Second- and Higher-Order Consumption Functions: A Precautionary Tale

Transcription

1 Second- and Higher-Order Conumpion Funcion: A Precauionary Tale Jame Feigenbaum Univeriy of Iowa November 20, 2002 Abrac A curren problem of inere in macroeconomic i he compuaion of conumpion and aving funcion for agen who experience uninurable income hock and engage in precauionary aving. Wih conan relaive rik averion uiliy, hi deciion problem canno be olved analyically. One popular mehod for compuing policy funcion ha been o ue perurbaion heory, Taylor-expanding around he pecial cae of perfec cerainy, where he problem i exacly olvable, and compuing correcion erm ha are linear in he variance of he income proce. Here, I how ha i i raighforward o exend hi procedure o compue he linear conribuion of he kewne. However, in hi dynamic environmen where perurbaion ge compounded over ime, for each momen of he income proce here will be a lower-bound on he inere rae below which he correcion erm generaed by ha momen diverge a large lifeime. In he limi of large order, hi lower bound on he inere rae converge o he equilibrium inere rae for he correponding infinie-horizon, conan-growh economy wihou uncerainy. Since in equilibrium, precauionary aving mu lower he inere rae below ha limi, perurbaion heory mu break down for ome momen of he income proce. Tha i, he Taylor erie will no converge, o adding erm of ucceively higher order o he erie will only improve he approximaion o a poin, afer which adding more erm will woren he approximaion. A an example, I preen a cae where a econd-order approximaion o he conumpion funcion, which include variance effec bu no kewne and higher-momen effec, perform ubanially wore han a zeroh-order approximaion, which enirely ignore he effec of uncerainy. I would like o hank Jame Bullard, Sefano Euepi, Harry Paarch, Chri Slee, and Charle Whieman for heir advice and uggeion. 1

2 A problem of inere in macroeconomic i he compuaion of parial and general equilibria in a bond economy where agen have conan relaive rik averion (CRRA) preference. Thi i a model in which here i no aggregae uncerainy, o coningen-claim ae which inure again hock o he economy a a whole are unneceary. Rik-free bond are a ufficien ae o achieve ineremporal ranfer need. I ha been hoped ha reamlined model of hi or migh help o unravel variou puzzle regarding conumpion and inere rae. Forherivialcaeofcompleemarke and complee cerainy, conumpion and bond demand funcion can be derived analyically. A more inereing problem arie when agen face an uncerain, uninurable income ream. In hi circumance, agen wihing o hedge again negaive income hock would ideally purchae ae ha are perfecly correlaed wih heir income. However, marke are incomplee in hi model, and uch ae are no preen. Therefore, agen mu inead ubiue rik-free bond for hee nonexien ae. Alhough rik-free bond offer reurn uncorrelaed wih income, agen are, neverhele, able o ue bond o parially alleviae he welfare lo aociaed wih heir idioyncraic rik. One can how qualiaively ha hi precauionary aving moive will increae he demand for bond (Leland (1968), Sandmo (1970)) and decreae inere rae (Aiyagari (1994), Hugge (1993)). However, conumpion and bond demand canno be compued analyically wih uninurable uncerainy if preference are CRRA. Thi complicae he quaniaive exercie of deermining he magniude of precauionary aving effec. While he original Lifecycle/Permanen-Income Hypohei (LCPIH), derived under perfec foreigh, i generally believed o be a poor decripion of empirical conumpion behavior (Browning and Croley (2001)), here i no conenu of opinion abou how much precauionary aving by ielf can accoun for oberved deviaion from he LCPIH. Several reearcher (a parial li would include Carroll (1997,2001a), Hall (1988), Leendre and Smih (2001), Skinner (1988), and Viceira (2001)) have inveigaed hi conumpion/aving model by perurbing conumpion funcion or Euler equaion 1. Alhough he erminology of perurbaion heory i fairly new o he economic lieraure, he echnology ha been in ue for a long a economic ha been a mahemaical cience. A perurbaive mehod i imply a mehod ha involve he approximaion of a funcion by i Taylor expanion. Any linearizaion procedure i an example of perurbaion heory in i mo elemenary form. More generally, hee mehod are an inegral componen of he macroeconomi oolki. They are ued regularly o expre endogenou variable a approximae funcion of exogenou variable. Alhough perurbaive mehod have been piched under hi name 1 Anoher mehod for olving hee kind of model i he inrinically numerical approach of compuing conumpion funcion eiher hrough value-funcion or Euler-equaion ieraion (Deaon (1992)), and aggregaing hee compued funcion o clear marke. Inead of perurbing around he known oluion o a olvable problem, hee mehod proceed by dicreizing he ae pace o a finie number of poin and compuing he exac value of he relevan funcion a each poin. 2

3 mo prominenly a an adjunc o numerical mehod (Judd (1999)), hey are fundamenally an analyical mehod wih he power o reveal funcional dependence ha can only be inferred when numerical value are plugged ino exogenou parameer from he ar. Thi paper ake a deailed look a how perurbaive mehod can be ued in a dynamic conex and wha new hing can go wrong ha do no arie in aic problem, epecially a one puhe on o higher order. Perurbaion heory expree endogenou variable a power erie in a dimenionle parameer, he perurbaion parameer, and provide a precripion for compuing each coefficien of he erie in erm of lower-order coefficien. For mall value of he perurbaion parameer, he power erie can hen be approximaed by he um of a finie number of erm. In aic problem, perurbaion calculaion can uually produce an enire infinie erie which will converge for any value of he perurbaion parameer below ome radiu of convergence. I hen i poible o achieve an approximaion o any deired accuracy a long a he perurbaion parameer i below hi radiu. In dynamic problem, ha will no alway be he cae, for pahologie can arie which limi he order o which a perurbaion erie can be compued. A a reul, here can be igher limionhowmuchperurbaionheorycanellyouhanwouldoccurinoher eing. The minde ha a fir or econd-order calculaion i an iniial ep in a equence of calculaion which will ulimaely converge o an exac anwer i overly opimiic. Under he label of a econd-order Taylor expanion, Skinner (1988) ha worked ou he conumpion funcion for a finie-horizon model wih income and inere-rae hock. 2 Given he aumpion we make regarding he income proce, hi reul are equivalen o a perurbaive expanion o econd order in he coefficien of variaion. 3 Thi capure he lowe-order effec of he variance and precauionary aving. Here, we compue policy funcion o hird order for a model wih a fixed inere rae and independenly diribued income hock. Noe ha here i nohing epecially remarkable abou he hird-order conribuion, which capure he lowe-order kewne effec. The main conribuion here i o how ha he co of performing he hird-order calculaion i he ame a for he econd-order calculaion. A uch, if one i going o compue he econd-order conribuion, one migh a well compue he hird-order conribuion alo. The marginal compuaional co only begin o increae again a fourh order, where cro erm fir appear. If we ignore cro erm and higher-order effec, i i raighforward o carry ou he procedure ued a econd and hird order o compue he loweorder conribuion of each momen of he income proce o value and policy funcion. In oher word, we compue a par of he nh-order erm in he perurbaion erie, noably he eaie par. Compuing hi par i enough o 2 Talmain (1998) ha exended hee reul for a general uiliy funcion in he pecial cae where he inere rae equal he dicoun rae. 3 The fir-order condiion of he opimizaion problem guaranee ha fir-order effec mu vanih. See Cochrane (1989). Thi reducion of he welfare lo from a fir-order o a econd-order effec i he exen o which agen can elf-inure wih bond. 3

