WORKING PAPERS IN ECONOMICS No 351 Rsk Averson and Expeed Uly of Consumpon over me Olof Johansson-Senman Aprl 29 ISSN 143-2473 (prn) ISSN 143-2465 (onlne) Deparmen of Eonoms Shool of Busness, Eonoms and Law a Unversy of Gohenburg Vasagaan 1, PO Box 64, SE 45 3 Göeborg, Sweden +46 31 786, +46 31 786 1326 (fax) www.handels.gu.se nfo@handels.gu.se
Rsk Averson and Expeed Uly of Consumpon over me Olof Johansson-Senman Deparmen of Eonoms, Unversy of Gohenburg, Box 64, SE-453 Göeborg, Sweden Phone: + 46 31 7862538, Fax: + 46 31 786143, E-mal: Olof.Johansson@eonoms.gu.se Absra he albraon heorem by Rabn (2) mples ha seemngly plausble smallsake hoes under rsk mply mplausble large-sake rsk averson. hs heorem s derved based on he expeed uly of wealh model. However, Cox and Sadraj (26) show ha suh mplaons do no follow from he expeed uly of nome model. One may hen wonder abou he mplaons for more appled onsumpon analyss. he presen paper herefore expresses uly as a funon of onsumpon n a sandard lfe yle model, and llusraes he mplaons of hs model wh expermenal small- and nermedae-sake rsk daa from Hol and Laury (22). he resuls sugges mplausble rsk averson parameers as well as unreasonable mplaons for long erm rsky hoes. hus, he onvenonal neremporal onsumpon model under rsk appears o be nonssen wh he daa. Key words: Expeed uly of nome, expeed uly of fnal wealh, dynam onsumpon heory, asse negraon, me nonsseny, narrow brakeng JEL lassfaon: D81, D91 Aknowledgemen: I am graeful for very onsruve ommens from wo anonymous referees, an advsory edor, Marn Dufwenberg, Kjell Arne Brekke, homas Aronsson, Frank Henemann, Rober Öslng, Rhard haler, Peer Wakker, Fredrk Carlsson, Mahas Suer and semnar parpans a Umeå Unversy, as well as for fnanal suppor from he Swedsh Researh Counl and he Swedsh Inernaonal Developmen Cooperaon Ageny (Sda). 1
1. Inroduon How well expeed uly (EU) heory desrbes human behavor n general, nludng n small- and nermedae-sake gambles, has reenly been dsussed nensvely. A he ore s wha expeed uly s expressed as a funon of. hs noe provdes a smple exenson of some mporan aspes of hs dsusson o a lfe yle seng where people derve uly from onsumpon (nsead of wealh or payoffs), and llusraes hs wh numeral mplaons based on expermenal daa from Hol and Laury (22). Rabn (2) presens an mporan heoreal onrbuon n erms of a albraon heorem ha mples onlusons of he followng knd: If for all wealh levels an expeed uly maxmzng person urns down a 5-5 lose $1/gan $2 gamble, he would also urn down a 5-5 lose $2/gan $2, gamble. Whle may seem plausble ha some people would urn down he frs gamble (for all wealh levels), seems muh less reasonable o urn down he seond. Aordng o Rabn and haler (21, 26): Even a lousy lawyer ould have you delared legally nsane for urnng down hs be. 1 An mporan feaure of hs albraon heorem s ha does no assume anyhng regardng he funonal form of he uly funon. However, he for all wealh levels par of he heorem s mporan. Alhough one an derve less exreme versons whou hs assumpon, one mus sll assume 1 Gven ha expeed uly refers o expeed uly of wealh, s aually sraghforward o derve an even sronger onluson, as follows: If for all wealh levels an expeed uly maxmzng person urns down a 5-5 lose $1/gan $2 gamble, he would also urn down a 5-5 lose $2/gan nfny gamble. Le K denoe he (ardnal) gan n uly U from a wealh nrease from w o w+ 2, where w s nal wealh. hen f he ndvdual urns down a 5-5 lose $1/gan $2 gamble, follows by onavy ha he uly loss from a wealh hange from w o w 2 s a leas 2K. Sne hs holds for all nal wealh levels would also hold for he nal wealh w+ 2; hene we know ha a wealh nrease from w+ 2 o w+ 4 mples a U nrease of less hen K/2, and ha a wealh nrease from w+ 2( r 1) o w+ 2r, where r s an arbrary 1 posve neger larger han 1, mples a uly nrease of less han ( K /2) r. Hene, he uly hange for a r 1 wealh nrease from w o w+ 2r s less han r K /2 = 2(1. 5 ) K. Consequenly, he expeed uly = r r hange of a 5-5 lose $2/gan $2r gamble s less han (1.5 ) K K =.5 K. hus, he expeed uly hange s negave rrespeve of r,.e. rrespeve of he gan. (One an easly oban less exreme versons by replang he for all wealh levels wh for wealh levels up o w+ Δ w. ) Moreover, by replang $1 wh an arbrary posve number A n he above analyss, follows more generally ha, If for all wealh levels an expeed uly maxmzng person urns down a 5-5 lose A/gan 2A gamble, he would also urn down a 5-5 lose 2A/gan nfny gamble. 2
ha he ndvdual would have made he same hoe had he been subsanally wealher han wha he aually s (see Rabn and haler and foonoe 1 n he presen paper). Largely based on he mplaons of hs heorem, Rabn (2a, b) and Rabn and haler (21) argue more generally ha EU heory anno explan behavor based on small-sake gambles, and hene ha we need some oher heory; hey sugges a ombnaon of loss averson and menal aounng. However, Cox and Sadraj (26) queson hs onluson n a reen paper. hey show ha for he small-sake rsk averson assumpon of Rabn (2), mplausble large-sake rsk averson would no follow for he expeed uly of nome (EUI) model, where uly s expressed as a funon of payoffs, n onras o he expeed uly of fnal wealh (EUW) model. 2 Moreover, sne he global small-sake rsk averson assumed by Rabn (2) has no mplaon for he EUI model, has no general mplaon for EU heory eher. I s lear ha Cox and Sadraj have a vald and mporan pon sne EU heory s very general and bulds on a se of axoms ha do no prelude ha uly may depend on wealh, nome, expermenal payoffs, or almos any sae varable. 3 In he lgh of he fndngs by Cox and Sadraj, one may be nlned o onlude ha wha has beome known as he Rabn rque s oversaed. Perhaps appled eonomss neresed n measurng people s rsk preferenes or analyzng behavor based on exsng esmaes an gnore he Rabn rque and onnue o nerpre her resuls n erms of he onavy of unversally vald uly funons? However, he resuls n hs paper sugges ha suh a onluson would be premaure. In appled eonom analyss people ofen make desons over me, dervng nsananeous uly based on her presen onsumpon level. Under rsk, he onvenonal 2 Alhough s lear from Rabn (2 a, b) and Rabn and haler (21, 22) ha hey fous on he EUW model n her analyses, some of her saemens may seem o mply (or a leas have been nerpreed o mply) a rsm of expeed uly heory more generally. Cox and Sadraj (26) also onsder a more general woargumen model where uly depends on boh nal wealh and payoff. 3 Samuelson (25), Rubnsen (26), and Harrson e al. (27) have provded smlar argumens. 3
