Tilburg University

Risk aversion, intertemporal substitution and consumption

van der Ploeg, F.

Publication date:1989

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Citation for published version (APA):van der Ploeg, F. (1989). Risk aversion, intertemporal substitution and consumption: The CARA-LQ problem.(CentER Discussion Paper; Vol. 1989-53). CentER.

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~a; ~~ DiscussionR for

~ mic Research a erzs3 i~~ iiiiiiiiiquuii u i iuiquniiuiiiiii iiii pi i

No. 8953RISK AVERSION, INTERTENIPORAL SUBSTITUTION

AND CONSUMPTION: THE CARA-LQ PROBLEM

by Frederick van der Ploeg ~~~~, r:

3~0.,~~:~a

November, 1989

RISIC AVERSION, INTERTEMPORAL

SUBSTITUTION AND CONSUMPTION:

THE CAii.A-LQ PROI3LEM

l~ rcderick vau der Ploeg

September, 1989Itevised October, 1989

CentlilC for Lconomic Itesearch, '1'ilburg University,

P.O. Box 90153, 5000 LE Tilburg, the Netherlands

AUstract

This paper employs the recursive utility approach, based on quad-

ratic Celicity functions and constant absolute risk aversion, to distin-

guish between risk avcrsion and intcrl.cmporal substitution. Stochastic

dynamic programming yields closeJ-loop linear decision rulca for the

~AR.A-LQ problem. Certainty equivalence no longer holds, but in-

stead the decision maker plays a min-max strategy against nature.

When applied to a li(e-cycle consumption problem, one finds a ratio-

nale for precautionary saving and a larger senaitivity of changes in

conaumption to income innovations.

The autitor thanks Itob Alessie, '1'on Bartmi, John llriffill and Bertrand

Melenberg fon c~lpful discussions.

Contents1 Introduction

2 Risk aversion and intertemporal preferences

3 The CARA-LQ problem

4 Precautionary saving aud consumption

b Conclusiona

1

2

b

10

íi

References í7

Appendix Z~

1

1 Introduction

'1'wo houaeholds with identical preferences over present and future consump-tion will under ccrtainty savc tbc same, but this doca not necessarily irnply

that these two households will save the same in uncertain environments.

1'his distinction Uetween intertemporal substitution and riak aversion has

recently received a great deal of attention (e.g., Kreps and Porteus, 1978,

1979; Epstein and Zin, 1989; Weil, 1989a). Consumers are, in contrast to the

timeless von Ncuman-Morgenstern utility theory, not indi[ferent about the

timing of the resolution o( uncertainty over temporal consumption lotteries,

hence the axiom of reduction of compound lotteties must be abandoned in

order not lo impose such an indifference. When consumers dislike riek more

(less) than intcrtemporal (luctuations, they prefer early (lale) resolution of

uncertainty. 'Che emphasis is on recursive preferences, because these lead

to tirne-consistad decisions. Most of the altention has bcen focussed on

stochastic interest rates, asset pricea and CAPM models with generalised

iso-elastic preferences (e.g., Epstein and Zin, 1989; Giovannini and Weil,

1989; Attanasio and Weber, 1989; Weil, 1989b), but relatively little atten-

tion has been paid to the implications of risk aversion and stochastic income

for the life-cycle hypothesis. The general result is that, if the third derivative

of the felicity function is positive, an increase in uncertainty about [uture

income increases saving (e.g., Leland, 19G8; Sandmo, 1970). No closed-form

analytical solutions are available, but numerical solutions [or the case of

a felicity function with constant relative aversion ahow that precautionary

saving is an important factor, that there is excess sensitivity of couswnption

to lransitory income aud lhal uncerlainty about uninsured medical expenses

Ieads Lo underspending of Lhc cldery ('l.cldes, 1989). Ilowever, previous work

on the effects of uncertain income on saving have all introduced risk aversion

in the felicity function and thus do not distinguish between intertemporal

subatitution on the aie hand and risk aversion on the other hand.

The objective of this paper is to consider risk aversion and intertemporal

subatitution in the life-cycle consumption problem when labour income is

stochastic and not fully diversifiable. Seclion 2 sets up the probletn of con-

sumption and savings. Section 3 solves the general problem with constant

absolute risk aversion and quadratic felicity functions. Section 4 applies it

to the life-cycle consumption problem and discussea "saving for retirement",

"saving for a rainy day" ancl "making I~ay while the sun shines". Section 5

concludes the paper. 'I'he Appendix considers tl~e continuous-time problem.

2

2 Risk aversion and intertemporal preferences

The life-cycle consumption problem can 6e formulated as:

T-tmax E[P~(Ce,...,CT-i) ~ 1a), Pi -~(1 f B)-I'-~)F(C.),

C,....,CT-, s-[

F'10,F"'c0 (2.1)

subject to the budget constraint

A,ti -(1 f r,)(A, f Y, - C,), s- t,...,T - 1, (2.2)

where B denotes the rate of time preference, A, denotes non-human wealth at

the beginning of pcriod t,C„Y, and r, denote consumption, labour income

and the intcrest rate o[ period s, and li denotes the information set at time

t. It yields the familiar Euler equation for the "tilt" of the consumption

profile ( e.g., Ilall, 1978; Illanchard and Fischer, 1989, Chapter 6):

