Chapter 8 Consumption and Portfolio Choice under Uncertainty

Transcription

1 George Alogoskoufis, Dynamic Macroeconomic Theory, 2015 Chaper 8 Consumpion and Porfolio Choice under Uncerainy In his chaper we examine dynamic models of consumer choice under uncerainy. We coninue, as in he Ramsey model, o ake he decision of he household wih regard o labor supply as given, assuming ha each household provides a uni of labor per period. We also assume ha he household can borrow and lend freely in compeiive capial markes. In hese iner-emporal models, consumpion does no depend on curren household income bu on oal household wealh, which consiss of is curren porfolio of asses, plus he presen value of expeced fuure labor income. In his sense, consumpion smoohs emporary changes in income, as i depends on he permanen income (Friedman 1957 or he life cycle income of he household (Modigliani Brumberg Analyzing he consumpion funcion of a represenaive household under cerainy in Chaper 2, we concluded ha he household consumes a proporion of is oal wealh ha depends on he evoluion of ineres raes, he pure rae of ime preference, he elasiciy of iner-emporal subsiuion in consumpion and he populaion growh rae. In his model, under cerainy, he impac of ineres raes on he propensiy o consume ou of oal wealh depends on he elasiciy of iner-emporal subsiuion. An increase in ineres raes has wo kinds of effecs on he propensiy o consume ou of oal wealh. Firs, i induces he household o subsiue fuure for curren consumpion, as i increases he cos of curren consumpion relaive o fuure consumpion. This is he resul of iner-emporal subsiuion in consumpion. Secondly, an increase in ineres raes increases household income from capial, inducing i o increase boh curren and fuure consumpion. This is he income effec. If he iner-emporal elasiciy of subsiuion is greaer han one, he propensiy o consume ou of oal wealh decreases when ineres raes rise, because he subsiuion effec prevails on he income effec. If he iner-emporal elasiciy of subsiuion is less han uniy, he propensiy o consume ou of oal wealh increases when ineres raes rise, because he income effec prevails upon he subsiuion effec. Finally, in he case in which he iner-emporal elasiciy of subsiuion of consumpion is equal o one, which corresponds o logarihmic preferences, he wo resuls cancel each oher ou, and he propensiy o consume ou of oal wealh is independen of he pah of real ineres raes, as i is equal o difference beween he pure rae of ime preference and he populaion growh rae. In addiion, an increase in real ineres raes leads o a decrease in he presen value of fuure labor income, reducing he overall wealh of he household and leading o lower consumpion, even if he case where he elasiciy of iner-emporal subsiuion is equal o one. Essenially, he wealh effec of real ineres raes on he presen value of income from employmen reinforces he subsiuion effec on curren consumpion.

2 The choice of consumpion under condiions of uncerainy is linked o he porfolio allocaion decisions of he household (Samuelson 1969, Meron Under uncerainy, consumpion generally depends on he same facors as under cerainy, only in he case of quadraic preferences, which guaranee cerainy equivalence. In he case of quadraic preferences we can derive he permanen income model of consumpion. In all oher cases, under uncerainy, we canno go beyond he firs order condiions and solve explicily for consumpion, unless we make furher resricive assumpions abou he preferences of households or he variabiliy of labor income. Wih regard o porfolio choice, his model resuls in he consumpion capial asse pricing model. This suggess ha, under quadraic preferences, he expeced reurn premium of a risky asse is proporional o he covariance of is reurn wih consumpion. This facor of proporionaliy is someimes referred o as a consumpion bea, and can be used o explain he valuaion of risky asses. I is worh keeping in mind ha, under uncerainy, he permanen income hypohesis and he consumpion capial asse pricing model rely on very resricive assumpions. In addiion, empirical sudies sugges wo puzzles ha hrow doub on he validiy of his model. One is he excess sensiiviy of consumpion o changes in curren income, and he second is he equiy premium puzzle. Thus, alhough he iner-emporal model of consumpion under uncerainy is a useful framework for modeling consumpion and porfolio allocaion decisions, one needs o go beyond he permanen income hypohesis and he consumpion capial asse pricing model and consider less resricive se ups, wih precauionary savings, incomplee markes for securiies and borrowing consrains. 8.1 Consumpion and Porfolio Choice under Uncerainy The problem of how consumers choose beween consumpion and saving under uncerainy, in conjuncion wih he allocaion of heir porfolio among differen asses, was firs analyzed in an iner-emporal seing by Samuelson (1969 and Meron (1969. In wha follows we shall presen he approach of Samuelson, who analyzed he problem in discree ime. Assume a household which a ime 0 maximizes an iner-emporal uiliy funcion of he form, T 1 1 E 0 1+ ρ u(c =0 (8.1 where E denoes a mahemaical expecaion based on he se of available informaion available a ime. ρ is he pure rae of ime preference of he household and u a one period uiliy funcion, depending on he level of curren consumpion. The household is uncerain regarding is fuure income from employmen and he fuure reurns of is porfolio. The evoluion of he value of he household porfolio is described by,!2