4 unearh a problem: for every n, here i a lower bound on he inere rae below which he nh-order perurbaion correcion will diverge for large lifeime. Thi lower bound increae wih n, converging o he equilibrium inere rae for a perfec-foreigh infinie-horizon economy in he limi of large n in he abence of growh, hi inere rae will be he dicoun rae. Since he equilibrium inere rae for a model wih precauionary aving mu be le han he perfecforeigh inere rae, in general equilibrium he perurbaion erie will only exi up o a finie number of erm in he infinie horizon. Conequenly, for inere rae below he dicoun rae, perurbaive mehod canno be ued o compue endogenou variable o arbirary accuracy. On he conrary, one can find example where a econd-order approximaion i ubanially wore han a zeroh-order calculaion. Forunaely, however, a divergence in he nhorder correcion doe no diminih he accuracy of lower-order perurbaion calculaion. Thee divergence are relaed o he ingulariy in he CRRA uiliy funcion a zero conumpion. Thi ingulariy caue he perurbaion erie o diverge for agen wih low wealh, bu one migh no expec i o caue a problem for high-wealh agen. In an economy wihou growh, if inere rae are below he dicoun rae, hen, in he abence of precauionary moive, agen will have no incenive o ave. Thu, wihou uncerainy, all agen will evenually pend heir wealh down o zero. Thi oluion wihou uncerainy correpond o he zeroh-order oluion o he problem, he oluion ha we are expanding around. Thi diipaion of wealh in he oluion we are expanding around ake agen ou of he convergence region of he ae pace, and ha i wha caue he perurbaion calculaion o go wrong. The paricular pahology idenified in hi paper doe no arie for inere rae above he dicoun rae, where he zeroh-order oluion ake agen away from zero conumpion. I hould be emphaized ha hi dynamic pahology i enirely a failure of perurbaion heory. The exac policy and value funcion are well-behaved. Moreover, hi failure doe no aler he common widom ha precauionary aving i a econd-order effec. The pahology arie becaue he value funcion ha we iner ino he Bellman equaion a each age i ielf he reul of a perurbaion calculaion. If we were omehow given he exac value funcion for an arbirary period, we could preumably ue perurbaion heory o compue he value funcion of he previou period wihou any difficuly, auming he agen wealh i large enough. I i he compounding of perurbaion correcion over ime ha produce hee arificial divergence. The paper proceed a follow. In Secion 1, we inroduce he model o be conidered here. In Secion 2, we dicu our choice of perurbaion parameer and how he validiy of perurbaive mehod depend on hi choice. In Secion 3, we derive he bond demand of agen o hird order in he coefficien of variaion. In Secion 4, we conider he behavior of he lowe-order conribuion of he nh momen of he income proce a large lifeime and examine how hi can diverge. In Secion 5, we conclude wih a dicuion of he generaliy of he pahologie found here o oher dynamic model and decribe oher applicaion of perurbaion heory o iue of conumpion and aving. 4

5 1 The Model Conider an economy wih one conumpion good in which agen live for T + 1 period. Alhough he economy will be aionary, we will allow for he poibiliy of conan economic growh, o behavioral quaniie can depend on boh abolue ime and he age of an agen. Since we will olve he agen opimizaion problem by backward inducion, i i convenien o meaure he age of an agen in erm of he number of period remaining in hi life raher han by how many period he ha lived. Behavioral quaniie like conumpion will, herefore, have wo ime indice: a ubcrip ha refer o abolue ime and a upercrip ha refer o he number of period remaining in life. For example, c i he conumpion of an agen a ime who ha period remaining. An agen born a τ maximize TX U τ = β u(c T τ+ ), (1) =0 where he dicoun rae β (0, 1) and he uiliy funcion u( ) ha he CRRA form. Given he coefficien of relaive rik averion γ 0, ½ ln c γ =1 u(c) = 1 1 γ c1 γ γ 6= 1. (2) (We will derive all reul for he cae γ 6= 1. I i eaily hown ha endogenou obervable vary coninuouly wih γ a γ = 1.) An agen a wih period lef will earn a ochaic income endowmen ey. Weaumehaey i nonnegaive and diribued independenly boh acro ime and acro agen. We will alo aume ha, for a given, ey and 0 G0 ey will have he ame diribuion for any and 0,whereG i he gro growh rae of he economy. In oher word, he diribuion of ey will cale a he growh facor G. (We will ay more abou he diribuion of he ey in he nex ecion.) One ineremporal ae, a rik-free bond, i available for invemen and pay a conan, exogenou gro inere rae R>1. The ne inere rae r = R 1. Bond holding are indexed according o when hey pay off: b i purchaed a 1, by an agen wih + 1 period remaining, pay off Rb a. The agen opimizaion problem can be expreed in erm of a recurive equence of Bellman equaion. Le v (b, y) denoehevaluefuncionofan agen a wih period lef, bond holding b, and curren income realizaion y. Conider an agen born a τ. In hi la period, he will conume any remaining wealh, o hi erminal value funcion i v 0 τ+t (b, y) =u (y + Rb). (3) Given he value funcion vτ+t (b, y), he value funcion for he agen when he ha + 1 period lef mu aify he Bellman equaion: v +1 τ+t 1 (b, y) =max b 0,c u(c)+βeτ+t 1 [v τ+t (b 0, ey 0 )] ª (4) 5