F"(~'~) - ~'~( ~f e )~"(~'~t~) I ~~1, (z.a)

or the marginal rate of substitution betwcen consumption in two periods

must equal the corresponding marginal rate of trans(ormation. When r~ - B,

consumption is a martingale. The problem with this approach is that the

felicity function, F(.), does double duty (e.g., Selden, 1978); for example,

on the one hand it defines the instantaneous elasticity of intertemporal sub-

stitution, ry(Ci) --F'(Ci)~F"'(Ci)Ci, and on the other hand it defines the

elaaticity of marginal utility or, in the presence of uncertainty, the cceffi-

cient of relative risk aversion, l~ry(Ci). 1[ence, the standard approach dces

not distinguish between intcrtemporal substitution and risk aversion. For

example, when ti~e felicity function satisfies constant absolute risk aversion,

F(C~) --exp(-pCi), r~ - B - 0, and income obeys a random walk,

y, - Y,-t f E~,E~ ~ IN(o,o~), then it follows that consumption satisfies

Ce - Ci-~ } ~poz } E~-i or Ci - (T - t)-~Ai f Yi - ~a(T - t - 1)0~ (see

3

Caballero, 1987; Kimball and Mankiw, 1989; Blancl~ard and Fischer, 1989,Chapter 6). 'fhis example shows that risk aversion in the face of stochas-tic shocks induces prudence and leads to precautionary saving. More un-

certainty leads to a lower level of consumption, given wealth and income.1}owever, this example is very specific and does not distinguish between two

distinct aspects of preferences, viz., risk aversion and intertemporal substi-

tution. In empirical work one finds very low elasticities of intertempocal

substitution (e.g, Ilall, 1988). This dces not contradict prior beliefs aboutconsumer behaviour, but the implied very high coefficiente of relative riskaversion do seem unrealistic. Hence, it ie important to cut the link betweenrisk averaion and intertemporal substitution.

An alternative approach is based on:

max E[U(Pi(Cr,...,CT-t)) ~ Ir],c,. -.~T-r

U'~O,U"~O,U"'10 (2.4)

subject to (2.2), where the preference function P,(.) describes the attitudetowards intertemporal substitution and the utility function U(.), together

with the preference function Pr(.), describes the attitude towards risk. t

A linear (concave) utility (unction corresponds to risk neutrality (aversion).

Even if the preference function is time separable, the function U(Pt(.)) will

in general not be time separable and the resulting consumption decisions will

be time inconsistent. }Iowever, if there is constant absolute risk aversion,

(J(Pr) - -exp(-pPt), one obtains tl~e dynamic programming relationship:

V,(A,) - max-U(F(C,))E[V,tr(A,tt)~ió ~!,], s - T- 1,...,t (2.5)

subject to (2.2), where the value function is defined as the expected utility

to go, evaluated along tlte optimal path,

T-t

V,(A,) --E[exp{-A[~(1 f B)-l' -')F(C;)]} ~!,], s- t,...,T- 1, (2.6)

,~-s

~For the caae that F(.) ia quadratic, the utility fundion alone is sufficient to deacribe

the attitude towards risk.

4

and VT(AT) --1. This corresponds to a special case o( recursive utility

functions (Epstein and Zin, 1989) and therefore alwaya yields time-consistent

consumption plans. W hen the axiom of reduction of compound lotteries used

in von Neumann-Morgenstern utility theory is abandoned to allow for non-

ditference about tl~e timing of the resolution of uncertainty and an axiom is

added to ensure the time consistency o( optimal plane, preferencea can be

represented recursively by

V~(A~) - maxW,(CnE[Vatt(A~tt) ~ f~))~ s -T - 1,...,t (2.7)

where W~(.) denotes the aggregator function ( Kteps and Porteus, 1978).

Epstein and Zin (1989) analyse various classes of aggregatorfunctiona and

discuss under what circumstances individuals prefer early or late resolu-

tion of uncertainty. Standard von Neumann-Morgenstern thmry emerges

when the aggregator function is linear in its second argument. Although

the preferences based on (2.4)-(2.6) distinguish between risk aversion and

intertemporal substitution, tl~ey assume indifference about the timing of the

resolution of uncertainty.In general, it is very difficult to obtain closed-form solutions for the prob-

]em (e.g., Farmer, 1989). Ilowever, when the felicity function is quadratic,

interest rates are known and income follows a linear model with normally

distributed error terms, linear risk-sensitive rules for consumption can be

found. The assumption of a quadratic felicity function is often used, but it

is nevertheless a bit awkward, for it implies a flnite marginal contribution of

consu~nption to felicity as consumption tends to zero and therefore dcea not

necessarily rule out negative or zero consumption. As an attitude towards

risk it is, however, much more problematic, because it implies increasing

absolute risk aversion as consumption increases (-F"~F' -(o - C,)-~).

]Ience, it is probably not too bad an approximation to have a quadratic

felicity function and an utility function with constant absolute risk aversion.

Section 3 derives some general results, which are then applied to the analysis

of consumption and saving in Section 4.

5

3 The CARA-LQ problem

The state at the beginning of period t is summarised by the vector xt.It summarises all information that occurred before period t as well. The

dynamic evolution o[ the state follows from the linear state equations:

x,tr - Ax, t Bu, } 6, f E„ e, ti IN(o, V), s- t,...,T- 1 (3.1)

where u, denotes the vector of policy instruments at time s and e, denotes

the vector of normally distributed and serially uncorrelated state distur-

bances at time s(dim(e,) - dim(x,) - n). For example, in the life-cycle

consumption x, includes non-human assets (A,) and u, includes consump-

tion (C,). Intertemporal preferences are described by the preference func-

tion:T-1

1i(xtiTlh...,TLT-t.Eti...ET-1) - ~(1 t B)-(~-t)F(x.,u,)

.-t

f (1 -F B)-IT-t)FT(xT) (3.2)

where T,B,F(.) and FT(.) denote the horizon, the pure rate of time pref-

erence, the felicity function and the final asset value function, respectively.

