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Precautionary savings under income uncertainty: across-sectional analysisValentino Dardanoni aa Department of Economics, University of York, York, UKYO1 5DDb Department of Economics, University of California, San Diego, La Jolla, Canada,92093, USAPublished online: 20 Oct 2008.

To cite this article: Valentino Dardanoni (1991): Precautionary savings under income uncertainty: a cross-sectionalanalysis, Applied Economics, 23:1, 153-160

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Applied Economics, 199 1 ,23 , 153-160

Precautionary savings under income uncertainty: a cross-sectional analysis

V A L E N T I N O D A R D A N O N I

Department of Economics, University of York, York YOI 5DD, U K , and Department of Economics, University of Calijornia, San Diego, La Jolla, C A 92093, USA

The first part of this paper derives a closed form solution for present consumption as a function of current and expected economic variables, which contains a precautionary component that is directly affected by future income risk. An estimating equation for the structural relationship between savings and their determinants is discussed and tested using cross-sectional data from the 1984 UK Family Expenditure Survey.

I. I N T R O D U C T I O N

The life-cycle hypothesis of intertemporal consumption is generally accepted as one of the better ways of modelling the determinants of savings and consumption decisions of indi- vidual agents over the course of their lifetime. While the life- cycle model is widely used as a framework for empirical analysis, both at the aggregate and at the household level, the empirical evidence has been, to date, mixed. Several recent surveys on the consistency of consumption data with the life-cycle theory (e.g. Blundell, 1988; King, 1985) con- clude that only a part of the variation in consumption data can be explained by life-cycle behaviour, while there are several phenomena (like the apparent excess sensitivity of consumption to current income, the saving-behaviour of the aged, the presence of liquidity-constrained households etc.) that are not easily explained by the 'simple' life-cycle theory.

It has been argued that the 'simple' model of life-cycle consumption omits some crucial real-world characteristics and that this seriously undermines its theoretical and empir- ical validity. A substantial amount of research has therefore been done on the consequences of relaxing the simplifying assumptions of the original model. The effects of random interest rates and labour income, non-separable preferences for endogenous labour supply, imperfect capital markets, and uncertain lifetimes are only a few examples of problems which are currently high on the research agenda.

In a recent paper, Skinner (1988) considers the effects of uncertainty on life-cycle consumption. By taking an ex- pansion of the Euler equation for the life-cycle consumption problem, the author derives an ingenious closed-form ap- proximation of the consumption solution, and, given plaus-

0003-6846191 $03.00+ . l 2 0 1991 Cllapman and Hall Ltd.

ible parameters, the extent of precautionary savings are then calculated. The author finds that savings that arise as a precaution against future uncertainty are a significant pro- portion of total savings, being over half the total life-cycle savings.

The fact that individuals facing different degrees of income uncertainty may have different saving attitudes has been recognized at least since Marshal1 (1920, p. 226) and Friedman's (1957, p. 17) Permanent Income Hypothesis (PIH). Furthermore, in the fifties and early sixties several empirical studies found different saving behaviour among people in different occupational groups: for example, Fisher (1956), Friedman (1957) and Liviatan (1968) all found that the self-employed saved substantially more than people in other occupational groups.

The importance of precautionary savings can hardly be overemphasized. Any change in income taxation under income uncertainty is bound to have effects not only on the expected value of present and future earnings, but also on their variability. Government measures that are neutral in a certain world may therefore have substantial effects in a stochastic environment (see, for example, Barsky et al., 1986 and Feldstein, 1988). Moreover, other crucial issues such as the effects of increased tax progressivity on savings and the efficient taxation of savings depend on the existence of a precautionary motive in a household's behaviour, as shown, among others, by Dardanoni (1988).

Given the importance of the issue, it is perhaps surprising that little empirical research on precautionary savings has been done following the studies quoted above. The paper by Skinner (1988) is a notable exception in so far as it contains a short empirical section on the estimation of precautionary

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V. Dardanoni

savings from household data. However, his results are such that little support is given for the view that precautionary savings are important.'

