Discounting and Confidence

Transcription

1 Chrisian P. Traeger Deparmen of Agriculural & Resource Economics, UC Berkeley CUDARE Working Paper No. 1117, June 2011 Absrac: The paper analyzes he social discoun rae under uncerainy. I employs a preference represenaion ha enriches he characerizaion of uncerainy by a degree of confidence ino probabilisic descripions of he world. Special cases of he model comprise discouning under smooh ambiguiy aversion as well as discouning under a disenanglemen of risk aversion and aversion o ineremporal subsiuion. I relae differen resuls in he lieraure swiching risk measures beween he classical Arrow Pra form and a measure of ineremporal risk aversion. I characerize he general class of preferences for which uncerainy implies a reducion of he social discoun rae. I also characerize he class of preferences ha lower he discoun rae compared o he sandard model. I derive a paricular parameric discouning formula under he assumpions of isoelasic preferences and normal growh raes. Apar from he usual characerisics of he growh process like expeced value and variance, he discoun rae depends on a measure of confidence ino fuure growh esimaes and a measure of aversion o he lack of confidence. JEL Codes: D61, Q54, D81, D90 Keywords: uncerainy, discouning, climae change, ambiguiy, confidence, subjecive beliefs, prudence, pessimism, expeced uiliy, ineremporal subsiuabiliy, ineremporal risk aversion Correspondence: Deparmen of Agriculural & Resource Economics 207 Giannini Hall #3310 Universiy of California Berkeley, CA raeger@berkeley.edu

2 1 Inroducion How does uncerainy abou he fuure affec opimaliy of projecs? Uncerainy abou economic growh and oher variables affec sociey s social discoun rae. This rae is highly relevan for he evaluaion of long-erm projecs including climae change relaed miigaion and adapaion projecs, invesmens in basic research, naional defense, infrasrucure projecs, projecs involving biodiversiy loss, and a long-erm energy sraegy including he appraisal of nuclear energy and wase. A he agen s level, he consumpion discoun rae analogously deermines ineremporal rade-offs and precauionary saving. The paper presens a new model for he social (or consumpion) discoun rae under uncerainy. The model unifies and generalizes a variey of previously derived discouning formulas. One of he main resuls is a general saemen when he social discoun rae decreases in he face of uncerainy. Uncerainy incorporaes risk, ambiguiy, and a more general form of confidence labeled beliefs. As a parameric special case, he paper develops a simple exension of he sochasic Ramsey equaion ha incorporaes general aversion o objecive risk, a confidence measure for non-objecive beliefs, and an aversion parameer o he lack of confidence. From Leland (1968) we learned ha a decision maker should increase his savings for he fuure under uncerainy, if his (absolue) Arrow-Pra risk aversion decreases in wealh. In Leland s (1968) analysis Arrow-Pra risk aversion simulaneously characerizes aversion o ineremporal subsiuion. Gollier(2002) disenangled risk aversion from he propensiy o smooh consumpion over ime. He showed ha Leland s reasoning sill goes hrough and a decision maker reduces he social discoun rae in he face of uncerainy if he, now disenangled, measure of Arrow-Pra risk aversion falls in wealh. Recenly, Gierlinger & Gollier (2008) conras hese findings wih a model of ambiguiy. They show in Klibanoff, Marinacci & Mukerji s (2005) model of smooh ambiguiy aversion ha a decreasing coefficien of absolue ambiguiy aversion is no sufficien o ensure a reducion of he discoun rae in he face of uncerainy. The auhors idenify a prudence erm ha does lower he discoun rae under uncerainy if absolue ambiguiy aversion is falling. However, hey also idenify a pessimism erm whose response o uncerainy can only be deermined by resricing he admissible loery domain. In he curren paper, I reconcile Gierlinger & Gollier s (2008) finding wih he earlier resuls in he lieraure and reconsiue 1

3 Leland s (1968) insigh regarding he effec of decreasing aversion on he discoun rae for he general seing, including ha of smooh ambiguiy aversion. I rack differences in he resuls o differences in he employed uncerainy measure. An appropriae change in risk measure can inroduce he pessimism effec in he original risk seing or eliminae i from he ambiguiy seing. Moreover, I derive a proposiion ha compares he uncerainy effec on he social discoun rae beween various models of generalized uncerainy and he ineremporally addiive expeced uiliy sandard model. The generalized uncerainy framework builds on Traeger(2010) and incorporaes as special cases he model based on Kreps & Poreus (1978), Epsein & Zin (1989), and Weil (1990) disenangling risk aversion from ineremporal subsiuabiliy and he model of smooh ambiguiy aversion by Klibanoff e al. (2005) and Klibanoff, Marinacci & Mukerji (2009). Limiing cases of he model also include he Arrow & Hurwicz (1972) crierion for decision making under ignorance and Gilboa & Schmeidler s (1989) maximin expeced uiliy. The general framework employs mulilayer probabilisic beliefs ha are indexed by a confidence measure. In paricular, he seing permis o model a decision maker who employs all (or any combinaion) of he above decision crieria condiional on he confidence in his descripion of he uncerainy he faces. Ever since Keynes (1921) and Ellsberg (1961) economiss and decision heoriss have expressed heir concern ha a sandard probabiliy disribuion canno capure uncerainy comprehensively. Some probabiliy disribuions are derived from long ime series and frequen observaions, while ohers are merely guessimaes. Consider wo muually exclusive evens. In absence of any informaion on he likelihood of hese evens, he principle of insufficien reason guides he decision maker o employ equal probabiliies. A he same ime, he flip of a fair coin is described by equal probabiliies on heads and ails. Ye, in he firs siuaion he decision maker s guess is based on complee ignorance, while in he second siuaion he faces well known objecive probabiliies. Boh uncerainies differ in a dimension ha is no capured by he probabiliy disribuions hemselves. I is a measure of confidence (or of subjeciviy) ha disinguishes he wo. Experimens based on he famous Ellsberg (1961) paradox have shown ha, indeed, people ofen ac differenly depending in on heir degree of confidence ino a probabiliy disribuion. Decision heoriss have explored his concern in deph over he las wo decades. I build on a recen exension of he widespread smooh ambiguiy model (Klibanoff, Marinacci, & Mukerji 2005, 2