Clcarly, it has been assumed that preferences are time separable and that

the felicity function describes intertemporal preferences. The attitude to-

wards risk is described by a separate utility function U(Pt), U' ~ 0, U" G 0

for risk aversion, U" 1 0 for risk loving, and U"' ~ 0. The optimal policy

instruments (or period t and tbe planned policy instruments planned during

pcriod t for period s, say u, - u,~t, s~ t, follow frotn:

max E[U(Pt(xt~uh...,uT-1,Eh...~ET-1)) I Il~ (3.3)

where tlie relevant information set includes xt. In the presence of shocks

planned policy instruments may be revised as time proceeds. A more natural

approach is based on stochastic dynamic programming. This yields the

following recurrence relationship:

6

V.(x,) - max exp(-AF(x„u,)~E(V,tt(x,tt)'~ ~ I,~, a-T - 1,...,t,u,

VT(xT) - - exp(-AFT(xT)) (3.4)

sttbject to (3.1), where V,(x,) denotes the value function for period a and

gives the maximum value o[ expected utility from period s onwards (as given

by (3.3)). Two crucial assumptions were required to obtain this relationship:

(i) time separability o[ the preference function; and (ii) a constant Arrow-

Pratt coe(ïtcient of absolute risk aversion. O[ course, the function U(P(.))

is not time separable. Nevertheless, equation (3.4) corresponds, in contrast

to the approach of Selden (1978; 1979) ~, to a special case of recursive

utility discussed in Kreps and Porteus (1978, 1979) and in Epstein and

Zin (1989) and therefore optimal policy actiona derived from (3.4) are time

consistent. Since -~ log(E(-U(P)j) ~ E(P)- z(3var(P), constant absolute

tisk aversion corresponds locally to the mean-variance approach.

In general it is very difficult to obtain a closed-form solution to the above

problem. However, if a quadratic preference function,

F(x„u,) - q'x, - 2x;Qx, } r'u, - 2u;Ru„ a- t,...,T - 1,

1 3.2'FT(xT) - xTxT - 2xTZT2T, ( )

where Q is a symmetric semi-positive definite matrix and R and ZT are

symmetric positive definite matrices, is used, a closed-loop solution with

linear policy feedback rules can be found (Whittle, 1982).

Theorem: For the problem (3.4) subject to (3.1), (3.2) and (3.2') the value

functions are given by

~Ordinal cerlainly-equivslent preferences have been developed for the two-period prob-

lem. Consumen firot ux s utility function, which describa lheir sltitude towards risk,

to convert future tonsumplion inlo its certainly-equivalent level and then use tkis in a

preference function over current and [uture consumplioa. The eaension lo multi-period

problems sulfers (rom time inconsistency (Johnsen snd I)analdson, 1985).

V.(x,) -- exp[-2R(x. - i,)~Z.(x. - x.)J, a - t,...,T - 1, (3.5)

and the optimal policy rules are linear and given by

u, - K,x, .} k„ a- t,...,T - 1, (3.6)

where the matrices of feedback coefficients are given by

K, - -(R } B'Z.t1B)-iB~Z.t1A

--R-~B'[(1 } t7)Z;}~ - pV f BR'1B~'~A, a- t,...,T- 1, (3.7)

the vectors of policy constants are given by

k, --(R f B'Z,tiB)~'[B~Z,ti(6, - i,ti) - r], a- t,...,T - 1, (3.8)

the positive definite and symmettic matrices Z, obey the Riccatti recursions

z,-tZfA'[(lte)z.~~-QvfBR-'B'J''A,s-T-1,...,t, (3.9)

the vectors x, obey the recursions

Z.y. - 9 f A~[(1 f B)Z;}i - PV f BR-1B'j-~

[z,ti - b, - BR-lr], a- T- 1,...,t, (3.10)

and the matrices Z, -[(1 fB)Z; 1-QV]-~, s- t~ 1,...,T, must be positive

definite for a well defined solution.Proof: Assume that the functional form of the value function (3.5) ie cor-

rect. When 2T e ZT~zT, this is up to a constant true tor a - T. Substitu-

tion of ( 3.1), (3.2') and (3.5) into ( 3.4) and taking expectations yields

8

V.(x.) - max exp[-AF(x.,u,)~~ f V.ti(G(xnu.,E,))'~

1 , -texp(-2E,V E,)dE,

- maX -~ exp[-J3S(u„ E,; x„ s) } constantjdE„L,

s-T-1,...,t, (3.11)

where S2 -((2x)" [ V ~)-? and S(.) is a positive definite quadratic form inu, and E,:

S(u„E,;x„s) - q'x, - 2x;Qx, t r'u, - 2u;Ru,

- 2(1tB)-~(Ax,tBu,tb,fE,-x,tl)'

Z,tl(Ax,fBu,tb,~E,-i,~t)

f :2E~(AV)-~Ee~ s - T - 1,...,t. (3.12)

A standard result on the maximisation and integration ofGaussian densitiessays that (3.11) is equal to

V,(x,) -- exp[-FiS(u„ E~; x„ s)), s- T - 1, ..., t, (3.13)

where u; and E; are the values of u, for which S(u„E,;x„s) attaine ite

maximum and the values ot E, for which S(u„ E,; x„ s) attains its minimum

(maximum) for p 1 0 (p ~ 0). This yields, upon use of the llouseholder

matrix identity, the optimal policy rules for period s, viz. (3.6)-(3.8), pro-

vided that the matrices Z,t~ are non-singular. Upon substitution of E; and

u; into (3.12) and (3.13), one obtains the recursions (3.9)-(3.10). O

It is clear from the prcw[ of the above theorem that the optimal policy

iuslruments for pcrivtl t, ui, and thc instruments planned during pcriod t for

9

period a, u,~~, a- t~ 1, ..., T- 1, can alternatively be found as the outcomeof a determiniatic, difference game againat nature (cf., Whittle, 1982):

T-l 1

max extE~...Er-~ ~(1tB)-1'-`1 [F(X„u,)f E,(AV)-tE.]...~U7~-1 J-( 2

}(1 f B)~T-'1 FT(xT) (3.14)

subject to (3.1) and (3.2'), where 'ext' stands for 'min' when p 1 0 and for'max' when p c 0. Thia sequential open-loop procedure leads to the sameoutcomes aa the closed-loop procedure of the above theorem. A risk-averae((i ~ 0) decision maker is pessimistic and treats nature as a non-cooperative,

belligerent player. A max-min strategy is therefore uaed to hedge against

shocks that may work to his disadvantage. Alternatively, Z, is replaced

by Z, in order to increase the effective shadow penalty on uncertain atatevariables as the decision maker is worried that ahocka will frustrate the

achievement o[ targets. On the other hand, a risk-loving deciaion maker isoptimiatic and asaumes that shocke work out to hie advantage. A cooperativemax-max atrategy is therefore used.

