Precautionary Saving and the Deaton Paradox

Applied Economics Letters

ISSN: 1350-4851 (Print) 1466-4291 (Online) Journal homepage: http://www.tandfonline.com/loi/rael20

Precautionary saving and the Deaton paradox Michel Normandin To cite this article: Michel Normandin (1997) Precautionary saving and the Deaton paradox, Applied Economics Letters, 4:3, 187-190, DOI: 10.1080/135048597355483 To link to this article: http://dx.doi.org/10.1080/135048597355483

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Date: 12 October 2015, At: 19:14

Applied Economics Letters, 1997, 4, 187–190

Precautionary saving and the Deaton paradox MICHEL NORMANDIN Department of Economics and Research Center on Employment and Economic Fluctuations, Universite´ du Que´ bec a` Montre´al, CP 8888, Succ. Centre-Ville, Montre´al, Que´bec, Canada H3C 3P8

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Received 12 March 1996

The Deaton paradox implies that the permanent income hypothesis (PIH) under certainty equivalence is rejected because observed consumption is excessively smooth. It is shown how several reasonable parametrization of the PIH under precautionary saving imply that consumption is smoother than labour income and that the relative smoothness matches that found in the data.

I.

INTRODUCTION

Deaton (1987) has shown that the permanent income hypothesis under certainty equivalence (the PIH-CE model) implies that the optimal consumption path is more volatile than the labour income path when labour income is positively autocorrelated in first differences. This prediction is inconsistent with one of the principal raisons d’eˆtre of the permanent income hypothesis, namely that consumption is smooth because permanent income is smoother than labour income. Moreover, this prediction is contradicted by a striking feature of consumption behaviour; observed consumption is smooth relative to labour income. Thus, the PIH-CE model is rejected because actual consumption is excessively smooth. This excess smoothness phenomenon is called the Deaton paradox. One potential way to resolve this paradox is to relax certainty equivalence by analysing precautionary saving behaviour. Such saving is used to self-insure against the uncertainty of future labour income. However, Caballero (1990) has found that some restrictive parametrizations of the permanent income hypothesis under precautionary saving (the PIH-PS model) yield the same excessive smoothness as obtained from the PIH-CE model. This letter demonstrates that several flexible, but reasonable, parametrizations of the PIHPS model fully account for the excess smoothness problem.

II.

THE DEATON PARADOX

It is assumed that a representative consumer solves the following; problem: 1350–5851 © 1997 Routledge

max Et

fCt ‡jg

s:t:

Et

1 X jˆ0

1 X …1 ¡ † jˆ0

Ct‡j ‡ 1¡

Ct‡j …1 ‡ r†¡j ˆ At ‡ Et

Yt ˆ

0

‡

1

…1 ‡ r†¡j

t‡j

1 X jˆ0

…1†

Yt‡j …1 ‡ r†¡j ˆ Wt

Yt¡1 ‡

…2† t

…3†

where Et and represent the conditional expectation and the first difference operators, r is the time preference rate (equal to the constant interest rate), Ct is consumption, At is financial wealth, Wt is expected total wealth, and Yt corresponds to a non-insurable stochastic (after-tax) labour income. Also, Equation 1 involves a hyperbolic absolute risk aversion (HARA) utility function where > 0; ‰Ct …1 ¡ †¡1 ‡ t Š > 0 for non-satiation, 6ˆ 1, and t is time-varying in order to accommodate consumption growth (Merton, 1971). This is a very flexible specification which nests quadratic ( ˆ 2), exponential ( ˆ ‡1 and t ˆ ), isoelastic ( t ˆ 0) and log ( ˆ t ˆ 0) utility functions. Equation 2 is the budget constraint. Equation 3 is the labour income process used by Deaton (1987). The quadratic utility function ( ˆ 2) yields the traditional PIH-CE model (Hall, 1978). For this model, the ratio ( u = ) can be derived analytically – where u and are standard deviations of the innovations ut in consumption and t in labour income. The smaller the ratio, the smoother the consumption relative to labour income. By calibrating the labour income process (Equation 3) from Blinder and 187