4 2009) by Traeger (2010). The laer analysis exends he classical von Neumann & Morgensern (1944) axioms, underlying he expeced uiliy model, o a seing where loeries are disinguished by heir degree of confidence. In he curren paper, I apply Traeger s (2010) framework o derive a discouning formula ha akes ino accoun no jus he riskiness of fuure payoffs, bu also he degree of confidence ino he probabiliy disribuions employed o describe he uncerainies. A comparable disincion of differen ypes of uncerainy has reached he applied policy arena in a field where recen research has proven he primordial imporance of selecing he righ discoun rae. The guidance noes of he Inergovernmenal Panel on Climae Change (AR4) ask he lead auhors o disinguish beween hree differen ypes of uncerainy: unpredicabiliy, srucural uncerainy, and value uncerainy. However, he subsequen economic assessmen is no, ye, able o incorporae hese disincions. In recen work, Gierlinger & Gollier (2008) and Traeger (2009) apply he smooh ambiguiy framework by Klibanoff e al (2005,2009) o models of social discouning. These models, however, can only capure wo ypes of uncerainy: objecive versus subjecive uncerainy. In real world applicaions, purely objecive probabiliies are rare and he degree of subjeciviy of, or informedness of, or confidence in a probabiliy disribuion varies widely. The curren paper suggess a framework in which decision makers, or climae scieniss, base heir uncerainy evaluaion no only on probabilisic esimaes of fuure uncerainy, bu also on he informedness of he probabilisic descripion. I ranslaes boh of hese informaions ino he social (or consumpion) discoun rae. The adoped preference framework builds on a behaviorally as well as normaively aracive se of axioms ha preserve ime consisency and a minimally modified version of von Neumann-Morgensern s independence axiom. In difference o Gierlinger & Gollier (2008), he framework also keeps separae wha is disinc: i disinguishes beween he desire o smooh consumpion over ime and he (various degrees of) uncerainy aversion, as is suggesed by he recen finance lieraure explaining he equiy premium and he risk free rae puzzles (Vissing-Jørgensen & Aanasio 2003, Basal & Yaron 2004, Basal, Kiku & Yaron 2010). 3

5 2 Background and Represenaion Curren consumpion is cerain and denoed by x 0. I will employ a unidimensional noaion u (x 0 ) for derivaives. However, oucomes can also be mulidimensional or elemens of a compac meric coninuously differeniable manifold unless saed oherwise. For he general case u (x 0 ) is shor for α 0u(x 0 ), denoing he direcional derivaivealongconsumpionchangeαaconsumpionpoinx 0. 1 Thedecisionmaker considers invesing a(marginal) uni of a good ino a producive projec ha pays one uni plus he yearly (average) ineres r in he fuure. The minimal ineres required o make he agen inves ino he projec is he (risk free) social discoun rae or consumpion discoun rae. Under cerainy i is characerized by he pure rae of ime preference δ and he raio of marginal uiliies in he fuure and in he presen: r = δ 1 u T ln (x T ), (1) u (x 0 ) where x T denoes fuure consumpion. Noe ha, in he one dimensional seing, he curvaure of u characerizes he desire o smooh consumpion over ime. This paper derives he modificaions of equaion (1) necessary o accoun for uncerainy over he fuure. In paricular, I accoun for differences in he informedness of probabiliies describing fuure uncerainy. Probabiliies can be objecive like for heossofacoinorhespinofarouleewheel. Buprobabiliiescanalsoderivefrom a small number of observaions (or simulaions) or base on he principle of insufficien reason. The laer principle saes ha if e.g. wo evens are possible and an agen has no informaion abou heir likelihood, he agen should assign equal probabiliy o boh evens. Qualiaively, hese seings differ in heir uncerainy, even when assigning he same probabiliies. Examples for lacking confidence in probabilisic descripions of long-erm fuure economic developmen include he possibiliy of faails in disribuions governing sock markes or, in he conex of climae change, he emperaure response o greenhouse gas emissions and heir GDP feedback. Similarly, probabiliy disribuions governing poliical sabiliy are ofen no objecive or known wih confidence. I employ a preference represenaion ha allows he agen o disinguish differen 1 Here, α is a curve deermining he consumpion change in he presen period. Similarly u (x 1 ) will be shor for β 1 u(x 1) denoing changes along curve β in period 1, and u (x T ) will be shor for γ T u(x T). 4

6 ypes of probabiliies by means of a degree of confidence (or subjeciviy). Traeger (2010) derives an according preference represenaion by enriching he well known von Neumann & Morgensern (1944) framework for decision making under uncerainy wih a dimension of confidence. Special cases of he model disenangle Arrow Pra risk aversion from ineremporal subsiuion like in Epsein & Zin (1989), Weil (1990), and Kreps & Poreus (1978) KP-model, or represen smooh ambiguiy aversion as in Klibanoff e al. (2005) and Klibanoff e al. (2009) KMM model. The represenaion builds on a se of increasing funcions {f s } s S ha characerize a measure of ineremporal risk aversion. Each index s S corresponds o a paricular degree of confidence (or subjeciviy), complemening he probabilisic measure of uncerainy. The se S is an arbirary finie se of confidence descripions. Throughou he paper I assume ha uiliy u and he risk aversion funcions f s are increasing and concave. In he case of wo periods and a single loery over fuure consumpion, which is characerized by he probabiliy measure p of degree of subjeciviy s, he welfare evaluaion wries as u(x 0 )+e δ f s 1 E p f s u(x 1 ), (2) where he superindex 1 denoes he inverse and denoes funcion composiion. The operaor M f p f 1 E p f akes a generalized mean of whaever follows o is righ. A concave funcion f implies ha he resul of he generalized mean M f p reurns a smaller value han he expeced value operaor iself (Hardy, Lilewood & Polya 1964). A concave funcion f s characerizes ineremporal risk aversion wih respec o uncerainy wih degree of confidence s. A way o hink abou ineremporal risk aversion is as a measure of risk aversion wih respec o uiliy gains and losses. Equaion (4) will give a choice heoreic inerpreaion. In general, many layers of uncerainy characerized by differen degrees of confidence can come ogeher in deermining fuure oucomes. An example of such a more general uncerainy descripion is depiced by he ree in Figure 1. The represenaion derived by Traeger (2010) implies ha a loery of degree of subjeciviy s wihin his uncerainy ree has o be evaluaed by he generalized mean ha is characerized by he ineremporal risk aversion funcion f s. A ree as in Figure 1 feaures loeries over loeries. I sar by labeling he roo loery o he lef as loery p 1. Loery p 1 is a loery over differen loeries p 2 in he nex uncerainy layer. The loeries in he las layer are loeries over fuure oucomes. The uncerain scenario depiced in Figure 1 involves 5

