met

Received

equation. Given plausible precautimary savings are of empirkal. savings rates

udget studies s found substantial differences in savin among occupations: isher (1956), for example, found the self-employe 12 percentage points more than managers.’

ue to differences in into precautionary savings against aN income ri

of aggregate capital accumullat f precautionary savin

may appear, ex JIOS~, ff and Summers (1981

hedge against future income uncertainty. rograms such as une

mcome risk, could have lower ecline in aggregate sa

ndrew Abel, Ch

sevier

238 J. Skinner, Risky income

rd, the dynastic model of intergenerational transfers [

on-specific income risk can be cushioned by the ap- of bequests, rendering life cycle precaution

ly, models that assume quadratic utilityr functions r by assumption precautionary savings against earnings risk. Finding si precautionary savings would cast doubt on the relevance of both t beauest model and quadratic utility functions. .a

recautionary savin s arise when individuals consume less (and hence save more) while young to ard against possible income downturns later in life.

hus the analysis of precautionary saving must begin with the analysis of how uncertain income affects consumption. There have been many studies of this topic, but most have been restricted to two-period models, or were so intracta- ble that precautionary savings could not be calculated. This paper closed-form multi-period life cycle model of consumption with

s and earnings. The true (but intractable) optimal consumption er Taylor-series expa

he somewhat surprising result of the theoretical model is that, given moderate levels of income uncertainty, precautionary savings are very small. The intuition is that precautionary savings depend on the proportion of lifetime resources at risk, but a given year’s earnings fluctuation is a small fraction of the present value of future income. It is only to the extent that annual variations in earnings signal a permanent change in future earnings

become imporrant. SW (1982) suggest that consumers do fetime resources; estimates from panel

e variation in annual earnings are a signal of a Precautionary savings are therefore r up to 56 percent of aggregate life

ive result - that precautionary savings are large - is con- sistent with the models of consumption under uncertainty developed by Zeldes

eldes (1986), the results presented below

result may re verse

numerical methods. e analytical closed-form

There is a substantial literature on t earnings uncertainty, on consumption and savings [

(1982)]. The analytic results derived below are, in o the exact solutions for con (1971), and, most recently, w (1987) because they allow for uncertainty both in interest rates and earnings. The major limitation of the derivations below is that they are approximations rather than exact solutions. An alternative approach to measuring the effect of uncertainty on consump- tion is to use numerical methods to solve rogramming problem for optimal consumption [Zeldes (1986), and Zeldes (1986)].

owever, it is often difficult to gain intuition from numerical solutions about why precautionary savings is important or unimportant.

The strategy in this section will be to derive an explicit uncertainty premium reflecting combi interest rate and earnings risk which can be implemented in a life cycle model. The consumer is assumed to maximize expected lifetime utility

EU= E, (1 -k- S)‘-‘U(Ci) ir4 1

0)

J( ,Si)=maX{U(Ci)+(l+S)-' cl

span add 4 fQllOW.

J. Skinner, Risky income

ue function which depends cm financial wealth age i and a vector of age and occupation-specific state variables Si.

ect differences among individuals in ea e functional relationship between Ci and

wi= (Wi-1 - ci_l)(l + ri) + F, (3)

d ri the net interest r

an

(&- C,)R;rO, (4) i=l

L i + rs)-l with j=1,

s=j+l

The first-order condition for (2) subject to (3) is written

e/‘(Ci) = (5)

equations in Grossman and Shiller (1982),

studies have used w, a second-order

es equ

expectation of weal

gs, but a zero (and certain) interest second-order Tayl

where Si is suppress and 3/=y+y* when the tion (or, equivalently, the value function) exhibits

second-period utility func- constant relative ri

sion.’ Just as uncertain wealth is discounted by y/20:, so also d parameter \k, a monotoni of future consu

ode1 to include interest rate uncertainty and multi-period consumption requires more structure and leads to greater analyti

ssumptions (i)-(iv) which follow facilitate the derivation of form solutions.

(i) Tke utility function d Letting y denote the Arrow-

constant relative risk aversion ( measure of G the utility function is

U(Ci) = -y(1 -Y), Y + 19

‘The value function a-is dS.

