Advanced and Contemporary Topics in Macroeconomics I. Consumption/Saving

Transcription

1 Advanced and onemporary opics in Macroeconomics I Alemayehu Geda ag56@gmail.com Web Page: lass Lecure Noe 6 onsumpion/saving Based on David Romer (0) Addis Ababa Universiy Deparemen of Economics PhD Program, 04

2 Lecure 6 onsumpion/saving

3 A. Inroducion: onsumpion/saving Aim: undersand he micro foundaion for grwoh. onsumpion and Invesmen decisions are crucial for growh and flucuaion Wih regard o grwoh he division of resources beween consumpion and various invesmen ( physical capilal, human capial and R&D) is crucial for growh hus i is imperaive o learn decisions abou such issues (such micro foundaions issues) I is also imperaive o noe ha a decision abou consumpion is closely relaed o saving which, in urn, is cenral o he grwoh process as we have seen in SS and RK as well as he Endogenous growh models.

4 Inroducion... on d In addiion here are wo oher major reason for sudying consumpion and invesmen: Firs, hey inroduce imporan issues involving financial markes (which in urn affec consumpion and invesmen) Second, insighful empirical work on macroeconomy in he las wo o hree decades is relaed o consumpion and invesmen Moreover in mos counries (in Ds) consumpion is abou 75% of GDP, and more in LDs hus he need o undersand his big aggregae.

5 B. onsumpion under erainiy:life- ycle/permanen-income Hypohesis Assumpions Assume an individual who lives years and her life ime uiliy is given by U u( ) u'( ) 0 u''( ) he individual has iniial wealh (A0), labour income Y o Y. She can borrow a an exogenous ineres rae (assumed zero; individual s discoun rae is also assumed zero) wih no Ponzi scheme. he budge consrain is hus given by 0 () A 0 Y ()

6 B. onsumpion under erainiy:life- ycle/permanen-income Hypohesis Behavior Since he marginal uiliy is always posiive, he individual saisfies he budge consrain wih equaliy and he Lagrangian for maximizaion is given by: L u( he FO for is u' ( ) ) A 0 0 Y u'( ) Since [4] holds every period he MU of consumpion is consan. Since consumpion uniquely deermines he MU, consumpion is, hence, consan. hus =..=, subsiuing his in he budge consrain offers Eqn [5] (3) (4)

7 onsumpion under erainiy...on d Y Eqn [5] saes ha he individual divides his/her life ime resources equally among each period of life** Implicaions A0 for all. his analysis says ha consumpion in a given period is deermined no by income in ha period bu by life ime income [ or permanen income] In he erminology of Friedman (957) he RH of Eqn [5] is he permanen income (PI) & he difference b/n PI and curren income is ransiory income. o see his disincion le us assume a windfall rise (Z) in curren income in he firs period. (5)

8 onsumpion under erainiy...on d Alhough Z raises curren income by Z, i raises permanen income only by Z/. If he hh s planning horizon [] is large he effec on PI and hence consumpion is negligible. {Ex, hus, he implicaion is say a emporary ax cu may no affec consumpion} We can also infer from his analysis ha alhough he ime paern of income is no imporan for consumpion, i is for saving, as give by [6] S S Y Y Y - A 0 Noe here ha S is higher when Y is higher relaive o is average ie when ransiory income is high (and he converse, negaive saving, if he reverse happens) (6)

9 onsumpion under erainiy...on d hus, individuals use saving and borrowing o smooh consumpion his is he key idea of he life-cycle/permanen income hypohesis of Modiglani and Brumberg (954) and Firedman (957). Wha is Saving? hen From boh heories above saving is simply fuure consumpion/or in classical language absinence from curren consumpion. hus, he decision beween saving and consumpion is driven by preference b/n presen and fuure consumpion. Given his view do you hink he poor saves less han he rich because is income is really low? Wha abou Keeping up wih joneses? Does i hold in his framework?**

10 onsumpion under erainiy...on d Empirical Applicaion: Undersanding Esimaed onsumpion Funcions he radiional Keynesian consumpion funcion posis ha consumpion is deermined by curren disposable income. And Keynes (936) claimed his relaionship is sable. Assered ha higher income leads o higher proporion of income o be saved. onrary o Keynes s claim esimaed consumpion funcion came up wih a differen sory. hese differen resuls are given by Figure, panel (a) o (c)

