Efficient Risk Sharing with Limited Commitment and Hidden Storage

Efficien Risk Sharing wih Limied Commimen and Hidden Sorage Árpád Ábrahám and Sarola Laczó March 30, 2012 Absrac We exend he model of risk sharing wih limied commimen e.g. Kocherlakoa, 1996) by inroducing boh a privae non-conracible and/or non-observable) and a public sorage echnology. We firs show ha agens Euler consrains are violaed a he consrained-efficien allocaion of he basic model under general condiions. We hen sudy a problem where agens defaul and saving incenives are boh aken ino accoun. We show ha when he planner and he agens have access o he same ineremporal echnology, agens no longer wan o sore a he consrained-efficien allocaion. The reason is ha he planner has more incenive o save, because she inernalizes he posiive effec of aggregae asses on fuure risk sharing by relaxing paricipaion consrains. Public sorage is posiive even when here is no aggregae uncerainy and he reurn on sorage is below he discoun rae. This is in conras o he case where hidden income or effor is he deep fricion ha limis risk sharing e.g. Cole and Kocherlakoa, 2001). We also show ha asses remain sochasic whenever only moderae risk sharing is implemenable in he long run, bu become consan if high bu sill imperfec risk sharing is he long-run oucome. However, if he reurn on he ineremporal echnolgy is as high as he discoun rae, perfec risk sharing is always self-enforcing in he long run. Furher, higher consumpion inequaliy implies higher public asse accumulaion. In erms of consumpion dynamics, we overurn hree counerfacual predicions of he basic limied commimen model. In paricular, he amnesia and persisence properies do no hold in our model when asses are sochasic in he long run. Furher, agens Euler inequaliies hold by consrucion. Keywords: risk sharing, limied commimen, hidden sorage, dynamic conracs JEL codes: E20 We hank Andy Akeson, Hugo Hopenhayn, Yang K. Lu, Alessandro Mennuni, Raffaele Rossi, and seminar and conference paricipans a SAET in Faro, Bank of Hungary, ESEM in Oslo, Universia Auònoma de Barcelona, UCLA, Universiy of Bonn, Insiue for Economic Analysis IAE-CSIC), RES PhD Meeing in London, Universia Pompeu Fabra, Paris School of Economics, Cardiff Business School, Federal Reserve Bank of S. Louis, and Universiy of Rocheser for useful commens and suggesions. All errors are our own. European Universiy Insiue, Deparmen of Economics, Villa San Paolo, Via della Piazzuola 43, 50133 Firenze FI), Ialy. Email: arpad.abraham@eui.eu. Universiy of California, Los Angeles, Deparmen of Economics, Bunche Hall 9367, Los Angeles, CA 90095, Unied Saes. Email: slaczo@econ.ucla.edu. 1

1 Inroducion In he exising models of risk sharing wih limied commimen, agens incenive o deviae from he consrained-efficien allocaion by saving is ignored. A he same ime, i has been shown ha hidden sorage reduces risk sharing in environmens wih privae informaion fricions, see Allen 1985), Cole and Kocherlakoa 2001), and Ábrahám, Koehne, and Pavoni 2011). To our knowledge, only observable and conracable saving has been considered in he limied commimen framework, see Ligon, Thomas, and Worrall 2000). In ha environmen, he opimal allocaion is condiioned on individual asse holdings, while his is no possible when savings are hidden. In several economic conexs where he model of risk sharing wih limied commimen has been applied, agens are likely o have a way o save, using some backyard echnology or simple sorage. In he conex of village economies Ligon, Thomas, and Worrall, 2002), households may keep grain or cash around he house for self-insure purposes. Households in he Unied Saes Krueger and Perri, 2006) may keep savings in cash or hide heir asses abroad. Spouses wihin a household Mazzocco, 2007) may also keep savings for personal use. In all hese examples he amoun of privae savings is no observable and/or conracible. This paper exends he lieraure on risk sharing wih limied commimen by inroducing a privae non-conracible and/or non-observable) sorage echnology ha agens have access o, as well as a public sorage echnology. Our saring poin is he wo-sided lack of commimen framework of Kocherlakoa 1996), ha we will ofen refer o as he basic model hereafer. Agens are infiniely lived, risk averse, and ex-ane idenical. They receive a risky endowmen each period. We assume ha here is no aggregae uncerainy. Agens may make ransfers o each oher in order o smooh heir consumpion. These ransfers are subjec o limied commimen, i.e. each agen mus be a leas as well off as in auarky a each ime and sae of he world. The sorage echnology we inroduce allows agens o ransfer consumpion from one period o he nex and earn a ne reurn r. 1 Borrowing is no allowed. Access o hidden sorage no only changes he value of auarky, bu i may also enlarge he se of possible deviaions along he equilibrium pah. Tha is, agens could defaul, or sore, or boh a he same ime or in differen periods. We firs sudy under wha condiions agens have an incenive o sore a he consrainedefficien allocaion of he basic model when sorage yields zero ineres. We refer o his 1 The reurn r can ake any value such ha 1 r 1 β 1, where β is he subjecive discoun facor. If r = 1, we are back o he basic model. We say ha he sorage echnology is efficien if r = 1 β 1. 2

ineremporal echnology as pure sorage. We show ha if income akes a leas hree values, he possibiliy of hidden pure sorage will affec he consrained-efficien allocaion direcly, i.e. agens Euler consrains will be violaed, whenever he soluion is no oo close o full insurance. Aferwards, we show ha whenever risk sharing is parial, here exiss a hreshold reurn on sorage, r < 1 1, above which he Euler consrain binds a he consrainedefficien allocaion. This means ha whenever he reurn on privae sorage is high β enough where some r 0 can be high enough), and he basic limied commimen model exhibis relaively lile risk sharing, he consrained-efficien allocaion in he basic model is no incenive compaible if agens have access o hidden sorage. In oher words, hidden sorage maers under general condiions. 2 The key inuiion is ha, in he consrained-opimal allocaion, agens wih high curren income and consumpion face a decreasing ineremporal paern of consumpion. canno be incenive compaible. If he reurn on sorage is high enough, his consumpion pah Aferwards, we describe our model, where agens incenive o defaul on ransfers and heir incenive o save are boh aken ino accoun, unlike in he previous lieraure. This means ha, in addiion o he paricipaion/enforceabiliy consrains ha make sure ha agens will no defaul, he Euler inequaliies of boh agens become consrains o rule ou deviaions by sorage, under he firs-order condiion approach. 3 We allow he social planner o save using he same ineremporal echnology as he agens. Wihou allowing he social planner o sore, he feasible se may be empy when agens have access o sorage in auarky. Public savings are also relevan for applicaions. Think of communiy grain sorage faciliies in developing counries or he European Financial Sabiliy Faciliy and he proposed European Sabiliy Mechanism for he euro area. In he res of his paper, we focus on he consrained-efficien allocaions in his environmen. Wih agens Euler inequaliies as consrains, he emporary Pareo weighs Marce and Marimon, 2011) or promised uiliies Abreu, Pearce, and Sacchei, 1990) are no longer sufficien saisics o summarize he pas hisory of income realizaions. In models wih a dynamic moral hazard problem, hidden sorage can be accommodaed in a recursive manner using he agen s marginal uiliy as a co-sae variable Werning, 2001; Ábrahám and Pavoni, 2008). In our environmen, his approach would raise serious racabiliy issues, because we would need wo more coninuous co-sae variables, in addiion o keeping rack of individ- 2 Noe ha his resul does no hinge on how exacly agens ouside opion is specified. In paricular, hey may or may no be allowed o sore in auarky, and hey may or no face addiional punishmen for defauling. 3 We will laer show ha he firs-order condiion approach is valid. 3

ual asse holdings and he co-sae variable needed o make he paricipaion consrains recursive. Insead of adding hese sae variables, we analyze he social planner s problem ignoring agens Euler consrains. This provides an imporan resul: if he social planner s Euler consrain is saisfied, which is a necessary condiion of opimaliy, agens no longer have an incenive o save. Therefore, he soluion of his simpler problem is idenical o he one where agens have a hidden sorage opporuniy. In oher words, hidden sorage no longer maers. The inuiion behind his resul is ha he planner has more incenive o save han he agens. Firs, he planner sores for he agens hereby eliminaing heir incenives o use he hidden ineremporal echnology, and second, sorage by he planner makes i easier o saisfy agens paricipaion consrains in he fuure. This is due o he fac ha, by increasing asses, he planner increases oal fuure consumpion available for agens if hey say in he risk sharing arrangemen. Since agens are excluded from he benefis of he public asse afer defaul, he higher he amoun of accumulaed public asses is, he lower agens incenives o defaul are. In oher words, he planner inernalizes he posiive effec of aggregae asses on fuure risk sharing, while he agens do no. I follows ha whenever hidden sorage maers, i.e. agens Euler consrains are violaed a he consrained-efficien soluion of he basic model wih sorage in auarky, he social planner uses he available sorage echnology, a leas when income inequaliy is highes. This happens even hough here is no aggregae uncerainy and he sorage echnology is inferior, i.e. he reurn on sorage is lower han he discoun rae or, β1 + r) < 1). Sorage by he planner makes he marke more complee in his environmen. Noe ha his is in conras o he case where privae informaion is he deep fricion ha limis risk sharing Cole and Kocherlakoa, 2001; Ábrahám and Pavoni, 2008). There, sorage by he planner would worsen he informaion revelaion problem, herefore i does no occur in equilibrium. We furher show ha whenever hidden sorage maers in he basic model wih sorage in auarky, he sock of public asses is someimes nonzero and is bounded in he long run. If β1 + r) < 1, asses remain sochasic in he long run if risk sharing is considerably limied in he sense ha paricipaion consrain of each agen binds a more han one income level. We show ha in his case here exiss an ergodic disribuion of asses, ha may or may no include zero. Oherwise, asses converge o a sricly posiive finie value. Furher, given inheried asses, sorage will always be higher when cross-secional consumpion inequaliy is higher. This paern holds for shor-run asse dynamics in general, i.e. also when asses are consan in he long run. The inuiion for hese resuls is ha he planner s and 4