3 A +1 = (A + Y C [(1+ r ω + (1+ x (1 ω ] (8.2 A is he value of he porfolio of asses of he household in he beginning of period. Y is labor income, which is assumed o be a random variable whose value is known in period. Gross savings of he household are defined by A+Y-C. The household allocaes is gross savings beween a safe asse, wih cerain reurn r, and a risky asse, wih uncerain reurn x. The porfolio allocaion decision of he household is deermined by he percenage ω of is asses ha is invesed in he safe asse. The erm in brackes in (8.2 hus denoes he average rae of reurn of he household s porfolio. The household chooses a consumpion and porfolio allocaion plan for period 0, in he knowledge ha i will be able o choose a new plan in he following period 1, a new plan in he following period 2, and so on, unil he penulimae period T-1. The easies mehod of solving of dynamic problems under uncerainy is he mehod of sochasic dynamic programming. 1 Dynamic programming convers muli-period problems ino a sequence of simpler wo period selecion problems. The firs sep is he inroducion of a value funcion V(A, which is defined as, T 1 s 1! V (A = max E, under he consrain (8.2 ( ρ u(c s s= The value funcion in period is he discouned presen value of he expeced uiliy of he household, calculaed under he assumpion ha he household follows he opimal program of consumpion and porfolio allocaion. This opimal value depends on he value of he porfolio of he household a he beginning of period, which is he only sae variable affecing he household. The value funcion depends of course on he condiional join probabiliy disribuion of he random variables ha describe he fuure labor income of he household and he uncerain rae of reurn of he risky asse, as well as he rae of reurn of he safe asse and he lengh of ime beween and T-1. This dependence is indicaed by he ime index for he value funcion, indicaing ha he value funcion may be changing over ime. From (8.3, he value funcion saisfies he following recursive equaion, which is known as he Bellman equaion. V (A = max u(c + 1 {C,ω } 1+ ρ E [ V +1 (A +1 ] (8.4 The value funcion in period is equal o he maximum uiliy of consumpion in period plus he discouned expeced value funcion in period +1. The firs order condiions for he maximizaion of he righ hand side of (8.4 under he consrain (8.2 are, 1 See Mahemaical Annex 3.!3

4 1 u (C = E ( 1+ ρ (1+ r ω + (1+ x (1 ω V (A E [ V +1 (A +1 (r x ] = 0 (8.5 (8.6 In (8.6 we have made use of he fac ha he discouned gross savings of he household, which are equal o (A-Y-C/(1+ρ, are known in period. Applying he envelope heorem o (8.4, i.e he effecs of a small change in he value of he porfolio of asses A on boh sides of (8.4, we ge ha, 1 V (A = E ( 1+ ρ (1+ r ω + (1+ x (1 ω V (A (8.7 From (8.5 and (8.7 i follows ha, V (A = u (C (8.8 The marginal value of he household porfolio of asses in he applicaion of he opimal program is equal o he marginal uiliy of consumpion. As a resul, we can use (8.8 o subsiue he marginal uiliy of fuure consumpion for he marginal value of he fuure porfolio of asses in he firs order condiions (8.5 and ( u (C = E ( 1+ ρ (1+ r ω + (1+ x (1 ω u (C [ ] = E [ u (C +1 (1+ x ] E u (C +1 (1+ r (8.9 (8.10 Subsiuing (8.10 in (8.9, he wo condiions ake he form, [ ] u (C = 1+ r 1+ ρ E u (C +1 (8.11 ( u (C = 1 1+ ρ E 1+ x u (C +1 (8.12 Condiions (8.11 and (8.12 have a simple inerpreaion, which is a generalizaion of he inerpreaion of he Euler equaion for consumpion in he Ramsey problem. Recall, ha he Euler equaion for consumpion in he Ramsey problem suggess ha he marginal rae of subsiuion beween he levels of consumpion in he wo periods mus be equal o he marginal rae of ransformaion. In he case of (8.11, assume ha he household reduces is consumpion by an infiniesimally small amoun dc in period, invess he amoun in he safe asse, and consumes he reurn in he nex!4