assumpon s hen ha people maxmze he expeed presen value of fuure nsananeous uly (e.g. Deaon 1992; Goller 21). We wll denoe hs model he expeed uly of onsumpon over me (EUC) model, and ake as our pon of deparure. An obvous example s how o bes nves reremen savngs; see e.g. Gomes and Mhaeldes (25). 4 In he EUC model, uly s expressed as a funon of a flow varable (unlke he EUW model),.e. onsumpon, and mples omplee asse negraon (as n he EUW model), meanng ha he gans from a rsky hoe wll be reaed n exaly he same way as nome or wealh obaned n any oher way. he man onrbuons of he presen paper an be summarzed as follows: Frs, he relaons beween he EUC model, on he one hand, and he EUW and EUI models, on he oher, are analyzed n Seon 2. I s onluded ha he EUC model s essenally equvalen o he EUW model when he wealh measure n he EUW model onsss of he presen value of all fuure onsumpon or nome. In addon, s shown ha he funonal form of he nsananeous uly funon, whh s expressed as a funon of urren onsumpon, arres over n a sraghforward way o a uly funon ha s expressed as a funon of he presen value of all fuure nomes, f and only f he nsananeous uly funon belongs o he lass of funons haraerzed by hyperbol absolue rsk averson (HARA), whh s a flexble funonal form ha nludes CRRA and CARA funons as speal ases (Meron, 1971). Seond, by usng daa from a areful expermenal sudy by Hol and Laury (22), Seon 3 analyzes wheher observed behavor n small- and nermedae-sake sze rsk expermens an be reonled by he EUC model. he answer s negave. he alulaed mpl rsk averson parameers are found o be unreasonably large, and herefore an no 4 Long-run envronmenal problems, suh as he greenhouse effe, onsue anoher mporan example where boh me and rsk are rual. I s also ypally shown ha he profably of exensve abaemen oday depends rally on he dsoun rae hosen, whh n urn depends srongly on he onavy of he nsananeous uly funon; see e.g. Sern (26) and Nordhaus (27). 4
onsue onavy measures of unversally vald nsananeous uly funons. Moreover, srong mplaons are derved wh respe o wha hese degrees of rsk averson would orrespond o for long-erm rsky hoes n erms of fuure nome or onsumpon levels. For example, wheher based on onsan relave rsk averson (CRRA) or onsan absolue rsk averson (CARA) preferenes, a majory of he subjes from Hol and Laury would (gven ha hey are EUC maxmzers) n he base ase prefer an nome level ha wh erany would enable hem o for he res of her lfeme onsume 36, USD annually, raher han a rsky alernave where hey wh a 1% probably would be able o onsume 35,99 USD annually and wh a 99% probably an nfne amoun. hs learly seems mplausble. Smlar mplausble resuls are hen obaned also for broader lass of HARA preferenes. Seon 4 generalzes and demonsraes ha he man onlusons hold also under uneran fuure nomes and for a me-nonssen formulaon. Compared o he resuls based on he EUW model by Rabn (2) and Rabn and haler (21), he resuls here are less general n he sense ha hey depend on spef funonal forms. On he oher hand, he resuls here are less resrve n he sense ha hey do no rely on any assumpon ha he hoes would have been he same for all lfeme wealh levels, or for any hgher lfeme wealh levels han he ndvduals urrenly have or expe o oban. 5 Seon 5 onludes ha he sandard EUC model appears nonssen wh avalable expermenal small- and nermedae-sake daa. 2. he EUC model he sandard approah when dealng wh neremporal hoes under unerany s o maxmze he expeed presen value of fuure nsananeous uly (e.g. Deaon 1992; Goller 5 However, noe ha he hoes for CARA preferenes would have been he same for all lfeme wealh levels. When uly s CRRA, by onras, we know ha an ndvdual ha s ndfferen beween aepng a rsky gamble would always aep for hgher wealh levels han he presen one. 5
21). Le us sar wh he neremporal onsumpon hoe under erany and n he nex sep ake rsky desons no aoun. 2.1 he neremporal hoe problem and HARA preferenes Here an ndvdual experenes he nsananeous uly ( ) u a me (from now), where u s nreasng and srly onave. Assume ha he ndvdual wll lve for more years, and, as s sandard, an addve and me onssen uly formulaon suh ha he ndvdual wll maxmze U = u( ) e ρ d, (1) where ρ s he pure rae of me preferene, somemes denoed he uly dsoun rae. We wll refer o U as uly. Under erany, U s purely ordnal, so ha any monoon ransformaon of U s permssble and hene onsues an equally vald measure of uly. Under rsk, however, eah possble uly ouome U mus be nerpreed n a ardnal sense, so ha only affne ransformaons are permssble. he expeed value of U, E(U), s neverheless sll ordnal. 6 he neremporal budge onsran mples ha he presen value of fuure onsumpon equals he presen value of fuure nome, so ha r r e d= ye d Y, (2) where r s he marke neres rae. he assoaed Lagrangean an hen be wren as ρ r L = u ( ) e d+ λ ( y ) e d, mplyng he orrespondng frs order ondons 6 hs means for example ha under erany, U lnu s an equally vald measure of uly as U, n he sense ha an ndvdual ha hooses a onsumpon pah n order o maxmze U, wll also maxmze U. However, he only ransformaons of u ha leaves he opmal onsumpon pah unaffeed are affne ransformaons; hene u s ardnal and unque only up o affne ransformaons. Under unerany, where we hoose beween dfferen loeres, we nsead wan o maxmze EU = p U. he opmal hoe would hen be unaffeed by any monoon ransformaon of EU, whereas only affne ransformaons of U, desrbng he uly n sae, leave he hoe unhanged generally, and are hene permssble. Consequenly, s ardnal. U 6