What is also worth noting is the fact that most of the empirical research on life-cycle consumption under income uncertainty has been carried out using certainty-equivalence solutions, where the optimal consumption is not affected by the degree of uncertainty of future income, therefore assum- ing away precautionary savings (see, for example, Hall, 1978; and Hall and Mishkin, 1982). However, as noted by Blanchard and Mankiw (1988), among others, the assump- tions yielding certainty equivalence are quite stringent and, in a stochastic setting, highly implausible. In particular, since the work of Arrow and Pratt, the quadratic utility employed to derive certainty-equivalence results is known to imply increasing absolute risk aversion, which is regarded as an unsatisfactory description of behaviour under risk.

On the other hand, given the increased knowledge of people's behaviour under risk caused by the dramatic im- provement in the techniques of expected utility maximiz- ation, the theory of optimal consumption under income uncertainty appears to be a complete and well specified body since the pioneering articles by Leland (1968) and Sandmo (1970). Studies by Sibley (1975) and Miller (1976) have extended the original analysis to a multi-period setting, the general Rothschild and Stiglitz (1970) characterization of increased income riskness has been applied, and a theoret- ical measure of the strength of the precautionary savings motive analogous to the Arrow-Pratt measure of risk aversion has recently been proposed by Kimball (1988).

The present paper-will extendskinner's empirical analysis by using a closed-form solution to the consumption problem and using it as a guide to the empirical analysis. This will demonstrate the role of the different variables in the estima- tion of the consumption function, and will help to explain why Skinner's empirical results seem to contradict the theory. By considering carefully the roles of income vari- ability, risk aversion and permanent income, the empirical estimation of the expected utility solution will show that precautionary savings are indeed an important part of total savings, as predicted by Skinner's theoretical analysis.

The detailed plan of this paper is as follows. Section I1 briefly surveys how expected utility theory has been used to derive a closed-form solution for present consumption as a function of current and expected economic variables, which contains a precautionary component that is directly affected by income risk, therefore disposing of certainty equivalence. An estimating equation for the structural relationship be- tween savings and their determinants having been derived, Section I11 explains the strategy used to implement the

reports the results of the estimation and Section V gives concluding remarks.

11. M O D E L

To keep the analysis and notation as simple as possible, a stationary infinite horizon model is used where the con- sumer maximizes the expected value of the sum of dis- counted utilities:

where E denotes the expectation taken with respect to the stochastic income stream, O c B < 1 is the time discount factor and C, is consumption at time t. The stochastic income stream is assumed to follow a random walk, Y, + , = Y, + et + , , where Y , denotes labour income at time t and e,, e,, . . . are independently and identically distrib- uted random variables with zero mean and a known dis- tribution. Note that a more realistic stochastic process could be used for the income stream (e.g. a Markov process) without affecting the basic results, but at the cost of clar- ity of exposition. This point will be considered again later. Assuming that the individual can borrow and lend at the constant riskless rate of return r > l , the amount of wealth owned at the beginning of period t + l, including current labour income, is denoted by W?+, =r(Wt-Cl)+ Y,+ , . The consumer's problem is to maximize Equation 1 subject to the solvency constraint

with W, given.

Note that for a bounded income process the solvency constraint may be rewritten in terms of bounds to the optimal consumption strategy, following Sibley (1975).

The stationary nature of the problem, in common with the previous literature (e.g. Yaari, 1976; Schechtman, 1976; Hey, 1980; Sibley and Levhari, 1986; Lam, 1987), is sufficient to ensure that the solution strategy is time-inde- pendent. This can be thought of as a simple analysis of how uncertain future income affects present consumption de- cisions. Given that the main objective is to derive a struc- tural equation for precautionary savings to be tested using cross-section data, this should not be too restrictive.

To solve the program, let V(W,) denote the maximum value of expression 1; V(.) will be a function of the indi- vidual wealth at the beginning of period 1 and will depend in general on the various parameters of the problem:

estimation of the derived equation using cross-sectional data m

from the 1984 U# Family Expenditure Survey. Section IV V(W,)=maxE 1 B'-'U(Ct)

t = 1

'Skinner correctly points out that this result may reflect self-selection of the least risk-averse into the most risky jobs.

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Precautionary savings under income uncertainty

subject to the solvency constraint. It is possible to show that V(.) is an increasing concave and differentiable function; see, for example, Benveniste and Scheinkman (1979).