5 period +1. The reducion in is uiliy in period is equal o u (C, ha is he lef hand side of (8.11. The increase in is expeced uiliy in period +1 is equal o he righ hand side of (8.11. In he applicaion of he opimal program, his infiniesimal reallocaion does no affec he value of he plan, and as a resul (8.11 holds. (8.12 holds for he same reason, under he assumpion ha he household invess in he risky asse, raher han he safe asse in is porfolio The Ramdom Walk Model of Consumpion Equaions (8.11 and (8.12 are jus firs order condiions, and do no describe he full soluion of he problem. Neverheless, hey sugges srong resricions for he dynamic behavior of consumpion. (8.11 implies ha, 1+ r, where ( ρ u (C = u (C + ε E (ε +1 = 0 (8.13 suggess ha given he marginal uiliy of consumpion u (C, here no addiional informaion available in period ha could help predic u (C+1, he fuure marginal uiliy of consumpion. Assuming ha he uiliy funcion is quadraic in consumpion, and ha he rae of reurn of he safe asse is equal o he pure rae of ime preference, hen (8.13 akes he form, C +1 = C + ε +1, where E (ε +1 = 0 (8.14 Consumpion follows a random walk. Given he level of consumpion in period, no oher variable known in period can help predic consumpion in period +1. This predicion of he model was firs highlighed by Hall (1978, who invesigaed i empirically. This has generaed a hos of heoreical and empirical follow up sudies of his predicion The Consumpion Capial Asse Pricing Model Condiions (8.11 and (8.12 can also be used o deermine he rae of reurn of he risk free asse and he expeced rae of reurn, and hence he price of he risky asse. This requires ha all individual households are alike, i.e. ha here is a represenaive household. From (8.11, he rae of reurn of he risk free asse will saisfy, u (C! 1+ r = (1+ ρ (8.15 E u (C +1 [ ] From (8.12, i follows ha,! u (C = 1 { ( ρ E ( 1+ x E ( u (C +1 + Cov ( 1+ x, u (C +1 } From (8.16, he expeced rae of reurn of he risky asse will saisfy,!5