6 ubjec o c + b 0 = y + Rb (5) c 0. For an agen ju born, b T τ = 0 ince he will have no iniial bond holding. 2 The Role of he Perurbaion Parameer Le δ be a parameer in a problem for which we know he oluion if δ =0. Perurbaive mehod involve Taylor expanion of he oluion wih repec o δ. We can view δ a a bookkeeping device. Calculaing he nh order effec amoun o calculaing all erm in he equence of order δ n or le. 4 The aumpion ha we can ignore higher-order erm in an nh-order approximaion depend on δ n+1 being ignificanly maller han δ n. Ideally hen, reearcher would like o conider iuaion where he value of δ i mall compared o 1. In he limi of very mall δ, even low-order perurbaion approximaion hould be very accurae. Converely, for δ on he order of 1, a reearcher will need o compue oluion o a high order o achieve ha ame level of accuracy. In he conex of he preen paper, we are ypically inereed in compuing expecaion of he form µ ew E (6) w for ome power ρ, wherew i oal wealh (he um of curren income, expeced fuure income, and bond holding) a. (We will uppre age upercrip for hi dicuion.) Expreion of hi ype arie becaue he Euler equaion for an agen in he model of Secion 1 i µec γ 1=βRE, (7) and conumpion i approximaely proporional o wealh. Le z = E [ ew ]and ew = w z. Then we can wrie (6) a µ ρ µ z E 1+ ew ρ. (8) w z 4 Le f(δ) be a funcion of he parameer δ [0, 1) and g(δ) a poiive funcion of δ. We ay ha f(δ) = O(g(δ)) ( f(δ) ioforderg(δ) ) if and only if here exi a real number M 0uchha f(δ) g(δ) M for all value of δ [0, 1). If f 1 (δ) =O(g 1 (δ)) and f 2 (δ) =O(g 2 (δ)), hen f 1 (δ)f 2 (δ) = O(g 1 (δ)g 2 (δ)). c ρ 6

7 Uing he Taylor expanion (1 + x) ρ =1+ρx + ρ(ρ 1) x ρ(ρ 1)(ρ 2) x 3 +, (9) 3! which i valid for x < 1, he expecaion in (8) can be replaced by µ 2 ew ρ(ρ 1) ew 1+ρE + E z 2 z + µ ρ(ρ 1)(ρ 2) ew E 3! z 3 +, (10) a long a he upremum of ew i le han z. 5 he only random componen of ew, Since curren income i and (10) become 1+ µ ρ(ρ 1) ey µ 2 E + 2 z ew = ey µ, µ ρ(ρ 1)(ρ 2) ey µ 3 E +. 3! z (11) If we can e up he problem o ha E [( ew /z ) n ]ioforderδ n for ome exogenou parameer δ, hen we are in a poiion o ue perurbaive mehod. For he purpoe of hi paper, we will achieve hi perurbaive iuaion by auming he nh momen of he income proce i of order δ n. Formally, we aume he momen of he diribuion for he income of an agen a wih period remaining, ey, aify he following condiion: le δ [0, 1) be an exogenou, dimenionle parameer, which we will impreciely call he coefficien of variaion, and uppoe ha E[ey ]=µ, (12) and h³ ni ey E µ µ = O(δ n ), n 2. (13) 5 The power erie (10) can be wrien P h ³ Γ(ρ+1) ew ni. n=0 E According o n!γ(ρ n+1) z 27.2 of Halmo (1974), if he power erie S = P Γ(ρ+1) n=0 E h n!γ(ρ n+1) ew n i i finie, z hen he ummaion and he expecaion operaor commue, o S equal he expecaion in ew n (??). Since E h i µ w up n,where w up z z i he upremum of he poible value of ew, S P µ Γ(ρ+1) w up n n=0 n!γ(ρ n+1). Sinceheeriein(9)hauniradiu z of convergence and ince power erie are aboluely convergen, hi bound on S will be finie if w up <z. 7

8 (For mo of he numerical example we conider in hi paper, δ will equal he raio of he andard deviaion o he mean in each period, and o he erminology ha δ i he coefficien of variaion would be preciely correc. Tha will no be rue for he general model, alhough δ will be proporional o he coefficien of variaion in he limi a δ 0.) Thi δ will play he role of he perurbaion parameer in our perurbaion expanion. We can hen define and ex = ey µ δ (14) k n, = E[(ex )n ] (15) for n 2, where k n, i finie a δ 0. A diribuion ha aifie (ey µ ) δµ for all poible realizaion of ey will aify hee momen condiion. More generally, we can view hee condiion a defining a compac diribuion in he ene ha Samuelon (1977) ued he erm (no o be confued wih he uual noion of a compac e). A a reul, under he momen condiion (13), (11) implifie o 1+ ρ(ρ 1) δ 2 k 2, 2 z 2 + ρ(ρ 1)(ρ 2) δ 3 k 3, 3! z 3 +. (16) Le x up beheupremumofheeofpoiblerealizaionof ex. If δx up < z, hen he erie (16) will converge. In he abence of exogenou borrowing conrain, Aiyagari (1994) howed ha here will be an endogenou borrowing conrain wih CRRA uiliy becaue agen will never borrow more han he minimum hey can be aured hey will be able o pay back. Suppoe we have a conan income proce independen of. If he realizaion of ex which i large in magniude i negaive, he lowe poible income realizaion will be µ δx up. In ha cae, he bond demand for an agen wih an infinie lifepan mu aify b µ δxup, r andexpecedwealha +1,z,muaify z = R r µ + Rb R r δxup. (17) Therefore, under hee condiion, he erie (16) will converge for any feaible wealh value. Noe, however, ha convergence of he erie doe no by ielf make perurbaion heory ueful. Truncaing (16) a order n will only give a decen 8

9 approximaion where erm of order higher han n are negligible relaive o he um up o he nh erm. Since each erm i a decreaing funcion of z,here will be a lower bound on he value of z for which he nh order approximaion achieve a given accuracy, and hi lower bound will generally be larger han he bound δx up. The condiion (13) i a very ringen condiion o place on he momen, and i would no be aified by many popular income parameerizaion. For example, Carroll (1997,2001b) ue an income proce in which here i a mall probabiliy p ha income will be zero in any period. One migh hink ha he probabiliy p would alo be a reaonable choice of perurbaion parameer ince i oo i a dimenionle parameer beween 0 and 1. However, hi i no he cae. Le u conider he behavior of he momen for a probabiliy diribuion ½ Y 1 p ey =, εy p where ε =1 δ > 0 i mall. The mean of hi diribuion i The nh order momen i µ = E[ey] =(1 p)y + pεy =(1 p(1 ε))y. E [(ey µ) n ] = (1 p)[1 (1 p(1 ε))] n Y n + p [ε (1 p(1 ε))] n Y n = (1 p)p(1 ε) n p n 1 +( 1) n (1 p) n 1 Y n. For 0 < ε < 1, hee momen will be of order δ n a we have aumed. However, if ε = 0, which would be he cae if we wan a diribuion wih a poibiliy of a zero realizaion, hen E[(ey µ) n ]=p + O(p 2 ) for all n 2. There i no diminihmen of he ucceive momen wih repec o p. Thu, p would no be a good choice of a perurbaion parameer if we are aemping o evaluae he momen expanion (11). Noice, however, ha while he momen do no diminih wih n, hey do cale a Y n. Thu, for mall Y/z, he erm of he erie (11) hould ill ge progreively maller, and we can ill do perurbaion heory uing Y/z a he perurbaion parameer. All he perurbaion calculaion involving value and policy funcion in hi paper will remain valid for a noncompac diribuion if we ue b δ = µ /z a he perurbaion parameer, where µ i ill mean income. 6 Unforunaely, hi alernae choice of perurbaion parameer ha wo downide. The fir i ha for comparable value of wealh he perurbaion parameer will be larger. If mean income i conan, average wealh will ypically be on he order of µ/r, wherer i he ne inere rae, o for an agen of average wealh he perurbaion parameer will be on he order of r. For ypical value of inere rae on he order of 5%, hi hould be mall enough o make low-order perurbaion calculaion accurae approximaion. On he 6 Thi i he approach ued in Feigenbaum (2001a). 9