As noted above, the maximum value function and the optimal strategy are time-independent; therefore, dropping the time subscript to avoid notational clutter, the dynamic recursive relation can be formed which will hold along the optimal path:

The optimal choice of C, denoted by C*(W), is deter- mined by the maximization of the term in braces in the right hand side of Equation 2. Assuming an interior solution, the first order condition is

The sufficiency condition for the unique maximum is ensured by the concavity of the problem.

Inserting the optimal value C *(W) in Equation 2, differ- entiating the resulting expression with respect to W, and using the envelope property to simplify gives

which, using the first order condition 3, implies that

Equation 4 states that the consumer, when choosing his optimal consumption plan, equates marginal utility of pre- sent consumption with expected discounted marginal utility of future consumption.

It is worth noting that Equation 4 is not the optimal consumption policy but simply an implication of it. While Euler's equation has been very popular for the study of the time-series behaviour of aggregate consumption, very often for the relative simplicity of its implementation under vari- ous stochastic assumptions, it must be stressed that it still only gives a relationship between two endogenous variables (present and future consumption), so that a more efficient strategy may be followed by solving Equation 4 in terms of the structural relationship between present consumption and its determinants.

The relative scarcity of empirical studies on the effect of income uncertainty on savings may be due to the difficulties in obtaining a closed-form solution for optimal consump- tion for Equation 4. Hayashi, for example, both in his study on the PIH and on liquidity constraints (Hayashi, 1982, 1985) states that it is in general impossible to derive a closed- form solution for the case of uncertain future labour income; similar remarks are made by Blanchard and Mankiw (1988) and Skinner (1988).

However, the studies of Merton (1971), Hey (1980, 1984), Cantor (1985), Sibley and Levhari (1986), Caballero (1987) and Lam (1987) all show how a closed-form solution may be obtained by assuming the utility function to display con-

stant absolute risk aversion. In such a case, they show that the optimal consumption strategy is linear in wealth and depends on the specific assumptions made on the distribu- tion of future income.

Substituting U(C) = -exp (- RC) in the first-order condi- tion 4, it can be verified that the optimal strategy is linear in wealth:

C*(W)=c,+c, W (5)

In particular, Equation 4 can be rewritten as

Given that this must hold for all values of W, the mar- ginal propensity to consume out of wealth must solve exp (- RC, W)= exp [- RC, r( l -c,)] for all W; therefore, in order for W to disappear, c, = (r - l ) / r = plr. This is a common result in the above-mentioned studies. Substituting c, =p/ r in Equation 6 gives the following equation for c,:

Taking logs and simplifying gives

c,= Ylr- Clog(rB)llR~-log CEex~(-RPe+,lr)llRP (7)

It is clear from examination of Equation 7 that the solution for c. will depend on the stochastic assumptions on the distribution of future income. In particular, assuming that e + , is normally distributed with variance V, we have

0

c, = Ylr - [log (rB)]/Rp - ~ p v / 2 r ~ (8)

If we note now that permanent income at any time t, P,, may be defined as the annuity value of total wealth, and total wealth as current wealth plus expected capitalized future earnings, we have that

so that, using Equations 5 and 8 and solving Equation 9, we have

Optimal consumption at time t will be equal to per- manent income 'adjusted' by two other components: a 'time preference' factor, which, as usual, will be negative if the rate of interest is greater than the time discount factor, and a risk premium, which represents precautionary savings and will be negative for non-degenerate distributions of future income and will be greater for riskier distributions and/or more risk-averse individuals. Thus, Equation 10 can be considered as a special case of Friedman's (1957) consumption function, explicitly derived by intertemporal expected utility maximization under income uncertainty.

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V. Dardanoni

This linear form of the stochastic permanent income hypothesis here derived, while depending on the constant absolute risk aversion utility function employed, does not depend on the particular stochastic process generating the income stream; using the results derived by Hey (1984) or Lam (1987) it is easy to show that consumption depends on permanent income minus the two other components exam- ined above for a general nth order Markov process.

The normality assumption employed is, strictly speaking, incorrect because it is unbounded from below. However, similar results may be obtained by approximating Equa- tion 10 using the formula of the moment-generating function for any distribution and neglecting higher moments for small R(r - l). Moreover, Charpin (1987) shows, with a simulation of a similar model using a normal distribution, that the probability of negative consumption is indeed negligible.