6 ( Cov ( 1+ x, ( u (C! E ( 1+ x = (1+ ρ u (C +1 (8.17 E u (C +1 E u (C +1 From (8.15 and (8.17, he expeced reurn premium of he risky asse is given by, ( (! E ( x r = Cov 1+ x, u (C +1 (8.18 E u (C +1 The expeced reurn premium of he risky asse depends on he covariance of he rae of reurn of he risky asse wih he marginal uiliy of consumpion. Given ha he marginal uiliy of consumpion is negaively correlaed wih consumpion, because of decreasing marginal uiliy, he expeced reurn premium of he risky asse will depend posiively on he covariance of he rae or reurn of he risky asse wih consumpion. Risky asses whose reurns are posiively correlaed wih consumpion, will end o have a higher expeced reurn relaive o he risk free asse. This model of he deerminaion of expeced asse reurns is known as he consumpion capial asse pricing model, or, consumpion CAPM. 8.2 From he Firs Order Condiions o he Full Analysis of Consumpion From he firs order condiions we canno fully describe he behavior of consumpion and savings, apar from specific cases. There are wo special cases where we can come up wih specific soluions. The firs is he case of insurable income risk, and he laer is he case of quadraic uiliy funcions. As demonsraed by Meron (1971, if labor income can be insured, we can deduce specific soluions for consumpion for a broad class of uiliy funcions, he so called hyperbolic absolue risk aversion, or HARA, uiliy funcions. This class includes isoelasic uiliy funcions wih consan relaive risk aversion (consan relaive risk aversion, or CRRA, he exponenial uiliy funcion wih consan absolue risk aversion (consan absolue risk aversion or CARA and quadraic uiliy funcions. One way o derive he specific soluion is o use he Belmann principle of opimaliy, which says ha for any value of he sae variable (he porfolio of asses in his case a a given ime period, he soluion for he fuure mus be opimal. Using his principle and he value funcion, he soluion can be found hrough backward inducion. For example, in period T-2, for any value of he porfolio of asses AT-2, he household faces a wo-period problem. Solving his problem, ake a sep back, and solve he problem of he period T-3, having already idenified he value of he value funcion of T-2. Then we move on and solve he same problem inducively for he T-4 period and so on. In he case of an infinie ime horizon we ake he limi of he soluion o he problem of T periods, as T ends o infiniy. Alernaively, we can also solve he problem of infinie periods direcly. For example, if he consumer has an infinie ime horizon, if he "safe" ineres rae is fixed and if he uncerain reurn x is disribued according o an independen, uniform, probabiliy disribuion, he value funcion is independen of ime and only depends on he sae variable A.!6

7 Therefore, we can presume is form, derive he consumpion funcion and verify if our presumpion was correc. The reason ha HARA ype uiliy funcions allow us o infer analyical soluions, is ha he value funcion belongs o he same family as he uiliy funcion, and all ha remains is o infer he parameers of he value funcion The Case of Logarihmic Preferences Consider he simple case in which, u(c = lnc (8.19 Using he resul of Meron ha he value funcion has he same funcional form as he uiliy funcion, for HARA ype uiliy funcions, we presume ha he value funcion akes he form, V(A = aln(a + b (8.20 where a and b are consan parameers o be deermined. This conjecure allows us o formulae he maximizaion problem in period as, maxln(c ρ E [ aln(a +1 + b] (8.21 under he consrain, 2 A +1 = (A C [(1+ rω + (1+ x (1 ω ] (8.22 Solving for consumpion C and he share of he porfolio invesed in he safe asse ω we ge, C = 1+ a 1+ ρ 1 A (8.23 ( 1 E (r x (1+ rω + (1+ x (1 ω = 0 (8.24 (8.23 deermines consumpion as a linear funcion of he value of he oal porfolio of asses of he household. (8.24 deermines indirecly he opimal proporion invesed in he safe asse ω as a consan, due o he assumpion ha he reurn of he risky asse x is disribued according o a uniform, independen probabiliy disribuion. The proporion ω is independen of he oal value of he porfolio of asses. In order o find a and b we subsiue (8.22, (8.23 and (8.24 in he value funcion (8.21 and compare coefficiens wih (8.20. b is a complex bu no economically significan consan, which 2 Since we assume ha labor income risk is insurable, we concenrae on he case in which labor income Y is zero.!7