10 oher hand, a relaively poor agen will have a larger value of b δ. For b δ cloe o uniy, one will have o go o very high order of perurbaion heory o obain a decen approximaion. Thi difficuly will plague any work involving Taylor expanion of he Euler equaion (7). Moreover, hee higher-order erm will be of ignifican magniude for poor agen. In he conex of Euler-equaion eimaion, if one neglec hird and higher-order effec in a regreion of conumpion growh on ae variable for a ample wih a ubanial proporion of low-wealh agen, eimae of rucural parameer will be inconien becaue hee higher-order effec are ignifican and depend on hoe rucural parameer. Thi i one of everal argumen ha Carroll (2001a) ha given for abandoning Euler-equaion eimaion of preference parameer, alhough hi paricular criicim hould no preen a problem if we reric our aenion o wealhy agen 7. The econd downide i ha we canno ue a perurbaion parameer ha i pecific o each agen o compue macroeconomic quaniie. Conequenly, if we wihed o clear marke and calculae inere rae under he approach of hi paper, we would have o ue a compac income diribuion. Imayalobepoibleoexploip a a perurbaion parameer, alhough i i no clear how ha could be done. 3 Value and Policy Funcion The conumpion/aving deciion decribed by he Bellman equaion (4) i a recurive problem ha mu be olved backward ieraively from he end of life. In hi ecion we demonrae how o compue he econd-order perurbaion correcion for he fir ieraion, when an agen ha one fuure period remaining in life and no exiing bond holding. The econd and hird-order correcion are fully worked ou for an agen wih T period remaining in Appendix B. Conider an agen who i born a = 0 and live unil =1. The conumer problem give rie o he Lagrangian L = (y 0 b 1 ) 1 γ 1 γ 1 + βe 0 1 γ (ey 1 + Rb 1 ) 1 γ. (18) (We will uppre he age upercrip in hi dicuion.) hechoicevariableb 1, we obain he fir-order condiion Differeniaing L by (y 0 b 1 ) γ = βre 0 h(µ 1 + δex 1 + Rb 1 ) γi, (19) 7 Here, we are perurbing around a linearizaion of he Euler equaion. The deail of hi dicuion will differ if we proceed from a log-linearizaion, bu he upho hould remain he ame. There will be a region of he wealh pace where he perurbaion erie converge, which will, however, be bounded boh above and below. A one ge cloer o he edge of hi region, one will have o go o ever higher order o mainain he ame level of accuracy. 10

11 where (14) define ex 1. The quaniy Eq. (19) can be implified o ³ µ1 à µ y 0 b 1 = R + b 1 E 0 1+ δex! γ 1/γ 1. (20) µ 1 + Rb 1 = βr 1 γ 1/γ (21) i he invere of he marginal propeniy o ave in he limi a T and all marginal propeniie o ave and conume are funcion of alone. Uing (9), we can Taylor-expand he expecaion facor of (20) o obain µ 1+ δex γ 1 =1 γδex 1 + γ(γ +1)δ2 (ex 1 ) 2 µ 1 + Rb 1 µ 1 + Rb 1 2(µ 1 + Rb 1 ) 2 + O(δ 3 ). (22) A we dicued in he previou ecion, hi Taylor expanion will be valid a long a δex 1 µ 1 + Rb 1 < 1 (23) for all poible realizaion of ex 1. For ufficienly mall δ, hi inequaliy hould be obeyed excep for very poor agen wih µ 1 + Rb 1 0. Noe ha large poiive realizaion of ex 1 can be ju a problemaic a large negaive realizaion. Given he momen aumpion (14) and (15), Eq. (22) ha expecaion µ E 0 1+ δex γ 1 =1+ γ(γ +1)δ2 k 2,1 µ 1 + Rb 1 2(µ 1 + Rb 1 ) 2 + O(δ3 ). Applying (9) a econd ime, we ge he reul à E 1 µ 1+ δex 1 µ 1 + Rb 1 γ! 1/γ =1 (γ +1)δ2 k 2,1 2(µ 1 + Rb 1 ) 2 + O(δ3 ). (24) Inering (24) ino he fir-order condiion (20) give µ1 y 0 b 1 = R + b 1 (γ +1)δ2 k 2,1 + O(δ 3 ). (25) 2R(µ 1 + Rb 1 ) Thi can be rewrien b 1 = 1 1+ ½ µ1 y 0 R (γ ¾ +1)δ2 k 2,1 + O(δ 3 ). (26) 2R(µ 1 + Rb 1 ) Noe ha we have no acually olved for he bond demand b 1 ince b 1 appear on he righhand ide. Alhough we could olve exacly for b 1 here if 11

12 we ignore he O(δ 3 ) erm, hi would no be feaible if we included any higher order erm. Perurbaion heory direc u o Taylor expand b 1 : b 1 = b (0) 1 + δb (1) 1 + δ 2 b (2) 1 + O δ 3. (27) Le oal wealh w be he um of curren income, expeced fuure income, and curren bond holding for an agen a when period remain: w = y + h 1 R + Rb. (28) Here, h i he expeced preen value of he curren and remaining ream of income for an agen a wih period remaining: h = X µ i +i R i=0 i. (29) (So Eq. (28) applie for =0,wedefine h 1 = 0 for all.) For an agen born a = 0 who live wo period, wealh a =0i w 0 = y 0 + µ 1 (30) R ince he agen will begin life wih no bond holding. We define he zeroh-order expecaion of wealh a =1a z 1 = µ 1 + Rb (0) 1. Then we can equae coefficien of power of δ in Eq. (26) o deermine he coefficien of he expanion (27): b (0) 1 = 1 h y 0 µ i 1 (31) 1+ R b (1) 1 = 0 (32) Since we can rewrie b (2) 1 = (γ +1)k 2,1 1+ 2Rz 1 z 1 = Rw 0 1+, b (2) 1 = (γ +1)k 2,1 2R 2 w 0. (33) 12