7 1 2 1 s s s s s s p 1 s 2 3 Figure 1 depics an example of muli-layer uncerainy in ree form. The indices s,s,s and s label he differen degrees of confidence of he uncerainy nodes. Collapsing he firs wo layers ino a single layer would no change he evaluaion as boh layers share he same degree of subjeciviy. Moreover, he degree of subjeciviys onhedegeneraenodeisirrelevanfor he evaluaion(a degenerae node expresses cerainy). four differen loeries p 3 wih hree differen degrees of subjeciviy. 2 In general, le here be N N layers of uncerainy. Moreover, le ŝ(p) denoe he degree of subjeciviy of a given loery. Then, he general preference represenaion can be wrien as u(x 0 )+e δ M fŝ(p1 ) p 1 M fŝ(p 2 ) p 2 M fŝ(p N ) p N u(x 1 )= u(x 0 )+e δ N M fŝ(pi ) u(x 1 ).(3) Inuiively, each generalized mean deducs a risk premium ha depends on he amoun of uncerainy (described by ) and on he degree of confidence ŝ( ). I denoe cerainy equivalen uiliy in he i-h layer by m i ( ) M fŝ(pi ) and m N+1 (x 1 ) u(x 1 ) or, dropping he argumen, simply by m i. M fŝ(p N ) p N u(x 1 ) The following characerizaion of ineremporal risk aversion adaped from Traeger (2010) gives a useful inuiion for he concep of ineremporal risk aversion. Le a decision maker be indifferen beween he wo combinaions of firs and second period oucomes (x 1,x 2 ) and (x 1,x 2 ), where u(x 1 ) > u(x 1 ) and u(x 2 ) > u(x 2 ). An ineremporalriskaverse decision maker prefershecerain consumpionpah(x 1,x 2 ) (or equivalenly (x 1,x 2 )) over a loery ha yields wih equal probabiliies eiher he pah (x 1,x 2 ) or he pah (x 1,x 2 ). Formally his condiion can be wrien as (x 1,x 2 ) (x 1,x 2 ) (x 1,x 2 ) (x 1,x 2 ) 1 2 s (x 1,x 2 ), (4) where 1 2 s denoes a probabiliy one half mixure wih degree of subjeciviy s. If his mixure represens an objecive loery, like in he case of a coin oss, equaion (4) 2 The hierarchical srucure of loeries over loeries can be visualized by indexing a loery in layer i wih θ 1,...,θ i 1, where θ j characerizes he risk saes going along wih loery p j. If he loery has a coninuous disribuion, he uncerainy layer feaures an uncounable se of loeries +1. A formal characerizaion of he general loery space is given in Traeger (2010) and employs Borel measures over disjoin unions of Borel algebras corresponding o he differen degrees of subjeciviy. 6

8 capures ineremporal risk aversion wih respec o objecive loeries. Alernaively, he loery can depic an even wih binary oucome and he complee lack of informaion on likelihood (principle of insufficien reason). Noe ha an agen described by he ineremporally addiive sandard model is always indifferen beween he cerain pah and he loery in equaion (4). I is useful o undersand ha, in an ineremporal seing, uncerainy affecs welfare in wo disinc ways. Firs, a sochasic variable generaes flucuaions over ime. A decision maker wih a preference for smooh consumpion pahs dislikes hese flucuaions. This effec of risk is capured by (ineracion wih) he uiliy funcion and is par of sandard model. Second, a decision maker can be inrinsically risk averse (averse o risk per se). This effec is capured by ineremporal risk aversion. A differen way o measure risk aversion in he general seing is as follows. Le he curvaure of u coninue o measure aversion o ineremporal subsiuion. Insead of measuring inrinsic risk aversion direcly wih a measure of ineremporal risk aversion, he overall risk aversion can be measured by he funcions g s = f s u (5) for all s S. Boh, he concaviy of f s and he concaviy of u ranslae ino he curvaure of he funcions g s. Hence, hey joinly capure boh sources of risk aversion. As discussed in Traeger (2007) and Traeger (2010) hese funcions g s characerize Arrow Pra risk aversion wih respec o loeries of degree of confidence s. Noe ha his inerpreaion in erms of Arrow Pra risk aversion only holds in he onecommodiy seing. Equaion (5) implies a furher characerizaion of ineremporal risk aversion. The saemen ha f s = g s u 1 is concave implies ha Arrow Pra risk aversion dominaes aversion o ineremporal subsiuion. Hence, an ineremporally risk averse agen prefers o subsiue ino he cerain fuure raher han ino an uncerain risk sae. The assumpions ha u and f s are increasing and concave imply he same characerisics also for g s. 3 Presen versus Fuure The model of his secion analyzes he case where fuure payoffs are colleced in a single uncerain fuure period. This is he seing of Leland (1968), Gollier (2002), 7

9 Gierlinger & Gollier (2008), and Traeger (2009) as well as mos analyic discussions of social discouning in he inegraed assessmens of climae change. I inroduce muliple layers of uncerainy characerized by differing degrees of confidence. The model relaes o he represenaions of he social discoun rae under KP preferences analyzed in Gollier (2002) and under smooh ambiguiy KMM preferences examined in Gierlinger & Gollier (2008). I compare, unify and generalize he resuls of hese papers. I derive a sufficien condiion under which he social discoun rae decreases wih uncerainy, and a sufficien condiion under which for he social discoun rae is smaller han in he sandard model. A special case of hese condiions characerizes when smooh ambiguiy aversion decreases he social discoun rae. The condiions depend on preferences only and hold for all uncerain scenarios (loeries). 3.1 The social discoun rae in erms of ineremporal risk aversion The following proposiion exends he expression for he social discoun rae in equaion (1) incorporaing uncerainy and uncerainy aiude. Proposiion 1: The social discoun rae under preferences of he form given in equaion (3) is r = δ ln = δ ln { N { N E } ) fŝ(pi (m ) u (x 1 ) fŝ(pi ) (m i ) u (x 0 ) E fŝ(pi ) (m ) fŝ(pi ) (m i ) }{{} prudence erm confidence level ŝ( ) E fŝ(pi ) (m ) E fŝ(pi ) (m ) }{{} pessimism erm confidence level ŝ( ) } u (x 1 ). u (x 0 ) The expeced value operaor E acs on everyhing carrying an index for he nexuncerainylayer: loeries+1 andcerainyequivalensm. Inparicular,he expeced value operaor prined in large also acs on he enries of he subsequen produc erm (wih E p N acing on u (x 1 )). I label he fracions wih he expeced value in he numeraor prudence erms. The name is based on Proposiion 2 below. The fracions wih he expeced value in he denominaor are weighs. These erm increase he weigh given o evens wih high marginals (generally low oucomes), and (6) 8