agens ) saving incenives depend on he consumpion disribuion. Whenever each agen s paricipaion consrain binds for more han one income level in he long run, hese incenives vary over ime as he consumpion disribuion varies over ime. This is no rue when he paricipaion consrain binds only a he highes income level for each agen, hence asses converge o a seady-sae level in ha case. If β1 + r) = 1, i is opimal for he planner o fully complee he marke by sorage in he long run. Tha is, perfec risk sharing is self-enforcing in he long run in his case. This is because he rade-off beween imperfec insurance and an inefficien ineremporal echnology is no longer presen. The inroducion of sorage has ineresing implicaions for he dynamics of consumpion prediced by he model. Firs, he amnesia propery, i.e. ha he consumpion disribuion only depends on he curren income of he agen wih a binding paricipaion consrain and is independen of he pas hisory of shocks whenever a paricipaion consrain binds Kocherlakoa, 1996), does no hold when asses are sochasic in he long run. The inuiion behind his resul is ha he curren consumpion disribuion depends on boh curren income and inheried asses, even when a paricipaion consrain binds. In oher words, he pas hisory of income realizaions affecs curren consumpions hrough aggregae asses. Second, he persisence propery of he basic model, i.e. ha he consumpion allocaion does no change for small changes in he income disribuion, does no hold eiher when asses are sochasic in he long run. Even hough he sharing rule deermining he consumpion allocaion is he same as in he previous period when neiher paricipaion consrain binds, consumpion is only consan if ne savings are idenical in he previous and he curren period. This does no happen when asses are sochasic in he long run.s Daa on household income and consumpion suppor neiher he amnesia, nor he srong persisence propery of he basic model see Broer 2011) for an exensive analysis). Hence, hese differences are seps in he righ direcion for his framework o explain consumpion dynamics. Third, he Euler consrain, a leas in is inequaliy form, canno be rejeced in micro daa from developed economies once labor supply decisions and demographics are appropriaely accouned for Aanasio, 1999). Since in our model he agens Euler inequaliies are saisfied by consrucion, we bring limied commimen models in line wih his observaion as well. In his paper, we also propose an algorihm o solve he model numerically. Using our algorihm, we presen some compued examples o illusrae he effecs of he availabiliy of sorage on risk sharing. We also invesigae he consequences of changing he discoun facor and he reurn on sorage on he dynamics of asses and consumpion. We, however, leave 5

sudying he quaniaive implicaions of sorage for he dynamics of consumpion o fuure work. The res of he paper is srucured as follows. Secion 2 inroduces and characerizes he basic model wihou access o any ineremporal echnology. Secion 3 sudies wheher and under wha condiions Euler consrains are binding a he consrained-efficien allocaion of he basic model. Secion 4 inroduces hidden and public sorage and derives he main resuls of he paper. Secion 5 presens some compued examples. Secion 6 concludes. 2 The basic model wihou sorage We consider an endowmen economy wih wo ypes of agens, i = {1, 2}, each of uni measure, who are infiniely lived and risk averse. All agens are ex-ane idenical in he sense ha hey have he same preferences and are endowed wih he same exogenous random income process. Agens in he same group are ex-pos idenical as well, meaning ha heir income realizaions are he same a each ime. 4 Le u) denoe he uiliy funcion, ha is sricly increasing, sricly concave, and coninuously differeniable. Assume ha he Inada condiions hold. Le s denoe he income sae realized a ime and s he hisory of income realizaions, ha is, s = s 1, s 2,..., s ). Given s, agen 1 has income ys ), while agen 2 has income equal o Y ys )), where Y is aggregae income. Noe ha here is no aggregae uncerainy by assumpion. We furher assume ha income has a discree suppor, ha is, s { s 1,..., s j,..., s N} wih ys j ) < ys j+1 ), and is independenly and idenically disribued i.i.d.), ha is, Pr s = s j ) = π j,. The wo ypes of agens framework ogeher wih he no aggregae uncerainy assumpion imposes some symmery on boh he income realizaions and he probabiliies. In paricular, ys j ) = Y ys N j+1 ) and π j = π N j+1. The i.i.d. assumpion can be relaxed as long as weak posiive dependence is mainained, i.e. he expeced ne presen value of fuure lifeime income is weakly increasing in curren income. Suppose ha risk sharing is limied by wo-sided lack of commimen o risk sharing conracs, i.e. insurance ransfers have o be volunary, or, self-enforcing, as in Thomas and Worrall 1988), Kocherlakoa 1996), and ohers. Each agen may decide a any ime and sae o defaul and rever o auarky. This means ha only hose risk sharing conracs are susainable which provide a lifeime uiliy a leas as grea as auarky afer any hisory 4 We will refer o agen 1 and agen 2 below. Equivalenly, we could say ype-1 and ype-2 agens, or agen belonging o group 1 and group 2. 6

of income realizaions for each agen. We assume ha he punishmen for deviaion is exclusion from risk sharing arrangemens in he fuure. This is he mos severe subgameperfec punishmen in his conex. In oher words, i is an opimal penal code in he sense of Abreu 1988). The consrained-efficien risk sharing conrac is he soluion o he following opimizaion problem: where λ i max c i s ) 2 i=1 λ i =1 s β Pr ) s u c )) i s, 1) is he iniial) Pareo-weigh of agen i, β is he discoun facor, Prs ) is he probabiliy of hisory s occurring, and c i s ) is he consumpion of agen i when hisory s has occurred; subjec o he resource consrains, and he paricipaion consrains, r= s r 2 c ) i s Y, s, 2) i=1 β r Pr s r s ) u c i s r )) Ui au s ), s, i, 3) where Prs r s ) is he condiional probabiliy of hisory s r occurring given ha hisory s occurred up o ime, and Ui au s ) is he expeced lifeime uiliy of agen i when in auarky if sae s has occurred oday. In mahemaical erms, and U au 1 s ) = u ys )) + β 1 β U au 2 s ) = u Y ys )) + β 1 β N π j u ys j ) ) 4) j=1 N π j u ys j ) ). Noe ha he main qualiaive resuls would remain he same under differen punishmens as long as he sric monooniciy of he auarky value is mainained. For example, agens could save in auarky as in Krueger and Perri, 2006), or hey migh endure addiional punishmen for defauling from he communiy as in Ligon, Thomas, and Worrall, 2002). j=1 2.1 Characerizaion of he basic model This model has been well sudied in he lieraure. We briefly derive and describe he main characerisics, in paricular hose which urn ou o be imporan for our analysis of he 7

ineracion beween limied commimen and hidden sorage. Our characerizaion is based on he recursive Lagrangian approach of Marce and Marimon 2011). However, he same resuls can be obained using he promised uiliy approach see a exbook descripion in Ljungqvis and Sargen, 2004). Le β Pr s ) µ i s ) denoe he Lagrange muliplier on he paricipaion consrain, 3), and le β Pr s ) γ s ) be he Lagrange muliplier on he resource consrain, 2), when hisory s has occurred. Then, he Lagrangian can be wrien as L = β Pr ) { [ 2 s λ i u c )) i s =1 s r= s r i=1 +µ ) i s β r Pr )] s r s ) u c i s r )) Ui au s ) =1 +γ s ) Y i c i s ))}. Using he ideas of Marce and Marimon 2011), we can wrie he Lagrangian in he form L = β Pr ) { 2 [ ) s Mi s u c )) i s µ ) i s Ui au s ) ] =1 s i=1 +γ s ) Y i c i s ))}, where M i s ) = M i s 1 ) + µ i s ) and M i s 0 ) = λ i. The necessary firs-order condiion 5 wih respec o agen i s consumpion when hisory s has occurred is L c i s ) = M ) i s u c )) i s γ s ) = 0. Combining such firs-order condiions for agen 1 and agen 2, we have x s ) M 1 s ) M 2 s ) = u c 2 s )) u c 1 s )). 5) Here x s ) is he emporary Pareo weigh of agen 1 relaive o agen 2. 6 Defining υ i s ) = µ i s ) /M i s ) and using he definiions of x s ) and M i s ), we can obain he law of moion of x as x s ) = xs 1 ) 1 υ 2 s ) 1 υ 1 s ). 6) 5 Under general condiions, hese condiions are also sufficien ogeher wih he paricipaion and resource consrains. 6 To reinforce his inerpreaion, noice ha if no paricipaion consrain binds in hisory s for eiher agen, i.e. µ 1 s τ ) = µ 2 s τ ) = 0 for all subhisories s τ s, hen x s ) = λ 1 /λ 2 is he iniial relaive Pareo weigh of agen i. 8

Noice ha given c 2 s ) = Y c 1 s ), he sric concaviy of he uiliy funcion implies ha boh consumpion levels are uniquely deermined by x s ) defined in equaion 5). In paricular, we have ha c 1 s ) is a sricly increasing funcion of x s ). Noe also ha whenever neiher agen s paricipaion consrain binds, i.e. υ 1 s ) = υ 2 s ) = 0, hen he emporary relaive Pareo weigh, x, and consequenly he consumpion levels, remain consan. This implies ha for small changes in he income disribuion which do no rigger a binding paricipaion consrain he consumpion allocaion does no respond. This propery is ofen called an exreme form of persisence. In conras, when a paricipaion consrain binds, he relaive Pareo weigh is adjused. In paricular, he relaive posiion of he agen wih a binding consrain will improve, i.e. υ 1 s ) > 0 implies x s ) > xs 1 ), and υ 2 s ) > 0 implies x s ) < xs 1 ). Moreover, noice ha a his poin we jump o a new relaive Pareo weigh which makes him jus indifferen beween defauling or no given his curren level of income. I is easy o see ha his relaive Pareo weigh only depends on his curren income and no he pas hisory. This implies ha when he Pareo weighs, and hence consumpions, change beween periods, he new levels are solely deermined by he income level of he agen whose paricipaion consrain is currenly binding. The lieraure ofen refers o his propery as amnesia. The soluion of he model is fully characerized by a se of sae-dependen inervals on he relaive Pareo weigh, x, which give he possible relaive weighs in each income sae for agen 1. Denoe he inerval for sae s j by [ x j, x j]. x j is defined as he lowes relaive Pareo weigh such ha agen 1, wih income ys j ), is willing o say in he risk sharing arrangemen, and x j is he highes relaive Pareo weigh such ha agen 2, wih income Y ys j )), is indifferen beween paricipaing and defauling. In oher words, a he lower limi of hese inervals agen 1 s paricipaion consrain is binding, while a he upper limi agen 2 s paricipaion consrain is binding. Denoe by x he new relaive Pareo weigh of agen 1 o be found a ime. Suppose ha las period he raio of marginal uiliies was x 1, and oday he income sae is s j. The relaive weigh of agen 1 oday is deermined by he following updaing rule: x j if x 1 > x j x = x 1 if x 1 [ x j, x j]. 7) x j if x 1 < x j This means ha he raio of marginal uiliies is kep consan for any wo agens whenever his does no violae he paricipaion consrain of eiher of hem. When he paricipaion consrain binds for agen 1, he relaive Pareo weigh moves o he lower limi of he opimal inerval, jus making sure ha his agen is indifferen beween saying and defauling. 9