8 depends on all he parameers of he model. Using (8.8, a is deermined as (1+ρ/ρ. Subsiuing his value in he consumpion funcion (8.23, we ge,! C = ρ ( ρ A In conclusion, assuming ha labor income is insurable and ha he uiliy funcion is logarihmic, we ge a consumpion funcion analogous o he case of cerainy. Consumpion is a linear funcion of he oal value of he porfolio of he household, and he marginal propensiy o consume ou of wealh depends only on he pure rae of ime preference and no on he real ineres rae or he rae of reurn of he risky asse. Changes in fuure non labor income (dividends and ineres affec consumpion only hrough heir impac on oal wealh Quadraic Preferences and Cerainy Equivalence The second case we shall examine is he case of quadraic preferences. We shall assume ha he porfolio consiss only of he safe asse. The maximizaion problem of he household is, T 1 1 max E 0 =0 1+ ρ 2 ( ac bc (8.26 under he consrain, A +1 = (1+ r A + Y C ( A T 0 From he firs order condiions for a maximum,, (8.27 E C +1 = r ρ 1+ r a 2b + 1+ r 1+ ρ C (8.28 In wha follows we shall assume ha r=ρ. In his case, (8.28 akes he form, E C +1 = C (8.29 (8.29 implies ha,! E 0 C = C 0 για =0, 1, 2,..., T-1 (8.30 Because of he equaliy beween he real ineres rae and he pure rae of ime preference, he opimal consumpion pah is such ha here is perfec consumpion smoohing. The opimal pah of consumpion is such ha expeced consumpion is consan along he opimal pah. The budge consrain (8.27 implies ha, T 1 A T = A 0 (1+ r T + (1+ r T (Y C =0 (8.31!8

9 Because he household does no derive uiliy from is porfolio direcly, bu only from is consumpion, is consumpion in he las period will be such ha AT=0. Using his ransversaliy condiion in (8.31, he iner-emporal budge consrain of he household will be equal o, T 1 1 T 1 1! (8.32 =0 1+ r C = A 0 + =0 1+ r Y According o (8.32, he presen value of consumpion of he household is equal o he value of is iniial porfolio, plus he presen value of expeced labor income. The household knows a ime 0 ha he budge consrain (8.32 should be saisfied, bu does no know fuure labor income, Accordingly, a ime 0 he household aims o saisfy, T 1 1 E 0 =0 1+ r C T 1 1 = A 0 + E 0 =0 1+ r Y (8.33 (8.33 suggess ha he expeced presen value of consumpion is equal o he value of he original porfolio of household, plus he expeced presen value of labor income. Subsiuing (8.30 in (8.33, and aking he limi as T ends o infiniy, C 0 = r 1 A 0 + E 0 1+ r =0 1+ r Y (8.34 Consumpion is consan and is a fixed percenage of he oal wealh of he household, including he presen value of expeced labor income. In every period, he household consumes a consan fracion of is oal wealh, depending on he real ineres rae (or he pure rae of ime preference, so ha expeced oal wealh remains consan The Permanen Income Hypohesis wih Quadraic Preferences From (8.30, he change in consumpion from period o period is deermined only by he revision of expecaions regarding labor income. C C 1 = r 1+ r i= r i ( E ( Y +i E 1 ( Y +i (8.35 For example, if labor income follows a saionary firs order auoregressive sochasic process, i.e. an AR(1 process of he form, Y = Y 0 + λy 1 + ε, 0 < λ < 1 (8.36 hen, from (8.31, he change in curren consumpion depends only on he curren innovaion in labor income ε. Thus, under he assumpion in (8.36, (8.35 akes he form,!9