111. E M P I R I C A L I M P L E M E N T A T I O N

The main difficulty in implementing the estimation of Equa- tion 10 is the unobservable nature of almost all the variables involved. Apart from the obvious difficulties in measuring a household's permanent income and time preference, which arise also in the non-stochastic case, the theory predicts that, under income risk, precautionary savings are a function of risk-aversion and the (subjective) variance of future labour income. The unobservable nature of both these variables and the difficulties in obtaining a structural equation for optimal consumptio~ under income risk may be one of the causes of the scarcity of empirical studies on precautionary savings.

In the present study, which is based on the 1984 UK Family Expenditure Survey (FES), it was decided to develop further the intuition of the early cross-sectional studies on savings behaviour by focusing on the different degrees of variability in earnings in the different occupations. However, instead of simply using intercept dummies for the different occupations as done by Skinner (1988), it was decided that a measure of the riskiness of income would be obtained for each occupational group as implied by the optimal consumption function (Equation 10).

The FES data contains three variables on the economic nature of the occupation of the head of the household: occupation of head, industry of head and economic position of head. The first variable classifies each household accord- ing to whether the head of the household is a professional or administrative worker, teacher, skilled, semi-skilled or un- skilled manual worker etc.; the industry variable contains 33 different values such as agriculture, chemical, food, construc- tion, education etc.; while economic position considers if the

head is self-employed, a full-time employee at work, a full- time employee temporarily away from work etc. By considering all the possible permutations of these three variables, each household can be classified in a group which contains only households with the head having the same occupation, working in the same industry and having the same economic position, as defined above. This approach enables accurate proxies to be obtained for the unobservable variables in Equation 10.

Under the assumption that the riskiness of future labour income is specific to each occupational group so identified, we can consider a measure of ex-post variability in labour income within the group as a proxy for ex-ante uncertainty. In particular, disposable income was divided into labour income (normal take-home pay) and non-labour income. The calculated variance of labour income levels within each group was then used as a proxy for the variance of future income in Equation 10.

Furthermore, the expected utility maximizing solution (Equation 10) implies that consumption should not be re- lated to disposable income, as in Skinner's analysis, but to permanent income. Average disposable income within each occupational group so identified was taken as a reasonable proxy for the value of permanent income of that group.2 Average total expenditure in each occupational group was employed as the consumption variable in the left hand side of Equation 10. This may also reduce measurement errors, typical of cross-sectional survey data.

For this 'grouped data' approach it is crucial to obtain groups which are as homogeneous as possible. To this end, certain households were eliminated from the original sample. First, households were eliminated where the head was retired, or where there were members other than the head of the household working. This ensures that each group contains only households with a comparable labour income source in cases where there are significant other labour income sources in the household. Secondly, house- holds were excluded where the head was unemployed. Although unemployment is surely one of the main sources of labour income uncertainty, so that groups with higher unemployment rates should have higher uncertainty, unfor- tunately the FES data do not specify the former occupa- tional group of a person currently unemployed. Finally, after calculation of all the possible permutations of the three occupational variables, there were a considerable number of empty cells, or cells containing only one household: in order to give reasonably precise grouped variables, all the cells containing less than five households were eliminated from the sample. The biggest cell contained 54 households.

' Some problems remain for the estimation of the structural consumption function. In Equation 10, the 'time preference' component depends on the rates of time preference and

'While this procedure does not take explicitly into account the timerseries properties of income within each group, it gives an unbiased proxy for permanent income if the age structure and future income prospects are not systematically different among different groups.

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Precautionary savings under income uncertainty 157

return on wealth, B and r, and the risk-aversion parameter R. Moreover, the 'risk premium-precautionary savings' component will also depend on the rate of interest and on risk aversion.

The crucial issue here is whether these variables vary systematically in the population. Using grouped occupa- tional data has the advantage that one has to worry only if time preferences, attitudes towards risk etc. are assumed to vary systematically among occupational groups, whereas they may vary among individuals.

In this context, given that the focus of the present paper is on the estimation of precautionary savings, it is noted that if risk attitudes systematically influence occupational choices, there may be the problem of self-selection of the more risk- tolerant to the more risky jobs. This emerges quite clearly by examination of the nature of the risk premium in Equa- tion 10, so that one could observe occupations with ceteris paribus higher income variances consuming more because of lower risk-aversion parameter R. As rightly recognized by Skinner, this may account for the fact that his empirical results provide little support for the theory.