10 reduces he weigh of evens wih low marginals(generally high oucomes). Therefore, hese weighs gain he name pessimism erm as hey effecively bias probabiliies o give more weigh o bad oucomes. Boh names were assigned by Gierlinger & Gollier (2008) in a special case described below. Proposiion 2: A prudence erm of confidence level s reduces he social discoun rae, if and only if, he funcion f s exhibis decreasing absolue risk aversion AIRA s = fs f s, which is equivalen o fs f s > fs f s. Onlyrelyingonsmoohnessofhefuncionf s,hecondiionisf s f s 1 convex. I call he erm fs f s absolue ineremporal prudence wih respec o confidence level s. The name prudence relaes o Kimball s(1990) work on hird order derivaives of uiliy funcions in he conex of precauionary savings. In his erminology he condiion in Proposiion 2 saes ha he prudence erm reduces he social discoun rae if prudence dominaes risk aversion (for a given confidence level). The condiion is equivalen o a falling degree of absolue ineremporal risk aversion AIRA S. Thus, he prudence erm conforms wih Leland (1968) finding in he sandard model. The following inuiion explains why he hird order derivaive or he change of risk aversion is crucial. Assume ha a decision maker is less risk averse a higher welfare levels. Then, saving for he fuure no only increases expeced fuure consumpion, bu also reduces he risk premium accouning for fuure uncerainy. Thus, he decision maker has an addiional incenive o save for he fuure under uncerainy. 3 Sufficien condiions for which he pessimism erm decreases he social discoun rae are more inricae. In a simplified version of he model, Gierlinger & Gollier (2008) analyze he erm. They use he KMM framework of smooh ambiguiy aversion by Klibanoff e al. (2005), which accouns for wo ypes of loeries: objecive (s = obj) and subjecive (s = subj). The decision maker exhibis ineremporal risk aversion only wih respec o subjecive loeries characerized by f subj, bu no wih respec o objecive loeries (f obj is he ideniy/absen from he model). Moreover, he smooh ambiguiy model assumes ha he decision maker faces a subjecive loery over an objecive loery. 3 A more echnical inuiion derives from he observaion ha he agen considers an opimal marginal rade-off under uncerainy. By Jensen s inequaliy concaviy of he welfare funcion deermines he welfare loss due o uncerainy. The discoun rae asks for he opimal ransfer beween he curren and he fuure period. Is change depends on he difference in welfare change under concave uncerainy aggregaion (a differen income levels). 9

11 Corollary 1: KMM model of smooh ambiguiy aversion In a seing wih subjecive over objecive loeries and ineremporal risk neuraliy wih respec o objecive risk, he social discoun rae collapses o he form r = δ ln = δ ln E p subj f subj (m obj ) E u (x 1 ) f subj (m subj p ) obj u (x 0 ) Ep subjf subj (m obj ) f subj (m subj ) }{{} amb. prudence erm E p subj f subj (m obj ) E p subjf subj (m obj ) }{{} pessimism erm E p obj u (x 1 ). u (x 0 ) HereE p obj akesheexpecaionwihrespecoanobjeciveloeryoveroucomesx 1, while E p subj akes expecaions over he objecive loeries p obj (and he objecive cerainyequivalenm obj = m obj (p obj )). Inhisframework,f subj correspondsoklibanoff e al. s (2005) measure of smooh ambiguiy aversion. In his conex, Gierlinger & Gollier(2008)alreadyderivedhadecreasingabsolueambiguiyaversion(AIRA subj ) implies ha he prudence erm reduces he social discoun rae. They also discuss in deail sufficien condiions for he pessimism erm o decrease he social discoun rae. In general, hese condiions are no longer mere preference resricions bu also involve resricions regarding he underlying loeries. Secion 3.2 discusses a differen formulaion of he social discoun rae ha avoids hese complicaions. Gierlinger & Gollier (2008) classify he pessimism effec as newly arising in he ambiguiy seing. However, he nex lemma shows ha he pessimism erm can already arise in a pure risk seing. Assume here is a single loery and no subjecive risk or ambiguiy. Then here is a unique funcion f characerizing ineremporal risk aversion. The seing is a special case of Kreps & Poreus (1978) and a generalizaion of Epsein & Zin s (1989) and Weil s (1990) model. Corollary 2: KP model In a seing wih a single loery (no disincion of confidence, no ambiguiy), 10

12 he social discoun rae collapses o he form r = δ ln E f (u(x 1 )) u (x 1 ) f (m) u (x 0 ) Ef (u(x 1 )) = δ ln E f (u(x 1 )) f (m) Ef }{{} (u(x 1 )) }{{} prudence erm pessimism erm u (x 1 ). u (x 0 ) Thus, already in a sandard seing wihou ambiguiy, an appropriae represenaion decomposes he social discoun rae ino a prudence and a pessimism erm. The smooh ambiguiy model and he KP model yield similar forms for he social discoun rae. Comparing he wo, he smooh ambiguiy model replaces oucomes in he KP seing by condiional expecaions (condiional wih respec o subjecive uncerainy). This similariy arises because he KP preference srucure allows for ineremporal risk aversion wih respec o objecive risk, while he KMM model allows for ineremporal risk aversion wih respec o subjecive risk (and assumes neuraliy wih respec o objecive risk). Combining he wo models yields prudence and pessimism effecs in boh uncerainy layers. Corollary 3: KMM merged wih KP model In a general seing wih subjecive over objecive loeries he social discoun rae collapses o he form f r = δ ln E subj (m obj ) E f obj (u(x 1 )) u (x 1 ) p subj f subj (m subj p ) obj f obj (m obj ) u (x 0 ) Ep subjf subj (m obj ) f = δ ln E subj (m obj ) f subj (m subj p subj ) E }{{} p subjf subj (m obj ) }{{} subj. prudence erm E p objf obj (u(x 1 )) f obj (m obj ) }{{} obj. prudence erm E p obj subj. pessimism erm f obj (u(x 1 )) E p objf obj (u(x 1 )) }{{} obj. pessimism erm u (x 1 ) u (x 0 ). Proposiion 2 hen saes ha he subjecive prudence erm reduces he social discoun rae if and only if subjecive prudence dominaes subjecive risk aversion and 11