Similarly, when agen 2 s paricipaion consrain binds, he relaive Pareo weigh moves o he upper limi of he opimal inerval. Thereby i is guaraneed ha as much risk sharing as possible is achieved while saisfying he paricipaion consrains. Given ha consumpion of agen 1 is a sricly increasing funcion of x, exacly he same characerizaion holds for consumpion dynamics. Hence, nex we analyze he dynamics of consumpion direcly. Define he limis of he opimal consumpion inervals as c j : x j = u Y c j) u c j) and cj : x j = u Y c j ). u c j ) I is easy o see ha, unless auarky is he only implemenable allocaion, we have ha c j > c j, for some j. 7 Moreover, given ha he value of auarky is increasing in curren income see 4)), i is easy o see ha c j > c j 1 and c j > c j 1, for all 1 < j N. Finally, noe ha symmery implies ha c j found in Ljungqvis and Sargen 2004). = c N j+1. The proofs of all hese saemens can be Given hese resuls, we will illusrae he dynamics of consumpion using a graphical represenaion. We will focus on scenarios where he long-run equilibrium is characerized by imperfec risk sharing. We do his boh because here is overwhelming evidence from several applicaions households in a village or in he Unied Saes, spouses in a household, counries) abou less han perfec risk sharing, and because ha case is heoreically no ineresing, as i is equivalen o he well-known unconsrained) efficien allocaion of consan individual consumpions over ime. Using our noaion, perfec risk sharing is obained in he long run if x 1 x N, or, equivalenly, if c 1 c N. In oher words, if here exiss a Pareo weigh where neiher agen s paricipaion consrain is violaed for all income saes. Hence, we assume from now on ha c 1 < c N, i.e. risk sharing is imperfec in he long run. I is no difficul o see ha he law of moion described by 7) will imply ha risk sharing arrangemens subjec o limied commimen are characerized by a finie se of consumpion values deermined by he limis of he opimal consumpion inervals. I urns ou ha, for boh he benchmark model and our model wih hidden sorage, considering wo disinc scenarios is enough o describe he general picure: i) each agen s paricipaion consrain is binding only when his income is highes, and ii) each agen s paricipaion consrain is binding in more han one sae. Furher, o describe he consrained-efficien allocaions in hese wo scenarios, i is sufficien o consider hree income saes, i.e. N = 3. 7 If in auarky agens simply consume heir curren income every period, hen some risk sharing implies c j > c j, j. If sorage is allowed in auarky, hen i migh be ha no ransfer is made when cross-secional income inequaliy is high, bu insurance ransfers do occur in saes when inequaliy is low. 10

Consider an endowmen process where each agen ges y h, y m, or y l unis of he consumpion good, wih y h > y m > y l, wih probabiliies π h, π m, and π l, respecively. Symmery implies ha y m = y h + y l )/2 and π e π h = π l = 1 π m )/2, where he upper index e refers o he mos exreme, i.e. mos unequal, income disribuion. We will refer o a sae s j when agen 1 has income y j. Given he uiliy funcion and he income process, he inervals for differen saes may overlap or no depending on he discoun facor, β. If β is sufficienly large, hen perfec risk sharing is self-enforcing by a sandard folk heorem Kimball, 1988). In his case, c l c h, and perfec risk sharing is implemenable in he long run. If β is sufficienly small, here does no exis any non-auarkic allocaion ha is susainable wih volunary paricipaion. In his case, each consumpion inerval collapses o one poin, y j, j = {l, m, h}. For inermediae levels of he discoun facor, parial insurance occurs. If parial insurance occurs, here are wo possible scenarios depending in he level of he discoun facor. For higher levels of β, c m c h > c l c m. This means ha he consumpion inerval for sae s m overlaps wih he inervals associaed wih boh he s h and he s l sae. This is he case where each agen s paricipaion consrain binds for he highes income level only. Figure 1 presens an example saisfying hese condiions. Suppose curren consumpion of agen 1 is below c h. When agen 1 draws a high income realizaion which occurs wih probabiliy 1 in he long run), his consumpion jumps o c h. Then i says a ha level unil his income jumps o he lowes level. A ha period, agen 2 s paricipaion consrain binds, because he has high income, and consumpion of agen 1 will drop o c l. Then we are back o where we sared from. This implies ha consumpion akes only wo values, c h and c l in he long run. When consumpion changes, i always moves beween hese wo levels, and he pas hisory of income realizaions does no maer. This is he way he aforemenioned amnesia propery presens iself in his case. When sae s m occurs afer sae s h or sae s l, he consumpion allocaion remains unchanged. Tha is, consumpion does no reac a all o his small change in income. This is he aforemenioned persisence propery. Noe ha consumpion also remains unchanged over ime if he sequence h, m, h) or he sequence l, m, l) akes place. Anoher key observaion here is ha, alhough individuals face consumpion changes over ime, he consumpion disribuion is consan over ime. In every period, half of he agens consume c h and he oher half consume c l. Finally, noe ha exacly his case occurs for any N if c 2 c N > c 1 c N 1. For lower levels of β, none of he hree inervals overlap, i.e. c h > c m > c m > c l. Figure 2 11

Figure 1: The inerval for sae s m overlaps wih he inervals for sae s h and sae s l shows an example of his second case. When all hree inervals are disjunc, consumpion akes four values in he long run. To see his, noice ha he paricipaion consrain of agen 1 binds boh for medium and high level of income. Tha is, whenever his income changes his consumpion will change as well, and similarly for agen 2. In his second case, in sae s m he pas hisory deermines which agen s paricipaion consrain binds, herefore consumpion is Markovian. Curren incomes and he ideniy of he agen wih a binding paricipaion consrain fully deermine he consumpion allocaion. The dynamics of consumpion exhibi amnesia in his sense here. Furher, consumpion responds o every income change, hence he persisence propery does no manifes iself. The key observaion for laer reference is ha he consumpion disribuion changes beween {c m, c m } and { c l, c h}. Tha is, he cross-secional disribuion of consumpion is differen whenever sae s m occurs from when an unequal income sae, s h or s l, occurs. If here are N > 3 income saes, he cross-secional consumpion disribuion changes over ime whenever c 2 < c N and c 1 < c N 1. Depending on he number of income saes and 12

Figure 2: The sae-dependen inervals are all disjunc he number of saes where a paricipaion consrain binds, he persisence propery may appear. 3 Does hidden sorage maer? In his secion, we sudy wheher and under wha condiions agens would save a he consrained-efficien soluion of he basic model. We assume ha parial insurance occurs a he soluion he ineresing case). If Euler consrains are violaed, he soluion is no robus o deviaions when hidden sorage is available. We firs sudy he case where he ineremporal echnology is pure sorage, defined as an ineremporal echnology ha yields zero ineres. Then, we consider he oher benchmark case where he ineres rae is as high as he rae of ime preference, i.e. β1 + r) = 1, as well as he general case of any reurn on sorage. 13

3.1 Pure sorage When he available ineremporal echnology is pure sorage, i.e. r = 0, we sudy he wo cases we have idenified in he previous secion: i) only wo consumpion levels occur in he long run, i.e. he level of risk sharing is high, and ii) a paricipaion consrain binds for more han one income level for each agens, or, more han wo consumpion levels occur in equilibrium, i.e he level of risk sharing is more moderae. We show ha hey are indeed qualiaively differen, as hidden pure sorage will maer in he laer bu no in he former case. 3.1.1 Two consumpion levels in he long run Assume ha we are in a siuaion where, alhough income can ake N values, he opimal allocaion of he basic model feaures only wo levels of consumpion, c h > c l. This means ha a paricipaion consrain binds only when income inequaliy is highes. In paricular, if he agen s curren consumpion is c h, hen i remains c h as long as he oher agen s income does no reach he highes level. This happens when agen 1 s income level is lowes. Hence, he probabiliy of swiching o c l is equal o he probabiliy of he lowes income sae, denoed again by π e. Therefore, given ha an agen s consumpion is c h oday, π e is he probabiliy of swiching o c l, and 1 π e ) is he probabiliy ha his consumpion remains c h. I is easy o see ha agens would have an incenive o save/sore only if hey have high consumpion oday, because a ha poin hey face a consumpion profile which is weakly decreasing. Hence, hidden sorage maers a he consrained-opimal allocaion if agens Euler consrains are violaed when hey consume c h, ha is, if we have u c h ) < β [ 1 π e ) u c h ) + π e u c l ) ]. 8) Given ha c l = Y c h, i is clear ha here exiss a level of high consumpion, ĉ h, where 8) is saisfied wih equaliy: u ĉ h ) = β [ 1 π e ) u ĉ h ) + π e u Y ĉ h ) ]. 9) This is he case, because, on he one hand, as c h is approaching Y/2 and consequenly c l ), he lef hand side of inequaliy 8) is higher, given β < 1. On he oher hand, as c h is geing close o Y and consequenly c l o 0), he righ hand side of inequaliy 8) is higher, because lim c l 0 u c l ) = by he relevan Inada condiion. Hence, agens would like o use he pure sorage echnology in equilibrium if and only if c h > ĉ h. In wha follows we will show ha his never happens. 14

In order o do his, we firs show ha ĉ h is also he level of high consumpion which provides he highes life-ime uiliy, given ha he agen sars wih high consumpion. Lemma 1. ĉ h maximizes welfare for he agen wih high consumpion oday across all possible consumpion values if consumpion akes only wo values. Proof. The expeced lifeime uiliy of an agen who receives he high consumpion, denoed by θ h, a ime is V θ h) = u θ h) + β [ τ Pr c +τ = θ h) u c h) + 1 Pr c +τ = θ h)) u Y θ h)] τ=1 = u θ h) + V θ h) [ β1 π e ) + ] β τ π e ) 2 1 π e ) τ 2 τ=2 + u Y θ h) β τ π e 1 π e ) τ 1. τ=1 Then, V θ h) can be expressed as V θ h) = 1 1 β 1 β1 π e )) u θ h) + βπ e u Y θ h). 1 β1 2π e ) The level of consumpion, denoed by c h, which maximizes her lifeime uiliy is given by c h = arg max θ h 1 1 β 1 β1 π e )) u θ h) + βπ e u Y θ h). 1 β1 2π e ) The necessary and sufficien firs-order condiion for his problem is u c h ) = β [ 1 π e )u c h ) + π e u Y c h ) ]. 10) Comparing 9) and 10) he resul follows. Given his resul, i is inuiive ha i should never be opimal o implemen any consumpion level higher han ĉ h = c h. The following proposiion proves his saemen formally. Proposiion 1. If consumpion akes only wo values in he long run a he soluion of he basic model, hen agens have no incenive o use a pure sorage echnology. Proof. The proof is by conradicion. Assume ha c h > c h. Then we can consruc an alernaive allocaion which makes boh agens beer off and does no violae any paricipaion consrain. The alernaive conrac is given by c h. By consrucion, his level of consumpion makes he agen wih high consumpion oday beer off, since c h maximizes his lifeime 15

uiliy. The agen wih low consumpion is also clearly beer off, because he enjoys boh higher consumpion currenly, since c h > c h implies ha Y c h > Y c h, and a reducion of risk in he fuure, since he allocaion c h, Y c h ) is a mean preserving spread of c h, Y c h ). We have increased he lifeime uiliy of boh agens, herefore he paricipaion consrains canno be violaed. This implies ha we canno have a consrained-efficien allocaion such ha c h > c h. Then Lemma 1 and equaion 9) imply ha agens will no have an incenive o use a pure sorage echnology. Proposiion 1 means ha he opimal allocaion will always implemen such a low level of consumpion variaion ha agens will have no incenive o use a pure sorage echnology. I is worh noing ha he N = 2 case will always fall ino his caegory, because here canno be more han wo levels of consumpion in he long-run equilibrium. Second, as we discussed above, more generally his case represens risk sharing siuaions where a considerable amoun of risk sharing is achieved. Proposiion 1 shows ha his amoun of risk sharing is sufficien o eliminae agens sorage incenives, and asses are a heir firs-bes level, zero, in his environmen wih no aggregae risk. 3.1.2 More han wo consumpion levels in he long run Le us now consider he case where he paricipaion consrain binds for a leas wo income levels for each agen. Considering hree income saes is sufficien o derive our main resul. We use he noaion which was inroduced in Secion 2. The key assumpion here is ha a paricipaion consrain is binding in he inermediae sae s m as well, in addiion o he exreme saes, s h and s l, in he long-run equilibrium, see Figure 2. Consider now he following hypoheical consumpion allocaion: he high low) income agen consumes θ h Y θ h ) and in sae s m boh agens consume ȳ = Y/2. Le us now define ĉ h as he high consumpion level for which he agen s Euler holds wih equaliy if he consumes θ h, ȳ, and Y θ h in saes s h, s m, and s l, respecively. In mahemaical erms, ĉ h is he soluion o u ĉ h ) = β [ π e u ĉ h ) + π m u ȳ) + π e u Y ĉ h ) ]. 11) Noe ha his also means ha agens would use he pure sorage echnology in auarky as long as y h > ĉ h. Similarly, le us now define c h as he high consumpion level ha maximizes he lifeime uiliy of his agen. The firs-order condiion ha characerizes c h is u c h ) = β [ π e u c h ) + π e u Y c h ) ]. 12) The following lemma will be useful o esablish wheher he Euler consrain may bind a he soluion of he basic model wih hree income saes. 16