u'( ) = λe = u'( ) e ( ρ r) ( ρ r), (3) whh ogeher wh he budge resron deermne he opmal onsumpon pah. 7 Sne he ndvdual maxmzes U gven a eran presen value of lfeme nome Y, we an alernavely wre U = V( Y), for a fxed neres rae. We wll subsequenly analyze mplaons of hoes beween small-sake loeres wh respe o wha hese hoes would mply n erms of rsk averson measures when people are EUC maxmzers, and also wha hey would mply n erms of large-sake hoes. In dong so, we would lke o know he relaonshp beween he nsananeous uly funon and our measure of uly as expressed as a funon of Y. More spefally, we would lke o know under whh ondons he funonal form arres over from u () ov (). For example, f u () s CRRA, an we hen know ha also V () s CRRA? If hs s he ase (and urns ou ha s), smplfes he analyss largely, sne s hen sragh forward o redue he dynam problem o a sa analogue and work wh he V-funon nsead of he u-funon. We wll sar by onsderng a more general resul on he relaon beween u () o V (), followed by a more spef resul whh s sragh forward o apply n he subsequen numeral analyss: Proposon 1. he funonal form of u arres over o V, n he sense ha we an eher * wre uly as U = V( Y) or as Uˆ = u( ( Y) ), where Uˆ s an affne ransformaon of U, f and only f any of he followng equvalen ondons are fulflled:. he opmal onsumpon n perod an be wren as an affne funon of Y, suh ha * = a () r + b() r Y, where a () r and b () r may depend on and r, bu are ndependen of Y.. he nsananeous uly funon u s HARA, suh ha u = ( ) ( β α + β 1)/ β 1 β. 7 For example, when ρ = r follows ha u '( ) = λ, mplyng ha also s onsan over me. Inuvely, people wan o smooh her onsumpon over her lfe yle n order o equalze her margnal nsananeous uly of nome, whh s a sandard resul n dynam onsumpon heory (e.g. Hall, 1978). 7
Proof: see Appendx. Noe ha sne Uˆ s an affne ransformaon of U, s by defnon an equally vald measure of uly. hs means ha as long as we know ha he onsumpon pah s opmal, he nsananeous uly n any pon n me (e.g. a presen or en years from now) s an equally vald measure of uly,.e. of he presen value of he nsananeous uly over he whole lfeme perod. Moreover, sne Uˆ s an affne ransformaon of U, and no jus a monoon ransformaon, follows ha for HARA preferenes Uˆ s also a vald measure of von Neumann-Morgensern uly under rsk. For our purposes, an even more useful resul follows drely from Proposon 1, expressed n erms of he annuy Y / S, where r 1 e S s he annuy faor: 8 r ( ) ( β α + β 1)/ Proposon 2. If he nsananeous uly funon s HARA, suh ha u =, β 1 follows ha an affne ransformaon of U, U ( ) ( β 1)/ β α β, s also HARA suh ha U + = β 1. Proof: see Appendx. Noe frs agan ha sne U s an affne ransformaon of U, U s boh a vald measure of uly under erany, and a vald measure of von Neumann-Morgensern uly under rsk. hs resul wll be used repeaedly n Seon 3. Noe also ha Proposon 2 holds wheher a onsan onsumpon pah s opmal or no,.e. wheher ρ = r or no. I s easy o verfy ha he Arrow-Pra oeffen of absolue rsk averson, defned based on he nsananeous uly funon, s n he HARA ase gven by β A u '' 1 = u ' α + β, and ha he orrespondng oeffen of relave rsk averson s gven by R u '' = u ' α + β. I follows ha he nsananeous uly funon s haraerzed by 8 hus, an ndvdual ould exaly afford he onsan onsumpon level for he res of hs lfeme. 8
CRRA for he speal ase when α =, mplyng ha u ( β 1)/ β 1 R = =, where R = 1/ β, β 1 1 R and hene ha we an wre U = ( o ) 1 R 1 R. Smlarly, u onverges owards CARA when β approahes, so u e A A / α = =, where A 1/ αe = α, and we may wre e U =. A A 9 hese resuls wll also be used n he numeral alulaons n Seon 3. 2.2 Inrodung rsk Consder now a loery wh he nome pah y for wh probably p, where he realzed nome pah s revealed before he onsumpon pah s hosen. Expeed uly s hen gven by EU n * = 1( u ( ) e d) ρ p = = = 1 n V ( Y + x ) p, (4) where eah elemen of he opmal onsumpon pah wll sasfy (3), and where s he * x loery gan. Agan, we see ha he EUC model s equvalen o he EUW model n he ase where wealh s defned as he presen value of all fuure nomes Y. Noe ha (17) holds generally, whereas n he ase of HARA preferenes we also have ha he funonal form arres over from u () o V (). I s also noeworhy ha Proposon 2 mples ha he hoe of an ndvdual wh HARA preferenes n a hoe beween loeres wh dfferen lfeme nomes, mplyng dfferen feasble fuure onsan onsumpon sreams, s ndependen of he ndvdual s me preferene ρ. Assumng ha he poenal gans x (whh an be posve or negave) from he loery our oday, we an wre expeed uly as EU n ( = V Y + x ) p. Aordng o he so- = 1 9 Noe agan ha any affne ransformaons are permssble. Noe also ha whle he parameer of relave rsk averson s dmenson free and sale ndependen, he parameer of absolue rsk averson s no dmenson free and an e.g. be expressed per dollar un. 9
alled Arrow-Pra approxmaon (see e.g. Goller 21, p. 22), for small rsks he rsk premum ψ s approxmaely gven by var( x) ψ ψ A, so ha A 2 and hene 2 var( x ) ψ S R 2, where A V ''/ V ' and R YV ''/ V ' are he assoaed oeffens of sd( x ) sd( x ) absolue and relave rsk averson, respevely. he leraure based on lfe yle onsumpon behavor ofen refers o values of R n he.5-3 range. 1 Aordng o Koherlakoa (1996, 52), A vas majory of eonomss beleve ha values above 1 (or, for ha maer, above 5) mply hghly mplausble behavor. he rao beween he presen value of all fuure onsumpon and he sandard devaon of he moneary ouome of a rsk expermen s ypally very large. hs mples ha he rsk premum mus be a ny fraon of he sandard devaon of he moneary ouome for he behavor n he rsk expermen no o be desrbed as hghly mplausble by he above quoaon, whh wll be llusraed furher n he nex seon. 3. Numeral llusraon based on daa from Hol and Laury (22) here are many suable expermenal sudes ha ould be used o llusrae he mplaons of he above model, bu le us here rely on he well-known and arefully underaken sudy by Hol and Laury (22), who eled he rsk preferenes of (manly) US unversy sudens by usng real money expermens wh dfferen sake szes. Eah suden made a number of parwze hoes beween one less rsky (Opon I) and one more rsky (Opon II) gamble; see able 1 for a relevan sub-se. Indfferene beween Opon I and Opon II hen mples a eran degree of rsk averson, and he hoes were ordered so ha ndfferene beween he opons mples larger and larger rsk averson. By observng a wha pon a subje swhed 1 For example, Blundell e al. (1994) and Aanaso and Brownng (1995) found, n mos of her esmaes, R o be n he order of magnude of 1 or slghly above. Vssng-Jørgensen (22) found ha R dffers beween sokholders (approx. 2.5 o 3) and bond holders (approx. 1 o 1.2). 1