13 ha he objecive prudence erm reduces he discoun rae if and only if objecive prudence dominaes objecive risk aversion. In he language of Leland (1968): If boh risk measures AIRA subj and AIRA obj are decreasing in heir argumens, hen (a leas) he prudence erms reduce he social discoun rae. Observe ha, in he represenaion of his secion, he argumens of he risk aversion funcions f s are no physical wealh, bu uiliy ha measures oucome appreciaion derived from ineremporal rade-offs. 3.2 The social discoun rae in erms of Arrow Pra risk aversion The previous secion decomposed he social discoun rae ino a prudence and a pessimism erm. While Gierlinger & Gollier (2008) have derived his decomposion for he ambiguiy seing, I have shown he analogous decomposiion for he more general seing as well as for he pure risk case of Kreps & Poreus (1978). Gollier (2002) has shown ha he effec of uncerainy on he social discoun rae in a Kreps- Poreus framework can be unambiguously deermined. His decomposiion does no involve he pessimism erm, whose sign generally depends on he underlying loery. This secion develops a similar represenaion of he social discoun rae for he general model and pins down he overall effecs of uncerainy on he social discoun rae. The key is o use a represenaion ha employs measures of Arrow Pra risk aversion raher han measures of ineremporal risk aversion (or smooh ambiguiy aversion). ThisapproachisonlypossibleinaonecommodiyseingandIassumehaoucomes are drawn from a closed subse of IR for he remainder of his secion. As discussed a he end of secion 2, he funcions g s = f s u characerize Arrow Pra risk aversion wih respec o loeries of degree of confidence s. The subsequen discouning formula builds on a preference represenaion ha eliminaes he f s funcion and inroduces he g s measures of risk aversion insead. In his represenaion, cerainy equivalens are measured in real erms raher han in cerainy equivalen uiliy. I denoe he i-h layer cerainy equivalen by n i n i ( ) M gŝ(pi ) M gŝ(p N ) x p N 1 and n N+1 x 1 (or equivalenly n i = u 1 (m i )). 12

14 Proposiion 3: In a one commodiy seing, he social discoun rae of Proposiion 1 is also characerized by r = δ ln u (n 1 ) u (x 0 ) }{{} Arrow Pra risk aversion (combined of all levels) N E gŝ(pi ) (n ) gŝ(pi ) (n i ) }{{} Arrow Pra prudence confidence level ŝ( ). (7) Again, heexpecedvalueoperaore acsonloeries+1 andcerainyequivalens n o he righ, including hose in he nex produc erm. The imporan difference beween equaion (6) and (7) is ha, in equaion (7), he marginal uiliy raio shows up on he lef of he expeced value operaors, evaluaing only he cerainy equivalen n 1. In consequence, he proof underlying Proposiion 2 can be applied recursively o all Arrow Pra prudence erms yielding he following resul. Proposiion 4: The social discoun rae under uncerainy is lower han I) under cerainy if he coefficien of absolue Arrow Pra risk aversion ARA s = gs g s is decreasing for all confidence levels, i.e. gs g s > gs g s for all s S. Only relying on smoohness of he funcions {g s } s S, he condiion is g s g s 1 convex for all s S. II) in he sandard model (where g s = u s S) if, in addiion o he condiions saed in par I, absolue Arrow Pra risk aversion dominaes uiliy prudence, i.e. gs g s > u u for all s S. Only relying on smoohness of he funcions {g s } s S and u, he condiion is u g s 1 concave for all s S. In he special case of KP preferences (#S = 1), par I of he proposiion was derived by Gollier (2002). If he decision maker is more Arrow Pra risk averse he lower his wealh, hen uncerainy over his fuure income will induce higher savings (hereby effecively reducing risk aversion). The corresponding special case of par II where 13

15 #S = 1 exends Gollier s finding by comparing he social discoun rae under KP preferences o he discoun rae in he sandard model. In he sandard model, he concaviy of marginal uiliy is he only ingredien ha reduces he social discoun rae under risk. In conras, under KP preferences he disenangled Arrow Pra risk aversion akes his role. If his disenangled risk aversion dominaes uiliy prudence Kreps Poreus preferences reduce he social discoun rae more. In paricular, a decision maker who does no exhibi uiliy prudence, bu exhibis Arrow Pra risk aversion, will always choose a lower discoun rae under Kreps Poreus preferences. The special case of he smooh ambiguiy model corresponds o S = {obj,subj} and g obj = u. The condiion of smooh ambiguiy aversion, i.e. ha f subj is concave, ranslaes ino g subj u 1 concave, which means ha he decision maker is more averse o subjecive risk han o objecive risk or ineremporal subsiuion. Ambiguiy iself relaes o second order subjecive uncerainy (over objecive firs order loeries). Corollary 4: KMM model I) The inroducion of uncerainy in erms of ambiguiy and/or objecive risk decreases he social discoun rae if ARA subj = gsubj and η = u g subj u are boh decreasing or, equivalenly, if gsubj g subj > gsubj g subj and u u > u u. Only relying on smoohness of u and g subj, he condiions are u u 1 and g subj g subj 1 concave. Translaed ino he f-represenaion of secion 3.1 hesecondiionsbecomeu u 1 and f subj f subj 1 u u 1 f subj 1 concave. II) In an uncerain world, he inroducion of ambiguiy aversion reduces he social discoun rae if in addiion o he condiions saed in par I subjecive Arrow Pra risk aversion ARA subj dominaes uiliy prudence: gsubj g subj > u u. Only relying on smoohness of u and g subj, his addiional requiremen is u g subj 1 concave. 14