Lemma 2. ĉ h < c h when income akes hree values. Proof. I is enough o verify ha he Euler inequaliy is no saisfied a c h. The righ hand side of equaion 11) includes an addiional posiive erm, π m u ȳ), compared o equaion 12), hus he Euler consrain is violaed a c h. Noe ha his resul differs from wha we have found in he case where consumpion akes only wo values. In paricular, agens would use he ineremporal echnology no jus for consumpion levels above he level ha yields he maximum lifeime uiliy, c h, bu also below i as long as θ h > ĉ h. Using his resul for he consumpion process θ h, π e ; ȳ, π m ; Y θ h, π e), we can now sae he main resul of his subsecion. Proposiion 2. Assume ha income akes hree values, parial insurance occurs, and y h is sufficienly close o c h. Then, he Euler consrain binds a he consrained-efficien soluion of he basic model, even when he ineremporal echnology yields no ineres. Proof. Parial insurance occurs if y h > c h and U au s h ) > uȳ)/1 β). This follows from Krueger and Perri 2006), whose resuls for he wo-saes case are easy o generalize o hree saes. For y h > c h close o c h, c h < c h is in a small neighborhood of c h, and consumpion in sae s m is in a small neighborhood of ȳ. The resul follows from Lemma 2 by coninuiy. I is naural ha, if for some high consumpion level agens would deviae from he consrained-efficien allocaion by using he pure sorage echnology, hen his happens when he soluion is close o auarky, i.e. when sae-coningen insurance ransfers are small, and agens bear a lo of consumpion risk. This resul concerning pure sorage is ineresing because i is hard o exclude ha such a echnology is available o agens in several economic conexs ha he risk sharing wih limied commimen model has been applied o. For example, hink of households in poor villages or members of a household who can hide grain or cash. 3.2 Sorage wih any reurn In his secion we firs consider he benchmark case where agens have access o an efficien ineremporal echnology, i.e. sorage earns a reurn r such ha β1+r) = 1. Aferwards, we sudy he general case. As above, we only examine wheher agens would use he available hidden ineremporal echnology a he consrained-efficien soluion of he basic model. We do no make any assumpion abou he number of income saes, excep ha income may ake a finie number of values, and he suppor of he income disribuion is bounded. 17

Lemma 3. Suppose ha parial insurance occurs and he hidden sorage echnology yields a reurn r such ha β1 + r) = 1. Then he Euler consrain is violaed a he consrainedefficien allocaion when an agen receives he highes possible income, y N. Proof. If parial insurance occurs, as opposed o full insurance, hen i mus be ha here exiss some sae s j where he agen consumes c j < c N. Then, u c N ) < Prs j )u c j ), s j ha is, he Euler consrain is violaed. Proposiion 3. There exiss r < 1/β 1 such ha for all r > r, agens Euler consrains are violaed a he consrained-efficien allocaion of he basic model. Proof. r is defined as he soluion o u c N ) = β1 + r) Prs j )u c j ). 13) s j For r close o 1, he righ hand side is close o zero. By Lemma 3, he righ hand side is greaer han he lef hand side if r = 1 1. I is obvious ha he righ hand side is increasing β in r. Therefore, here is a unique r solving equaion 13), and agens Euler consrain is violaed for higher values of r. Having esablished ha Euler consrains bind for a sufficienly high r, where high can mean r = 0, in he nex secion we urn o he general problem o find he consrained-efficien risk sharing conrac saisfying boh paricipaion and Euler consrains. 4 The model wih sorage In his secion, we provide he formulaion and analyical characerizaion of our model wih limied commimen and hidden sorage. We add agens Euler consrains o he problem given by he objecive funcion 1) and he consrains 2) and 3). We also modify he resource consrain o allow he social planner o use he same ineremporal echnology as he agens. We mainain he assumpion of no aggregae uncerainy o exclude he precauionary moive for saving a he aggregae level, and o hus isolae he saving incenive in limied commimen models due o endogenously incomplee markes. 18

The social planner s problem is P1 ) max {c i s ),Bs )} s.. 2 i=1 λ i =1 s 2 c ) i s i=1 r= s r β Pr ) s u c )) i s 14) 2 y i s ) + 1 + r)b s 1) B s ), s, 15) i=1 β r Pr s r s ) u c i s r )) Ũ i au s ), s, i, 16) u c i s )) β1 + r) s +1 Pr s +1 s ) u c i s +1 )), s, i, 17) B s ) 0, s. 18) The social planner chooses he consumpion of each agen given any possible hisory of income saes, c i s ), i, and curren gross sorage given any possible hisory of income saes, B s ), and maximizes a weighed sum of agens lifeime uiliies. The firs consrain, 15), is he resource consrain, where B s 1 ) denoes asses inheried from he previous period. The nex consrain, 16), is he paricipaion consrain, where Ũ i au s ) is he value funcion of auarky when sorage is allowed. Equaion 17) is agens Euler consrain. Finally, 18) is he planner s borrowing consrain. A few remarks are in order abou his srucure before we urn o he characerizaion of consrained-efficien allocaions. Firs, agens can sore in auarky, bu hey lose access o he benefis of he public asse. 8 This implies ha Ũ i au defined as Vi au s j, b ) = max b { uy i s j ) + 1 + r)b b ) + β s j ) = V au i N k=1 s j, 0), where Vi au s j, b) is π k Vi au s k, b )}, 19) where b denoes privae savings. Since V au i s j, 0) is increasing decreasing) in j for agen 1 2), i is obvious ha if we replace for he auarky value in he basic model inroduced in Secion 2 wih he one defined here, he same characerizaion holds, given ha a soluion exiss. Second, we allow for public sorage exacly in order o assure ha a soluion exiss. In oher words, his assumpion makes sure ha he feasible se is nonempy. If neiher he social planner nor he agens can sore in he opimal conrac, bu agens can and would sore in auarky, hen no feasible soluion exiss for sufficienly low βs, i.e. when he soluion 8 This is he same assumpion as in Krueger and Perri 2006), where agens lose access o he benefis of a ree afer defauling. 19

of he basic model is close o auarky. Noe, however, ha since he reurn on public and privae sorage is he same, public sorage can replicae any allocaion which agens can achieve using privae sorage. Furher, equaion 19) implicily assumes ha public sorage does no affec he value of auarky. As menioned above, when agens defaul hey are no only excluded from fuure risk sharing bu also from he benefis of he public asse. This implies ha public sorage has an addiional benefi: i makes defaul a relaively less aracive opion. Las bu no leas, public sorage is relevan for applicaions as well. One example is he exising communiy grain sorage faciliies in low-income villages. Anoher example is he European Financial Sabiliy Faciliy and he European Sabiliy Mechanism o be launched), which can faciliae risk sharing across counries wihin he euro area. If a counry lef he euro area, i would lose access o hese resources. Third, we use a version of he firs-order condiion approach FOCA) here. Tha is, we only check single deviaions. In paricular, we check wheher he agen is beer off saying in he risk arrangemen or defauling given ha he does no sore condiion 16)), and we check wheher he is happy wih no soring given ha he does no defaul condiion 17)). I is no obvious wheher hese condiions are sufficien. 9 In principle, i is possible ha he agen sores in he curren period o increase his value of auarky in fuure periods, and defauls in a laer period. Our consrains do no ake such deviaions ino accoun. For now his is an assumpion. Given his assumpion, we will characerize he soluion. Aferwards, we will show ha agens indeed have no incenive o use hese more complex deviaions in Secion 4.2. Fourh, boh he paricipaion consrains 16) and he Euler consrains 17) involve fuure decision variables. Given hese wo ypes of forward-looking consrains, a recursive formulaion using eiher he promised uiliies approach Abreu, Pearce, and Sacchei, 1990) or he Lagrange mulipliers approach Marce and Marimon, 2011) is difficul. Euler consrains have been deal wih in models wih moral hazard and hidden sorage using he agen s marginal uiliy as a co-sae variable, see Werning 2001) and Ábrahám and Pavoni 2008). In our environmen, his could raise serious racabiliy issues, since we would need wo more coninuous co-sae variables, in addiion o he sae variables o keep rack of individual asse holdings. In his paper, we follow a differen approach ha avoids hese complicaions. In paricular, we solve he problem ignoring agens Euler consrains firs. Then we verify ha he soluion of he simplified problem saisfies hose Euler consrains. Tha is, insead of Problem P1, 9 In fac, Kocherlakoa 2004) shows ha in an economy wih privae informaion and hidden sorage he firs-order condiion approach can be invalid. 20

we solve he following simpler problem: max {c i s ),Bs )} P2 ) s.. 2 i=1 λ i =1 s 2 c ) i s i=1 r= s r β Pr ) s u c )) i s 2 y i s ) + 1 + r)b s 1) B s ), s, i=1 β r Pr s r s ) u c i s r )) Ũ i au s ), s, i, B s ) 0, s. Nex, we wrie Problem P2 in a recursive form. Le β Pr s ) µ i s ) denoe he Lagrange muliplier on he paricipaion consrain, 16), and le β Pr s ) γ s ) be he Lagrange muliplier on he resource consrain, 15), when hisory s has occurred, as in Secion 2. Then, he Lagrangian is L = β Pr ) { [ 2 s λ i u c )) 2 i s =1 s r= s r i=1 +µ ) i s β r Pr )] s r s ) u c i s r )) Ũ i au s ) +γ s ) 2 yi s ) c )) i s + 1 + r)b s 1) B s ))} i=1 Similarly as for he basic model, we can rewrie he Lagrangian as L = β Pr ) { 2 [ s M ) i s u c )) i s µ ) ] i s Ũi au s ) =1 s i=1 +γ s ) 2 yi s ) c )) i s + 1 + r)b s 1) B s ))}. i=1 where M i s ) = M i s 1 ) + µ i s ) and M i s 0 ) = λ i, as before. This Lagrangian is recursive wih curren income realizaions, inheried asses, and he relaive Pareo weigh from las period serving as sae variables, and he curren relaive Pareo weigh and sorage serving as conrols. 4.1 Characerizaion The firs-order condiion wih respec o agen i s consumpion when hisory s has occurred is i=1 L c i s ) = M ) i s u c )) i s γ s ) = 0. 20) 21