13 Noice ha any fir-order conribuion o he bond demand vanihe. Thi i a fairly general reul ha will hold whenever he choice of he bond demand i unconrained. Eenially, he fir-order condiion i a rericion ha any fir-order perurbaion mu vanih 8. Conequenly, he econd-order effec of Eq. (33) i he lowe-order conribuion o he precauionary aving effec. A poiive variance induce agen o ave more han hey would if heir income wa cerain. Becaue he coefficien of w0 1 in (33) i poiive, he bond demand funcion will be ricly convex. Thi i he flip ide of Carroll and Kimball (1996) reul ha he conumpion funcion will be ricly concave. To implify he more general expreion derived in Appendix B ariing from a model where agen live for T + 1 period, we make ue of he noaion of q-deformed number, a concep fir inroduced by Heine. For q 0andn boh real number, we define (n) q = 1 qn 1 q. (34) For n a poiive ineger, hi i equivalen o he geomeric erie (n) q =1+q + + q n 1. Appendix A conain ome ueful reul from q-arihmeic ha help o implify compuaion. For general and 0 T, we can wrie he value funcion ha olve (4) a v (b,y ) = ( +1)γ 1 (w ) 1 γ 1 δ 2 (1 γ)γ 1 γ 2 where K 0 n, = k 0 n,, and +δ 3 (1 γ 2 )γ 6 K3, k3, (w ) 3 + O(δ 4 ). K 2, k 2, (w ) 2 (35) for >0. µ n 1 ( Kn, = kn, +1) Kn, 1 + () R n (36) The bond demand a for 0 T 1i b (b +1,y +1 ) = ( +1) ( +2) (y +1 + Rb ( +2) + ) h R γ +1 δ 2 (K2, +1 k2, +1 ) 2 w +1 (γ +1)(γ +2) δ 3 (K3, +1 k +1 6 (w +1 ) 2 3, ) + O(δ 4 ). (37) 8 In a model wih binding liquidiy conrain, where he fir derivaive of he Lagrangian i no zero, fir-order perurbaion correcion need no vanih. 13

14 We carry ou hee calculaion up o hird order becaue he only conribuion o hee expreion a econd and hird order come from he lowe-order (i.e. linear) conribuion of he variance and kewne repecively. Beginning wih fourh order, we would ee he effec of quare and produc of variance, cro erm ha complicae he algebra enough o foreall heir calculaion in hi hi paper. Cro erm do no appear a econd or hird order becaue of he cancelling of fir-order correcion. If here wa a nonzero fir-order correcion, he quare of hi fir-order correcion would appear in he equaion for he econd-order correcion, bu he fir-order condiion eliminae hoe erm. Indeed, i will generally be rue ha cro erm will no emerge unil fourh order in perurbaion calculaion where fir-order correcion vanih. Thu, he relaive impliciy of he hird-order calculaion for hi problem i no a pecial reul. Thevarianceeffec, refleced in erm involving K2, +1, behave a in he wo-period cae. Increae in he variance of income lower uiliy and induce agen o increae heir bond holding. Thi i he precauionary aving effec. In conra, he kewne effec, refleced in he K3, +1 erm, increae uiliy and decreae aving. Agen wih CRRA preference are rik avere bu kewne loving. If we wen on o work ou he fourh-order correcion, hee would have an ambiguou ign. Term deriving from he fourh-order momen will lead o greaer aving, bu cro-erm involving produc of variance will have he oppoie ign. Which force win ou will depend on wheher he raw kuroi erm are greaer or le han ome funcion of he variance erm. Similar conideraion would apply o fifh and higher-order momen. By ubiuing (37) ino he budge conrain (5), we obain he conumpion funcion c +1 (b +1,y +1 ) = +1 ( +2) w +1 γ +1 2 δ 2 (K +1 2, k +1 2, ) w +1 (γ +1)(γ +2) δ 3 (K3, +1 k3, +1 + ) 6 (w +1 + O(δ 4 ). (38) ) 2 Le c n, denoe he nh-order perurbaion approximaion o he conumpion funcion c n, = y + Rb nx i=0 b 1 (i) (o (38) correpond o c +1 3, ). A an example of how much he econd and hird-order correcion o he conumpion funcion, conider he cae where he income proce ha a aionary diribuion ½ ey 1+δ = probabiliy p 1 δ probabiliy 1 p (39) for all and, o he income will have variance k 2 =4p(1 p)δ 2 and kewne 8p(1 p)(1 2p)δ 3, which will be poiive or negaive depending on wheher p 14

15 i le or greaer han 1/2. In Fig. 1, we plo c n, /bc for n = 0, 2 and 3 a a funcion of wealh w for an agen wih β =0.96, γ =2,p =0.9, δ =0.5, and 41 period remaining, where c n, = c n, bc and bc i a conumpion funcion evaluaed numerically uing value-funcion ieraion and Schumaker hape-preerving pline a decribed in Judd.(1999). 9 The choen parameer p and δ for he income proce give a variance k 2 =0.09 and a negaive kewne k 3 = Thi i a iuaion where he agen ypically receive a conan income, bu ha a 10% probabiliy of receiving a emporary negaive income hock correponding o a lo of 2/3 of hi uual income. For mall value of w, none of he hree perurbed conumpion funcion doe a good job of approximaing he numerical conumpion funcion, which i o be expeced from he dicuion of Secion 2. In he low-wealh regime, he conumpion funcion i alway much le han i prediced by perurbaion heory. In par, hi reflec addiional precauionary aving induced by he Aiyagari borrowing conrain, which doe no how up in low-order perurbaion calculaion. On he oher hand, for large value of wealh, he perurbaion approximaion work quie well, improving wih each ucceive order. The econd-order conumpion funcion, which begin o accoun for precauionary aving, improve upon he zeroh-order, cerainy-equivalen conumpion funcion by abou a facor of en in erm of he relaive diance beween he perurbed and numerical conumpion funcion. Meanwhile, he relaively large negaive kewne of hi example lead o addiional precauionary aving, and he hird-order conumpion funcion improve upon he econd-order approximaion by a facor of five. To ee wha happen wih poiive kewne, conider he revere exampleoffig. 2wherep =0.1, correponding o a iuaion where an agen ha a en percen chance of receiving a windfall ha increae hi income by a facor of 3. The relaive performance of he hree perurbaive conumpion funcion i abou he ame for high wealh. In hi cae, however, he hird-order approximaion doe wore han he zeroh- and econd-order conumpion funcion in he low wealh regime. Thi i becaue he poiive kewne, inead of adding o precauionary aving, reduce i. Since he hird-order correcion goe a w 3 while he econd-order correcion goe a w 2, he kewne effec will acually ouweigh he precauionary aving effec from he variance erm, o ha hird-order conumpion i even greaer han he cerainy-equivalen conumpion. However, he effec of he Aiyagari borrowing conrain, which do no appear in hee perurbaion calculaion, will overwhelm he kewne effec, reducing he acual conumpion funcion far below even he econd-order approximaion. To cloe hi ecion, we can ue hird-order conumpion funcion (38) o derive an expreion for he expeced rae of conumpion growh (ee Ap- 9 For more deail on hi procedure, ee Feigenbaum (2001a). 15

