives j Z e js yks = k Z Y = s yks R j Ys y js Ys y k Z Rk k Z k Z Y s k Z k Z js = 0, which shows ha he insurance company will face no risk wih he adjused paymens. A hird ype of conrac o help individuals replicae heir opimal conrac receips are spending-sharing conracs. These conracs are needed o handle individual uiliy shocks ha cause eiher increases or decreases in an individual s spending needs. From he previous secion, we learned ha in a pure-exchange economy wihou sorage, one s Pareo-efficien consumpion varies wih respec o only wo facors: real aggregae supply and individual uiliy shocks. Normal conracs in conjuncion wih RASRT conracs handle variaions in real aggregae supply as we will discuss laer. Spending-sharing conracs will enable consumers o handle individual uiliy shocks. Since individual j s Pareo-efficien consumpion a ime varies across saes of naure solely wih real aggregae supply and individual uiliy shocks, we can wrie j s Pareo-efficien - 7 -

20 consumpion as he following funcion of real aggregae supply and individual uiliy shocks: c~ ( j Y s, ξ, ξ,..., ξ, where ξ ( 0, is he uiliy shock o individual j s uiliy s 2s ms consruced so ha ξ = represens no uiliy shock. 6 js js Noe ha j s Pareo-efficien consumpion is no only affeced by his/her individual uiliy shocks bu by oher individual uiliy shocks as well. For example if individual s experienced a posiive uiliy shock a ime causing individual o need o spend more ha period; he Pareoefficien soluion is for oher individuals such as individual j (assuming j o spend less so ha individual can spend more. A real life example of such a uiliy shock would be if individual had a healh-care emergency causing her spending o sharply increase. The Pareoefficien soluion is for individuals who did no have healh-care issues o help individual. To do so, oher individuals will need o spend less so ha individual can spend more on he healhcare emergency. Theoreically, an individual j s spending-sharing conrac receips will equal: ω = c~ ( Y, ξ, ξ,..., ξ c~ ( Y,,,..., (5 js j s s 2s ms j s where ω is j's spending-sharing conrac receips in sae s a ime, and c ~ (,,,..., is j s js Pareo-efficien consumpion wih no uiliy shocks. (This assumes ha he uiliy shocks are defined so ha a uiliy shock of one means no uiliy shocks. j Y s Figures 2 and 3 help show some differences beween endowmen-sharing conracs and spending-sharing conracs. Figure 2 shows he effec of he endowmen-sharing conrac receips or paymens which move he individual from he kinked curve wih he endowmen shocks o he sraigh line wihou he endowmen shocks. The endowmen-sharing conrac 6 In order for hese uiliy shocks o be well defined, some condiion needs o exis. I surmise ha his condiion would be somehing like ha ] E Ω ~ mus be he same wheher wih or wihou uiliy shocks. [ c * j* - 8 -

21 receips or paymens resul wih he resources of he individual being moved o where hose resources are proporional o real values j s endowmen C real aggregae supply. The reason for his is ha unexpeced increases in individual A B slope = y E Y j endowmens is no a jusificaion for an individual o consume more or less in Pareo-efficien sense. If markes are real aggregae supply Figure 2: Endowmen-Sharing Conrac Receips (Paymens complee, individuals will agree in advance o make some paymens when heir endowmens in some saes are greaer han expeced so ha hey will receive paymens when heir endowmens in oher saes are less han expeced. The effec of he endowmen-sharing conrac receips or paymens is o make he sum of one s endowmen wih hese receips or paymens o be proporional o real aggregae supply. On he oher hand, spending-sharing conrac receips move one from smooh consumpion o kinked consumpion as shown in Figure 3. Someimes an individual s Pareoefficien consumpion will be greaer han oher imes because of uiliy shocks. If markes are complee, an individual would real values j s P.E. consumpion be willing o make some paymens when his/her Pareo-efficien consumpion is less han normal so ha he/she can receive slope = c j E Y paymens when his/her Pareo-efficien consumpion is greaer han normal. real aggregae supply Figure 3: Spending-Sharing Conrac Receips (Paymens - 9 -