16 In he saemen relaing o he f-represenaion in par I of he corollary, he expression is a muliplicaion ( ) of wo composed funcions. The corresponding condiion can be ranslaed ino a hird order condiion, however, he resuling expression exends over several lines and is of lile insigh. Gierlinger & Gollier (2008) have analyzed he quesions answered in Corollary 4 in he f-represenaion. For general funcional forms, hey only found join condiions on preferences and loeries in order o idenify when ambiguiy aversion reduces he discoun rae. In conras, he above corollary holds for all well defined ambiguous loeries. Noe ha he corollary also holds for he seing where he decision maker faces objecive over subjecive loeries, which is no par of he KMM seing. Finally, observe ha i would be misleading o inerpre uiliy prudence in par I of Corollary 4 as a decreasing absolue aversion o ineremporal subsiuion. The comparison wih he general resul in Proposiion 4 shows ha aversion o ineremporal subsiuion only plays a role as enangled aversion o objecive risk. 4 The muliperiod case This secion exends he discouning formula o seings wih an arbirary ime horizon. In he discouned expeced uiliy sandard model he discoun rae only depends on consumpion and uncerainy in he invesmen and he payoff period. This simplificaion no longer holds in he curren seing, or in he special cases of Kreps-Poreus or smooh ambiguiy preferences. Here, uncerainy resoluion beween he invesmen and he payoff period influences he discoun rae, and so does he uncerainy governing he pos-payoff fuure. In general, Arrow Pra prudence no longer characerizes fully he overall effec of uncerainy on he discoun rae. The second par of his secion derives a paricularly simple discouning formula under he assumpions of normal growh raes and homoheic preferences. I discuss how he familiar Ramsey rule saed in equaion 1 changes under aversion o he lack of confidence in probabiliy esimaes and a decrease of confidence in he fuuriy of forecass. 4.1 The general case Le he decision maker evaluae a projec wih payoffs in period T. In general, he ime horizon influences he discoun rae, even if i surpasses he ime of he projec 15

17 payoffs. I assume a planning horizon T > T. 4 The social discoun rae corresponds o he equilibrium ineres rae on a zero coupon bond wih mauriy in period T. Uncerainy in period is capured by N layers of uncerainy. Loery p 1 is a loery over loeries p 2 in he nex lower uncerainy layer, coninuing down o loeries p N 1 over p N. The final layer of uncerainy in period, p N, characerizes uncerainy over oucomes x and over he remaining fuure p A degree of confidence ŝ( ) S characerizes each loery. This consrucion generalizes Kreps & Poreus s (1978) concep of emporal loeries. For a deailed descripion see Traeger (2010). Preferences are exended recursively o he muliperiod case. They are saionary giving rise o he exisence of consan pure rae of ime preference δ. If he decision maker adops a finie planning horizon T hen W T = u(x T) capures welfare in he las period afer all uncerainy resolved. If he decision maker s planning horizon coincides wih he ime of payoff T, hen W T = u(x T ). Welfare in earlier periods is obained by recursively calculaing 5 N W 1 (x 1,p 1 ) = u(x 1 )+e δ for 1,...,T. M fŝ(pi ) W (x,p 1 +1) (8) I denoe cerainy equivalen welfare in uncerainy layer i of period for some j given loery by m i ( N ) ) W (x,p 1 +1) or, dropping he argumen, Mfŝ(p j=i p j simply by m i. Moreover, I use m N+1 rae in he muliperiod seing was defined as he yearly average. W (x,p 1 +1). Recall ha he social discoun 4 In he case of an infinie ime horizon, I assume ha fuure consumpion grows sufficienly slow ha he welfare funcional converges. Here, welfare is generally obained as a fix poin of equaion (8) under some saionariy assumpion, or by explicily spelling ou he equaions of moion and he formulaing he Bellman equaion. Assuming a posiive ime preference and a saionary consumpion process beyond some poin in ime ˆT > T would permi he decision maker o calculae he value WˆT in he infinie ime horizon seing and hen o simply work hrough he relevan years of he projec recursively. 5 The recursion calculaes welfare a he end of a given period when uncerainy only remains abou fuure oucomes. To obain welfare a he onse of period 1 simply apply N 1 o equaion (8). M f ŝ( 1 )

18 Proposiion 5: The social discoun rae in he muliperiod seing for payoffs in period T is r = δ 1 T ln { T =1 = δ 1 T ln { T =1 N N E fŝ(pi ) (m ) fŝ(pi ) (m i ) E fŝ(pi ) (m ) fŝ(pi ) (m i ) }{{} prudence erm confidence level ŝ( ) } u (x T ) u (x 0 ) E fŝ(pi ) (m ) E fŝ(pi ) (m ) }{{} pessimism erm confidence level ŝ( ) } u (x T ) u (x 0 ) For i < N T he expeced value operaor E acs on loeries +1 and cerainy equivalens m. The expeced value operaor E N p acs on m N+1 = x and, for < T, on he loery p 1 +1 characerizing uncerainy in he nex period. The form for he discoun rae in Proposiion 5 does no depend on wheher he decision maker applies a finie planning horizon of ime T, some larger finie horizon, or an infinie planning horizon. However, he evaluaion of he cerainy equivalen uiliy levels do depend on he ime horizon. In he case of an infinie ime horizon he m i ( ) depend on an infinie consumpion process. Proposiion 2 also applies in he general seing. Proposiion 2 : A prudence erm of confidence level s reduces he social discoun rae, if and only if, he funcion f s exhibis decreasing absolue risk aversion AIRA s = fs f s : fs f s > fs f s. Only relying on smoohness of he funcions {f s } s S, he condiion is f s f s 1 convex for all s S. Once more, he assumpion of a one-commodiy seing permis a ranslaion of he risk aversion measures ino Arrow Pra erms by using g s = f s u for s S. The corresponding represenaion of he social discoun rae employs he definiions of he cerainy equivalens in real erms n i ( ) = u 1 (m i ( )) including n N x 0. Proposiion 6: In a one-commodiy seing, he social discoun rae expressed by means of Arrow Pra risk aversion for I) a decision maker adoping he ime horizon T = T coinciding wih he ime of he payoff is r = δ 1 T ln T =1 u (n 1 ) u (n N ) 17 N E gŝ(pi ) (n ) gŝ(pi ) (n i ). (9)