10 C C 1 = r 1+ r λ ε (8.37 The coefficien of he ransiory innovaion in income ε is smaller han uniy. (8.37 incorporaes he predicions of he permanen income hypohesis of Friedman (1957 and he life cycle hypohesis of Modigliani και Brumberg (1954, ha consumpion smoohs ou ransiory changes in income. If λ is equal o uniy, hen disurbances in labor income ε are of a permanen naure, and he coefficien of (8.33 is also equal o uniy. Permanen changes in labor income lead o equivalen permanen changes in household consumpion. Empirical sudies of he permanen income hypohesis sugges ha aggregae consumpion displays excess sensiiviy o changes in curren income, indicaing ha one may have o go beyond he permanen income hypohesis in explaining aggregae consumpion The Consumpion Capial Asse Pricing Model wih Quadraic Preferences Assuming quadraic preferences, as we have in (8.26, he marginal uiliy of consumpion is given by,! u (C = a bc (8.38 The marginal uiliy of consumpion is hus a negaive linear funcion of consumpion. Subsiuing for he marginal uiliy of consumpion in (8.18, we ge, ( r = 2bCov ( 1+ x,c +1! E x (8.39 a be C +1 This confirms ha under quadraic preferences he expeced reurn premium of a risky asse is proporional o he covariance of is reurn wih consumpion. This facor of proporionaliy is someimes referred o as a consumpion bea, from a regression of consumpion growh on asse reurns. Thus, a cenral predicion of he consumpion CAPM is ha he reurn premium of a risky asse is proporional o is consumpion bea. 4 However, empirical sudies sugges a so called equiy premium puzzle, i.e a much bigger difference beween he average reurn of equiies (he risky asse and governmen bonds (he safe asse han would be suggesed by he consumpion CAPM. 5 3 The empirical lieraure on he permanen income hypohesis is huge. Tess based on aggregae daa include he original paper of Hall (1978, Flavin (1981, 1985, Hansen and Singleon (1982, 1983, Campbell and Mankiw (1989, 1991 and ohers. Aanasio (1999 surveys boh aggregae and disaggregaed sudies. 4 See Meron (1973 and Breeden (1979. The original capial asse pricing model (CAPM of Sharpe (1964 and Linner (1965 assumed ha invesors are concerned wih he mean and variance of he reurn of heir porfolio, raher han he mean and variance of consumpion. Tha version of he model hus focused on so-called marke beas, ha is coefficiens from regressions of asse reurns on he reurns of a marke porfolio. 5 See he imporan paper of Mehra and Presco (1985 ha has generaed a large heoreical and empirical lieraure.!10

11 & George Alogoskoufis, Dynamic Macroeconomic Theory, 2015 Chaper Precauionary Savings and Borrowing Consrains The assumpion of quadraic preferences is quie resricive. When preferences are no quadraic, cerainy equivalence does no hold, and uncerainy abou non-insurable labor income generally affecs consumpion. To see his, consider for example he firs order condiion (8.11 wih a consan risk free rae rae r. [ ] u (C = 1+ r 1+ ρ E u (C +1 (8.11 So long as consumers are risk averse, i.e. so long as u <0, increased uncerainy in he form of an increase in he variance of consumpion, decreases expeced uiliy. Bu he effec of increased uncerainy on behavior depends on wheher i affecs consumers expeced marginal uiliy, hrough he firs order condiion (8.11. As long as uiliy is quadraic, marginal uiliy is linear in consumpion, and u =0. Thus, in he case of quadraic uiliy, he variance of consumpion has no effec on expeced marginal uiliy, and hus no effec on opimal behavior. Afer all, his is why cerainy equivalence holds wih quadraic preferences. In he general case however, for mos plausible uiliy funcions, u >0. This means ha marginal uiliy is convex in consumpion, and an increase in uncerainy increases expeced marginal uiliy. Thus, from (8.11, increased variabiliy of fuure consumpion would require an increase in expeced fuure consumpion relaive o curren consumpion. Uncerainy leads consumers o defer consumpion, and hus be more pruden. The effecs of prudence on savings were firs analyzed by Leland (1968, and subsequenly by Sandmo (1970 and Dreze and Modigliani (1972. However, in general i is almos impossible o solve for opimal consumpion in he presence of pruden behavior. A case ha can be solved analyically is he case of consan absolue risk aversion, which implies a uiliy funcion of he form,! u(c = 1 (8.40 θ e θc where θ is he coefficien of absolue risk aversion. This case has been analyzed in Caballero (1990. Non quadraic preferences and precauionary savings are no he only deviaions from he permanen income hypohesis ha have been considered in he lieraure. Oher deviaions ha have been considered are incomplee markes and liquidiy (borrowing consrains, as well as deparures from full opimizaion. These exensions have been moivaed by empirical weaknesses of he permanen income hypohesis and he consumpion capial asses pricing model, and are surveyed exensively in Aanasio ( Conclusions!11