o Opon II, hey obaned a rsk averson range n whh he subje belongs. Hol and Laury used several dfferen funonal forms, nludng he flexble expo-power funonal form ha nludes CRRA and CARA as speal ases, bu dd no negrae he gans wh oher expeed lfeme nomes,.e. n lne wh he EUC model. 3.1 CARA and CRRA preferenes In order o es he mplaons of he EUC model wh real daa, le us frs fous on he wo mos ommonly used funonal forms, CRRA and CARA, 11 where he nsananeous uly funon an hene be wren as A u = /(1 R) and u = e, respevely. From 1 R Proposon 2 ogeher wh (4) we have ha when an ndvdual s ndfferen beween wo loeres, I and II, we have: 1 R n ( + / ) = ( + / ) n I I II II = 1 = 1 p x S p x S n I Ax I / S n II Ax II / S p 1 e = p 1 e = = 1 R, (5). (6) From (5) and (6) we an easly solve numerally for R and A. Consder now for omparson he EUI model where he loeres are evaluaed n solaon, and hene ndependen of oher nomes. he EUI model s herefore of ourse n general no onssen wh EUC. In he CRRA ase we have: 1 R n ( ) = ( ) n I I II II p 1 x p 1 x = = 1 R. (7) Clearly, sne x s ypally small ompared o, (7) should generally resul n a smaller R han (5), when ndfferen beween he wo loeres. However, n he CARA ase, where nal wealh does no affe hoes, (6) sll holds (orreed for he sale of A). he reason s of ourse ha he expeed uly hange of a loery s here ndependen on he nal S 11 Followng onvenon, hese names jus refle he funonal form of he nsananeous uly funon. Wha hese funonal forms mply n erms of aual hoes under rsk depends of ourse also abou oher assumpons of he model. 11
wealh level, whh s only rue for CARA preferene. Consequenly, he EUI model s equvalen o he EUC model for he CARA, and only he CARA, nsananeous uly funon. Consder frs for omparson he resul of he EUI model, where he expermenal gans are evaluaed ndependenly of people s baselne nome and wealh levels. I an be observed from able 1 ha, based on he CARA preferenes as expressed n (6), he medan parameer of absolue rsk averson A s beween.11 and.299 based on he low-sake loery, and beween.15 and.26 based on he hgh-sake loery. Based on CRRA preferenes, he medan parameer of relave rsk averson R s alulaed from (7) o be beween.146 and.411 based on he low-sake loery, and beween.411 and.676 based on he hgh-sake loery. Consder now he onvenonal EUC model. In he CARA ase, A of ourse remans he same, sne wh CARA preferenes he hoe beween rsky opons are ndependen of nal wealh; f. e.g. Rabn and Wezsäker (27). Sne he parameer esmaes dffer largely beween he hgh- and low-sake loeres, hs suggess ha CARA does no onsue a good approxmaon of subje preferenes. However, he man onern here s wheher he orders of magnude onsue reasonable refleons of globally vald nsananeous uly funons. In he CRRA ase, we learly need esmaes of S and n order o solve for R n (5). Le us herefore assume ha he subjes are 2 years old, ha hey expe o lve unl hey are 8 (.e. ha hey have 6 years lef), ha he real marke neres rae s 5% annually, and ha hey que pessmsally wll earn fuure nomes ha wll enable hem o onsume = 1, USD per year (a oday s pre level). For example, he seond hgh payoff loery n he Hol and Laury expermen orresponds hen o a loery beween he presen values of fuure nomes, suh ha he subjes n opon I an afford a onsan annual onsumpon of 12
12.148 USD wh probably.6 and 11.6839 USD wh probably.4, and n opon II 14.518 USD wh probably.6 and 1.1524 USD wh probably.4. As an be seen n able 1, he medan R s now larger han 19, based on he lowsake loery, and larger han 2,8 based on he hgh-sake loery. hese are learly values way above wha s generally onsdered o be plausble,.e. values n he range of.5 o 3, or n any ase onsderably smaller han 1. 12 Noe ha we have made no assumpon regardng he pure rae of me preferene ρ, and all resuls are ndependen of wheher he sudens aually would prefer o have a fuure nreasng or dereasng onsumpon pah over me. If he fuure annual onsumpon of he subjes would be larger han 1, USD, hen he mpl parameers of relave rsk averson would of ourse be even larger. However, one may also beleve ha sudens have lqudy onsrans and hene fae a larger real neres rae han ohers. Le us herefore make he exreme assumpons of an annual real neres rae of 5% (nsead of 5%). Solvng for R n (5) neverheless agan reveals absurdly large values, as he las olumn of able 1 shows. <<able 1 abou here>> hus, we have seen ha he hoes n Hol and Laury mply absurdly large rsk averson oeffens f based on CRRA preferenes, whereas he oeffens are denal beween he EUI model and he EUC model n he ase of CARA preferenes. However. sne A s no dmenson free, may be dfful o have a good nuon abou wha a reasonable range of A s. One perhaps empng nerpreaon ould be ha he EUC model works perfely fne, bu ha people have CARA preferenes (or smlar) raher han CRRA preferenes. However, even f one s wllng o gnore he A dsrepanes beween he small- and large-sake expermens, hs s no a plausble onluson. o see hs, onsder he followng gamble: In a safe alernave he ndvdual would oban he presen 12 Independen from hs sudy, Sheher (27) also obaned absurdly large parameers of relave rsk averson n a rsk expermens based on a sample n rural Paraguay. 13
value of all fuure nome equal o 5 mllon USD. In a rsky alernave, he ndvdual would nsead wh he probably of 1 % oban 4.9999 mllon USD, and wh 99 % oban an nfne amoun. Presumably, mos people would prefer he rsky alernave. However, an ndvdual wh A =.11 would aually prefer he safe alernave. 13 Hene, A =.11 s ndeed unreasonably large. We wll nex more sysemaally look no he mplaons of he hoes n he Hol and Laury loeres, for mpled hoes n large sake loeres expressed n erms of fuure onsumpon possbles. Consder he hoe beween a safe and a rsky opon onernng a subje s fuure nome. In he safe opon he wll wh erany for he res of hs lfe earn an amoun orrespondng o a onsan annual onsumpon of S per year. In he rsky opon he wll wh probably p oban he hgh fuure nome level ha orresponds o a onsan annual onsumpon level of H, and wh probably 1 p a low fuure nome orrespondng o L L he onsan annual onsumpon level. We an hen solve for from (5) and (6) for he CRRA and he CARA ases as follows: L 1 1 R 1 R 1 R S H ( ) p( ) = 1 p, (8) L 1 1 p = ln S A A e pe A H. (9) In he speal ase where he luky ouome mples an nfne onsumpon level, and where R > 1 and A >, (8) and (9) redue o: L = (1 p) 1/(1 R) S, (1) L S 1 = + ln(1 p ). (11) A <<able 2 abou here>> 13.11 5 1 hs s beause 6.11 4.9999 1 e >.1e 6.99. 14