In order to tackle this serious problem, it was decided that every household whose head was self-employed or an em- ployer should be eliminated from the sample. Thus, the 'economic position' variable contains only full-time and part-time employees. The implicit assumption here is that risk aversion does not influence systematically the choice, for example, of being employed as a manual or as a professional worker, which depends more on other charac- teristics (education, skills etc.), while it may influence the choice of 'being one's own boss'. This will also reduce measurement errors if self-employed and employers sys- tematically under-report their disposable incomes, or if they are able to attribute a part of their total expenditure to their business, as reported by Friedman (1957)."

The assumption that the unobservable variables r, B and R are constant among the occupational groups implies that they are not statistically identified and are buried in the intercept and in the coefficient of the variance variable of the consumption function. This assumption, however, will not be left unchallenged, and a test will be made for variations in structural coefficient among occupational groups.

Finally, note that the use of grouped data, while reducing dramatically the influence of measurement errors (most obvious in the permanent-income variable but also in total expenditure, which is notoriously inaccurate in cross- sectional survey data) and the possibility of heteroskedas- ticity caused by random coefficient variations, will cause the variance of the error term to be inversely proportional to the

group size (see, for example, Judge et al. (1986)). This may be corrected by using an appropriate generalized least squares (GLS) estimation; this amounts to giving greater weights to cells containing more observations. The data employed are contained in the Appendix.

IV. RESULTS

Before the GLS parameter estimates are reported, some diagnostic tests are applied on the various statistical assumptions used in the estimation. First of all, a check is made for any remaining heteroskedasticity after the correc- tion for group size.4 Note that the presence of heteroskedas- ticity would also invalidate the results of other diagnostic tests based on the classical assumptions. The Breush-Pagan test (Breush and Pagan, 1979) is therefore applied to the unrestricted regression of total expenditure on permanent income, the variance of labour income and an intercept, each multiplied by the square root of group size. The test statistic of 3.9 may be compared with the critical value of the x 2 distribution with two degrees of freedom and confidently accepts the null at the 5% level.

After the test for heteroskedasticity, a Ramsey's (1969) RESET test may be carried out to detect possible mis- specifications or omitted variables. This may be constructed by means of a regression of the dependent variable on the original independent variables and the powers of the predic- ted values of the dependent variable. There is some evidence from simulations (e.g. Thursby, 1979) that the optimal number of added variables is the square, cube and fourth power of the ordinary least squares predictions. Under the null of no specification errors, the RESET test amounts to an F-test that these powers are not significant. Adding these variables to the regression, an F statistic of 1,07 is obtained, whose significance level with 3, 71 degrees of freedom is >36%, thus accepting the null.

Next, as discussed in the previous section, a test is made for the constancy of the coefficient in Equation 10 with respect to the risk-aversion parameter. Moreover, if riskier occupations face different rates of interest, this would affect the constancy of the coefficient of the constant term in Equation 10. A Chow test was carried out by splitting the sample in two subsamples according to the 'riskiness' of labour income in each group. To this end, an estimate was first made of the (measure free) coefficient of variation of labour income in each group. Then after calculation of the median value of the estimated coefficient, the dummy vari- able 'risky income' was created, which takes value 1 when

'A referee has pointed out that it has been suggested (e.g. Marglin, 1987; Pitelis, 1987) that the standard life-cycle hypothesis can better describe the behaviour of high-income groups (employers, self-employed) than that of employees, whose saving could be better explained in terms of compulsory saving (pension funds, mortgage repayments) and precautionary saving. This line of reasoning would give further backing to the exclusion of employers and the self-employed. 'Skinner (1988) does not report whether a heteroskedasticity test has been performed in his sample, and whether the results in Table 1 are accordingly corrected if the null is rejected.

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158

Table 1. GLS parameter estimates

Dependent variable: total expenditure

Parameters: Intercept - 3.1283 (1.039)

t-ratios in Variance - 0.00193 l (2.608)

Permanent income 1.00

N =77, R2=0.91 Breush-Pagan Test for Heteroskedasticity: 3.9 (2) RESET specification test: 1.07 (3, 71) Chow test for structural change: 0.17 (2, 72) F test for coeficient restriction: 2.15 (1, 74)

Table 2. Descriptive statistics P- P-

Mean SD Min. Max.