16 Figure 1: Relaive diance beween he nh-order conumpion funcion c n and he numerically compued conumpion funcion bc for an agen wih 41 period remaining, given R = β 1 = , γ = 2, and he income diribuion (39) wih p =.9 andδ =0.5, o he variance i k 2 =0.09 and he kewne i k 3 =

17 Figure 2: Relaive diance beween he nh-order conumpion funcion c n and he numerically compued conumpion funcion bc for an agen wih 41 period remaining, given R = β 1 = , γ = 2, and he income diribuion (39) wih p =.1 andδ =0.5, o he variance i k 2 =0.09 and he kewne i k 3 =

18 pendix C): E [ec ] =(βr) 1/γ c γ +1 2 δ 2 k +1 2, E [ ew +1 ] (γ +1)(γ +2) 2 6 δ 3 k +1 3, E [ ew +1 ] 3 + O(δ 4 ). (40) Noice ha only he momen of income a period, ey, ener hi relaive expreion. Unlike he abolue conumpion funcion, conumpion growh doe no depend on he momen of income a laer period, o he cumulaive momen K n, do no appear in (40). If we reric our aenion o econd order, we can obain a imilar expreion for he acual rae of conumpion growh and no ju i expecaion 10. ec c +1 =(βr) 1/γ ew w γ +1 δ 2 k2, +1 2 E [ ew +1 + O(δ 3 ). (41) ] 2 Given he aumpion we have made regarding he income proce, hi i equivalen o he expreion for he rae of conumpion growh obained by Skinner (1988). 4 Large Lifeime Limi of Policy and Value Funcion In Appendix B we compue he lowe-order conribuion of he nh momen o boh he value funcion and he bond demand funcion for n 2: v (n),+1 Γ(γ + n 1) =( 1)n+1 δ n (Kn, +1 kn, +1 )( +2) γ (w +1 n!γ(γ) 1 ) 1 γ n + O(δ n+1 ) (42) b (n) Γ(γ + n), =( 1)n n!γ(γ +1) +1 δ n (K +1 ( +2) n, k +1 w n 1 n, ) + O(δ n+1 ), (43) where he K aify he difference equaion (36). Le u focu on he cae where he only variaion in income over ime come from he preence of economic growh i.e. he diribuion of ex i independen of. By aumpion hen, ex 0 and 0 G0 ex will have he ame diribuion for all, 0,, and 0. Thi will imply ha k 0 n, 0 = Gn(0 ) k n, 10 A hird order, cro erm involving produc of Σ 2,+1 and ex+1 will occur. Thee vanih upon aking expecaion. However, he acual conumpion growh rae will depend on Σ 2,+1 and herefore momen of income a laer period. 18

19 for all, 0,, and 0 and n 2. If we define bk n = G n K n, and bk n = G n k n,, we can rewrie (36) a bk n = ( +1)n 1 () n 1 µ n G bk 1 n + R b k n. (44) Now conider wha happen o K b n in he limi a,whichcorrepond o he limi of a large lifeime. We will aume ha R>Go ha neinereraearepoiiveandhepreenvalueoffuureincomeifinie. We can alo aume ha > 1 becaue hi i he uual condiion ha mu be aified in order o guaranee ha lifeime uiliy, defined by (1)-(2), i finie for all feaible conumpion pah. In ha cae, he limi lim ( +1) = lim () 1 1 = lim ( 1) +1 1 = 1 exi. Conequenly, he finiene of (42)-(43) depend enirely on he behavior of he coefficien K n, (and hereby b K n). For large, o Eq. (44) approximae o ( +1) +1 1 lim = lim () 1 =, bk n n 1 µ G R n bk 1 n + b k n (45) in ha limi. The oluion o (45) will be finie a large if and only if n 1 µ G R n < 1. (46) Le R G = β 1 G γ. (47) Thi i he inere rae for which conumpion will grow a he conan rae G in an economy wihou uncerainy, and i i he unique equilibrium inere rae for an infinie-horizon deerminiic economy whereincomegrowahe rae G. Thi alo correpond o he inere rae which aifie G = R. If R R G, hen we will have G R. Since we have alo aumed G<R,he 19

20 condiion (46) mu hold in ha cae. So for R R G, he expreion (42)-(43) will be well-defined for all n. Onheoherhand,ifweolveheinequaliy(46)oiolaeR, wefind he condiion hold only for R>R n = hβ 1 n γ G n i γ γ+n 1. The hrehold inere rae for order n, R n, i an increaing funcion of n and converge o R G a n. 11 Thi i a problem becaue when we inroduce precauionary aving ino an infinie-horizon economy, he increaed demand for bond will puh he inere rae below he inere rae, R G, for he deerminiic economy. If R<R G, hen here mu be ome n for which R>R n. Tha i, for ome n, henh-order value and policy funcion mu diverge. We have o aume ha R>G, which mean he economy mu be dynamically efficien, in order for he zeroh-order policy and value funcion ha we perurb around o exi. A long a > 1, which i alo needed for he zeroh-order funcion o exi, he nh-order bound will be a ronger bound han he bound required by dynamic efficiency ince if G R and > 1hen he condiion (46) will be violaed. 12 Moreover, ince i ricly greaer han 1, here will be a neighborhood of G uch ha all R in ha neighborhood will be below he econd-order hrehold R 2. Thu, we are no imply replicaing he rivial reul ha he economy mu be dynamically efficien in order for perurbaion heory o work becaue, oherwie, he oluion we are expanding around doe no exi. Wha do hee divergence in he perurbaion calculaion mean for he pracical applicaion of perurbaive mehod? Fir, we hould noe ha hee divergence are no real. If we olve for value and policy funcion numerically, we do no ee any ignifican change when he inere rae croe any of he hrehold pecified by (46). In Fig. 3, we compare he econd-order perurbaion calculaion of he conumpion funcion c 2 o a numerical calculaion bc for a cae where 41 period remain and G 2 /R 2 =1.16. Noice ha he numerical conumpion funcion i well-behaved, deviaing only lighly from a raigh line, and everywhere poiive. The effec of he divergence i beer een in Fig. 4, which compare he relaive diance beween he nh-order conumpion funcion c n and he numerical conumpion funcion bc for n =0 and n = 2. Noice ha he zeroh-order conumpion funcion i everywhere a beer approximaion han he econd-order conumpion funcion. If we adju he rik averion coefficien γ from 1.1 o 0.5, G 2 /R 2 increae o 1.43, and we 11 Becaue hee inere rae hrehold are cale-invarian, i i no poible o ecape or alleviae hi pahology by changing he ime cale. Suppoe he lengh of a period i τ. We define R = e qτ, G = e gτ,andβ = e ρτ,whereq, g, andρ are he he coninuou-ime inere, growh, and dicoun rae repecively. The condiion n 1 (G/R) n < 1holdif and only if q>( n 1 n 1 ρ + ng)/(1 + ), independen of τ. 12 γ γ Aufficien condiion for (R) > 1forR [G, R G ]ihag < R G ince ha will guaranee boh (G) and(r G ) > 1 and ince (R) i monoonic. 20