11 onsumpion under erainiy...on d Panel (a): Across hh a a poin in ime he relaionship is he ype ha Keynes posulaed. Panel (b): Bu, wihin a counry over ime, aggregae consumpion is essenially proporional o aggregae income. Panel (c): Furher he cross-secion consumpion funcion differ across groups (eg he slop of he consumpion funcion is he same for blacks and whies bu he inercep is higher for whies in USA) As noed by Friedman (957) a permanen income hypohesis provides an explanaion for hese findings and hence can reconcile hem.

12 onsumpion under erainiy...on d panel (a) Panel (b) Y Y Panel (c)

13 onsumpion under erainiy...on d onsider he regression of consumpion on curren income. In a univariae regression his implies bˆ i a by ov( Y, ) Var( Y) P ov( Y Y, Y P Var( Y Y ) P Var( Y ) P Var( Y ) Var( Y i e i P ) ) We have used here consumpion equals o PI & in he las expression PI and I are uncorrelaed. (7) (8)

14 onsumpion under erainiy...on d he regression resul implies ha he consan is given by (for he mean I is zero): aˆ by ˆ P Y bˆ Y P Y hus, he PI hypohesis predics ha he key deerminan of he slope of an esimaed consumpion funcion, b, is relaive variaion of permanen and ransiory income. An increase in curren income (I), increase consumpion only o exen i increase PI subsanially (hence I impacs) If I impac on PI is small, hence negligible impac on consumpion. ˆ P b Y (9)

15 onsumpion under erainiy...on d his explains he previous graphs. Across hh, much of he variaion in income reflecs I (such as unemploymen, hh being in differen life-cycle ec) hence he esimaed slope is subsanially less han. Over ime, in conras, variaion in income shows long-run growh/permanen increase in he economy s resources, hus he slope coefficien is closer o. For black and whie he relaive variaion of PI and I income is similar hence similar slope. Bu blacks average income is lower and hence lower inercep erm, a. [So difference in average income of blacks and whies is he reason for differen consumpion paern, no es or anyhing] Noe also he finding of Kuznes (94) where AP is consan over he long-run (agains Keynes where AP declines as Y increase [AP>MP] Recall he graph (similar o panel c)!.

16 . onsumpion under Uncerainiy: he Random-Walk Hypohesis Le us exend our analysis so far o accoun for uncerainly We will assume he insananeous uiliy funcion, u( ), is quadraic. hus, individuals maximizes: E( U) E a (0) As before no-ponzi scheme hus he budge consrain is given by Eqn []. o invesigae he hh behavior we use he Euler equaion approach of he RK lecure. Specifically suppose he individual choses he firs period consumpion opimally, given he informaion available (and do so in fuure periods oo). a 0

17 onsumpion under Uncerainiy...on d Now consider, a reducion in of d from he value he individual has chosen and an equal increase in consumpion a some fuure dae from he value he/she would have chosen. If he individual is opimizing, a marginal change of his ype doesn affec expeced uiliy. Since he MU of consumpion in period is -a, he change has a uiliy cos of (-a)d. Sine he MU of period- consumpion is -a, he change has an expeced uiliy benefi of E[-a]d, where E( ) is expecaion condiional on informaion available a period. hus if he individual is opimizing we have Eqn[] below

18 onsumpion under Uncerainiy...on d a Since E E [ E [ a ] [ a ] ] for equals,3...a ae Since he individual knows her life ime consumpion saisfies he budge consrain [] wih equaliy, he expecaion of he wo sides of he consrain will be, [ ] [] [] E [ ] A 0 Y E [ ] [3] [] implies ha he LHS of [3] is, subsiuing his in [3] and dividing by gives, A0 E[ Y ] he individual consumes / of her expeced life ime resources [4]

19 onsumpion under Uncerainiy...on d Implicaions Eqn showed ha expecaion a period of equals, generalizing his, expeced nex period consumpion equals curren consumpion. hus, where E [ [ expecaion of ] e ] e e as of period -is 0, huse [5] - [6] [ ] his is he famous Hall s (978) resul ha he life-cycle/ permanen income hypohesis implies ha consumpion follows a random walk. he inuiion of his resul is ha if consumpion is expeced o change, he individual can do beer job of smoohing consumpion hus he individual adjus his or her curren consumpion o he poin where consumpion is no expeced o change