When risk aversion increases, the intensity of the feedback rule (mea-sured by the norm of K,) increases, the norm of the closed-loop transitionmatrix A f BK, reduces and there is a possibility that Z, may become

negative definite and there is an infinite expected losa of utility. ltence, such

a neurotic policy can lead to a"nervous break-down". On tlie other hand,

when risk loving tenda to infinity, tlie decision maker becomes "complacent"and considers discretionary policy inappropriate (the norm of K, tends to

zero aa (i -. -oo).The optimal decision rules presented in the above Theorem incorparate

the risk-neutral decision rulea developed in the literature on optimal eco-

nomic planning (e.g, Chow, 1975; Kendrick, 1981; Preston and Pagan, 1982)

as special cases and thus relax multi-period certainty equivalence. Section

4 diacussea an application to the life-cycle consumption problem and vander Plceg (1984) develops the theory in continuous time and presents ap-

plications to the supply of a monopolist and the depletion of exhaustibleresources. Extensiona to allow for imperfect observations of the state vector

and Kalman filters can be found in IIenaouason and van Schuppen (1983)

and Whittle (1982). Just as the linear decision rulea depend on the covari-

ance matrices (V), the Kalman filter now depends on the penalty matrices

10

of the preference function (Q). A risk-averse decision maker increases theeffective variance of the incoming observations, since he ie afraid to incor-porate possibly faulty information that may lead to large lossea in utility.1'his lcads to the use of biassed estimates of the state vector.

4 Precautionary saving and consumption

The method discussed in the previous section can be applied to the life-cycle consumption problem discussed in Section 2. A constantinterest rate,r, - r, s- t, ..., T- I„ a quadratic felicity function with a denoting thebliss level oí consumption, a constant ccefficient of absolute risk aversion,p, and an AR(1) process for shocks to income are assumed. More generalstochastic processes for income are considered at the end of this section.The information set in period t includes At and Yr-i so that current income

is assumed to be unknown when current consumption is decided upon 3.}[ence, the problem for the consumer in period t is:

r (T -~tnax E I-exp{-(i I~(1 ~B)-1'-`)(aC, - 2C;) } ~ An}'i-r (4.1)

C~,...,CT-~ ` l t

subject to the consolidated budget constraint,

T-1 T-t~(1 f r)-1'-t1C~ - At f~(1 f r)-i'-`lYa, (4.2)s-t

and the stochastic process for income,

,-t

- PÍl:-t - 1;-t) f c„ c, ~ IN(O,o~), s- t,...,T - 1 (4.3)

where Y„ s? t, are constants. Transitory shocks to income correspondto p- 0 and permanent shocks to p- 1. It is assumed that p C 1 f r is

~If current income is seaumed to be known when conaumption ie decided upon, lhe

qualitative reeults sre unsRected

11

satisfied, so that human wealth is finite as T y oo. From ( 3.14) and (4.3)it is clear that current consumption, Ct, and consumption planned duringperiod t for period s, C,It, s- t} 1, ...,T- 1, can be found from:

T-~ j 1 1 1max extt~....,eT-t ~ I(1 } B)-('-e)(aC, - 2C.) } 2(E;Iila~) J (4.4)

C,,...,CT-t , `

subject to ihe exp~cted consolidated budget cnnstraint,T-1

~(1 } r)-l'-t)C, - At } Ilt},-t

T-t ~~(1 } r)-l'-`)~PI'-`-')(Yc-i - Yt-t) } ~ p'-'~E,~],-t ,~-t

where the determinístic component of human wealth is defined as

T-t

flt - ~(1 } r)-l'-t)Y,.,-e

This yields

(4.6)

(1 } bl'-tC,~t-E;-l]}rJ

~tGtr, s-1,...,T-1 (4.7)

and

jT-1

E,It - -QO~ IIL~(I -F r)-1'~-e1P'~-' ~ts~-,

z r(1 } r)-Ia-i-O -( 1 i. r)-lT't-t)Pr-'--Qo I` at, s ~ t(4.s)

1}r-p

wherc Ct - Ct~t and -((iatVt) ~ 0 denotes thc marginal utility of non-

human wealth at the beginning of period t. Equation (4.7) shows the familiaz

12

"tilt" of the planned consumption stream; if the interest rate exceeds (is be-

low) the rate o( time pre[etence, planned consumption increases (decreases)

over time and the consumer initially saves (dissaves). Also, consumptiondecreases as the marginal utility of non-human wea)th increases. Equation

(4.8) shows that a risk-averse (risk-loving) consumer is pessimistic (opti-mistic) in the sense that he takes into account that shocks to income may

turn out to be negative (positive). When there are no stochastic shocks orwhen the consumer is risk neutral (po~ - 0), the perceived shocks to future

income are zero (e,~i - 0, e~ t). In general (po~ ~ 0), however, thepresent approach departs ftom certainty equivalence.

Upon substitution of (4.7) and (4.8) into the expected consolidated bud-

get constraint and letting T y oo, one obtains a relatively simple expression

for ~i:

o - ~tt.~ (A~ t H~) - ~tt~) (Y~-~ - Ye-t)1a~- ~ ~ ,~t. ~-á J.2t.) e

where

(4.9)

z(1 t r)~ [r(z f r) - P(1 ~ r- P)]a lo

~-~o ~ (ltr-P)3(2fr) ( ~ )

A higher degree of absolute risk aversion and a higher variance of shocks

raise p and thus increase the marginal utility of wealth and reduce planned

consumption. Henee, risk aversion leads to precautionary saving and the

building up of financial assets for fear of a rainy day. Since it can be shown

that p is larger for p- 1 than for p - 0, persistence in income shocks

reinforces this rationale for precautionary saving. The sign of 8(3~ap corre-

sponds to the sign of [2(1 f r)~ t p~ - 3], so that as p increases from 0 to

1(3 first talls ( when r is not too high) and then increases. Hence, for inter-

mediate degrees of autocorrelation in income the opposite of precautionary

saving occurs. Textbook theory assumes risk neuttality and leads to the

certainty-equivalent level of consumption ( associated with ~3 - 0), which is

unaffected by the variance of income shocks ( when the felicity function is

quadratic).In order to gain a better understanding of our bivariate model of con-

sumption and income, it is from now on assumed that the interest rate equals

13

the rate of time prPference (r - B). This leads to a flat planned consumption

prulilr (cf., I~lavin (I~JRI) for ~) - 0):