Combining such firs-order condiions for agen 1 and agen 2, and using he definiions of M i s ) and of x s ), we have x s ) = M 1 s ) M 2 s ) = u c 2 s )) u c 1 s )). 21) Remember ha υ i s ) = µ i s ) /M i s ). Then, he law of moion of x is as in he basic model. x s ) = xs 1 ) 1 υ 2 s ) 1 υ 1 s ), 22) The planner s Euler consrain, i.e. he opimaliy condiion for B s ) is γ s ) β1 + r) s +1 β Pr s +1 s ) γ s +1), 23) which, using 20), can also be wrien as M i s ) u c i s )) β1 + r) s +1 Pr s +1 s ) M i s +1 ) u c i s +1 )). Then, using 21) and 22), he planner s Euler becomes u c i s )) β1 + r) s +1 Pr s +1 s ) u c i s +1 )) 1 υ i s +1 ), 24) where 0 υ i s +1 ) 1. Given he definiion of υ i s +1 ) and equaion 22), i is easy o see ha 23) represens exacly he same mahemaical relaionship for boh agens. Comparing he planner s Euler, 24), o he sandard Euler consrains for he agens gives he following resul: Proposiion 4. When he planner s Euler is saisfied, he agens Eulers are saisfied as well. Therefore, he soluion of he model wih hidden sorage, P1, corresponds o he soluion of he simplified problem, P2. Proof. The planner s Euler is 24), while he agens Euler is u c i s )) β1 + r) s +1 Pr s +1 s ) u c i s +1 )). 25) The righ hand side of 24) is bigger han he righ hand side of 25), for i = {1, 2}, since 0 υ i s +1 ) 1, s +1. Therefore, 24) implies 25). Corollary 1. The planner sores in equilibrium whenever an agen s Euler consrain is violaed a he consrained-efficien allocaion of he basic model wih sorage in auarky. 22

Noe ha he public sorage echnology is used even hough here is no aggregae uncerainy and he echnology is inferior r < 1/β 1). Inuiively, his is opimal for wo inerrelaed reasons. Firs, given ha an agen would have an incenive o sore, he planner can eliminae i by public sorage. Noe ha when an agen has he highes level of consumpion, he planner s Euler is idenical o he agen s Euler. This is easy o see comparing 24) and 25). Hence, in his case, he only reason for public asse accumulaion is o sore for he high-income agen. This also implies ha his will be he only reason for using he available ineremporal echnology in he case wih a high level of risk sharing, i.e. where only wo consumpion levels occur in he long run. Second, sorage by he planner makes i easier o saisfy he paricipaion consrains nex period, hus i improves risk sharing in he fuure. This is he case whenever a paricipaion consrain binds in oher saes as well, no jus when income inequaliy is highes. Comparing 24) and 25) again, i is obvious ha he planner has more incenive o save han he agens in he oher saes. In paricular, he presence of 1/ 1 υ i s +1 ) 1 in he planner s Euler exacly indicaes how increasing asses helps he planner o relax fuure paricipaion consrains, and hereby improve fuure risk sharing. Given Proposiion 4, we can focus on solving problem P2, which will also give us he soluion of problem P1. Nex, we inroduce some useful noaion and show more precisely he recursive formulaion of P2, which we will use o solve he model numerically. Le y denoe he curren income of agen 1, and V ) denoe he value funcion. The following sysem is recursive using X = y, B, x) as sae variables: x X) = u Y + 1 + r)b B X) c 1 X)) u c 1 X)) x X) = x 1 υ 2X) 1 υ 1 X) u c 1 X)) β1 + r) y Pr y ) u c 1 X )) 1 υ 1 X ) 26) 27) 28) u c 1 X)) + β y Pr y ) V X ) Ũ au y) 29) u Y + 1 + r)b B X) c 1 X)) + β y Pr y ) V Y y, B, 1/x ) Ũ au Y y) 30) B X) 0. 31) The firs equaion, 26), says ha he raio of marginal uiliies beween he wo agens has o be equal o he curren relaive Pareo weigh, and where we have used he resource 23

consrain o subsiue for c 2 X). Equaion 27) is he law of moion of he co-sae variable, x. Equaion 28) is he social planner s Euler consrain, which we have derived above. Equaions 29) and 30) are he paricipaion consrains of agen 1 and agen 2, respecively, where Ũ au y) = Ũ au 1 s j ) and Ũ au Y y) = Ũ au 2 s j ), wih y = y j = y 1 s j ). Finally, equaion 31) is he planner s borrowing consrain. Given he recursive formulaion above, and noing ha he ouside opions Ũ au y) and Ũ au Y y) are monoone in y and ake a finie se of values, he soluion can be characerized by a se of sae-dependen inervals, as in he basic model. This is sraighforward in he for now hypoheical) case of consan and posiive level of public savings, B > 0. In his case, we can repea he same analysis as in Secion 2 wih only wo differences: i) he auarky value is Ũ i au ) insead of Ui au ), and ii) aggregae consumpion is given by Y + rb insead of Y. This implies ha he same updaing rule applies as in he basic model, see 7). In he case where public asses are no consan, he consumpion inervals naurally become a funcion of he changing level of available aggregae resources, Y + 1 + r)b. Neverheless, opimal sae-dependen inervals on he relaive Pareo weigh sill characerize he soluion, bu hey depend on B as well, no jus on curren income realizaions. To see his, noe ha we can express he value funcion in erms of he end of he period relaive Pareo weigh and he inheried asse level only. This is wihou loss of generaliy because of equaion 27). I follows ha he following condiions define he lower and upper bound of hese inervals: V y j, x j B), B) = Ũ au y j) and V ) Y y j 1, x j B), B = Ũ au Y y j). Hence, given he inheried Pareo weigh, x 1, and accumulaed asses, B, he updaing rule 7) needs o be modified as follows: x j B) if x 1 > x j B) x = x 1 if x 1 [ x j B), x j B) ] x j B) if x 1 < x j B). 32) Since his updaing rule is pracically he same as he one for he basic model compare 7) and 32)), we can deermine he level of risk sharing in his model as well, given B, by checking how much hese inervals overlap. Despie he fac ha he ransiion rules are remarkably similar o he benchmark case, here is a very imporan difference: as he aggregae sorage level changes, hese inervals also change over ime, which implies ha we do no necessary say a he limis of he opimal inervals even in he long run. To see his, noice ha whenever B > B, he relaive araciveness of auarky is reduced, hence 24

x j B) > x j B ) and x j B) < x j B ). This implies ha if he income sae is he same oday as yeserday, we have ha he x j B ) < x 1 < x j B ), hus x = x 1, and x may no be a he bound of any sae-dependen inerval given ha inheried asses are B. 4.1.1 The dynamics of aggregae asses Nex, we furher characerize he evoluion of aggregae asses. Given he law of moion of he relaive Pareo weigh defined above, his will no only provide us wih he dynamics of aggregae consumpion, bu also of individual consumpions. Recall ha he only reason for which he planner saves in he high risk sharing wo consumpion levels only) case is o preven he high consumpion agen from using privae sorage. In conras, when risk sharing is more limied, he paricipaion consrain binds a several levels of income. The planner s saving incenives vary across hese income levels boh because agens privae saving incenives vary and because a lower levels of income she has more incenives o save han he agens in order o relax fuure paricipaion consrains. The nex proposiion will show ha his difference is no so imporan for he shor-run dynamics of asses bu has very significan implicaions for he long-run dynamics. Before he main proposiion, we provide some useful properies of he policy funcion for sorage. Firs, noice ha 24) implies ha he opimal choice of asses is solely deermined by he end of period Pareo weigh and inheried asses, hence we can wrie B B, x ). The following lemma makes his saemen more precise: Lemma 4. B s j, B, x) = B B, x ). Tha is, for deermining aggregae sorage, he curren relaive Pareo weigh x is a sufficien saisic for he curren income sae, s j, and las period s relaive Pareo weigh, x. Proof. Once we know x, equaions 26) and 28), which do no depend on x, give c 1 and B. The nex lemma provides a key propery of he aggregae sorage decision rule. Lemma 5. B B, x ) is sricly increasing in x for x 1 and B B, x ) > 0. Tha is, he higher cross-secional consumpion inequaliy is, he higher public asse accumulaion is. Proof. A higher x 1 increases public sorage for wo complimenary reasons. Firs, higher consumpion inequaliy, which is uniquely deermined by he curren relaive Pareo weigh x, increases he privae saving incenive of he agen wih high curren consumpion. To make sure ha he high-consumpion agen does no deviae by hidden sorage, he planner has 25

o save more when x > 1 is higher. Second, remember ha he planner s addiional saving incenives come from he fac ha she aims o improve risk sharing reduce consumpion inequaliy) in he fuure. A higher x > 1 implies higher consumpion inequaliy omorrow. To see his, noe ha hree hings can happen omorrow wih respec o he paern of binding paricipaion consrains: i) no paricipaion consrain binds omorrow, hus consumpion inequaliy remains he same as oday, ii) agen 1 s paricipaion consrain is binding, hus consumpion inequaliy eiher remains higher for a higher x > 1 or no longer depends on x when comparing a higher and a lower x, and iii) agen 2 paricipaion consrain is binding, and eiher x > 1 and he same cases are possible as for ii), or x < 1 in which case consumpion inequaliy omorrow does no depend on x. Therefore, o reduce inequaliy omorrow, he planner has higher incenives o save when x > 1 is higher as well. We are now ready o characerize he long-run behavior of aggregae asses. Proposiion 5. i) B converges almos surely o a sricly posiive consan in he long run whenever he agens Euler consrains are violaed a B = B = 0 for some income disribuion, r < 1/β 1, and each agen s paricipaion consrain binds only when his income is highes in he long run. ii) B is sochasic and bounded in he long run whenever he agens Euler consrains are violaed a B = B = 0 for some income disribuion, r < 1/β 1, and each agen s paricipaion consrain binds in more han one income sae in he long run. iii) B converges almos surely o a sricly posiive consan B in he long run such ha B B perfec-risk sharing is self-enforcing, whenever he agens Euler consrains are violaed a B = B = 0 for some income disribuion and r = 1/β 1. If he iniial level of asses is above B, hen aggregae asses say consan a ha level. Proof. Firs, from Corollary 1 we know ha he planner s Euler consrain, 24), is violaed under he assumpion ha he agens Euler consrains are violaed a B = B = 0 for some x = ˆx, where ˆx is included in he long run ergodic se of relaive Pareo weighs. Therefore, B 0, ˆx) > 0. I is easy o see ha here exiss a high level of inheried asses, denoed ) B, such ha ) perfec ineremporal insurance is a leas emporarily enforceable, ha is, x 1 B x N B. Therefore, B B, x ) < B for all B B and x 1 B) x x N B), i.e. asses have o opimally decrease, because r < 1/β 1 by assumpion. This implies ha asses are bounded above in he long run. 26