able 2 llusraes he ase where he luky onsumpon level s nfne, and where moreover he probably of a luky ouome s as hgh as 99%. Consder frs he CRRA ase wh a 5% neres rae. he frs lne of able 2 reveals ha ndfferene beween he safe and he rsky opon mples he same R as ndfferene beween Opon I and Opon II n able 1. Consequenly, f people s behavor an be desrbed by he EUC model wh CRRA preferenes, he same fraon (66%) would prefer he less rsky opon. hs means ha 66% of he subjes n Hol and Laury would aually prefer beng able o onsume 36, USD annually wh erany raher han beng able o onsume an nfne amoun wh a 99% probably and 35,991 USD annually wh a 1% probably. If we nsead draw on he resuls from he hgh payoff loery n Hol and Laury, he resuls beome less exreme, alhough only slghly. Indeed, as shown from he ffh lne, as many as 62% would prefer he safe opon (36, USD annually) before a rsky one wh a 1% probably of beng able o onsume 35,942 USD per year and a 99% probably of ganng nfne onsumpon. If we onsder he exreme ase of 5% neres per year, he mpled hoes are sll absurd. Moreover, as observed n he hrd and fourh olumn of able 2, when onsderng CARA (nsead of CRRA) preferenes he resuls are onssenly even more exreme. 14 3.2 More general HARA preferenes Whle HARA s he mosly used flexble funonal form of he uly funon, he seond mos used s he so-alled Expo-power uly funon (Saha, 1993). Boh of hese flexble forms nlude CRRA and CARA as speal ases. However, sne he Expo-power funon has some unarave haraerss, n parular n regons of exreme rsk averson, we wll here fous on he HARA funon. Sll, we wll brefly desrbe some feaures of he Expopower funon, and how n prnple an be used, n he Appendx. 14 An mporan reason for hs s he pessms assumpon regardng he subjes fuure nome ha underles he R esmaes n able 1. 15
he HARA nsananeous uly funon, u = ( ) ( β α + β 1)/ β 1 β, mples dereasng absolue rsk averson ( A < ) for β >, and nreasng absolue rsk averson for β < ; we also observe dereasng relave rsk averson for α < and nreasng relave rsk averson for α >. I s also sraghforward o see ha hs nsananeous uly funon s globally onave as long as α < β, and ha boh A and R are everywhere nreasng n α and β. When an ndvdual s ndfferen beween wo loeres, I and II, we have n I I ( β 1)/ β n II II ( β 1)/ β p ( α β( x / S) ) + + = p ( α + β( + x / S) ). (12) = 1 = 1 From (12) we an solve for β for a gven value of α, and ve versa, or solve for eher α or β for a spefed relaonshp beween hem. I s onvenen for presenaonal purposes o rewre (12) for β > as n I I ( β 1)/ β n II II ( β 1)/ β p ( α / β x / S) + + = p ( α / β + + x / S), (13a) = 1 = 1 whereas we for β < nsead have n I I ( β 1)/ β n II II ( β 1)/ β p ( α / β x / S) = p ( α / β x / S). (13b) = 1 = 1 Moreover, suppose now ha α and β have been denfed based on a rsk expermen, suh as he one by Hol and Laury, for an ndvdual. Le he same ndvdual hoose beween a safe and a rsky opon regardng all fuure nome levels, as n he prevous ase for CRRA and CARA preferenes. Gven ndfferene beween he opons we an solve for follows: L as β/( β 1) ( β 1)/ β ( β 1)/ β L 1 α S p α H α = + + 1 p β 1 p β β (14) Le us now agan fous on he exreme ase where he hgh nome ouome mples an nfne onsumpon level. For β < 1, (14) hen onverges owards 16
α α = ( 1 ) +. (15) β β L β/( β 1) S p In able 3 below, we alulae L for a very wde range of α / β. 15 As observed, he mpled hoes are sll absurd for almos all values of α / β. Consder for example he ase where α / β = 7. he number 35,977 n he fourh olumn should hen be nerpreed as follows: Assume ha a suden makes a hoe beween Opon I and Opon II n he frs low payoff loery hoe desrbed n able 1, and ha he has HARA preferenes where he relaon beween α and β s suh ha α = 7β, where β s a posve number. Based on he EUC model wh a 5% annual neres rae, hs mples ha f he hooses Opon I, he would prefer a fuure nome sream allowng hm o for he res of hs lfe onsume 36, USD annually wh erany before a rsky alernave where he wh a 99% probably would be able o onsume an nfne amoun and wh a 1% probably would be able o onsume 35,977 USD annually. hs learly seems mplausble. <<able 3 abou here>> he only exepon ours where α / β s very lose o he negave of he baselne nome level, whh n our ase ours where = 1,. Indeed, when α / β = 1, we an wre uly of a loery ouome a sae as U ( x ) = ( x ) 5.85 5.85 =.17 1, + +.17, where x here represen he possble onsan onsumpon level on op of model a hs value of. Hene, hs funon s equvalen o he CRRA EUI. hs also means ha he oeffen of he relave rsk averson 15 Noe ha for he nsananeous uly funon o be defned we mus for β >,.e. where we have dereasng absolue rsk averson, have ha k > ( + x / for all S) x. In he loery abou fuure wages we mus hen L have ha k >. When β <,.e. where we have nreasng absolue rsk averson, we mus have ha k < ( + x. hs means ha he nsananeous uly funon n hs range s no defned for a suffenly / S) H large onsumpon level. In our fuure wage loery we mus hen have ha k <. In order o sll llusrae hs H (raher unreals) range of he HARA uly funon, we hoose = 1 here. 17