19 II) a decision maker adoping a ime horizon T > T (possibly infinie) is { r = δ 1 T ln u (n 1 N } ) E gŝ(pi ) (n ) u (x T ) =1 u (n N ) gŝ(pi ) (n i ) u (n N T+1 T ) { = δ 1 T ln N } E gŝ(pi ) (n ) u (n 1 ) u (x T ). gŝ(pi ) (n i ) u (n N+1 ) u (x 0 ) =1 Noe ha, once more, also he evaluaion of he cerainy equivalens n i depend on he ime horizon. In equaion (9) he use of Arrow Pra risk aversion shifs marginal uiliy again o he lef of he expeced value operaor in he respecive period. In a wo period seing, his shif enabled he powerful saemens of Proposiion 4. However, in he muliperiod seing hese marginal uiliies also depend on uncerainy and consumpion levels in oher periods. Because of his inerdependence of welfare he proof underlying Proposiion 4 no longer applies. Moreover, if he planning horizon exceeds ha of he projec, he inerdependence wih fuure welfare inroduces an addiional marginal uiliy erm o he righ of he las expecaion operaor. Assuming ha uncerainy only resolves in period T = T recovers a separable welfare funcion and Proposiion 4 applies again. Corollary 5: Uncerainy resolves only in period T = T If he planning horizon of he agen coincides wih he ime of he payoff T = T and here is no uncerainy resolving in earlier periods, hen Proposiion 4 also holds for he muliperiod seing. 4.2 Isoelasic preferences and normal uncerainy Isoelasic preferences are arguably he mos prominen specificaion in economics. They imply decreasing absolue coefficiens of aversion and underly he usual parameric formulaion of he Ramsey discouning formula. This secion exends he parameric Ramsey rule o he seing of his paper, assuming isoelasic preferences and normal growh raes. Le X IR describe an aggregae consumpion commodiy. Assume ha consumpion growh is uncerain and described by a normal disribuion. Growh from one period o he nex is capured by a single uncerainy layer. The 18

20 growh rae g = ln x +1 x N(µ,σ ;s ) is disribued normally, where s S labels confidence. I assume isoelasic preferences ha can be represened by he funcions u(x ) = xρ ρ and f s (z) = (ρz) αs ρ, implying g s = x α. In a seing wihou confidence, hese preferences correspond o hose used by Epsein & Zin (1989) and Weil (1990) o disenangle risk preferences from ineremporal subsiuabiliy. I employ he inverse of he ineremporal elasiciy of subsiuion η = 1 ρ = u u for measuring he decision maker s propensiy o smooh consumpion over ime. Epsein & Zin (1989) measure Arrow Pra risk aversion as a funcion of α only. Insead, I employ a measure proporional o he coefficien of relaive ineremporal risk aversion RIRA s = fs (z) f s (z) z = { 1 α s ρ for ρ > 0 α s ρ 1 for ρ < 0 (10) for all s S and ρ 0 η 1. The absolue in he definiion arises because he funcionsf aredecreasingforρ < 0, acaseiavoidedabovewihoulossofgeneraliy. 6 Noe ha in he Epsein-Zin specificaion he measure RIRA s of relaive ineremporal risk aversion goes o infiniy ρ 0. I herefore inroduce a renormalized measure { RIRAs 1 η 2 for η 1 ζ s = 2α s for η = 1 which is coninous a η = 1 and posiive if and only if he decision maker is ineremporal risk averse. Proposiion 7: The social discoun rae for payoffs in period T is T r = δ + 1 ηµ η 2σ2 T 2 ζ σ 2 s. (11) 2 =1 The firs erm in he sum capures how expeced growh reduces fuure marginal uiliy deriving from a uni of consumpion. The second erm corresponds o he risk erm of he sandard model. I is caused by aversion o ineremporal flucuaions generaed by he sochasic process. The las erm capures ineremporal risk aversion. Common values for η in he sandard model are beween 1 and 2. However, he 6 The generalized mean M f and M f coincide, hus, I can always pick an increasing version of f. 19

21 disenangled approach generally esimaes values of η smaller han uniy: Aversion o ineremporal subsiuion urns ou smaller when i no longer has o simulaneously play he role of risk aversion. Vissing-Jørgensen & Aanasio (2003), Basal & Yaron (2004), and Basal e al. (2010) idenify η = 2 as a reasonable esimae. Assuming 3 an expeced growh rae of 2%, he growh erm resuls in he range 1.3% 4%. Assuming a yearly sandard deviaion of 4%, he second erm ranges in 0.04% 0.3%, which is negligible. 7 Thus, ineremporal growh rends are highly significan for he social discoun rae, while wiggles are no. Relaive risk aversion, measured in he Arrow Pra sense, generally is assumed o range 5 10 giving rise o ζ 7 2 /9,15 5 /9 for η = 2, ζ 8,18 for η = 1, and ζ 9,27 for η = 2. The las conribuion in 3 equaion (11) hen ranges 0.6% 1.9%. Thus, ineremporal risk aversion reduces he social discoun rae significanly, while risk has a negligible effec in he sandard model. The numerical reasoning above assumes consan growh expecaions and disregards confidence. In he following, I keep µ and σ fix a no necessarily consan levels. I analyze he siuaion where confidence ino he normal disribuions decreases he furher he agen looks ino he fuure. I assume ha he decision maker is averse o subjeciviy or, equivalenly, o he lack of confidence in beliefs. The formal definiion of aversion o subjeciviy requires a non-degenerae complee order S 2 on he se of degrees of subjeciviy. A ranking where loeries labeled s are considered more subjecive (or less confiden) hen loeries labeled s is denoed s s. Following Traeger (2010), I define a decision maker as sricly averse o he subjeciviy of belief or he lack of confidence in beliefs if for all s,s S s s (x 1,...,x ) 1 2 s (x 1,...,x ) (x 1,...,x ) 1 2 s (x 1,...,x ) x 1,...,x,x 1,...,x X wih non-indifferen pahs f s (f s ) 1 sricly concave for all or, equivalenly, some {1,..., T}. The equivalence of he wo lines on he righ hand side is shown in he cied paper. Noe ha he definiion of smooh ambiguiy aversion is a special case of aversion o he subjeciviy of belief (or lack 7 These values are a rounded adopion of Kocherlakoa s (1996) esimaes µ = 1.8% and σ = 3.6% based on 90 year annual ime series for he US. The given ranges correspond o η 2 3,2. 20