22 Wih rue spending-sharing conracs, adverse selecion is no heoreically an issue. Because a spending-sharing conrac is based on one s Pareo-efficien consumpion wih and wihou uiliy shocks, if hose Pareo-efficien consumpion levels are accuraely assessed, all risk averse individuals will choose o paricipae in he insurance. Such universal spendingsharing conracs would pose no risk o he insurance company. To see his, noe: m j= ω js m = ~ (,,,..., ~ c j Ys ξs ξ 2s ξ ms c j ( Y j= m j= s,,,..., = Y s Y s = 0. Thus, if one insurance company provided everyone wih hese spending-sharing conracs, he aggregae spending-sharing-conrac receips would equal 0. This resul is based on he equilibrium condiion ha he sum of Pareo-efficien consumpion over all consumers equals real aggregae supply regardless wheher ha Pareo-efficien consumpion is wih or wihou uiliy shocks. IV. RASRT conracs When consumers have differen relaive risk aversion, no longer will heir Pareo-efficien consumpion unis 45 line c B y B consumpion be proporional o real aggregae supply. Figure 5 shows how he Pareo-efficien y A consumpion will vary for individual A and B when c A A has greaer han average relaive risk aversion and B has less han average relaive risk aversion. (More precisely, his example assumes A and B have coefficiens of relaive risk aversion of 2 and ½ respecively. As equaion ( predics, A s real aggregae supply Figure 5: A s and B s consumpion when A has greaer han average relaive risk aversion and B has less han average relaive risk aversion

23 Pareo-efficien consumpion will change less han proporionally o changes in real aggregae supply whereas B s Pareo-efficien consumpion will change more han proporionally o changes in real aggregae supply. This is cleares a he inersecion of A s and B s consumpion curves. If real aggregae supply decreases, B decreases his consumpion more han proporionaely so ha A can decrease her consumpion less han proporionaely. Because of her high relaive risk aversion, individual A appreciaes ha B is willing o do his. In reurn for having o decrease his consumpion more han proporionaely when real aggregae supply decreases, B will increase his consumpion more han proporionaely when real aggregae supply increases. Basically, A has agreed hrough complee markes o benefi less han proporionaely when real aggregae supply increases in reurn for her being able o decrease her consumpion less han proporionaely when real aggregae supply increases. Since A has higher relaive risk aversion han does B, boh A and B are beer off wih his arrangemen han if hey proporionaely shared in changes in real aggregae supply. To ry o enable consumers o ransfer real-aggregae-supply risk among hemselves, his paper invens RASRT conracs. (RASRT sands for Real-Aggregae-Supply-Risk-Transfer. We worked wih wo differen ypes of RASRT conracs, sraigh and curved. Because hese RASRT conracs do no perfecly mee each individual s needs, we will discuss wo mehods consumers could use o approximae heir risk-ransfer needs. If an individual uses he angency mehod, hey will ener ino RASRT conracs so heir resuling consumpion is angen o heir Pareo-efficien consumpion. A second mehod an individual could use is he minimumvariance mehod, which minimizes he expeced squared deviaions of heir resuling consumpion from heir Pareo-efficien consumpion

24 The simples RASRT conrac is a sraigh RASRT conrac. The seller of he sraigh RASRT conrac agrees o pay he buyer an amoun equal o b( F Y where b is some posiive consan specified in he conrac, F is he price of he RASRT conrac, and Y is real aggregae supply. If b( F Y <0, hen he buyer will pay he seller he amoun. The amoun, ( F Y b, is in real erms since opimal conrac receips are in real erms. The sraigh RASRT conrac has similariies wih many fuures conracs currenly in use, excep ha no fuures conracs currenly deal wih real aggregae supply. There are some problems wih sraigh RASRT conracs. The firs problem is bankrupcy. In Figure 5, we would ploed he Pareo-efficien consumpion as a funcion of real aggregae supply for individuals A and B who had coefficiens of relaive risk aversion of wo and ½ respecively. Figures 6, 7, and 8 coninue wih his example, bu incorporae differen RASRT conracs. Figure 6 shows A s consumpion for boh he angency mehod and he minimum-variance mehod of deermining he number of RASRT conracs. Regardless wheher she uses he angency or minimum-variance mehods, individual A will consume a posiive amoun when real aggregae supply is zero if she relies on sraigh RASRT conracs. However, ha canno be possible in an economy wihou sorage. When we look a Figure 7 we see ha consumer B s use of a sraigh RASRT conrac will cause him o consume a negaive amoun when real aggregae supply equals zero also regardless if he uses he angency mehod or he minimum-variance mehod. Since i is impossible o consume a negaive amoun, his means ha B will be unable o fulfill his sraigh RASRT conracual obligaions when real aggregae supply is quie small. In oher words, sraigh RASRT conracs will lead o B becoming bankrup a low levels of real aggregae supply, which will hen impac A as well