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188

M. Normandin

Deaton’s (1985) data set and by fixing r to 1% (per quarter), Deaton (1987) has found that the PIH-CE model yields … u = † > 1. Thus, this model predicts that consumption is more volatile than labour income. However, this model is rejected because observed consumption is excessively smooth: the data indicate that … u = † < 1. Non-quadratic utility functions with convex marginal utility ( < 1 or > 2) yield the PIH-PS model (Leland, 1968). For this model, an explicit solution for ( u = ) can be obtained only with an exponential utility function. Caballero (1990) has shown that such preferences produce the same smoothness as induced by the PIH-CE model. Thus, this particular parametrization of the PIH-PS model cannot resolve the Deaton paradox. For the HARA utility function, ( u = ) must be found numerically. This is done by applying the method developed by Den Haan and Marcet (1990). More precisely, the expected future marginal utility, the non-resolvable term, is first approximated by an exponential function of a second-order polynomial in Wt , so the Euler equation becomes Ct ‡ 1¡

t

¡1

ˆ Et

Ct‡1 ‡ 1¡

exp

0‡

t‡1

¡1

1 log Wt ‡

2 2 …log Wt †

…4†

Then, Equation 4 yields the following approximation of the consumption function (for which the analytical solution is unknown): ( " # ) 2 0 ‡ 1 log Wt ‡ 2 …log Wt † Ct ˆ …1 ¡ † exp ¡ t ¡1 …5†

Also, t ˆ …1 ‡ †t ; and the s are found numerically by applying the Marcet (1991) procedure for the same data set,

Table 1.

interest rate, and labour income process (required to construct expected future labour incomes involved in Wt ) as in Deaton (1987). Finally, ( u = ) is computed by calibrating from the OLS estimate of Equation 3 and by ev al uating u from the co nstructed tim e series ut ˆ Ct ¡ Et¡1 Ct where Et¡1 Ct is obtained from a secondorder Taylor series expansion (evaluated at known variables in t ¡ 1) on Equation 5. This exercise is performed for the reasonable parametrization ( < 1; 0 < r 10) of the HARA utility function, where is recovered from and the relative risk aversion r . The restrictions on r are consistent with most empirical work. The restriction on produces precautionary saving and decreasing absolute risk aversion (i. e. the consumer is willing to pay less to avoid a given bet as wealth increases). For all the reasonable parametrizations, the Den Haan and Marcet (1994) test indicates that Equation 5, is an accurate approximation of the consumption function, i.e. the induced innovations in marginal utility and in consumption are orthogonal to the agent’s past information. Table 1 reveals that the isoelastic utility function ( ˆ ¡9; r ˆ 10) increases the volatility of consumption since … u = † ˆ 3:65 is larger than obtained from the PIH-CE model, which is 1.77. This result holds for all isoelastic utility functions, for the log-utility function and for all HARA utility functions having 0. Consequently, these parametrizations accentuate the Deaton paradox. In contrast, HARA utility functions involving 0 < < 1 almost always yield smooth consumption paths relative to labour income (i.e. … u = † < 1). Next it is verified whether the smoothness induced by the most promising parametrizations (0 < < 1; 0 < r 10) matches that found in the data. This exercise requires the ratio ( u = ) associated with Blinder and Deaton’s (1985) data set. Once again, is calibrated by estimating Equation 3 using OLS. And u is evaluated from the time series ut , which

Smoothness induced by the PIH-PS model r

¡90 ¡40 ¡1:0 0.0 0.1 0.5 0.9

0.1

0.5

0.9

1.0

2.0

– –

– – – – – 1.79 (1.00) 1.01 (0.31)

– – – – 1.89 (1.00) 1.16 (0.59) 0.93 (0.22)

– – – 1.92 (1.00) 1.83 (1.00) 1.11 (0.53) 0.89 (0.18)