would be he same as for he EUI ase, repored n able 1. Hene, we do no oban he absurd hoes n he example of fuure wages here. However, as shown below, w e wll sll oban unreasonable large sake hoes lose o he baselne onsumpon level. So far we have drawn mplaons based on a sngle par wse hoe based on eher he low payoff or hgh payoff loeres of Hol and Laury. However, sne we have wo parameers n he HARA ase we an aually esmae he parameers onssen wh beng ndfferen n he frs low-payoff parwse loery hoe as well as he seond hgh-payoff parwse loery hoe. When dong so we oban parameer values ha are raher lose o he ase desrbed above. Indeed, for he ase where r = 5% annually, we an wre uly as ( ) ( ) 1.9 1.9 U =.92 9999.84 + + x =.92.16 + x. Here oo, here are no exreme rsk averse hoes wh respe o he above hough expermen of fuure wages. he reason s ha n order o mah ndfferene n boh he frs low-payoff parwse hoe and he seond hgh-payoff parwse hoe of Hol and Laury, he uly funon has o have an exreme urvaure n hs regon. hs, n urn, mples ha he loal rsk averson for small hanges around = 1, wll be exremely large, whereas wll derease rapdly fo r larger levels. For example, he relave rsk averson a he benhmark onsumpon level =1, s here equal o / β 1,.479 R = = = 29937.5, whereas a he onsumpon α / β + 9999.84 + 1, level 36, we have 36,.479 R = =.18. hs mples ha we wll oban absurd 9999.84 + 36, large sake rsk averson here oo, bu n anoher nerval, namely lose o he benhmark onsumpon level. Indeed, wh hese preferenes an ndvdual would prefer a safe opon wh a fuure nome orrespondng o a onsan onsumpon level of 1, per year, nsead of a rsky opon where he wh 99.99% probably would oban an nfne amoun 18
and wh a.1% probably would oban an amoun orrespondng o 9999.84 per year (sne uly onverges o mnus nfny a hs level). hus, we observe absurd large sake mplaons based on he hoe behavor n he Hol and Laury expermens for all HARA uly funons onssen wh eher he behavor n he small sakes expermen, he large sake expermen, or boh. Overall, an hen be onluded ha he major onlusons abou absurd large sake mplaons based on he hoe behavor n he Hol and Laury expermen hold muh more generally han for CRRA and CARA preferenes. 4. Generalzaons In Seon 3 we found ha he EUC model does no seem o be onssen wh he expermenal daa provded by Hol and Laury (22), and ha absurd onlusons follow also when based on pessms foreass regardng fuure nome and wh an exremely hgh neres rae. In hs seon we provde furher generalzaons. Frs we onsder a seng where we ake fuure nome unerany no aoun. hen we onsder possble self-onrol problems n erms of me-nonssen presen-bas preferenes. As wll be shown, he man fndngs are robus o hese generalzaons. 4.1 Unerany n fuure nome So far we have assumed perfe foresgh,.e. where fuure nome s known, whh an be quesoned. Le us herefore now assume ha fuure nome s uneran. Le us moreover assume ha he rue fuure nome wll be revealed only one, drely afer he expermenal ouome s o bserved. In hs way, he mpa of he unerany s maxmzed, sne hs makes mpossble o pool any fuure nome rsks. An ndvdual would hen maxmze u ( ) e ρ d as before, bu hs me subje o he budge onsran ha r e d= Y, 19
where Y s a sohas varable represenng he presen value of all fuure nome. he frs order ondons are he same as before and gven by (3), and expeed uly s now gven by EU Y max n = p V ( Y + x ) f ( Y ) dy. (16) = 1 Y mn Moreover, le he dsrbuon of he fuure nome per me un be unform, mplyng a unform dsrbuon also of (sne = Y / S). Besdes falang smple expressons, hs also mples hk als n order o ge large effes from he unerany of presen rsky hoes. Combnng hs wh HARA preferenes, f. eq. (12), we oban max EU = p( α + β( + x / S) ) d mn ( β 1)/ β max mn 1 = + + max (2β 1)/ β p mn ( α β( x / S) ) mn (2β 1) (2β 1)/ β (2β 1)/ β max mn (( ( / )) ( ( / )) ) = θ p α + β + x S α + β + x S max, (17) whereθ max mn 1 (2β 1) s a onsan. Indfferene beween he wo loeres I and II hen mples ha (2β 1)/ β (2β 1)/ β ( α + β( + / )) ( α + β( + / )) (2β 1)/ β (2β 1)/ β II max II mn II (( α β( x / S) ) ( α β( x / S) ) ) p x S x S I max I mn I = p + + + + ( ). (18) In order o oban some numeral esmaes, le us make he exreme assumpons ha mn = 1 USD and max = 19, per year, so ha he expeed fuure nome s he same as before (.e. equally pessms). However, numeral alulaons of he knd provded n able 1 reveal n he CRRA ase ha he mpled parameers of relave rsk averson are sll absurdly large, alhough abou a faor en smaller han n he base ase. he medan values are beween 1,926 and 5,675 and beween 285 and 492 n he small- and large-sake loeres, respevely, wh a 5% annual neres rae. he orrespondngly mpled hoes as repored 2
n able 2 would of ourse hen also be exreme. Consequenly, he absurd resuls are no drven by our deermns fuure nome assumpons. 4.3 me nonssen preferenes So far, we have made he sandard assumpon ha people have me-onssen neremporal preferenes. However, here s muh empral evdene ha people n fa ofen do no make neremporal hoes n a me-onssen way (Frederk e al., 22). Here we herefore generalze he model n order o ake me nonsseny no aoun, by means of a presen bas model smlar o he one used n he semnal work by Labson (1997), and many ohers. In our onnuous me framework, he ndvdual would hen a eah me perod maxmze ρτ ( ) U = σu( ) d+ u( τ ) e d + d τ, (19) where σ s he rao beween he wegh aahed o he presen onsumpon and ha for he near fuure, whereas he relave weghs gven o fuure me perods are gven by he onsan dsoun rae as n he sandard exponenal dsounng ase. In order o fous on he mos exreme ouomes, we wll solely fous on naïve ndvduals ha wll no oday ake no aoun ha hey wll be me nonssen also omorrow; f. O Donoghue and Rabn (1999). he Lagrangean assoaed wh he maxmzaon a me an hen be wren ρτ ( ) e d Y r r τ L = σu ( ) d+ u ( τ ) τ + λ e e τ dτ, + d where r rs Y = e Y se ds s he presen value of lfeme nome mnus he presen value of onsumpon unl me, mplyng he orrespondng frs order ondon for onsumpon a me σu'( ) = λ, (2) 21