~ s~~~, f F~~(l ~ B)-la-nF(I~. I f~ ~)1 - ri,~Íi~o J

~ ~..~~ - - - - -.- .. - 1 - ~1

- r(B)(Ai t H,) t~1~) (Yi-i - 1'a-i) - Qa s~ t. (4.11)IL ife1-p

It follows from Ci - Ci~i that the change in actual consumption satisfies

B ~(1 f B)-' [E(Yit~ ~~i) - E(1'et~ ~ fi-~)]1Ca-C~-i - (1-p)~-o

pB a-Ce-i), (4.12)f ~1-p (

so that marginal felicity obeys

F'(Ce) - a- Ci - I 1-( Ba ~ JF'(C~-t)l 1-á

( e(1 t e) (a.la)-l(1-P)(lte-v) ~`~

As nsual "news" docs nnt matLM, so that the change in ronsumption docs

not depend on the par,t history or on previously anticipated changesin in-

come ( Hall, 1978). The marginal propensity to consume out of income

shocks equals [B(1 t B)~(1 -(3)(1 t B - p)], so that the marginal propensity

to consume out of unanticipated transitory shocks is B~(1 - p ) and out of

permanent shocks is (1 t B)~(1 - p). Hence, consumption teacts as usual

much stronger to permanent than to transitory innovations in labour in-

come. It should be noted that risk aversion raises the sensitivity ot changes

in consumption to unanticipated changes in income. This excess sensitivity

could be reterred to as "making hay while the sun shinesn. This excess sen-

sitivity does uot occur with risk-neutral consumers and a felicity function

14

witL constant absolute risk aversion ( Kimball and Mankiw, 1989), but itdoes occur with risk-neutral consumers and felicity functions with constant

relative risk aversion ( Barsky et al., 1986). Empirical research indicates that

the observed sensitivity of consumption to curtent income is greater than

~s warranted by the permanent-income, life-cycle hypothesis, even when the

roie ot current income in signal.mg changes in permanent income is taken

into account ( Flavin,1981).Risk aversion also introduces a degree of persistence in the stochastic

process for consumption. This can be seen from solving (4.12).

(',s - (T } ~t - ~ 1 B~~ 1 J '-t (~ ( - ry)

l ~ 1- 1- p c,,, s~ t, (4.14)}( 9(1 f B)1 s l ` Ba ~1'-~.(1 - p)(1 -F B - V) ,~-atr

where Ci is less than the certainty-equivalent or risk-neutral level of con-

sumption, C~E. Figure 1 illustrates what happens. Given a known variable

future income stream, the risk-neutral level of consumption is completely

smoothed in a way that the consolidated budget constraint is satisfied.

F'igure 1: kisk aversion and preeautionary saving ~~~~'

However, when thcrc is risk aversion, consumption is initially below its

cerlainty-equivalent level as the consumer is "saving [or a rainy day". In the

15

absence of shocks, consumption subsequently rises to its bliss level becauseeach period the consumer is pleasantly surprised to find out that bad shocksdid not materialise after all and that consequently his total (human andnon-human) wealth has risen. In this sense risk aversion has a similar effectas depressing the rate of time preference or increasing the return on assets,because this would alsu leaa to an upward-sloping consumption profile. Itis useful to derive an expression for saving, S~:

( 8 l lS~ - `1fB)Ai-FYi-ci- Ipp [o-(1t9IA~-YeJ

1 r ~t ( l IEi -~(1 f B)-~DE(Yatc ~ fe-t) , s~ t. (4.15)

`1 - AI l ;-t

With risk neutrality (~i - 0), saving corresponds to the present discountedvalue of expected future declines in labour income. This is what Campbell(198 ï) calls "saving for a rainy day". Since [uture rain fall is uncertain, this

seems a misnomer given that Campbell's expression applies to a determinis-tic or certainty-equivalent environment. It may be better to refer to "saving

for retirement", because the only thing one can be sure of in life is that onebecomes older and has to retire and suffer a loss in income. It is easy to

show that saving under risk aversion exceeds the certainty-equivalent levelot saving and it seems more appropriate to reserve the expression "saving

for a rainy day" tor this type of prudent behaviour. If future labout incomeis fully predictable or the consumer is risk neutral, "saving for a rainy day"

would not and "saving for retirement" would occur. Atkinson and Stiglitz(1980, p.73) distinguisli between the life-c~cle motive for saving, giving tiseto a transfer ot purchasing power from one period to another period when

the time profiles oí income and desired consumption do not coincide, andthe precautionary motive, relevant when individuals want to save in order

to provide insurance against future periods in which their incomes are low or

their needs (e.g., uninsured medical expenses) are high. The former motivecorresponds to "saving for retirment" (or for the finance of education or ofthe purcha-ce of home) and the Iatter to "saving for a rainy day".

Finally, it is worthwhile to consider more stochastic processes for shocksto income such as the AR(p) modeL

16

d'(L)(Ye - Y~) - 4, ti~(L) - 1 }~ J~;L', ci ~ IN(O,o~). (4.16);-i

F.quation (4.3) corresponds to p- 1 and ~~ -- p. It follows that (4.8)becomes c,~i --Ao2ry,-iai, s~ t, where

~ ( (1 } r)L l - ~.

r(L ) - `I(1 } r)L - 1)~(L)I - -L 7; (4.17)

Upuu tinhstitutiun tugether with (4.7) into the exprcted cunsulidated budK~"tconstraint, one can solve for ai and Ci. In particular, when changes inincome shocks follow an AR(1) process, as is popular in the literature (e.g.,Campbell and Deaton, 1989), one has p- 2, ~r --(1 } r,i), ~s - rG, ry; -ttr ~-~~}~-~, r,: ~ 0, so that the first equation of (4.11), ( 4.12), and (4.15) hold

w~th

1}rQ-ilo~~(1}r-~)(1-~)

f~ 1}r ~'s

(2}r)r~~-~~(1}r)~-d~~~(1}r)rl'-1~~(4.18)

Note that with ~i - 0 and p- 1, expressions ( 4.10) and ( 4.15) coincide.