25 Anoher problem wih sraigh RASRT conracs is ha sraigh RASRT conracs usually resul wih an individual s consumpion differing significanly from one s Pareo-efficien consumpion. Figures 6 and 7 shows ha ha he consumpion resuling from he sraigh RASRT conracs for A and B depar significanly from heir A s consumpion wih sraigh RASRT conrac wih angency mehod or minimum variance mehod consumpion unis A s Pareo-efficien consumpion real aggregae supply Figure 6: A s consumpion wih sraigh RASRT conrac compared o Pareo-efficien consumpion under angency mehod Pareo-efficien consumpion for boh mehods.. The curved RASRT conracs deal wih boh of hese problems. In general, a curved RASRT conrac akes he form of ( f F f ( Y b where b is a posiive consan and f is an ( increasing funcion. A special case is f ( Y = Y which is he sraigh RASRT conrac. Two oher special cases are f ( Y = Y, which we will call he SQRT RASRT conrac, and is Y = + f ( Y LN, which we will call he LOG RASRT conrac, where Y is iniially se equal Y o E Y ]. [ Figure 8 shows A s consumpion using he SQRT RASRT conrac using he angency mehod of consumpion unis B s Pareo-efficien consumpion B s consumpion wih sraigh RASTR conracs under minimumvariance mehod and angency mehod deermining he amoun of hese RASRT conracs o demand. Also shown is A s consumpion using he real aggregae supply Figure 7: B s consumpion wih sraigh RASRT conrac compared o Pareo-efficien consumpion under angency mehod

26 LOG RASRT conrac. While he SQRT RASRT conrac sill resuls wih A consuming more han Pareo-efficien when real aggregae supply differs from is expeced value, he oversaemen is subsanially reduced. The LOG RASRT conrac does even beer han he SQRT RASRT conrac when real aggregae supply is less han is expeced value, bu does very poorly when real aggregae supply is greaer han expeced. To see how differen RASRT conracs can perform for a variey of consumpion unis A s consumpion wih sraigh RASRT conrac A s consumpion wih Y RASRT conrac A s Pareoefficien consumpion A s consumpion wih LN(+Y /E[Y ] RASRT conrac real aggregae supply Figure 8: Pareo-efficien consumpion compared wih consumpion using Sraigh and curved RASRT conracs differen disribuions of endowmen and relaive-risk aversions, I sudied several examples involving 8 individuals each of whom had CRRA uiliy funcions, 7 bu wih differen coefficiens of relaive risk aversion and wih differen endowmen raios. 8 I looked a six differen scenarios, which are described in Table 2. For all scenarios, he possible values of real aggregae supply were 3, 6, 9,, 47, 50 each wih a 0.02 probabiliy. I used numerical echniques o deermine he expeced variance of he combinaion of he RASRT conrac receips (or paymens from he opimal conrac receips (or paymens. These resuls are presened in Table 3. 7 Some may criicize my use of CRRA uiliy funcions. However, wha is imporan according o equaion ( is how one s relaive risk aversion compares o average relaive risk aversion. Tha comparison based on individuals having CRRA uiliy funcions is likely o be similar o wha would occur if consumers have oher reasonably behaved uiliy funcions. 8 Tha I assumed individuals had endowmens ha were a consan proporional of real aggregae supply was necessary so ha individuals no need endowmen-sharing conracs o obain heir opimal conrac receips. The issue of endowmen-sharing conracs will be discussed in he nex chaper

27 For each RASRT conrac, he percenages represen he expeced resuling variance as a percenage of he variance ha would have resuled if no RASRT conracs exised; only nominal conracs exised. The firs percenage wihou he Scenario I Scenario III end. coefficien of relaive risk aversion coeff. of endowmen raio raio r.r.a A B C J K L 0.8 A D G J M P D E F M N O B E H K N Q.2 G H I P Q R.2 C F I L O R Scenario II Scenario IV end. coefficien of relaive risk aversion coeff. of endowmen raio raio r.r.a A B C J K L 0.8 A D G J M P D E F M N O B E H K N Q.2 G H I P Q R.2 C F I L O R Scenario VI Scenario V end. coefficien of relaive risk aversion coeff. of endowmen raio raio r.r.a A B C J K L 0.5 A D G J M P D E F M N O B E H K N Q.4 G H I P Q R 2 C F I L O R Table 2: Scenario Deails parenheses is he resul when I used he minimum variance mehod o deermine he conracs purchased or sold. The percenage in parenheses is he expeced variance when I used he angency mehod. For example, if following he minimum variance mehod and consumers use only sraigh RASRT conracs, hey will be able o reduce he expeced variance o abou 27% of wha i would have been wih no RASRT conracs. On he oher hand, he individuals following he angency mehod using sraigh RASRT conracs would have only reduced his expeced variance o 59% of wha ha expeced variance would have been wihou any RASRT conracs. The SQRT RASRT conrac was able o do much beer wih he expeced variance falling o 6.84% under he minimum variance mehod and 3.08% under he angency mehod. Beer sill were he RASRT conracs where Y = + f ( Y LN and Y f =. For boh of 0.99 ( Y Y hese conracs, individuals in Scenario I were able o reduce he expeced variance beween heir consumpion wih RASRT conracs and heir Pareo-efficien consumpion o less han % he