and for all fuure me afer u'( ) ( r ) λe ρ τ + τ =. (21) Consequenly, he neremporal margnal rae of subsuon beween wo fuure me perods j s gven by u'( + τ ) / u'( ) e as n he me onssen ase, whereas he + τ = j ( r ρ )( τ τ ) orrespondng neremporal margnal rae of subsuon beween he presen and any fuure r me perod s gven by u'( ) / u'( ) e ( σ ρ ) τ + =. τ Consder for example he ase where he nsananeous uly funon s haraerzed by CRRA, so ha 1 R u ( ) =. hen follows ha 1 R η = σ = σ Y, where 1 e η s 1/ R 1/ R he me onssen onsumpon a me,.e. he onsumpon an ndvdual would have f 1/ R σ = 1, and where η = (rr + ρ r) R. Smlarly, = σ, where s he onsumpon hosen by an ndvdual who has been me nonssen up o me bu who wll be me onssen from hen on. hus, 1/ R σ η = Y ( ), where agan Y 1 e η s he presen value of lfeme nome mnus he presen value of onsumpon unl me. hen we have 1/ R σ η r rs = e ( ) s 1 e η Y e ds. (22) For example, when preferenes are logarhm (R = 1), σ = 1.1 and ρ = r, so ha a me onssen ndvdual would prefer a onsan onsumpon level, follows ha an ndvdual wll always onsume 1% more per me un han he an afford o onsume per me un durng he res of hs lfeme. By dfferenang he negral equaon (or more presely he Volerra negral equaon of he seond knd) n (22) by, and hen subsung n (22), we oban he followng separable dfferenal equaon 22
1/ R 1/ R 2 1/ R d σ η r r rs σ η η ( ) + r σ η r = ee ( ) + Y e s e re η + 2 ( ) ( ) d 1 e η η ds ( 1 e ) 1 e.(23) ( = re ) + r η ( ) 1 e η ( ) 1/ R η σ η I s sraghforward o solve (23), for η >, as mplyng or η ( ) 1/ ( η r) e + r ησ R d = d ( ), η 1 e 1/ ( R 1/ ) ( R η 1) ln ( η ησ σ ) ln = r + e e + K, 1/ R 1/ R ( r ησ ) σ 1 η η = e e e e ( ) ( ) K, (24) where K s he onsan of negraon. K an be denfed by he nal ondon 1/ R = Y, mplyng 1 ( ) ( R ησ 1 ) K η σ = 1 e η 1/ 1/ R ησ e e Y, whh subsued n (24) mples η 1 e ( ) ( 1/ R σ 1 ) η η e e 1/R ( r ησ ) ( ) ( 1/ R σ 1 1 ) η e 1/ R ησ = e Y. (25) η 1 e A person who repeaedly and onssenly behaves n hs way wll hen end up wh muh lower onsumpon when old even for a σ ha s relavely lose o one. Indeed, n hs ase follows ha he onsumpon wll onverge owards zero a he end of lfe when σ > 1,.e. rrespeve of ρ and r, sne ( ) ( 1/ R σ 1 ) η η e e wll hen onverge owards zero. 17 We an 16 16 For η < we oban nsead he opmal onsumpon pah as follows: ( ) 1/ 1/ R 2+ σ ησ = ( ) ( ) ( 1/ R 1 ) r η σ η η e e e 1/ R η ( e 1) R 1/ R r 1 σ Smlarly, for η = we ge = σ e Y. he same reasonng an be used for he more general ase of HARA preferenes, alhough he negral and dfferenal equaons naurally beome subsanally more omplaed n hs ase (resuls are avalable from he auhor upon reques). 17 See Damond and Köszeg (23) for a dsree me applaon o penson savngs. σ 1/ R Y. 23
hen oban non-monoon onsumpon pahs,.e. pahs ha are nally nreasng up o a eran pon n me and hereafer dereasng, when ρ < r. For example, a 2 year old ndvdual who expes o lve unl he age of 8, wh R = 1, σ = 1.3, ρ =.3, and where r =.5, would reah a peak onsumpon a abou he age of 6. Le us nex onsder he dre mplaons of me-nonssen preferenes for rsky hoes by agan wrng he expeed uly when an ndvdual faes a loery wh n dfferen pres. Leng G 1/ R ησ = η 1 e ( ) ( 1/ R σ 1 ) η η e e 1/R ( r ησ ) ( ) ( 1/ R 1 1 ) e σ η e we an wre (25) as = G Y, so ha we may wre expeed uly as R ( G ( Y + x )) ( G ( Y + x )) 1 1 R n ρ EU = σ d + e d p = 1 1 R 1 R ( Y + x ) G G = + 1 R 1 R 1 R 1 R 1 R 1 R n ρ σ d e d p = 1, (26) 1 R n ( Y + x) n ( + x / S) =ϒ p ' 1 =ϒ p = = 1 1 R 1 R where ϒ and 1 R ϒ ' are arbrary posve onsans. hus, hese fnal expressons look exaly as n he me onssen ase n eq. (12),.e. hey do no depend a all on σ! Hene, alhough me nonsseny n he form of presen bas preferenes an have very large mplaons on he onsumpon pah and savng desons, he mplaons for he behavor n rsky hoes are muh smaller, and n our ase none. Consequenly, our sandard, bu perhaps unreals, assumpon abou me onsseny n our nal EUC formulaon s no wha drves he absurd mplaons of he hoes made n he Hol and Laury (22) daa. 4. Dsusson and Conluson he explanaory power of EU heory has reenly been dsussed nensvely (e.g. Rabn 2a; Rabn and haler 21; Cox and Sadraj 26; Rubnsen 26). he presen paper s 24
onerned wh he mplaons of hs dsusson for appled eonoms researh under rsk. In suh researh people ofen make desons over me, where he sandard model (here denoed he EUC model) assumes ha people maxmze he expeed presen value of fuure nsananeous uly. he relaons beween he EUC model and he EUW and EUI models, respevely, are analyzed. I s shown ha he EUC model s equvalen o he EUW model when wealh s measured by he presen value of all fuure onsumpon or nome, whereas he EUC model s equvalen o he EUI model only for CARA preferenes. Moreover, s shown ha he funonal form of he nsananeous uly funon, whh s expressed as a funon of urren onsumpon, arres over n a sraghforward way o a uly funon ha s expressed as a funon of he presen value of all fuure nomes, f and only f he nsananeous uly funon belongs o he lass of funons haraerzed by hyperbol absolue rsk averson, whh nludes CRRA and CARA as speal ases. However, as argued by Palaos-Huera and Serrano (26), he mporan queson s no wheher he EU model, or n our ase he EUC model, s lerally orre. We know ha s no. Wha s mporan for appled eonoms s he exen o whh he model an provde a reasonable approxmaon of aual behavor. Here he mplaons of he EUC model have herefore been nvesgaed based on daa from a areful rsk expermen repored n Hol and Laury (22). he resul suggess ha he EUC model s ll-sued o explan expermenal behavor n suh small- and nermedae-sake gambles. he alulaed mpl rsk-averson parameers are found o be unreasonably large, and herefore an no onsue onavy measures of unversally vald nsananeous uly funons. For example, n he base ase wh CRRA preferenes, makng onservave or reals assumpons regardng fuure wages, e., he medan oeffen of relave rsk averson s above 2, even based on he hgh-sake loery hoes, despe ha mos analyss seem o agree ha R should be n he.5-3 range, or a leas no larger han 1. 25