1'able 1 shows that the extent o[ "saving for a raiuy day" depends crucially

on the characteristics of the stochastic process for income. The first point

to note is that, when income shocks follow an AR(1) process with p-

0.5, p G 0 and thus consumption is initially above its risk-neutral level

and no precautionary saving occurs. In fact, the opposite of precautionary

saving occurs. Since the term in the small square brackets of the right-hand

side of (4.10) attains a minimum value of ~(r - 0.1547)(r {. 2.1547) when

p- Z(1 } r), Q is negative unless r~ 0.1547. The second point to note

is Lhat permanent shocks Ln inromr (P I,~~ - 0) Icad to murh moro

prccautiouary saviug than lransitory shucks [o iucume (p -( 1). '1'hc liual

point to note is that persistence in the shocks to changes in income lead to

even greater values of ,0 and thus to even more precautionary saving.

17

Table 1:Gfiects of stochastic rocess for income on "savin for a rain day"

((3~~30~) p-0 p-0.5 p-1or1G-0 ~-0.5r- 0.02 0.01961 0.80440 1313.3762 4956.103r- 0.04 0.03L~-tb -(1.63.13Fi 3-14.62ï5 1230.755

5 Conclusions

The life-cycle consumption problem with known interest rates and a stochas-tic process for income has been considered. Intertemporal substitution andconstant absolute risk aversion aze distinguished in order to get a bettergrasp of precautionary saving. When the felicity function is quadratic andincome follows a general autoregressive process, analytical decision rules forconsumption and saving can be found. In general, consumption dces not(ollow a random walk. Two motives for precautionary saving can be distin-guislred: ( i) "saving for retirement" in view of anticipated future declines inlabour income, ( ii) "saving [or a rainy day" when consumers are risk averseand hedge against unanticipated future declines in labour income. In ad-dition, risk aversion increases the sensitivity of changes in consumption tounanticipated changes in income which can be referred to as "making haywhile the sun shines". This is consistent with the evidence reported in Flavin(1981). However, Campbell and Deaton ( 1989) argue that there is excesssensitivity of consumption to anticipated changes in income and too littlesensitivity of consumption to unanticipated changes in income and suggestthat this is one of the reasons why consumption is so smooth. The presenceot "making hay while the sun shines" therefnre makes it more difTicult too~pla.in the puzzlc of why ronsumption is so smooth. Futurc work will con-sider the insurance aspects of future taxes and its e(fect on precautionarysaving and the break-down of Ricardian equivalence ( cf. Barsky et al., 1986;Kimball and Mankiw, 1989).

The solution of the general CARA-LQ problem has also been given andcan be applied to other problems, such as the problem of investment underuncertainty or the problem of inventory management, as well.

18

References

Atkinson, Anthony B. and Joseph E. Stiglitz (1980). Lectureson Public F.conomics, McGraw-Hill, Maidenhead, Berkshire.

ALI,au:4ein, Orario P. :uid (:nQlieltno Wobor (19R9). Intrrt~,m-

pural substiLUtiun, risk avrrsiun and thi~ I:uler oyual.i~~u fur

consumption, h,'cnnomic Jounral, Supplement, 99, fi9-73.

Barsky, Robert B., N. Gregory Mankiw and Stephcn P. 'Leldes(1986). Ricardian consumers with Keynesian propensities,American Economic Review, 76, 676-691.

Bensoussan, A. and Jan H. van Schuppen (1983). Optimal con-trol of a partially observable stochastic system with an expo-nential- of -integral performance index, BW194~83, Mathe-matisch Centrum, University oí Amsterdam.

Blanchard, Olivier Jean and Stanley Fischer (1989). Lectures onMacroeconomics, MIT Press, Cambridge, MA.

Caballero, Ricardo (1987). Consumption and precautionary sav-ings: Empirical implications, mimeo, MIT, Cambridge, MA.

Campbell, John (1987). Does saving anticipate declining labourincome? An alternative test of the permanent income hy-pothesis, Econometrica, 55, 1249-1273.

Campbell, John and Angus Deaton (1989). Why is consumptionso smooth, Review oJ Economic Studies, 56, 357-374.

Chow, Gregory C. (1975). Analysis and Control oj DynamicEconomic Systems, Wiley, New York.

Epstein, Larry G. and Stanley E. Zin (1989). Substitution, risk

aversion, and the temporal behaviot of consumption and as-set returns: A theoretical framework, Econometrtica, 57, 4,

9:57-969.F'anner, Roger F;.A. (1989). RINCE preferences, mimeo, l1CLA,

Los Angeles, CA 90024.Flavin, Marjorie (1981). The adjustment of consumption to

changing expectations about future income, Journaf oj Po-litical Economy, 89, 5, 974-1009.

Giovannini, Alberto and Phillippe Weil (1989). Risk aversionand intertemporal substitution in the capital asset pricingmodel, discussion paper no. 1421, Harvard University, Cam-bridge, MA.