21 Figure 3: Second-order conumpion funcion c 2 and numerical conumpion funcion bc a a funcion of wealh w for β =0.8, γ =1.1, R =1.03, and an income diribuion given by (39) wih p =0.5 andδ =0.1. ge an even more pecacular failure of perurbaion heory a een in Fig Clearly, however, ince bc i well-behaved in hi cae alo, he divergence arie becaue perurbaive mehod have broken down, no becaue here i anyhing divergen abou he exac problem. Wha happen if G 2 /R 2 < 1? Alhough he econd-order correcion will no blow up, we know ha if R<R G hen he nh-order conumpion funcion mu diverge a large lifeime for ome n. Doehiinerferewih he approximaion abiliy of lower-order conumpion funcion? Conider Fig. 6, in which we compare he zeroh-order conumpion funcion o he econdorder conumpion funcion for a cae where 249 period remain, R =1.03 < β 1 = , and G 2 /R 2 =0.98. A uch a large lifeime, if he econdorder approximaion i going o be hrown off by divergence a higher order, we hould ee i here. However, he econd-order calculaion i alway beer han he zeroh-order calculaion and uually i ignificanly beer. Thu, even hough perurbaion calculaion mu diverge a ome finie order if R<β 1, hi doe no invalidae perurbaion calculaion a lower order. Noice ha Kn, will diverge if (46) i violaed no maer how large δ i. The coefficien of variaion could be le han one par in a billion, and Kn, 13 Acually, Fig. 5 omewha exaggerae he difference beween he econd-order conumpion funcion and he acual conumpion funcion becaue we have no enforced he nonnegaiviy conrain on conumpion in he perurbaion calculaion, and he econd-order conumpion funcion i negaive for all picured wealh value. 21

22 Figure 4: Relaive diance beween he nh-order conumpion funcion c n and he numerically compued conumpion funcion bc for an agen wih 41 period remaining, given β =0.8, γ =1.1, R =1.03 and he income diribuion (39) wih p =.5 andδ =0.1. In hi cae, G 2 /R 2 =1.16, o he econd-order conumpion funcion diverge a large lifeime. 22

23 Figure 5: Relaive diance beween he nh-order conumpion funcion c n and he numerically compued conumpion funcion bc for an agen wih 41 period remaining, given β =0.8, γ =0.5, R =1.03 and he income diribuion (39) wih p =.5 andδ =0.1. In hi cae, G 2 /R 2 =

24 Figure 6: Relaive diance beween he nh-order conumpion funcion c n and he numerically compued conumpion funcion bc for an agen wih 249 period remaining, given R =1.03 < β 1 = , γ =1.1, and he income diribuion (39) wih p =.5 andδ =0.1. In hi cae, G 2 /R 2 =

25 will ill grow wihou bound, alhough hi divergence migh only affec he value and policy funcion a giganic value of he remaining lifeime. Wha caue hi divergence? A we dicued in Secion 2, he perurbaion erie of expeced uiliy will only converge for value of wealh w above ome cuoff relaed o he variance of income. Given a conan income proce, hi cuoff hould no vary wih ime. Ye here we find ha for any value of w, here will be ome a which perurbaion correcion explode if he inere rae i le han R G. The dependence of hi effec on he inere rae ugge i i relaed o how he long-erm rend in conumpion growh depend on he inere rae. Coniderhecaewihougrowh. 14 From Eq. (40), we ee ha in he abence of uncerainy, conumpion will decay exponenially if he inere rae R i le han he dicoun rae β 1. In hi cae, conumer will pend heir wealh down o i minimum value, zero, which i le han he minimum value allowed if here i uncerainy (on accoun of he Aiyagari (1994) endogenou borrowing conrain). Converely, if R>β 1, conumpion will increae wihou bound. Conumpion will remain able over ime only if R = β 1. When uncerainy i urned on, conumpion will ill grow wihou bound almo urely if R>β 1 or if R = β 1 and income i ufficienly ochaic (Chamberlain and Wilon (2000)). If R<β 1, hen he convexiy of he aving funcion will preven agen from pending heir wealh down o i minimum. Inead, here will be a unique eady ae in he mapping of curren wealh o expeced nex-period wealh (Carroll (1997, 2001b)), and conumer will be araced o hi value of wealh, known a he buffer-ock wealh. Perurbaive mehod canno properly repreen he uiliy deriving from a fuure evenualiy where he agen ha near-zero wealh. Neverhele, if he buffer-ock wealh i high enough, one migh hope ha i would be exremely unlikely for an agen a or above he buffer ock o approach he minimum wealh and ha he conribuion o he value funcion from hi poible evenualiy would, hu, be negligible. However, he buffer ock i no a propery of he zeroh-order oluion ha we are expanding around. Evidenly, he zeroh-order behavior of pending wealh down o zero ge refleced in perurbaion correcion. A R approache β 1 from below, he rae a which agen pend down heir wealh in he zeroh-order oluion will decreae o nil, bu a low income hock can peed up hi proce. Since informaion abou low-income hock i conveyed by he higher-order momen of he income proce, i make ene ha he conribuion of each ucceive momen hould become pahological a a higher hrehold inere rae. While hee divergence arie in expreion of abolue conumpion and abolue wealh, i i worh noing ha hey cancel ou of ome relaive expreion. For example, he expreion for he rae of conumpion growh given by Eq. (40)-(41) depend on k2, and k 3, bu no on he cumulaive momen K2, and K3, where he divergence arie. Thi may be a pecial propery 14 Thecaewihgrowhwilleeniallybeheameifweubiueheconumpiono income raio and he wealh o income raio for conumpion and wealh repecively. 25