Even more srkngly, he resuls also sugges unreasonable mplaons n erms of wha hese degrees of rsk averson would orrespond o for long-erm rsky hoes n erms of he subjes fuure onsumpon levels. Wheher based on CRRA or CARA preferenes, mos subjes from Hol and Laury would n he base ase EUC model prefer a eran nome enablng hem o for he res of her lfeme onsume 36, USD annually nsead of a rsky alernave where hey wh a 1% probably would be able o onsume 35,99 USD annually and wh a 99% probably would be able o onsume an nfne amoun. Moreover, sne he resuls n Hol and Laury (22) are n no way unque, bu are n lne wh mos small and nermedae sake expermenal rsk sudes, 18 an be onluded ha he EUC model appears nonssen wh observed expermenal small- and nermedaesake daa. he same apples o a leas some knds of aual onsumpon behavor, suh as addonal nsuranes for eleron equpmen. However, a avea regardng he numeral resuls s n order. he numeral analyss here mplly assumes ha he hoes repored n Hol and Laury are whou errors, whh s of ourse no he ase n realy. Moreover, when exrapolang o mpled hoes a a muh larger sale, as s done here, suh errors wll of ourse nrease oo, as poned ou by Cox and Harrson (28). hs means ha he numeral resuls here should aordngly be aken wh some gran of sal. Sll, n order o esape he man onlusons, one has o assume ha he hoes n Hol and Laury are almos solely drven by random, whh seems unlkely. Consequenly, we need anoher model o explan small- and nermedae-sake rsk behavor. here are several suggesons well worh onsderng n he leraure; see e.g. Rabn and haler (21), Köbberlng and Wakker (25), Barbers e al. (26), Cox e al. (28), Henemann (28), and Rabn and Wezsäker (27). Alhough s beyond he sope of he presen sudy o dsrmnae beween hese and oher models, wo remarks appear lear based 18 See e.g. Cox and Sadraj (28) for a dsusson and analyss of oher reen rsk expermens. 26
on he fndngs here: 1. Loss averson anno explan he hoe behavor n Hol and Laury, and hene no he numeral fndngs n hs paper, sne all ouomes were n he gan doman (unlke he hough expermens by Rabn and haler). 2. I appears ha a suessful model should nlude he elemen ha a deson-maker who faes mulple desons n eah ase end o make a deson wh nsuffen regard o he oher desons, and hene wh nsuffen regard o nome or wealh from oher soures. 19 Fnally, sne repeaed games also add payoffs over me, one may worry ha repeaed game heory would be n rouble oo. However, as noed by an advsory edor, he resuls from hs paper do no jusfy suh a onluson. An mporan dfferene s ha n repeaed game heory, uly n a eran perod depends only on he hoes made n ha perod whle n he EUC model an nreased onsumpon redues fuure uly. he exen by whh people negrae he payoffs from a eran sage n a game wh he payoffs from oher sages, as well as wh bakground lfeme wealh, s a separae and mporan ssue ha s lef for fuure researh. Referenes Aanaso, O. P., Brownng, M., 1995. Consumpon over he lfe yle and over he busness yle. Ameran Eonom Revew 85, 1118 1137. Barbers, N., Huang, M., haler, R.H., 26. Indvdual Preferenes, Moneary Gambles, and Sok Marke Parpaon: A Case for Narrow Framng. Ameran Eonom Revew 96, 169-19. Blundell, R., Brownng, M., Meghr, C., 1994, Consumer demand and he lfe-yle alloaon of household expendures. Revew of Eonom Sudes 61, 57 8. Cox, J.C., Harrson, G.W., 28. Rsk Averson n Expermens: An Inroduon. Ch. 1 n J. C. Cox and G. W. Harrson (eds.), Rsk Averson n Expermens, Researh n Expermenal Eonoms, Vol. 12, Cornwall: Emerald JAI Press, pp. 1-7. 19 hs and smlar phenomena have dfferen names n he leraure, nludng deson solaon, menal aounng, narrow brakeng, narrow framng and paral asse negraon. 27
Cox, J.C., Sadraj, V., 26. Small- and large-sakes rsk averson: Implaons of onavy albraon for deson heory. Games and Eonom Behavor 56, 45-6. Cox, J.C., Sadraj, V., 28. Rsky Desons n he Large and n he Small: heory and Expermen, Ch. 2 n J. C. Cox and G. W. Harrson (eds.), Rsk Averson n Expermens, Researh n Expermenal Eonoms, Vol. 12, Cornwall: Emerald JAI Press, pp. 9-4. Cox, J.C., Sadraj, V., Vog, B., Dasgupa, U., 29. Is here a Plausble heory for Rsky Desons? Updaed verson of Expermenal Eonoms Cener Workng Paper 28 4, Georga Sae Unversy. Deaon, A., 1992. Undersandng Consumpon. Oxford: Oxford Unversy Press. Damond, P., Köszeg, B., 23. Quas-hyperbol dsounng and reremen. Journal of Publ Eonoms 87, 1839-1872. Frederk, S., Loewensen, G., O Donoghue,., 22. me dsounng and me preferene: a ral revew. Journal of Eonom Leraure 4, 351 41. Goller, C., 21. he Eonoms of Rsk and me, Boson: MI Press. Gomes F., Mhaeldes, A., 25. Opmal Lfe-Cyle Asse Alloaon: Undersandng he Empral Evdene. Journal of Fnane 6, 869-94. Hall, R.E., 1978. Sohas mplaons of he lfe yle/permanen nome hypohess; heory and evdene. Journal of Polal Eonomy 86, 971-987. Harrson, G.W., Lau, M.I., Rusröm, E.E., 27. Esmang Rsk Audes n Denmark: A Feld Expermen. Sandnavan Journal of Eonoms 19, 341-368. Henemann, F., 28. Measurng rsk averson and he wealh effe, n: James C. Cox and Glenn W. Harrson (eds.), Rsk Averson n Expermens, Researh n Expermenal Eonoms, Vol. 12, Cornwall: Emerald JAI Press, pp. 293-313. Hol, C.A., Laury, S.K., 22. Rsk averson and nenve effes. Ameran Eonom Revew 92, 1644-1655. Koherlakoa, N., 1996. he Equy Premum: I s Sll a Puzzle. Journal of Eonom Leraure 34, 42-71. Köbberlng, V., Wakker, P.P., 25. An ndex of loss averson. Journal of Eonom heory 122, 119 131. Labson, D., 1997. Golden Eggs and Hyperbol Dsounng. Quarerly Journal of Eonoms 112, 443-477. Meron, R.C., 1971. Opmum Consumpon and Porfolo Rules n a Connuous me Model. Journal of Eonom heory 3, 373-413. 28
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Appendx Proof of Proposon 1. Le * denoe he opmal onsumpon a me and onsder he welfare effe of a small lfeme nome nrease: * * du V * ρ * r u'( ) '( e * = = d = u ) e d = u'( ) dy Y Y, (A1) Y where we have used (3) and * e r d = 1 from dfferenang he budge resron (2). Y Inuvely, he uly nrease of onsumng everyhng of a small nome nrease mmedaely s equally large as any oher paern of he onsumpon nrease, as long as he nal onsumpon paern s opmal. By ombnng (3) and (A1) we also have λ = V Y. When r s gven we an wre U ( ) = V( Y), and from (3) we hen have dv u * ( '( r ( Y )) e ρ ) = so dy ( r ρ ) * VY ( ) = e u' ( Y) dy+ K, (A2) where K s ndependen of Y. Now, sne * ( ( ( ))) d u Y dy * ( Y ) = u' ( ) Y * we have ha * ( ( )) u Y * u' ( ( Y) ) dy = + K, where K s ndependen of Y, f and only f Y * ndependen of Y,.e. when we an wre Y * s * = a () r + b() r Y, (A3) so ha wre Y * = b() r. Hene, here s an affne relaonshp beween V and u suh ha we may ( ) 1 * = ( ) = ( ) + 2, U V Y K u Y K 3