2 =

- .=

1 i9

co

b 2 2i= 2

i iS is et

244 J. Skinner, Risky income

uncertainty about lifetime resources Li in the next period. Like eq. (7) above, the ‘income’ uncertainty premium is simply one-half # times the proportional V ante of next period full wealth Li. The second term in (11) i ‘substitution’ effect, which reflects the covariance between the error te the asset yield, (ri - i;)/(l + i’), and the unexpected change (or ‘error term’) in the proportional realization of lifetime resources, (Li - Li)/Zi. Note that vi >< 0, depending on the relative magnitude of the income and substitution effectsF

Eq. (10) is simplified by taking the logarithm of both sides, noting that (1 + M) = x for x = r,&v, and expressing ln[ Ci/Ci_ 11 as 6,

c =~~~-~+.il+ln[Li,~i]. Y

(12)

certainty model, the consumption growth rate is simply n income is uncertain, there are two additional terms. The

first is the risk premium vi. The second term, ln[L,/Zi], represents the revaluation of lifetime resources following the new realization of interest rates

e consumption is a linear fu in consumption is equal ime resources. g the solutio; for optimal consumption, it is important to

note sources of error in the Taylor-series approximation. First, the approxima- tion assumes that the unconditional expectation of the uncertainty premium

J vi}, j < i, is equal to the correct conditional expectation E,_,( Pi}. This may not hold for two reasons. As earnings and interest rates are

e expectation of the asset share (~7) and the present value of rags share (pi) at age i may shift, the anging the expectation of vi.

source of error is that a depend (marginally) on the since increasin e, will reduce assets at age i.’

se sources of err d in section 5, where the cal solutions.

type of error by iterating over the vectors C = fixed-point solution for optimal consumption. This

wed, rather than

The general expressio ctiv

Cj = Lj

where

(13)

Eq. (13) is a forward-looking representation of consumption, in which Li encompasses all information about future earnings and current assets. It is also useful for econometric purposes to express Ci as the geometric sum of consumption originally planned for age i plus the cumulative revaluations in Ei { Li } since age 1. That is, using logarithmic approximations,

i-l

ln(Cj)= [(i-l)(?-S)]/y+ vj/U+In(Cl)

j-l

This expression for consumption is similar to the estimating equation in ‘X-constant’ models of labor supply [e.g., acurdy (1980)], and is straightforward to implement for applied work. The final term of (14) will qualify as an ‘error’ term, since its mean is zero and, assuming rational expectations, is independent of other terms on the S [Flavin (19Sl)].

Finally, will the optimizing consumer ever b The theoretical model predicts that the individual will only borrow on the certain component of future earnings. Since marginal utility is infinite when consumption is zero, any positive probability that Ci+j = 0, j > 0, would violate the first-order conditions. As long as current wealth is positive, the consumer will never borrow against the random element of earnings and thereby risk consu nothing at age i + j. The Taylor-series approximati condition [since it does not account for the asy Ci = 0] uce error if it pre section

246 J. Skinner, Risky income

earnings process to compare capital accumulation in a certain and uncertai regime. Turning first to the parameters of the utility fu

tion in measures of risk aversion [for e lume (1975), Landskroner (1977), ansen and Sin

and Shiller (1982), and Skinner (1985)], a central measure of y = 3.0 app reasonable, while the time preference rate 6 is assumed to be 1.5 percent. degree of interest rate uncertaintv is measured by the v nce of the return on Aaa bonds, adjusted by the G&P deflator, over the pe Report of the President, 198T). The average PX FCW real in percent, with a standard error of 2.9 percentage points. Finally, a,,. was assumed to be zero.

The structure of earnings uncertainty is a key factor in affecting precau- tionary savings. A general expression for earnings is an ARMA(l, 2) process,

yj = Xjfi + 0 + Ui, (15) and

ui = pUj_1 + Ej + mlej_1 + m2Ej-2,

where yi is log earnings at age i, Xi is a vector of exogenous factors such as experience and education, o the individual-specific effect, Ui the error term, Ei an iid variable, and year dummy variables have been suppressed.