Hall, Robert E. (1978). Stochastic implicationa of the lifecycle -

19

p~rmanent ineome hypothesis: Thmry ancl evidence, Journnt

uf I'u(itiral l;rortnrn,y, 7i(ï, (i, 971-!ltili.

ilall, Kohcrt h;. (1988). Intcrtemporal substitution in cunsump-

tion, .lourna! of Pnlitical Econorny, 9fi, 2, 3:59-3.57.Johnsen, 'Chore 11. and John lh. Uonaldson ( 1985). ['he struc-

turn of intertemporal preferences under uncertainty and time

consistent plans, Ernnometrira, 53, 6, 1451-1458.Kendrick, Uavid (1981). Control theory with applications to

economics, in Kenneth J. Arrow and Michael D. lntrilligator

(eds.), Handbook of Mathematical Economics, VoI.F, North-

Holland, Amsterdam.Kimball, Miles and N. Gregory Mankiw (1989). Precautionary

saving and the timing of taxes, Journa! of Politicat Economy,

97, 4, 863-879.Kreps, David M. and Evan L. Porteus (1978). Temporal resolu-

tion of uncertainty and dynamic choice theory, Econometrica,

46, 1, 185-200.Kreps, David M. and Evan L. Porteus (1979). Dynamic choice

theory and dynamic programming, Econametrica, 47, 1, 91-

100.Leland, Hayne E. (1968). Saving and uncertainty: The precau-

tionary demand for saving, Quarterly Journal oj Economica,82,465-473.

Plceg, Frederick van der (1984). Economic policy rules for risk-

sensitive decision making, Zeitschrift jur Nationalókonomie,

44, 3, 207-235.Yreston, Alan J. and Adrian R. Pagan (1982). The Theory of

Economic Policy, Statics and Dynamics, Cambridge Univer-

sity Press, Cambridge.Sandmo, Agner (1970). The effect of uncertainty on saving de-

cisions, Review of Economic Studies, 37, 353-360.Selden, L. (1978). A new representation of preference over 'cer-

tain x uncertain' consumption pairs: The 'ordinal certainty

equivalent' hypothesis, Econometrica, 46, 1045-1060.

Selden, L. (1979). An OCE analysis of the effect of uncertainty

on saving under risk preference intecdependence, Revieul of

Economic Studies, 45, 73-82.Weil, Philippe (1989a). Non-expected utility in macrceconomics,

Quarterly Journal oJ Economics, forthcoming.

20

Weil, Yhilippe (1989b). The equity premium puzzle and therisk-free rate puzzle, Journal of Monetary Economics, forth-

coming.Whittle, Peter (1982). Optimization over Time. Dynamic Pro-

gramming and Stochastic Control, Wiley, Chichester.Zeldes, Stephen P. (1989). Optimal consumption with stochas-

tic income: Deviations from certainty equivalence, QuarterlyJournal of Economics, 104, 2, 275-298.

Appendix

For completeness a brief discussion of the continuous-time problem is given.

The consumer's preference function is now given by

Pe - r~ exP[-9(s - t)]F(C(s))ds.le

and the budget constraint is given by

dA(s) -(r(s)A(s) t Y(s) - C(s)]di ~- adW(s), s~ t

where ) '(.v) denotes the deterministic component of labour income and W(s)

denotes a standard Wiener process. Effectively, it has been assumed that

income at time s is serially uncorrelated and normally distributed with mE .

Y, and variance o~. Hence, non-human wealth is driven by Brownian motion

with drift. The results given in van der Ploeg ( 1984) show that this problem

(for (i ~ 0) is equivalent to a deterministic, differential, max-min game

against nature ( cf., Section 4):

m~c min I ~{exp[-B(s - t)]F(C(9)) -} 2(E(s)~~po~)}ds

subject to

A(s) - rA(s) f Y(s) t E(s) - C(s), s~ i.

'1'his vi~lds tlu~ following solutions:

21

~~(s, t) - rr - exp[-(r - d)(y - l)]~(t). s] t

e(s,t) --(3a~exp[-r(s - t)]a(t), s 1 t

where C(s, t) and c(s, t) denote the planned or (perceived) level of consump-

tíon and income shock at time s given information at time t and a(t) de-notes the marginal value of wealth at time t. Hence, a risk-averse consumerensures prudence by having a negative bias in the expectation on incomeshocks. This bias diminishes for more distant income ahocks. Substitutioninto the intertemporal budget constraint gives

a - r[A(t) ~ H(i)]~(t) - ~ i~r-e~ - ~Rp7

where the deterministic component of human wealth is defined by

H(t) - I ~ exp[-r(s - t)]Y(s)ds.!a

When the market rate of interest equals the pure rate ot time preference(r - B), one has

rB[A(t) } ll(t)] - ~o(3oI~~'(,,t)- 1` 1 - 2p~s

t CCe(t) - B[A(t) t H(t)J, s~ i

where CcE(t) denotes the certainty-equivalent ( or risk-neutral) level of con-sumption. Hence, risk aversion leads consumers to engage in precautionarysaving and as a result the expected profile of consumption over the life cycle

is upward-sloping and there is excess sensitivity of consumption to unantic-

ipated changes in income.

Discussion Paper Series, CentFït, Tilburg

No. Author(s)

8801 Th. van de Klundertand F. van der Plceg

8802 J.R. Magnus andB. Pesaran

8803 A.A. Weber

8804 F. van àer Ploeg andA.J. de Zeeuw

8805 M.F.J. Steel

8806 Th. Ten Ras andE.N. Wolff

880~ F. van der Ploeg

8901 Th. Ten Raa endP. Kop Jansen

8902 Th. Nijman and F. Palm

8903 A. van Soest.I. Woittiez, A. Kapteyn

8904 F. van der Ploeg

8905 Th. van de Klundert andA. van Schaik

8906 A.J. Markink andF. van der Plceg

8907 J. Osiewalski

8908 M.F.J. Steel

Title

University, The Netherlands:

Fiscal Policy and Finite Lives in Interde-pendent Economies with Real and Nominal WageRigidity

The Bias of Forecasts from a First-orderAutoregression

The Credibility of Monetary Policies, Policy-makera' Reputation end the EMS-Hypothesis:Empirical Evidence from 13 Countries

Perfect Equilibrium in e Model of CompetitiveArms Accumulation

Seemingly Unrelated Regression EquationSystems under Diffuse Stochastic PriorInformation: A Recursive Analytical Approach

Secondary Products and the Measurement ofProductivity Growth

Monetary and Fiscal Policy in InterdependentEconomies with Capitel Accumulation, Deathand Population Growth