Total expenditure 142.4 53 56.6 290.3 Permanent income 148.3 52.5 64.6 303.6 Labour income 105.1 46.2 14.2 220.6 Variance 2248 3803 61.2 24890 Savings 5.9 24.2 - 72.3 72.7 Group size 11.8 9.2 5.0 54.0 Estimated risk 4.3 7.3 0.12 48.1

premium

the coefficient is greater than the median, and otherwise. Multiplying this dummy by the intercept and the variance coefficient, one can test for structural variations due to self- selection or varying interest rates. The F statistic of0.17 with 2,72 degrees of freedom has a significance level of 84%, and therefore the null hypothesis is again accepted.

Finally, a test is made for the theoretical restriction from the intertemporal maximization problem that the coefficient of permanent income equals 1. The F statistic of 2.15 with 1,74 degrees of freedom has significance level of 14%, so that the null is again accepted.

The results of the GLS estimation (where the coefficient of permanent income is restricted to take the value 1) are presented in Table 1. Both the intercept and the coefficient of the variance variable have the sign predicted by the theory, but while the coefficient of the intercept is not significant, the coefficient of the variance term is quite sharply estimated. The value of R2 of 0.91 is quite high for a cross-section estimation, but this may depend on the use of grouped data, which are bound to increase its value, and on the relatively small number of observations.

In order that the economic significance of these estimation results may be better appreciated the quantitative import, ance of the precautionary savings component of optimal consumption can be calculated. This may be done by

multiplying the estimated coefficient of the variance term in Equation 10 by the value of the variance of labour income in the sample. Table 2 shows that the estimated average amount of the risk premium-precautionary component of consumption is f4.34 per week. This may be compared to the average value of total expenditure and, more interestingly, to the average value of savings, defined as the average differ- ence between disposable (permanent) income and total expenditure in the sample. In particular, the average value of total expenditure of £142 per week indicates a risk premium of about 4% of total consumption; while average savings are £5.8 per week, indicating that more than 60% of savings in the sample arise as a precaution against future income risk. This is very similar to the results of Skinner (1988), who found by a simulation exercise that 56% of total savings were of a precautionary nature. Finally, note that the estimated value of the risk premium-precautionary savings ranges from 12 p to £48 per week, while total expenditure and total savings range respectively from £56.6 to E290.3 per week and from -E72.3 to £72.7 per week.

V. C O N C L U S I O N S

This paper has shown how expected utility maximization techniques may be employed as a guide to the empirical estimation of life-cycle precautionary savings. This has enabled Skinner's (1988) empirical analysis of precautionary savings to be extended.

Using data from the 1984 UK FES, aggregated in different economic occupations, the paper has found that pre- cautionary savings are a significant proportion of total savings. In order to reduce the bias due to the self-selection of the more risk-prone to the riskier occupations, the self- employed and employers have been excluded from the sample; given the fact that these occupations are thought to face greater labour income risk than employees, it is likely that the results are conservative estimates of the true effect of income uncertainty in the whole population. Including these observations would render a proper estimation of risk attitudes in the occupational groups obligatory; while there is no obvious way in which this may be done, this leaves an important avenue for future research.

Even though this paper suffers from many limitations due to the simplifying assumptions needed to derive the estim- ating equation and the choice of the proxies used in the empirical analysis, this is, to the best of the author's know- ledge, the first attempt to estimate from survey data on consumption behaviour the precautionary component of saving using a theoretical model of expected-utility maxim- ization as a guide to the empirical analysis. The findings are similar to those of other studies on the effects of income uncertainty on savings, based on simulations of life-cycle consumption under plausible parametrizations (e.g. Zeldes, 1986; Caballero, 1987; Skinner, 1988).

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Precautionary savings under income uncertainty

Given the importance of precautionary savings in the analysis of the effects of policy measures such as stabilization policies which involve the taxation of risky incomes, these results should be of considerable policy interest and should stimulate further research on this topic.