28 expeced variance had no RASRT conracs exised, assuming hey followed he minimum variance approach. Even if hey followed he angency approach, Scenario Scenario Scenario Scenario Scenario I II III IV V Sraigh Fuures 27.0% 34.76% 22.95% 22.95% 24.34% (59.0% (83.4% (49.5% (49.5% (53.5% SQRT(Y 6.84% 2.56% 4.30% 4.30% 5.45% (3.08% (23.4% (7.88% (7.88% (8.94% Y + LN Y 0.92% 2.74%.3%.3% 2.37% (2.06% (7.84% (2.6% (2.6% (6.03% Y^ % 4.35% 0.4% 0.4%.36% (2.3% (8.93% (0.30% (0.30% (2.86% Table 3: Minimum Variance Compared o Variance Wih Only Normal Conracs (Percenage in Parenheses represens raio under Tangency Mehod hey would be able o reduce ha expeced variance o less han 2.5% of wha would have exised wihou RASRT conracs. For he oher Scenarios, he resuls were similar, alhough for Scenario II he reducion of he expeced variance was less and for Scenarios III and IV, he reducion was more especially for he RASRT conrac where f = ( Y Y Table 3 shows ha when we look a he aggregae of hese expeced variances, curved RASRT conracs can be very successful a enabling individuals o approximaely replicae heir opimal conrac receips and herefore heir Pareo-efficien consumpion. However, i is imporan o consider how each individual was able o use hese RASRT conracs o mee heir needs. Table 4 shows how each individual fared wih sraigh RASRT conracs or wih he Y^0.99 RASRT conracs compared o no RASRT conracs a all when he individuals used he minimum variance mehod. The curved RASRT conracs enabled mos individuals o reduce heir expeced variance beween heir wih-rasrt consumpion and heir Pareo-efficien consumpion. One excepion is individual B in Scenario II, who was able o use sraigh RASRT conracs o reduce his expeced variance o 22.08% of wha i would have been wihou any

29 Sraigh RASRT conracs Y^0.99 RASRT Conracs Scenario Scenario Scenario Scenario Scenario Scenario Scenario Scenario Scenario Scenario ind. I II III IV V I II III IV V A 29.97% 37.54% 24.00% 24.00% 25.88% 0.95% 3.33% 0.06% 0.06% 0.29% B 3.65% 22.08% 44.42% 44.42% 57.59% 2.23% 39.83% 6.4% 6.40% 4.8% C 45.23% 60.04% 20.72% 20.72% 5.96% 6.8% 6.60% 0.04% 0.04%.25% D 29.97% 37.54% 24.00% 24.00% 25.88% 0.95% 3.33% 0.06% 0.06% 0.29% E 3.65% 22.08% 44.42% 44.42% 57.59% 2.23% 39.83% 6.4% 6.40% 4.8% F 45.23% 60.04% 20.72% 20.72% 5.96% 6.8% 6.60% 0.04% 0.04%.25% G 29.97% 37.54% 24.00% 24.00% 25.88% 0.95% 3.33% 0.06% 0.06% 0.29% H 3.65% 22.08% 44.42% 44.42% 57.59% 2.23% 39.83% 6.4% 6.39% 4.8% I 45.23% 60.04% 20.72% 20.72% 5.96% 6.8% 6.60% 0.04% 0.04%.25% J 45.23% 32.90% 24.00% 24.00% 25.88% 6.8%.87% 0.06% 0.06% 0.29% K 24.35% 2.89% 44.42% 44.42% 57.59% 0.08% 0.64% 6.4% 6.39% 4.8% L 8.29% 5.78% 20.72% 20.72% 5.96% 0.30%.53% 0.04% 0.04%.25% M 45.23% 32.90% 24.00% 24.00% 25.88% 6.8%.87% 0.06% 0.06% 0.29% N 24.35% 2.89% 44.42% 44.42% 57.59% 0.08% 0.64% 6.4% 6.39% 4.8% O 8.29% 5.78% 20.72% 20.72% 5.96% 0.30%.53% 0.04% 0.04%.25% P 45.23% 32.90% 24.00% 24.00% 25.88% 6.8%.87% 0.06% 0.06% 0.29% Q 24.35% 2.89% 44.42% 44.42% 57.59% 0.08% 0.64% 6.4% 6.39% 4.8% R 8.29% 5.78% 20.72% 20.72% 5.96% 0.30%.53% 0.04% 0.04%.25% Table 4: Individual Minimum Variance Resuls Compared o Normal Conracs Only RASRT conracs. Individual B did even worse in Scenario II when he used he Y^0.99 RASRT conrac as he was only able o reduce his expeced variance o 39.83% of he no-rasrt level. Figure 9 plos several individual s opimal conrac receips including individual B s. The curvaure of B s opimal conrac receips changes. For low levels of real aggregae supply, B s opimal conrac receips are convex, bu for higher levels hey are concave. I raced he reason o consumpion unis A how he average relaive risk aversion for he economy changed as real aggregae supply changed. Remember ha he average relaive 0 Figure 9: Seleced Individual s Opimal Conrac Receips in Scenario II C B K Y