26 ha only hold o hird order hough. I i no known wheher i will peri a fourh and higher order where cro erm appear. Neverhele, hee divergence are one perurbaion pahology ha may no bee Euler-equaion regreion of conumpion growh. The finding above ha he economy mu be dynamically efficien for perurbaion heory o work may, however, relae o anoher perurbaion pahology involving Euler equaion. Carroll (2001a) ha found in imulaed economie ha he eimaion of econd-order approximaion o he Euler equaion give inconien eimae for he known preference parameer of hee arificial economie. However, all of he agen in hi example experience an inere rae le han or equal o heir income growh rae. Conceivably, hi may accoun for hi finding ha he econd-order Euler equaion i uch a poor approximaion o he rue Euler equaion. 5 Concluding Remark Perurbaive mehod can be a very powerful ool. In phyic, hey are commonplace, and heori have ued hem o accuraely predic he magneic momen of he elecron o a unning nine decimal place. Neverhele, perurbaive mehod are no foolproof. They need o be ued wih ome care. Applied blindly, hey can produce nonene ju like any oher mehod. Unforunaely, he heory underlying perurbaive mehod i limied. The Taylor heorem provide he foundaional bai for perurbaive mehod bu offer lile pracical guidance regarding heir uage. Given a funcion of he perurbaion parameer which i C a a choen poin, i eablihe ha a Taylor erie for hi funcion will converge wihin an open ball of ha poin, and he radiu of ha ball will depend on he analyiciy properie of he funcion. However, hi doe no help much if our knowledge of he funcion i limied o a finie number of erm in i Taylor expanion or if higher-order erm do no exi. In he rare iuaion whereweknowanenireperurbaion erie, we can probably olve he relevan problem analyically wihou any need for perurbaive mehod. If a radiu of convergence exi, for value of he perurbaion parameer near bu le han hi radiu, Taylor approximaion will only be accurae when compued o large order. For value of he perurbaion parameer above he radiu, Taylor approximaion are compleely uninformaive. Conequenly, i i of anamoun imporance ha any reearch involving perurbaive mehod clearly pecify wha he perurbaion parameer i. Similar conideraion apply if a radiu of convergence doe no exi. A he example of hi paper demonrae, ha may ofen be he cae in a dynamic model. The ingulariy in he CRRA uiliy funcion implie ha perurbaion expanion of expeced uiliy will only be valid a wealh level above ome lower bound. Thi rericion reflec he above dicuion and will hold boh in a aic model where agen make deciion involving only a ingle period of uncerainy and in a dynamic model where agen mu make a equence 26

27 of deciion involving muliple period of uncerainy. In he econd cae, however, he poenial exi for he agen o evenually leave he convergen region of he ae pace, and hi can caue individual erm in he perurbaion erie o diverge a he number of fuure period increae no maer how mall he perurbaion parameer i. The poibiliy of uch dynamic pahologie will have o be conidered in any model involving CRRA preference and ochaic hiorie. In fac, hey are a propery of mo of he hyperbolic abolue rik averion (HARA) cla of uiliy funcion (Feigenbaum (2001b)). (Quadraic and CARA uiliy are immune o he pecific pahology examined here.) Mo likely, hee pahologie will how up in any dynamic model involving perurbaive approximaion where he zeroh-order dynamic imply he ae variable hould ineviably leave he convergence region for he perurbaion erie. Tha aid, we have only conidered emporary income hock in hi model. Viceira (2001) i anoher perurbaive reamen of a conumpion/aving problem in which here are only permanen income hock. Thi i a omewha more racable problem under CRRA uiliy ince i ha more of he rucure of a andard porfolio-allocaion problem (Koo (1999)). Viceira doe no conider he poibiliy of dynamic pahologie, o i remain o be een wheher dynamic pahologie will arie in he abence of emporary income hock. I hould emphaize here ha while we eablihed a lower bound on he e of inere rae a which he perurbaion erie o a given order will remain finie a large lifeime, we only conidered divergence ha arie from he lowe-order conribuion of each momen of he income proce. Higherorder conribuion may induce igher bound. For example, a fourh order, he cro erm ariing from produc of variance may produce a igher bound on he inere rae han he bound we compued which come from he pure kuroi erm. Depie hee word of cauion, I do no wih o overly dicourage reearcher from exploiing perurbaive mehod. In Feigenbaum (2001a), I cloe he economy in he preen paper by inroducing an overlapping generaion (OLG) rucure. If we coninue wih he aumpion ha he income proce ha a compac diribuion, hen we can ubiue our perurbed bond demand ino he marke-clearing equaion and olve for perurbaion correcion o he inere rae. The reul i an approximae expreion for he inere rae ha i a funcion of exogenou parameer, and, in fac, he econd-order expreion expreion i quie imple in he limi of large lifeime 15. I find ha, a lea for hor lifeime, hi econd-order predicion compare favorably o numerical reul. A a numerical mehod, hybrid perurbaive mehod may be uperior o value-funcion ieraion for he purpoe of compuing inere rae in an economy wihou borrowing conrain if hey can more efficienly approximae bond demand funcion. Auming only a mall meaure of agen fall ouide he region of he ae pace where perurbaive mehod are inaccurae, one 15 For realiic value of he exogenou parameer, he bond demand hould no diverge a large lifeime o econd order. However, even if hey do, hee divergence will no how up inheinereraeunilfourhorder. 27

28 can aggregae hee perurbed demand funcion and olve for he equilibraing inere rae wih a numerical equaion olver. I would no advocae uing perurbaive mehod o compue inere rae direcly a a numerical mehod for wo reaon. Fir, hi would be inefficien ince a much greaer effor i needed o compue perurbaion correcion han i required o inpu funcion ino an equaion olver. Second, direc compuaion of inere rae would only be poible if he income diribuion i compac. However, a we have een, a compac income diribuion i no required o accuraely perurb policy funcion for agen a high enough level of wealh. Throughou hi paper, we have mainained he aumpion of no exogenou borrowing conrain. Thi aumpion wa made becaue even he zeroh-order problem wihou uncerainy become much more complicaed when borrowing conrain are inroduced. Neverhele, hi problem i ill analyically olvable for finie lifeime. Wih borrowing conrain, value and policy funcion will be piecewie mooh funcion. The number of inerval required o define hee funcion will grow a he lifeime increae, and for ha reaon he problem quickly loe racabiliy a large lifeime. Ye for mall lifeime, he zeroh-order problem will be racable and can be perurbed around. In Feigenbaum (2002), I ue perurbaive mehod o udy how borrowing conrain and precauionary aving inerac in a imple general-equilibrium model. A a heoreical ool, I believe hi la manner of applicaion i where perurbaive mehod will be mo beneficial. They can be ued quie effecively o analyze model which are imple enough o afford inigh abou he working of more complicaed model and ye which are complex enough hemelve o be unolvable by exac mehod. A A Brief Inroducion o q-arihmeic For real number q 0andn, wedefine he q-analog of n a (n) q = 1 qn 1 q. In he limi a q 1, we can apply l Hôpial rule o evaluae (n) q : lim (n) nq n 1 q = lim = n. q 1 q 1 1 Thu a q deviae from 1, (n) q i a deformaion of n. Noe ha (0) q = 0 and (1) q = 1 for all value of q. ineger, For n a poiive n 1 X (n) q = q i =1+q q n 1. i=0 28