– – 1.95 (1.00) 2.69 (1.00) 1.01 (0.38) 0.98 (0.23) 0.69 (0.05)

– – – 1.64 (1.00)

5.0 2.01 5.99 4.29 0.86 0.81 0.54

– (1.00) (1.00) (1.00) (0.17) (0.07) (0.01)

10.0 3.65 7.59 6.07 4.62 0.61 0.57 0.51

(1.00) (1.00) (1.00) (1.00) (0.05) (0.01) (0.00)

Notes: An en dash indicates the irrelevant cases due to negative consumption. The numbers represent the ratios ( u = ) associated with the PIH-PS model. Entries in parentheses are the p-values that the PIH-PS model induces a consumption that is more volatile than labour income, i.e. ( u = † > 1. These p-values are computed by performing a Monte Carlo experiment with T ˆ 127 (the actual sample size) and 1200 replications. Numbers in bold correspond to isoelastic utility functions for 6ˆ 0 and to the log-utility function for ˆ 0. For the PIH-CE model. ( u = † ˆ 1:77.

189

Precautionary saving and the Deaton paradox is approximated via the following ARIMAX process (estimated by OLS):

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^t ˆ ‡ C

X

^

j Ct¡j

jˆ0

‡

X

^

j Wt¡j

jˆ0

‡ ut

…6†

^ t and W ^ t are stationary variables and, 0 ˆ 0 ˆ 0. As where C usual, consumption is measured by expenditures on nondurables and services. Expected total wealth is computed by summing the financial wealth …1 ‡ r†=r times capital income (the difference between total income and labour income) – and the present value of expected future labour incomes, constructed using Equation 3. Four specifications of Equation ^ t ˆ Ct ; W ^ t ˆ Wt and 6 are analysed. S1 : C ˆ 0; ^ ^ ^ t ˆ Ct =…1 ‡ †t ; S2 : Ct ˆ Ct ; Wt ˆ Wt and ˆ 5; S3 : C ^ t ˆ Wt =…1 ‡ †t and ^ t ˆ Ct =…1 ‡ †t ; W ˆ 0; and S4 : C t ^ Wt ˆ Wt =…1 ‡ † and ˆ 5. These various specifications suggest that the ratios ( u = ) associated with the data are between 0.32 and 0.48. For comparability, simulated consumption and expected total wealth are also used to estimate (by OLS) each specification of Equation 6 in order to compute the ratios ( u = ) associated with the PIH-PS. Table 2 indicates that the ratios ( u = ) obtained from the data do not fall in the 95% probability intervals of the ratios induced by the isoelastic utility functions ( ˆ 0:1; r ˆ 0:9), ( ˆ 0:5; r ˆ 0:5), and ( ˆ 0:9; r ˆ 0:1). On the other hand, the parametrization ( ˆ 0:1; r 2:0), ( ˆ 0:5; 0:9), and ( ˆ 09; 0:5 r r < 10:0) of the HARA utility function induce a smoothness which matches that found in the data. These results are robust to the various specifications of the ARIMAX process (Equation 6). Table 2.

III.

CONCLUSION

The analysis of the smoothness of consumption was performed for the environment studied by Deaton (1987) with a modification to the structure of preferences. This exercise reveals that log and isoelastic utility functions cannot resolve the Deaton paradox. In contrast, several reasonable parametrizations of the HARA utility function imply that the PIH-PS model yields consumption paths which are smoother than labour income and the relative smoothness statistically matches that found in the data.

ACKNOWLEDGEMENTS I am grateful to John Galbraith, Allan Gregory, Thomas McCurdy, Gregor Smith, Tony Smith and Paul Storer for helpful suggestions. I would like to thank Alan Blinder and Angus Deaton for supplying me with data. I acknowledge financial support from Fonds pour la Formation de Chercheurs et l’Aide a´ la Recherche (FCAR). I retain full responsibility for any errors.