Consider first the specii case of an AR(l) process (m, = m2 = 0) estimated ilk (1978). Using data from the Panel Study of Income und that p = 0.406 and CJ: = 0.069. The persistence of

over time implies that simply measuring var( Ui) will understate otal risk to lifetime resources. For the purposes of estimating precau-

tionary savings one requires the variance of an iid process which causes the same uncertainty to lifetime resources as the empirically observed structure of earnings. To calculate such a measure, note that for a given realization of

“i = ui - pt~~_~, the impact on the present value of expected and future resources is

20

26

24

22

20

10

16

14

12

19

50 55 60 t35 70 75 20 25 30 35 40 45

his

Fig. 1

estimates imply that this equivalent white percent of average earnings.*

Adopting the earnings regression from

noise variance is approximately 43

Lillard and Willis for whites with high school education (column 1, table A2) and assuming an annual real growth rate in wages of 0.5 percent and continuous employment, yields an average age-earnings profile (A) as shown in fig. 1. In this model, period 1 corresponds to age 2 individuals retire at age 65 and die

ows at a constant rate 0

248 J. Skinner, Risky income

ulation growth rate, is estimated to be only 12 percent of aggregate gs.9

s estimated usin larger degree of persistence equation of log earnings from m, = - 0.390, and m2 = - 0.094. year, new information out future lifetime e gs is substantial, leading to measures of Vi in excess of 4 percent.

Path (D) in fig. 1 presents the calculated Taylor-series measure of consump- life cycle, using the aCurdy earnings structure and given that

tions of earnings and erest rates occur. The substantially higher measures of vi lead to a steeply sloped consumption path, with both lower

mption in early periods and higher co ption at later periods (this r level of consumptio reflects the ‘spe down’ of the precautionary

g retirement). recautionary savings are calculated to be 56 per- cent of aggregate savings.

The degree of risk aversion s an important role in determining precau- tionary savings. lIncreasing the ow-Pratt measure of risk aversion y to 6.0 (\t = 32) increases precautionary savings to 76 percent of aggregate savings,

1, then variations in t

dure continues i

ue of Uf(CD_J is n

approximation tracks b&e

ion subject to certain income (CF) explained by th proxkation (C:). The R* is written:

ains more than 99 percent of its earnings process, path (D) seems

to diverge from the exact solution (E) most strongly at early ages, owing to the strict borrowing const less, the accuracy of t accuracy of restrictive.

ve would predict that average savings rates should be

3co J. Skinner, Risky income

pension contributions an

resented in table 1. It is not surpri

income is also lower. and farmers, opposite of

es, while the second equation e occupational dummy variable

coefficients are generally small; except for the self-employed, managers, and sale workers, the ence in savings rates are less than 2.5 percentage points, or a difference 0 percent of average savings. The savings rates of the self-employed and sales workers, those generally thought to receive riskier

than the benchmark group of craftsmen. ta refute the oft-cited stylized fact that the self-employed and

ers save more than others, a number of other factors could explain the titular, there may be problems in measuring differences of attitudes towards risk among e most accepting of ris so chose sales or

there would be no retical presump- re.

rofession

Manageri

Sales

0.144 795

0.143

0.130

.2P7 345

0.221 286

Craftsmen

Operatives

0.341

0.138

0.093

0.100

1322

1072

.222

0.219

- -

0.019 (2.27)

0.003 (0.22)

- 0.017 (1.36)

0.204 (25.94)

- 0.016 (7.36)

0.005 (1.51)

- 1.772 (20.60)

0.122

0.021 (2.95)

0. (0.

- 0.024 (2.43)

0.232 (35.86)

- 0.021 (12.31)

(1.64)

OX3E.-3 (0-W

- 1.708 (24.15)

0.211

Laborers 0.228 284

Service workers 0.220 368

Log(net income)

Family size

49

Age-squared

Constallt

RZ

e of t-statistics in parentheses; IV = 5685; craftsmen are the excluded occu me: Consumer Expenditure Survey 1972-73. .

2 =

9

-Y

-1 -1

=

J. grimmer, isky imcome