20 onsumpion under Uncerainiy...on d Our analysis can be used o find wha deermined he change in consumpion, or he e? onsider he change from period o. Using he same reasoning as in Eqn [4] implies ha =/(-) of he individual's expeced remaining life ime resources. A E [ Y ] A 0 Y E [ Y ] [7] Where he nd line uses A=A0+Y-. We can rewrie he expecaion as of period of income over he reminder of life, = E [Y ], = as expecaion of his quaniy as of period, = E [Y ], plus he informaion learned beween period &. = E [Y ] = E [Y ],Wih his we can wrie Eqn [7] as

21 onsumpion under Uncerainiy...on d [ie. Using his reasoning & A=A0+Y-, similar o Eqn [4]] Eqn [7] becomes, [8] ] [ ] [ ] [ 0 Y E Y E Y E Y A [9] ] [ ] [ ] [ ] [ Y E Y E Y E Y E

22 onsumpion under Uncerainiy...on d Eqn 9 shows ha he change in consumpion beween he wo periods equals he change in individual's esimae of her life ime income divided by he # periods of life remaining Eqn [9] shows he individual consumes he amoun she would if her fuure income were cerain o equal is means as shown in eqn 4 (ie uncerainy has no impac on consumpion). o see he inuiion for his cerainy-equivalen behaviour consider he Euler equaion relaing consumpion in he wo periods u ( ) E [ u'( )] [0] ' When uiliy is quadraic, MU is linear, hus E[MU()] is he MU(E[]) [ie E[-a]=-aE[], hus Eqn [0] becomes** [] u' ( ) u' E[( )] his implies [ =E [ ]. hus he analysis shows he quadraic uiliy is he source of he cerainy-equivalen behaviour

23 . Emperical Applicaion: Some ess on he Rndom-Walk Hypoheis he radiional view of consumpion over he business cycle implies ha when oupu declines, consumpion declines bu is expeced o recover implies ha here are predicable movemens in consumpion. Hall s exension of he permanen-income hypohesis, in conras, predics ha when oupu declines unexpecedly, consumpion declines only by he amoun of he fall in permanen income; as a resul, i is no expeced o recover and fall wih each change in income he hypohesis ha consumpion responds o predicable income movemens (which is in line wih Keynes) is referred o as excess sensiiviy of consumpion proposiion (Flavin, 98)

24 Emperical Applicaion...on d A) ampbell and Mankiw s es Using Aggregae Daa he RW hypohesis implies ha consumpion is unpredicable hence no variable a ime is useful for predicion: One approach o es he random-walk hypohesis is herefore o regress he change in consumpion on variables ha are known a. If he random-walk hypohesis is correc, he coefficiens on he variables[eg income] should no differ sysemaically from zero. his is he approach ha Hall ook in his original work. He found ha neiher income nor consumpion can predic he change in consumpion. He did find, however, ha lagged sock-price movemens have saisically significan predicive power for he change in consumpion.

25 Emperical Applicaion...on d he disadvanage of his approach is ha he resuls are hard o inerpre For example, Hall s resul ha lagged income does no have srong predicive power for consumpion could arise no because predicable changes in income do no produce predicable changes in consumpion, bu because lagged values of income are of lile use in predicing income movemens. ampbell and Mankiw (989) herefore use an insrumenal-variables approach o es Hall s hypohesis agains a specific alernaive. ( Y Y ) ( ) e Z v [] Z and v are almos surely correlaed [ie imes when income increases grealy are usually also imes when households receive

26 Emperical Applicaion...on d hus esimaing [] by OLS leads o esimaes of λ ha is biased upward. he soluion is o use IV raher han OLS. ˆ ˆ Z ( Z Z ) v Using his approach hey found ha: [3] lagged changes in income have almos no predicive power for fuure changes [in income]. his suggess ha Hall s failure o find predicive power of lagged income movemens for consumpion is no srong evidence agains he radiional view of consumpion. As a base case, hey herefore used lagged values of he change in consumpion as insrumens. When hree lags are used, he esimae of λ is 0.4, wih a sandard error of 0.6; when five lags are used, he esimae is 0.5, wih a sandard error of 0.3. Oher specificaions yield similar resuls.