A P P E N D I X

Listing of raw

Total expenditure

1 204.19 2 220.65 3 148.45 4 147.78 5 157.03 6 264.42 7 226.93 8 205.38 9 202.87

10 249.84 1 1 262.55 12 137.42 13 112.06 14 170.93 15 290.32 16 217.06 17 227.31 18 210.18 19 182.43 20 204.35 21 270.29 22 150.43 23 150.74 24 175.57 25 136.61 26 106.83 27 108.22 28 102.08 29 182.36 30 135.16 31 121.82 32 103.46 33 96.333 34 102.90 35 82.909 36 102.38 37 74.779 38 119.25 39 158.20 40 176.29 41 151.13 42 129.10 43 122.32 44 133.19 45 135.66 46 163.24 47 118.74 48 137.61 49 112.86 50 149.08

data

Permaner; income

238.76 182.55 194.75 176.93 178.80 210.41 233.38 214.06 219.53 177.56 202.70 155.24 161.77 243.64 280.32 2 17.75 267.06 186.95 2 19.70 208.75 303.64 197.53 167.67 180.23 143.33 93.323 98.920

104.99 135.45 133.68 135.97 108.70 1 15.64 103.93 90.738 98.474 75.306

148.59 156.31 156.91 147.43 142.49 136.73 147.37 138.26 149.97 131.98 140.20 110.59 152.97

~t Labour income

180.55 124.19 150.2 1 135.96 140.42 152.53 187.76 157.5 1 191.42 134.49 1 14.8 1 122.12 50.849

185.70 220.64 158.84 126.55 141.78 146.95 137.32 202.93 140.74 127.82 103.03 107.30 79.523 32.271 74.775

124.12 102.23 46.183 88.696 88.701 36.008 80.816 72.262 24.166 98.208

129.33 125.03 127.63 129.08 1 10.94 116.72 104.72 105.34 120.66 120.29 9 1.965

11 1.79

Variance

5606.8 381.00

3321.1 637.63

1 124.3 9409.2 4029.5 3424.8

12373.0 2308.0 1062.6 5136.0 792.20

24893.0 15784.0 1233.5 2727.0 67 16.1 7034.6 644.67

7567.3 1716.1 903.13

2005.1 1747.0 645.16 177.01 301.58

1343.3 1549.7 1981.0 871.89 345.23 522.47

1817.1 493.68

84.589 2044.8 1570.3 807.65

1406.1 4015.7 1601.7 761.65 973.06 550.70

5605.8 2956.0 429.21 861.43

Group size

7 8

15 5 9 5 5

27 5

12 6

54 10 6 9 7 5

10 35 6

20 10 30 27 12 11 7 6 8

33 6

11 9 6 6

12 9 7

13 7

13 39 12 18 13 8

16 40 15 7

Total Permanent expenditure income

110.06 115.87 121.31 133.71 128.16 133.46 162.05 181.68 169.70 190.12 124.61 136.27 149.34 128.36 97.052 89.309

122.28 151.21 99.219 118.94 93.683 86.002

1 12.26 145.28 143.40 134.58 128.60 121.41 94.152 73.802

1 15.54 129.30 79.623 99.602 86.074 84.73 1

131.13 106.63 66.961 64.596 78.610 71.858 92.416 115.15 56.632 64.728 69.962 91.923 66.574 96.300 62.429 66.321

150.38 166.20

Labour income

93.808 1 12.99 109.75 134.28 147.82 91.217 88.567 29.101 98.772 83.128 67.591

117.18 119.61 9 1.263 24.876

111.71 80.745 22.601 89.786 29.322 14.192 82.106 18.684 68.618 29.705 19.983

131.79

Variance

677.58 1210.0 1 1 13.0 914.55

1589.5 396.83 786.70 283.21 858.51 25 1.63 569.92

1629.0 1351.2 313.86 216.38 687.69 178.03 140.35 747.24 126.00 170.50 868.23 61.258

316.48 227.04 235.92 882.93

159

Group size

10 6

19 14 13 8 8 7 7 8 5

10 9

12 10 9 5 5 7

10 7

10 5 6 5 6

10

A C K N O W L E D G E M E N T S

The author would like to thank John Hey, Peter Simmons and an anonymous referee for extremely helpful comments on a previous draft of this paper. Financial support from CNR-Rome is' also gratefully acknowledged.

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