For par i), we firs show ha here exiss a unique consan level of asses, B, such ha all opimaliy condiions are saisfied. Aferwards, we will show ha asses converge almos surely o B saring from any iniial level, B 0. Firs, recall ha if aggregae asses are consan, he opimal inervals for he relaive Pareo weigh are ime invarian. Given ha each agen s paricipaion consrain binds only for he highes income level in he long run, he opimaliy condiion 26) and x N B ) uniquely deermine c h B ), he ime-invarian high consumpion level. Then, using he planner s Euler, we can deermine he unique level of B such ha all opimaliy condiions are saisfied. The planner s Euler is u c h B ) ) = β1 + r) [ 1 π e ) u c h B ) ) + π e u c l B ) )]. Dividing boh sides by u c h B ) ), we obain 1 = β1 + r) [1 π e ) + π e u c l B ) ) ] u c h B )) = β1 + r) [ 1 π e ) + π e x N B ) ], 33) where we have used 26). Noe ha x N B ) is monoone and coninuous in B. Furher, a B = 0 he righ hand side of equaion 33) is larger han 1 by assumpion, and a B = B he righ hand side of 33) is smaller han 1, because x N B) = 1 and B < B. Therefore, we know ha here exiss a unique B where he planner s Euler is exacly saisfied by seing B = B = B. Nex, we show ha asses converge almos surely o B saring from any iniial asse level, B 0. We already know ha B B 0, x ) < B 0 for he ergodic range of x when B 0 > B, i.e. when perfec risk sharing is emporarily) self-enforcing, and B 0, x ) > 0 for some x in he ergodic range of x, since we have assumed ha agens Euler consrains are binding in he basic model. Consider B < B 0 < B firs, and assume ha sae N occurs, and agen 1 s paricipaion consrain is binding. This is wihou loss of generaliy, because his occurs wih probabiliy 1 in he long run, and he problem is symmeric across he wo agens. We know ha he righ hand side of 33) is smaller han 1, because x N B 0 ) < x N B ). Therefore, marginal uiliy omorrow has o increase relaive o marginal uiliy oday o saisfy he planner s Euler, herefore B B 0 ) < B 0. Wha happens nex period? The paricipaion consrain will bind again even if he same sae occurs. 10 This is because B B 0 ) < B 0 implies x N B B 0 )) > x N B 0 ). Then asses will decrease again. Wha if some s j wih 10 Noe ha his never happens in he basic model. 27

2 j N 1 occurs? We know ha he paricipaion consrains in hese saes are no binding for any B B, because hey are no binding for B. This means ha now x = x = x N B 0 ) < x N B B 0 )). Then, by Lemma 5, sorage will be lower han when he paricipaion consrain is binding. Noe ha if saes s 2,..., S N 1 occur repeaedly, asses will converge o a level below B. Then we are in he case where B 0 < B, ha we sudy nex. Consider 0 B 0 < B now, and suppose again ha sae N occurs, and agen 1 s paricipaion consrain is binding. We know ha x N B 0 ) > x N B ) in his case. Using 33) again, i follows ha B B 0 ) > B 0. Now, if he same sae occurs omorrow in fac, any s j wih j 2), hen he paricipaion consrain will be slack. This means ha now x = x = x N B 0 ) > x N B B 0 )). Then, by Lemma 5, sorage will be higher han when he paricipaion consrain is binding. This also implies ha if sae s 1 does no occur for many periods, asses converge o a level above B. Then once s 1 occurs, which happens wih probabiliy 1 in he long run, we are back o he case B 0 > B, and asses sar decreasing. 11 To see par ii), consider he case where in he long run here is a hird sae in which a paricipaion consrain binds. In his case, each agen s consumpion akes a leas four differen values in he long run. These have o saisfy an addiional paricipaion consrain, an addiional resource consrain, and an addiional Euler, which is generically impossible for consan B. To see par iii), noe ha when β1 + r) = 1, he only way o saisfy agens Euler consrains in all saes is o provide hem wih a perfecly smooh consumpion sream over ime. Furher, as long as a paricipaion consrain binds given B, he planner has an incenive o save more, because she does no face a rade-off beween improving risk sharing and using an inefficien ineremporal echnology. If he B 0 is higher han he minimum level necessary o saisfy all paricipaion consrains, B = B clearly saisfies boh he agens and he planner s Euler consrains, hus asses will remain consan a B 0 B. Proposiion 5 says ha aggregae asses will be consan in he long run if he crosssecional consumpion disribuion does no change over ime. This is wha happens no jus when consumpion akes only one value, bu also when i akes wo values in he long run. In he laer case half he agens consume c h and he oher half consume c l in each period, only he ideniy of he agen wih c h changes over ime. The social planner rades off wo 11 Paricipaion consrains in more saes may be binding when B is low, even if hey only bind in saes s 1 and s N for B. We know ha asses will increase in he wo mos unequal saes when B < B, herefore wih probabiliy 1 asses will reach a level where he paricipaion consrains of he oher saes are no longer binding. 28

effecs of increasing aggregae sorage: i is cosly because β1 + r) < 1, bu i is beneficial because i reduces consumpion dispersion in he fuure and discourages privae sorage. The seady-sae level of asses jus balances hese wo opposing forces. When paricipaion consrains bind in more saes, and consumpion has o ake more han wo values, he cross-secional consumpion disribuion will change over ime. Thus, he relaive srengh of hese forces also changes over ime, implying ha asses remain sochasic in he long run. The following proposiion shows how exacly public sorage varies wih he income and consumpion disribuion. Proposiion 6. B s j, B, x) B s k, B, x), B, x), where j N/2 + 1, k N/2, and j > k. The inequaliy is sric, i.e. B s j, B, x) > B s k, B, x), if he opimal inervals for saes s j and s k do no overlap given B. Tha is, given inheried asses and he relaive Pareo weigh las period, aggregae sorage is greaer when consumpion inequaliy is higher. Proof. From Lemma 4 we know ha B s j, B, x) = B B, x ). If j > k, and he opimal inervals for hese wo saes do no overlap given B, hen x mus be higher in sae s j han in sae s k. Then, he resul follows from Lemma 5. If he opimal inervals overlap given B, hen here exiss x for which x = x in boh saes s j and s k. I is clear ha aggregae savings are idenical in he wo saes in his case. Noe ha Proposiion 6 is relevan boh in he long run in he case where asses remain sochasic, and in he shor run in all cases before asses converge o B. Figure 3 illusraes he shor-run dynamics of asses in he case where asses are consan in he long run. The solid blue) line represens B B, x N B) ), i.e. we compue B assuming ha he relevan paricipaion consrain is binding. Suppose ha sae N occurs when inheried asses are a he level B 0 < B. Then public sorage will be B B 0, x N B 0 ) ). Nex period, if any sae s j wih j 2 occurs, no paricipaion consrain is binding, hus asses will be B B, x N B 0 ) ) > B B, x N B) ). This is represened by he do-dashed red) line. As long as sae s 1 does no occur, asses say on his line and evenually converge o he level B > B. Now, assume ha )) sae s 1 occurs when inheried asses are B. By symmery, sorage will be B B, x N B. Nex period, if eiher sae s 1 or s N occurs, he paricipaion consrain binds again, and asses follow he solid blue) line. If any oher sae occurs, asses will decrease more, and if his coninues happening, asses will approach a level lower han B no represened). We now characerize he bounds of he saionary disribuion of asses when hey are sochasic in he long run. disribuion of asses. Le B B ) denoe he lower upper) limi of he saionary 29

Figure 3: Shor-run asse dynamics when asses are consan in he long run Proposiion 7. The lower limi of he saionary disribuion of aggregae asses, B, is eiher sricly posiive and is implicily given by u c m B)) = β1 + r) N j=1 π j u c j B, x m B))) 1 υ j B, x m B)), 34) where he upper index m refers o he leas unequal income sae, or is zero, and he Euler consrain above holds as sric inequaliy. The upper limi of he saionary disribuion of aggregae asses, B, is implicily given by u c N B, x N B) )) = β1 + r) N π j u c j B, x N B) )). 35) Proof. From Proposiion 6 i is clear ha B will be approached if he leas unequal income sae, denoed s m, 12 happens repeaedly, while B can only be approached wih sae s N or 12 Noe ha s m refers o wo saes when N is even, s N/2 and s N/2+1. j=1 30

s 1 ) happening many imes in a row. If B is par of he saionary disribuion, hen i mus be ha B B. This means ha here are less and less resources available over ime while asses approach B, hus he relevan paricipaion consrain will always bind along his pah. Therefore, x = x m B) along his pah, and he planner s Euler is 34) if B > 0, or B = 0. 13 The upper limi of he saionary disribuion, B, is approached from below, hus along ha pah, he relevan paricipaion consrain is slack. 14 As a resul, when B converges o is upper limi, x x = x h B 1 ) where B 1 is he level of inheried asses when we swiched o sae s N or s 1 ). Denoe by B a level of asses where B migh converge o from below when sae s N occurs many imes in a row. B is he soluion o he following sysem: )) u c 1 B, x u c N B, x )) = x ) ) c N B, x + c 1 B, x = Ȳ + r B u c N B, x )) = β1 + r) N j=1 π j u c j B, x )). 36) When is B equal o B, he upper limi of he saionary disribuion? Using Lemma 5, we know ha B B, x) is highes when x is highes. A which asse level B 1 wihin he saionary disribuion of asses should we swich o sae s N in order o have x = x N B 1 ) he highes possible? This happens when B 1 is a he lower limi of he saionary disribuion, i.e. when B 1 = B. In ha case, x = x N B). Then, replacing for x in 36) gives 35). Figure 4 illusraes he dynamics of aggregae asses in he case where hey are sochasic in he long run. For simpliciy, we consider hree income saes. This means ha here are wo ypes of saes: wo wih high income and consumpion inequaliy saes s h and s l ) and one wih low income and consumpion inequaliy sae s m ). The solid red) line represens B B, x h B) ), i.e. sorage in sae s h or s l ) when he relevan paricipaion consrain is binding. Similarly, he do-dashed blue) line represens B B, x m B)), i.e. sorage in sae s m when he relevan paricipaion consrain is binding. Saring from B 0, if sae s m occurs repeaedly, asses converge o he lower limi of he saionary disribuion, B. The relevan paricipaion consrain is always binding along his pah, because inheried asses 13 We will give an example for each of hese cases in he nex secion, where we presen some compued examples. 14 I will become clear ha he upper limi is reached if sae s N or sae s 1 occurs many imes in a row, bu no if he economy alernaes beween hese wo saes. 31