22 of confidence) corresponding o he case #S = 2. Finally, le S = {S :S 0,1 S (s) > S (s ) s s and s,s S s.h. S (s)=0,s (s)=1} denoe he space of all order preserving maps from he absrac space of confidence descripions ono he uni inerval, which map he label indicaing mos confidence o zero and he label indicaing leas confidence o uniy. The following proposiion expresses he social discoun rae in erms of subjeciviy and aversion o subjeciviy (lack of confidence). Proposiion 8: Le a decision maker exhibi isoelasic preferences, ineremporal risk aversion o objecive loeries, and aversion o he lack of confidence. Le he growh rae be normally disribued as laid ou above. Then here exis parameers η IR, λ 0,1), and ζ 0 and a map S S such ha he discoun rae for a payoff in period T is T r = δ 1 ηµ η 2σ2 T 2 ζ σ 2, (12) 1 λs 2 =1 where s = S (s ) 0,1. For a decision maker who is sricly ineremporal risk averse wih respec o objecive loeries and saisfies sric aversion o he lack of confidence in beliefs i is ζ,λ > 0. The parameer λ capures aversion o he subjeciviy of belief. Assume ha confidence ino he normal disribuions describing uncerainy decreases he furher he agen looks ino he fuure: s +1 > s {1,...,T}. A decision maker whose evaluaion is independen of confidence (and, hus, obeys he KP model) is characerized by λ = 0. For him, he conribuion of ineremporal risk aversion o he social discoun rae is capured by ζ. For general decision makers, however, ζ only reflecs ineremporal risk aversion o objecive loeries (ζ = ζ s ). A decision maker wih sric aversion o he subjeciviy of beliefs exhibis λ > 0. Such a decision maker is relaively more willing o inves ino he fuure, in order o increase and ensure his fuure consumpion level, han is a decision maker described by he sandard model. This difference increases he furher he agen looks ino he fuure, increasing baseline consumpion furher in response o an increasing lack of confidence. An example 21

23 is a decision maker who akes ino consideraion ha climae change, revoluionary research, or social ensions migh make fuure growh less (and less) predicable. Given he agen s lack of confidence in his abiliy o describe he fuure adequaely, he invess more ino projecs wih fuure payoffs. More likely han no, a decision maker in such a siuaion would no only pick s o be increasing over ime, bu also σ, boh lowering he social discoun rae for long-erm payoffs. Noe ha he combinaion of a complee lack of confidence s = 1 and an exreme aversion o he lack of confidence λ 1 resuls in he Arrow & Hurwicz (1972) crierion for decision making under ignorance. Tha is, such a decision maker would only pay aenion o he wors possible oucome. 8 5 Conclusions I derived a generalized discouning formula for a seing where a decision maker disinguishes he degree of confidence for differen probabilisic descripions of he fuure. As special cases I obained he social discoun raes in he smooh ambiguiy model and in he seing of Kreps-Poreus (or Epsein-Zin) preferences. I have shown ha a previously derived decomposiion of he social discoun rae under ambiguiy ino a prudence and a pessimism effec already obains in he Kreps-Poreus seing wih objecive risk. Moreover, he ambiguiy prudence effec is a special case of a prudence effec capured by ineremporal risk aversion wih respec o subjecive loeries. Expressing he preference represenaion in erms of (generalized) Arrow Pra risk aversion I eliminaed he pessimism effec. In he wo period seing, I have shown ha decreasing Arrow Pra risk aversion (for differing degrees of confidence) is a sufficien condiion ensuring ha general forms of uncerainy reduce he social discoun rae. The inuiion behind his finding is he same as already esablished 8 The limi λ 1 corresponds o γ s for s 1, giving rise o full weigh on he minimal elemen carrying posiive probabiliy mass. The normal disribuion has full suppor on IR so ha he decision maker pus all weigh on however small possibiliy of dying of hunger (or worse). If we offer such an agen a zero coupon bond enabling a sure ransfer ino he fuure and allowing him no o worry abou sarvaion in ha period he would pay an infinie amoun for he firs marginal ransfer. This is refleced by he discoun rae going o infiniy if boh, λ and s, approach uniy. Obviously, he underlying growh model in combinaion wih he offer of a cerain ransfer would be oo simple a model in order o suppor he decisions of such an agen. A dismal heorem inerpreaion of equaion (12) would be a misinerpreaion (or exrapolaion beyond applicabiliy) of he model. 22

24 by Leland (1968). A decision maker whose risk aversion decreases in wealh has an addiional incenive o increase fuure baseline consumpion in order o reduce he harm of uncerainy. The condiion is equivalen o absolue prudence dominaing absolue risk aversion, boh measured in Arrow Pra erms for given confidence levels. Moreover, he social discoun rae is lower han in he sandard model, if in addiion absolue Arrow Pra risk aversion dominaes he absolue prudence of he uiliy funcion. A special case of hese findings is a sufficien condiion esablishing when smooh ambiguiy aversion decreases he social discoun rae, wihou resricing he domain of admissible loeries. I derived he parameric special case resuling from he assumpions of normally disribued growh raes and isoelasic preferences. Here, ineremporal risk aversion always reduces he discoun rae. I proved a paricularly convenien represenaion of he discoun rae ha employs a degree of ineremporal risk aversion wih respec o objecive loeries, a normalized numerical degree of subjeciviy of loeries, and a parameer characerizing aversion o he lack of confidence in beliefs. In his seing, I discussed how he assumpion ha confidence decreases in fuuriy reduces he discoun rae over ime. The furher he agen looks ino he fuure and he lower his confidence, he higher is his incenive o increase baseline consumpion. Many of he examples cied in he inroducion exhibi long-run benefis of curren acions - or long-run coss of curren inacion. Here, a reduced social discoun rae implies ha more projecs should be carried ou, e.g. miigaion of greenhouse gas emissions. Falling confidence in fuuriy implies hyperbolic dicoun raes reducing he exreme devaluaion of long-run consequences of curren acion implied by exponenial discouning, while mainaining sandard discoun raes for he shor erm (even for iid uncerainy). A slighly differen perspecive on his finding is as follows. The sandard model conains he implici assumpion ha long-run uncerainies are of he same ype as flipping a coin. This implici assumpion can resul in a bias agains precauionary acion ensuring fuure consumpion levels. 23