The Choice of Model in the Conatruction ofInput-Output Ccefficients Matrices

Generalized Least Squares Estimation ofLinear Models Containing Rational FutureExpectations

Labour Supply, Income Taxes and HoursRestrictions in The Netherlands

Capital Accumulation, Inflation and Long-Run Conflict in International Objectives

Unemployment Persistence and Loss ofProductive Capacity: A Keynesian Approach

Dynemic Policy Simulation of Linear Modelswith Rational Expectations of Future Events:A Computer Package

Posterior Densities for Nonlinear Regressionwith Equicorrelated Errors

A Bayesian Analysís of Slmultaneous EquationModels by Combining Recursive Analytical andNumerical Approaches

No. Author(s)

8909 F. van der Plceg

8910 R. Gradus andA. de Zeeuw

8911 A.P. Barten

8912 K. Kamiya ar.dA.J.J. Talman

8913 G. van der Laan andA.J.J. Talman

8914 J. Osiewalski andM.F.J. Steel

8915 R.P. Gilles, P.H. Ruysand J. Shou

8916 A. Kapteyn, P. Kooremanand A. van Soest

8917 F. Canova

8918 F. van der Ploeg

8919 W. Bossert andF. Stehling

892o F. van der Ploeg

8921 D. Canning

8922 C. Fershtman andA. Fishman

8923 M.B. Canzoneri andC.A. Rogers

8924 F. Groot, C. Withagenand A. de Zeeuw

Title

Two Essays on Political Economy(i) The Political Economy of Overvaluation(ii) Election Outcomes and the Stockmarket

Corporate Tax Rate Policy and Publicand Private Employment

Allais Characterisation of PreferenceStructures and the Structure of Demand

Simplicisl Algorithm to Find Zero Pointsof a Function with Special Structure on aSimplotope

Price Rigidities and Rationing

A Bayesian Analysis of Exogeneity in ModelsPooling Time-Series and Cross-Section Data

On the Existence of Networks in RelationalModels

Quantity Rationing and Concavity in aFlexible Household Labor Supply Model

Seasonalities in Foreign Exchange Markets

Monetary Disinflation, Fiscal Expansion andthe Current Account in an InterdependentWorld

On the Uniqueness of Cardinally InterpretedUtility Functions

Monetary Interdependence under AlternativeExchange-Rate Regimes

Bottlenecks and Persistent Unemployment:Why Do Booms End?

Price Cycles and Booms: Dynamic SearchEquilibrium

Is the European Community an Optimal CurrencyArea? Gptimal Tax Smoothing versus the Costof Multiple Currencies

Theory of Natural Exhaustible Resources:The Cartel-Versus-Fringe Model Reconsidered

No. Author(s)

8925 O.P. Attanasio andG. Weber

8926 N. Rankin

8927 Th. van de Klundert

8928 C. Dang

8929 M.F.J. Steel andJ.F. Richard

8930 F, van der Plceg

8931 H.A. Keuzenkamp

8932 E. van Damme, R. Seltenand E. Winter

8933 H. Carlsson andE. van Damme

8934 H. Huizinga

8935 C. Dang andD. Talman

8936 Tn. Nijman anaM. Verbeek

8937 A.P. Barten

8938 G. Marini

8939 w. cuth anaE. van Damme

894G G. Marini andP. Scaramozzíno

8941 J.K. Dagsvik

Title

Consumption, Productivity Growth and theInterest RateMonetary end Fiscal Policy in a'Hartien'Model of Imperfect Competition

Reducing External Debt in a World withImperfect Asset and Imperfect CommoditySubstitution

The D-Triangulation of Rn for SimplicislAlgor~thms for Computing Solutions ofNonlinear Equations

Bayesian Multivariate Exogeneity Analysis:An Application to a UK Money Demand Equation

Fiscal Aspects of Monetary Integration inEurope

The Prehistory of Rational Expectations

Alternating Bid Bargaining with a SmallestMoney Unit

Global Payoff Uncertainty and Risk Dominance

National Tax Policies towards Product-Innovating Multinational Enterprises

A New Triangulation of the Unit Simplex forComputing Economic Equilibria

The Nonresponse Bias in the Analysis of theDeterminants of Total Annual Expendituresof Households Based on Panel Data

The Estimation of Mixed Demand Systems

Monetary Shocks and the Nominal Interest Rate

Equilibrium Selection in the Spence SignalingGame

Monopolistic Competition, Expected Inflationand Contract Length

The Generalized Extreme Value Random UtilityModel for Continuous Choice

No. Author(s)

8942 M.F.J. Steel

8943 A. Roell

8944 C. Hsiao

8945 R.P. Gilles

8946 W.B. MacLeod andJ.M. Malcomson

8947 A. van Soest endA. Kapteyn

8948 P. Kooreman andB. Melenberg

8949 C. Dang

8950 M. Cripps

8951 T. Wansbeek andA. Kapteyn

8952 Y. Dai, G. van der Laan,D. Talman andY. Yamamoto

8953 F. van der Ploeg

Title

Weak Exogenity in Misspecified SequentialModels

Dual Capacity Trading and the Quality of theMarket

Identification and Estimation of DichotomousLatent Variables Models Using Panel Data

Equilibrium in a Pure Exchange Economy withan Arbitrary Communication Structure

Efficient Specific Investments, IncompleteContracts, and the Role of Market Alterna-tives

The Impact of Minimum Wage Regulations onEmployment and the Wage Rate Distribution

Maximum Score Estimation in the OrderedResponse Model

The D -Triangulation for SimplicialDefor~ation Algorithms for ComputingSolutions of Nonlinear Equations

Dealer Behaviour and Price Volatility inAsset Markets

Simple Estimators for Dynamic Panel DataModels with Errors in Variablea

A Simplicial Algorithm for the NonlinearStationary Point Problem on an UnboundedPolyhedron

Risk Aversion, Intertemporal Substitution andConsumption: The CARA-LQ Problem

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