30 risk aversion is a weighed average of he individuals coefficiens of relaive risk aversion. My invesigaion found ha he weighed average relaive risk aversion fell from.88 o 0.43 as real aggregae supply increased from 6 o 47. This resuled in hose individuals wih coefficiens of relaive risk aversion of 0.5,.0, and.5 changing from being below average relaive risk aversion o being above average relaive risk aversion. This was paricularly difficul for individual B as he swiched from having his 0.5 coefficien of relaive risk aversion iniially being well below average o becoming above average where real aggregae supply exceeded 09. This changed his opimal conrac receips from being convex o being concave wih respec o real aggregae supply. Since he curved RASRT conracs are eiher concave or convex depending wheher one sells or buys hem, he curved RASRT conracs worked worse for individual B han would have sraigh RASRT conracs. This problem of changing from being below average o above average relaive risk aversion was also rue for individuals C and J who had coefficiens of relaive risk aversion Individual C also experiences he swich bu a a lower level of real aggregae supply (40, which means ha he range of real aggregae supply for which C s opimal conrac receips are convex is relaively insignifican. This insignificance is even more for individual j whose swich occurs when real aggregae supply is 9. The issue of swiching is less an issue for Scenarios I, III, IV and V. For scenario I, he average relaive risk aversion drops from.09 o 0.87 resuling wih individuals C, F, I, J, M, and P experiencing only modes shifing. For Scenarios III and IV, he relaive risk aversion only drops from.04 o 0.95, again causing only modes shifing among individuals B, E, H, K, N, and Q. For Scenario V, he relaive risk aversion does drop from.6 o 0.70, which causes

31 consumpion unis shifing o affec individuals B, E, H, K, N, and Q causing hese individuals o reain 4.8% of he expeced variance beween RASRT consumpion and Pareo-efficien consumpion. To furher invesigae he performance of curved A real aggregae supply Figure 0: Individual Opimal Conrac Receips in Scenario VI B C L Sraigh Y^0.99 RASRT RASRT individual conracs Conracs A 4.5% 5.47% B 82.88% 43.37% C 59.38% 6.83% D 4.5% 5.47% E 82.88% 43.37% F 59.38% 6.83% G 4.5% 5.47% H 82.88% 43.37% I 59.38% 6.83% J 44.84% 6.90% K 5.84%.88% L 7.38% 6.02% M 44.84% 6.90% N 5.84%.88% O 7.38% 6.02% P 44.84% 6.90% Q 5.84%.88% R 7.38% 6.02% Overall 4.95% 7.25% Table 5: Variance beween consumpion wih RASTR conrac and Pareo-efficien consumpion compared o no RASTR conrac under Scenario VI RASRT conracs, I consruced anoher scenario, Scenario VI, which is defined in Table 2 a he end of his chaper. Here he range of coefficiens of relaive risk aversion was even greaer han in Scenario II, ranging from 0. o 5. Table 5: shows how he expeced variance beween wih- RASRT consumpion and Pareo-efficien consumpion compare beween sraigh RASRT conracs and he Y^0.99 RASRT conrac. In Scenario VI, he curved RASRT conrac ouperforms he sraigh RASRT conrac for all individuals, alhough individuals B and C, E and F, and H, and individual I do experience problems of he shifing average relaive risk aversion. Figure 0 shows how he opimal conrac receips for B and C change wih real aggregae supply