27 Emperical Applicaion...on d hus ampbell and Mankiw s esimaes sugges significan deparures from he predicions of he randomwalk model: consumpion appears o increase by abou fify cens in response o an anicipaed one-dollar increase in income, and he null hypohesis of no effec of change income on consumpion is srongly rejeced. A he same ime, he esimaes of λ are far below. hus he resuls also sugges ha he permanen-income hypohesis is imporan o undersanding consumpion.

28 Emperical Applicaion...on d B) Shea s es using Household Daa Shea regressed consumpion growh on a measure of expeced wage growh; he permanen-income hypohesis predics ha he coefficien should be 0.9 he esimaed coefficien is in fac 0.89, wih a sandard error of hus Shea also finds a quaniaively large (hough only marginally saisically significan) deparure from he random-walk predicion. One reason ha consumpion migh no follow a random walk is ha he assumpion of borrowing as you wish migh fail ha is, ha households migh face liquidiy consrains.[if u no borrow curren income could be less han permanen income; so consn is deermine by curren income implies predicable change in income brings abou predicable change in consumpion.]

29 Emperical Applicaion...on d Many oher researchers have obained findings similar o ampbell and Mankiw s and Shea s. For example, Parker (999), Souleles (999), Shapiro and Slemrod (003), and Johnson, Parker, and Souleles (006) idenify feaures of governmen policy ha cause predicable income movemens. All hese auhors find ha he predicable changes in income resuling from he policies are associaed wih subsanial predicable changes in consumpion. his paern appears o break down, however, when he predicable movemens in income are large and regular Ec ec Read! more such aricles

30 D. he Iners Rae and Saving If we exend our consumpion wih cerainy o a consan ineres rae (r), he individuals budge consrain, discouned o period 0 will be given by ( r) A 0 ( r) Y [4] Similarly, as before, he consan-relaive-riskaversion uiliy wih a personal discoun rae ρ funcion could be given by [5] U ( ) [5]

31 D. he Iners Rae and Saving Doing our usual experimen of decreasing consumpion a and increasing i a +, wih ineres rae (r+) and he MU for &+gives [6], which could be simplified as [7] ( ) ( r / r) ( ) [6] his analysis implies ha once we allow for he possibiliy ha he real ineres rae and he discoun rae are no equal, consumpion need no be a random walk: consumpion is rising over ime if r exceeds ρ and falling if r is less han ρ.. [7]

32 he Iners Rae and Saving...on d Hansen and Singleon (983), Hall (988b), ampbell and Mankiw (989), and ohers herefore examine how much consumpion growh responds o variaions in he real ineres rae. For he mos par hey find ha i responds relaively lile, which suggess ha he ineremporal elasiciy of subsiuion is low (ha is, θ is high). I hink similar resuls are found in Ehiopian and African counries (check some MSc hesis)

33 E. onsumpion and Risky Asses :he onsumpion apial-asse Price Model [APM] Individuals can inves in many asses which have uncerain reurns. If we exend our model o muliple asses and risk, new issues abou hh behavior and asse markes will emerge his is discussed here he ondiion for Individual Opimizaion onsider an individual reducing consumpion in period by an infiniesimal amoun and using he resuling saving o buy an asse, i, ha produces a poenially uncerain sream of payoffs, Di +, Di +,.... If he hh is opimizing, he marginal uiliy she forgo from he reduced consumpion in period mus equal he expeced sum of he discouned marginal uiliies of he fuure consumpion provided by he asse s payoffs. u' ( ) E i ( r ) u'( ) for alli [8]

34 onsumpion and Risky Asses...on d Since he expecaion of he produc of wo variables equals he produc of heir expecaions plus heir covariance, we can rewrie his expression as i i E [ r ] E [ u'( )] ov ( r, u'( ) for all i [9] u' ( ) If we assume ha uiliy is quadraic, u() = a²/, hen he marginal uiliy of consumpion is a. Using his o subsiue for he covariance erm in [9], we obain i i E [ r ] E [ u'( )] aov ( r, ) [30] u' ( )