keep decreasing. The dashed green) line represens he scenario when sae s h or sae s l ) occurs when inheried asses are a he lower limi of he saionary disribuion, B, and hen he same sae occurs repeaedly. This is when asses will approach he upper limi of he saionary disribuion, B. The relevan paricipaion consrain is no binding from he period afer he swich o s h, herefore sorage given inheried asses is described by he funcion B B, x h B) ). Finally, wihou loss of generaliy assume ha sae s l occurred many imes while approaching B, and suppose ha sae s h occurs when inheried asses are close o) B. In his case, x = x h B ) < x h B), and asses will decrease. They will hen converge o a level B from above wih he relevan paricipaion consrain binding along his pah. The same will happen whenever B > B when we swich o sae s h or s l ). B is implicily given by )) u c h B = β1 + r) N j=1 ))) π j u c j B, x h B. 4.1.2 The dynamics of individual consumpions Having characerized asses, we now urn o he dynamics of consumpion. One key propery of he basic model is ha whenever eiher agen s paricipaion consrain binds υ 1 X) > 0 or υ 2 X) > 0), he resuling allocaion is independen of he preceding hisory. In our formulaion, his implies ha x is only a funcion of s j and he ideniy of he agen wih a binding paricipaion consrain. This is ofen called he amnesia propery Kocherlakoa, 1996), and ypically daa do no suppor his paern, see Broer 2011) for he Unied Saes and Kinnan 2011) for Thai villages. Allowing for sorage helps o bring he model closer o he daa in his respec. Proposiion 8. The amnesia propery does no hold when aggregae asses are sochasic in he long run. Proof. x and hence curren consumpion depend on boh curren income and inheried asses, B, when a paricipaion consrain binds. This implies ha he pas hisory of income realizaions affecs curren consumpion hrough B. Anoher propery of he basic model is ha whenever neiher paricipaion consrain binds υ 1 X) = υ 2 X) = 0), he consumpion allocaion is consan and hence exhibis an exreme form of persisence. This can be seen easily: 27) gives x = x, and he consumpion allocaion is only a funcion of x wih consan aggregae income. This implies ha for 32

Figure 4: Asse dynamics when asses are sochasic in he long run small income changes which do no rigger a paricipaion consrain o bind, we do no see any change in individual consumpions. I is again no easy o find evidence for his paern in he daa. In our model, even if he relaive Pareo weigh does no change, 21) does no imply ha individual consumpions will be he same omorrow as oday. This is because 1 + r)b B X) is generically no equal o 1 + r)b B X ) when asses are sochasic in he long run. Proposiion 9. The persisence propery does no hold when aggregae asses are sochasic in he long run. Proof. Even hough x = x, when neiher paricipaion consrain binds, consumpion is only consan if ne savings are idenical in he pas and he curren period. This is no he case when B is sochasic. The las wo proposiions imply ha he dynamics of consumpion in he our model are 33

richer and closer o he daa han in he basic model. I is also imporan o noe ha, for developed economies, he Euler consrain canno be rejeced in micro daa, a leas in is inequaliy form, once labor supply decisions and demographics are appropriaely accouned for see Aanasio 1999) for a comprehensive review of he lieraure). Since in our model agens Euler inequaliy is saisfied by consrucion, we bring limied commimen models in line wih his observaion as well. 4.2 Validiy of he firs-order condiion approach Unil now we have assumed ha by inroducing he agens paricipaion and Euler consrains equaions 16) and 17), respecively) in Problem P1, we guaranee incenive compaibiliy. In oher words, we have assumed ha he consrained-opimal allocaion can be obained by checking ha agens have no incenive o defaul given ha hey do no save, and ha hey have no incenive o save given ha hey do no defaul. In principle, hey may sill find i opimal o use more complicaed double deviaions involving boh sorage and defaul, poenially in differen ime periods, given some hisory of income shocks. Firs, we need o consider conemporaneous join deviaions: he agen defauls and saves a he same ime. 15 Since in he paricipaion consrain 16) we use Ũ i au s ), he value of auarky when he agen can save see equaion 19)), his conemporaneous double deviaion is already aken ino accoun. Furher, noe ha in auarky he agen is allowed o save whenever his makes him beer off. Therefore, he defaul oday and save laer -ype of double deviaions are already aken care of as well. This implies ha he only poenially profiable double deviaions we sill need o consider are hose which involve privae asse accumulaion firs and defaul in a laer period. We demonsrae ha such deviaions canno be profiable in he simples possible case: only wo consumpion levels occur in he long run, c h and c l wih c h > c l and swiching probabiliy π e. I is no difficul o generalize he argumen for more consumpion levels. Le V h denoe he expeced lifeime uiliy of an agen who consumes c h oday. Since c h is pinned down by he binding paricipaion consrain of agens when heir income reaches is highes level, we know ha V h = V1 au s N, 0 ), where Vi au ) is defined in equaion 19). Now, we formally define he problem of an agen who is facing his consumpion process and has he opion of soring oday and defauling laer. We denoe he value funcion 15 In he lieraure wih privae informaion, a similar join deviaion, shirking or reporing a lower income) and saving, is he relevan deviaion. Deailed discussion of hese join deviaions can be found for he hidden acion dynamic moral hazard) case in Kocherlakoa 2004) and Ábrahám, Koehne, and Pavoni 2011), and for he hidden income case in Cole and Kocherlakoa 2001). 34

for an agen who is eniled o receive c h c l ) in he curren period, has b unis of asses accumulaed, and decides no o defaul oday by W h b) W l b)). These value funcions are defined recursively as and W h b) = max b 0 1 2 W l b) = max b 0 1 2 { uc h + 1 + r)b b ) + β [ N j=2 + π 1 max{w l b ), V1 au s 1, b ) ]} } { uc l + 1 + r)b b ) + β [ N 1 j=1 π j max{w h b ), V1 au s j, b ) } π j max{w l b ), V1 au s j, b ) } 37) + π N max{w h b ), V1 au s N, b ) ]} }. 38) We define he soluion of he above opimizaion problems as g h b) and g l b), respecively. Lemma 6. g h 0) = 0. Tha is, he agen assigned o consume c h oday will no save, even if defauling laer is an opion. Proof. Assume indirecly ha g h 0) > 0, ha is, he agen saves oday bu does no defaul. Two cases are possible: eiher i) he agen defauls in some saes) omorrow, or ii) he agen does no defaul in any sae omorrow bu he does laer. In case i), he agen mus defaul when his income is he highes possible omorrow, i.e. when he earns y N. Le c au y j, b) denoe he consumpion level chosen by he agen in auarky given ha his income is y j and he has accumulaed savings b. Remember ha π e = π N = π 1. Soring g h 0) oday and defauling omorrow if his income is y N, he agen s Euler inequaliy is u c h g h 0) ) = β1 + r) [ π e u c au y N, g h 0) )) + 1 2π e ) u c h + 1 + r)g h 0) g h g h 0) )) 39) +π e u c l + 1 + r)g h 0) g l g h 0) ))], We will show ha he agen would wan o borrow given his consumpion pah, which he can do by reducing g h 0). Given ha u c h) = β1 + r) [ 1 π e ) u c h) + π e u c l)], a sufficien condiion for his is ha c au y N, g h 0) ) > c h, c h + 1 + r)g h 0) g h g h 0) ) > c h, and c l +1+r)g h 0) g l g h 0) ) > c l. Consumpion canno decrease in he agen s income, i.e. 35

i canno be ha he chooses a consumpion lower han c j when he has access o c j +1+r)g h 0) unis of he consumpion good raher han only c j unis. To see he firs condiion, we firs show ha c au y N, 0 ) > c h. Assume indirecly ha his is no rue. Given ha he paricipaion consrain holds wih equaliy when he agen s income is y N, his implies ha he benefis of being in he risk sharing arrangemen occur oday while is coss occur in he fuure relaive o auarky. This in urn implies ha risk sharing mus increase when he discoun facor decreases. This conradics he folk heorem. Recall ha in Proposiion 5 we have shown ha as β increases o a level such ha β1 + r) = 1, perfec risk sharing is he long-run oucome. Inuiively, a higher β means a beer enforcemen echnology in models of risk sharing wih limied commimen. Now, clearly, c au ) is increasing in is second argumen, herefore we also know ha c au y N, g h 0) ) > c h holds for any g h 0) 0. This means ha 39) is a sric inequaliy, hus he agen wishes o increase curren consumpion, which he can do by reducing g h 0). A similar argumen can be used if he agen would wan o defaul in more saes omorrow. In case ii), subsiuing in he fuure Euler equaions, we can use an almos idenical argumen as above. For example, ake he case when he agen would save in period 0 and 1 and defaul in he high sae in period 2 only if he income delivered by he opimal allocaion c h ) remains high in boh periods. The Euler equaion in period 0 is u c h g h 0) ) = β1 + r) [ 1 π e ) u c h + 1 + r)g h 0) g h g h 0) )) 40) +π e u c l + 1 + r)g h 0) g l g h 0) ))]. When he curren sae is h, he Euler equaion in period 1 is u c h + 1 + r)g h 0) g h g h 0) )) = β1 + r) [ π e u c au y N, g h g h 0) ))) 41) + 1 2π e ) u c h + 1 + r)g h g h 0) ) g h g h g h 0) ))) and when he curren sae is l, i is +π e u c l + 1 + r)g h g h 0) ) g l g h g h 0) )))], u c l + 1 + r)g h 0) g l g h 0) )) = β1 + r) [ π e u c h + 1 + r)g l g h 0) ) g h g l g h 0) ))) 42) + 1 π e ) u c l + 1 + r)g l g h 0) ) g l g l g h 0) )))]. Using equaions 41) and 42) o subsiue for he marginal uiliies on he righ hand side of 40) gives he wo-period ahead Euler equaion in his case. Noe ha when he agen 36

neiher saves nor defauls for wo periods, he wo-period ahead Euler equaion is given by u c h) = β1 + r) [ 1 π e ) β1 + r) 1 π e ) u c h) + π e u c l)) 43) +π e β1 + r) 1 π e ) u c l) + π e u c h))]. Now, comparing he righ hand sides of 40) afer subsiuion and 43) erm by erm we can use pracically he same argumen as above o show ha g h 0) = 0. Proposiion 10. The firs-order condiion approach is valid. Proof. The firs-order condiion approach is valid if V h = W h 0), V l = W l 0), and g h 0) = g l 0) = 0. I is easy o see ha g l 0) = 0. Lemma 6 shows ha g h 0) = 0. Replacing hese soluions ino 37) and 38), he firs wo condiions follow. 4.3 Compuaion We use he recursive sysem given by equaions 26)-31) o solve he model numerically. We discreize x and B y is assumed o ake a finie number of values). We have o deermine x and B on a 3-dimensional grid on X. The iniial values for V X ), c 1 X ), and υ 1 X ) are from he soluion of a model where he paricipaion consrains are ignored. We ierae unil he value and policy funcions converge. As we proceed, we use he characerisics of he soluion. In paricular, we know ha if agen 1 s paricipaion consrain binds a x, i will bind a all x < x. Similarly, if agen 2 s paricipaion consrain binds a ˆx, i will bind a all x > ˆx. A each ieraion, a each income sae and for each B, we solve direcly for hese limis, using 29) and 30) wih equaliy in urn, firs assuming ha B = 0. Aferwards, we check wheher he planner s Euler is saisfied a he limis. If no, we solve a 2-equaion sysem of 29) or 30)) and 28) wih equaliy, wih unknowns x, B ). Finally, we solve for a new B a poins on he x grid where neiher paricipaion consrain binds, i.e. a he inerior of he opimal inerval of he curren ieraion. 4.4 Decenralizaion Noe ha public sorage can be hough of as a form of capial, B unis of which produce Y + 1 + r)b unis of oupu omorrow and which fully depreciaes. Ábrahám and Cárceles-Poveda 2006) show how o decenralize a limied commimen economy wih capial accumulaion which is very similar o he one sudied in his paper. In paricular, hey inroduce compeiive inermediaries and show ha a decenralizaion wih endogenous deb 37