REFERENCES Blinder, A.S. and Deaton, A. (1985) The time series consumption function revisited, Brookings Papers on Economic Activity, 2, 465–511. Caballero, R.J.(1990) Consumption puzzles and precautionary savings, Journal of Monetary Economics, 25, 113–36..

The 95% probability intervals of the smoothness induced by the PIH-PS model r

Specifications

0.1

0.5

0.1

S1 S2 S3 S4

– – – –

– – – –

0.5

S1 S2 S3 S4

– – – –

0.9

S1 S2 S3 S4

(1.54, (1.40, (1.16, (1.10,

1.90) 1.90) 1.45) 1.40)

0.9

2.0

5.0

10.0

(2.21, (2.11, (1.66, (1.59,

2.76) 2.65) 2.11) 2.02)

(0.20, (0.17, (0.13, (0.12,

2.01) 1.66) 1.15) 1.04)

(0.17, (0.13, (0.10, (0.09,

1.14) 1.05) 0.71) 0.67)

(0.14, (0.09, (0.07, (0.07,

0.90) 0.82) 0.57) 0.53)

(1.55, (1.72, (1.34, (1.28,

1.91) 1.82) 1.67) 1.62)

(0.23, (0.21, (0.15, (0.14,

2.25) 1.93) 1.29) 1.22)

(0.21, (0.17, (0.14, (0.13,

1.08) 1.03) 0.75) 1.22)

(0.17, (0.12, (0.10, (0.10,

0.77) 0.74) 0.54) 0.51)

(0.15, (0.10, (0.08, (0.08,

0.50) 0.48) 0.36) 0.34)

(0.38, (0.25, (0.29, (0.28,

1.03) 1.24) 0.75) 0.72)

(0.27, (0.24, (0.20, (0.19,

0.91) 0.88) 0.66) 0.63)

(0.18, 0.67) (0.10, 0.89) (0.12, 0.48) (0.11, 0.46)

(0.14, (0.09, (0.08, (0.07,

0.54) 0.51) 0.39) 0.38)

(0.14, (0.09, (0.08, (0.07,

0.25) 0.22) 0.17) 0.17)

Notes: An en dash indicates the irrelevant cases due to negative consumption . Numbers in parentheses refer to the lower and the upper bounds of the 95% probability interval of the ratio ( u = ) induced by the PIH-PS model. These bounds are obtained by choosing the appropriate quantile of the ratios generated by estimating (by OLS) the ARIMAX process (Equation 6) from the simulated consumption and expected total wealth for each replicate of a Monte Carlo experiment (with T ˆ 127 and 1200 replicates). Entries in bold correspond to isoelastic utility functions. For the PIH-CE model, … u = † ˆ 1:77. For the data, ( u = ) is 0.48, 0.44, 0.35 and 0.32 for S1, S2, S3, and S4.

190

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Deaton, A. (1987) Life-cycle models of consumption: is the evidence consistent with the theory, in Advances in Econometrics Fifth World Congress, Vol. 2, ed. T.F. Bewley, Cambridge University Press, Cambridge, pp. 121–48. Den Haan, W.J. and Marcet, A. (1990) Solving the stochastic growth model by parameterizing expectations, Journal of Business and Economic Statistics, 8, 31–34. Den Haan, W.J. and Marcet, A. (1994) Accuracy in simulations, Review of Economic Studies, 61, 3–17. Hall, R.E. (1978) Stochastic implications of the life cycle–permanent income hypothesis: theory and evidence, Journal of Political Economy, 86, 971–87.

M. Normandin Leland, H.E., (968) Saving and uncertainty: the precautionary demand for saving, Quarterly Journal of Economics, 82, 465– 73. Marcet, A., (1991) Solving non-linear stochastic models by parameterizing expectations: an application to asset pricing with production, Economics Working Paper 5, Universitat Pompeu Fabra. Merton, R.C., (1971) Optimum consumption and portfolio rules in a continuous time model, Journal of Economic Theory, 3, 373– 413.