35 onsumpion and Risky Asses...on d Eqn [30] implies ha in deciding wheher o hold more of an asse, he individual is no concerned wih how risky he asse is: he variance of he asse s reurn does no appear in Eqn [30]. Eqn [30] implies ha he aspec of riskiness ha maers o he decision of wheher o hold more of an asse is he relaion beween he asse s payoff and consumpion. Suppose he hh can buy a [risky] asse whose expeced reurn is equal o ha of risk-free asse ha she already could, hen In buying his new [risky] asses, if he payoff o he new asse is ypically high when he marginal uiliy of consumpion is high (ha is, when consumpion is low), buying one uni of he asse raises expeced uiliy by more han buying one uni of he risk-free asse.

36 onsumpion and Risky Asses...on d hus, he hh can raise her expeced uiliy by buying he new asse As he hh invess more in he new asse his consumpion comes o depend on he asses payoff and he covariance b/n consumpion and his payoff becomes less negaive In he example we are considering, since he asse s expeced reurn equals he risk-free rae, he individual will invess in he asse unil he covariance of is reurn wih consumpion reaches zero his discussion implies ha hedging risks is crucial o opimal porfolio choices A seelworker whose fuure labor income depends on he healh of he U.S. seel indusry should avoid asses whose reurns are posiively correlaed wih U.S. seel companies. Insead he worker should inves in asses whose reurns move inversely wih he healh of he U.S. seel indusry, such as foreign companies.

37 onsumpion and Risky Asses...on d One implicaion of his analysis is ha individuals should exhibi no paricular endency o hold shares of companies ha operae in he individuals own counries. In realiy hough, individuals porfolios are very heavily skewed oward domesic companies (French and Poerba, 99). his paern is known as home bias. hus, his discussion abou consumpion and equiy leads o he consumpion capial-asse pricing model, or consumpion APM which may shed ligh on his issue.

38 onsumpion and Risky Asses...on d he onsumpion APM he discussion hus far akes asses expeced reurns as given. Bu individuals demand for asses deermine hese expeced reurns. Solving Eqn [30] for expeced reurn, E i ( r ) E [ u'( )] i ( ) u'( ) aov ( r, ) [3] Equaion [3] saes ha he higher he covariance of an asse s payoff wih consumpion, he higher is expeced reurn mus be. We can simplify [3] by considering he reurn on a riskfree asse. If he payoff o an asse is cerain, hen he covariance of is payoff wih consumpion is zero. hus he risk-free rae, saisfies r

39 onsumpion and Risky Asses...on d E [ r r i ] r ( ) u'( ) E [ u'( )] aov ( r E [ u'( i, )] ) [3] Eqn [33] saes ha he expeced-reurn premium ha an asse mus offer relaive o he risk-free rae is proporional o he covariance of is reurn wih consumpion. his model of he deerminaion of expeced asse reurns is known as he consumpion capial-asse pricing model, or consumpion APM [33] he coefficien from a regression of an asse s reurn on consumpion growh [ or heir covariance] is known as is consumpion bea

40 onsumpion and Risky Asses...on d hus he cenral predicion of he consumpion APM is ha he premiums ha asses offer are proporional o heir consumpion beas (Breeden, 979; see also Meron, 973, and Rubinsein, 976) Empirical Applicaion: he Equiy-Premium Puzzle One of he mos imporan implicaions of his analysis of asses expeced reurns concerns he case where he risky asse is a broad porfolio of socks. Using Eqn [8], and assuming RRA uiliy, he Euler Eqn becomes** Eqn [34]; which could be simplified o Eqn [35] E ( ) i E ( r ) i ( r ) [34] [35]

41 onsumpion and Risky Asses...on d For convenience le g c denoe he growh rae of + consumpion from o +, (+/), and omi he ime subscrips. Wih his we will have E i c ( r )( g ) ( ) [36] o see he implicaions of Eqn [36], we ake a secondorder aylor approximaion of he lef-hand side around r = g = 0. ompuing he relevan derivaives yields: ( r)( g) r g gr ( ) g [37]