consrains ha are no oo igh ha make he agens jus indifferen beween defauling and paricipaing), as in Alvarez and Jermann 2000), is possible. In ha environmen, however, hey use a neoclassical producion funcion where wages depend on aggregae capial. This implies ha here he value of auarky depends on aggregae capial as well. 16 Ábrahám and Cárceles-Poveda 2006) show ha if he inermediaries are subjec o endogenously deermined capial accumulaion consrains, hen his exernaliy can be aken ino accoun, and he consrained-efficien allocaion can be decenralized as a compeiive equilibrium. 17 In our model, however, public sorage does no affec he ouside opion of he agens. Hence, he resuls above direcly imply ha a compeiive equilibrium corresponding o he consrained-efficien allocaion exiss. In paricular, households rade in Arrow securiies subjec o endogenous borrowing consrains ha preven defaul, and inermediaries also sell hese Arrow securiies o build up public sorage. The key inuiion is ha equilibrium Arrow securiy prices ake ino accoun binding fuure paricipaion consrains, as hese prices are given by he usual pricing kernel. Moreover, agens will no hold any shares in public sorage, hence heir auarky value will be unaffeced. Finally, no arbirage or perfec compeiion will make sure ha he inermediaries make zero profis in equilibrium. 5 Compued examples In his secion we use he above algorihm o solve for efficien allocaions in economies wih limied commimen and access o hidden sorage. We show ha aggregae savings due o he planner s desire o complee markes can be significan in magniude. We also illusrae he role of he discoun facor, β, and he reurn on sorage, r. To do his, we consider five scenarios, four wih β1 + r) < 1 and one wih β1 + r) = 1 o sudy he benchmark of an efficien sorage echnology as well. In each case agens perperiod uiliy funcion is of he he CRRA form wih a coefficien of relaive risk aversion equal o 1, i.e. u) = ln). Income of boh agens is i.i.d. over ime, and may ake hree values, {0.2, 0.5, 0.8}, wih equal probabiliies. Remember ha income is perfecly negaively correlaed across he wo agens, hus aggregae income is 1 in all hree saes. When β1 + r) < 1, he discoun facor may ake wo values, 0.8 and 0.7, and he ineres rae may ake wo values as well, 0.16 and 0.11. In addiion, we consider he case β = 0.8 and r = 0.25. For he basic model, we deermine agens value of auarky allowing hem o save. This 16 This is also he case in he wo-counry producion economy model of Kehoe and Perri 2004). 17 Chien and Lee 2010) achieves he same objecive by axing capial insead of using a capial accumulaion consrain. 38

means ha agens ouside opion is he same in he basic model and our model, hereby we can focus on he effec of sorage in equilibrium. In all five scenarios, he soluion of he basic model is characerized by parial insurance. In one scenario β = 0.8, r = 0.11), consumpion akes only wo values in he long run, while i akes four values in he oher four scenarios. Table 1 shows he saionary disribuion of consumpion for he basic model. Furher, in all five cases, agens would wan o save a he soluion of he basic model. Tha is, hidden sorage maers. Therefore, he planner will save in our model. Table 1: Saionary disribuion of consumpion in he basic model wih sorage in auarky, percen of aggregae income β 0.8 0.8 0.8 0.7 0.7 r 0.25 0.16 0.11 0.16 0.11 c h 80.00 80.00 59.51 80.00 64.84 c m 54.30 51.85 59.51 50.07 57.96 c m 45.70 48.15 40.49 49.93 42.04 c l 20.00 20.00 40.49 20.00 35.16 In all cases u) = ln) and income may ake hree values, {0.2, 0.5, 0.8}, wih equal probabiliies. Now we urn o he resuls from our model wih sorage. Firs, le us look a he behavior of asses in he long run. Table 2 shows he limis of he saionary disribuion of asses. As noed above, asses reach heir minimum if sae s m occurs repeaedly, while hey reach heir maximum if sae s h or sae s l ) occurs many imes in a row, and given ha when we swich o eiher of he unequal saes asses were a heir minimum. Table 2: Saionary disribuion of asses, percen of aggregae income β 0.8 0.8 0.8 0.7 0.7 r 0.25 0.16 0.11 0.16 0.11 B max {B 0, 36.97} 16.56 2.90 7.41 1.38 B 36.97 16.56 2.90 6.43 0.00 In all cases u) = ln) and income may ake hree values, {0.2, 0.5, 0.8}, wih equal probabiliies. B 0 is he iniial level of aggregae asses. When he discoun facor is high β = 0.8), he paricipaion consrains in sae s m do no bind in he long run for any of he ineres raes considered here. As a resul asses are 39

consan in he long run, as Proposiion 5 saes. When he sorage echnology is efficien r = 0.25), asses reach 36.97 percen of aggregae income in he long run, if he iniial level of asses B 0 36.97. Else, asses say consan a heir iniial level. When β1 + r) < 1 bu he ineres rae is sill relaively high r = 0.16), he planner s savings amoun o 16.56 percen of aggregae income. Noe ha he presence of sorage relaxes he paricipaion consrains sufficienly so ha hey do no bind in sae s m, unlike in he basic model. When he ineres rae is relaively low r = 0.11), savings are 2.90 percen of aggregae income. When he discoun facor is low β = 0.7), paricipaion consrains bind in all hree saes, and asses remain sochasic in he long run. When he ineres rae is relaively high r = 0.16), savings by he planner o make sure agens do no wan o save and o improve risk sharing in he fuure vary beween 6.43 and 7.41 percen of aggregae income. When he ineres rae is relaively low r = 0.11), savings vary beween 0 and 1.38 percen. This las example shows ha 0 can be par of he saionary disribuion of aggregae asses. In erms of how much risk sharing is achieved, an increase in he ineres rae reduces risk sharing in he basic model, since i raises he value of auarky, see Table 1. In conras, a higher reurn on sorage improves risk sharing in our model, see Table 3. The benefis from aggregae sorage ouweigh he negaive effec of he increase in he value of agens ouside opion. Table 3: Saionary disribuion of consumpion, percen of aggregae income β 0.8 0.8 0.8 0.7 0.7 r 0.25 0.16 0.11 0.16 0.11 c h 54.62 56.67 58.14 [62.94, 63.41] [63.97, 64.61] c m 54.62 56.67 58.14 [59.73, 60.57] [57.81, 59.24] c m 54.62 45.97 42.18 [41.20, 41.30] [41.91, 42.19] c l 54.62 45.97 42.18 [37.50, 38.48] [35.19, 36.18] In all cases u) = ln) and he income may ake hree values, {0.2, 0.5, 0.8}, wih equal probabiliies. The role of he discoun facor is he same in our model wih sorage as in he basic model. In paricular, a higher β implies more risk sharing. In he benchmark case where β = 0.8 and r = 0.25, i.e. when he sorage echnology is efficien, perfec risk sharing is self-enforcing in he long run. This is because in his case here is no rade-off beween he wo inefficiencies, namely, imperfec risk sharing and an inferior ineremporal echnology. When he discoun facor is low β = 0.7), consumpion depends on inheried asses, 40

which in urn depend on he hisory of income shocks. As a resul, he amnesia and persisen properies do no hold in he long run in hese examples. The variaion of consumpion is small wih a maximum of 2.8 percen in sae s l when β = 0.7 and r = 0.11. 6 Concluding remarks This paper has shown ha some implicaions of he basic limied commimen model wih no privae or public sorage are no robus o hidden sorage. When public sorage is allowed hough, he incenive for privae sorage is eliminaed in he consrained-opimal allocaion. The ineremporal echnology is used in equilibrium even hough here is no aggregae uncerainy and he reurn is lower han he discoun rae, i.e. β1 + r) < 1. The planner saves for wo reasons. She saves o eliminae agens incenives o use heir hidden sorage echnology. When income inequaliy is no he highes, he planner has more incenive o save han he agens. The reason for addiional sorage by he planner is ha public asses relax fuure paricipaion consrains, and hus improve risk sharing. We have also shown ha aggregae asses may be sochasic in he long run. This happens when each agen s paricipaion consrain binds a more han one income level, i.e. when cross-secional consumpion inequaliy varies over ime. Given inheried asses, public sorage is higher when consumpion inequaliy is higher. Finally, he dynamics of consumpion is richer in our model compared o he basic model wihou sorage. In paricular, he amnesia and persisence properies do no hold in general, which brings limied commimen models closer o he daa Broer, 2011). Furher, in our model agens Euler consrains in inequaliy form hold by consrucion, which is consisen wih empirical evidence from developed counries Aanasio, 1999). The lieraure on incomplee markes eiher exogenously resric asse rade, mos prominenly by allowing only a risk-free bond o be raded Hugge, 1993; Aiyagari, 1994), or considers a deep fricion ha limis risk sharing endogenously. The lieraure has focused on wo such fricions, namely limied commimen and privae informaion. This paper merges hese wo srands of he incomplee markes lieraure by allowing for sae-coningen rading subjec o a deep fricion and a self-insurance opporuniy a he same ime. This has been done when he deep fricion is hidden income or effor Allen, 1985; Cole and Kocherlakoa, 2001; Ábrahám, Koehne, and Pavoni, 2011), bu no in he case where he deep fricion is lack of commimen, o our knowledge. Comparing our model wih limied commimen and hidden sorage o models wih hidden income or effor and hidden sorage poins o remarkable differences. In our model, if he 41

same sorage echnology is available o he planner and he agens, welfare is improved. 18 In conras, in models wih hidden income/effor, public sorage is never used and he presence of he hidden sorage opporuniy reduces welfare. This is because in our model saving by he planner boh improves insurance and relaxes he incenive problem, by relaxing fuure paricipaion consrains; while in he hidden income/effor conex he incenive problem is made more severe by addiional self-insurance, and aggregae asse accumulaion makes incenive provision more expensive. Our model could be applied in several economic conexs. The model predics ha risk sharing among households in villages can be improved by a public grain sorage faciliy. Cooperaion among parners in a law firm, for example, is faciliaed by common asses ha someone quiing he parnership has no access o. Marriage conracs may specify ha some commonly held asses are los by he spouse who files for divorce. Finally, supranaional organizaions may help inernaional risk sharing by simply having a joinly held sock of asses. The European Financial Sabiliy Faciliy and he European Sabiliy Mechanism may serve his purpose. Fuure work should sudy he quaniaive implicaions of saving using some of hese applicaions. 18 The planner and he agens can always choose zero asses. Given ha in he consrained-opimal allocaion he planner has sricly posiive asses a leas in some periods), welfare sricly improves. 42

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