12 In his chaper we have examined he deerminaion of household consumpion under condiions of uncerainy, in conjuncion wih he deerminaion of he allocaion of he porfolio of he household among alernaive asses (Samuelson 1969, Meron Under condiions of uncerainy, for a household ha can borrow and lend freely in he capial marke, consumpion generally depends on he same facors as under cerainy. The curren and expeced fuure raes of asses, he curren and expeced fuure labor income and he oal value of he porfolio and human wealh of he household. Consumpion does no depend on curren household income bu on oal wealh, which consiss of he value of is porfolio, plus he presen value of curren and expeced fuure labor income In his sense, consumpion smoohs ou emporary changes in income, as i depends on permanen or life cycle income. (Friedman 1957, Modigliani Brumberg However, he permanen income hypohesis canno explain many of he feaures of individual or aggregae consumpion paerns, especially he excess sensiiviy of consumpion o changes in curren income. In addiion, he consumpion capial asse pricing model, which is an associaed predicion of sochasic iner-emporal models of consumpion, seems o be refued by he equiy premium puzzle. The lieraure has hus examined models ha resul in precauionary savings, or in which markes are incomplee and households are also bound by borrowing consrains (see Aanasio 1999.!12

13 References Aanasio O. (1999, Consumpion, in Taylor J.B. and Woodford M. (eds, Handbook of Macroeconomics, Volume 1B, Amserdam, Norh-Holland. Breeden D. (1979, An Ineremporal Asse Pricing Model wih Sochasic Consumpion and Invesmen, Journal of Financial Economics, 7, pp Caballero R.J. (1990, Consumpion Puzzles and Precauionary Savings, Journal of Moneary Economics, 25, pp Campbell J.Y. and Mankiw N.G. (1989, Consumpion, Income and Ineres Raes: Reinerpreing he Time Series Evidence, in Blanchard O.J and Fischer S. (eds, NBER Macroeconomics Annual, 1989, pp Campbell J.Y. and Mankiw N.G. (1991, The Response of Consumpion o Income: A Cross Counry Invesigaion, European Economic Review, 35, pp Dreze J. and Modigliani F. (1972, Consumpion Decisions under Uncerainy, Journal of Economic Theory, 5, pp Flavin M. (1981, The Adjusmen of Consumpion o Changing Expecaions abou Fuure Income, Journal of Poliical Economy, 89, pp Flavin M. (1985, Excess Sensiiviy of Consumpion o Curren Income: Liquidiy Consrains or Myopia?, The Canadian Journal of Economics, 18, pp Friedman M. (1957, A Theory of he Consumpion Funcion, Princeon N.J., Princeon Universiy Press. Hall R. (1978, Sochasic Implicaions of he Life Cycle Permanen Income Hypohesis: Theory and Evidence, Journal of Poliical Economy, 86, pp Hansen L.P. and Singleon K.J. (1982, Generalized Insrumenal Variables Esimaion of Nonlinear Raional Expecaions Models, Economerica, 50, pp Hansen L.P. and Singleon K.J. (1983, Sochasic Consumpion, Risk Aversion and he Temporal Behavior of Asse Reurns, Journal of Poliical Economy, 91, pp Leland H. (1968, Saving and Uncerainy: The Precauionary Demand for Saving, Quarerly Journal of Economics, 82, pp Linner J. (1965, The Valuaion of Risky Asses and he Selecion of Risky Invesmens in Sock Porfolios and Capial Budges, Review of Economics and Saisics, 47, pp Mehra R. and Presco E.C. (1985, The Equiy Premium: A Puzzle, Journal of Moneary Economics, 15, pp Meron R.C. (1969, Lifeime Porfolio Selecion under Uncerainy: The Coninuous Time Case, The Review of Economics and Saisics, 51, pp Meron R.C. (1973, An Ineremporal Capial Asse Pricing Model, Economerica, 41, pp Modigliani F. and Brumberg R. (1954, Uiliy Analysis and he Consumpion Funcion: An Inerpreaion of Cross Secion Daa., in Kurihara K. (ed., Pos Keynesian Economics, New Brunswick N.J., Rugers Universiy Press. Samuelson P.A. (1969, Lifeime Porfolio Selecion by Dynamic Sochasic Programming, The Review of Economics and Saisics, 51, pp Sandmo A. (1970, The Effec of Uncerainy on Saving Decisions, Review of Economic Sudies, 37, pp Sharpe W.F. (1964, Capial Asse Prices: A Theory of Marke Equilibrium under Condiions of Risk, Journal of Finance, 19, pp !13