42 onsumpion and Risky Asses...on d We can rewrie [36] as, E[r i ] E[ g c ] {E[r ( ) i ] E[ g c ] ov(r c c E g Var( g ) [38] i c When he ime period involved is shor he E[ r ] E[ g ] and erms are small relaive o he ohers, omiing his erms and solving he resul for yields For risk free asse [39] simplifies o [40]; also subracing 40 form [39] yields [4] r E[ g E[r i ]- r c ] ov(r i ( )Var( g g c ) E[ r i i c i c E[r ] E[ g ] ov(r g ) - ( )Var( g ] c ) i g c ) c ) c E[ g ] [39] [40] [4]

43 onsumpion and Risky Asses...on d In a famous paper, Mehra and Presco (985) show ha i is difficul o reconcile observed reurns on socks and bonds wih Eqn [4]. Mankiw and Zeldes (99) repor a simple calculaion ha shows he essence of he problem. [see nex slide abou Mankiew and Zeldes and reead hese wo classic aricles which are in your consumpion folder]

44 onsumpion and Risky Asses...on d According o Mankiw and Zeldes (99) sudy: For he Unied Saes during he period (which is he sample ha Mehra and Presco consider), he difference beween he average reurn on he sock marke and he reurn on shor-erm governmen deb he equiy premium is abou 6 percenage poins. Over he same period, he sandard deviaion of he growh of consumpion (as measured by real purchases of nondurables and services) is 3.6 percenage poins, and he sandard deviaion of he excess reurn on he marke is 6.7 percenage poins; he correlaion beween hese wo quaniies is hese figures imply ha he covariance of consumpion growh and he excess reurn on he marke is 0.40(0.036)(0.67), or Eqn [4] herefore implies ha he coefficien of relaive risk aversion needed o accoun for he equiy premium is he soluion o 0.06 = θ(0.004), or θ = 5.. his is an exraordinary level of risk aversion; i implies, for example, ha individuals would raher accep a 7 percen reducion in consumpion wih cerainy han risk of a chance of a 0 percen reducion.

45 onsumpion and Risky Asses...on d As Mehra and Presco describe, & oher evidences suggess ha risk aversion is much lower han his. Such a high degree of aversion o variaions in consumpion makes i puzzling ha he average risk-free rae is close o zero despie he fac ha consumpion is growing over ime. Furhermore, he equiy-premium puzzle has become more severe in he period since Mehra and Presco idenified i. From 979 o 008, he average equiy premium is 7 percenage poins, which is slighly higher han in Mehra and Presco s sample period. More imporanly, consumpion growh has become more sable and less correlaed wih reurns (Romer, 0). he large equiy premium, when coupled wih he low risk free rae, is hus difficul o reconcile wih household opimizaion behavior we are working wih. his equiy-premium puzzle has simulaed a large amoun of research, and many explanaions for i wih no clear resoluion of he puzzle has been provided, however, so far.

46 F. Beyond he Permanen Income Hyphoesis/Alernaive Views of onsumpion (A). Background and he Buffer-Sock Saving he permanen-income hypohesis provides appealing explanaions for many imporan feaures of consumpion. For example, i explains why emporary ax cus appear o have much smaller effecs han permanen ones, and i accouns for many feaures of he relaionship beween curren income and consumpion, Ye here are also imporan feaures of consumpion ha appear inconsisen wih permanen-income hypohesis. Eg: boh macroeconomic & microeconomic evidence sugges consumpion ofen responds o predicable changes in income. And as we jus saw, simple models of consumer opimizaion canno accoun for he equiy premium.

47 Beyond he Permanen Income... on d Permanen-income hypohesis also fails o explain some cenral feaures of consumpion behavior: Eg According o PI hypohesis consumpion growh is deermined by he real ineres rae and he discoun rae, no by he ime paern of income [no reln b/n E(&Y growh)]. arroll and Summers (99) presen exensive evidence ha his predicion of he permanen-income hypohesis is incorrec More generally, mos households have lile wealh (see, Wolff, 998). heir consumpion approximaely racks heir income and hence heir income has large role in he deerminaion of heir consumpion ec However, hhs have a small amoun of saving ha hey use in he even of sharp falls in income/emergency needs: Deaon (99) calls his buffer-sock savingpo

48 Beyond he Permanen Income... on d In he erminology of Deaon (99), mos households exhibi buffer-sock saving behavior. As a resul, a small fracion of households hold he vas majoriy of wealh hese failings of he permanen-income hypohesis have moivaed a large amoun of work on exensions or alernaives o he heory. hree ideas ha have received paricular aenion are. (A) Precauionary Saving (B) Liquidiy onsrain () Deparures from omplee Opimizaion We will briefly ouches on some of he issues raised by hese ideas o finalize our lecure.[hopefully you will do an in-deph reading and ask are hey relevan o Africa?]

49 Beyond he Permanen Income...on d Precauionary Saving Our derivaion of he random-walk resul in Secion was based on he assumpion ha uiliy is quadraic. Quadraic uiliy implies: marginal uiliy reaches zero a some finie level of consumpion and hen becomes negaive. he uiliy cos of a given variance of consumpion is independen of he level of consumpion his means ha, since he marginal uiliy of consumpion is declining, individuals have increasing absolue risk aversion: he amoun of consumpion hey are willing o give up [saving!] o avoid a given amoun of uncerainy abou he level of consumpion rises as hey become wealhier.

50 Beyond he Permanen Income...on d onsidering he Euler Eqn given in Eqn[], if uiliy is quadraic, MU is linear and so E[u (+)=u E[+] In his case he Euler Eqn reduces o =E[+] Bu if u ( ) is posiive, hen u () is a convex funcion of. In his case, E [u (+)] exceeds u (E [+]). Bu his means ha if and E [+] are equal, E [u (+)] is greaer han u (), and so a marginal reducion in increases expeced uiliy. [see figure in he nex slide] hus he combinaion of a posiive hird derivaive of he uiliy funcion and uncerainy abou fuure income reduces curren consumpion, and hus raises saving. his saving is known as precauionary saving (Leland, 968).

51 Beyond he Permanen Income...on d Since u () is negaive, u () is decreasing in. And since u () is posiive, u () declines less rapidly as rises. If consumpion akes on only wo possible values, L and H, each wih probabiliy /, he expeced marginal uiliy of consumpion is he average of marginal uiliy a hese wo values. In erms of he diagram, his is shown by he midpoin of he line connecing u (L) and u (H).

52 Beyond he Permanen Income...on d Liquidiy onsrain he PI hypohesis assumes ha individuals can borrow a he same ineres rae a which hey can save as long as hey evenually repay heir loans. Ye he ineres rae ha households pay on borrowing are ofen much higher han he saving rae Some are even unable o borrow more a any ineres rae. Liquidiy consrains can raise saving in wo ways. Firs, whenever a liquidiy consrain is binding, i causes he individual o consume less han he/she oherwise would. Second, even if he consrains are no currenly binding, he fac ha hey may bind in he fuure reduces curren consumpion [Zeldes (989)]

53 Beyond he Permanen Income...on d Example:. Suppose, here is some chance of low income in he nex period. If here are no liquidiy consrains, and if income in fac urns ou o be low, he individual can borrow o avoid a sharp fall in consumpion. If here are liquidiy consrains, however, he fall in income causes a large fall in consumpion unless he individual has savings. hus he presence of liquidiy consrains causes individuals o save as insurance agains he effecs of fuure falls in income hese poins can be seen in a hree-period model.

54 Beyond he Permanen Income...on d o disinguish he effecs of liquidiy consrains from precauionary saving, assume ha he insananeous uiliy funcion is quadraic. In addiion, coninue o assume ha he real ineres rae and he discoun rae are zero Begin by considering he individual s behavior in period. Le A denoe asses a he end of period. Since he individual lives for only hree periods, 3 equals A + Y3, which in urn equals A + Y + Y3. he individual s expeced uiliy over he las wo periods of life as a funcion of his or her choice of is herefore:

55 Beyond he Permanen Income...on d ** he derivaive of his expression wr o is: his expression is posiive for < (A + Y + E[Y3])/, and negaive oherwise. [44] ) ] [ ( ]) [( ( U [43] ] ) ( ) [( ) - ( U Y E Y A a Y Y A ae a Y Y A a Y Y A E a

56 Beyond he Permanen Income...on d hus, as we know from our earlier analysis, if he liquidiy consrain does no bind, he individual chooses = (A + Y + E[Y3])/. Bu if i does bind, he or she ses consumpion o he maximum aainable level, which is A + Y. hus, A Y E[ Y3 ] min, A Y [45] hus he liquidiy consrain reduces curren consumpion if i is binding

57 Beyond he Permanen Income...on d onsider he firs period. If he liquidiy consrain is no binding, he individual has he opion of marginally raising and paying for his by reducing. hus if he individual s asses are no zero, he usual Euler equaion holds. Wih he specific assumpions we are making, his means ha equals he expecaion of. Bu he Euler equaion holds does no mean ha he liquidiy consrains do no affec consumpion: Eqn [45] implies ha if he probabiliy ha he liquidiy consrain will bind in he nd period is sricly posiive, he expecaion of as of period is sricly less han he expecaion of (A +Y + E[Y3])/. A is given by A0 +Y, and he law of ieraed projecions implies ha E[E[Y3]] equals E[Y3]. hus,

58 Beyond he Permanen Income...on d A0 Y E[ Y ] E[ Y3 ] [46] c Adding o boh sidesof Eqn [46]and hen dividing by 3/ yields A0 Y E[ Y ] E[ Y3 ] [47] 3 hus even when he liquidiy consrain does no bind currenly, he possibiliy ha i will bind in he fuure reduces consumpion. Finally, if he value of ha saisfies = E[] (given ha is deermined by [45]) is greaer han he individual s period- resources, A0+Y, he s period liquidiy consrain is binding; in his case he individual consumes A0 + Y(no as envisaged by he Euler/opimizaion condiion) hus liquidiy consrains alone, like precauionary saving alone, raise saving [ie less consumpion]

59 Beyond he Permanen Income...on d Deparure form omplee Opimizaion he assumpion of cosless opimizaion is a powerful modeling device, and i provides a good firs approximaion of agen s responses o many changes. However, i does no provide a perfec descripion of how people behave. here are well-documened cases of quaniaively significan, consisen and sysemaic deparure from he predicions of sandard models of uiliy maximizaion we jus learned his may be he case wih choices beween consumpion and saving. Because, iner alia: (a)he calculaions involved are complex, (b) he ime periods are long, and (c) here is a grea deal of uncerainy ha is difficul o quanify.

60 Beyond he Permanen Income...on d So insead of complee opimizing, individuals may follow rules of humb in choosing heir consumpion Such rules of humb may be he raional response o problems noed earlier Relying on such rules may lead households o use saving and borrowing o smooh shor-run income flucuaions; hus hey will ypically have some savings, and consumpion will follow he predicions of he PI hypohesis reasonably. hus, consumpion will rack income fairly closely over long horizons; hus savings will ypically be small. One specific deparure ha has received considerable aenion is ime-inconsisen preferences (eg Laibson, 997). here is considerable evidence ha individuals (and animals as well) are impaien a shor horizons bu paien a long horizons hus ime inconsisency

61 Beyond he Permanen Income...on d ime inconsisency alone, like he oher deparures from he baseline model alone, canno accoun for he puzzling feaures of consumpion, however. By iself, ime inconsisency makes consumers ac as hough hey are impaien: a each poin in ime, individuals value curren consumpion grealy relaive o fuure consumpion, and so heir consumpion is high (Barro, 999). And ime inconsisency alone provides no reason for consumpion o approximaely rack income for a large number of households, so ha heir savings are close o zero. Oher facors liquidiy consrains, he abiliy o save in illiquid forms (may be as a resrain), and perhaps a precauionary saving moivaion seem o be needed for models wih ime inconsisency o fi he facs ( see Angeleos, Laibson, Repeo, obacman, and Weinberg, 00).

62 Summary & onclusion wo hemes emerge from his discussion. Firs, no single facor can accoun for he main deparures from he permanen-income hypohesis. Second, here is considerable agreemen on he broad facors ha mus be presen: a high degree of impaience (from eiher a high discoun rae or ime inconsisency wih a perpeually high weigh on curren consumpion); some force prevening consumpion from running far ahead of income (eiher liquidiy consrains or rules of humb ha sress he imporance of avoiding deb); and a precauionary-saving moive. I am sure i will also be a lo more complicaed in LDs/Africa so a lo of work/research for your guys/!!